DRUG OVERDOSE TREATMEN T BY NANOPARTICLES By MARISSA S. FALLON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005
ii ACKNOWLEDGMENTS I would like to take this opportunity to acknowledge thos e that have personally and professionally helped me get to this juncture of my life. First and foremost, I want to thank my advisor, Anuj Cha uhan, for his guidance and suppor t throughout my time at the University of Florida. He has been both an advisor and a friend and has taught me concepts and skills that will benefit me throughout my career. His guidance has greatly contributed to my professional development, and his passion and enthusiasm for research have inspired me. I also want to thank Donn Dennis and Tim Morey from the Department of Anesthesiology for their helpful research di scussions and for allowing me to perform experiments in their laboratory. I would also like to thank Jason Flint and Manoj Varshney for their assistance with experiment al work. I would like to acknowledge the Particle Engineering Research Center at the Un iversity of Florida fo r its financial support and for providing me with numerous opportunitie s to present my research. I want to express my sincere gratitude to my s upervisory committee members (Atul Narang, Dinesh Shah, Donn Dennis, and Anuj Chauhan) for their guidance and time. I am also grateful to my lab coworkers for their friendship and for providing such a positive working environment. Finally, I would like to thank all of my friends and family for their support throughout the years. I especially would lik e to thank my parents, Joseph and Maxine
iii Fallon, for their love and encouragement, and Karl Dockendorf, who has brought so much joy and happiness into my life.
iv TABLE OF CONTENTS page ACKNOWLEDGMENTS..................................................................................................ii LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION........................................................................................................1 2 SEQUESTRATION OF AMITRI PTYLINE BY LIPOSOMES..................................8 Introduction................................................................................................................... 8 Materials and Methods...............................................................................................11 Materials..............................................................................................................11 Liposome Preparation..........................................................................................12 Preparation of Buffers.........................................................................................13 Drug Uptake Experiments...................................................................................13 Additivity Check.................................................................................................15 Results and Discussion...............................................................................................15 Time Scale of the Precipitate Formation after pH Change..................................15 Absorbance Results for the Control Solutions....................................................17 Results for the Solutions after Liposome Addition.............................................18 Qualitative Results.......................................................................................20 Quantitative Results.....................................................................................22 Absorbance Results for Solutions Containing DMPC Liposomes.........23 Absorbance Results for Solu tions Containing DMPC:DOPG Liposomes...............................................................................................24 Quantitative Evaluation of Drug Uptake by Liposomes.....................................28 Area Per Molecule of the Adsorbed Drug...........................................................33 Effectiveness of Liposomes U nder Physiological Conditions............................34 Conclusions.................................................................................................................36 3 PHYSIOLOGICALLY-BASED PHAR MACOKINETIC MODEL OF DRUG DETOXIFICATION...................................................................................................38
v Introduction.................................................................................................................38 Modeling.....................................................................................................................39 Model 1: Impermeable Capillary Walls, 1 > > 2................................................41 Model 2: Impermeable Capillary Walls, 1 ~ 2..................................................43 Model 3: Permeable Capillary Walls, 1 ~ 2......................................................45 Results and Discussion...............................................................................................47 NP Injected a Long Time After Overdose...........................................................47 NP Injected a Short Time After Overdose..........................................................52 Conclusions.................................................................................................................59 Notation......................................................................................................................60 4 DISPERSION IN THE CAPILLARIES.....................................................................63 Introduction.................................................................................................................63 Dispersion Analysis....................................................................................................65 Governing Equations...........................................................................................65 Method of Solution..............................................................................................66 Small Interfacial Mass Transf er Resistance: O(1) Bi...................................66 Large Interfacial Mass Transfer Reisistance: O( ) Bi..................................72 Results and Discussion...............................................................................................79 Range of Validity of the Average Equations.......................................................79 Dispersion Coefficient for Large Permeability: O(1) Bi.....................................80 Contribution to Dispersion from Molecular Diffusivity (* 0D )...................80 Contribution to Dispersion from Interfacial Resistance (* RD )....................80 Contribution to Dispersi on from Convection (* cD ).....................................82 Mechanism of Dispersion.............................................................................82 Asymptotic Limits of * cD ............................................................................84 Flow Through a Channel..............................................................................86 Dependency of * cD on Parameters................................................................89 Effect of Velocity Profile in the Tube on Dispersion...................................92 Dispersion Coefficient for Small Permeability: O( ) Bi.....................................93 Application to Transport in Tissues....................................................................95 Comparison of D* Values with Experimental Values.........................................97 Conclusions.................................................................................................................99 Notation....................................................................................................................100 5 INCLUSION OF DISPERSION IN PHARMACOKINETIC MODELS FOR BOTH FLOW RATE LIMITED AND MEMBRANE RESISTANCE LIMITED DRUGS.....................................................................................................................104 Introduction...............................................................................................................104 Modeling...................................................................................................................108 Derivation of Average Ma ss Transfer Equations..............................................108 Incorporation of Dispersion into Pharmacokinetic Models...............................115 Single Tissue Dispersi on Model Equations...............................................116
vi Whole-Body Dispersion Model Equations................................................119 Results and Discussion.............................................................................................121 Calculation of D*...............................................................................................122 Comparison of Calculated D* Values With Experimentally Determined Values............................................................................................................126 Comparison of Well-Mixed and Di spersion Single Tissue Models..................128 Flow Rate Limited Case.............................................................................128 Membrane Resistance Limited Case..........................................................129 Comparison of Dispersion and We ll-Mixed Whole-Body Models...................133 Conclusions...............................................................................................................137 Notation....................................................................................................................139 6 CONCLUSIONS......................................................................................................144 APPENDIX A DERIVATION OF THE DISPERSION COEFFICIENT FOR THE CASE OF FLOW RATE LIMITED DRUGS............................................................................151 B DERIVATION OF THE DISPERSION EQUATIONS FOR THE CASE OF MEMBRANE RESISTANC E LIMITED DRUGS..................................................155 C WELL-MIXED SINGLE TISSUE M ODEL AND WHOLE-BODY MODEL EQUATIONS FOR FLOW RATE LIMITED DRUGS...........................................158 D WELL-MIXED SINGLE TISSUE MOD EL EQUATIONS FOR MEMBRANE RESISTANCE LIMITED DRUGS..........................................................................160 E SEQUESTRATION OF DRUG BY MICROEMULSIONS....................................161 Experimental Methods..............................................................................................161 Drug Binding by ME.........................................................................................161 Drug Partitioning Experiments..........................................................................162 Data Interpretation....................................................................................................163 Drug Partitioning Experiments..........................................................................163 Drug Binding by ME.........................................................................................164 Determination of X............................................................................................168 LIST OF REFERENCES.................................................................................................170 BIOGRAPHICAL SKETCH...........................................................................................176
vii LIST OF TABLES Table page 2-1. Summary of the appearance of each solu tion in the drug uptake experiments after liposome addition.....................................................................................................21 2-2. Fraction of the drug sequestered by the DMPC liposomes at different pH values and liposome concentrations, calculated at = 285 nm..........................................31 3-1. Physiological and pharmacokinetic parameters for various human tissues...............47 4-1. Comparison of tissue dispersion nu mbers predicted from our model to experimentally determined values for cyclosporine in a rat.....................................99 5-1. Calculation of *D and PeL -1 in the tissue compartments.........................................125 5-2. Calculation of *D and PeL -1 in the vessels of the blood compartment.....................126 5-3. Comparison of tissue dispersion numbers predicted from our model (incorporating RBCs) to experimentally dete rmined values for cyclosporine in a rat............................................................................................................................ 128
viii LIST OF FIGURES Figure page 2-1. Structure of amitriptyline............................................................................................11 2-2. Structures of the lipids used to make the liposomes...................................................11 2-3. Absorbance vs. time at = 285 nm for 1.82 mM amitriptyline at pH 12 (9% control solution).......................................................................................................17 2-4. Absorbance vs. wavelength for the control solutions.................................................19 2-5. Qualitative look at the drug uptake by liposomes......................................................21 2-6. Check of the absorbance additivity assumption.........................................................23 2-7. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 9..........................................................................................................................2 5 2-8. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 10.7.....................................................................................................................26 2-9. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 12........................................................................................................................27 2-10. Absorbance vs. wavelength for the drug solutions containing 70:30 DMPC:DOPG liposomes at pH 10.7........................................................................28 2-11. Absorbance of the drug on the liposome vs. wavelength at the solubility limit of the drug for three different pH values (9, 10.7, and 12) and two different DMPC liposome concentrations (9% and 18% by volume).................................................30 2-12. Absorbance vs. wavelength for 1 mM amitriptyline in water..................................30 2-13. Absorbance of the drug on the liposome vs. wavelength at the solubility limit of the drug for DMPC and 70:30 DMPC: DOPG liposomes at two different liposome concentrations (9% a nd 18% by volume) at pH 10.7...............................32 3-1. Amitriptyline concentration as a functi on of time following an injection of NP after all tissues have reached their respective equilibrium values............................51 3-2. Amitriptyline concentrati on in the blood for an NP injection at 5 minutes...............52
ix 3-3. Amitriptyline concentration in the heart tissue for NP addition at different times....54 3-4. Amitriptyline concentra tion in the liver for NP injection at 5 minutes......................57 3-5. Amitriptyline concentration in the liver tissue for NP addition at 5 minutes for an X=10 NP...................................................................................................................59 4-1. Tube-annulus geometry..............................................................................................66 4-2. A pulse of solute is introduced at z = 0 and then disperses at it travels down the length of the tube......................................................................................................67 4-3. Depiction of the spreading of a solute pulse inside the tube-annulus geometry........84 4-4. * cD~ for Poiseuille flow vs. a for different K values where Dt/D = 0.1......................90 4-5. * cD~ for Poiseuille flow vs. K for different Dt/D values where a = 5.........................91 4-6. * cD~ for Poiseuille flow vs. Dt/D for different a values where K = 5.........................92 4-7. Ratio of of * cD~ for plug and Poiseuille flow vs. a for different K values where Dt/D = 0.1.................................................................................................................93 5-1. Krogh tissue cylinder................................................................................................106 5-2. A schematic representation of RBC flow through the capillary...............................107 5-3. Model depiction of RBCs within the capillary.........................................................109 5-4. Single tissue model...................................................................................................116 5-5. Whole-body model...................................................................................................121 5-6. Dynamic dimensionless mean drug concentr ation in the capillary as predicted by the flow rate limited single tissue models for different values of Rt......................130 5-7. Dimensionless drug concentration profile s in the capillary as a function of axial distance in the capillary as predicte d by the flow rate limited single tissue models....................................................................................................................131 5-8. Dynamic dimensionless mean drug concentr ation in the capillary as predicted by the membrane resistance limited single tissue models for Rt = 0.001...................132 5-9. Dynamic dimensionless mean drug concentr ation in the capillary as predicted by the membrane resistance limited single tissue models for Rt = 0.01.....................132
x 5-10. Plots of effeffKC and tC as a function of t as predicted by the membrane resistance limited single tissue dispersion model for Rt = 0.001...........................134 5-11. Dimensionless drug concentration profiles in the blood compartment as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.................................................................................................135 5-12. Dimensionless drug concentration profile s in the capillaries of the fat tissue as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.................................................................................................136 5-13. Dimensionless drug concentration pr ofiles in the capillaries of the gut as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.................................................................................................136 5-14. Dimensionless drug concentration prof iles in the capillaries of the heart as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.................................................................................................137 E-1. Fraction of amitriptyline sequestered by the microemulsion, as experimentally determined and as predicted by the adsorption model for microemulsion formulations containing 0.05% and 0.10% oil.......................................................162 E-2. Fraction of amitriptyline that could be sequestered by the microemulsion if the mechanism was purely absorption of the drug into the ethyl butyrate core of the microemulsion for a microemulsion containing 0.05% and 0.10% oil..................164 E-3. Area of drug/molecule at the surface of the microemulsion as calculated from the experimental data assuming an adso rption mechanism for a microemulsion containing 0.05% and 0.10% oil............................................................................167
xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DRUG OVERDOSE TREATMEN T BY NANOPARTICLES By Marissa S. Fallon December 2005 Chair: Anuj Chauhan Major Department: Chemical Engineering Drug overdose is a major health care problem , as a number of widely used drugs can cause life-threatening toxic ities and are without antidotes. This dissertation focuses on the development of nanoparticulate plat forms for treatment of drug overdoses. The essential idea is to design biocompatible nanoparticles (NPs) that can rapidly sequester significant amounts of drugs, such that an injection of NPs may lead to a reduction in the free drug concentration in vivo. In this dissertation we perform in vitro experiments to study the uptake of the drug amitriptyline by liposomes composed of dimyristoyl phosphatidylcholine (DMPC), a zwitterionic lipid, and of a mixture of DMPC and 1,2-di oleoyl-sn-glycero-3-[phosphorac-(1-glycerol)] (DOPG), an anionic lipid. The results show that both liposomes can rapidly sequester almost all of the drug pr esent in solution and suggest that DMPC liposomes adsorb the drug on the surface as a bilayer, while the DMPC:DOPG liposomes sequester drug through a partial adsorp tion, partial absorption mechanism.
xii For NPs to be successful in treating an ove rdose it is imperative that they reduce the drug concentration in th e body in time scales of fewer than 30-60 minutes. In vivo, these time scales depend not only on the time scale of drug uptake by the NPs, but also on the time scales of drug equili bration between the tissues. While in vivo tests to determine the time scale of concentration re duction in the body after NP injection are desirable, these are not always possible. Herein, a pharmacokinetic model is developed for drug detoxification by NPs that can predict these time scales. This model assumes the tissues to be well-mixed, a common assu mption in pharmacokinetic models; however, this assumption is usually invalid. We use th e method of multiple time scales to solve the drug mass transfer problem in the tissues a nd obtain the effective dispersion equations. These equations are then incorporated into the pharmacokinetic model such that the axial concentration gradients in the tissues are predicted. To summarize, we design NPs that can sequester amitriptyline, study the mechanisms of drug sequestration, and then incorporate the drug detoxification into a pharmacokinetic model that can predict the efficacy of the NPs for drug overdose treatment.
1 CHAPTER 1 INTRODUCTION Drug overdose is a major health care probl em, as a number of widely used drugs can cause life-threatening toxi cities and are without antidotes. There are over 300,000 emergency room patients per year due to dr ug toxicity, and over $10 billion each year is spent on hospitalization and lost productivity as a result of drug overdoses, and this does not include the $80 billion lost as a result of alcohol abuse (1 , 2). Two examples of drugs for which overdoses may be lethal and for which no specific pharmacological antidotes exist are amitriptyline and bupi vacaine. Amitriptyline is used for the treatment of depression and other psychiatric disorders and is the most widely prescribed tricyclic antidepressant in the United States, but is al so a common vehicle for suicide (3). Side effects of amitriptyline include cardiovascular complications and respiratory suppression. Because amitriptyline has a narrow therapeu tic window, only a slight overdose of this drug can be toxic. Bupivacaine is an am ide type local anesthetic. Overdoses of bupivacaine can occur by accidental rapid intr avenous injection, rapid absorption, or excessive dosage (4). High concentrations of this drug can cause changes in blood pressure and heart rhythm and may result in cardiac arrest (4, 5). In many cases, treatment may be given for the side effects caused by a drug overdose, but no specific pharm acological antidote exists for treatment of the overdose itself. In such instances, the only effective way to treat overdoses of these drugs is by reducing the free drug concentration in the body. Current techniques for doing this include gastric lavage, admini stration of cathartic drugs, administration of activated
2 charcoal, hemodialysis, and admi nistration of an IV (6). Gastric lavage, also known as the stomach pump, involves emptying the conten ts of the stomach. This method is only effective in treating drug over doses for which the drug was ad ministered orally and if performed before the drug exits the stomach (about a half hour after administration). Another method of treatment involves admini stering cathartic drugs, such as ipecac, which induce vomiting; however, this method of treatment is s ubject to the same limitations as gastric lavage. Activated char coal, a fine, black, odor less, tasteless, and nontoxic powder, may be administered for the treatment for drug overdoses. The activated charcoal adsorbs drugs inside the stomach and large intestine; however, once the drug has been absorbed by the gastrointes tinal tract, the activated charcoal can no longer retrieve the toxic ingestion. Thus, gastric lavage, catharti c drugs, and activated charcoal are only effective in instances when the overdose event occurred fewer than 30 minutes prior to treatment via oral administrati on. In the event that the drug has entered the bodyâ€™s circulation, hemodialysis, whic h involves removing the drug from the blood outside of the body, or an IV, which involve s infusing saline to the blood in order to dilute the drug concentration in the body, may be used; however, these methods are invasive and are require long amounts of time to reduce drug concentrations. Furthermore, hemodialysis is not feasible fo r drugs with large volum es of distribution, such as amitriptyline. Thus, a medical need exists for a drug overdose treatment which can rapidly remove drugs from all areas of the body. Nanoparticles (NPs), such as microemuls ions (MEs) (7), core-shell NPs (8), microgels, liposomes (9, 10), and nanotubes, may provide a solution to this problem. These particles can sequester drugs through ei ther an adsorption mechanism, whereby the
3 drug is sequestered onto the surface of the NP, or an absorption mechanism, whereby the drug is absorbed into the NP. If these particles are able to sequester drugs in vivo they may serve as drug overdose treatments by re ducing the free drug concentration in the blood and in the tissues. The small size and large surface area of these NPs make them particularly advantageous for drug overdos e treatment because they can flow through even the smallest capillaries and have the poten tial to rapidly sequest er large quantities of drugs. Furthermore, these particles may be lo aded with specific enzymes that act on the adsorbed/absorbed drug molecules and br eak them down into harmless byproducts. Before these NPs may be commercialized for use as drug overdose treatments, several steps must be taken. First, in vitro proof of concept experiments must be conducted to demonstrate that the NPs can sequester a significan t amount of drug in buffered solution as well as complex media, such as plasma. Furthermore, in vitro testing may be done using isolated animal organs su ch as the heart to further demonstrate the ability of the NP to sequester drug. Following the design of NPs that exhibit significant uptake efficiency in vitro, a safety analysis must be conducted on the NPs to ensure that there are no adverse effects asso ciated with intravenously inje cting the NPs into the body. This safety analysis includes thromboelastography (TEG) to ensure that the NPs do not affect the normal clotting characteristics of the blood, and hemolysis testing, to verify that the components of the NPs do not lyse red blood cells (RBCs). Subsequent to establishing the safety of the NPs, animal and human tests must be performed to demonstrate the detoxification effect of the NPs in vivo. Finally, if each of these stages of testing is successful, the FDA may appr ove the product for co mmercialization.
4 The steps toward commercialization described above represent a complex process, and to quickly and successfully accomplish this process, it is imperative that a thorough understanding of the interactions among the NP, the drug, and the body is achieved. Such an understanding allows for superior expe rimental design of in vitro, human, and animal tests and may lead to the design of more efficacious NP formulations. A model for predicting drug uptake by NPs may e licit some key attributes of the NPs that make them effective at drug overdose trea tment and may help us to understand the mechanism of drug sequestration. Such a model can be developed using in vitro experimental data. Furthermore, an understanding of the e ffects that these nanoparticulate systems have on the dynamic drug concentrations in th e blood and in various essential tissues is critical, because when a patient is overdosing on a drug, time is of the essence. Thus, in order for an overdose treatment to be effectiv e, it must not only be able to significantly reduce the drug concentrations in the body, but also do so rapidly. The project of developing detoxification platform s is a collaborative project of the Particle Engineering Research Center at the University of Florid a. Several groups have developed NPs that have shown potential for drug overdose tr eatment and have been proven safe and efficient in sequestering drugs ; however, no human testing has been performed thus far. A model that predicts the kinetic effects th e NPs will have on the drug concentrations throughout the body may be developed before human tests are performed by combining the model for drug uptake by NPs with pharmac okinetic modeling principles. This model will serve as a valuable aid in determining if a NP may be suitable for drug overdose treatment in vivo.
5 Pharmacokinetic models are used to pred ict drug concentrations in the body as a function of time. Physiologically-based pha rmacokinetic (PBPK) models divide the body into a number of compartments, each compartment representing either a specific tissue or the arterial and venous blood vesse ls (11). These models have the ability to incorporate physiological, anatomical, and other physioch emical data into the model (11-13) and utilize mass balance equations to determin e the dynamic drug dist ribution throughout the body. In this dissertation, drug overdose treatment, also referred to as drug detoxification, by NPs is explored. In Chapter 2, we perform in vitro experiments to study the sequestration of amitriptyline by liposomes. While other research ers have studied a number of other types of NPs that have shown promise in serving as effective drug overdose treatments, there are some concerns as to their biocompatibility. Liposomes are an attractive platform for drug detoxification because they are known to be biocompatible and have been used for a number of drug delivery applications. We study the drug uptake by liposomes composed of dimy ristoyl phosphatidylcholine (DMPC), a zwitterionic lipid, as well as liposomes composed of a mixture of DMPC and 1,2dioleoyl-sn-glycero-3-[phospho-rac-(1-glycero l)] (DOPG), a negatively charged lipid. Comparisons of drug uptake between these two types of liposomes allow us to see the effect of surface char ge on the drug uptake. In Chapter 3, we develop a pharmacokinetic model for drug overdose treatment by NPs. This model predicts the dynamic drug con centrations in the blood and in the tissues of the body both before and after addition of NPs. This model is developed assuming that all the tissue compartmen ts are well-mixed. This assumption is commonly used in
6 pharmacokinetic models; however, investigation in to the time scales of the dispersion and convection time scales in the capillary shows th at the capillary is not well-mixed. Thus, to accurately predict the drug concentration in the body, dispersion of the drug as it traverses the capillary should be in cluded into pharmacokinetic models. In order to develop a model that takes into account the fact the axial concentration gradients exist inside each capillary, the details of the microcirculation must be taken into account. While dispersion models have been developed in the past, these models neglected the presence of the red blood cells (RBCs) in the capillaries and have also focused on drugs that have large permeabilities across the capillary walls. In Chapter 4, a model for drug dispersion in the capillaries is developed for the cases of both large and small capillary wall permeabilities. These models are developed by solving the mass transfer equations in the tissues using th e method of multiple time scales and regular expansions in the aspect rati o of the capillary. While developing the mass transfer model in Chapter 3, the presence of RBCs in the capillaries is neglected. In Chapter 5, the dispersion model is expanded to include th e presence of the RBCs, and the dispersion models for both large permeability drugs (flow rate limited drugs) and small permeability drugs (membrane resistance limited drugs) are incorporated in to a pharmacokinetic model. The model for drug detoxification by NPs th at is developed in Chapter 3 can be modified by incorporating the results of Chapters 4 and 5 to account for the axial concentration variations for both the drug and the NPs. Development of this modified model requires knowledge of a large number of parameters, e.g., permeabilities of the capillary walls to drug and NP transport, equilibrium drug and NP partition coefficients,
7 degradation parameters for the NP, etc. Becau se the values of many of these parameters are unknown and difficult to experimentally de termine, this modified model is not developed herein; however, development of such a model will be advantageous if these parameters become known. Finally, Chapter 6 includes a summary of th e work included in this dissertation as well as some concluding remarks regarding th e findings of this res earch. Furthermore, suggestions for future studies are made.
8 CHAPTER 2 SEQUESTRATION OF AMITRIPTYLINE BY LIPOSOMES Introduction NPs, such as liposomes, can potentia lly separate compounds from aqueous solutions by either adsorption on the surface or absorption into the particle. Such separations can be important in a variety of applications, such as removal of toxic substances from the blood stream. Their small size and large surface area make NPs particularly advantageous for drug overdos e treatment because they can flow through even the smallest capillaries and have the poten tial to rapidly sequest er large quantities of drugs. NPs that are able to sequester drugs in vivo can reduce the free drug concentration in the body and therefore are possible drug ove rdose treatments (7, 14). NPs such as MEs (7), core-shell NPs (8), microgels, lipos omes (9, 10), and nanotubes may be used for detoxification. Varshney et al. have used Pluronic-based oil-in-wat er MEs to reduce the free concentration of bupivacaine and have found th at these MEs can extract up to 90% of bupivacaine from normal saline solution (7). These MEs ha ve also been studied for amitriptyline sequestration, and the results of some of these studies are given in Appendix E. Further studies have demonstrated the ab ility of these MEs to sequester drugs in blood and whole animal systems. Underhill et al. have synthesized oil-in-water MEs with a polysiloxane/silicate shell at the surface of the oil (8). These nanocapsules have been shown to sequester the drugs quinoline a nd bupivacaine from saline, but with less efficiency than the ME used by Varshney et al . Thus, the potential of NPs to reduce free
9 drug concentrations has been demonstrated. Herein, we will investigate the potential of liposomes, composed of the lipids 1,2-dimy ristoyl-sn-glycero-3-phosphocholine (DMPC) and 1,2-dioleoyl-sn-glycer o-3-[phospho-rac-(1-glycerol)] (DOPG), at sequestering amitriptyline. Liposomes may offer significant advantages over other types of NPs when used as drug overdose treatment for several reasons. Liposomes have been widely investigated for drug delivery applications (15-19) and ar e already currently being used on the market for some of these applicati ons (20). Thus, liposomes ar e proven to be biocompatible systems. The bilayer configuration of the lipid s in the liposome is similar in structure to that of the cell membrane, making liposom es less recognizable by the bodyâ€™s immune system. Furthermore, liposomes may be st erically stabilized by coating with inert hydrophilic polymers, allowing for even longer circulation time in vivo, and targeted interactions are possible with attachment of ligands to the liposomeâ€™s surface (15, 18, 19, 21). In this chapter, we demonstrate the ability of liposomes to sequester drug molecules. We aim to understand the m echanism of drug sequestration by liposomes, determine the effect of pH and charge on drug uptake by liposomes, and quantify the amount of drug liposomes sequester. We study the separation of amitriptyline, which is a common cause of overdose-related fatalities , from aqueous solutions at high pH by dimyristoyl phosphatidylcholine (DMPC) li posomes and by liposomes composed of DMPC and 1,2-dioleoyl-sn-gl ycero-3-[phospho-rac-(1-glyce rol)] (DOPG). The DMPC liposomes have no net charge, whereas th e liposomes containing DOPG have a net negative surface charge. The structures amitr iptyline and of the DMPC and DOPG lipids
10 are given in Figures 2-1 a nd 2-2. We use UV-Vis spectrophotometry to quantify the amount of the drug that is sequestered by the li posomes for pH values 9, 10.7, and 12. Some studies on the potential of liposomes for drug overdose treatment have been done by other researchers (9, 10). Petrikovics et al. have used sterically stabilized liposomes containing a water soluble enzy me, phosphotriesterase, to absorb and chemically degrade the toxin, paraoxon (9 ). In their study, they showed that administering the encapsulated enzyme prior to administering parathion eliminated gross toxic side effects in mice; however, these experiments involved a specific enzyme-drug reaction, which will not work for other dr ug types. Deo et al. have investigated liposomes composed of phosphatidyl choline an d phosphatidic acid for their potential to sequester amitriptyline (10). They have demo nstrated the capability of liposomes to sequester amitriptyline in pH range of 6 to 7. 4, and have suggested that amitriptyline may partition into the bilayer of the liposomes. Ou r study differs from that of Deo et al. in that we use liposomes composed of DMPC and DOPG lipids, which allows us to investigate the effect of charge on liposom e surface on the drug sequestration. We note that the mechanisms of uptake are different in the case of the liposomal systems used in our study compared to those of Deo et al., and these differences become apparent in subsequent sections of this chapter.
11 Figure 2-1. Structure of amitriptyline. Figure 2-2. Structures of the lipids used to make the liposomes. a) 1,2-Dimyristoyl-snGlycero-3-Phosphocholine (DMPC), b) 1,2-Dioleoyl-snGlycero-3-[Phosphorac-(1-glycerol)] (Sodium Salt) (DOPG). Materials and Methods Materials Methanol, chloroform, bicine, sodium hydr oxide, and amitriptyline were purchased from Sigma Aldrich. Sodium monobasic monohydrate, sodium tribasic dodecahydrate, a) b) N CH3CH3
12 and 13 mm, 0.45 m nylon filters were purchased from Fisher Scientific. The lipids 1,2Dimyristoyl-sn-Glycero-3-Phosphocholine (DMPC) and 1,2-Dioleoyl-sn-Glycero-3[Phospho-rac-(1-glycerol)] (Sodium Salt) ( DOPG) were purchased from Avanti Polar Lipids, Inc. The DMPC was purchased from the manufacturer in powder form, and the DOPG was purchased alrea dy dissolved in cholor oform (25 mg/mL). Liposome Preparation Liposomes containing DMPC lipid (4 mg/m L) as well as liposomes containing a mixture of DMPC and DOPG (4 mg/mL) were prepared using a reverse phase evaporation procedure. For the liposom es containing only DMPC, the lipid was dissolved in a 9:1 mixture ( by volume) of chloroform:methanol, such that a 10 mg/mL concentration was obtained. For the lipos omes containing both DMPC and DOPG, the DMPC was first dissolved in a 9:1 mixture of chloroform:methanol such that a 10 mg/mL concentration was obtained, and DOPG (25 mg/m L chloroform) was then added such that a 70:30 molar ratio of DMPC:DOPG was obtai ned. The organic solvent was then evaporated under a stream of nitrogen. After an even and uniformly dried lipid film was obtained, the dried lipid laye r was hydrated in distilled water, such that the lipid concentration was 40 mg/mL, and sonicated in a bath sonicator (G112SP1 Special Ultrasonic Cleaner, Avanti Polar Lipids, Inc.) at room temperature to form lipid vesicles. The solution was then diluted to 4 mg/mL with more distilled water and sonicated using a probe sonicator (Fisher Scientific Sonic Dismembrator Model 100) for 40 minutes at room temperature to reduce the vesicle size. The liposomes were then filtered using a 0.45 m filter.
13 Preparation of Buffers Solutions of 100 mM pH 9 buffer were prep ared by dissolving bicine in water such that a 111.11 mM concentration was obtained. The solution was then titrated to pH 9 with 1 M NaOH and diluted with water to 100 mM bicine. Solutions of 100 mM pH 10.7 were prepared by preparing an aqueous solution of 46.3 mol% sodium monobasic monohydrate and 53.7 mol % sodium tribasic dodecahydrate. Solutions of 100 mM pH 12 were prepared by dissolving sodium m onobasic monohydrate and sodium tribasic dodecahydrate in water. The solution was th en titrated to pH 12 with 1M NaOH and diluted with water to 100 mM. Drug Uptake Experiments Traditionally, quantification of drug uptak e by NPs in vitro has been done by first separating the NP from the solution and measurement of the free drug concentration using some type of spectroscopic technique, e.g., HPLC or UV-Vis spectrophotometry. Separation of the NPs from solution usually involves ultracentrifugation with filtered centrifuge tubes. We have attempted this methodology for quantification of drug uptake with the types of liposomes used in this pa per; however, these liposomes were breaking during the ultracentrifugation pr ocess, and thus, the drug upt ake could not be measured using this type of methodology. Herein, we employ an alternate method of drug uptake quantification by using UV-Vis spectr ophotometry as described below. Amitriptyline exists in both a charged, hydrophilic form (AH+) and an uncharged hydrophobic form (A). The pKa value of am itriptyline is 9.4, and at high pH amitriptyline exists mostly in the hydrophobi c form. Since the hydrophobic form of the drug in relatively insoluble in water, at high pH the drug will form a precipitate in aqueous solutions, causing these solutions to be cloudy in app earance. We have
14 performed experiments in which drug solutions above the solubility limit were prepared in a high pH buffer and then liposomes were then added to these cloudy solutions. Depending on the concentration and the amount of liposomes added, in some cases, the solutions changed from cloudy to clear in a ppearance after the addition of liposomes. This visual observation provi des direct evidence of dr ug uptake by liposomes. Some drug solutions stayed cloudy even after lip osome addition because the amount of unsequestered drug was above the solubility limit. In order to obtain quantitative measurement of the drug uptake, after equilib ration, the solutions were filtered with a 0.45 m filter to remove the precipitated drug. The absorbances of the filtered solutions, which are visually clear, were measured with a UV-Vis spectrophotometer over the wavelength ( ) range of 270 to 290 nm. If desired, this method of quantification could have been done at only one wavelength value; however, we have chosen to quantify the drug uptake over a range of wavelengths (270 to 290 nm) to ensure consistency in our measurements. Control solutions were also prepared by adding water rather than liposomes to the high pH drug solutions, and the absorbances of these solutions were measured using same methodology. The abso rbances of the high pH buffers and the absorbances of the liposomes in the buffers were also measured. The experiments described above were c onducted at pH 9, 10.7, and 12 using 9% and 18% (by volume) DMPC liposome a nd at pH 10.7 using 9% and 18% 70:30 DMPC:DOPG liposomes. Aqueous solutions of amitriptyline were added to high pH buffer, followed by the addition of either lipos omes or water (for control). The solutions were prepared such that the amitriptyline co ncentrations were 0.23, 0.45, 0.92, and 1.82 mM amitriptyline and contained either 9 or 18% liposomes (or water in the case of the
15 control) by volume. Solutions of 9 and 18% liposomes in high pH buffer were also prepared. After equilibration of the dr ug (approximately 20.5 hours), the control solutions, the solutions containing liposomes, and the buffer were filtered using a 0.45 m filter and the absorbance was measured with a Genesys 10 UV-Vis spectrophotometer over the wavele ngth range of 270 to 290 nm. Additivity Check The absorbance spectra in the filtered solutions have c ontribution from the buffer, liposomes and both free and sequestered forms of the drug. If the drug adsorbs on the surface of the liposomes rather than penetr ate the bilayer, its spectra may remain unchanged. In this case, if the solution is dilute, the absorbance of a solution could simply be sum of the contributions from each co mponent. To check if this is the case, we measured the absorbance of an aqueous 0. 23 mM amitriptyline solution, an aqueous solution containing 9% DMPC liposomes by vol ume, an aqueous solution containing 18% DMPC liposomes by volume, an aqueous 0.23 mM amitriptyline solution containing 9% DMPC liposomes by volume, and an aqueous 0.23 mM amitriptyline solution containing 18% DMPC liposomes by volume using a Genesys 10 UV-Vis spectrophotometer over the wavele ngth range of 270 to 290 nm. Results and Discussion Time Scale of the Precipitat e Formation after pH Change As explained in the Materials and Methods section, the drug amitriptyline was first dissolved in pH 7 distilled wa ter and was then added to a high pH buffer. The drug had a reasonably large solubility in water at pH 7 because a majority of it ionized into the charged form; however, after addition of the high pH buffer, the equilibrium between the charged and the uncharged form was altered, and a majority of the drug was converted to
16 the uncharged form. Since the uncharg ed form is hydrophobic and the initial concentrations were chosen such that the co ncentration of the uncharged form after the pH change was above the solubility limit, the drug began to preci pitate out of the solution. The process of conversion of th e charged form into the uncharged form occurred almost instantaneously as evident from the rapid change in the solution from clear to cloudy; however, in order to filter ou t the precipitate it was useful to let the insoluble form of the drug aggregate into larger particles. The kinetics of the aggregation process were monitored by measuring the ab sorbance of the soluti on as a function of time. The results of one such precipitation reaction are shown in Figure 2-3 for an initial drug concentration of 1.82 mM after additi on of a pH 12 buffer. The absorbance was measured at a wavelength of 285 nm. The absorbance began to decrease because the precipitates began to settle, but it took almost a day for the absorbance values to level off. It is noted that, as st ated above, this time scale is th e time required for settling of the precipitate and not for the pro cess of the formation of the pr ecipitate, which is a much faster process.
17 Figure 2-3. Absorbance vs. time at = 285 nm for 1.82 mM amitriptyline at pH 12 (9% control solution). Absorbance Results for the Control Solutions Since the settling occurred on a slow time scale of a few hours, we chose to let the aggregation and settling proceed for 20.5 hours and then filter the solution. Since the precipated particles were large at this time, the filtration removed all the precipitated form of the drug. Thus, for the controls, th e filtrate contained only the soluble form of the drug. In the filtrate the uncharged hydr ophobic form must be at the solubility limit and the charged form must be at the equilibr ium concentration, which was dictated only by the ionization equilibrium of the drug. Thus, the concentration of both the charged and the uncharged form should have been inde pendent of the initia l drug concentration, and should have only depended on the pH. Accordingly the absorbance of the filtrate of the controls should have been independent of the starting concentra tion. The absorbance results for the controls, shown in Figure 2-4, demonstrate that with the experimental error 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0246810121416182022 Time (hrs)Absorbance
18 the absorbances of the controls were indepe ndent of the beginning concentration for the pH 9 buffer. There were differences at othe r pH values; however, we point out that the values of the absorbances for the controls we re very small, and thus, the concentration dependence of the absorbances was due to the experimental error. Since these absorbances of the control solutions were much less than those of the solutions containing liposomes, these errors did not sign ificantly affect the accuracy of our results. Results for the Solutions after Liposome Addition In the experiments described below, a fixed amount of aqueous solution of liposomes was added to the aqueous solution of the drug at high pH af ter the precipitation was completed, i.e., 20.5 hours after the initia l addition of the high pH buffer to the drug solution. It was expected that after the a ddition of liposomes, some of the drug molecules would be sequestered by the liposomes. Th ere are two possible mechanisms of drug uptake: (1) absorption of the drug into the co re of the liposome or (2) adsorption of the drug onto the liposomeâ€™s surface. The process of adsorption is expected to be rapid, whereas, it may take a signifi cantly longer time for the drug to cross the lipid layer and partition in the annulus. Thus, the time scales of the uptake could be used as a gauge to determine the mechanism of drug uptake. As the drug began to be sequestered on to/into the liposome, the precipitated form was expected to dissolve to maintain the concentration of the uncharged form at the solu bility limit and that of the charged form at the concentration dictated by the ionization equilibrium. Furthermore, if the liposome was able to sequester more of the drug than the amount of the precipitate formed, the solution was expected to turn clear, implyi ng that the concentration of the uncharged form of the drug was below the solubility limit. By determining the critical drug
19 a) b) c) d) e) f) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.00 0.01 0.02 0.03 0.04 0.05 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.00 0.01 0.02 0.03 0.04 0.05 0.06 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.00 0.02 0.04 0.06 0.08 0.10 270275280285290 WavelengthAbsorbance buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT Figure 2-4. Absorbance vs. wavelength for the c ontrol solutions. a) pH 9, 9% control, b) pH 9, 18% control, c) pH 10.7, 9% cont rol, d) pH 10.7, 18% control, e) pH 12, 9% control, f) pH 12, 18% control. concentration below which the solution b ecomes clear on liposome addition and above which it remains opaque, one can determine the amount of the drug that is sequestered by the liposomes; however, the change in appear ance is rather gradual, and thus, it is difficult to accurately determine the drug upt ake by this rather qua litative observation. Below we first describe the qualitative obser vations of changes in the appearance of the solutions after liposome addition and then desc ribe an approach which can be used to obtain quantitative information on the drug uptake by liposomes.
20 Qualitative Results Figure 2-5 shows a sequence of images th at show the transition in the visual appearance of a cloudy drug solution of 0.45 mM amitriptyline at pH 12 after the addition of 18% DMPC liposomes. The cloudy solutio n turned clear in about 5 seconds after addition of liposomes. This clearly proves th at the liposomes sequester the drug. Table 21 summarizes which solutions turned clea r and which solutions remained cloudy upon addition of liposomes. For the 18% liposom es concentrations, the 0.23 and 0.45 mM drug solutions turned clear after DMPC lipos ome addition at all pH values tested and after 70:30 DMPC:DOPG liposome addition at pH 10.7. For the 9% liposome concentrations, the 0.23 mM drug solution turn ed clear after DMPC liposome addition at all pH values tested, whereas both the 0.23 and 0.45 mM drug solutions turned clear after 70:30 DMPC:DOPG liposome addition at pH 10.7 . This suggests that the DMPC:DOPG liposomes could be sequestering more of the drug than the DMPC liposomes. As shown in Figure 2-5, the uptake of the drug by the DMPC liposomes occurred in about 5 seconds, but it took a much larger time s cale of a few minutes for the 0.45 mM, 9% DMPC:DOPG liposome solutions to turn comp letely clear. This observation coupled with the fact that the DMPC:DOPG liposomes sequester more of the drug than the pure DMPC liposomes suggest that the liposom es containing DOPG c ould be sequestering some of the drug through an absorption mechanism. These liposomes have a negative surface charge, and electrostatic repulsion between the negatively charged lipids may create gaps in the lipid bilaye r, allowing the drug to cross.
21 0.9 s 2.1 s 3.5 s 5.4 s a)b) c)d) Figure 2-5. Qualitative look at the drug uptak e by liposomes. Both vials contain a solution of amitriptyline at pH 12. The vi al on the left is the control solution, to which water (18% by volume) has been added. DMPC liposomes (18% by volume) are added to the vial on the right. The amitriptyline concentration after liposome or water addition is 0.45 mM. Initially, both solutions are cloudy in appearance. The water has been added to the control solution prior to the pictures being taken, and no visi ble change in appearance was observed. A change in appearance from cloudy to cl ear is observed in the vial on the right after DMPC liposomes are added. The pictures show the solutions a) 0.9 seconds, b) 2.1 seconds, c) 3.5 second s, and d) 5.4 seconds after liposome addition. Table 2-1. Summary of the appearance of each solution in the drug uptake experiments after liposome addition. 70:30 DMPC:DOPG pH 9pH 10.7pH 12pH 9 0.27 mM Amitriptylineclearclearclearclear 0.45 mM Amitriptylinecloudycloudycloudyclear 0.92 mM Amitriptylinecloudycloudycloudycloudy 1.82 mM Amitriptylinecloudycloudycloudycloudy 0.27 mM Amitriptylineclearclearclearclear 0.45 mM Amitriptylineclearclearclearclear 0.92 mM Amitriptylinecloudycloudycloudycloudy 1.82 mM Amitriptylinecloudycloudycloudycloudy 9% Liposomes 18% Liposomes DMPC
22 Quantitative Results In our quantitative analysis of the data obtained from the drug uptake experiments, we have assumed that the absorbances of each component in the solution (i.e., buffer, liposome, and drug) are additive. To check the validity of this assumption, we measured the absorbances of a 0.23 mM amitriptyline solution in water (Adrug), a solution of 9% DMPC liposomes by volume in water (A9%lip), a solution of 18% DMPC liposomes by volume in water (A18%lip), a solution of 0.23 mM amitrip tyline containing 9% DMPC liposomes by volume, and a solution of 0. 23 mM amitriptyline containing 18% DMPC liposomes by volume over the wavelength rang e of 270 to 290 nm and also calculated the expected absorbances of the 0.23 mM amitr iptyline solutions containing 9 and 18% DMPC liposomes (A9%exp and A18%exp), according to the following equations. A9%exp() = Adrug() + A9%lip() (2-1) A18%exp() = Adrug() + A18%lip() (2-2) The results are shown in Figure 2-6, and it can be seen from this figure that our calculated predictions for the absorbance values of the 0.23 mM amitriptyline solutions containing 9 and 18% DMPC liposomes matched the measured values. Thus, our assumption that absorbances are additive is valid. The absorbance of the 70:30 DMPC:DOPG liposomes is similar to that of DOPG liposomes, and thus, we have assumed absorbances are additive for both types of liposomes.
23 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 270275280285290 WavelengthAbsorbance 9% Lip 18% Lip 0.23 mM Control Measured 0.23 mM + 9% Lip Measured 0.23 mM + 18% Lip Calculated 0.23 mM + 9% Lip Calculated 0.23 mM + 9% Lip Figure 2-6. Check of the absorbance additiv ity assumption. Absorbance vs. wavelength for DMPC liposomes in water (9 and 18% by volume), 0.23 mM amitriptyline in water, 0.23 mM amitriptyline in wa ter containing DMPC liposomes (9 and 18% by volume), and the calculated ab sorbance values of the 0.23 mM amitriptyline in water containing DM PC liposomes (9 and 18% by volume). Below we first describe the absorbance measurements for the solutions containing both types of liposomes (DMPC and DMPC:DOP G) and then evaluate the amount of the drug sequestered by the liposomes. Absorbance Results for Solutions Containing DMPC Liposomes Figures 2-7 2-9 shows th e absorbances for the solutions containing 9 and 18% DMPC liposomes at pH values 9, 10.7, and 12, respectively. For each pH value tested, the 0.23 mM amitriptyline solutions containi ng 9% DMPC liposomes turned clear after liposome addition, and the 0.45, 0.92, and 1.82 mM amitriptyline solutions remained cloudy. At the 18% DMPC liposome concentration, the 0.23 and 0.45 mM solutions turned clear after liposome addition, while the 0.92 and 1.82 mM solutions remained cloudy. It can be seen in Figures 27 2-9 for each pH and DMPC liposome concentration, all of the solutions that remained cloudy had approximately the same absorbance values after filtra tion. This suggests that equilibrium has been attained
24 between the bulk concentra tion of the drug and the drug adsorbed/absorbed by the liposomes. After filtration, the concentration of the drug in the bulk for these solutions was at the solubility limit and the drug con centration on/in the liposomes was at the equilibrium value, such that the total amount of the drug present in these solutions was fixed. The results also showed that the clea r solutions had an absorbance value less than the absorbances of those solutions that remained cloudy, which was also expected because the concentration of the drug in the bu lk was less than the solubility limit for the solutions that turned clear. Furthermore, for each pH value at the 18% DMPC liposome concentration, the 0.45 mM solution had appr oximately twice the absorbance of the 0.23 mM drug solution once the absorbance of the liposomes in the buffer was subtracted. Since there was twice the amount of the drug present in the 0.45 mM solution as there was in the 0.23 mM solution, this finding was expected. Absorbance Results for Solutions Containing DMPC:DOPG Liposomes Figure 2-10 shows the absorbances of th e solutions containing 9 and 18% 70:30 DMPC:DOPG liposomes at pH 10.7. In this case, the 0.23 and 0.45 mM solutions turned clear at both the 9 and 18% liposome concen trations. Similar trends were found with these liposomes as were found in the DMPC liposomes, for which the cloudy solutions had approximately the same absorbance values after filtration and, after subtracting the absorbance of the liposomes in the buffer, the 0.45 mM solutions had approximately twice the absorbance of the 0.23 mM solutions. The liposomes containing DOPG sequestered more of the drug than those cont aining only DMPC. This was visually seen with the fact that the 9% liposome 0.45 mM amitriptyline solution turned clear once 70:30 DMPC:DOPG liposomes were adde d, but remained cloudy when DMPC liposomes were added. Additionally, even though the absorban ces of the DMPC
25 a) b) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 270275280285290 WavelengthAbsorbance 9% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 270275280285290 WavelengthAbsorbance 18% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT Figure 2-7. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 9. a) 9% DMPC liposomes, b) 18 % DMPC liposomes.
26 a) b) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 270275280285290 WavelengthAbsorbance 9% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.0 0.2 0.4 0.6 0.8 1.0 1.2 270275280285290 WavelengthAbsorbance 18% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT Figure 2-8. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 10.7. a) 9% DMPC liposomes, b) 18 % DMPC liposomes.
27 a) b) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 270275280285290 WavelengthAbsorbance 9% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 270275280285290 WavelengthAbsorbance 18% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT Figure 2-9. Absorbance vs. wavelength for the drug solutions containing DMPC liposomes at pH 12. a) 9% DMPC liposomes, b) 18 % DMPC liposomes. liposomes were about the same as the ab sorbance values of the 70:30 DMPC:DOPG liposomes, the absorbance values of the filt ered solutions contai ning 70:30 DMPC:DOPG liposomes were higher than the absorbance values of the filtered solutions containing the DMPC liposomes at the same pH.
28 a) b) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 270275280285290 WavelengthAbsorbance 9% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 270275280285290 WavelengthAbsorbance 18% lip in buffer 0.23 mM AMT 0.45 mM AMT 0.91 mM AMT 1.82 mM AMT Figure 2-10. Absorbance vs. wavelength for the drug solutions containing 70:30 DMPC:DOPG liposomes at pH 10.7. a) 9% liposomes by volume, b) 18% liposomes by volume Quantitative Evaluation of Drug Uptake by Liposomes As shown above, it is reasonable to assume that the absorbances of each component in the solutions (liposomes, drug, and buffer) are additive, and thus the absorbance values in the experiments Aexp() can be expressed as
29 Aexp() = Alip() +Acontrol() + (Alip in buffer() Abuffer()) (2-3) where Acontrol is the absorbance of the control soluti on which can be attributed to the drug at the solubility limit and the buffer, Alip in buffer is the absorbance of the liposomes in buffer at the same concentration as in the experiment and Abuffer is the absorbance of the buffer, and thus (Alip in buffer Abuffer) is the absorbance of the liposomes. Finally, Alip is the absorbance due to the drug se questered by the liposomes. All the variables in the above equation repr esent absorbances after filtration. The above equation was used for a given , pH, liposomes type (DMPC or 70:30 DMPC:DOPG), and liposome concentration (9 or 18%). The absorbance of the drug on the liposomes for pH values 9, 10.7, and 12 for DMPC liposome concentrations of 9 and 18% liposome by volume are shown in Figure 2-11. The results show that the amount of the drug on the liposomes was relatively indepe ndent of the pH. After determining the absorbance of the sequestered drug, the amount of the drug taken up by the liposomes was determined by using the absorbance spectrum of 1 mM amitriptyline in water, which is shown in Figure 2-12. The spectrum of the drug on the liposomes, shown in Figure 211, followed a similar trend to that shown in Figure 2-12, because all of this absorbance came from the drug itself (and not from the liposome or buffer), and this result again reinforces the fact that the absorption sp ectra of the drug does not change significantly after uptake by liposomes. Additionally, the fraction of the drug sequestered by the liposome (f) at each pH and liposome concentration was calculated as buffer in lip exp lipA A A f (2-4)
30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 270275280285290 WavelengthAbsorbance pH 12, 18% pH 10.7, 18% pH 9, 18% pH 9, 9%pH 10.7, 9% pH 12, 9% Figure 2-11. Absorbance of the drug on the liposome vs. wavelength at the solubility limit of the drug for three different pH values (9, 10.7, and 12) and two different DMPC liposome concentr ations (9% and 18% by volume). 0.0 0.2 0.4 0.6 0.8 1.0 1.2 270275280285290 WavelengthAbsorbance Figure 2-12. Absorbance vs. wavelength for 1 mM amitriptyline in water.
31 The results for the fractiona l uptake as calculated at = 285 nm are shown in Table 2-2. This fraction is the ratio of the am ount of the drug sequestered by the liposome to the total amount of the drug presen t in the system after filtrati on. As is shown in Table 22, approximately 70 to almost 100% of the drug present in the solution was sequestered by the liposomes. The fraction sequestered decreased with a reduction in pH because even though the amount of the drug sequestered by the liposome was relatively the same at each pH for a given liposome concentrati on, the amount in the solution was higher at smaller pH values because a larger fraction of the drug was ionized. Table 2-2. Fraction of the drug sequestere d by the DMPC liposomes at different pH values and liposome concentrations, calculated at = 285 nm. 0.98 0.94 pH 10.7 0.91 0.81 18% Liposome 0.92 0.71 9% Liposome pH 12 pH 9 0.98 0.94 pH 10.7 0.91 0.81 18% Liposome 0.92 0.71 9% Liposome pH 12 pH 9 The absorbance of the drug sequestered by th e liposome at the solubility limit of the drug at pH 10.7 was also found for 9% and 18% 70:30 DMPC:DOPG liposomes using Eqn. (2-3) and is shown in Figure 2-13, along with the absorbances absorbance of the drug sequestered by the liposome at the solubility limit of the drug at pH 10.7 for the 9% and 18% DMPC liposomes. The 70: 30 DMPC:DOPG solutions had larger absorbances than those solutions containi ng DMPC liposomes, and we conclude that more of the drug was sequestered by the li posomes containing DOP G. Possible reasons for this could be that DOPG is a negatively charged lipid, which may have an electrostatic interaction with the AH+ form of the drug. Another possible explanation for the increased drug uptake is that some of th e drug may be absorbing into the liposome containing DOPG. Electrostatic repulsion between the negatively charged head groups of the DOPG may cause gaps in the lipid bilayer, which is tightly packed in the case of the
32 DMPC liposomes. These gaps may give the dr ug the ability to enter the liposome. As stated earlier, when the experiments were performed with the DMPC:DOPG liposomes, we did notice that the 0.45 mM solution at th e 9% liposome concentration took a time of a few minutes to turn completely clear in comparison to the few seconds it took for the case of the DMPC liposomal systems. This observation is consistent with the hypothesis that for DMPC liposomes all of the seque stered drug was adsorbed on the surface, whereas for the DMPC:DOPG liposomes a fract ion of the sequestered drug entered the lipid bilayer. Additionally, we calcula ted the fraction of drug uptake by the 70:30 DMPC:DOPG liposomes at pH 10.7 for = 285 nm using Eqn. (2-4). This value was 0.99 for both the 9% and 18% liposome concentr ations, and thus, prac tically all of the drug present in the solution wa s sequestered by the liposome. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 270275280285290 WavelengthAbsorbance 9% DMPC 18% DMPC 9% 70:30 DMPC:DOPG 18% 70:30 DMPC:DOPG Figure 2-13. Absorbance of the drug on the liposome vs. wavelength at the solubility limit of the drug for DMPC and 70:30 DMPC:DOPG liposomes at two different liposome concentrations (9 % and 18% by volume) at pH 10.7.
33 Area Per Molecule of the Adsorbed Drug Assuming an adsorption mechanism, we calculated the area of the drug adsorbed to the liposome per molecule. We did this by fi rst converting the abso rbance of the drug on the liposome to concentration using abso rbance vs. wavelength calibration curve constructed at the intermediate wavelength of 285 nm and by assuming that the lipids pack on the surface of the liposome at 50 2/molecule. The calculated areas were about 25 2/molecule for the DMPC liposomes at all three pH values. This value of area per molecule is too small to be plausible, a nd suggests that the drug molecules may have been adsorbing as a bilayer. Thus, the area per molecule was approximately 50 2/molecule. We postulate that the first la yer adsorbed on the surface of the liposomes due to the electrostatic interaction between the positively charged drug and the negative charge on the zwitterionic DMPC lipid. The adsorption of the first layer exposed the relatively hydrophobic segment of the drug e xposed to water and this led to the adsorption of the second layer. This hypothe sis requires that the charged form of the drug adsorbs on the liposome surface, the c oncentration of which is small at high pH; however, the time scale of the dr ug ionization is very rapid, a nd thus, as soon as some of the charged molecules adsorb on the liposome, more are created in the bulk to establish the ionization equilibrium. For the case of DMPC:DOPG liposomes, ba sed on the assumption that all the drug adsorbs as a monolayer, the calculated area per molecule was 5.21 and 8.36 2/molecule for the 9 and 18% liposome concentrations, resp ectively, at pH 10.7. This result also supports the hypothesis that the DMPC:DOPG liposome may sequester the drug partially through an absorption mechanism. The lipos omes containing DOPG exhibit a negative surface charge, and thus, these liposomes may ha ve less tightly packed lipid bilayers due
34 to electrostatic repulsion between the lipids. This may create gaps in the bilayer through which the drug can penetrate into the core of the liposome. Based on the assumption that the adsorbed amount on the surface of the DMPC:DOPG liposomes is the same as that on the DMPC liposomes, the ratio of the absorbed and the adsorbed drug amounts was about 1.5. Futhermore, although the values of th e area per molecule calculated for the DMPC liposomes at each wavelength were about the sa me for a given pH, there were significant differences between the values for DMPC: DOPG liposomes. These differences could partially be attributed to the fact that the absorption spectra of the absorbed fraction of the drug is expected to be different fr om that of the adsorbed fraction. Effectiveness of Liposomes Under Physiological Conditions Since we propose to use liposomes as drug overdose treatments, it is important to determine their efficacy at drug sequestration under physiological conditions or in vivo. Aside from the presence of plasma proteins and other blood components, there are two major differences between the conditions under which our experiments were performed and the conditions in vivo. The first differen ce is that our experiments were conducted at pH values in the range of 9 to 12, while physio logical pH is 7.4. The second difference is that the methodology used herein quantifies th e drug uptake at the so lubility limit of the drug, which is on the order of 100 M, whereas the concentrations experienced in vivo will be less than 1 M. While the methodology presented in this chapter cannot quantify the drug uptake at lower concentrations or at physiological pH valu es, we conducted a set of two experiments that provided a qualitativ e indication of the exp ected behavior under physiological conditions. In these experiment s, 9% DMPC liposomes were added to a solution of 0.23 mM amitriptyline solution (1) after the pH was raised to 11.5 with NaOH addition and (2) before the pH was raised to 11.5 with NaOH addition. In the first
35 experiment, i.e., the case where the pH was raised before the liposome addition, the solution turned cloudy after the pH change and cleared up after addition of liposomes, which is consistent with the experiments desc ribed above. In the ot her case, i.e., the one in which pH was reduced after the liposome a ddition, the solution remained clear at all times. The time scale for the 0.23 mM drug so lution to turn cloudy wa s very short, and thus, the fact that the drug solution remained clear in the case where liposomes were added before the pH change indicates that the liposomes must have sequestered the drug before the pH change. This qualitative obs ervation again suggests that the charged form of the drug adsorbed on the liposome surface and that the amount of the drug on the liposome surface was relatively independent of the pH, and will thus remain relatively similar even at physiological pH values. Furthermore, the experiments described herein were conducted at large drug concentrations at which the surface of the liposomes is saturated as evident by the small valu es of area per molecule; however, under physiological conditions or in vivo, the concentration of the drug will be small and will presumably not saturate the surface of the liposomes. Based upon classical Langmuir adsorption behavior, the fraction of the drug will remain cons tant at drug concentrations less than the saturation limit and will then de crease as the drug concentration starts to exceed the saturation limit. While the adsorpti on of liposomes is unlikely to be governed by the Languir isotherm, the qualita tive trends will be similar, a nd thus, it is expected that the fraction of the drug uptake in vivo may be larger than the uptake in our experiments. The DMPC:DOPG liposomes showed an even larger amount of drug uptake than the DMPC liposomes, and we therefore expect a large amount of drug uptake with these
36 liposomes in vivo. Furthermore, the liposomes may be tailored to incorporate more negatively charged lipid, which may lead to even larger drug uptake in vivo. Conclusions Herein, we have developed and employed a methodology for quantifying the amount of the drug sequestered by liposomes at hi gh pH and at the solubility of the drug. We used UV-Vis spectrophotometry to measure the absorbance of the drug solutions with and without liposomes, and based on the diffe rences of these absorbances, we have calculated the absorbance of the drug se questered by the liposome. Two types of liposomes were investigated, (1) DMPC li posomes, containing no net charge and (2) 70:30 DMPC:DOPG liposomes, containing a negativ e surface charge. In some cases for the DMPC liposomes, liposome addition at hi gh pH caused the drug solutions to change from cloudy to clear in color. This color changed occurred within a few seconds and demonstrates that the mechanism of drug uptake must, at least partially, occur via adsorption of the drug onto the liposomes surf ace, as penetration of the drug into the bilayer of the liposome would require a much larger time scale. For the case of the liposomes containing DOPG, a rapid color ch ange from cloudy to clear was observed for the lower drug concentrations; however, in so me cases, a color change was observed over a larger time scale. For these types of liposomes, a partial adsorption and partial absorption mechanism is speculated. In th is case, the negatively charged lipids may exhibit electrostatic repulsions , creating gaps in the lipid bi layer, through which the drug may cross. From the high pH studies, we determine that at the solubility limit, DMPC liposomes sequester approximately 70% of the drug present in the solution, while the liposomes containing DOPG sequester approximately all of the drug present in the
37 solution. We conclude that the amount of drug uptake by DMPC liposomes is relatively independent of pH, and we observe a large upt ake in the case of the liposomes containing DOPG. The area of a drug molecule on the DM PC liposomeâ€™s surface is calculated to be about 25 2/molecule, which implies the drug may be adsorbing to the surface of the liposomes in a bilayer configuration, with an actual area per molecule of about 50 2/molecule. For the liposomes containing DOP G, the calculated are per molecule is much smaller, which may be explaine d by a partial absorption mechanism. The methodology used herein is subject to the limitation that it may only be used to quantify the drug uptake at high pH and at the solubility limit of the drug. While future work may include direct calculati on of the drug uptake at physiological and at different drug concentrations using a different methodology, the results of the experiments conducted using this me thodology suggest that the DMPC and DMPC:DOPG liposomes will be effective under physiological conditions. It is important to determine the efficacy of the liposomes at drug sequestration in the presence of complex media such as plasma, where the li posomes have to compete with the plasma proteins for drug binding. While further in vitro experiments needs to be performed to gauge the effectiveness of the DMPC and DMPC:DOPG liposomes for use in treatment of amitriptyline drug overdos e, the results of this st udy clearly demonstrate the significant potential of these systems, particular ly in view of the fact that liposomes are highly biocompatible and are already being used for other drug delivery applications.
38 CHAPTER 3 PHYSIOLOGICALLY-BASED PHARMA COKINETIC MODEL OF DRUG DETOXIFICATION Introduction It is relatively straightforward to qua ntify the effectiveness of nanoparticulate systems at sequestering drugs by conducting in vitro experiments; howev er, the uptake of the drug by the NPs is only one of the variab les that determine the efficacy of these systems in treating overdoses. Inside the body, the dynamic changes in the drug concentration will also depend on a number of physiological factors such as the blood flowrates and the partition coefficients of the drug in various tissues. Thus, in order to determine the effectiveness of NPs at treat ing overdoses, one needs to perform animal and human testing. It is not possible to conduct testing on humans until the safety and efficacy of these nanoparticulate systems is firmly established. An alternate means to gauge the effectiveness of NPs is to deve lop a physiologically based pharmacokinetic (PBPK) model to predict the dynamic drug c oncentration in various tissues of an overdosed patient after an injection of NPs. In this chapter, we propose such a model. While numerous others have developed PBPK m odels (22-29), this report is the first to extend these models to include the drug de toxification from the bodyâ€™s blood and tissues following an intravenous injection of NPs. The main objective of this chapter is to illustrate how the addition of NPs to the body may be included into a PBPK model. As an illustration of our model, we show the
39 dynamic drug concentration profiles using partit ion coefficient values for amitriptyline, an antidepressant drug for which there is a need for drug overdose treatment. Modeling We use an eight compartment PBPK model that includes blood, liver, gut, heart, brain, kidneys, muscle, and fat compartments. In general every organ that is expected to have a finite drug uptake should be included in the PBPK model, but since this chapter only intends to illustrate how to include NPs in to a PBPK model, we restrict the model to only eight compartments. The PBPK model for this eight compartment system is available in literature (p. 364 of 30). Below, we develop mass balance equations for the inclusion of NPs into this model. The cl earance of the drug by th e body is neglected in the PBPK model because for many drugs (suc h as amitriptyline) it does not play a significant role in the short time scales of about 15-30 minutes which are relevant in drug overdose treatment. In order to develop a model for drug de toxification by any type of NP, it is important to know whether the NPs will cross the capillary wall and enter the tissue space or remain only in the blood. The capillary wa ll consists of a singl e layer of endothelial cells and a basement membrane. Between adjacent endothelial cells there are intercellular clefts which are approximately 6-7 nm in size (31-33). Any solute that moves between the vascular and tissue spaces must pass through this capillary wall, either by convection through th e intercellular clefts, di ffusion through the endothelial cells, or pinocytosis (33). Most NPs will be too large to pass through the intercellular clefts; however, lipid soluble NPs may diffuse directly through the e ndothelial cells (31). Additionally, some NPs may cro ss the capillary wall via pinocy tosis. This leads to the possibility that some types of NPs will traverse the capillary wall and enter the interstitial
40 space of the tissue. Because some types of NPs will cross the capillary wall and others will not, we develop separate models to simu late the drug detoxification by NP in each case. In developing a drug detoxification PBPK model, there are two important time scales that must be compared. The first time scale, 1, is the time scale for equilibrium to be established between the blood and the NP. Th is time scale is essentially the inverse of the rate constant for drug transport from the blood to the NP. For an effective drug overdose treatment system, 1 needs to be on the order of seconds. The second time scale, 2, is the time scale for equilibrium to be established between the blood and the tissue. This is essentially the inverse of the resistance offered by the capillary walls to drug transport. Since most PBPK models a ssume that the tissue and blood compartments are in equilibrium (12, 30), 2 is of the same order as the time required for blood to flow through the capillaries, which is about 1 to 3 seconds (31). It follows that if 1>> 2, then the NP requires more time to equilibrate w ith the blood than the time the blood (and thus, the NP) is actually inside the capillary space. In this case, we can neglect the uptake of the drug by NP inside the capilla ry blood. Alternatively, if 1 is comparable to 2, there is fast equilibration between the NP and blood, and we must include the drug uptake by NP in the capillary. Since different NP systems may have different 2 values (larger or smaller than 1), models are developed fo r each of these two cases. Based on the above discussion, the addition of the NPs may fall in four different categories depending on whether they enter the tissue space and whether they sequester drugs from within the capillaries. To take into account these po ssible scenarios, three models are developed. Models 1 and 2 repres ent the case in which the NP does not cross
41 the capillary walls and, consequently, does not enter the tissue space. Amongst these, Model 1 reflects the case of slow drug e quilibration between the NP and the blood ( 1>> 2), and Model 2 reflects the ca se of rapid equilibration ( 1~ 2). In Model 3, the NP is assumed to diffuse through the capillary wa lls into the interstitium, and rapid drug equilibration between the blood and the NP ( 1~ 2) is also assumed. For this model, we neglect any barriers offered by th e capillary walls to NP trans port. As a result, the NPâ€™s drug concentration in the inters titial space is assumed to be the same as that in the capillary blood. Though not shown herein, a model in which the NPs enter the tissue space and 1>> 2 can easily be developed by combin ing Models 1 and 3. Below we present mass balance equations for each of the three models discussed above. Model 1: Impermeable Capillary Walls, 1 > >2 In Model 1, the NP sequesters the drug only from within the non-capillary blood vessels. While there is rapid equilibration of the NP with the blood, it is assumed in this model that this equilibration time scale ( 1) is large when compared to the even more rapid 2. In this model, the NP does not enter th e interstitial space and does not sequester the drug while in the capillaries. Thus, the NP effectively stays confined to the blood compartment. In this situation, the drug c oncentration in the tissu es and the NPâ€™s drug concentration individually e quilibrate with the drug concentration in the blood, Cb. The NP may sequester the drug through either an adsorption or absorption mechanism. In the case of an adsorpti on mechanism, we denote the surface drug concentration on the NP as . In the case of an abso rption mechanism, we denote the drug concentration absorbed in the NP as CNP. In all the models, we assume the NPâ€™s drug concentration is in equilibrium with Cb, such that for an adsorption mechanism
42 NPb RC and for an absorption mechanism NPNPbCRC where RNP is the partition coefficient between the NP and the blood. For an adsorption mechanism, the total amount of the drug present in the blood compartment is bbbbVCS VCX (3-1) where Vb is the volume of the blood, S is the to tal surface area of the NPs, and X is a parameter that characterizes the effectivene ss of the NP at binding a particular drug, defined for an absorption mechanism as NP b1S/VR where (S/V)b is the ratio of the surface area of NP to blood volume. For an absorption mechanism, the total amount of the drug present in the blood compartment is bbbNPbbVCVCVCX (3-2) where is the volume fraction of NP in the blood, and X is defined as NP1R . This parameter X may be found from in vitro experimental data and will vary with the type and formulation of NP. In Appendix E, we find X for a specific formulation of oil-inwater MEs. The mass balance for the blood compartment is as follows b i b ibb iL,H,B, i K,M,FdC C XVQQC dtR (3-3) In the above equation, the accumulation term, i. e., the term involving the time derivative, includes both the drug in the blood and also the drug sequestered by the NP. Since in Model 1, the NP does not sequester drugs inside the tissue compartments, the mass balances for the tissue compartments are rather straight forward (the same as in a normal PBPK model) and are as follows.
43 For the gut, heart, brain, kidneys, muscle, a nd fat compartments (i = G, H, B, K, M, and F, respectively) the mass balance equation is ii iib idCC VQC dtR , (3-4) and for the liver compartment the mass balance equation is LG LL LLGbGL GC dCC VQQCQQ dtRR (3-5) In the above equations, the subscripts L, G, H, B, K, M, F, and b represent the liver, gut, heart, brain, kidney, muscle, fat, a nd blood respectively. For a specific tissue i, Ci is the drug concentration in that tissue, Ri is the tissueâ€™s drug partition coefficient, Vi is the volume of tissue, and Qi is the blood flow rate to the tissu e. It is important to note that although the NP does not sequester the drug inside the capillary spaces of the tissues, the drug concentrations of each tissue are affected by the a ddition of NP since the blood compartment is directly linked to all the tissue compartments. Since we assume that the NP and the bl ood are always in equilibrium inside the blood compartment, Cb instantaneously drops to a lower value after the injection. In all three models, if Cb prior to NP injection is Cb,0, then the drug concentration in the blood immediately following NP injection, Cb,I, is such that b ,0 b,IC C X (3-6) Model 2: Impermeable Capillary Walls, 1 ~ 2 In Model 2, the NP is dispersed within th e blood and also sequesters the drug inside the capillary. Thus, the blood with the NP can be treated as a single phase with an overall drug concentration, Cb *, that includes the free drug in the blood, Cb, and the drug concentration adsorbed to/absorbed in the NP . This overall concentration is given by Cb *
44 = CbX. Prior to NP injection Cb * = Cb, and while Cb changes instantaneously once NP is injected, there is no discontinuity in the profiles of Cb * with time. Even after the NP injection, the drug concentration in the tissue, Ci for tissue i, equilibrates with Cb, so that * i iibbR CRCC X (3-7) We define a new effective part ition coefficient for each tissue i, Ri *, as the ratio between the drug concentration of tissue i, Ci, and the total drug concentration in the blood, Cb *. X R Ri i (3-8) The mass balance equations for Model 2 are the same as that of the PBPK model in the absence of NP addition, but Ri and Cb are now replaced by Ri * and Cb *, respectively. The Model 2 equations are as follows. For the blood compartment * * b i b ibb * iL,H,B, i K,M,FdC C VQQC dtR (3-9) For the gut, heart, brain, kidneys, muscle, and fat compartments (i = G, H, B, K, M, and F, respectively) * ii iib * idCC VQC dtR (3-10) For the liver compartment L* G LL LLGbGL ** GC dCC VQQCQQ dtRR (3-11)
45 Model 3: Permeable Capillary Walls, 1 ~ 2 Model 3 reflects the situa tion where the NP sequester s the drug in all the blood vessels, including the capillaries, and permeat es the capillary wall so that is also sequesters the drug from within the interst itial space of the tissue. Since NP is continuously entering and leaving the blood an d tissue compartments, the concentration of NP is constantly changing. Consequen tly, in addition to the drug mass balance, we also need a NP mass balance in each of the eight compartments. We assume equilibrium between the blood leaving the tissue and the NP inside the tissue, so that each tissue i is characterized by a partition coefficient, Ki. For an adsorption mechanism, Ki is defined as the ratio of S/V of NP in tissue i (given by (S/V)i) to S/V of NP in the blood leaving the tissue i. For an absorption mechanism, Ki is defined as the ratio between the volume fraction of NP in tissue i (given by i) to the volume fraction of NP leaving tissue i. As Ki approaches zero, less NP accumulates inside tissue i. We additionally assume the concentration of the drug sequestered by the NP inside the tissue is equal to the NPâ€™s equilibrium drug concentrati on in the blood leaving tissue i, i.e., RNPCi/Ri. As an illustration of our model, the M odel 3 mass balance equations are shown below for the case in which the NP se questers the drug through an adsorption mechanism. In the case of an absorption mechanism, (S/V)i should be replaced by i in the equations below. For the gut, heart, brain, kidneys, muscle, and fat compartments (i=G, H, B, K, M, and F, respectively): i ii b i idS/V 1 VQS/VS/V dtK (3-12)
46 i b NPb b i ii iNPi i NP i i i iiC CS/VRC R dCC d VRS/VQ R C dtdtR S/V KR (3-13) For the liver compartment: G LL LLG GbL GLdS/V Q Q VS/VQQS/VS/V dtKK (3-14) GGNP LL LNP LG LGG LGbNP b NP LL L LLLQCR dCC d VRS/V1S/V dtdtRRK Q_QC1S/VR R QC 1S/V RKR (3-15) For the blood compartment: b i bb ib iL,H,B, i K,M,FdS/V Q VS/VQS/V dtK (3-16) b iii bMEbiME bi iL,H,B, iii K,M,F bbME bdC CQC d VRS/VCQS/VR dtdtRKR QC1S/VR (3-17) We wish to use X as a parameter rather than specify RNP and an initial (S/V)b separately. We therefore multiply E qns. (3-12), (3-14), and (3-16) by RNP and treat RNP(S/V)i and Ci as the unknown variables. On substituting X for NP b1S/VR, RNP(S/V)b simply becomes X-1. Knowing the init ial conditions for all 16 variables, we can solve these equations simultaneously to determine the drug and NP concentrations in various tissues as a function of time.
47 Results and Discussion We now discuss the model predictions for the dynamic concentration profiles in the tissues after a NP injection. Assuming th e toxic drugs donâ€™t signi ficantly deplete organ function, the values used for the volumes and flow rates of each tissue are independent of the type of drug and are given in Table 3-1. Amitriptyline, an antidepressant drug, to used illustrate the model. Ri values for amitriptyline are not explicitly known, so we have calculated them by scaling up thiopental Ri values by a constant f actor, which is obtained by matching amitriptylineâ€™s calculated volume of distribution with th e literature value. These calculated Ri values for amitriptyline are given in Table 3-1. In the remainder of this section, we discuss two cases of drug overd ose treatment. In the first case, the NP is injected a long time after the overdose has occu rred, such that the drug concentrations in the tissues have already reached steady-state. While this is not a very realistic scenario, it helps to identify the key diffe rences between the three models. In the second case, a more realistic scenario is presented, in which the NP is injected a short time after the overdose. Table 3-1. Physiological and pharmacokinetic parameters for various human tissues. Organ Organ Volume1, V (L) Blood Flow Rate1, Q (L/min) Drug-Tissue Partition Coefficient for Amitriptyline2, R Blood 5.4 5.14 -Liver 1.5 1.5 23.58 Gut 2.4 1.2 13.33 Heart 0.3 0.24 11.28 Brain 1.5 0.75 7.18 Kidneys 0.3 1.25 31.79 Muscle 30 1.2 5.13 Fat 10 0.2 79.98 1Values reflect that of a standard man, 30-39 years of age, weighing 70 kg (30) 2Calculated from Reference (24)
48 NP Injected a Long Time After Overdose In the simulations presented below, the PB PK model in the absence of NP was used to determine the steady-state drug concentratio ns for each tissue following a bolus drug dosage which caused Cb0 to be 100 M. Subsequently, a bolus of the NP was introduced. The amount of NP introduces in the bolus is re flected in the parameter X. The equations presented above were solved for each model to determine Ci as a function of time. Three of these plots are shown in Figure 3-1. The zero time on these plots corresponds to the time at which the NP is added. Please note that the drug concentration shown in these plots does not include the drug sequestered by the NP. For comparison, two values of Ki are shown (Ki = 1 and Ki = 100) for Model 3. All three mo dels are expected to yield the same steady-state because in all three models, Ci for each tissue is in equilibrium with Cb, and Cb0 as well as the dosage of NP are the same for each of the model. At the time of NP addition, it is assumed that the drug in the blood immediately equilibrates with the drug on the NP, and thus, Cb drops to a lower value (equal to Cb/X) for all three models, as is shown in Figure 3-1a. Cb then begins to rise as the tissues start losing the drug to establish equilibrium with the blood. As is seen in the Figure 3-1a, Model 2 shows the quickest concentration re sponse; its predicted Cb increases faster than in either of the other two models. The concen tration predicted by Model 2 reaches the steady-state value the fastest, while the Model 1 profiles take th e longest to equilibrate. This is reasonable because the NP in Model 1 is not equilibrati ng with the blood inside the capillaries, and thus it takes a longer time for the blood, tissues , and NP to equilibrate with each other. The fact that Model 3 always has a slower response than Model 2 may be counterintuitive because one may expect that if the NP is al so taking up the drug from inside the tissue
49 then the concentration responses will be fa ster; however, this is not the case. A distinction is not made between the tissue space and the capillary sp ace in the models; thus in both Models 2 and 3, the NP in the tissue compartment equilibrates with the blood. In Model 2, after the NP sequesters the drug in the capillaries, it leaves the tissue compartment and enters the blood compartmen t. Once inside the blood compartment, it releases some of the drug that was sequester ed in the tissue to the blood, which has a drug concentration lower than the equilibrium value. Thus, the NP transfers the drug from the tissue to the blood compartment. After rele asing the drug in the blood compartment, the NP can then circulate through the body, ente r another tissue compar tment, and transfer more of the drug to the blood compartment. Consequently, the NP in Model 2 acts as a â€œtransfer agentâ€, which speeds up the transfer of the drug from the tissue to the blood. In Model 3, some of the NP ente rs the interstitial space of the tissue, causing less NP to leave the tissue compartment and enter the blood . It follows that in Model 3 the NP is not as effective a transfer agent, and the time to reach steady-state is longer. As the Ki values in Model 3 increase, the amount of NP that enters the inte rstitial space of the tissues increases, and a longer amount of time is needed to attain steady-state. As the Ki values approach zero, the plots for Model 3 approach that of Model 2. The dynamic drug concentration in the hear t tissue, starting from its equilibrium value (Figure 3-1b), follows trends similar to those seen in the other tissues, with the exception of the fat tissue (Figure 3-1c). Th e drug concentration in the heart tissue, CH, decreases immediately following NP administration, reaches a minimum value, and then starts to increase. All other tissues except the fat follow this trend. The drug concentration in the fat tissue, CF, decreases extremely slowly until the steady-state value
50 is reached. The long equilibration time of the fat tissue (about 500 minutes for Model 1 and 4000 minutes for Models 2 and 3) can be attributed to the large value of RFVF/QF. At the time CH has reached its minimum values, CF is about the same as its initial value. Thus, the minimum CH value represents the equilibriu m value of the drug concentration in the heart in the absence of the fat tissue. Beyond this ti me the fat tissue continues to release the drug to the blood. A fraction of the drug that is released by the fat to the blood compartment is taken up by the other tissues to re-establish equilibrium with the blood. Beyond the time at which the minimum occurs in the concentration profiles, the tissues (except fat), the blood, and the NP are essentially in equilibrium. This dynamic behavior is a pseudo-steady-st ate driven by the slowest time scale of the fat tissue.
51 0 5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Time (min)Amitriptyline Concentraion (M)Model 1 No NP Model 3 K=100 Model 2 Model 3 K=1 0 5 10 15 20 25 30 3.5 4 4.5 5 5.5 6 Time (min)Amitriptyline Concentraion (M)Model 1 No NP Model 2 Model 3 K=1 Model 3 K=100 0 5 10 15 20 25 30 40.6 40.7 40.8 40.9 41 41.1 41.2 41.3 Time (min)Amitriptyline Concentraion (M)No NP Model 2 Model 3 K=1 Model 3 K=100 Model 1 a) b) c) Figure 3-1. Amitriptyline concentration as a f unction of time following an injection of NP after all tissues have reached their respective equilibrium values. The figures shows the drug concentration in (a) the blood, (b) the heart tissue, and (c) the fat tissue for an X=10 NP.
52 NP Injected a Short Time After Overdose In the simulations shown below, at time zero all Ci values are zero and Cb0 is 10 M to reflect a bolus overdose. The NP is then introduced at a finite time. The Ki values in Model 3 will vary with the type of NP, but as an illustration of our model, unless otherwise specified, we have taken the Ki values to be equal to the Ri values shown in Table 3-1. In the simulations shown in Figures 3-2 â€“ 3-4, the profiles from all the models show a decrease in the drug concentration at long time scales and reach the same steadystate concentration. Additionally, the Model 2 profiles reach steady-state the fastest. We are primarily concerned with th e trends in the drug concentr ations in the blood and in the heart, as Cb directly affects all the tissue concen trations, and the heart is the primary organ that determines whether or not an overdose of amitriptyline is lethal. 0 5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Amitriptyline Concentration (M)Time (min) Model 2 X=10 Model 2 X=2.2 Model 1 X=10 Model 3 X=10 Model 3 X=2.2 Model 1 X=2.2 No NP Figure 3-2. Amitriptyline concen tration in the blood for an NP injection at 5 minutes.
53 Figure 3-2 shows the dynamic Cb for NP addition at 5 minutes for all the three models and for two different values of X (X = 2.2 and X = 10). An X value of 1.5 was calculated for one formulation of ME, shown in Appendix E. As is shown in Figure 3-2, Cb immediately drops to a lowe r value at the time of the NP addition due to the drug sequestration. The concentration drop is the same for all the three models, but larger for larger values of X. This sudden drop reduces Cb to below the value of the equilibrium drug concentration for most tissues. Accordi ngly, the tissues begin to release the drug into the blood, and Cb begins to increase. The loss of the drug from the tissues stops and a maximum value of Cb is reached. At this point, all the tissues (with the exception of the tissues with a very large equilibration time, su ch as the fat) are in equilibrium with the blood. Beyond this time, the blood and all the other tissues lose some of the drug to the fat, causing the drug concentrations to decrease slowly. These trends are the same for all the models and for both values of X. As expected, the Model 2 profiles reach the maximum and final steady-state values faster than the Model 1 and 3 profiles. The CH profiles are shown in Figure 33a and 3-3b. The only differe nce between the two graphs is the time at which the NP is added. In the absence of the NP addition, CH initially increases after the overdose and then begins to decrease as the slowly equilibrating organs begin to absorb the drug. The prof iles in Figure 3-3a correspond to NP addition at 1 minute when CH is still increasing, and the simulations shown in Figure 3-3b correspond to the case when NP addition occurs at 5 minutes, at which time CH has already started to decrease. These plots show the importance of both the values of X and the time at which the NP is added. At 1 mi nute, the heart tissue is still taking up the drug from the blood (Figure 3-3a). When NP is a dded at this point in time and if the X value
54 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 Amitriptyline Concentration (M)Time (min) No NP Model 1 X=2.2 Model 2 X=10 Model 3 X=10 Model 1 X=10 Model 3 X=2.2 Model 2 X=2.2 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 Amitriptyline Concentration (M)Time (min) Model 3 X=2.2 No NP Model 1 X=2.2 Model 1 X=10 Model 2 X=2.2 Model 3 X=10 Model 2 X=10 a) b) Figure 3-3. Amitriptyline concentration in the heart tissue for NP addition at different times. The NP is added at (a) 1 minute and (b) 5 minutes.
55 is small, such as 2.2, CH increases after NP addition, though at a slower rate than in the absence of NP. CH reaches a maximum value and then decreases as the heart loses the drug to the blood. With an X value of 2.2, there is not a significant decrease in CH for Models 1 and 3, while Model 2 shows about a 1 M reduction at the end of 30 minutes. This happens because concentrations in Model 2 equilibrate quicker than those in Models 1 and 3. Even though CH is still increasing at the time of NP addition (1 minute), if the X value is large enough, such as a value of 10, then the dr ug concentration begins to decrease immediately following NP administra tion. All models show a decrease in CH immediately following NP addition for X= 10, with Model 2 showing the fastest reduction. Thus, the concentration reduction is quicker for higher X values. In the absence of NP, CH has already started to decrease at 5 minutes. When NP is added at this point (Figure 3-3b), the concentration reduction is faster after NP addition than in Figure 3-3a for all values of X. Larg er values of X lead to larger concentration reductions. It is important to note that for each X value, all three models will reach the same steady-state in each tissue; however, only the short time responses are important in drug overdose patients. Even though Models 1 and 3 will even tually lead to considerable decreases in the drug concentrations for an X value of 2.2, the response time is longer than 30 minutes. Consequently, a signifi cant decrease in the drug co ncentration is not shown in Figure 3-3 for Models 1 and 3 at X = 2.2. A comparison of Figures 3-3a and 3-3b shows that, despite these opposing tre nds between NP addition at 1 and 5 minutes for X values of 2.2, the resulting concentr ation values at 30 minutes are relatively the same. Depending on when the NP is added, an interesting behavior sometimes occurs in the liver, muscle, and fat tissues for Models 2 and 3; the addition of NP sometimes causes
56 an increase in Ci beyond the value of the Ci value in the absence of NP. This trend occurs for Model 2 and for low values of K in Model 3. An example of this trend is illustrated in Figure 3-4, which shows the amitriptyline concentration inside the liver tissue, CL, for NP addition at 5 minutes. CL for Model 2 exceeds that of the case with no NP. It is important to note that while the drug concentrations in Model 2 and 3 are sometimes larger than the case in which the NP is not administered, the steady-state drug concentrations are always lower. Moreover, the increased concentra tion value as a result of NP addition never exceeds the maximum valu e of concentration for the case when the NP is not added. In order to explain the reasons behind th is drug concentration increase after NP addition, the tissue mass balance for the case wh ere no NP is present (Eqn. (3-18)) must be compared with the tissue mass balance for Model 2 (Eqn. (3-19)). For simplicity, we will refer to Ci as the drug concentration inside tissue i for the case where no NP is present and i C as the drug concentration in tissue i when a Model 2 NP is present. These equations are ii iib idCC VQC dtR (3-18) and * ii iib i dCCX VQC dtR (3-19) In Eqn. (3-19), Cb * is the effective drug concentr ation in the blood, equal to b CX, i.e., the sum of the free and the NP-seque stered drug in the blood. As discussed previously, the drug concentrati on in the blood changes from Cb,0 to Cb,0/X, i.e., Cb,I, at
57 the time of NP injection. If the X value is large enough, Cb,I will be lower than the equilibrium concentration for certain tissues, i.e., Cb,I < Ci/Ri. This is true for tissues that take up the drug quick ly, i.e., where RiVi/Qi is small. As a result, the blood will start to sequester the drug from these tissues. These ti ssues may lose so much of the drug to the blood that Cb * becomes larger than Cb. The tissues with small drug concentrations, i.e., those with large RiVi/Qi values, will then take up more of the drug than what they would have taken in the absence of the NP, causing i C to become larger than Ci (Figure 3-4). Mathematically, this behavior may be expl ained by a comparison of Eqns. (3-18) and (3-19). If Cb * becomes larger than Cb, there will be a tendency for the concentration inside the tissue to increase at a faster rate. This is compensated by the fact that i i CX R is Figure 3-4. Amitriptyline concentration in the liver for NP injection at 5 minutes. 0 5 10 15 20 25 30 0 1 2 3 4 5 6 Amitriptyline Concentration (M)Time (min) Model 2 X=2.2 No NP Model 3 X=2.2 Model 2 X=2.2 Model 2 X=10 Model 1 X=10 Model 3 X=10
58 larger than Ci/Ri, since X is a positive value larger than 1. In tissues with large RiVi/Qi values (such as the fat, liver, and muscle), Ci is approximately zero at the time of NP addition, meaning i i CX R and Ci/Ri are both about zero. Thus, i C is larger than Ci for a short time span following the NP administration. This same trend occurs for small values of K in Model 3. It can be seen in Figure 3-4 that at 20 minutes, i C for the X = 10 NP was about 2.8 M below Ci; however, i C for the X = 2.2 NP was slightly above Ci. At the end of 60 minutes, both values of X result in a decreased i C inside the liver. We now discuss the effects of variations in the Ki values on the predictions of Model 3. To evaluate the eff ect of the magnitude of the Ki values, simulations are performed in which all the tissues have the same Ki value. These simulations were repeated for different values of Ki (10, 100, and 1000) and show that as Ki get smaller, Ci changes faster. This is demonstrated in Figure 3-5, which shows the amitriptyline concentration in the liver tissue, CL, for NP addition at 5 minutes for an X value of 10. It is important to note that, for a constant X va lue, all the Model 3 concentration profiles reach the same steady-state inside each ti ssue, regardless of the values of Ki, with the smaller Ki values reaching steady-state quicker.
59 Figure 3-5. Amitriptyline concentration in the liver tissue for NP addition at 5 minutes for an X=10 NP. Conclusions The goal of this chapter is to develop a PB PK model that can be used to predict the kinetic aspects of drug-overdose treatment by NPs. Depending on the properties of the NPs, such as the size and the surface characteri stics, the NPs may enter the tissue space. Furthermore, depending on the time scales of the drug-uptake by the NPs, they may be assumed to be confined to the blood compar tment. Corresponding to the two properties listed above, four different scen arios are envisaged and models are developed for three of these cases. Since comparison with anim al experiments has not yet been done, the models should only be used as guidelines for determining the efficacy of NPs on detoxification. This chapter illustrates th e methodology of inclusion of NPs in a PBPK 5 10 15 20 25 30 3 3.5 4 4.5 5 5.5 6 6.5 Amitriptyline Concentration (M)Time (min) No NP K=10 K=100 K=1000
60 model, and the dynamic concentration profiles presented here help us to understand the effect of various parameters on detoxification. The simulations show that the detoxificati on process is quicker if the NP does not enter the interstitial space of the tissue a nd if the NP equilibrates with blood in the capillaries, i.e., 1 is comparable to 2. Thus, it may be preferable to design NPs that do not cross the capillary wall. It is important to note that we have not included the kinetics of drug transport across the capillary wall in our models. If the mass transfer across the capillary wall slows down the dr ug transport between the tissue and blood spaces, quicker detoxification may occur using a NP that easi ly traverses the capillary wall and directly sequesters the drug from the interstitium. We have also determined that the effectiv eness of the NP at detoxification can be characterized by the parameter X, and larger X values lead to a more effective and rapid detoxification processes. For each drug, th e X value depends on the dosage of the NP administered and the drugâ€™s partition coeffi cient between the NP and the blood. The maximum allowable dosage of NP will be a fixed quantity, but developing NPs that have a larger affinity for the drug may increase th e effectiveness of the NP at treating drug overdoses. The model predicts that for amitriptyline, a NP with an X value of at least 10 is required to treat an overdose. Notation = concentration of the drug ad sorbed on the surface of the NP i = surface drug concentration of the NP in tissue i, 1 = time scale for the drug sequestered by the NP to equilibrate with the drug in the blood
61 2 = time scale for the drug in the blood to e quilibrate with the drug concentration in the tissue = volume fraction of the NP in the blood i = volume fraction of the NP in tissue i Cb = drug concentration in the blood Cb,0 = drug concentration in the blood prior to NP injection Cb * = overall drug concentration in the blood which includes the free drug in the blood and the concentration of the drug sequestered by the NP Cb,I = drug concentration in the blood immediately following NP injection Ci = drug concentration in tissue i i C = drug concentration in tissue i when a Model 2 NP is present CNP = drug concentration absorbed in the NP Ki = ratio of the equilibrium concentration of NP in tissue i to the equilibrium concentration of NP in the blood Qb = volumetric flowrate of the bl ood leaving the blood compartment Qi = volumetric flowrate of the bl ood leaving tissue compartment i Ri = drug-tissue partition coefficient of tissue i Ri * = effective drug-tissue partition coe fficient of tissue i used in Model 2 RNP = drug partition coefficient be tween the NP and the blood S = total surface area of the NPs (S/V)b = surface area of NP per unit volume in the blood (S/V)i = surface area of NP per unit volume in tissue i Vb = volume of blood
62 Vi = volume of tissue i X = drug-dependent property of the NP describing effectiveness at drug uptake
63 CHAPTER 4 DISPERSION IN THE CAPILLARIES Introduction The transport of solutes in geometries with large aspect ratios, such as the capillary, can usually be described in terms of an effective dispersion coefficient, frequently referred to as the Taylor disp ersion coefficient, which quantifies the spread of a pulse of solute in the axial direction. Aris first m odeled the problem of dispersion in a tube for a Newtonian fluid and obtained the classical result that the dispersion coefficient, D*, is given by 2 2uR D+ 48D (34), where D, u, and R are the molecular diffusivity, the mean fluid velocity, and the tube radius, respect ively. Since then, many researchers have studied dispersion of various cla sses of fluids in several geom etries and in different time regimes. It has since been shown that the Taylor dispersion coeffi cient is the long time asymptotic limit of the generalized dispersion coefficient and is valid only at times that are much larger than the time the solute requ ires for equilibration in the lateral direction (35-37). A number of impor tant problems in chemical and biomedical engineering concern geometries with aspect ratios in wh ich the convective time scales for axial flow are much larger than the latera l equilibration time. In such problems, the Taylor limit of the dispersion coefficient is of relevance. In this chapter we determine the Taylor dispersion coefficient for a solute in a coreannular geometry, where the solute diffuses in to the solid annulus. This geometry is simlar to that of a capillary with surrounding tissue. Aris fi rst solved this problem in a
64 core-annular geometry by using the method of moments. He developed the general expressions that are valid for any velocity profile in the core and the annulus, and developed the expression for so me special cases including the case of Poiseuille flow in the core and no-flow in the annulus (38). Levitt recognized th at the core-annular geometry considered by Aris is identical to the geometry of the Krogh cylinder that is used to model the microcirculation and capillary-tissue exchange kinetics. Levitt modeled the capillary-tissue exchange of solutes by using Arisâ€™s results and also recognized that in certain cases Arisâ€™s mode l might not be appli cable in the capillarytissue system because of finite capillary le ngths and relatively small capillary wall permeabilities to certain solutes (39). Ba sed on the perturbation approach proposed by Gill and Sankaras (36), Tepper et al. modi fied Arisâ€™s results to include the timedependence of the dispersion coefficient for plug flow in the capillary (40). These results can be applied in cases where Arisâ€™s results are not valid because of differences between the geometry of the Krogh cylinder and the in finite geometry considered by Aris. In particular Tepper et al. suggested that for cap illary wall permeabilitie s smaller than about 10-5 cm/s, the time dependency of the dispersi on coefficient is important, and Arisâ€™s results cannot be used. In their analysis, Tepper et al. pointed out that the physical significance of the perturbation parameters is not simple, and that for a given set of parameters it is a priori not clear whether Arisâ€™s results are sufficient or the timedependent coefficients need to be used. In this chapter we obtain the dispersi on coefficient for a Krogh tissue cylinder geometry with an interfacial mass-transfer resistance by using a perturbation expansion in the aspect ratio. This method allows for eas y determination of the regimes in which the
65 results are applicable. We first consider solutes with small interfacial mass-transfer resistance and recover Arisâ€™s results. A dditionally, we consider the case of large interfacial mass-transfer resi stance and show that this problem can also be solved effectively by the method of perturba tion expansion in the aspect ratio. In the next section, the expressions for the average mass transfer equations are obtained for the entire range of interfacial mass transfer resistances. Two different velocity profiles in the capillary are considered ; Poiseuille and plug. Next, for the case of small interfacial mass transfer resistance, th e dispersion coefficients are separated into contributions from effective molecular diffusi on, interfacial mass tran sfer resistance, and convection. The mechanisms that contribute to each of these are discussed. Asymptotic expressions for the dispersi on coefficients are derived to elucidate the various mechanisms and the associated time scales th at contribute to disp ersion. Additionally, the dispersion coefficient resulting from fl ow through a channel surrounded on each side by a solid, which is the two-dimensional (2 D) equivalent of co re-annular flow is determined. Exact and asymptotic expressions are also derived for this geometry in order to understand the time scales that lead to the dispersion. Next, the effect of various physical parameters on dispersion is investigat ed for core-annular Poiseuille flow, and the differences in dispersion between Poiseuille and plug flows are examined. Finally, the relevance of core-annular flow to transport of solutes in the capillar ies is discussed, and the theoretical results are compared with some experimental values. Dispersion Analysis Governing Equations The core-annular geometry consisting of a tube surrounded by a solid annulus is shown in Figure 4-1. The inner and outer cylinders are of radii R and aR, respectively.
66 The diffusion coefficients of the solute in th e fluid and in the solid annulus are D and Dt, respectively. ANNULUS a R TUBE L r z R ANNULUS Figure 4-1. Tube-annulus geometry. The transport of solute in the tube an d the annulus is governed by the convectiondiffusion equations, which are 2 z 2CC1CC uDr tzrrr z (4-1) and 2 t 2 t t tz C r C r r r 1 D t C (4-2) In the above equations uz is the velocity of fluid in the tube in the z-direction, and C and Ct are the solute concentrations in the fluid in the tube and in the annulus, respectively. Herein we assume that there is no convection in the annulus a nd that the radial flux of the solute at the outer bound ary (that is, at r = aR) is zero. Method of Solution Small Interfacial Mass Tran sfer Resistance: O(1) Bi To facilitate solving Eqns. (4-1) and (4-2), it is conveni ent to introduce a pulse of solute at t = 0 at a certain spatial location a nd follow the pulse in time. The pulse moves in the axial direction with a velocity, u, and disperses into a Gaussian shape with a
67 dispersion coefficient, D* (Figure 4-2). We reformulate the problem in a reference frame moving in the z-direction with a mean velocity, u, so that *zzut , where z* is the axial coordinate in the moving reference frame. Figure 4-2. A pulse of solute is introduced at z = 0 and then disperses at it travels down the length of the tube. We de-dimensionalize the governing equations as follows: ** ** t zt 2** zu C tDzruCD t,u,z,r,u,C,C,D LuLRuCCD (4-3) where L is the length of the tube , R is the radius of the tube, u is the average velocity of the fluid in the tube, and C* is a reference concentration. Because we are only interested in the longtime asymptotic limit of the dispersion, time has been dedimensionalized by L2/D. Additionally, we define a small dimensionless perturbation parameter, L R , and the Peclet number as uR Pe D . The dimensionless forms of Eqns. (4-1) and (4-2) in the moving reference frame are 2 * 2 2 * zz ~ C ~ r ~ C ~ r ~ r ~ r ~ 1 1 z ~ C ~ u ~ u ~ Pe t C ~ (4-4)
68 and 2 tttttt *2*2CCDCDC Pe11 ur t z DrrrDz (4-5) In the above equations, u ~ is an unknown constant, which must be found before D* can be determined. Equations (4-1) and (4-2) ar e subject to the follo wing boundary conditions: 1. At the center of the tube, r ~ = 0, the concentration is finite, or equivalently 0 r ~ C ~ . 2. We assume that the tube-annulus interface offers a resistance to solute transport characterized by a mass transfer coefficient, . We neglect solute adsorption at this interface, so that the radial flux of solu te is continuous. It follows that the boundary condition at r ~ = 1 is tt tDC C Bi(KCC) rDr , where K is the soluteâ€™s partition coefficien t between the tube and surrounding annulus, and Bi is the dimensionless masstran sfer coefficient, given by R D . 3. There is no flux out of the annulus at the perimeter of the tube, i.e., 0 r ~ C ~ t at r ~ = a. Because <<1, we expand C ~ and tC ~ in regular expansions as follows 2 2 1 0C ~ C ~ C ~ C ~ (4-6) t2 2 t1 t0 tC ~ C ~ C ~ C ~ (4-7) We now substitute the above expansions into Eqns. (4-4) and (4-5), and use the boundary conditions to solve these equations to various orders of . O(1/2) To 21 O , Eqns. (4-4) and (4-5) are r ~ C ~ r ~ r ~ r ~ 1 00 (4-8)
69 r ~ C ~ r ~ r ~ r ~ 1 0t0 (4-9) Equations (4-8) and (4-9) along w ith the boundary conditions yield t0C ~ = K0C ~ . Thus, the leading order concentrations in the tube and the annulus do not vary in the radial direction and are in equilibrium. O(1/) To 1 O , Eqns. (4-4) and (4-5) become r ~ C ~ r ~ r ~ r ~ 1 z ~ C ~ u ~ u ~ Pe1 * 0 z (4-10) t0tt1 *CDC 1 -Peur zDrrr (4-11) The constant u ~ is obtained by multiplying Eqns. (4-10) and (4-11) by r ~ , integrating Eqn. (4-10) from r ~ = 0 to r ~ = 1, integrating Eqn. (4-11) from r ~ =1 to r ~ = a, and then adding the two resulting equations. By fo llowing these steps and applying the boundary conditions, we obtain th e following equation. 1 2 z 01 urdr1K1u 2a (4-12) We use the fact that 1 z 01 urdr 2 , and obtain the following expression for the dimensionless mean velocity. 21 u 1K1a (4-13)
70 This result is independent of the velocity profile in the tube. The *z dependency of Eqns. (4-10) and (4-11) can be satisfied by letting * 0 1z ~ C ~ ) r ~ B( C ~ and * 0 t t1z ~ C ~ ) r ~ ( B C ~ , where B and Bt are functions of r . B and Bt can be determined by substituting the assumed expressions for 1C ~ and t1C ~ into Eqns. (4-10) and (4-1 1) and solving the resulting equations for B and Bt using the appropriate bounda ry conditions. The resulting expressions for B and Bt are 4 2 1 2PerPe11 B1rc 822 1K1 a (4-14) 42 2 t 2 2 2 2 tK1K Dr BPeKlnrc D 41K1 21K1 aa a a (4-15) where c1 and c2 are constants. Applying the boundary condition that tB Bi(KB-B) r at r ~ = 1, we obtain the following rela tionship between the two constants. 1 2 t 2 2 1 22D PeK3K121 D PeK1 cKc 81K12Bi1K1 a a aa (4-16) The two constants c1 and c2 cannot be determined from the last boundary condition (1C ~ = Kt1C ~ at r ~ = 1) because the choice of mean velocity ensures that it is already satisfied; however, separate values of c1 and c2 are not needed for the calculation of D*. If desired, c1 and c2 can be determined by using a normalizing condition 1 1t1 01CrdrCrdr0a which
71 is equivalent to stating that the total mass in the system at O( ) should be zero because the incoming and the outgoing masses at the boundaries are exactly O(1). O(1) To O(1) Eqns. (4-4) and (4-5) become 2 * 0 2 2 * 1 z 0z ~ C ~ r ~ C ~ r ~ r ~ r ~ 1 z ~ C ~ u ~ u ~ Pe t ~ C ~ (4-17) and 2 t0t1tt2tt0 **2CCDCDC 1 Peur tzDrrrDz (4-18) We substitute for 1C ~ in terms of B, for t1C ~ in terms of Bt, and for t0C ~ as K0C ~ . Next, we multiply both equations by r ~ , integrate Eqn. (4-17) from r ~ = 0 to r ~ = 1, and integrate Eqn. (4-18) from r ~ = 1 to r ~ = a. Adding the integrated E qns. (4-17) and (4-18) gives 2 * 0 2 * 0z ~ C ~ D ~ t ~ C ~ (4-19) where the mean dimensionless dispersion coefficients, *D ~ , for Poiseuille (* p oD) and plug flow (* p lD) are given by
72 2 t * t po 2 111 42 ttt 2 2 22 111 42 ttt 2 22D 11Pe4848 D D D D 481K(1) DDD 12ln912838 DDD Pe 24 11K(1) DDD 4ln3411 DDD Pe 8 11K(a aaa aa aaa aa 3 2 22 3 21) KPe1 2Bi1K(1)a a (4-20) and 2 t * t pl 2 111 42 ttt 2 2 22 111 42 ttt 2 22D Pe88 D D D D 81K(1) DDD 4ln3422 DDD Pe 8 11K(1) DDD 4ln3411 DDD Pe 8 11K(a aaa aa aaa aa 3 2 22 3 21) KPe1 + 2Bi1K(1)a a (4-21) The above results agree with the results of Aris for Poiseuille flow (29) and that of Levitt for plug flow (39). Large Interfacial Mass Transfer Reisistance: O( ) Bi We now solve the problem for solutes that have a large interfacial mass-transfer resistance, where Bi Bi In this case the time scale for transport across the tube-
73 annulus interface (R/ ) is much slower than the time scale for lateral diffusion (R2/D) but is still faster than the axial diffusion time scale (L2/D). Accordingly, at the shortest time scale of R2/D, the tube and the annulus are not ex pected to be in equilibrium, and the intermediate time scale of R/ must be introduced into the problem. We do this assuming the concentration in the tube and the annul us are functions of st t/R,2 ltDt/L, rr/R , and L / z z ~ . It is noted that this problem is solved in a stationary reference frame. Given that slCC(t,t,z,r) and ttslCC(t,t,z,r) , it follows that 2 lsCDC C + tLtRt (4-22) and ttt 2 lsCCC D tLtRt (4-23) Thus, the dimensionless forms of the govern ing equations Eqns. (4-1) and (4-2) are 2 z 22 lsC1CPeC11CC Biur t t z rrrz (4-24) and 2 tttttt 22 lsCCDCDC 111 Bir t t DrrrDz (4-25) The dimensionless boundary conditions are: 1. C 0 r at r0 2. tt tDC C Bi(KC-C) rDr at 1 r ~ 3. tC 0 r at r a
74 A set of equations for various orders of are obtained by substituting the regular expansions of C ~ and tC ~ into Eqns. (4-24) and (425) and applying the boundary conditions. O(1/2) To 21 O Eqns. (4-24) and (4-25) are r ~ C ~ r ~ r ~ r ~ 1 00 (4-26) r ~ C ~ r ~ r ~ r ~ 1 0t0 (4-27) Equations (4-26) and (4-27) along with the boundary conditions yield t0t0lsCC(t,t,z) and 00lsCC(t,t,z) . This shows that, similar to the small resistance case, the leading order concentrations in the tube and the a nnulus do not vary in th e radial direction; however, these concentrations are no longer in equilibrium as they were in the O(1) Bi case. O(1/) To 1 O Eqns. (4-24) and (4-25) are 00 1 z sCC C 1 Bi+Peu=r tzrrr (4-28) and t0tt1 sCDC 1 Bi=r tDrrr (4-29)
75 Averaging Eqns. (4-28) and (4-29) (that is , multiplying Eqns. (4-28) and (4-29) by r and then integrating Eqn. (4-28) from r = 0 to r = 1 and Eqn. (4-29) from r = 1 to r = a) and applying the boundary conditions gives the followi ng expressions: 00 0t0 sCC BiPe Bi(KCC) 2t2z (4-30) and 2 t0 0t0 sC 1 (KCC) 2ta (4-31) Rearranging Eqns. (4-30) and (4-31) to eliminate s 0t ~ C ~ and s 0 tt ~ C ~ from Eqns. (4-28) and (4-29) gives 0 1 0t0zC C 1 2Bi(KC-C)Pe(u1)r zrrr (4-32) and tt1 0t0 2DC 2Bi1 (KC-C)r 1Drrra (4-33) Based on the above differential equations we postulate the following form for 1C ~ and t1C ~ : 0 120t01C CBiB(r)(KC-C)+PeB(r) z (4-34) t12t0t0CBiB(r)(KC-C) (4-35) Substituting these postulated forms into Eqns. (4-32) and (4-33), we get the following differential equations for B1, B2 and B2t.
76 r ~ B r ~ r ~ r ~ 1 ) 1 u ~ (1 z (4-36) 2B 1 2r rrr (4-37) t2t 2DB 21 r 1Drrr a (4-38) The boundary conditions are 1B 0 r at r = 0 , 2B 0 r at r = 0 , and 2tB 0 r at ra . For Poiseuille flow 2 r ~ 2 u ~ 2 z . The above equations can be integrated to give the following results: 3 4 2 18 r ~ 4 r ~ B (4-39) 2 21r B 2 (4-40) 11 22 tt 2t2 22DD r Blnr D2(-1)D(-1) a aa (4-41) where 1, 2, and 3 are constants of integration. Conservation of mass at O( ) requires 1 1 0Brdr0 and 1 22t 01BrdrBrdr0a . Thus, 42 1 2 t 12 24ln()321 D 11 1 4D14aaa a a (4-42) and 3 = -1/12. O(1) To O(1) Eqns. (424) and (4-25) are
77 2 00 112 z 2 lsCC CCC 1 BiPeur ttzrrrz (4-43) and 12 t t0tt2tt0 2 lsC CDCDC 1 Bir ttDrrrDz (4-44) Plugging in for 1 sC t , 1C z , and t1 sC t , and applying the boundary conditions, Eqns. (4-43) and (4-44) become 2 2 00 2 t00 10t01 lCC Pe1BiKPe 962z12z CC BiPe11 MKCC 0 26z2t (4-45) and 2 2 tt00 1 2 2 t0 10t0 l1 DCC PeKBi 3 1 D2z6z 1 C MKCC0 2t a a (4-46) where M1 is a constant given by -1 42 2 t 1 2 24ln()341 D Bi MK 4D -1 aaa a (4-47) Recall, 0C and t0C depend on two time scales, the shorter time scale (ts) and the longer time scale (tl). We are now interested in comb ining the expressions for these two time scales to obtain an expression which is va lid at all times. We do this by using Eqns. (4-22) and (4-23) to determin e the derivatives of the leading order concentrations with
78 respect to the total time scale (t), that is, 0C t and t0C t . The expressions for 0 sC t and t0 sC t are found from Eqns. (4-30) and (4-31), and the expressions for 0 lC t and t0 lC t are found from Eqns. (4-45) and (4-46). The expression for 0C t is 2 2 000t0 10t01 22 0 0t0CCCC DPeBiKPe1 12MKCCBiPe tL48z6z6z C Pe 2(KCC) RBiz (4-48) Redimensionalization gives 00t0 1 2 2 0 1 0t0 22CCC Bi K1 uPe t6z6z C 2DM Pe2 D1KC-C 48zLR (4-49) The coefficient of 0C z in Eqn. (4-49) simply reduces to -u because is a small parameter. Similarly, the coefficient of t0C z can be written as 11 uBi 6 , which is a small quantity and can be neglec ted. Additionally, the term 2DM1/L2 can be expressed as 1M Bi 2 R and is therefore negligible in comparison to 2 R . The coefficient in front of (KC0 â€“ Ct0) can be reduced to 2 R , and Eqn. (4-49) becomes 2 2 000 0t0 2CCC Pe2 uD1KCC tz48zR (4-50)
79 By a similar approach the equation for t0C t becomes 2 t0t0 t0t0 2 2CC 2 DKCC tz R1 a (4-51) The above derivation was for the case of Poiseu ille flow. For the case of plug flow and O( ) Bi, the average equations are the same as Eqns. (4-50) and (4-51) except that in Eqn. (4-50) the term Pe2/48 is eliminated. Results and Discussion Range of Validity of the Average Equations Because the average equations have been derived by using regular expansion, it is clear that all the results are strictly valid onl y if each of the dimensionless parameter is O(1), that is, they should be much smaller than 1/ and much larger than . Furthermore, if any parameter become too small or too la rge, it should be scal ed by the appropriate order in . To demonstrate this, let us consider the case of a typical capillary with a value of 10-7 m/s. As noted by Tepper et al., the re sults of Aris are no longer accurate for this value of . The typical capillary radius is about 5 m and the typical capillary length is about 1 mm. Thus, the value of is about 5x10-3. Using a value of 10-9 m2/s for D, a value of 10-7 m/s corresponds to a Bi value of 0.5x10-3, which is clearly O( ). Thus, it is clear that Arisâ€™s results will not be valid for this value of Instead the average equations for O( ) Bi which are derived herein can be used to find the concentration profiles.
80 Dispersion Coefficient for La rge Permeability: O(1) Bi To investigate the dependence of *D ~ on the parameters Pe, Bi, K, a , and tD D , it is useful to separate *D ~ into the contributions from effective molecular diffusivity (* 0D ~ ), from the interfacial resistance to mass transfer (* RD ~ ), and from convection (* c 2D ~ Pe) as follows ***2* 0RcDDDPeD (4-52) We note that the expressions for * 0D ~ and * RD ~ are the same for both Poiseuille and plug flow. The dimensionless quantities in Eqn. (4 -52) can be converted to the dimensional form by multiplying them by D. Contribution to Dispersion from Molecular Diffusivity (* 0D) For both Poiseuille and plug flow, * 0D ~ is given by t * t 0 2D 1 D D D D1K(-1) a (4-53) As K( a2-1) approaches infinity, there is no annulus surrounding the tube, and * 0D ~ will approach D Dt. Alternatively, as K( a2-1) approaches 0, the tube is of a negligible size compared to the annulus, and * 0D ~ will approach 1. Thus, D0 *, the dimensional * 0D ~ , varies from D to Dt as 2K(-1) a varies from 0 to infinity. Contribution to Dispersion fr om Interfacial Resistance (* RD) The dispersion caused by the interfacial resistance, * RD ~ , is given by
81 2 22 * R 3 2KPe-1 D= 2Bi1+K(-1) a a (4-54) * RD ~ is expected to only depend on the mean velocity, u, and not on the velocity profile. To whether this is the case, we obtain the dispersion coefficient for the same geometry in the limiting case when both the fluid in the tube and the solid annulus are radially well mixed. In this limit, the dime nsionless governing equatio ns for the tube and the annulus are 2 t *2*2CPeC2BiC 1u(KCC) t z z (4-55) and 2 tttt t *22*2CCDC Pe2Bi u(KCC) t z(-1) Dz a (4-56) The above equations are also solved by th e regular expansion method to different orders in to determine the mean velocity and the dispersion coefficient. The results to different orders are O(1/ ): 00CC(z,t) and t,0t,0CC(z,t) O(1/ ): 21 u 1+K(-1) a and 0 1t,1C Pe-2Bi(KC-C) z O(1): 2 ** 0 R0 *2C C DD tz Thus, the dispersion coefficient for the case of radially well mixed tube and annulus is simply the sum of the contributions to di spersion from diffusion and interfacial mass transfer resistance. This shows that * RD is simply an additive term that does not depend on the velocity profile.
82 Contribution to Dispersion from Convection (* cD) The convective contribution to dispersion, * cD ~ , is the only term that is affected by the velocity profile. For Po iseuille flow it is given by -1 24 t -1-1 222 tt * c 3 2D 11KK24ln-18 D DD -22KK24611K-6K11 DD D 481K(-1) aa a a (4-57) Below we discuss the physical mechanisms that contribute to * cD ~ and the dependency of * cD ~ on various parameters. These discussions are limited to Poiseuille flow, although the differences in dispersion between the Pois euille and the plug flow are discussed later in the paper. Mechanism of Dispersion The mechanism of dispersion that leads to * cD ~ for Poiseuille flow is illustrated in Figure 4-3. Once a pulse of solute is introduced into the tube, it spreads into the annulus (Figure 4-3a). The flow inside the tube then stretches the pulse into a paraboloid, creating radial concentration gradients (Figure 4-3b). Acco rdingly, the solute diffuses from the tube to the annulus at the front end of the pulse and from the annulus to the tube at the back end. After neglect ing interfacial mass tr ansfer resistance, th ere are four time scales that are involved in th e process described above: (1) 1, the time scale for transfer of solute from the tube to the annulus at the front end of the parabaloid, (2) 2, the time scale for transfer of solute from the annulus to the tube at the back end of the parabaloid,
83 (3) 3, the time scale for radial diffu sion within the tube, and (4) 4, the time scale for radial diffusion within the annulus. Each of the four time scales discussed above contributes to the dispersion, and it is useful to determine their scalings. Before we do this, let us point out that the dimensional convective contribution to the dispersion coefficient is Pe2 Dc *, which scales as 2l/ t, where l is the axial distance traveled by a pulse during t, so that t u ~ l. It follows that 2Pe 2 *2 c 2 2u D~u t t 1K(1) a . Because, to leading order, the concentrations in the tube and the annul us are in equilibrium, the appropriate t is the time needed to achieve equilibrium in the radial direction. This t is essentially the sum of the four time scales that are discussed above. We now de termine the scalings for each of these four time scales for Poiseuille flow. The time scales 3 and 4 are simply D R2 and 22 tR(1) D a , respectively. To determine 1, we first find the mass of the solute that diffuses from the tube to the annulus at the front end of the paraboloid. At the front end of the parabaloid, the concentration in the annulus changes from appr oximately zero to KC, where C is the concentration scale in the tube. Accordingly, the mass transferre d between the tube and annulus at the front end of the parabaloid per unit length is KC R2( a2-1). The radial solu te flux in the tube scales as DC/R, so the solute transpor t in the tube per unit length is DC. 1 is the ratio of these two quantities, that is, 22KR(1) D a . To determine 2, we first find the mass that diffuses from the annulus to the tube at the back end. Because the tube and the annulus
84 are always in equilibrium, this is equal to the mass that diffuses from the tube to the annulus at the front end, KC R2( a2-1). The diffusive flux from the annulus to the tube scales as tDKC R(1) a . Accordingly, the solute transpor t in the annulus per unit length is tDKC (1) a . 2 scales as the ratio of these two quantities, 22 tR(-1)(+1) D aa . The contribution to Dc * from each of the four time scales beco mes clearly evident in the asymptotic expressions for * cD, which are found below. capillary tissue region pulse of solute u 1234 capillary tissue region pulse of solute u 1234 (b) (a) Figure 4-3. Depiction of the spreading of a solu te pulse inside the tube-annulus geometry. (a) A pulse of solute is introduced in to the tube, which immediately spreads into the annular region. (b) The pulse of solute acquires a parabolic shape as a result of convection inside the tube . The solute diffuses as a result of concentration gradients. Asymptotic Limits of * cD The derivation above assume that al l the dimensionless parameters ( a , K, and tD D ) are O(1) quantities in hus, in the asymptotic expansions of the Poiseuille flow * cD ~
85 below, the large and the small quantities are supposed to be <1/ and > , respectively. Below we obtain the asymptotic expressions and also identify the time scale that dominates the dispersion in each case. The expression for * cD ~ is linear in 1 tD D , so the asymptotes for small and large values of tD D are obvious. As tD D approaches 0, the dominant time scales are 2 and 4, which contribute to the term proportional to tD D . For large tD D , the dominant time scales are 1 and 3, which contribute to th e term independent of tD D . In the limit of a approaching 1, the asymptotic result obtained by expanding Eqn. (4-57) is * c 11K(1) limD 488aa (4-58) In the limit of K approaching 0, the asympto tic result obtained by expanding Eqn. (4-57) is 1 4 t * c 11 K0 2 ttD 13 ln() 2D4 1 limDK 48 DD 1111 2D88D2 aa a (4-59) In both of the above cases the longest time scale is 3, which contributes to the leading order term for * cD ~ in the above expressions. This term is identical to the dispersion coefficient for flow through a tube (35). Thus, the dispersion coefficient in our system reduces to that for flow thr ough a tube as the thickness of the annulus, a -1, or
86 the partition coefficient, K, approach zer o. Equation (4-57) is only valid when t 41 D K< (ln)D aa , so the region of validity becomes small as the value of a increases. For the limit of * cD ~ as K approaches infinity, the longest time scale is 1. Because the mean velocity scales as 1/K and t scales as K, * cD ~ scales as 1/K. The exact expression for * cD ~ as K approaches infinity is 42 * c 23 K112211 limD 48(1)K aa a (4-60) As a becomes large, the mean velocity scales as 1/ a2, and 2 scales as a3. The dispersion coefficient is thus expected to scale as 1/ a ; however, the exact e xpansion shows that * cD ~ scales as ln( a )/ a2. -1-1 2 tt * c 324 aDD 11K24ln()K-18K 11 DD limDO 48K a aa (4-61) The cylindrical curvature of the tube -annulus geometry was neglected while determining the scalings and that may be the reason for the disagreement between the scaling and the exact expansion as a becomes large. To verify that neglect of the cylindrical curvature led to this discrepa ncy, we find the dispersion coefficient for Poiseuille flow in the 2D equivalent of core-a nnular flow, that is, 2D planar fluid flow in a channel of height 2h surrounded by a solid of thickness ( a -1)h. We also develop scalings and the exact solutions for this geometry. Flow Through a Channel The results for flow through a channel, neglecting interfaci al mass transfer resistance, are given below.
87 1 u 1K(1) a (4-62) 22 tt 2 * t 2 2 t 2 3DD 17Pe3535510512 D Pe DD D D351K(1)15 1K(1) D 211 Pe D 3 1K(1) aa a a aa a (4-63) The time scales for equilibration in th e channel and in the surrounding solid are 2 3h = D and 22 4 t(-1)h = D a . For a channel width W, the mass transferred per unit length is KC( a 1)hW. The lateral flux in the solid scales as DC/h, and accordingly, the solute transport per unit length is DC/hW. 1 scales as the ratio of these two quantities, 2h K(-1) D a . The mass that diffuses from the annulus to the tube at the back end is equal to the mass that diffuses from the tube to the annulus at the front end, and is thus equal to KC( a -1)hW. The diffusive flux from the annul us to the tube scales as tDKC R(1) a . Accordingly, the solute transport per unit length is tDKC (1) a . 2 scales as the ratio of these two quantities, 22 t(-1)h D a . Thus, for the planar problem both and are the same. Similar to previous cases, the dispersion coefficient is linear in 1 tD D . As tD D approaches 0, the dominant time scales are 2 and 4, and for large tD D , the dominant time scales are 1 and 3.
88 The asymptotic expressions of * cD ~ for Poiseuille flow in this geometry for large and small values of a and K are * c 124K(1) limD 10535aa (4-64) 1 1 t t 332 * c 22K1 D 1217 D D3K3K3K35K D limD 3Kaa (4-65) 1 *32 t c K0D 21144 limDK 105D333535 aaaa (4-66) * c K17 limD 35(1)K a (4-67) As a approaches infinity the dominant time scale is 2 and the convective contribution to dispersion, * cD ~ , scales as 2 2u ~ 2 22 2 tu (-1)h D(1+K(-1)) a a ~ 2 1 2 t 22uh D 1 D DDK ~ 1 2 t 2D DPe KD . This matches the asymptotic scaling given in Eqn. (4-65). Similarly, as a approaches 1, the dominant time scale is 3 and 2 2 22 3 2 tu h u ~~DPe D(1+K(-1)) a , which matches the asymptotic scaling given in Eqn. (4-64). As K approaches infinity, the dominant scale is 1 and 2 1u ~ 2 22 2 tu K(-1)hDPe ~ D(1+K(-1))K(-1) a aa , which is the same as the asymptotic limit obtained in Eqn. (4-67). As K approaches zero, both 3 and 4 are of the same order; however, 4 does not contribute to dispersi on because in this limit, to leading order, no solute diffuses in to the annulus. Accordingly, only 3 is the dominant
89 time scale and 2 2 2 t 2 3 2DPe ~ )) 1 a ( K 1 ( u D h ~ u , which matches the asymptotic scaling from Eqn. (4-66). It follows that this si mple scaling analysis can indeed predict the asymptotic forms of the dispersion coefficien t; however, in the scaling analysis for the cylindrical tube-annulus geometry, curvature effects were neglected and that caused a discrepancy between the scaling anal ysis and asymptotic expansion for * cD ~ in the limit of large a . Dependency of * cD on Parameters The dependency of * cD ~ on a , K and tD D for the case of cylindrical Poiseuille flow surrounded by the solid annulus is illustrated in Figures 4-4 4-6. Figure 4-4 shows the dependency of * cD ~ on a for four different values of K. At a = 1, * cD ~ goes to 1/48 as is expected from the asymptotic results. Although not evident in the gra ph for all values of K, * cD ~ increases linearly with a slope of K/8 as a becomes >1. As a becomes large, * cD ~ decreases as ln( a )/ a2. Accordingly, there is a maximum value of * cD ~ for an intermediate value of a , which is larger for smaller K values. The values of a at which the maxima are reached become smaller on increasing K. In fact for K = 10 and 100, the maxima are not evident in the figure because they occur at very small a values.
90 Figure 4-4. * cD ~ for Poiseuille flow vs. a for different K values where Dt/D = 0.1. Figure 4-5 shows the dependency of * cD ~ on K for different values of tD D . These curves show the linear dependency of * cD ~ for small K values and the 1/K dependency for large K values, as evident by slopes of 1 and -1, respectively, on the log-log plot. The region of linear behavior in the small K regime is restricted to very small K values for Dt/D values of 10 and 100, and accordingly, the linear regime is not visible in the figure for these two cases. It is noted that our analysis is strictly valid for * cD ~ values that are O(1) in . Thus, in the small K and large a regimes, the trends shown in Figures 4-4 and 4-5 are valid only for long tubes, which correspond to very small values. As expected, * cD ~ approaches 1/48 as K approaches 0. 10 20 30 40 50 60 70 80 90 100 10-8 10-6 10-4 10-2 100 102 104 K=0.01 K=1 K=10 K=100 a* cD
91 Figure 4-5. * cD ~ for Poiseuille flow vs. K for different Dt/D values where a = 5. Figure 4-6 illustrates the behavior of * cD ~ as a function of tD D for three different values of a . * cD ~ behaves as 1 tD D in the small tD D regime as evident by a slope of -1 on the log-log plot and appro aches a constant value as tD D approaches infinity. K* cD Dt/D=0.1 Dt/D=1 Dt/D=10 Dt/D=100 10-4 10-3 10-2 10-1 100 101 10-4 10-3 10-2 10-1 100 101 102 Dt/D=0.1 Dt/D=1 Dt/D=10 Dt/D=100 K* cD Dt/D=0.1 Dt/D=1 Dt/D=10 Dt/D=100
92 Figure 4-6. * cD ~ for Poiseuille flow vs. Dt/D for different a values where K = 5. Effect of Velocity Profile in the Tube on Dispersion The difference between the dispersion coeffi cients for Poiseuille (Eqn. (4-20)) and plug flow (Eqn. (4-21)) is given by 22 ** popl 2 2Pe15K1 D-D 481K(1) a a (4-68) Interestingly, this difference is independent of the soluteâ€™s diffusivity in the annulus. Each of the four time scales discussed above fo r Poiseuille flow are also pertinent to plug flow. Thus, qualitative ly, the dependency of *D on various parameters for plug flow is similar to that for Poiseuille flow. Th e ratio of the convect ive contribution of * p lD 10-3 10-2 10-1 100 101 102 10-4 10-3 10-2 10-1 100 101 a=2 a=5 a=10 * cD D t /D a = 2 a = 5 a = 10
93 (denoted * c plD ) to the convective contribution of * p oD (denoted * c poD ) is shown as a function of a in Figure 4-7. This ratio is zero for a = 1 because there is no convective contribution to dispersion for plug flow in th is limit. The ratio then increases with increasing a and levels off at values th at range from about 0.5 to 1 for a wide range of K. The value of the ratio at large a varies inversely with K. Figure 4-7. Ratio of of * cD ~ for plug and Poiseuille flow vs. a for different K values where Dt/D = 0.1. Dispersion Coefficient fo r Small Permeability: O( ) Bi The expressions for the dispersion coeffi cient given by Eqns. (4-20) and (4-21) become unbounded as Bi approaches zero. This is clearly an incorre ct result because as the interfacial mass transfer resistance appr oaches infinity, the mass transfer problem should reduce to the classical pr oblem of Taylor dispersion in a tube. This suggests that 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 K=0.01 K=1 K=10 K=100 a * c pl * c poD D
94 Eqns. (4-20) and (4-21) are not un iformly valid. On scaling Bi as O( the average equations reduce to Eqns. (4-50) and (4-51), which represent a set of two coupled partial differential equations. The equation for the tu be essentially becomes identical to flow through a tube with a source term that accounts for the mass loss to the annulus. The annulus equation is similar excep t that there is no effect of convection. Also, on taking the limit as Bi goes to zero, the tube equation correctly becomes identical to the problem of flow through a tube. The set of coupled equations can be comb ined to yield the following differential equation for the tube concentration: 2 22 000 t 2 2 2 43 2 t 0t00 43 2 2 00 t 2 22CCC R1RPe KD1D 2 tt2 48zt 1 uRDuR CDDRCC Pe 1 2 zt2 48z2 z u CC DPe KD1 48zz 11 a aa (4-69) Following the approach of Sankarasubramanian and Gill (37), we use the following expansion, 23 0000 123 23CCCC ... tzzz (4-70) By substituting the above expansion in Eqn. (4-69), the various coefficients can be determined. The resulting expressions for these coefficients are: 1 2u 1K1 a (4-71)
95 2 2 2 22 t 2 3 2 2Pe D1KD1 KPeD1 48 1K1 2Bi1K1 a a a a (4-72) 3 32 2 3 5 22 22 2 2 t t 32 2 22PeKRD1 1K1 4Bi1K1 PePe 2D1KD12D1 PeR1 4848 D 2Bi 1K1 1K11K1 a a a a a a aa (4-73) As is shown by the equations, is independent of Bi and is simply the mean velocity of the solute for the case of O(1) Bi; however, and 3 behave as 1/Bi and 1/Bi2, respectively. Thus, the contribu tion from the higher order terms is very large at small Bi. This shows that series given by E qn. (4-70) does not converge for O( ) Bi. The average equations cannot be expressed as a single convection-dispersion equation, and Eqns. (4-50) and (4-51) need to be solved simulta neously to predict the average concentration profiles in the tube and the annulus. Application to Transport in Tissues Exchange of nutrients, drug, etc. between vascular and tissue regions occurs in the capillaries by transport across the capillary wall . In most capillary-transport models it is commonly assumed that each capillary supplie s nutrients to an annular tissue region around the capillary, referred to as the Krogh tissue cylinder (33, 41, 42). As noted by a number of researchers (Tepper et al. (40), Levitt (39), and ot hers), the problem explored in this paper can be thought of as a blood capillary surrounded by a Krogh tissue cylinder. Our model is thus of relevance for understanding and quantifying dispersion of
96 solutes in the capillary blood vessels, which is important in the fields of physiology, toxicology and pharmacokinetics. Although the geometry of our model mimics the Krogh tissue cylinder, a number of our assumptions are not consistent with the phys iological conditions in the capillaries. It is well known that blood is a non-Newtonian fluid and that the blood flow through the blood vessels is best descri bed by a Casson fluid model ( 43). The velocity profiles (Poiseuille and plug) used in our model do not accurately represent the blood flow in the capillaries. In blood flow thr ough the capillary, the tube diamet er is smaller than the size of the red blood cells (RBCs). Consequently , the RBCs must deform in order to flow through the capillaries. Plasma fills the sp ace between successive RBCs, and there is an extremely thin layer of plasma between th e RBCs and the capillary wall. Blood flow inside the capillaries is si milar to plug flow; however, the region between successive RBCs is kept well-mixed by a pair of counter -rotating vortices that convect with the flow (43-45). Aside from the velocity profiles, there are a number of ot her important issues that have been neglected from our model, such as convective flow of blood into the tissue (filtration), geometric factors (s uch as variations in capillary length and diameter), protein binding, and consumption of solutes by reac tions. Thus, our model cannot be directly applied to quantify dispersion in the tissues; ho wever, it elucidates some of the important mechanisms that lead to dispersion. Our current model may be more valid to assess dispersion in isolated organ experiments, in which a buffer solution may be used in place of blood (46-49). In such a case, RBCs are not present, and thus, the fluid velocity profile is expected to be Poiseuille.
97 Although, for the reasons mentioned above, our model predictions are not expected to match the experimentally measured values of dispersion coefficients in the capillaries of tissues, it is nonetheless usef ul to make such a comparison. This allows us to gauge the importance of the proposed mechanisms in causing dispersion in the capillaries. Below we compare the calculated dispersion coefficients for both Poiseuille and plug flow for the O(1) Bi case, neglecting interf acial mass transfer-res istance, with those experimentally determined in animal tissues. Comparison of D* Values with Experimental Values Oliver et al. have include d dispersion in a whole-body pharmacokinetic model, in which they characterize the dispersi on in the tissues by a parameter DN (50). They reviewed a number of experimental studies and determined DN for the drug cyclosporine in a 250 g rat. We neglect capill ary wall resistance and calculate *D ~ for cyclosporine in a rat from Eqns. (4-20) and (421). These values are then compared to those determined by Oliver et al. Although values of the flow rates (Q), tissue volum es (V), and partition coefficients (K) for cyclosporin e in a rat are given in the li terature (50), the values of parameters a , D, Dt, R, and L are not. We therefore assume L to be 0.3 mm (31) and tD D to be 0.1 in all tissu es (51). Calculation *D ~ requires determination of Pe and thus u for each tissue. The following equation is used to determine u. c 2n u R Q (4-74) The volumetric flowrate of blood exiting each tissue (Q), a known parameter, is assumed to be equal to the product of the area of a single capillary ( R2), u, and the number of capillaries in the tissue (nc). The number of capillaries in the tissue is found
98 by setting the total volume of the tissue comp artment (V) equal to the sum of the volume of the capillaries in that tissue (Vc) and the volume of tissue space in that compartment (Vt). This results in the following equations: 222222 tccccVVVn RLRLn RLn RL aaa (4-75) c 22V n RL a (4-76) By combining Eqns. (4-76) and (4-74), u is found for each tissue according to the following equation: 2QL u V a (4-77) The parameters a and R were calculated for each tissu e from experimentally determined values of capillary number density (nc/A) and capillary surface area to tissue volume ratio (Sc/V) (52) according to the following equations. cS 1 R 2 V (4-78) 1 cn 11 R Aa (4-79) These calculated values are used in Eqns. (4-20) and (4-21) to find *D ~ for Poiseuille and plug flow. For tissues in which nc/A and Sc/V data was not available, a was assumed to be 10 and R was assumed to be 5 m. Because the DN value proposed by Oliver et al. was de-dimensionalized differently than our *D ~ , we use the following relationship to make the comparison with our calculated *D ~ . * * N 2DDVK D LQ (4-80)
99 To avoid confusion, we have designated our comparable calculated dispersion number as DN *, which will be compared to the experimental DN values for each tissue. The calculated DN * values (for both Poiseuille and plug flow) and the experimentally measured DN values are shown in Table 4-1, along with the K, V, Q, a, and R values. In most tissues, the model predictions are of th e same order as the experimental values, suggesting that the mechanism of dispersion pr oposed in this paper may be important in tissues. The Krogh cylinder model is expected to most closely resemble the actual physiology of the muscle (41), and the ag reement is the best for this tissue. Table 4-1. Comparison of tissue dispersi on numbers predicted from our model to experimentally determined values for cyclosporine in a rat Tissue Kb Qb (mL/min) Vb (mL) a R (m) DN b DN * Poiseuille flow DN * Plug flow Lungs 5.70 44.50 1.60 10.005.00 0.60 1.91 1.79 Muscle 1.30 6.80 125.003.06a 3.61a 3.00 2.80 1.96 Skin 3.90 4.50 43.80 10.005.00 12.002.60 2.60 Adipose 13.90 1.80 10.00 11.28a3.02a 12.005.18 5.28 Kidney 7.43 14.27 2.00 2.80a 3.01a 0.48 0.09 0.09 Liver 12.43 14.70 11.00 3.03a 2.87a 0.60 0.68 0.68 Dt/D = 0.1, L = 0.3 mm, D = 10-9 m2/s aCalculated using information from Reference (52) bFrom Reference (50) Conclusions The dispersion of a solute in a core-annu lar flow with a solid annulus has been studied for both Poiseuille and plug flow fo r small and large interfacial mass transfer resistances. The dispersion coefficient for small resistance is the sum of contributions from axial diffusion, interfacial mass tran sfer resistance, and convection. The contribution from interfacial mass transfer resi stance is independent of the fluid velocity profile. The convective contribution to dispersi on arises from four different mechanisms which can be expressed in te rms of four time scales: (1) 1, the time scale for transfer of
100 solute from the tube to the annulus at the front end of the solute pulse, (2) 2, the time scale for transfer of solute from the annulus to the tube at the back end of the solute pulse, (3) 3, the time scale for radial diffusion within the tube, and (4) 4, the time scale for radial diffusion within the annulus. For the cases of vanishing annulus thickness and small solute solubility in the annulus, the di spersion coefficient becomes the limit of that for flow through a tube. The dispersion coeffi cients are similar for the cases of Poiseuille and plug flow, but the difference between these tw o dispersion coefficients is larger for a thin annulus. For the case of large inte rfacial mass transfer re sistance, the average equations cannot be expressed as the convect ion-diffusion equation. The results of this study are of relevance to dispersion in tissu es, where the mechanisms proposed in this paper can augment the dispersion of solute s. The calculated values of dispersion coefficients for certain tissues are of the same order of ma gnitude as the experimentally determined values. This suggests that the proposed mechanism for dispersion is relevant to solute transport in capillaries. Notation a = parameter which describes the radius of the combined tube and surrounding annulus Bi = Biot number, dimensionles s mass transfer coefficient, R D Bi = scaled Biot number, L D C = total solute concentration in the tube C* = reference drug concentration Ct = solute concentration in the annulus C = dimensionless solute concentration in the tube
101 tC = dimensionless solute concentration inside the annulus D = solute diffusion coefficient in the tube D* = dispersion coefficient in the tube Dt = diffusion coefficient within the annulus DN = Oliver et al.â€™s dispersion number for a specific tissue DN * = calculated dispersion number for a specif ic tissue used for comparison with Oliver et al.â€™s dispersion number *D = dimensionless dispersion coefficient * 0D ~ = contribution of the effective molecular diffusivity to dispersion * p oD = dimensionless dispersion coefficient for Poiseuille flow * p lD = dimensionless dispersion coefficient for plug flow * cD ~ = part of the contribution of conv ection to dispersion (dimensionless) * c poD = convective contribution to dispersion for Poiseuille flow (dimensionless) * c plD = convective contribution to disp ersion for plug flow (dimensionless) * RD ~ = contributions of the interfacial re sistance to mass transfer to dispersion (dimensionless) = small dimensionless perturbation parameter, R/L = mass-transfer resistance to solute transport at the tube-annulus interface K = soluteâ€™s tube:annulus partition coefficient L = length of the tube nc/A = capillary number density
102 Pe = Peclet number, uR D r = radial coordinate r = dimensionless radial coordinate Q = flow rate of blood leaving the tissue R = radius of the tube t = time t = dimensionless time st = dimensionless time scale for tr ansport across the capillary wall lt = dimensionless time scale for axial diffusion uz = velocity of fluid in the tube in the z-direction u = mean velocity of a pulse of solute in the tube u = dimensionless mean velocity of a pulse of solute in the tube u = average velocity of fluid in the tube zu = dimensionless velocity of fluid in the z-direction Sc/V = capillary surface ar ea to tissue volume ratio 1 = time scale for transfer of solute from the tube to the annulus at the front end of the solute pulse 2 = time scale for transfer of solute from th e annulus to the tube at the back end of the solute pulse 3 = time scale for radial diffusion within the tube 4 = time scale for radial diffusion within the annulus V = volume of the tissue
103 z = axial coordinate z* = axial coordinate in the moving reference frame *z = dimensionless axial coordinate in the moving reference frame
104 CHAPTER 5 INCLUSION OF DISPERSION IN PHAR MACOKINETIC MODELS FOR BOTH FLOW RATE LIMITED AND MEMBRA NE RESISTANCE LIMITED DRUGS Introduction Physiologically based pharmacokinetic (PBPK) models divide the body into a number of blood and tissue compartments, with the tissue compartments including the intracellular and lipid bilayer regions of the tissue cells, the interstitial fluid, and the capillary blood. Most pharmacokinetic models neglect axial drug concentration gradients within the capillaries by assuming the tissue compartments are well-mixed (11, 12, 30, 33); however, a simple scaling analysis of th e convective and dispersive time scales of the drug inside the capillary space shows that the capillaries in most tissues are not wellmixed. To more precisely describe this di stribution, the axial de pendence of the drug concentration in the capillaries should be in cluded in PBPK models. This may be done through use of an effective dispersion coefficient, D*, which characterizes the extent of axial spreading of the drug as it flows through the capillaries (50, 53, 54). As a condition for a well-mixed blood vessel, the time scale for drug dispersion within the vessel (L2/D*) should be less than the time scale for convection (L/ u), where L is the length of the blood vessel, D* is the dispersion coefficient of the drug within the blood vessel, and u is the mean velocity of a pulse of the drug. It follows that u L D*>>1 is the condition for a well-mixed vessel. We will refer to the dimensionless quantity *Lu D as PeL. It follow that as D* (and thus, PeL -1) approaches infinity, the blood vessel becomes
105 well-mixed. Herein, we de rive and expression for D* for the capillaries of each tissue. We then calculate PeL -1 for multiple tissues and show that the well-mixed condition is not satisfied in most tissues. While the majority of pharmacokinetic models use the well-mixed assumption, there are a few researchers that have attemp ted to include dispersion principles into pharmacokinetic models (50, 53, 55-60). These studies focused on obtaining D* by fitting experimental pharmacokinetic data to dispersion models. Our study is different from these in two main regards. First, previous researchers obtained D* by fitting experimental data to models, but we obtain D* in a mechanistic and predictive manner by solving the mass transfer problem in the tissue. Second, while the previous researchers focused on flow rate limited drugs (those drugs with small interfacial mass transfer resistances at the capillary wall), in this chapter we also develop a pharmacokinetic model for the membrane resistance limited drugs (those drugs with large interfacial mass transfer resistances at the capillary wall). In our derivation of D*, we assume the tissue compartment to be a collection of Krogh tissue cylinders (33, 41), wherein each capillary supplies a sp ecific annular region of tissue, as shown in Figure 5-1. Othe r researchers have developed expressions for dispersion coefficients in similar geometries (36, 38-40). In Chapter 4, we investigated dispersion in tissues for drugs with both small and large interfacial mass-transfer resistances while neglecting th e red blood cells (RBCs) in th e capillaries. Herein, we extend the model developed in Chapter 4 to account for the presence of RBCs within the capillary space for both the cases of flow rate limited drugs and membrane resistance limited drugs.
106 a) TISSUE SPACE aR CAPILLARY L r z R A rter y A rteriole Venule Vein Capillaries Tissue Space Figure 5-1. Krogh tissue cylinder. (a) Netw orked arrangement of capillaries, with each capillary going from an arteriole to a venule, (b) Representation of the capillary and surrounding tissue sp ace as a Krogh tissue cylinder Because the capillary diameter is smaller than the size of the RBCs, the RBCs must deform their shape and flow in single file th rough the capillary. This is much different than blood flow through larger blood vessels, where the RBCs flow freely with the blood. The spaces between successive RBCs in the ca pillary are filled with plasma, and there is also an extremely thin layer of plasma be tween the RBCs and the capillary wall. Blood flow inside the capillaries is similar to plug flow; however, the region between successive RBCs is kept well-mixed by a pair of counter -rotating vortices that convect with the flow (43-45). The flow of RBCs through the capill aries is depicted in Figure 5-2. To account for the RBCs in the capillary in our model, we describe the capillary as having regions of equally spaced RBCS, with regions of plasma in between the RBCs. The thin plasma layers between the RBCs and the capillary wall is neglected in our model. Figure 5-3 shows the blood flow through the capill aries as depicted by our model. In this chapter, we have included the axial dependence of drug concentrations inside the capillary regions into pharmac okinetic models for both membrane resistance limited and flow rate limited drugs. The pr esence of the RBCs in the capillaries is included in both of these models. In the case of the flow rate limited drugs, an expression for D* is developed. We calculate D* for human tissues for one particular drug to show
107 that the well-mixed assumption is invalid fo r most tissues. Calculations of D* for cyclosporin in rat tissues are executed and the results are compared to experimental values (50). The flow rate limited D* is then incorporated into a single tissue model and then into a 7 tissue whole-body PBPK model. The concentration pr ofiles obtained from these dispersion models are compared with those obtained from their respective wellmixed models. We then solve the pharmaco kinetic problem for the case of membrane resistance limited drugs, and show that in th is case the drug concentrations in the tissue and capillary are not in equilibrium and th e average mass transfer problem cannot be reduced to a single convec tion-dispersion equation. Thus, an expression for D* cannot be found. We develop a single tissue pharmaco kinetic model for this case and compare the results of this model to a well-mixed model. While a whole-body mo del is not developed for membrane resistance limited drugs, it ma y be developed using the principles shown herein. TISSUE RBC TISSUE Figure 5-2. A schematic representation of RBC flow through the capillary. The RBCs flow through the capillary in single f ile, and the plasma spaces in between successive RBCs are kept well-mixed by a pair of counter-rotating vortices that convect with the flow (43-45).
108 Modeling Derivation of Average Mass Transfer Equations In order to calculate D* within the capillaries, we re present each capillary and the surrounding tissue it supplies as a Krogh tissu e cylinder (33, 41), shown in Figure 5-1, where R is the capillary radius and aR is the radius of the Krough tissue cylinder. Each tissue compartment in our mo del is a collection of nc Krogh tissue cylinders, where nc is the number of active capillaries in that tissu e. For simplicity, our model assumes each tissue cell within the Krogh ti ssue cylinder is supplied nu trients by only one capillary; however, capillaries in physiological system s actually have a network arrangement in which a region of tissue may be supplied by mu ltiple capillaries. Fu rthermore, we have assumed a constant value of nc, although capillaries in physio logical systems will have a time-dependent nc as the metabolic state of the tiss ue changes and capillaries alternate between closed and open states (32, 61). In the development of the model we ne glect the resistances to drug transport offered by the RBC and tissue cell walls and as sume there is no convection in the tissue space. We do not explicitly include pr otein binding in our model. The drug concentration in the blood is the sum of th e free and protein-bound drug, and similarly, the drug concentration in the tissue includes th e drug bound to the proteins and tissue cell walls as well as the free drug in the intracellu lar and interstitial spaces. Essentially, we assume that all these concentrations are in equilibrium and can be combined. If kinetic information describing the protein binding or cell wall resistances becomes known, these effects can be included within the framework of our proposed model. We describe the capillary as having re gions of RBCs and plasma, as shown in Figure 5-3. The RBC re gions are of length lc and the regions between the RBCs are of
109 length lp. Below we perform a differential mass balance over a segment (shown as a dotted box in Figure 5-3) that is much larger than the size of the plasma and the RBC but much smaller than the capill ary length. The concentration in the plasma space in between any two RBCs does not depend on posit ion because as mentioned above, the two counter-rotating vortic es keep this space well-mixed ; however, the RBC is not well mixed, and thus, there are concentration gr adients across each RBC. In our model, the RBC is assumed to be in equilibrium with the plasma space. Since there are concentration gradients inside each RBC, the concentration in two adjacent plasma spaces are different, and one can define average concentration gradients in the plasma space. A schematic of the concentration profil e in the differential segment is also shown in Figure 5-3. z z+ z z plasma Tissue RBC lplcExchange z z+ z z plasma Tissue RBC lplcExchange Figure 5-3. Model depiction of RBCs within the capillary (not to scale). The RBCs and the plasma space are moving to the right with mean velocity u. The values of lp and lc are about 12 m (62) and 2 m (63, 64). The fractional time for which the inlet, i.e., z = z, is occupied by an RBC is 0.14, and for the remaining fractional time, 0.86, the inlet is occupied by plasma. The solid line shows the expected drug concen tration profile in the capillary. The diffusive flux, j, into the dashed differential segment is
110 c ccC j-Df z (5-1) where Dc is the diffusion coefficient of the drug inside the RBC, Cc is the drug concentration inside the RBC, and fc is the fraction of time th at the inlet to the box is occupied by an RBC. In the remaining time, a plasma space occupies the inlet and since the plasma is well-mixed, it does not contri bute to diffusive flux. The concentration gradient inside the cell can be related to the concentration gradient inside the plasma as follows: c cC C K1f zz (5-2) where Kc is the drug partition coefficient be tween the RBC and the plasma, f is the ratio between lp and lc, C is the drug concentration inside the plasma, and z C is the average concentration gradient over a length scale of a few plasma spaces. The fraction of time that the inlet to the dashed box is occupied by an RBC, fc, is 1/(1 + f). Combining Eqns. (5-1) and (5-2) gives the following result: ccC jDK z (5-3) Based on the above equation, we now define an effective diffusion coefficient within the capillary space, Deff, as follows: effccDDK (5-4) To describe the drug concentration in the capillary region, we now write a mass balance on the dashed segment as follows.
111 2 zczzzczz 2 trReffeff zzz 2 cf1f1 u RCKCCKC f1f1f1f1 fCC 2 Rz CKC RDD f1zz f1C RzK f1f1t (5-5) In the above equation, the first term represen ts the net convective flux into the differential box. It is composed of four terms; the firs t and the second terms represent the convective flux entering the differential box with th e plasma and the RBCs, respectively, and the third and the forth terms represent the convectiv e flux leaving the box. The second term in Eqn. (5-5) represents the flux exchange between the tissue and the capillary. In order to derive this term, it has been assumed that the direct exchange of the drug between the RBC and the tissue is negligible because only about 14% of the area available for exchange between the tissue and the capillary is occupied by the RBCs (62-64). Furthermore, the drug has to overcome an ex tra barrier of the RBC wall in order to cross into the tissue directly from the RBC. The third term on the left hand side of Eqn. (5-5) represents the net diffusive flux, and is base d on Eqn. (5-4) derived above. Finally, the right hand side of Eqn. (5-5) represents the accumulation of mass in the box, which includes the accumulation of the drug in the plasma and the RBCs, respectively. In the above equation, u is the average blood velocity in the capillary, R is the capillary radius, Ct is the drug concentration in the tissue space, K is the drug partition coefficient between the tissue and capillary spaces, is the permeability of the drug across the capillary wall, t is time, r is the coordinate of the radial direction, and z is the coordinate of the axial direction. Dividing Eqn. (5-5) by 2 Rz results in the following equation.
112 2 c c trReff 2fKfK C2 fCC uCKCD f1zRf1zf1t (5-6) To describe the drug concentration in the tissue region, we use the convection-diffusion equation (65), given by 2 t 2 t t tz C r C r r r 1 D t C (5-7) where Dt is the diffusion coefficient of the drug within the tissue space. To facilitate solving Eqns. (5 -6) and (5-7), it is conveni ent to introduce a pulse of the drug at a certain spatial location at t = 0 and follow the pulse in time (as was done in Chapter 4). The pulse moves in the axial direction with a velocity u and disperses into a Gaussian shape, as shown in Figure 4-3. We reformulate the problem in a reference frame moving in the z-direc tion with the mean velocity u, so that t u z z* , where z* is the axial coordinate in the moving reference frame. In the moving reference frame Eqns. (5-6) and (5-7) respectively become 2 c c trReff **2*fKfK C2 fCCC uCKCDu f1zRf1zf1tz (5-8) and 2 * t 2 t t * t tz C r C r r r 1 D z C u t C (5-9) The boundary conditions for Eqns. (5-8) and (5-9) are: 1. The capillary wall offers a resistance to solute transport characterized by a mass transfer coefficient, . We assume that only the drug in the plasma regions crosses the capillary wall and neglect drug adsorption to the capillary wall. It follows that the flux of the drug into the tissue sp ace is equal to the mass of the drug that
113 crosses the capillary wall, and that the boundary condition at r = 1 is t tt r = RC f D KCC rf1 . 2. We assume that a specific area of ti ssue is supplied by only one capillary. Consequently, there is no flux out of the tissue at the perimeter of the Krogh tissue cylinder, and the boundary condition at r = a is tC 0 r . We now de-dimensionalize the above equations as follows: * * efft t 2 0t0tDC zruC t,z,r,u,C,C LLRuCC where L is the capillary length and C0 is a reference concentration. Additionally, we define , a small dimensionless perturbation parameter, as L R , the Peclet number as effuR Pe D , and the Biot number as eff R Bi D . The dimensionless forms of Eqns. (5-8) and (5-9), respectively, are 2 tr1 *2*2 ccCPeC2Biff1C 1uCKC t z fKfKz (5-10) and 2 tttttt *2*2 effeffCCDCDC Pe11 ur t z DrrrDz (5-11) Eqns. (5-10) and (5-11) are the dimensionl ess governing mass tran sfer equations. These equations are solved for two classes of drugs. The first class of drugs contains those drugs with large permeabilitie s across the capillary wall; these drugs are referred to flow rate limited. The second class of drugs contain those drugs with small permeabilities across the capillary wall; thes e drugs and are referred to as membrane resistance limited. In the case of the membrane resistance limited drugs, is small and
114 the Biot number becomes an order quantity ( eff RR Bi~ DL ). For this class of drugs, we solve the dimensional mass transfer equations in the stationary reference frame (Eqns. (5-6) and (5-7)), as shown in Appendix B. These equations cannot be reduced to a single convection-dispersion equation because the drug concentrations in the tissue and capillary are not in equilibrium. For the cas e of flow rate limited drugs, the Biot number is order 1, and the dimensionless mass transf er equations in the moving reference frame (Eqns. (5-10) and (5-11)) are solved, as shown in Appendix A. For this case, the mass transfer equations reduce to a single equation, which is of the same form as a one dimensional convection-diffusion equation. In this derived equation th ere is a dispersion coefficient, D*, which replaces the diffusion coef ficient in the convection-diffusion equation. The same form of equation was used earlier by Rowland et. al. for transport of cyclosporin (47). For the case of flow rate limited drugs, the mass transfer equations reduce to the following equation: 2 * 2CCC uuDD tzz (5-12) where 2 2 22 c * 3 2 c 1 2 242 t c eff 3 2 c 2 t eff 2 cPeK-1f1fK D 2BifKK-1f1 D Pef1fKK4ln341 D 8fKK-1f1 D f1K-11 D fKK-1f1 a a aaa a a a (5-13)
115 and c 2 cfK u fKf1K1a (5-14) For the case of the membrane resistance limited drugs, the radially averaged differential equations for the capillary and the tissue are: t cCC2 f =u+CKC tzRfK (5-15) and t t 2C 2 f =KCC tf1 R1a (5-16) Incorporation of Dispersion into Pharmacokinetic Models We first incorporate dispersion of the drug inside the capillaries using a single tissue model, composed of a single tissue compartment and the blood compartment, shown in Figure 5-4. While in actuality there are many more tissue compartments affecting drug concentrations in the blood and the tissue, this simple model allows us to better assess the effect of dispersion. We develop single tissue models for both the cases of flow rate limited and membrane resistance limited drugs. Next, for the case of flow rate limited drugs, we incorporate dispersion into a whole-body PB PK model, shown in Figure 5-5, containing a blood compartment a nd 7 tissue compartments. This is a more accurate depiction of drug distribution than th e single tissue model. While we present the dispersion whole-body model only for the case of flow rate limited drugs, this model may also be developed for the case of membrane resistance limited drugs. Finally, in the results section, we compare the dynamic drug concentration profiles obtained from the
116 dispersion models to those obtained from th e respective well-mixed models (Appendices C and D). BLOOD CbVb TISSUE CV Cb Bolus Dose Q Figure 5-4. Single tissue model. Single Tissue Dispersion Model Equations Flow Rate Limited For the case of flow rate limited drugs, the mean convection-dispersion equation in terms is given by: 2 * effeffeff 2CCC uD tzz (5-17) where Ceff is the effective drug concentration in the capillary, which includes both the plasma and RBC spaces, and is defined as c efffK CC f1 (5-18) The drug concentration in the tissue space, Ct, is given by Ct = KC = KeffCeff, where Keff is the effective partition coefficient of th e drug between the tissue and capillary, defined as eff c1f KK fK . The mass balance for the blood compartment, which is assumed to be well-mixed, is as follows.
117 b L z b bQC N dt dC V (5-19) In the above equation, Q is the volumetric flowrate, Vb is the volume of the blood compartment, Cb is the drug concentration in the blood compartment., and N is the total moles of the drug per unit time moving in th e axial direction at a certain distance z, defined by the following equation. * eff eff zz zC D NQC uz (5-20) At the entrance of the capillary, N is e qual to the moles of the drug per unit time entering the capillary from the blood compartment, i.e. b 0 zQC N . Plugging this into Eqn.(5-20), we can now describe the drug concentration in the blood compartment as * eff beff z0 z0C D CC uz (5-21) We impose a boundary condition at z = L, such that there is no diffusive flux at the capillary exit. This assumes that all of th e drug that leaves the capillary enters the blood compartment as a result of convection. It follows that eff zLC 0 z , eff zLzLNQC, and Eqn. (5-19) becomes b b effb zLdC VQ(CC) dt (5-22) The initial conditions are that at time 0, Ceff is 0 and Cb is Cb0. We will now de-dimensionalize the above equations as follows: effb effb b 0b0CC tuz t,z,C,C LLCC
118 where Rt is a ratio of the time scales in the tissue and blood compartments, defined as t b QL R uV . The dimensionless forms of Eqns (5-17) and (5-19) are 2 effeffeff 2 LCCC 1 tzPez (5-23) and b teffb 1dC RCC dtz (5-24) subject to the BCs eff z1C 0 z and eff beff z0 L z0C 1 CC Pez and the initial conditions effC = 0 and bC ~ = 1 at t ~ = 0. Equations (5-23) and (5 -24) are solved numerically by finite difference using a semi-implicit numerical scheme to determine the drug concentrations in the blood and tissue compartments. A backward difference approximation is used for the first order time derivatives, and the central difference approximation is used for the first and s econd order spatial derivative terms. Membrane Resistance Limited To develop a single tissue model for the case of membrane resistance limited drugs, we first convert Eqns. (5-15) and (5-16) to dimensionless form, rescaling time as ut L . This results in the following equations: effeff teffeffCC =2 CKC z (5-25) and
119 t effefft 2C 2 =KCC 1a (5-26) where f1Bi f1 Pe . Additionally, we write a mass balance on the blood compartment as follows: b teffb z1dC RCC d (5-27) where in this case, t b QL R uV . These equations are subject to the boundary condition that b effCC at z = 0 and the initial conditions at that = 0, bC ~ = 1 and effC = tC = 0. Whole-Body Dispersion Model Equations Our whole-body PBPK model includes 7 tis sue compartments (liver, gut, heart, brain, kidneys, muscle, and fat) and a blood compartment (Figure 5-5). This model is developed for the case of flow rate limited drugs, and the equations used in this model are similar to those used in the single tissue model. iC is used to represent the dimensionless effective drug concentration in the capillary sp ace of tissue i. The subscript i denotes the specific tissue compartment, with i = L, G, H, B, K, M, and F representing the liver, gut, heart, brain, kidneys, muscle, and fat compartments, respectively. Equations for this model are de-dimensionalized as follows. b0 b b b0 i i KC C C ~ , C C C ~ , L z z ~ , L u t t ~ We choose the velocity in the kidneys, Ku, to de-dimensionalize time, because the kidneys have the shortest residence time. We also define the dimensionless parameters
120 * i i i L,D u L Pe , b K b b t,V u L Q R (a ratio of the time scal es between the blood and kidney compartments), k K i t,u u R (a ratio of the time scales between tissue i and the kidney compartment), and b i i R,Q Q Q (a ratio of the flowrates between tissue i and the blood). The dimensionless equations for this model are as follows. For the blood compartment: b F M, K, B, H, G, i i i R, b t, bC ~ C ~ Q R t ~ d C ~ d (5-28) subject to the initial condition bC ~ = 1 at t ~ = 0. For the tissue compartments: 2 i 2 i L, i i i t,z ~ C ~ Pe 1 z ~ C ~ t ~ C ~ R (5-29) subject to the initial condition iC ~ = 0 at t ~ = 0, and the boundary conditions 0 z ~ C ~ 1 z ~ i , and b 0 z ~ i i L, 0 z ~ iC ~ z ~ C ~ Pe 1 C ~ in all tissues except for the liver, and 1 z ~ G L G b L G 0 z ~ L L L, 0 z ~ LC ~ Q Q C ~ Q Q 1 z ~ C ~ Pe 1 C ~ in the liver tissue.
121 BLOOD CbVb GUT CGVG LIVER CLVL HEART CHVH BRAIN CBVB KIDNEYS CKVK MUSCLE CMVM FAT CFVF CbQbQLQFQMQKQBQHCbCbCbCb Bolus Dose QG CbCb Figure 5-5. Whole-body model (30). Results and Discussion In this section, we calculate *D ~ for each tissue to show that the well-mixed assumption is invalid for most tissues and D* in the vessels of the blood compartment to verify our assumption that the blood compartment is well-mixed. We then compare *D ~ values calculated from our model to those found by Oliver et al. (50) for cyclosporin in a 250 g rat. Finally, we compare the concentrat ion profiles obtained from the single tissue and whole-body dispersion models to those obtained from their respective well-mixed
122 models. Firstly, we note that upon investigation of the expression for *D ~ for the flow rate limited drugs, we can be separated into contributions from effective molecular diffusivity (* 0D ~ ), from the interfacial re sistance to mass transfer (* RD ~ ), and from convection (* cD) as follows **** 0RcDDDD (5-30) where 2 t eff * 0 2 cD f1K-11 D D fKK-1f1 a a (5-31) 2 2 22 c * R 3 2 cPeK-1f1fK D 2BifKK-1f1 a a (5-32) and 1 2 242 t c eff * c 3 2 cD Pef1fKK4ln341 D D 8fKK-1f1 aaa a (5-33) Calculation of D* For the flow rate limited drugs, we calculate *D ~ for the capillaries of various tissues in the human body (liver, gut, heart, brain, kidney, muscle , and fat tissues) and check whether the well-mixed condition, PeL -1>>1, is satisfied for these tissues. For these calculations, the parameters of a, K, Pe, and tD D , Bi, f, and Kc must be known. Some estimations have been used in our calcu lations, as some of these parameters have not been measured. As in Chapter 4, we have estimated a and R from experimental data
123 (52), and these values are shown in Table 5-1. The parameter K is both a tissue and drugdependent. For these calculations, we use th e partition coefficients for Thiopental in a Wistar rat (24), given in Table 5-1, and neglect the interfacial mass transfer resistance at the capillary wall such that Bi is equal to infi nity. We assume the value of the partition coefficient of the drug between the RBC and the plasma, Kc, is similar to that of a protein, such that Kc = 2.5 (45). The drug diffusion coefficient in the plasma spaces of the capillary, D, is a drug-dependent paramete r that is expected to be on the order of 10-9 m2/s (60). The diffusion coefficient of the drug in the tissue space, Dt, is tissuedependent parameter which is expected to have a value on the order of 10-10 m2/s (51, 66). The diffusion coefficient of the the drug inside the RBCs (Dc) will be on the order of D, such that effccDDK = 2.5*10-9 m2/s. Accordingly, we have set Dt/Deff equal to 0.04 for each tissue. We use an f value of 6 for all tissues, which is based on an lc value of 2 m (63, 64) and an lp value of 12 m (62). Based on these values of lc and lp, the hematocrit value, that is, the fraction of RBCs in the capilla ry, is about 14%, which is consistent with the literature values (63, 64). Pe is both a tissue and drug-dependent parameter, given by effuR Pe D . The volumetric flowrate of blood to exiting each tissue (Q), a known parameter, and can be described as c 2n u R Q (5-34) where nc is the number of capillaries in the tissue. We use the fact that the volume of the tissue compartment, V, is equal to the sum of the volume of the capillaries in that tissue, Vc, and the volume of tissue space in that compartment, Vt to determine an expression for nc as follows
124 222222 tccccVVVn RL RLn RLn RLaaa (5-35) c 22V n RLa (5-36) We combine Eqns. 5-34 and 5-36 to find the following expression for u. 2QL u Va (5-37) The parameters Q and V are known and given in Table 5-1. The capillary length, L, is assumed to be 1 mm (31, 67) for each tissue. Pe and *D ~ are calculated from these quantities, and these values are shown in Table 5-1. To calculate PeL -1 and check if the well-mixed condition is satisfied, we must know u, which is found by multiplying the calculated u ~ (from Eqn. (5-14)) by u (from Eqn. (5-37)). The value of PeL -1 can now be calculated and the well-mixed condition checked. The values used for V, Q, and K as well as the calculated values of *D ~ and PeL -1 are given in Table 5-1 for each tissue. As seen in Table 5-1, the value of PeL -1 is smaller than 1 in all the tissues, except for the fat tissue. This means that in every tissue, with the exception of the fat tissue, the well-mixed condition is not satisfied for this particular drug, indicating that a well-mixed pharmacokinetic model is not sufficient. Fo r these tissues, a disp ersion model must be used to accurately predict the drug concentration in the tissue space.
125 Table 5-1. Calculation of *D and PeL -1 in the tissue compartments. Tissue Ka Vb (L) Qb (L/min)Rc (m)ac PeL -1 Liver 2.3 1.5 1.5 3.03 2.87 0.099 0.03 Gut 1.3 2.4 1.2 3.05 3.78 0.102 0.03 Heart 1.1 0.3 0.24 3.00 2.58 0.176 0.03 Brain 0.7 1.5 0.75 3.02 10.77 0.761 0.13 Kidneys 3.1 0.3 1.25 3.01 2.80 0.136 0.01 Muscle 0.5 30 1.2 3.61 3.06 0.217 0.39 Fat 7.8 10 0.2 3.02 11.28 0.041 1.96 aSteady state partition coefficients for Thiopental in Wistar Rats (24) bTissue volumes for a standard man, 3039 yrs of age, weighing 70 kg (30) cCalculated from Ref. (52) In our dispersion models, we assume the blood compartment, containing the aorta, vena cava, arteries, veins, arterioles, and ve nules, is well-mixed. We check the validity of this assumption by calculating PeL -1 for the blood compartment. The blood vessels of the blood compartment are larg er in diameter than the RBCs , and therefore, the RBCs do not have to deform in shape as they do in the capillaries. In our calculation of *D ~ in the blood compartment, we have neglected the presence of the RBCs, since they are not as critical to dispersion in these large vessels . The vessels of the blood compartment have no surrounding tissue, meaning a = 1, 48 Pe 1 D ~2 * , and u ~ = 1. Each type of blood vessel in the blood compartment has different radii and velocities. Consequently, Pe and *D ~ will vary amongst these blood vessels in the blood compartment. Approximate values for R, u, and L (31, 68, 69) for different types of blood vessels are given in Table 5-2 along with the calculated values of Pe, *D ~ , and PeL -1. For these calculations, D was assumed to be 1*10-9 m2/s. The values of PeL -1 are indeed greater th an 1 in the aorta, vena cava, arteries, and veins, indicating well -mixed blood vessels. In the arterioles and *D
126 venules, PeL -1 is less than 1, which implies the ar terioles and venules are not well-mixed; however, since these blood vesse ls are lumped into a singl e compartment along with the other well-mixed blood vessels, we assume the entire blood compartment to be wellmixed. Table 5-2. Calculation of *D and PeL -1 in the vessels of the blood compartment. Blood Vessel Ra (cm) L (m) Pe PeL -1 Aorta 1.25 0.330 0.15c 41250003.5411 7160 Vena Cava 1.50 0.103 0.15 15450004.9710 3219 Artery 0.20 0.041 0.01a 82400 1.418 343 Vein 0.25 0.002 0.01a 5150 5.535 26.8 Arteriole 0.0015 0.021 0.001b 309 1990 0.097 Venule 0.0010 0.003 0.001 33 23.7 0.007 aFrom Ref. (68) bFrom Ref. (31) cFrom Ref. (69) Comparison of Calculated D* Values With Experimentally Determined Values To get an idea of the accuracy of our mode l, as in Chapter 4, we calculate the dispersion coefficient for different tissues in the body and compare them to the dispersion numbers used by Oliver et al. in a whole-b ody PBPK dispersion model (50). There are two major differences between the dispersion coefficient values we have calculated and those used by Oliver et al. The first differe nce is that Oliver et al., have not obtained their dispersion coefficients by direct calculati on, as we have done. Rather, Oliver et al. have used a dispersion number, DN, that was obtained by fitting experimental data for cyclosporin in a 250 g rat to a dispersion mode l. Secondly, the model that they have used to fit their data includes separate compartments for the arterial and venous blood, thereby distinguishing between the blood before it enters and after it exits the lungs. In our bu(m/s)*D
127 model, we use a single blood compartment, wh ich includes both the arterial and venous blood. As a way of checking our *D ~ equation, we will insert the values of the flow rates, tissue volumes, and partition coefficients us ed by Oliver et al. (reflecting that of cyclosporin in a 250 g rat) into Eqn. (5-13) to find *D ~ . Since the DN used by Oliver et al. was de-dimensionalized differently than our *D ~ , we calculate a comparative quantity, DN *, according to the following equation. * * effeff N 2DDVK D LQ (5-38) To avoid confusion, we have designat ed our calculated dispersion number as DN *, which is compared to Oliver et al.â€™s DN values for each tissue. The results of this comparison are shown in Table 5-3, along with the parameters used in the calculations. This is a rough comparison, as the values used for a, D, Dt, f, Kc, and L in our calculation are estimated values, which we believe to be reasonable but may differ from that of the rats used in Oliver et al.â€™s study. In most tissu es, the model predictions are of the same order as the experimental values, suggesting that the mechanism of dispersion proposed in this chapter may be important in tissues.
128 Table 5-3. Comparison of tissue disper sion numbers predicted from our model (incorporating RBCs) to experimentally de termined values for cyclosporine in a rat Tissue Kb Qb (mL/min) Vb (mL) a R ( m) DN b DN *c Lungs 5.7 44.5 1.6 10 5 0.6 3.55 Muscle 1.3 6.8 125 3.06a 3.61a 3 5.32 Skin 3.9 4.5 43.8 10 5 12 2.74 Adipose 13.9 1.8 10 11.28a 3.02a 12 5.24 Kidney 7.43 14.27 2 2.80a 3.01a 0.48 0.11 Liver 12.43 14.7 11 3.03a 2.87a 0.6 0.79 aCalculated from Ref. (52) bFrom Ref. (50) cCalculated using the following parameters : f = 6 (62-64), L = 0.3 mm (31, 67), Kc = 2.5 (45), D = 1*109 m/s (60), Dt/Deff = 0.4 (51, 66) Comparison of Well-Mixed and Di spersion Single Tissue Models Flow Rate Limited Case We compare the drug concentration-time profiles obtained from the single tissue flow rate limited dispersion model to those obtained from a similar well-mixed single tissue model. The equations for the well-mixed model can be found in Appendix C. We average effC for the dispersion model over z ~ to get the dimensionless mean drug concentration in the capillaries,meanC, which is used for comparison with the drug concentrations obtained from the well-mixed model. We note that both PeL and Rt are parameters of the dispersion model, while the well-mixed model equations contain the parameter Rt but not PeL. We first show that as PeL of the dispersion model becomes very small, the drug concentration profiles of the dispersion model approach the wellmixed profiles. Secondly, we use the models to determine the effect of Rt on the limiting value of PeL, which is the value of PeL at which the well-mixed model is a good approximation. Figure 5-6 shows meanC ~ as a function of t ~ at constant Rt values and
129 varying PeL values. In these curves varying PeL is equivalent to varying D* by changing Dt. Figure 5-6 shows that if PeL is about 0.1, the well-mixed model is a good approximation. Furthermore, the critical value of PeL is relatively insensitive to Rt. Thus, a simple calculation of PeL based on our model can be used to determine whether the simple well-mixed model is appropriate for a given tissue and a given drug, or whether the more complicated dispersion model should be used. It should be noted that while PeL affects the dynamic drug concentration, it does not affect the steady-state concentration value; the steady-state concen tration is affected only by the value of Rt. In the dispersion model, the drug concentr ation in the capillary varies with both time and position. Figure 5-7 shows the dimensionless drug concentration in the capillary, effC, as a function of dimensionless position, z , for given values of t. These plots show that as PeL gets smaller, the axial concentration gradients vanish, signifying a well-mixed behavior, and the drug concentratio n in the capillary approaches the wellmixed concentration. Membrane Resistance Limited Case We now compare the drug concentration-time profiles obtained from the single tissue membrane resistance dispersion model to those obtained from a similar well-mixed single tissue model. The equations for the we ll-mixed model can be found in Appendix D. Figures. 5-8 and 5-9 show meanC ~ as a function of t ~ for constant Rt values and varying values of , where a = 5 and Keff = 10. The first thing to note is that larger values of Rt will reach steady-state more rapidly, and that changing Rt does not significantly affect the trends observed in the concentration profiles as is changed. As becomes small, meanC ~ becomes almost identical to the concentration predicted by the well-
130 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Rt = 0.25 tCmeanPeL = 0.1 well-mixed PeL = 1 PeL = 10 PeL = 100 t mC 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 tCmeanRt = 10 well-mixed PeL = 0.1 PeL = 1 PeL = 10 PeL = 100 t meanC a) b) Figure 5-6. Dynamic dimensionless mean drug concentration in the capillary as predicted by the flow rate limited single tissue models for different values of Rt. a) Rt = 0.25, b) Rt = 10. meanC
131 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 0.70 0.75 0.80 0.85 0.90 Ceffz Ceff vs, z for Rt = 0.25, t = 1.516 Dimensionless well-mixed PeL = 0.1 PeL = 1 PeL = 10 PeL = 100 effC z efftC vs. z for R0.25, t = 1.516 a) effC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 zCeffCeff vs. z for Rt = 10, t = 0.3 Dimensionless PeL = 1 PeL = 10 PeL = 0.1 PeL = 100 well-mixed effC z efftC vs. z for R10, t = 0.3 b) Figure 5-7. Dimensionless drug concentratio n profiles in the capillary as a function of axial distance in the capillary as pr edicted by the flow rate limited single tissue models. a) Rt = 0.25 and t = 1.516, b) Rt = 10 and t = 0.3.
132 Figure 5-8. Dynamic dimensionless mean drug concentration in the capillary as predicted by the membrane resistance limited single tissue models for Rt = 0.001. Figure 5-9. Dynamic dimensionless mean drug concentration in the capillary as predicted by the membrane resistance limited single tissue models for Rt = 0.01.
133 mixed model. This is because at small values, the average ve locity of blood in the capillary becomes large, and thus, the capilla ries become well-mixed. Another thing to note is that the concentrations predicted by the well-mixed be come relatively independent of at large values. This results from the fact that at large values of , the drug concentrations in the tissue and capillary a pproach equilibrium, and can be seen by combining Eqns. (D-2) and (D-3) in Appendi x D at this limit. Figure 5-10 shows tC and effeffKC for Rt = 0.001 and varying values of . As becomes large, the drug concentrations in the capillary and tissue regions reach equilibrium and tC becomes equal to effeffKC. Furthermore, at small values, the drugâ€™s permeability across the capillary walls is small and most of the drug stays in the capillaries. This results in a rapid rise in the drug concentrations in side the capillary, as seen in the = 0.01 plot of effeffKC. Comparison of Dispersion and Well-Mixed Whole-Body Models Using the parameters given in Table 5-1, the drug concentration-time profiles for the whole-body dispersion model for flow rate limited drugs were compared to a similar well-mixed whole-body model, shown in Appendix C. Similar to the single tissue model, we calculate mean concentrations, averaging the drug concentrations in the capillaries over z ~ , to get mean,iC, the mean drug concentration in th e capillaries of tissue i. This mean,iC is used for comparison with the well-mixed model. The drug concentrations in the blood compartment for both models are shown in Figure 5-11. The concentration profiles of the dispersion and well-mixed models are similar, with only slight variations between
134 Figure 5-10. Plots of effeffKC and tC as a function of t as predicted by the membrane resistance limited single tissue dispersion model for Rt = 0.001. them. Even though the blood compartments in both models are assumed to be wellmixed, these variations were expected, becau se inclusion of dispersion affects the drug concentrations of the capillaries in the indi vidual tissues, which in turn affect the drug concentration in the blood compartment. Fi gure 5-12 shows the drug concentration in the fat capillaries as a function of time. At sm all times, the concentration profiles of the two models are similar. Beyond 5 minutes, th e dispersion model has a slightly higher concentration than the well-mix ed model. Because the fat tissue is a well-mixed tissue, (PeL -1>1), we expect the capillary concentration in the dispersion model, with respect to the dispersion modelâ€™s blood concentration, to be similar to that of the well-mixed model, with respect to that well-mixed modelâ€™ s blood concentration. Since the blood concentration profiles are similar, the fat cap illary concentration profiles are similar as
135 well. Figures 5-13 and 5-14 show the drug concentration profiles in the gut and heart capillaries. In these non-well-mixed tissues , there is a greater difference between the concentration profiles of the two models. The dispersion model predicts much larger drug concentrations at shorter times than the well-mixed model does for these tissues. Such a difference demonstrates the need for the inclusion of dispersion in pharmacokinetic modeling. For each tissue, the dispersion and well-mixed models will eventually reach the same steady state concentr ation value; however, this will take a large amount of time. The fat and muscle tissues in particular take a l ong time to equilibrate with the blood. 0 5 10 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dispersion model well-mixed model t b C Figure 5-11. Dimensionless drug concentra tion profiles in the blood compartment as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.
136 0 5 10 15 0 1 2 3 4 5 6 7 8 x 10-3 dispersion model well-mixed model t F,meanC Figure 5-12. Dimensionless drug concentration profiles in the capillaries of the fat tissue as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models. 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 dispersion model well-mixed model G,meanC t Figure 5-13. Dimensionless drug concentration profiles in the capillaries of the gut as a function of time as predicted by the flow rate limited whole-body dispersion and well-mixed models.
137 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 dispersion model well-mixed model t H,meanC Figure 5-14. Dimensionless drug concentration profiles in the capillaries of the heart as a function of time as predicted by the fl ow rate limited whole-body dispersion and well-mixed models. Conclusions This chapter investigates dispersion of both flow rate and membrane resistance limited drugs in the capillaries and represents the first attempt to directly calculate D* in the capillaries with the RBCs present. Inclusion of disp ersion in pharmacokinetic models is desirable, because in most cases, the wellmixed assumption is not valid. Experimental determination of D* is a tedious and time consuming process. Furthermore, D* values will vary with physiological factors and drug characteristics, and experimental determination of D* would require new experiments to be performed for changes in these variables. The predictive models for D* developed in this chapter can provide estimations of D* without performing such tedious experiments and can easily be applied to any drug in any tissue if some basic pharmacokinetic data is available.
138 In the derivation of D* presented in this chapter, we have made a number of simplifying assumptions. We have assumed a cylindrical tissue arrangement, whereas in actuality, there is a networked capillary arrangement (31). Other factors, such as clearance, existence of anastomoses, and solute trapping may become important in certain tissues can affect the dispersion coe fficient (53). These factors have not been included in our current formulation of D* due to lack of available data, but can be included within the framework proposed in this chapter. We have compared our calculated D* values, neglecting mass transfer resistances, with Oliver et al.â€™s dispersion numbers, wh ich were determined by comparison of their model with experimental data (50) (Table 5-3). Since D* the exact values of some of the tissue-dependent and drug-dependent parameters for the rats and drug used in Oliver et al.â€™s study are not known, our comparison s can only be viewed as approximations; however, the comparisons matched to a reasonable extent, especially considering the variation of properties amon gst different tissues and drugs and the exclusion of the aforementioned issues. We have developed a single tissue and a whole-body dispersion model for the case of flow rate limited drugs and a single tissue model for membrane resistance limited drugs. As expected, the drug concentrations of the of the flow rate limited single tissue models approach well-mixed behavior as the PeL value decreases (i.e., the D* value increases). The minimum value of D* for which a tissue can be considered well-mixed does not strongly depe nd on the value of Rt. For some tissues, the whole-body dispersion model predicted va stly different drug concentr ation profiles than the wellmixed model, verifying the importance of in cluding dispersion effects in pharmacokinetic
139 models. For the membrane resistance limited single tissue model, the drug concentrations approach well-mixed behavior at smaller values of . At large values of , the drug concentration in the tissues of th e dispersion model becomes identical to the equilibrium drug concentration and the we ll-mixed results become independent of . We find that Rt does not affect these trends , but larger values of Rt result in a quicker time to reach steady state. Notation a = parameter which describes the tissue cylinder radius Ac = area of the capillary space At = area of the tissue space B = function of r that allow us to determine 1C Bt = function of r that allow us to t1C C = total drug concentration in the blood Ci = total drug concentration in the capilla ries of tissue i in the whole-body model C0 = reference drug concen tration in the capillary Cb = drug concentration in the blood compartment Cb0 = initial drug concentra tion in the blood compartment Ct = total drug concentration in the tissue space Cti = total drug concentration in the tissue space of tissue i in the whole-body model Ct0 = reference drug concentr ation in the tissue space C = dimensionless drug concentration in the capillary 0C ~ = dimensionless drug concentration in the capillary to order 0
140 1C ~ = dimensionless drug concentration in the capillary to order -1 2C = dimensionless drug concentration in the capillary to order -2 b C = dimensionless drug concentration in the blood compartment tC = dimensionless drug concentr ation inside the tissue space iC = dimensionless drug concentration in the capillaries of tissue i in the whole-body model t0C ~ = dimensionless drug concentration in the tissue space to order 0 t1C ~ = dimensionless drug concentration in the tissue space to order -1 t2C = dimensionless drug concentration in the tissue space to order -2 meanC = mean drug concentration in the capillaries mean,bC = mean drug concentration in the blood compartment mean,iC = mean drug concentration in the capill aries of tissue i in the whole-body model D = drug diffusion coefficient in the capillary D* = drug dispersion coe fficient in the capillary Di * = drug dispersion coefficient in the capilla ries of tissue i in the whole-body dispersion model Dt = drug diffusion coefficient within the tissue space DN = dispersion number used by Oliver et al. DN * = calculated dispersion number for comparison to Oliver et al.â€™s DN *D = dimensionless drug dispersion coefficient in the capillaries * 0D ~ = contribution of the effective molecular diffusivity to dispersion
141 * iD = dimensionless drug dispersion coefficient in the capillaries of tissu e i in the wholebody dispersion model * cD ~ = part of the contribution of convection to dispersion = small dimensionless perturbation parameter f = ratio between lpand lc jc = flux in the capillary space jt = flux in the tissue space = mass transfer coefficient at the capillary wall K = drug partition coefficient between the tissue space and the capillary blood Kc = drug partition coefficient be tween the RBC and the capillary blood Ki = drug partition coefficient between the ti ssue space and the capilla ry blood for tissue i in the whole-body model = dimensionless parameter used in the sing le tissue models for membrane resistance limited drugs, equal to f1Bi f1 Pe L = capillary length lc = length of one RBC in the capillary lp = length of the plasma region be tween successive RBCs in the capillaries nc = number of active capillaries in a tissue N = total moles of drug per unit time moving in the axial direction at a certain distance z Ni = total moles of drug per unit time in tissue compartment i moving in the axial direction at a certain distance z in the whole-body dispersion model Pe = Peclet number
142 PeL = dimensionless quantity used in the flow rate limited single tissue and whole-body models, also used to determine if a blood vessel is well-mixed Q = volumetric flowrate of blo od exiting the tissue compartment Qi = volumetric flowrate of blood exiting ti ssue compartment i in the whole-body model QR,i = ratio of the flowrate s between tissue i and the blood in the whole-body model r = radial coordinate r = dimensionless radial coordinate R = capillary radius Rt = a ratio of the time scales in the tissue and blood compartments Rt,i = a ratio of the time scales between tissue i and the kidney compartment in the wholebody model Rt,b = a ratio of the time scales between the blood and kidney compartments in the wholebody model t = time t = dimensionless time uz = velocity of capillary blood in the z-direction u = mean velocity of a pulse of the drug in the capillary iu = mean velocity of a pulse of the drug in the capillaries of tissu e i in the whole-body model u = dimensionless mean velocity of a pulse of the drug in the capillary u = average velocity of blood in the capillary zu = dimensionless velocity of capillary blood in the z-direction V = volume of the tissue compartment in the single tissue model
143 Vb = volume of the blood compartment Vc = capillary volume Vt = volume of tissue space in that compartment Vi = volume of tissue compartment i in the whole-body model z = axial coordinate z* = axial coordinate in the moving reference frame *z = dimensionless axial coordinate in the moving reference frame
144 CHAPTER 6 CONCLUSIONS This dissertation focuses on the development of nan oparticulate platforms for treatment drug overdoses. The essential idea is to design biocompatible NPs that can rapidly sequester significant amounts of drugs, such that an injection of NPs may lead to a reduction in the free drug concentration in vivo. The specific aims of this dissertation were to first design suitable types of NPs, quantify the drug uptake of the NPs by in vitro testing, and then to use modeling to predict th e effect of the NP addition to a human body after an overdose. In Chapter 2 we focus on the design of the NPs, and in Chapter 3 we develop a model for the drug detoxification. Chapters 4 and 5 focus on improvements to the model by accounting for the presence of axial concentration gradients inside the capillary. In Chapter 2, we investigated the potential of liposomes for drug overdose treatment. These systems were chosen b ecause of their known biocompatibility and because the surface properties of these sy stems may easily be altered by synthesizing them with different types of lipids. We performed in vitro experiments to study the uptake of the drug amitriptyline by liposomes composed of dimyristoyl phosphatidylcholine (DMPC), a zwitterionic lipid, and by liposomes composed of a mixture of DMPC and 1,2-di oleoyl-sn-glycero-3-[phospho-r ac-(1-glycerol)] (DOPG), an anionic lipid, in a 70:30 molar ratio. Thus, we were able to investigate the effect of surface charge on the drug uptake. The drug up take was investigated at pH 9, 10.7, and 12 for the DMPC liposomes and at pH 10.7 for the DMPC:DOPG liposomes. We
145 concluded that the drug uptake was occurrin g through an adsorption mechanism in the case of the DMPC (uncharged) liposomes. The drug uptake was quantified at the solubility limit of the drug for each pH studied, and we foun d that the amount of the drug sequestered by the DMPC liposomes was not si gnificantly affected by pH. Furthermore, we calculated the area per molecule of the se questered drug and foun d that the drug is possibly being sequestered in a bilayer conf iguration, which implies that the charged form of the drug is sequestered. We foun d that approximately 70% of the drug was sequestered by the DMPC liposomes. In the case of the liposomes containing DOPG (negatively charged liposomes), we obtained qualitative evidence that while so me of the drug was sequestered through an adsorption mechanism, a fraction of the drug was being absorbed into the annulus of the liposome. Quantitative tests further validated this hypothesis. We concluded that the ionic repulsion between the negatively charged lipids creates gaps in the lipid bilayer of the liposome, which may allow the drug to penetrate into the liposome. At the solubility limit of the drug at pH 10.7, the negatively charged liposomes sequestered almost all of the drug present in the solutions, and sequeste red more of the drug than the uncharged DMPC liposomes sequestered. Although th e drug uptake was not quantified at the physiological pH of 7.4, we obtained qualit ative evidence suggesting that significant drug uptake will occur at this pH. Thus, we ha ve shown that liposomes are potential drug overdose treatments. For NPs to be successful in treating an ove rdose, it is not enough to simply prove their efficacy in sequestering drugs in vitro and their safety for injection into the body. For these particles to be effective drug over dose treatments, it is imperative that they
146 reduce the drug concentration in the body in time scales of fewer than 30-60 minutes. In vivo, these time scales depend not only on the time scale of drug uptake by the NPs, but also on the time scales of drug equilibrat ion between the tissues, which involve a number of physiological factors such as the blood fl owrates and the partition coefficients of the drug in various tissues. While in vivo tests to determine the ti me scale of concentration reduction in the body after NP injection are de sirable, these are not always possible. In Chapter 3, a pharmacokinetic model was developed for drug detoxification by NPs that can predict the dynamic change s in drug concentration inside the body following an intravenous injection of NP. This model may be applied to any type of NP and considers the fact that some types of NPs will sequester drugs by adsorbing the drug to the surface of the particle, while others types of NPs will absorb the drug into the particle. Once inside the body, different types of NPs will have different behaviors. Some types of NPs may cross th e capillary walls and enter th e interstitial spaces of the tissue, while others may not be able to cross these biological barriers. Furthermore, the time scale for equilibrium sequestration of the drug by the NPs may be different for different types of NPs. If this equilibration time is much larger than the residence time of blood in the capillaries, 1-3 seconds (31), then there is essentially no uptake of the drug by NPs inside the capillaries. We have considered all of these possibilities in the development of the model. The model predictions show that the detoxification process is quicker if the NP does not enter the interstitia l space of the tissue and if the NP equilibrates with blood in the capillaries. Th us, it may be preferable to design NPs that do not cross the capillary wall.
147 We also determined that the effectiven ess of the NP at detoxification can be characterized by the parameter X, which depe nds on the dosage of the NP and the drugâ€™s partition coefficient between the NP and the blood. In Appendix E, we have demonstrated how to use in vitro experiments to determine X for a specific drug and NP. In our illustration, we focused on uptake of amitriptyline by a particular formulation of oil-in-water MEs and calculate d a value of 1.5 for the parameter X. Other MEs have been developed by our collaborators that se quester more of the drug and have larger X values. The model predictions show that larg er X values lead to a more effective and rapid detoxification processes and that for the drug amitriptyline, a NP with an X value of at least 10 is required for effec tive drug overdose treatment. The pharmacokinetic model developed in Chapter 3 assumes that the tissues are well-mixed compartments. Thus, the axial de pendence of the drug co ncentration inside the tissues is not included in the model. Although most pharmacoki netic models assume well-mixed tissues, a scaling analysis inside th e capillary shows that this assumption is in most cases invalid; therefore, these models may be inaccurately predicting, and possibly under-predicting, drug concentrations in the ti ssues. In Chapter 4, we have developed a model for drug dispersion inside the capillaries. In this model, we have treated the tissue as a collection of Krogh tissue cylinder s and have solved the convection-diffusion equations in the capillary and tissue regions. A perturbation expansion in the aspect ratio of the capillary was used to obtain the effec tive dispersion equations. We first considered drugs with small interfacial mass transfer resistances, flow rate limited drugs, and recovered Arisâ€™s results for the dispersion coefficient, D* (38). This D* may be expressed as the sum of contributions from axial diffusi on, interfacial mass transfer resistance, and
148 convection, and the convective contribution to dispersion was determined to arise from four different mechanisms. While in some instances, D* may also be experimentally determined, this is a tedious and time consum ing process, and the value will vary with physiological factors and drug characteristics. Thus, the predictive model for calculation of D* developed in Chapter 4 is extremely useful in providing estimations of D* without performing such tedious experiments. This model may easily be applied to any drug in any tissue if some basic pharmacokinetic data ar e available. The values of the predicted dispersion coefficient match reasonably well w ith the values reported in literature (50). A number of other researchers have develo ped equations for dispersion coefficients that are valid for mass transfer in capillaries (38-40) but a scaling analysis shows that these expressions are not valid when the perm eability of the drug across the capillary wall is small. We have also investigated this case of drugs with small permeabilities across the capillary wall, membrane resistance limited drugs, and determined that in this case the average equations cannot be expressed as a single convection-diffusion equation. In this limit the mass transfer problem reduces to two coupled equati ons, one each for the tissue and the capillary. We al so show that there is a smooth transition from the regime of flow limited drugs to the membrane resi stance limited drugs and in the intermediate regime of the permeability, both models will give the same results. In Chapter 5, we have extended the model developed in Chapter 4, describing drug dispersion inside the capillaries, to incl ude the presence of the RBCs. Because the capillary diameter is smaller than the size of the RBCs, the manner in which the RBCs flow through the capillary is complex and may significantly impact the dispersion of the drug inside the capillaries. As in Chapter 4, we have investigated both flow rate and
149 membrane resistance limited drugs in Chapter 5. For the case of flow rate limited drugs, we have developed an expression for D* and have shown that calcu lated values using this model are close to the experimentally determined values. The average equations for both classes of drugs represen t the solutions to the mass transfer problem in a single tissue, and in order to determine the drug concentration in an y tissue, the average equations for all the tissues have to be so lved simultaneously. We have accomplished this in Chapter 5 by incorporating the model fo r drug dispersion in the capillaries into a pharmacokinetic model so that the effects of drug dispersion on the time scales of drug equilibration between tissues could be examined. We have numerically solved the dispersion equations using the method of fin ite difference so that the drug concentration inside the tissues could be predicted. Ideally, one would like to incorporate a ll of the concepts discussed in this dissertation into a comprehensive model for drug detoxification by NPs in vivo. For such a model, the pharmacokinetic models incorporating dispersion presented in Chapter 5 would be combined with the pharmacokinetic model for drug detoxification presented in Chapter 3, which assumes well-mixed tissue compartments. This could in principle be done; however, this requires a number of drug and NP-dependent and physiological parameters, some of which are currently unavail able. Thus, we have chosen not to create such a model, although this may be done in th e future using the principles and concepts shown in this dissertation. Furthermore, if the model is to be a truly accurate representation of drug detoxification by NPs in vivo, a number of other factors should be included. For example, the binding of the dr ug to proteins and the drug clearance should be explicitly included into the model. Add itionally, binding of proteins and other ions to
150 the NPs should be investigated and included . Factors such as br eakup and clearance of NPs, existence of anastomoses, and solute tr apping were also neglected due to lack of available data, but may become important in the drug dispersion in certain tissues (53). Such factors may be included within the fram ework proposed in this dissertation if such data become available. On the experimental side, future work shou ld include use of other techniques such as pulsed field gradient NMR to quantify th e drug uptake by liposomes at physiological pH. Also studies could be conducted with ot her types of liposomes to further investigate the effects of surface charge and lipid pack ing on the drug uptake. Furthermore, the liposomes could be stabilized through the use of polymeric lipids to ensure that there is no disintegration upon injection into the body. Additionally, a safety analysis of the liposomes should be conducted by TEG and hemolysis testing. To summarize, in this dissertation we have shown that liposomes have significant potential for use in drug overdose treatment and have developed a ri gorous and detailed theoretical framework that can be used to evaluate the efficacy of specific NPs for drug overdose treatment.
151 APPENDIX A DERIVATION OF THE DISPERSION COEF FICIENT FOR THE CASE OF FLOW RATE LIMITED DRUGS We find the dispersion coefficient for the case of large permeability drugs, discussed in Chapter 5, by using Eqns. (5-10) and (5-11), in which u ~ is an unknown constant that must be found before we can determine D*. Since <<1, we can assume a regular perturbation expansion for C ~ and tC ~ as follows 2 2 1 0C ~ C ~ C ~ C ~ (A-1) t2 2 t1 t0 tC ~ C ~ C ~ C ~ (A-2) We now substitute these expansions into Eqns . (5-10) and (5-11), and the BCs to solve the equations to various orders of . Solving to the order 1/2 To the order 2 1 , Eqns. (5-10) and (5-11) are t0r10 cf 02BiCKC fK (A-3) and tt0 effDC 1 0r Drrr (A-4)
152 Eqn. (A-3) becomes t0r10CKC. Eqn. (A-4) implies that the leading order concentration in the tissue, t0C ~ , does not vary in the radial di rection. It follows that t00CKC everywhere. Solving to the order 1/ To the order 1 , Eqns. (5-10) and (5-11) become 0 t0r10 * cC f Pe1u2BiCKC zfK (A-5) t0tt1 * effCDC 1 -Peur zDrrr (A-6) The constant u ~ is obtained by multiplying Eqn. (A-6) by r ~ and integrating it from r ~ =1 to r ~ = a, and then adding this result to Eqn. (A-5). Doing this, and using the BCs we determine c 2 cfK u fKf1K1a (A-7) We note that the *z dependency of Eqns. (A-5) and (A -6) can be satisfied by letting * 0 1z ~ C ~ ) r ~ B( C ~ and * 0 t t1z ~ C ~ ) r ~ ( B C ~ , where B and Bt are functions of r. B and Bt can be determined by substituting the assumed expressions for 1C ~ and t1C ~ in Eqns. (A-5) and (A-6) and solving the resulting equations for B and Bt along with the following BCs. t teff t tdB 0 at r dr dBD f BiKB-B at r0 drDf1a This results in the following expressions.
153 -1 22 eff 2 cDt Bif1--1f1 D Pe B 2Bi fKK-1f1aa a (A-8) and -1 222 ccc eff t 2 cDt K2lnrfK-rfK+2ffK D Pe B 4 fKK-1f1aa a (A-9) Solving to the order 0 Solving to the order 0, Eqns. (5-10) and (5-11) become 2 0 0 1 t22 **2 ccCC C ff1 Pe1u2BiC-KC tzfKfKz (A-10) and 2 t0t1tt2tt0 **2 effeffCCDCDC 1 Peur tzDrrrDz (A-11) We substitute for1C ~ in terms of B, for t1C ~ in terms of Bt, and for t0C ~ as K0C ~ . Next, we multiply Eqn. (A-11) by r ~ and integrate from r ~ = 1 to r ~ = a. Adding Eqn. (A-10) to the integrated Eqn. (A-11) gives 2 * 0 2 * 0z ~ C ~ D ~ t ~ C ~ (A-12) where the mean dimensionless dispersion coefficient, *D ~ , is given by
154 2 2 22 c * 3 2 c 1 2 242 t c eff 3 2 c 2 t eff 2 cPeK-1f1fK D 2BifKK-1f1 D Pef1fKK4ln341 D 8fKK-1f1 D f1K-11 D fKK-1f1 a a aaa a a a (A-13) *D ~ is a function of the dimensionless parameters Pe, Bi, K, Kc, f, a, and t effD D . Knowing these parameters, Eqn. (A -13) can be used to calculate *D ~ for each tissue. It can be seen from re-dimension alization of Eqn. (A-12) that the dimensional dispersion coefficient, D*, is given by ** effDDD .
155 APPENDIX B DERIVATION OF THE DISPERSION EQUATIONS FOR THE CASE OF MEMBRANE RESISTANC E LIMITED DRUGS In this appendix, we solve the proble m for membrane resistance limited drugs as discussed in Chapter 5, where Bi Bi In this case the time scale for transport across the capillary wall (R/ ) is much slower than the time scale for lateral diffusion (R2/D) but is still faster than the axial diffusion time scale (L2/D). Accordingly, at the shortest time scale of R2/D, the capillary and tissue are not exp ected to be in equilibrium, and the intermediate time scale of R/ must be introduced into the problem. We do this assuming the concentration in the capillary and the tissue are functions of 2 slt t/R, tDt/L, rr/R , and L / z z ~ . This problem is solved in a stationary reference frame. Given that slCC(t,t,z,r) and ttslCC(t,t,z,r) , it follows that 2 lsslslCDC C CDRC C C +=+=+ tLtRtRt LLtRtBit (B-1) and ttttttt 2 lsslslCCCCCCC D RD =+=+ tLtRtRtL LtRtBit (B-2) Thus, the dimensionless forms of the govern ing equations Eqns. (5-6) and (5-7) are 2 tr=1 2 lsccCBiCPeCBiff1C 2CKC t t z fKfKz (B-3) and
156 2 tttttt 22 lseffCCDCDC 111 Bir t t DrrrDz (B-4) The dimensionless boundary conditions are: 1. t t effD Cf Bi(KC-C) Drf1 at 1 r ~ 2. tC 0 r at r a A set of equations for various orders of are obtained by substituting the regular expansions of C ~ and tC ~ into Eqns. (B-3) and (B-4 ) and applying the boundary conditions. O(1/2) To 21 O Eqns. (B-3) and (B-4) are r ~ C ~ r ~ r ~ r ~ 1 00 (B-5) r ~ C ~ r ~ r ~ r ~ 1 0t0 (B-6) Equations (B-5) and (B-6) along w ith the boundary conditions yield t0t0lsCC(t,t,z) and 00lsCC(t,t,z) . This shows that, similar to the small resistance case, the leading order concentrations in the t ube and the annulus do not vary in the radial direction; however, these concentrations are no longer in equilibrium as they were in the O(1) Bi case. O(1/)
157 To 1 O Eqns. (B-3) and (B-4) are 00 t00 scCC f Bi+Pe=2BiC-KC tzfK (B-7) and t0tt1 seffCDC 1 Bi=r tDrrr (B-8) Averaging the tissue equation, that is, multiplying Eqn. (B-8) by r and then integrating from r= 1 to r = a, and applying the bound ary conditions gives the following expression 2 t0 0t0 sC 1f (KCC) 2tf1a (B-9) Rearranging Eqns. (B-7) and (B-9) to eliminate s 0t ~ C ~ and s 0 tt ~ C ~ from Eqns. (B-1) and (B-2) gives 000 t00 cl 0 t00 cCCC fPe =2C-KCtRfKBizRBit C 2 f u+CKC zRfK (B-10) and t00 0t0 2 0t0 2CC 2 f1 =KCC tRf11RBitl 2 f KCC f1 R1a a (B-11)
116 dispersion models to those obtained from th e respective well-mixed models (Appendices C and D). BLOOD CbVb TISSUE CV Cb Bolus Dose Q Figure 5-4. Single tissue model. Single Tissue Dispersion Model Equations Flow Rate Limited For the case of flow rate limited drugs, the mean convection-dispersion equation in terms is given by: 2 * effeffeff 2CCC uD tzz (5-17) where Ceff is the effective drug concentration in the capillary, which includes both the plasma and RBC spaces, and is defined as c efffK CC f1 (5-18) The drug concentration in the tissue space, Ct, is given by Ct = KC = KeffCeff, where Keff is the effective partition coefficient of th e drug between the tissue and capillary, defined as eff c1f KK fK . The mass balance for the blood compartment, which is assumed to be well-mixed, is as follows.
159 and c eff b t b1 V1V K 1CC V (C-4) Following the same principles, the dimensionless equations for the well-mixed whole-body model are as follows. For the gut, heart, brain, kidneys, muscle, a nd fat compartments (i=G, H, B, K, M, and F, respectively) LK C ~ Q K C ~ Q C ~ Q Q V u L t ~ d C ~ dtL L G tG G b G L L K tL (C-5) For the liver compartment titi i b KiidCC QL C dtuVK (C-6) For the blood compartment b b t, F M, K, B, H, L, i i ti i b K bC ~ R K C ~ Q V u L t ~ d C ~ d (C-7) These equations are subject to the initial conditions bC ~ = 1 and iC ~ = 0 at t ~ = 0. In the above equations, the subscripts L, G, H, B, K, M, F, and b represent the liver, gut, heart, brain, kidney, muscle, fat, and blo od respectively. These equations were dedimensionalized in the same manner as the whole-body dispersion model for flow rate limited drugs.
160 APPENDIX D WELL-MIXED SINGLE TISSUE MOD EL EQUATIONS FOR MEMBRANE RESISTANCE LIMITED DRUGS This appendix gives the equations used in the well-mixed single tissue model for the case of membrane resistance limited drugs discussed in Chapter 5. The mass balance equations for this model are de-dimensionali zed in the same manner as the dispersion single tissue model for membrane resistance lim ited drugs in Chapter 5. The equation for the blood compartment is: b teffbdC RCC d (D-1) In the tissue compartments the mass bala nce equations are as follows for the capillary and tissue spaces: eff b effeffefftdC CC2 KCC d (D-2) and t effefft 2dC 2 KCC d 1a (D-3) where f1Bi f1 Pe .
161 APPENDIX E SEQUESTRATION OF DRUG BY MICROEMULSIONS In this appendix, we investigate the upta ke of amitriptyline by oil-in water MEs, stabilized by a Pluronic surf actant and develop a model for th e drug uptake. In the next section we describe the in vitro experiments that were done to measure the uptake of amitriptyline by ME. We also develop a model to interpret the data and to relate the uptake of the drug by ME to the bulk drug concentration. This uptake model may be subsequently incorporated into the PB PK model described in Chapter 3. Experimental Methods Drug Binding by ME In vitro experiments were conducted to determ ine the binding efficiency of oil-inwater ME for amitriptyline (Figure 2-1). The ME contained ethyl butyrate as the oil phase (0.9% by weight), Pluronic F-127 as the surfactant (10.6% by weight), sodium caprylate as the co-surfactant (0.1% by weight ), and normal saline (0.85% NaCl) as the continuous phase (88.4% by weight). To conduct the binding experiments, the ME was first diluted with normal saline such that th e diluted solution contai ned either 0.05% or 0.10% oil by weight. Solutions of amitriptyline in normal saline at different drug concentrations were prepared, and the diluted ME was then added to these drug solutions. The ME and the drug solution were stirred fo r about 15 minutes, which is much larger than the equilibration time (determined in a separate experiment to be less than 10 seconds). The solutions were then centrifuged at 3000 rpms for 30 minutes to separate the ME from the bulk solution, and the free drug concentration was measured using an
162 HPLC. The fraction of drug removed, , was then calculated at each drug concentration. The values of are plotted as a function of initial drug concentration, C0 in Figure E-1. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 10100100010000 Initial Drug Concentration, C0 ( M)Fraction Drug Uptake, Experiment Model Figure E-1. Fraction of amitriptyline se questered by the microemulsion, as experimentally determined and as predicted by the adsorption model for microemulsion formulations containing 0.05% and 0.10% oil. Drug Partitioning Experiments To determine whether the drug sequestered by ME is adsorbed on the surface or absorbed in the oil core, we measured the partition coefficient of amitriptyline between the oil phase (ethyl butyrate) and the a queous phase (normal saline). The partition coefficient is a measure of the maximum amount of the drug that can be absorbed into the oil phase of the ME. To measure the partition coefficient, we prepared drug solutions of amitriptyline in normal saline at different concentrations, and then added an equal volume of ethyl butyrate. The solutions were stirred for 30 minutes, and then allowed the phases to separate for 12 hours. The fina l drug concentrations in each phase were measured using an HPLC.
163 Data Interpretation Below we develop a model to interpret th e data for drug sequestration by the ME. The model assumes that the drug uptake occu rs entirely due to adsorption/binding on the surface of the oil drops and that the drug absorbed by the oil core of the ME is negligible. It is shown in the next secti on that the maximum amount of drug that can be absorbed by the oil phase is indeed negligible, and thus , the assumption of sequestration by adsorption is justified. Drug Partitioning Experiments To confirm that absorption does not sign ificantly contribute to drug uptake by ME, we analyze the results of the drug partiti oning experiment. The oil-water partition coefficient, K, is defined as ratio between the final drug concentration in the oil phase (Coil) to the final concentratio n in the aqueous phase (Cw,f), i.e., K = Coil/Cw,f. Since both Coil and Cw,f were measured in the drug partitioning experiments, the value of K can be determined. This value of K can then be us ed to calculate the ma ximum fraction of drug (abs) that could be absorbed by ME if th e mechanism of drug extraction were a pure absorption into the oil phase. A simple mass balance for the drug in the presence of ME gives 1 V V K 1 1 oil w abs (E-1) In the above equation, Vw/Voil is the aqueous to oil volume ratio in the drug binding by ME experiments. As shown in Figure E-2, abs values for amitriptyline are on the order of 10-4, which are significantly smaller than the values that were experimentally determined from the microemulsion experiments (shown in Figure E-1). Additionally,
164 the predictions for abs show a decreasing trend with increasing C0, whereas Figure E-1 shows a relatively constant at low C0. Thus, absorption cannot be the primary mechanism for drug extraction by ME, and the assumption of negligible drug uptake by absorption is justified. 0 0.1 0.2 0.3 0.4 0.5 0.6 020040060080010001200Initial Drug Concentration in Aqueous Phase, C0 (mM)abs x 10-3 0.05% oil 0.10% oil Figure E-2. Fraction of amitriptyline that could be sequestered by the microemulsion if the mechanism was purely absorption of the drug into the ethyl butyrate core of the microemulsion for a microemulsion containing 0.05% and 0.10% oil. Drug Binding by ME In vitro experiments showed that the equilibrium between the bulk drug concentration and the surface drug concentra tion on ME drops is reached within a few seconds, which suggests that drug is being ad sorbed to the surfac e of the surfactantcovered ME drops. Furthermore, the results of the absorption studi es discussed in the previous section also suggest an adsorption m echanism. We thus assume an adsorption mechanism of drug sequestration and use a Lang muir adsorption isotherm to describe the drug uptake by the ME, given by
165 C C (E-2) where is the surface drug concentration, C is the drug concentration in the bulk, is the surface drug concentration at maximum packing, is the drugâ€™s ad sorption constant, and is the drugâ€™s desorption consta nt. It should be noted that and are surface drug concentrations, and thus have units of mole s of adsorbed drug per unit area (moles/m2). We note that this model does not differentiate between different mechanisms of adsorption or binding, such as hydrophobic effects, electrostatics , and specific binding. The drug amitriptyline is administered in the hydrochloride form (AHCl). In normal saline, a majority of the drug is present in the acidic form (AH+). Thus, it is reasonable to assume the existence of only one species at the inte rface, as is implicitly assumed in the above model. At pHs close to the pKa of drugs, we will need to modify the model to include all forms of the drug. Th e final concentration of drug in saline, C, can be related to the initial concentration, C0, by 1 C C0 (E-3) where is the fraction of drug removed from th e normal saline by ME. We assume that the drug must either remain in solution or adso rb to the surfactant, so that the drug mass balance is S VC VC0 (E-4) where V is the volume of continuous media ( normal saline) and S is the total surface area of the ME oil droplets. Combining Eqns. (E-2 ), (E-3), and (E-4), we obtain the following implicit equation for .
166 1 C 1 V S 0 (E-5) We fit the experimental data to Eqn. (E-5) to obtain values for and / , and the results of this fit are shown by the solid lines in Figure E-1. It should be noted that the value of S depends upon the diameter of the ME oil droplets. The diameter of the MEs was determined by dynamic light scattering to be about 25 nm; however, because the Pluronic F-127 is a large surfact ant, we have assumed the diameter of the oil droplets to be 5 nm in our calculations. The best fit values of and / for amitriptyline at the microemulsion formulation c ontaining 0.05% oil are 3.5 mol/m2 and 9174 mol, respectively. Similarly, the best fit values of and / for amitriptyline at the microemulsion formulation c ontaining 0.10% oil are 2.9 mol/m2 and 6759 mol, respectively. From the fitted values, we calculate the adsorbed area for each drug molecule at maximum drug pack ing to be between 48 and 57 2/molecule. This number appears to be reasonable based on looking at the structure of amitriptyline (Figure 2-1) and also matches the area of adsorbed dr ug per molecule determined in the liposome experiments described in Chapter 2. Furtherm ore, we calculate the area per molecule of sequestered drug directly from the experiment al data. The results of these calculations are shown in Figure E-3 and show that as dr ug concentration is incr eased, more drug is sequestered, and thus, the surface becomes more tightly packed. The calculated values of the area per molecule from the microemuls ion experiment in which 0.10% oil was present lie along the same lin e as the calculated values from the microemulsion experiment in which 0.05% oil was present. If this line is then extrapolated to higher
167 final drug concentrations, the area per molecule will eventually become constant. From Figure E-3, it appears this le veling off could occur around 50 2/molecule, which matches the value predicted from the adsorption model. 1 10 100 1000 10000 100000 10100100010000 Final Drug Concentration, Cf (M)area of drug/molecule at M E surface (2/molec) 0.05% oil 0.10% oil Figure E-3. Area of drug/molecule at the su rface of the microemulsion as calculated from the experimental data assuming an adsorption mechanism for a microemulsion containing 0.05% and 0.10% oil. From Eqn. (E-2), it can be seen th at at small drug concentrations (C<< / ), which are expected inside the body, C . Accordingly, we defi ne a partition coefficient for the ME, RME . The RME value for amitriptyline is between 3.78-4 L/m2 (calculated from the experiment in which the microemulsion contained 0.05% oil) and 4.31-4 L/m2 (calculated from the experiment in which the microemulsion contained 0.10% oil).
168 Determination of X As described in Chapter 3, the effectiv eness of ME at bi nding drugs can be characterized by a single parameter X, given by ME bSR X1 V (E-6) where S/Vb is the surface area per unit volume of the ME inside the human body. Thus, the dilution effects of the ME inside the body af ter injection must be considered and Eqn. (E-6) may be re-written as I ME stock bV S X1R VV (E-7) where stockS V is the surface areas of oil per unit vol ume in the stock ME solution, V is the volume of ME injected into the body, and Vb is the volume of the blood in the body. The ratio of VI to Vb is the dilution in the ME concen tration due to mixing with the blood volume and can be 0.1 at most fo r the largest safe dose of ME. From the in vitro data shown in Figure E-1, we cal culated an X value of 1.5 for a VI /Vb value of 0.1. The value of X will be lower in presence of complex media like plasma and blood due to protein binding. Thus , to determine the accurate value of X, the drug binding experiments must be performed in complex media. It should be noted that X values of about 10 have been attained in complex media by some of our collaboraters for other ME formulations. The model shown in this appendix may be used for any NP which sequester drug through an adsorption mechanism and may be used to determine the value of X for a particular NP and drug. This X value may then be incorporated into
169 the model described in Chapter 3 to predict drug uptake by the NP in vivo. A similar model may be developed for NPs which sequest er drug by an absorption mechanism.
170 LIST OF REFERENCES 1. Litovitz T. The TESS Database: Use in Product Safety Assesment. Drug Saf. 1998; 18 :9-19. 2. Litovitz T., Holm K., Clancy C., Schmitz B., Clark L., Oderda G. 1992 Annual Report of the American Association of Poison Control Centers Toxic Exposure Surveillance System. Am J Emerg Med 1993; 11 :494-555. 3. Kapur S., Mieczkowski T., Mann J. J. An tidepressant medications and the relative risk of suicide attempt and suicide. J. Am. Med. Assoc. 1992; 268 :3441-3445. 4. Bukbirwa H., Conn D. A. Toxicity from Local Anaesthetic Drugs. Update in Anesthesia 1999; 10 :1. 5. McEvoy G. K., ed. AHFS Drug Information, 2000 . Bethesda: American Society of Health-System Pharmacists, 2000. 6. Beck L. Overdose. In: Krapp K, ed. Gale Encylopedia of Nursing and Allied Health : Thomson Gale, 2002. 7. Varshney M., Morey T. E., Shah D. O., F lint J. A., Moudgil B. M., Seubert C. N., Dennis D. M. Pluronic microemulsions as nanoreservoirs for extraction of bupivacaine from normal saline. J. Am. Chem. Soc. 2004; 126 :5108-5112. 8. Underhill R. S., Jovanovic A. V., Carino S. R., Varshney M., Shah D. O., Dennis D. M., Morey T. E., Duran R. S. Oilfilled silica nanocapsules for lipophilic drug uptake: implications for drug detoxification therapy. Chem. Mater. 2002; 14 :4919-4925. 9. Petrikovics I., Hong K., Om buro G., Hu Q. Z., Pei L., McGuinn W. D., Sylvester D., Tamulinas C., Papahadjopoulos D., Jasz berenyi J. C., Way J. L. Antagonism of paraoxon intoxication by recombinant phos photriesterase encapsulated within sterically stabilized liposomes. Toxicol. Appl. Pharmacol. 1999; 156 :56-63. 10. Deo N., Somasundaran T., Somasundaran P. Solution properties of amitriptyline and its partitioning in to lipid bilayers. Colloids Surf. B 2004; 34 :155-159. 11. Bourne D. W. A. Mathematical Modeling of Pharmacokinetic Data . Lancaster: Technomic Publishing Co., 1995.
171 12. Saltzman W. M. Drug Delivery: Engineering Pr inciples for Drug Delivery . Oxford ; New York: Oxford University Press, 2001. 13. Welling P. G. Pharmacokinetics: Regulatory, Industrial, Academic Perspectives. In: Tse FLS, ed. Drugs and the Pharmaceutical Sciences . Vol 67. New York: Marcel Dekker, 1995. 14. Morey T. E., Varshney M., Flint J. A., Ra jasekaran S., Shah D. O., Dennis D. M. Treatment of local anesthetic-induced cardiotoxicity using drug scavenging nanoparticles. Nano Lett. 2004; 4 :757-759. 15. Sharma A., Sharma U. S. Liposomes in drug delivery: progress and limitations. Int. J. Pharm. 1997; 154 :123-140. 16. Rui Y., Wang S., Low P. S., Thompson D. H. Diplasmenylcholine-folate liposomes: an efficient vehicle for intracellular drug delivery. J. Am. Chem. Soc. 1998; 120 :11213-11218. 17. Weinstein J. N., Leserman L. D. Liposomes as drug carriers in cancer chemotherapy. Pharmac. Ther. 1984; 24 :207-233. 18. Lasic D. D. Novel applications of liposomes. Trends Biotechnol. 1998; 16 :307321. 19. Lasic D. D., Papahadjopoulos D. Liposomes revisited. Science 1995; 267 :12751276. 20. Lasic D. D., Papahadjopoulos D. Medical Applications of Liposomes . Amsterdam: Elsevier Science B.V., 1998. 21. Ostro M. Liposomes From Biophysics to Therapeutics . New York: Marcel Dekker, Inc., 1987. 22. Cole C. E., Tran H. T., Schlosser P. M. Physiologically based pharmacokinetic modeling of benzene metabolism in mice thr ough extrapolation from in vitro to in vivo. J. Toxicol. Environ. Health, Part A 2001; 62 :439-65. 23. Clewell H. J., III, Gentry P. R., Covington T. R., Gearhart J. M. Development of a physiologically based pharmacokinetic model of trichloroethylene and its metabolites for use in risk assessment. Environ. Health Perspect. 2000; 108 Suppl 2 :283-305. 24. Ebling W. F., Wada D. R., Stanski D. R. From piecewise to full physiologic pharmacokinetic modeling: applied to th iopental disposition in the rat. J. Pharmacokinet. Biopharm. 1994; 22 :259-92.
172 25. Fisher J. W. Physiologically based phar macokinetic models for trichloroethylene and its oxidative metabolites. Environ. Health Persp. Supplements 2000; 108 :265273. 26. Lutz R. J., Galbraith W. M., Dedrick R. L., Shrager R., Mellett L. B. A model for the kinetics of distri bution of actinomycin-D in the beagle dog. J. Pharmacol. Exp. Ther. 1977; 200 :469-78. 27. Poulin P., Theil F. P. A priori prediction of tissue:plasma partition coefficients of drugs to facilitate the use of physiologi cally-based pharmacokinetic models in drug discovery. J. Pharm. Sci. 2000; 89 :16-35. 28. Thrall K. D., Soelberg J. J., Weitz K. K., Woodstock A. D. Development of a physiologically based pharmacokinetic m odel for methyl ethyl ketone in F344 rats. J. Toxicol. Environ. Health A 2002; 65 :881-96. 29. Wada D. R., Bjorkman S., Ebling W. F., Harashima H., Harapat S. R., Stanski D. R. Computer simulation of the effect s of alterations in blood flows and body composition on thiopental pharmacokinetics in humans. Anesthesiology 1997; 87 :884-99. 30. Gibaldi M., Perrier D. Pharmacokinetics, 2nd ed. New York: Marcel Dekker, 1982. 31. Guyton A. C. Textbook of Medical Physiology, 7th ed. Philadelphia: Saunders, 1986. 32. Smith J. J., Kampine J. P. Circulatory Physiology : The Essentials, 2nd ed. Baltimore: Williams & Wilkins, 1984. 33. Fournier R. L. Basic Transport Phenomena in Biomedical Engineering . Philadelphia: Taylor & Francis, 1999. 34. Taylor G. I. Dispersion of soluble ma tter in solvent flowi ng slowly through a tube. Proc. R. Soc. Lon. Ser. A 1953; 219 :186-203. 35. Ananthkrishnan V., Gill W. N., Barduhn A. J. Laminar dispersion in capillaries: Part I mathematical analysis. AICHE J. 1965; 11 :1063-1072. 36. Gill W. N., Sankaras R. Exact anal ysis of unsteady convective diffusion. Proc. R. Soc. London, Ser. A 1970; 316 :341-350. 37. Sankarasubramanian R., Gill W. N. Unsteady convective diffusion with interphase mass transfer. Proc. Roy. Soc. London, Ser. A 1973; 333 :115-132.
173 38. Aris R. On the dispersion of a solute by diffusion, convection and exchange between phases. Proc. Roy. Soc. A 1959; 252 :538-550. 39. Levitt D. G. Capillary-tissue exchange ki netics an analysis of the Krogh cylinder model. J. Theor. Biol. 1972; 34 :103-124. 40. Tepper R. S., Lee H. L., Lightfoot E. N. Transient convective mass transfer in Krogh tissue cylinders. Ann. Biomed. Eng. 1978; 6 :506-530. 41. Krogh A. The number and distribution of cap illaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol. 1919; 52 :409-415. 42. Popel A. S., Goldman D., Vadapalli A. Modeling of oxygen diffusion from the blood vessels to intracellular organelles. Adv. Exp. Med. Biol. 2003; 530 :485-495. 43. Charm S. E., Kurland G. S. Blood Flow and Microcirculation . New York: Wiley, 1974. 44. Bloor M. I. The flow of blood in the capillaries. Phys. Med. Biol. 1968; 13 :44350. 45. Burton A. C. Physiology and Biophysics of the Ci rculation: An Introductory Text . Chicago: Year Book Medical Publishers, 1965. 46. Alphin R. S., Martens J. R., Dennis D. M. Frequency-dependent effects of propofol on atrioventricula r nodal conduction in guinea pig isolated heart. Mechanism and potential antidysrhythmic properties. Anesthesiology 1995; 83 :382-94. 47. Morey T. E., Martynyuk A. E., Napolitano C. A., Raatikainen M. J., Guyton T. S., Dennis D. M. Ionic basis of the different ial effects of intrav enous anesthetics on erythromycin-induced prolongation of ventri cular repolarization in the guinea pig heart. Anesthesiology 1997; 87 :1172-81. 48. Napolitano C. A., Raatikainen M. J., Ma rtens J. R., Dennis D. M. Effects of intravenous anesthetics on atrial wavelength and atri oventricular nodal conduction in guinea pig heart. Potentia l antidysrhythmic properties and clinical implications. Anesthesiology 1996; 85 :393-402. 49. Novalija E., Fujita S., Kampine J. P., St owe D. F. Sevoflurane mimics ischemic preconditioning effects on coronary flow and nitric oxide release in isolated hearts. Anesthesiology 1999; 91 :701-12.
174 50. Oliver R. E., Jones A. F., Rowland M. A whole-body physiologically based pharmacokinetic model incorporating di spersion concepts: Short and long time characteristics. J. Pharmacokinet. Pharmocodyn. 2001; 28 :27-55. 51. Stroh M., Zipfel W. R., Williams R. M., Webb W. W., Saltz man W. M. Diffusion of nerve growth factor in rat striatum as determined by multiphoton microscopy. Biophys. J. 2003; 85 :581-588. 52. Altman P. L., Katz D. D., Grebe R. M. Handbook of Circulation . Philadelphia: Saunders, 1959. 53. Oliver R. E., Heatherington A. C., Jo nes A. F., Rowland M. A physiologically based pharmacokinetic model incorporati ng dispersion principles to describe solute distribution in the perf used rat hindlimb preparation. J. Pharmacokinet. Biopharm. 1997; 25 :389-412. 54. Levenspiel O. Chemical Reaction Engineering, 2d ed. New York: Wiley, 1972. 55. Diaz-Garcia J. M., Evans A. M., Rowland M. Application of the axial dispersion model of hepatic drug elimination to th e kinetics of diazepam in the isolated perfused rat liver. J. Pharmacokinet. Biopharm. 1992; 20 :171-93. 56. Evans A. M., Hussein Z., Rowland M. In fluence of albumin on the distribution and elimination kinetics of diclofenac in the isolated perfused-rat-liver-analysis by the impulse-response techniqu e and the dispersion model. J. Pharm. Sci. 1993; 82 :421-428. 57. Yano Y., Yamaoka K., Aoyama Y., Ta naka H. Two-compartment dispersion model for analysis of organ perfusion system of drugs by fast inverse Laplace transform (FILT). J. Pharmacokinet. Biopharm. 1989; 17 :179-202. 58. Banks H. T., Musante C. J., Tran H. T. A dispersion model for the hepatic uptake and elimination of 2,3,7,8-te trachlorodibenzo-p-dioxin. Math. Comput. Model. 1998; 28 :9-29. 59. Hisaka A., Sugiyama Y. Analysis of nonlinear and nonsteady state hepatic extraction with the dispersion model using the finite difference method. J. Pharmacokinet. Biopharm. 1998; 26 :495-519. 60. Lafrance P., Lemaire V., Varin F., Donati F ., Belair J., Nekka F. Spatial effects in modeling pharmacokinetics of rapid action drugs. Bull. Math. Biol. 2002; 64 :483500. 61. Milnor R. R. Capillaries and Lym phatic Vessels. In: Mountcastle VB, ed. Medical Physiology, 14th ed. Saint Louis: C.V. Mosly Company, 1980;1085 1093.
175 62. Rakusan K., Cicutti, N., Kolar, F. Effect of anemia on cardiac function, microvascular structure, and capi llary hematocrit in rat hearts. Am. J. Physiol. Heart Circ. Physiol. 2001; 280 :H1407-H1414. 63. Dzwinel W., Boryczko, K., Yuen, D. A. A discrete-particle model of blood dynamics in capillary vessels. J.C.I.S. 2003; 258 :163-173. 64. Boryczko K., Dzwinel, W., Yuen, D. A. Dynamical clustering of red blood cells in capillary vessels. J. Mol. Model. 2003; 9 :16-33. 65. Bird R. B., Stewart W. E., Lightfoot E. N. Transport Phenomena . New York: Wiley, 1960. 66. Boderke P., Schittkowski K., Wolf M., Merkle H. P. Modeling of diffusion and concurrent metabolism in cutaneous tissue. J. Theor. Biol. 2000; 204 :393-407. 67. Marieb E. N. Human Anatomy & Physiology, 4th ed. Menlo Park: Benjamin/Cummings, 1998. 68. Mountcastle V. B. Medical Physiology, 14th ed. Saint Louis: C. V. Mosby Co., 1980. 69. Shin H., Chavan A., Witthus F., Selle D ., Stamm G., Peitgen H. O., Galanski M. Precise determination of aortic length in patients with aortic stent grafts: in vivo evaluation of a thinning algorithm a pplied to CT angiography data. Eur. Radiol. 2001; 11 :733-738.
176 BIOGRAPHICAL SKETCH Marissa S. Fallon was born on June 13, 1979, on Clark Air Base, Philippines. She graduated from Kadena High School in Okinaw a, Japan, in 1997. Marissa began her college education at the Illinois Institute of Technology in Chicago, Illinois, as a Camras National Excellence in Technology Scholar. In May 2001, she received her Bachelor of Science in chemical engineering with a sp ecialization in biotec hnology. Marissa then entered graduate school at the University of Florida in Gainesville, Florida. As a graduate student she began performing research in the areas of nanoparticle science and technology and pharmacokinetic modeling under the guidance of Dr. Anuj Chauhan. Marissa received her Master of Science in biomedical e ngineering in December 2004 and her Doctor of Philosophy in chemical engineering in December 2005 from the University of Florida.