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1 MAGNETIC RESONANCE IMAGING BASED COMPUTATIONAL MODELS OF SOLID TUMORS By GREGORY PISHKO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGRE E OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011
2 2011 Gregory Pishko
3 To my mom and dad
4 ACKNOWLEDGMENTS There are so many people to acknowledge and thank. Most importantly, I would like to thank my parents for sacrificing and provid ing me everything that I needed to be where I am today. I appreciate all the support that my aunts, uncles, and cousin s provided My girlfriend provided me wi th a lot of emotional support a nd food to get me through some tough times As far as research goes I would like to thank Dr. Malisa Sarntinoranont for giving me the opportunity to work in her lab with financial assistance. Garrett Astary deserves a very special thank you for his dedication to imaging mouse tumors with me sometimes very late at night. Without his help this project would have not been possible. years. An extra special thanks is deserved to Dr. Jung Hwan Kim, Sung Jin Lee, Magdoom Mohammed, and Jin gya Zhang. I would like to extend a big thanks to my committee members : Dr. Brian Sorg, Dr. Hugh Fan, and Dr. Tran Son Tay. Additionally, I would like to thank Dr. Thomas Mareci for assistance with MR imaging, Marda Jorgensen at MBI Cell & Tissue Analysis Core (CTAC), and Dr. Dietmar Siemann and his lab: Dr. Lori Rice, Sharon Lepler, and Chris Pampo. The research in this dissertation Opportunity Fund.
5 TABLE OF CONTENTS pag e ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 1.1 Background ................................ ................................ ................................ ....... 13 1.2 Previous Studies and Motivation ................................ ................................ ....... 15 2 DEVELOPMENT AND SENSITIVITY ANALYSIS OF COMPUTATIONAL MODEL WITH HETEROGENEOUS VASCULATURE AND POROSITY ................ 20 2.1 Introduction ................................ ................................ ................................ ....... 20 2.2 Materials and Methods ................................ ................................ ...................... 24 2.2.1 Animal Preparation ................................ ................................ .................. 24 2.2.2 Magnetic Resonance Imaging (MRI) ................................ ....................... 24 2.2.3 Vascular Permeability and Porosity Maps ................................ ............... 25 2.2.4 3D Porous Media Model ................................ ................................ .......... 27 2.2.5 Computational Model ................................ ................................ ............... 30 2.2.6 Sensitivity Analysis ................................ ................................ .................. 31 18.104.22.168 Hydraulic c onductivity of t issue ( K ) ................................ ................ 33 22.214.171.124 Vascular h ydraulic p ermeability ( L p ) ................................ ............... 34 2.2.6. 3 Surface a rea p er u nit v olume ( S / V ) ................................ ................ 35 126.96.36.199 Porosity ( ) ................................ ................................ ..................... 36 188.8.131.52 Diffusion c oefficient ( D ) ................................ ................................ .. 36 2.3 Results ................................ ................................ ................................ .............. 37 2.3.1 Interstitial Fluid Transport ................................ ................................ ........ 37 2.3.2 Tracer Transport ................................ ................................ ...................... 39 2.4 Discussion and Conclusions ................................ ................................ ............. 41 2.5 Acknowledgements ................................ ................................ ........................... 49 3 PATIENT SPECIFIC COMPUTATIONAL MODELS OF SMALL SOLID TUMORS: A VALIDATION STUDY ................................ ................................ ........ 61 3.1 Introduction ................................ ................................ ................................ ....... 61 3.2 Materials and Methods ................................ ................................ ...................... 66 3.2.1 Animal Preparation ................................ ................................ .................. 66 3.2.2 Magnetic Resonance Imaging (MRI) ................................ ....................... 66
6 3.2.3 Histology ................................ ................................ ................................ .. 68 3.2.4 Image Segmentation ................................ ................................ ............... 69 3.2.5 Arterial Input Function, Vascular Permeability Maps, and Porosity Maps ................................ ................................ ................................ ............. 69 3.2.6 3D Porous Media Mathematical and Computational Model ..................... 70 3.3 Results ................................ ................................ ................................ .............. 72 3.3.1 Segmentation, Leakiness, and Histology ................................ ................. 7 2 3.3.2 Interstitial Fluid Transport ................................ ................................ ........ 73 3.3.3 Tracer Transport ................................ ................................ ...................... 73 3.4 Discussion and Conclusions ................................ ................................ ............. 76 4 PATIENT SPECIFIC COMPUTATIONAL MODELS OF LARGE SOLID TUMORS: AN EXPERIMENTAL COMPARISON ................................ ................... 93 4.1 Introduction ................................ ................................ ................................ ....... 93 4.2 Materials and Methods ................................ ................................ ...................... 96 4.2.1 Animal Preparation ................................ ................................ .................. 96 4.2.2 Magnetic Resonance Imaging (MRI) ................................ ....................... 97 4.2.3 Histology ................................ ................................ ................................ .. 98 4.2.4 Image Segmentation ................................ ................................ ............... 98 4.2.5 Arterial Input Function, Vasc ular Permeability Maps, and Porosity Maps ................................ ................................ ................................ ............. 99 4.2.6 3D Porous Media Mathematical and Computational Model ..................... 99 4.3 Results ................................ ................................ ................................ ............ 100 4.3.1 Segmentation, Leakiness, and Histology ................................ ............... 100 4.3.2 Interstitial Fluid Transport ................................ ................................ ...... 101 4.3.3 Tracer Transport ................................ ................................ .................... 101 4.4 Discussion and Conclusions ................................ ................................ ........... 104 5 CONCLUSIONS AND FUTURE WORK ................................ ............................... 115 LIST OF REFERENCES ................................ ................................ ............................. 121 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 130
7 LIST OF TABLES Table page 2 1A Tissue and vascular parameters used in simulations ................................ ......... 50 2 1B Tissue and vascular parameters used in simulations ................................ ......... 50 2 2 Summary of chapter 2 sensitivity analysis ................................ .......................... 50 4 1 Dynamic contrast enhanced magnetic resonance imaging parameters ........... 106 4 2 Diffusion weighted imaging parameters ................................ ............................ 106
8 LIST OF FIGURES Figure page 1 1 Norm al vs. tumor microvasculature ................................ ................................ .... 18 1 2 Examples of tumor necrosis ................................ ................................ ............... 19 2 1 Block diagram of the mathematical methodology and experimental verification for the patient s pecific tumor model. ................................ ................. 51 2 2 Geometric reconstr uction of tumor and host tissue ................................ ............ 52 2 3 Uptake and washout of average Gd DTPA c oncentration, C t,avg ( t ), in tumor tissue at seven t ime points for the first 19 min ................................ .................... 53 2 4. Signal enhancement in vessel and tumor periphery ................................ ........... 54 2 5 Maps from slice 5 of MR data set of A) K trans and B) with histograms .............. 55 2 6 IFP sensitivity analysis for varying transport properties (Table 2 1B) ................. 56 2 7 Tissue averaged IFV sensitivity analysis for varying transport properties (Table 2 1B) ................................ ................................ ................................ ........ 57 2 8 Comparison of experimental and simulated Gd DTPA distribution patterns in slice 5 for t = 5, 30, and 60 min for baseline transport values in Table 2 1B. ..... 58 2 9 Gd DTPA distribution along bisecting tumor line at early, int ermediate, a nd late time points ................................ ................................ ................................ ... 59 2 10 Gd DTPA distribution along bisecting line thro ugh two additional KHT sarcomas ................................ ................................ ................................ ............ 60 3 1 Orientation of MR slices with respect to the mouse anatomy (top view) ............. 81 3 2 CFD compatible meshes ................................ ................................ .................... 82 3 4 Tumor segmentation of MRI dat a for small KHT sarcomas. ............................... 83 3 5 H&E staining and example of estimating tumor area ................................ .......... 84 3 6 Leakiness ( K trans ) maps of small KH T sarcomas ................................ ................. 85 3 7 Example of MECA 32 staining for KHT 3. ................................ .......................... 86 3 8 Interstitial fluid pressure for small KHT sarcomas ................................ ............... 87 3 9 Interstitial fluid velocity for small KHT sarcomas ................................ ................. 88
9 3 10 Comparison of simulated and experimental average Gd DTPA in tumor ........... 89 3 11 Comparison of error versus time quantified for simulations and experiment for KHT sarcomas ................................ ................................ ............................... 90 3 12 Simulated tracer t ransport for small KHT sarcoma (KHT 1) .............................. 91 3 1 3 Comparison of experimental and simulated concentration profiles across small KHT sarcomas ................................ ................................ .......................... 92 4 1 Tumor segmentation with corresponding H&E staining of small KHT sarcomas. ................................ ................................ ................................ ......... 107 4 2 H&E staining of large KHT sarcomas for A) KHT 4, B) KHT 5, C) KHT 6, and D) KHT 7 ................................ ................................ ................................ .......... 108 4 3 Example of MECA 32 staining for KHT 6 ................................ ......................... 109 4 4 Leakiness ( K trans ) maps of large KHT sarcomas. ................................ .............. 110 4 5 Interstitial fluid pressure for large KHT sarcomas ................................ ............. 111 4 6 Interstitial fluid velocity for large KHT sarcomas. ................................ .............. 112 4 7 Experimental Gd DTPA concentration, C t distribution in large KHT sarcomas at time equals 30 min. ................................ ................................ ...................... 113 4 8 Simulated tracer t ransport for large KHT sar comas ................................ .......... 114
10 LIST OF ABBREVIATIONS ADC Apparent Diffusion Coefficient AIF Arterial Input Function CA Contrast Agent CFD Computational Fluid Dynamics DCE MRI Dynamic Contrast Enhanced Magnetic Resonance Imaging DTI Diffusion Tensor Imaging DWI Diffusion Weighted Imaging Gd DTPA Gadolinium Diethylene Triamine Penta Acetic Acid IFP Interstitial Fluid Pressure IFV Interstitial Fluid Velocity MR Magnetic Resonance MW Molecular Weight SE Spin Echo TE Time of Echo TR Time for Recov ery
11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MAGNETIC RESONANCE IMAGING BASED COMPUTATIONAL MODELS OF SOLI D TUMORS By Gregory Pishko August 2011 Chair: Malisa Sarntinoranont Major: Mechanical Engineering The s olid tumor environment is characterized by inherently unique heterogeneities. Specifically, tumors can have hyperpermeable, chaotic vessel networks t hat result in the non un iform distribution of drug. These network s further restrict drug penetration by imposing a high pressure within the tumor due to increased efflux of plasma fluid Regions of necrosis can play a role in reducing the uniformity of dru g distribution since these regions tend to have altered tissue properties and lack vasculature. The purpose of this study was to provide a methodology for an image based computational model of a solid tumor that could be used to predict distribution of a tracer for specific tumors at three stages of tumor growth. Simulations were compared to experimental concentrations in tissue measured from magnetic resonance imaging ( MRI ). First, an image based computational model was developed that incorporates 3D ma ps of vascular leakiness and porosity for small non necrotic murine KHT sarcomas with high leakiness (n = 3). Maps were constructed using early temporal MRI data ( t = 19 min). These models provided descriptions of extravasation and delivery of an MR
12 visibl e tracer (Gd DTPA) at later time points ( t = 1 hr). A s ensitivity analysis of the image based computational model showed the high leakiness tumors to be dominated by transvascular exchange The strength of the model to capture distribution of Gd DTPA conce ntration within the tumor was demonstrated. Second, the computational model was applied to small tumors with low leakiness for a validation study (n = 3). The low leakiness prevented the creation of porosity maps using early time points, so a constant poro sity was assumed. The porous media model was determined to be a superior model to the two compartment model for predicting uptake and washout behavior of average Gd DTPA concentration in tumor. The porous media can account for the role of convection in tra cer clearance in tumors with low leakiness. Third, a computational model was developed to account for heterogeneous perfusion of Gd DTPA due to both vasculature and fluid filled, avascular necrotic regions within large, mature tumors (n = 4). The porous me tracer transport was limited by the sparseness of the leakiness maps, which reduced the tracer sources in the model and complicated accurate prediction of transient behavior and distribution of Gd DTPA. Because of the slow le akiness, DCE MRI data can be collected at a lower temporal resolution and higher number of averages to provide more accurate mapping and transport predictions. This study provided a framework for an MRI based tumor model that predicts the heterogeneity of Gd DTPA perfusion based on vasculature and necrosis for individual tumors. A predictive patient specific computational model of solid tumors could be useful for individualized chemotherapeutic dose optimization and cardiotoxic limitation.
13 CHAPTER 1 INTROD UCTION 1.1 Background Solid tumors describe a particular type of neoplasm that exist as firm tissue in reference to a particular cancerous disease that takes the form of a solid tumor such as breast, prostate, colon, or lung cancers. These cancers alone represent the estimated 712,100 of the 1,479,350 new cases and 277, 280 of the 562,340 deaths in 2010  Because soli d tumors represent a significant portion of cancer cases, more effective treatment could make a noticeable reduction of the negative impacts emotional, physical, and financial of cancer at large. S olid tumors are understood to have transport obstacles that hinder systemically delivered drugs and can stymie the overall effectiveness of the treatment since it is difficult to deliver to all cells in a tumor The inherent uniqueness of each tumor, created in part by tumor angiogenesis, compound treatment diffic ulties. Uniqueness can be witnessed with t umor angiogene sis, which is responsible for the creation of vessel network s that are abnormal when compared to vessel networks of healthy tissue (Figure 1 1) The abnormalities are marked by a disorganized vessel n etwork as well as non uni form microvascular permeability or vessel leakiness  These abnormalities along with impaired lymphatic function are partly responsible for creatin g a macroscopically observable elevated tumor pressure. Elevated tumor pressures can work against drugs exiting the vessel and entering into the tissue en route to destroy cancer cells. While pressure in normal tissue is approximately 0 kPa, researchers ha ve measured elevated intratumoral pressure [3 7] Also, intratumoral pressure has been shown to vary within
14 a given tumor line. For example, Gutmann, R., et al.  found tumor pressure to range from 0.5 4.4 kPa in squamous cell carcinoma. Each tumor has unique microvascular blood flow, interstitial hydraulic conductivity, and inte rstitial matrix elasticity, which contribute to variations of intratumoral pressures  Heldin et al. showed that high intratumoral pressure inhibited drug penetration  Also, elevated tumor pressures have been shown to be predictors of disease progression after radiotherapy in patients with cervix cancer  Long term studies of cervical cancer have reported that intratumoral pressures have greater impact on predicting survival than tumor hy poxia (significantly lower cellular oxygen levels)  It is necessary for tumor blood vessels to supply surrounding cells in solid tumors with proper nutrients. When th e concentration of nutrients falls below a critical value, tumor cells can no longer remain viable. This can result in a region of necrosis within the tumor or an area where cells slowly swell and disintegrate. This process creates distinct regions of viab le and necrotic cells within solid tumors (Figure 1 2). Essentially, these regions can become avascular in addition to the changes in the tissue matrix due to cellular alterations. Thus, many solid tumors exhibit tissue heterogeneity in addition to vascula r heterogeneity. Systemically delivered drugs are typically distributed non uniformly due to the intricacies of the heterogeneous tumor environment. As mentioned earlier, high intratumoral pressure results in poor uptake. Regions closer to the tumor core, where interstitial pressure approaches the microvascular pressure, are more likely to exhibit poor uptake. In addition, tumor angiogenesis inherently creates non uniform vessel density and leakiness, which increases the likelihood of non uniform drug dist ribution.
15 Furthermore, necrotic regions can lack vasculature altogether and act as a reservoir for drugs, reducing the exposure of viable tumor cells to therapeutics. The extent of said non uniformities can vary from patient to patient based on these pheno mena. This dissertation focuses on incorporating these features into an image based model. 1.2 Previous Studies and Motivation Mathematical tumor models have provided the groundwork by which transvascular exchange, transport of interstitial fluid (extracel lular plasma fluid), and transport of macromolecules can be noninvasively investigated. Macromolecular therapeutic agents such as monoclonal antibodies and encapsulated drugs have provided impetus for exploration of transport barriers in the tumor environm ent. For example, previous tumor models have been used to understand the role of increased interstitial pressure on the 2 and Fab) [11, 12] In this porous media model, Baxter and Jain used an idealized spherical tumor geometry with homogeneous vessel density and leakiness properties to elucidate the difficulty of uniform macromolecular delivery. Baxter and Jain accounted for heterogeneous tissue prope rties by including a necrotic core to study the case of a non uniformly perfused tumor  The tumor was modeled with a necrotic core that did not have a vascular network. Their model noted that a necrotic core had an effect on solute distribution of large, slow diffusing molecules. One of the limitations of this model is its inability to capture vessel these models have corroborated experimental dat a and elucidated other tumor phenomena, they lack the ability to describe tumors on a patient by patient basis.
16 This dissertation focuses on the creation of a 3D computational tumor model to predict interstitial solute transport. The model essentially inte grates more reality into solid tumor models than previously achieved by using an image based approach. Reality is sought in the form of tissue boundaries as well as heterogeneous vessel and tissue properties. Magnetic resonance imaging (MRI) can be used to non invasively acquire these properties for specific tumors, but there exists some limitations to obtaining heterogeneous vessel and tissue properties as the tumor grows and matures. Vascular remodeling can reduce the vessel leakiness (or permeability) ov er time and high fluid filled regions can develop due to cell death at more advanced stages of tumor growth. In order to adapt the model to these scenarios, the model is investigated at three stages of tumor maturation: small tumors with high leakiness, sm all tumors with low leakiness, and large tumors with low leakiness and necrosis. Additionally, reduced vessel leakiness may increase the importance of interstitial convection and diffusion relative to transvascular exchange of systemically injected solute. This results from transvascular exchange occurring over a much longer time scale when vessel permeability is low. Vessel and tissue properties are obtained via dynamic MRI data. It is the goal of this dissertation to use less than 20 minutes of the MRI da ta to acquire model inputs: heterogeneous permeability of vessels and extracellular volume fraction of the tissue. These inputs acquired from early time point data are used to predict solute distribution with the 3D computational model at later time points For small high leakiness tumors, MRI can be used to account for both spatially varying permeability of the vessels and extracellular volume fraction of the tissue. Further along in tumor development, the
17 remodeled vasculature may result in less permeable vessels. At this stage only spatially varying permeability can be obtained and a constant extracellular volume fraction must be assumed. In more advanced tumors with necrosis, only spatially varying permeability can be obtained due to the low leakiness of the tumor. However, high fluid filled regions can be detected with MRI and can be accounted for in the 3D porous media model as regions of higher diffusivity and extracellular volume fraction. It is necessary to investigate the model throughout these stag es in order to develop a model that can be used for these typical tumor environments. Doing this will improve upon previous tumor models by creating a methodology of a range of realistic tumor environments thereby providing a framework by which drug doses specific tumor environment.
18 Figure 1 1 Normal vs. tumor microvasculature. A) Normal microvasculature is organized with relatively constant vessel diameter. B) Tumor vasculature is marked by disor ganization, chaotic branches and varying diameter  SEM corrosion cast of C) normal microvasculature and D) tumor microvasculature  C D
19 Figure 1 2 Examples of tumor necrosis. A) Necrosis surrounded by well vasculari zed site of 1.5 mm human adenocarcinoma of colon. B) Histological section through center of CaSki spheroid that shows viable rim surrounding necrotic center . C) MR image of tumor that shows heterogeneous uptake. D) A region with necrosis and a high co ncentration of viable cells. E) Viable cells well outside the necrotic region . A B C D E
20 CHAPTER 2 DEVELOPMENT AND SENS ITIVITY ANALYSIS OF COMPUTATIONAL MODEL WITH HETEROGENEOUS VASCUL ATURE AND POROSITY 2.1 Introduction Solid tumors are understood to have tra nsport obstacles that hinder systemic deliver y of drugs and can stymie the over all effectiveness of treatment since it is difficult to deliver drugs to all cells in a tumor  The inherent uniqueness and heterogeneity of each tumor vascular network, created in part by tumor angiogenesis, compound treatment difficulties. Tumor angiogenesis is responsible for the creation of vessel networks that are abnormal when compared to networks in healthy tissue. The abnormalities a re marked by a disorganized vessel network as well as non uniform microvascular p ermeability or vessel leakiness  In this dissertation and commonly in literature, permeab ility and leakiness are synonymous These abnormalities along with impaired lymphatic function are partly responsible for creating an elevated interstitial internal tumor pr essure. Elevated tumor pressures can work against drugs exiting blood vessels and e ntering into the parenchyma en route to destroy cancer cells. Furthermore, each tumor has unique microvascular blood flow, interstitial hydraulic conductivity, and interstitial matrix elasticity, which contribute additionally to variations of intratumoral pressures and flows  The net result is heterogeneous extravasation and distribution of therapeutic agents into tissue, dependent upon specific tumor environments, following systemic injection Early mathematical tumor models have provided the groundwork by which transvascular exchange, transport of interstitial fluid (extracellular plasma fluid), and transport of macromolecules can b e noninvasively investigated. For example, tumor models have been us ed to understand the role of interstitial pressure on the transport of
21 a systemically delivered solute [12, 13] Baxter and Jain (1989) used a continuum porous media approach to tumor tissue t o solve interstitial and solute transport (IgG, or an idealized spherical tumor, they found elevated interstitial fluid pressure (IFP) to be a significant barrier: (1) elevated IFP reduced the driving force for extravasation of flu id and macromolecules in tumors; (2) spatially varying IFP resulted in no n uniform filtration of fluid and ma cromolecules from blood vessels; (3) elevated IFP lead to an experimentally verifiable, r adially outward interstitial fluid velocity (IFV) which opposes the inward diffusion of interstitial solute such as tracers, macrom olecules, or drugs [11, 12] Other researchers have further investigated tumor models using a continuum approach. El Kareh and Secomb studied the effects of uniformly increased vessel permeability on transport of m acromolecules within a spherical tumor geometry  In later studies, Baxter and Jain expanded upon their own spherical tumor model by including a necrotic core without functioning vessels  The goal was to e stablish a more realistic case of a non uniformly perfused tumor by including the concept of heterogeneous tissue properties. They found that the necrotic core did not reduce the central interstitial pressure in the tumor and had an effect on solute distri bution of large, slow diffusing molecules. Mathematical descriptions of tumor transport have been adapted to numerous treatment situations and have shown the potential to impact clinical practices. Eikenberry modeled delivery of doxorubicin to a solid tum or comprised of multiple tumor cords and showed the potential for predictability of dose efficacy and cardiotoxicity  Smith and Humphrey elucidated the role of transvascular exchange during the
22 administration of drug via intratumoral convective enhanced delivery (CED)  Their model suggested that transvascular exchange reduction procedures would increase convective flow thereby aiding infusion therapies. Recent computational fluid dynamics (CFD) approaches to tumor drug delivery, by integrating more patient specificity with the aid of medical imaging data, have created a new realm of personalized clinical possibilities. T an, et al. used patient MRI data to reconstruct the 3D geometry of a resected brain region where the main part of the tumor was removed  In this simulation, the resected region was implanted with poly (lactide co glycolide) (PLGA) wafers loaded with 1% Etanidazole. Elevated pressure was predicted in t he resected region, as well as in the remaining tumor tissue. Our group has also presented an image based model, which includes in vivo heterogeneous vasculature data since the extent of vessel non uniformity ca n vary from patient to patient  This current study has two aims: (1) to expand upon the CFD modeling approach by incorporating spatially variable tissue porosity (volume fraction of extracellular ext ravascular space (EES) over total tissue volume) in addition to variable vessel permeability and (2) to compare simulated tracer distribution to experimental concentration data derived from MRI. In this study, heterogeneous tissue and vascular properties a re modeled by incorporating spatially varying porosity and vascular permeability maps created from two compartment analysis of early time point (19 min) dynamic cont rast enhanced (DCE) MRI data. DCE MRI allow s signal enhancement measurements of MR contras t agent (CA) uptake and washout in the KHT sarcoma bearing hind limbs of mice (n = 3). This chapter looks at small sarcomas with highly permeable vessels. Signal
23 enhancement was measured following the systemic delivery of the low molecular weight, extracel lular CA, gadolinium diethylenetriam ine pentaacetic acid (Gd DTPA), and was used to quantify concentration of Gd DTPA in tumor and healthy tissue in each voxel of MR data on a voxel by voxel basis. Porosity and permeability maps, derived from these data, w ere incorporated into a 3D porous media model to pr edict interstitial fluid and tracer transport in tumo r and host tissue at later time points. A sensitivity analysis was conducted based on a range of literature values using the data of one mouse: (1) to s elect an arterial input function (AIF) ( the time dependent CA concentration in arterial blood plasma) based on consistency of the first 19 min of experimental and simulated tracer transport and (2) to investigate the effects of baselin e tissue and vascular transport properties on interstitial fluid and tracer transport. The optimal AIF and baseline transport properties were used to simulate interstitial fluid and tracer transport for two additional KHT sarcomas. Magnitude and distribution of experimental an d simulated tracer concentration in tumor was compared for all three tumors. An image based solid tumor model may be used to optimiz e and evaluat e treatment strategies for patient specific therapies. The computational model presented here provides the nece ssary first steps toward that application. First, the simulation of CA uptake and washout in the tumor environment provides a foundation from which the model can be expanded to investigate transport of reactive drugs. Second, sensitivity analyses of these models provide a better understanding of the underlying transport processes and the effects of vascular and tissue properties within realistic tumor
24 geometry. Third, by comparing simulations to experimental data, elements of the tumor model can be refined to create more accurate image based models of drug delivery. Content in this chapter is based on a paper submitted to the Annals of Biomedical Engineering in collaboration with Garrett Astary (Department of Biomedical Engineering), Thomas Mareci (Departmen t of Biochemistry and Molecular Biology), and Malisa Sarntinoranont (Department of Mechanical and Aerospace Engineering). 2.2 Materials and Methods 2.2.1 Animal Preparation Animal models provide means to investigate in vivo systemic delivery of CA and crea te permeability and porosity maps. The patient specific model was based on three C3H female mouse (Jackson Laboratories, Bar Harbor, ME) that was inoculated with 10 5 murine KHT sarcoma cell s in the gastrocnemius muscle. Tumors were grown for 7 days to a hi nd limb diameter between 6.0 7.5 mm. During imaging, mice were anesthetized using a gas mixture of 98% oxygen, 2% isoflurane and given a 0.1 0.2 mmol kg 1 of body weight (bw) bolus tail vein injection of Gd DTPA (Omniscan, GE Healthcare Inc., Princeton, NJ ) CA at a constant rate (~0.4 mL min 1 0.23 mL ). Animal experiments were performed within the principles of the Guide for the Care and Use of Laboratory Animals and approved by the University of Florida Institutional Animal Care and Use Committee (IACUC). 2.2.2 Magnetic Resonance Imaging (MRI) The MRI experiment was performed using a Bruker Avance imaging console (Bruker NMR Instruments, Billerica, MA) connected to Magnex Scientific 11.1 T horizontal bore magnet system (Varian, Inc., Magnex Scientific Pr oducts, Walnut Creek California) The hind limb of the mouse was placed in a 1.5 cm inside diameter, transmit
25 and receive, volume coil. Multiple T 2 weighted spin echo (SE) scan s (TR = 2000 ms, TE = 15, 30, 45, 60 and 75 ms) were acquired for the ca lculatio n of T 2 values. Tumor boundaries could be manually segmented for each slice  Segmented images were used for ge ometric reconstruction of the tumor and host tissue volumes A variable time for recovery ( TR ) SE se quence (TR = 5000, 2000, 1000, 500 and 250 ms, TE = 15 ms, field of view (FOV) = 2 cm 1 cm 1 cm, 11 slices, matrix size = 192 96, 2 averages 2), was acquired for calculation of native tissue T 1 values used in CA concentration calculations Serial D CE MR images, consisting of T 1 weighted SE sequence (TR/TE = 330 ms/9.4 ms, FOV = 2 cm 1cm 1 cm, 11 slices, matrix = 192 96, NA = 6, total acquisition time = 2 min 6 s) were collected before and after CA administration. T 1 weighted SE sequence with a s hort TE was used to avoid noise in the measured signal that can occur under hypoxic conditions  with sequences sensitive to T 2 effects. 2.2.3 Vascular Permeability and Porosity Maps Vascular leakiness was descri bed using endothelial transfer cofficient ( K trans ) values K trans values were estimated using a two compartment model approach, where the two compartments are the blood plasma and EES. From DCE MRI data, K trans values were quantified on a voxel by voxel ba sis by: (1 ) estimating CA concentration in tissue via the relationship between concentration and signal intensity  and (2) using a Tofts and Kermode model to fi t to the CA kinetics within the first 19 min Previous DCE MRI data showed that peak tissue concentration was reached within 5 min post injection for fast enhancing breast tumor ( K trans > 0.1 min 1 ) and that this time window was acceptable to properly fit K trans  The 19 minute time window was chosen for our study because of slower enhancing peripheral tumor tissue (peak time at ~17 min). It was assumed that change in CA concentration in tissue is dominated by a fast
26 transvascular exchange at early time points. Though, over longer time scales interstitial convection and diffusion may play an important role in the distribution of CA within the tumor space requiring a continuum approach to make accurate predictions. A linear relationship between contrast agent concentration and T 1 was assumed. An average of the pre contrast images was used to solve for baseline (pre injection) signal values, an d the transverse relaxation contribution to signal was assumed to be unity. Then, concentrations of Gd DTPA in tissue were solved at each time point using the standard spin echo signal equation  (2 1) C t is the volume averaged tissue concentration of Gd DTPA determined by MRI; R 1 is the longitudinal relaxivity of the contrast agent; TR is the recovery time; T 10 is the relaxation time without contrast agent; S ( C t ) and S (0) are signal intensities at CA concentrations C t and zero respectively; TR T 10 S (0), and S ( C t ) are known values; and R 1 was approximated as 2.7 L mmol 1 s 1 (based on G d DTPA relaxivity in rat muscle  at 6.3 T). T ofts Kermode two compartment model  incorporates basic physiological phenomena to describe the transport of systemically injected contrast agents into tissue spac e. The two compartment model can be described by (2 2) where t is time; C p is the AIF that describes the time course of Gd DTPA concentration in the blood plasma; is the volume fraction of EES or porosity (also
27 referred to as v e in tumor perfusion studies). The volume fraction of plasma in each voxel was assumed small compared to T ransvascular transport of Gd DTPA solely into the EES at early post infusion times was assumed C p following a bolus injection was represented by a biexponential decay  that was normalized: (2 3) where a 1 and m 1 represent th e amplitude and rate constant, respectively, of the fast equilibrium between plasma and extracellular space; a 2 and m 2 represent the amplitude and rate constant of the slow component of the clearance; d is the dose of the bolus injection. Solving Equation 2 2 by substitution of Equation 2 3 in terms of concentration of CA in tissue, C t produces (2 4) MR experimental concentrations determined by Equation 2 1 were used to fit Equation 2 4 in order to solve K trans and at each voxel within the tumor boundary to generate 3D maps. K trans and parameters were fit using a nonlinear regression in MATLAB (MATLAB Version 7.1, The MathWorks, Inc., Natick, MA). Only physically relevant ranges were considered ( K trans 0 and 2 > 0.5. Values of K trans were set to zero and = 0.3 in voxels where the fit was poor (R 2 occurred in ~5% of voxels within the tumor. 2.2.4 3D Porous Media Model The 3D porous media tumor model accounts for the CA distribut ion time dependence on the underlying plasma flow and diffusion within the interstitial space of
28 tissue. Each point in space in th is porous media model contains a tissue and vascular component. The vascular components provide a source or sink for both plas ma fluid and tracer per unit tissue volume The filtration rate of plasma fluid per unit volume ( J V / V ) across the vessel w aw  (2 5) where L p is the vessel permeability, S / V is microvascular surface area per unit volume, p V is the microvascular pressure, p i is the IFP, T is the average osmotic reflection for plasma protein, V is osmotic pressure in microvasculature, and i is osmotic pressure in interstitial space. Since we were interested in capturing vascular heterogeneity, it was necessary to include vessels that were non uniformly permeable to both plasma fluid and CA. K trans values directly describe the leakiness of v essels to CA. In order to account for heterogeneous plasma fluid leakiness, K trans values were normalized by the average K trans within the tumor, and this normalized value was used to scale J V / V in order account for plasma leaki ness heterogeneities This was done assuming that patterns of leakiness are similar for both tracer and plasma fluid. The tissue continuum was mo deled as a porous media. In tissue the continuity equation  tiss ue averaged IFV ( v ) and were given by (2 6 ) and (2 7 )
29 respe ctively. The first term on the right hand side of Equation 2 6 describes a volumetric source due to the vasculature; the second term on the right hand side is a sink due to lymphatic vessels. L pL S L / V i s the lymphatic filtration coefficient, p L is pressure in the lymphatic vessels, and K is the tissue hydraulic conductivity. L ymphatic function was assumed in host tissue only with p L = 0 kPa applied to tumors, perfused muscle tissues, and flow in soft conne ctive tissues  K is small in these types of tissues such that the role of viscosity is more important at the fluid solid interface of the porous media than within the fluid. Transport of interstitial Gd DTPA was solved using the convection and diffusion equation for porous media  : (2 8) where D eff is th e effective diffusion coefficient for Gd DTPA in tissue The second and third term on the left describe convective flux and diffusive flux, respectively. The transvascular source term on the right is a modified form of the Kedem Katchalsky equation where d iffusion dominates the transfer of solute across the microvessel wall; the sink term on the right accounts for lymphatic drainage of solute in normal tissue. A block diagram (Figure 2 1) shows the flow of the mathematical methodology and experimental comp arison, which attempts to summarize the fundamental goals of the chapter: 1) test the sensitivity of interstitial fluid transport (Equations 2 6 and 2 7) to changes in interstitial fluid transport parameters, 2) test the sensitivity of tracer transport (Eq uation 2 8) to changes in tracer transport parameters, and 3) compare the simulated tracer transport results with experimental tracer measurements. The first step in accomplishing these goals was to obtain DCE MRI signal enhancement data for in vivo
30 tumor. That provided in vivo Gd DTPA concentration in tissue over the course of the experiment (60 min) using Equation 2 1. The first 19 min of the data was used to select an AIF from a range of common literature AIFs (slow, intermediate, and fast). The model re lies on this input, which described the decay of Gd DTPA concentration in the blood plasma following a bolus systemic injection over time using Equation 2 3. This function can affect the magnitude and distribution of K trans and maps, which in turn affects interstitial fluid and tracer transport. The first 19 min of average concentration of simulated Gd DTPA in tumor (tracer transport) was compared with its experimental counterpart. The slow AIF provided the most consistent resu lts between simulated and experimental average concentration of Gd DTPA in tumor; therefore, this AIF was used in the sensitivity analysis of Equations 2 6, 2 7, and 2 8. The results of modeled tracer transport (Equation 2 8) was compared with experimental tracer transport by comparing 1) Gd DTPA distribution along a line bisecting the tumor at early ( t = 5 min), intermediate ( t = 30 min), and later ( t = 60 min) time points, and 2) uptake and washout of average Gd DTPA within the tumor. 2.2.5 Computational Model From MR slices, tumor and hind limb geometries were manually segmented (MATLAB v.7, Natick, MA) based on contrast differences in T 2 values (Figure 2 2A). Triangulated mesh surfaces were generated to create tumor and host tissue boundaries (Amira 4.1 .1, San Diego, CA) and were converted to parametric representation, non uniform rational B spline surfaces (NURBS) (Geomagic Studio, Research Triangle Park, NC). Meshes for the volumes of host tissue and tumor were created (Gambit, Fluent, Lebanon, NH; Fig ure 2 2B). The meshes were composed of ~ 2,700,000 4 node
31 tetrahedral elements. The large number of elements aided the porous media solver with convergence for solutions with steep pressure gradients at the tumor boundary in the CFD software package (FLUEN T 6.3, Fluent, Lebanon, NH). Prediction of tumor tracer distribution was a two step process. First, steady state interstitial plasma fluid flow was solved (Equation 2 6). Then, tracer transport (Equation 2 8) was solved over 1 h Boundary conditions were s uch that the solute was assumed to exit at the cut ends at the same rate as the interstitial plasma fluid. Additionally, t he skin boundary was assumed to be impermeable to tracer and plasma while the cut ends of the leg were considered far enough away from the tumor boundary and therefore set to a normal tissue pressure [6, 11] p i = 0 kPa Initial tissue concentration was zero for the tracer solution. 2.2.6 Sensitivity Analysis The model was dependent on vascular an d tissue transport parameters, and AIF. In this chapter, animal specific AIF was not known, so a sensitivity of AIF using three test cases from literature was first conducted to determine a model AIF that best represents the experiments AIF. Simulated aver age tracer uptake and washout in the tumor is compared to experiment for the three test cases. The modeled AIF that provided similar average tracer uptake and washout in the tumor to experimental observation was selected as the AIF in the interstitial tran sport sensitivity analysis (Equations 2 6, 2 7, & 2 8). In the interstitial transport analysis, known vascular and tissue transport parameters ( L p K L pL S L / V D eff and initial concentration of Gd DTPA in blood plasma) were varied to determine the effects on unknowns: interstitial pressure ( p i ) and velocity ( v ). Values in Table 2 1A were considered known values and used for all simulations; these values were not varied in the sensitivity analysis. Some of these
32 values were selected based on the model prese nted by Baxter and Jain (1989); these particular parameters are not well measured in literature and are often used by other models of interstitial flow within tumors [19, 20, 29] Table 2 1B contains the nominal bas eline values for t he sensitivity analysis that were selected based on or near the ranges obtained from literature. Three AIFs were considered in order to estimate the time course of CA c once ntration in blood plasma: a fast decay  where a 1 = 9.2 kg bw L 1 a 2 = 4.2 kg bw L 1 m 1 = 0.004 s 1 and m 2 = 0.0008 s 1 ; an intermediate decay  where a 1 = 13.0 kg bw L 1 a 2 = 16.0 kg bw L 1 m 1 = 0.005 s 1 and m 2 = 0.0004 3 s 1 ; a slow decay [26, 32] where a 1 = 4.0 kg bw L 1 a 2 = 4.8 kg bw L 1 m 1 = 0.002 s 1 and m 2 = 0.0002 s 1 Given that d ose was 0.2 mmol kg 1 a 1 and a 2 were scaled so that C p ( t = 0), C p0 was 2.0 mM based on the dilution calculation (from the measured concentration of the bolus injection of Gd DTPA and the estimated blood volume of the mouse based on body weight)  K trans and maps were created for each AIF and used to simulate interstitial fluid flow and the first 19 min of tracer transport. The slow AIF was selected for the sensitivity analysis because the average Gd DTPA concentration in tumor tissue, C t ,avg behavior was m ost similar to the experimental data (Figure 2 3). K trans and maps created based on the slow AIF were used for sensitivity analysis (Figure 2 5). Other parameters of interest in this sensitivity analysis were as follows: tumor vessel permeability ( L p,t ), normal tissue permeability ( L p,n ), L pL S L / V D eff and ratio of tumor hydraulic conductivity to normal tissue hydraulic conductivity ( K t / K n ). K t / K n was chosen to account for the relative difference in hydraulic conductivity of normal and tumor tissue as me asured experimentally. C p0 was also analyzed to observe the effects
33 of over or under estimating this value. Sensitivity analysis was conducted by using high and low values with respect to th e baseline values. Permeability, L pL S L / V and K t / K n values were s et such that the high case was 2 baseline and the low case was 0.5 baseline. D eff were set such that high case 10 1 baseline and low case was 10 1 baseline. The high case for C p0 was 2.5 mM and low case was 1.5 mM. 184.108.40.206 Hydraulic c onductivity of t issue ( K ) Hydraulic conductivity describes the ease at which a fluid can pass through a saturated porous material. The hydraulic conductivity of several types of tumors as well as various types of healthy tissue has been experimentally measured. Tumor tissue ( K t ) Hydraulic conductivity has been found to vary across tumor lines, within a specific tumor line, and can depend on the experimental setup. The values for tumor tissue range from 1 10 15 to 1 10 12 m 2 /Pa s. In a 1D experiment using a permeab ility chamber (P = 392 to 8332 Pa) with one O ring, K t was measured to be 1.8 10 13 m 2 /Pa s in human colon adenocarcinoma (LS174T)  For t he same tumor line, K t = 1.3 10 13 to 2.3 10 13 m 2 /Pa s in a low flow rate experiment with in vivo flow rates of 0.1 to 0.14 l/min  Netti and cohorts found K t = 3.9 10 13 m 2 /Pa s in vitro for LS174T  K t for rat fibrosarcoma MCA R tissue ex vivo was found to be 1.0 10 14 m 2 /Pa s for a pressure of 2000 Pa [37 ] In a 1D experiment using a permeability chamber (P = 392 to 8332 Pa) with one O ring, Morris hepatoma 5123 tissue found K t = 3.0 10 14 m 2 /Pa s. They also found K t = 3.0 10 15 to 8.0 10 15 m 2 /Pa s for the same tumor line with in vivo data using an unsteady flow model  Dog squamous cell carcinoma was K t found to be 1.8 10 12 m 2 /Pa s  Rat hepatacarcinoma tissue in vitro and in vivo found K t to be m 2 /Pa s  Netti et al.
34 completed a series of experiments to determine K t of multiple tumor lines. K t was estimated from the transient stress relaxation rate using a poroviscoelastic model. Each specimen was compressed 25 m in ramps of 15 s and allowed to relax for 20 min. For h uman soft tissue sarcoma (STS 26T) K t = 7 10 14 m 2 /Pa s; for mouse mammary carcinoma (MCa IV) K t = 1.9 10 12 m 2 /Pa s; for human glioblastoma (U87), K t = 4.9 10 13 m 2 /Pa s  Healthy tissue ( K n ) The values for healthy tissue hydraulic conductivity range fro m 1.5 10 15 to 1.8 10 12 m 2 /Pa s. Hydraulic conductivity in rat stratum corneum (outer most layer of the epidermis) was measured to be 6.4 10 15 m 2 /Pa s  Zakaria et al. measured K n by applying an interstitial pr essure gradient across unsupported abdominal wall muscle over a pressure range : 200 Pa ( K n = 1.1 10 13 m 2 /Pa s) to 1066 Pa ( K n = 5.9 10 13 m 2 /Pa s)  K n for in vivo subcutaneous connective tissue was measured to be 1.8 10 12 m 2 /Pa s  Fleischman et al. noted K n as low as 1.5 10 15 m 2 /Pa s  220.127.116.11 Vascular h ydraulic p ermeability ( L p ) Vascular hydraulic permeability describes the transvascular filtration rate of plasma. Measurements of vascular hydraulic permeability are less extensive than hydraulic conductivity o f tissue. Tumor vascular hydraulic permeability ( L p,t ) Vessels from rat mammary adenocarcinoma (R3230AC) tumors were measured by Sevick and Jain. This was accomplished by directly measuring capillary filtration coefficient along with vascular surface area to back out the values of L p,t between 1.0 10 10 and 3.5 10 10 m/Pa s
35  Jain et al. measured L p,t of mouse mammary carcinoma MCaIV vessels to be 1.9 10 10 m/Pa s  Healthy vascular hy draulic permeability ( L p,n ) In a normal vessel whole organ experiment for rabbit, L p,n was measured to be 0.78 10 13 m/Pa s  In a normal vessel whole organ experiment for human, L p,n was measured to be 0.726 10 13 m/Pa s  A normal vessel sing le frog pial microvessel L p was fitted to a monoexponential function that described rising dye concentration within occluded segment. The fit provided L p,n = 2.06 10 13 m/Pa s  Permeability for normal single rat brain vessels was measured by Kimura et al.  Specifically, they measured the volume flux of water per unit surface area across the arteriole wall occluded vessel. Pressures were varied from 2666 Pa to 9333 Pa. L p was derived from slope of J v / S versus ap plied pressure. They found: mean L p = 13.5 10 13 m/Pa s and median L p = 6.0 10 13 m/Pa s  Normal capillaries in skeletal muscle of rats was found: L p = 2.7 10 12 m/Pa s  Lymph vessel rat mesenteric postcapillary venules were cannulated with micropipettes attached to a water manometer to control hydrostatic perfusion to find L p = 3.6 10 11  18.104.22.168 Surface a rea p er u nit v olume ( S / V ) Surface area per unit volume is purely a dimensional quantity, which is calculated as the vessel surface area to volume o f vessel contained within the surface area. One of the earliest studies of this parameter found S / V of peripheral capillary membranes to be 7000 m 1  For vessels in human brain tissue, S / V was found to be 6000 14000 m 1  S / V was found to be 5700 m 1 and 19000 m 1 in white and grey matter, respectively  M urine mammary carcinomas C3H/Bi small tumors (0.035
36 cm 3 ) S / V was measured to be 26500 m 1 and 15000 m 1 for large tumors (>0.5 cm 3 )  S / V for human colon adenocarcinoma (LS174T) in vivo was meas ured at 24000 m 1  22.214.171.124 Porosity ( ) Porosity is a dimensionless quantity that describes the fraction of void space in a representative volume. It is exactly calculated as the volume of void space over the total volume of the representative volume. While porosity was not explicitly a par t of this sensitivity analysis, the literature search provides some context for the values obtained using DCE MRI in this chapter. Porosity of renal allograft interstitial fibrosis was established by using a histomorphometric method. Approximately 200 poin ts were randomly selected in a field for each biopsy. Ratio of overlying interstitium points were then calculated and ranged from 0.1 to 0.6  Using a DCE MRI approach with a two compartment model porosity was found to be 0.28 in MDA PCa 2b tumor and 0.16 in skeletal muscle  For normal rat brain tissue, mean was measured to be between 0.20 and 0.22  Porosity in m uscle tissue was measured to be 0.08 using DCE MRI methods [58, 59] 126.96.36.199 Diffusion c oefficient ( D ) The diffusion coefficient is a constant of proportionality that describes the transport of molecules (solute) through an area due to Brownian motion wit h a driving force that is the concentration gradient of solute. Diffusion coefficient for Gd DTPA in healthy human articular cartilage was measured to be 1.4 10 10 m 2 /s  For a tissue surrogate, 10% PVA C hydrogel, D for Gd DTPA was measured to be 2 .6 10 10 m 2 /s 
37 2.3 Results 2.3.1 Interstitial Fluid Transport The solution of Equations 2 6 and 2 7 results in interstitial fluid transport; this provides two unknowns, p i and v throughout tissue space. For the baseline simulation of interstitial fluid transport, pressure was predicted to be higher within the tumor than outside the tumor. The pressure contour in Figure 2 6A shows higher IFP ( p i, max = 1.4 kPa) within the tumor than the surrounding host tissue for the baseline simulation. Pressure varied within the tumor from 0.86 kPa at the boundary to 1.4 kPa in the tumor core (slightly skewed towards the left from mid tumor in Figure 2 6A). The pressure fell p i, max | = 0.79 kPa mm 1 ) at the tumor boundaries Figure 2 6B. Outward flow of interstitial fluid from the tumor was observed (Figure 2 7A). The magnitude of volume averaged flui d velocity was predicted to be highest at or near the boundary of the tumor 1 1 at both ends of the bisecting line (Figure 2 7A & Figure 2 7B). Approximately 1 mm into the host tissue, fluid velocity was reduced by 75% to 0.1 1 1 ). In both the high and low cases of the sensitivity analysis, interstitial hypertension was observed within the tumor with a decrease in pressure at the tumor tissue interface (Figure 2 6B). The h ig h L p,t L p,n and low L pL S L / V K t / K n cases all resulted in intratumoral pressures that were greater than the baseline simulation. Percent age i ncrease in intratumoral pressures from the baseline was highest for the high L p,t case. Decreases in the pressure gradient at the tumor tissue interface from the baseline simulation were observed in the low K t / K n L pL S L / V L p,t and high L p n cases. The low L p,t and high K t / K n resulted in a flattening of intratumoral IFP profile. Percent difference in intratumoral
38 pres sure and pressure gradient at the tumor tissue interface for all cases with respect to the baseline deviations are calculated in Table 2 2. Volume averaged velocity of interstitial fluid was predicted to be highest near the boundary of the tumor in all ca ses of the sensitivity analysis (Figure 2 7B). Volume averaged velocity at the boundary was reduced by > 100% near the cut ends in all cases. High L p,t K t / K n and L p,n and low L p,n and K t / K n cases all resulted in tumor boundary tissue averaged IFVs that w ere greater than the baseline simulation. The highest increase in velocity was witnessed for the high L p,t case, which corresponded to a peak volume s 1 at the tumor tissue interface. L p,n had little effect on the velocity at t he tumor boundary (< 2 % difference from baseline ). Tumor vessel permeability, L p,t had the greatest impact on the interstitial fluid solution. Changes in L p,t resulted in a >30% difference from the baseline when looking at three behaviors: intratumoral pressure, tumor tissue pressure gradient, and magnitude of velocity at the tumor tissue interface. Normal vessel permeability, L p,n was the least sensitive parameter in terms of impact on the interstitial fluid solution. Though the low and high C p0 cases resulted in respectively higher and lower K trans values than the baseline (within the tumor), interstitial fluid transport for the low and high C p0 cases was identical to the baseline. Baseline simulations for the two additional KHT sarcomas resulted in lo wer interstitial peak pressures (0.50 p i,max | = 0.15 kPa mm 1 ) than the data set used for the sensitivity analysis; however, the pressure distribution pattern was similar. Subsequently, outward flow of interstit ial fluid
39 was observed with the highest magnitude of volume averaged fluid velocity near the 1 ). 2.3.2 Tracer Transport The effect of AIF upon average Gd DTPA concentration, C t ,avg ( t ), in tumor tissue was examined and compa red to experimental data obtained from MRI for the first 19 min (Figure 2 3). Three simulations of tracer transport were conducted which corresponded to three AIFs. The aim was to determine the best AIF based on agreement between experimental and simulated C t ,avg ( t ) behavior at early time points. The simulated curves showed higher peak concentration than the MR measured concentration. Peak concentrations were 11%, 30%, and 30% higher than the experimental peak for slow, intermediate, and fast respectively. Assuming monoexponential decay for the tumor volume averaged washout behavior, decay rates were as follows: 0.029 min 1 (experimental), 0.050 min 1 (fast), 0.034 min 1 (slow), and 0.045 min 1 (intermediate). The slow AIF produced peak concentration an d washout behavior in tumor similar to experimental observation (Figure 2 4). Signal enhancement in the lateral marginal vein (Figure 2 4A) was detected in one preliminary data set (mouse with xenografted renal cell carcinoma). The vessel was indicated by high signal enhancement (>500 %) in the first scan after the bolus injection of contrast agent in the serial DCE MR imaging sequence (Figure 2 4B). The transient behavior of signal enhancement in this ROI showed to be similar to the biexponential decay of the slow AIF. This clearance behavior was quite different from the behavior seen in fast enhancing, high uptake regions in the tumor periphery (Figure 2 4C). The baseline simulation showed increased Gd DTPA deposition on the left half of the tumor (Figure 2 8), which corresponded to the leakier, left side of the tumor (Figure
40 2 5A). Clearance of Gd DTPA due to transvascular exchange was observed in the tumor from 5 to 60 min. Tracer transport simulations showed qualitative spatial distribution and transien t behavior similar to that of the experimental data. Also, tracer transport was less sensitive than interstitial fluid transport to changes in model parameters in Table 2 1B (Figure 2 9). However, tracer transport was sensitive to changes in C p0 Consider able overlap along the bisecting tumor line was observed for the baseline simulation and simulations of high and low model parameters from Table 2 1B (Figure 2 9). The consistency of the distribution pattern along the tumor line resulted in a high Pearson product moment correlation coefficient (PMCC), r for all simulated cases ( r > 0.99 for t = 5, 30, 60 min). Concentration deviations from the baseline within the tumor were estimated; root mean squared (RMS) errors, were found to be small for all cases (0.03 10 5 mM < < 4.4 10 5 mM), but for the high and low C p0 cases (80 10 5 mM < < 260 10 5 mM) over the 1 hr simulation (Table 2 3). The high and low C p0 cases along the tumor line resulted in C t lower and higher than the baseline case at t = 5 min, respectively. High and low L p,t L p,n L pL S L / V K t / K n and D eff cases exhibited a similar washout behavior to C t ,avg ( t ) of the baseline case (~ 0.03 min 1 ). Variations in experimental and numerical error mad e it difficult to correlate the small changes in to relative sensitivity for the tissue and vascular parameters. The results of the sensitivity analysis showed agreement between simulated and experimental C t for the first 30 min ( = 0.46 10 3 mM at t = 5 min; = 0.58 10 3 mM at t = 30 min) for a ll cases but high and low C p0 (1.8 10 3 mM < < 3.9 10 3 mM). At t = 60 min, all cases over predicted C t (0.5 10 3 mM < < 1.8 10 3 mM) within the
41 tumor (Figure 2 9C). Even though C t was over predicted, sensitivity analysis simulations showed si milar patterns of Gd DTPA concentration distribution to experimental patterns ( r > 0.89) within the tumor. Including the host tissue in the correlation reduced r = 0.42 at t = 60 min. This was due to later peak C t (> 17 min) in the experimental data observ ed at two regions of the host tissue (2 3 mm and 16 20 mm; Figure 2 9). Because tracer transport was less sensitive to changes in transport parameters, baseline values were used to predict transport in the t wo additional KHT sarcomas (Figure 2 10 ). Similar simulated and experimental distribution patterns of Gd DTPA concentration were observed at t = 5 and 30 min ( r > 0.78) within the tumor (Figure 2 10A D ); however, the correlation in the tumor was not as strong ( r > 0.58) at t = 60 min (Figure 2 10E F ). Ma gnitude of simulated Gd DTPA concentration along the bisecting line through a tumor was greater than experimental magnitudes at t = 60 min (0.5 10 3 < < 2.1 10 3 mM; Figure 2 10E F ). Furthermore, the average Gd DTPA concentration within the entire tumor volume for the simulations were greater than experiments at t = 60 min for all three tumors. The largest percent difference of average Gd DTPA concen tration within the tumor volume (700%) occ urred in the tumor shown in Figure 2 10B F Relative concentration error was < 38% in the other two tumors. The results of the sensitivity analysis in chapter 2 have been summarized in Table 2 2. 2.4 Discussion and Conclusions This study presents the first image based tumor model, with both heterogeneous vasculature and tissue porosity, which predicts interstitial fluid and solute transport within tissue. The model exhibits two important interstitial transport chara cteristics: it predicts interstitial hypertension, and increased velocities at the host tissue tumor interface. A high correlation was observed ( r > 0.89) between predicted and
42 experimental spatial deposition of Gd DTPA along a bisecting line within the tu mor after 1 h in the tumor used to develop the model. Also, transient behavior (washout rates) of Gd DTPA in the tumor showed good agreement between the simulation and experiment. The two additional KHT sarcomas used to test the developed model showed vari ed outcomes in terms of predicting the average concentration of Gd DTPA at 1 h. Namely, experimental and simulated concentration distribution and magnitude correlated well for one modeled tumor while errors were significant in the other. In previous studie s, intratumoral IFP has been investigated and hypertension has been simulated [12, 19, 20, 62] and measured [3 7] IFP values have been shown to vary within a tumor line and to be proportional to tumor volume by Gutmann et al.  In their study, a pressure range of 0.5 4.4 kPa for squamous cell carcinomas was observed. The baseline intratumoral pressure simulated in this image based model fell within the low end of this range. This could be in part due to tumor size. The tumor volumes in the current study were smaller (~95%) than the non met astatic smallest expected to have a lower intratumoral IFP. Smaller tumors have fewer vessels than larger tumors so they have fewer available fluid sources. Previous models of i dealized tumors predicted increased velocities at the tumor boundary  For an isolated tumor with a diameter of 2 cm, fluid velocity at the boundary was approximately 0.1 1 In the presented model, velocity at tumor boundary in the baseline simulation was predicted to be higher (0.35 1 ). It should be noted that solute transport in this study was limited to a low molecular weight MR visible tracer Gd DTPA. The transport of low molecular weight tracers is
43 less likely to be convective when compared to macromolecular tracers or therapeutic agents. Despite these limitations, image based models using Gd DTPA as an interstitial tracer offe rs a few advantages : (1) DCE MRI using Gd DTPA is well understood, documented, and it is a clinically used approach to understand the tumor environment; (2) DCE MRI using Gd DTPA provides qualitative and quantitative ( e.g., vascular leakiness maps ) data; ( 3) Gd DTPA is similar molecular weight to some chemotherapeutic drugs (e.g. doxorubic in = 543 g mol 1 ); (4) Gd DTPA extravasates more effectively than macromolecules and does not bind hence it provides a best case scenario for the evaluation of smaller ma cromolecular drug transport For solute transport simulations, the model presented in this study exhibited a similar extent of heterogeneity (70 80% difference in maximum and minimum concentrations) at early time points (5 6 min) along a bisecting tumor l ine as a previous Gd DTPA transport model from our group  However, there are two markedly different tracer transport behaviors between our two studies. In the previous study, along the entire extent of the bisecting tumor line, Gd DTPA concentration increased and the concentration profile flattened over the course of 30 min. The current study observed an overall decrease across the bisecting line for this sa me time course and a less dramatic flattening. These differences are due to improvements upon the previous model that were the result of three key refinements made in the present model: (1) a faster decaying AIF than the previous model provided for transva scular sink terms during the first 30 min of the simulation; (2) the incorporation of both spatially varying porosity and K trans allowed for more finely tuned kinetic behavior than the previous fixed
44 porosity model; (3) lower effective diffusivity contribu ted to less dramatic flattening of the tracer concentration profile. The sensitivity analysis pointed out aspects of the model parameters, source terms, and measurements that most effect transport. This knowledge can be used to simplify the model by reduci ng the number of input parameters. Also, the sensitivity analysis was conducted with a priori knowledge of the experimental data. This allowed the investigation of changes in model parameters with respect to the experimental concentration data, which made it possible to measure model error and indicate possible reasons for the error. Interstitial fluid flow was most sensitive to vessel permeability with L p,t having the most profound impact on IFP and boundary fluid velocity. These permeability values scale pressure differences that exist across vessel walls and directly affect volumetric source and sink terms of the continuity equation. Non invasive, patient specific measurements of L p,t might be necessary to in order to make more accurate predictions of in tratumoral pressure and tumor boundary convection using an image based modeling approach. Alternatively, non invasive measurements of IFP distribution in tumors  could be imported into this model to better estimate transport parameters such as L p,t and calculate transport. With the validation of predicted intratumoral IFP, these models could be used to predict disease progression after radiotherapy in patients with cervix cancer  L ong term studies of cervical cancer have reported that intratumoral pressures have greater impact on predicting survival than tumor hypoxia 
45 Gd DTPA tracer transport was relatively insensitive to changes in vascular and tissue transport parameters. When modeling low molecular weight tracers or drugs in smaller tumors (~6 7 days for KHT sarcoma cells), it is likely that tracer transport is dominate d by transvascular exchange, rather than convection or diffusion, at the simulated time scale. In the case of transvascular exchange dominated tumors, accurate AIF and C p0 measurements are the most important parameters for accurate prediction of concentrat ion in tissue at later time points. For larger tumors, this may not be the case due to the increase of tumor tissue heterogeneity, which will occur as the tumor develops necrotic regions Because the transvascular exchange term is dominant, due to the freq uency of high K trans ( 0.1 min 1 ) values in the tumor, it might not be important to have a continuum porous media approach which accounts for convection and diffusion in the interstitial space of smaller high leakiness tumors at the simulated time scale; a two compartment model may suffice (Equation 2 2). However, it should be noted that the low average porosity in all three tumors (~0.1) could have masked the role of convection and diffusion. It has been suggested by other researchers that low porosity va lues (< 0.2) in the tumor obtained from DCE MRI may be indicative of convective tracer clearance and not physiologically low extracellular volume fraction  A continuum porous media approach is likely necessary to model cases of lower leakiness where the transvascular exchange term is less dominant and porosity values less obtainable using early DCE MRI time poi nts. In these cases diffusion and convection can have greater impact on interstitial molecules and porosity values would have to be obtained DCE MRI independent methods or fixed. Moreover,
46 the continuum model would be necessary to model transport of macrom olecular agents that require convective transport over smaller time scales. The insensitivity of vascular and tissue transport parameters resulted in no considerable spatial changes in time, yet temporal and spatial distribution of baseline simulation and experiment Gd DTPA in tissue was comparable. This underscores the importance of spatially heterogeneous leakiness and porosity in the transvascular exchange mechanism for solute transport in patient specific tumor models as well as proper estimation of bas eline C p0 The low and high C p0 case illustrated the importance of proper dose measurement. There are two potential sources of error, which can lead to incorrect estimation of original values of C p0 First, Omniscan Gd DTPA is highly concentrated (287 mg m L 1 ) and requires a ~30 fold dilution for one dose. Second, the mouse blood volume per weight requires estimations based on the mean literature value from a range of values (63 80 mL kg 1 ). For example, the combination of these two factors could lead to an over estimation of C p0 if Gd DTPA is over diluted and/or the actual mouse blood volume per weight is greater than the estimated mean value. When comparing predicted concentrations with MR data, it should be noted that sources of error also exist in the MR I derived in vivo concentration values. R 1 which inversely scales the concentration estimated from MRI, was based a literature value that was measured in rat muscle at 6.3 T. This could be a direct source of error when comparing magnitudes of simulated (b aseline) and experimental concentration, if the actual R 1 varies from implemented literature value. Also, concentration measurements are dependent upon T 1 0 maps that were created by fitting a monoexponential curve to 5 data points measured from a variable TR SE sequence. Errors can occur from the fit as
47 well as motion between TRs. Motion was minimal since movement was not observed between scans. Additionally, there is background noise that comes from all of the electrical components used for imaging. The ba ckground noise can lead to less accurate measurements of low concentrations (< 0.01 mM). Furthermore, the noise has the potential to reduce the number of properly K trans fitted voxels, which happened infrequently (< 5% of tumor voxels). Another limitation of the model was the use of an AIF based on literature values. This potentially lowers the patient specificity of the model. In this study, AIF was selected from a range of literature values based on a comparison of simulated and experimental transient beh avior of C t ,avg ( t ) in tumor tissue at early time points. Though the slow AIF resulted in an acceptable correlation for one data set, it may not have been the experimental AIF for all data sets. This fact was likely the cause of the large simulated and expe rimental concentration discrepancies at later time points seen in Figure 2 10B F. K trans magnitudes and patterns, to a lesser degree, can be altered based on the type of AIF used. This could lead to variations of tracer deposition, since K trans is essentia lly a kinetic parameter that drives the tracer transport source term. In the future, more accurate models can be obtained by measuring patient specific AIFs to create more accurate 3D K trans and maps, as well as source terms in tracer transport which inc lude a C p term. The image based patient specific framework of this study offers opportunities to fine tune the model for a variety of applications. The model can be adjusted to incorporate a wider variety of tumor environments. For example, the image base d model can be expanded to include necrotic regions. A larger range of uptake and
48 washout behaviors can be observed in larger necrotic tumors which cannot be characterized by K trans and used in the standard two compartment model  In addition, diffusion weighted imaging could be used to detect regions of increased diffusion, which have been shown to be correlated with regions of necrosis [67 69] These regions can be modeled as avascular and highly diffusive porous regions within the framework of the presented computational model. Additional patient specificity can be captured with tensorial descriptions of K and D eff to account for preferen tial interstitial fluid and tracer transport. This method has been employed to predict transport in central nervous system tissue with the aid of diffusion tensor imaging [70 72] The prediction of patient specific interstitial fluid flow could also be incorporated into a multiscale model to microenvironment beca use it allows for a no slip boundary condition  Since shear stresses on tumor cells have been found to induce a G 2 /M cell cycle arrest and inhibit cell di fferentiation  tissue remodeling could potentially be predicted with this type of multiscale model. In order to predict transport of a therapeutic drug, the model could also be expanded to account for binding a nd degradation of drug by including a rate of binding term in the tracer transport equation. The model presented in this study could account for binding or degradation of extravasated solu te with the knowledge of its  For example, a first C t can be subtracted from the right hand side of Equation 2 8.
49 2.5 Acknowledgements We would like to thank Dr. Dietmar Siemann, Dr. Lori Rice, and Chris Pampo for providing the murine KHT sar coma cells and tumor inoculation Th is research was Opportunity Fund. MR data was obtained at the Advanced Magnetic Resonance Imaging and Spectroscopy facility in the McKnight Brain Insti tute of the U niversity of F lorida
50 Table 2 1A Tissue and vascular parameters used in simulations Variable Parameter Value S / V (m 1 ) Microvascular surface area per unit volume 20000 t ; 7000 n [12, 1 9] p V (Pa) Microvascular pressure 2300  i (Pa) Osmotic pressure in interstitial space 3230 t  ; 1330 n [12, 19] V (Pa) Osmotic pressure in microvasculature 2670  T Average osmotic reflection coefficient for plasma 0.82 t 0.91 n [12, 13] Table 2 1B Tissue and vascular parameters used in simulations Variable Parameter Value Literature range L p (m Pa 1 s 1 ) Vessel permeability 2 10 11 t 3 10 12 n 2.1 10 11  3.5 10 10  t 7.3 10 14  3.6 10 11  n L pL S L /V ( Pa 1 s 1 ) Lymphatic filtration coefficient 1 1 0 7 8 10 8 2 10 6  K (m 2 Pa 1 s 1 ) Hydraulic conductivity 1.9 10 12 t 3.8 10 13 n 7 .0 10 14  1.8 10 12  t 6.4 10 15  1.8 10 12  n D eff (m 2 s 1 ) Effective coefficient of diffusion 1 10 9 6.9 10 9  2.6 10 10  t: tumor tissue; n: normal host tissue Table 2 2 Summary of chapter 2 sensitivity analysis Pa rameter Pressure Boundary Velocity Concentration L p ,t Increase Increase No change L p ,n No change No change No change L pL S L / V Decrease Increase No change K t / K n Decrease Increase No change C p0 No change No change Decrease D eff No change No change No c hange The table describes the result of pressure, boundary vel ocity, and concentration within the tumor (increase, decrease, or no change) due to an increase in the parameter from the baseline value.
51 Figure 2 1 Block diagram of the mathematical met hodology and experimental verification for the patient specific tumor model.
5 2 Figure 2 2 Geometric reconstruction of tumor and host tissue. Based on A) T 2 map s (s lice 5 middle of tumor of MR data set ). Segmented tumor boundar y is outlined in black. B) CFD compatible mesh of reconstructed hind limb that includes tumor ( light blue ), skin boundary ( green ), cut ends ( yellow ), and representation of slice 5 of the MR data ( dark blue ).
53 Figure 2 3 Uptake and washout of average G d DTPA concentration, C t ,avg ( t ), in tumor tissue at seven time points for the first 19 min. Comparison between C t ,avg ( t ) based on experimental ( uptake rate = 0.071 mM min 1 ; monoexponential washout rate = 0.029 min 1 ) and simulations implementing slow (0. 076 mM min 1 ; 0.034 min 1 ), intermediate (0.089 mM min 1 ; 0.045 min 1 ), and fast (0.089 mM min 1 ; 0.050 min 1 ) AIFs.
54 Figure 2 4 Signal enhancement in vessel and tumor periphery A) Dorsal surface of right hind limb of mouse [ The Anatom y of the Laboratory Mouse Margaret J. Cook, http://www.informatics.jax.org/cookbook/chapters/foreword.shtml] B) First post scan of T 1 weighted DCE MR image and experimental signal enhancement in vessel region of interest (ROI) compared with the modeled s low AIF. C) Post scan (2500 s) of T 1 weighted DCE MR image and corresponding signal enhancement in tumor periphery ROI. A B C t = 20 s t = 2500 s
55 Figure 2 5 Maps from slice 5 of MR data set of A ) K trans and B ) with histograms. Maps were implemented in the CFD compatible mes h based on the slow AIF. Areas of greatest leakiness and porosity, respectively, are in red.
56 Figure 2 6 IFP sensitivity analysis for varying transport properties (Table 2 1B); A ) IFP contours for the baseline case; B ) IFP along the black li ne bisecting the tumor in the direction of the arrow in Figure 2 6A (dashed lines indicates tumor boundary).
57 Figure 2 7 Tissue averaged IFV sensitivity analysis for varying transport properties (Table 2 1B); A ) Vector field for the baseline case; B ) IFV along the black line bisecting the tumor in the direction of the arrow in Figure 2 7A (d ashed line indicates tumor boundary )
58 Figure 2 8 Comparison of experimental and simulated Gd DTPA distribution patterns in slice 5 for t = 5, 30, and 60 min for baseline transport values in Table 2 1B.
59 Figure 2 9 Gd DTPA distribution along bisecting tumor line at early, intermediate, and late time points. The distribution is along the bisecting tumor line in the direction of the arrow as defined in Figure 2 8. Simulation from sensitivity analysis of parameters in Table 2 1B and C p0 as compared with experimental distribution for A) t = 5, B) t = 30, and C) t = 60 min. Simulated data shows considerable overlap except for simulations based on high and low C p0
60 Figure 2 10 Gd DTPA distribution along bisecting line through two additional KHT sarcomas. Simulated Gd DTPA distribution ( grey ) compared with experimental distribution ( black ) for two tumors at A) and B) t = 5, C) and D) t = 30, and E) and F) t = 60 min, respectively. Dashed lines indicate tumor boundary.
61 CHAPTER 3 PATIENT SPECIFIC COMPUTATION AL MODELS OF SMALL S OLID TUMORS: A VALIDATION STUDY 3.1 Introduction The previous chapter presented a patient specific methodolog y for predicting interstitial fluid and tracer transport within a young, non necrotic tumor. Transport was predicted using a porous media computational model and parameters that govern transport were investigated, via a sensitivity analysis, in order to do cument their effects on the transport solution. More specifically, tracer transport sensitivity analysis was conducted in the context of experimental data. The sensitivity analysis showed the dominance of transvascular exchange due to the high leakiness in the young tumors. The developed model was then applied to two additional young tumors in order to strength of the model is the ability to capture a distribution profile which compares well to the experimental profile an advancement on our previously presented work  This is largely a product of the porosity maps. A correlat ion was observed between predicted and experimental spatial deposition of Gd DTPA within the tumor over the course of the experiment. Also, transient behavior of Gd DTPA in the tumor showed good agreement between the simulation and experiment for two of th e data sets. The degree of mismatches in late time point transient behavior, resulting in concentration magnitude differences, was likely proportional to errors in modeled and experimental AIF and interstitial flow. The results and conclusions of the young low interstitial fluid pressure (IFP) tumors with high leakiness presented in chapter 2 bolster those of low IFP tumors in the work of Hassid and cohorts (2006)  Their study presented a simple two
62 compartment model to estimate the transvascular exchange rate and porosity of low and high IFP tumors: MCF7 human breast and H460 lung cancer tumor, respectively, f rom magnetic resonance imaging (MRI) data. For lower IFP tumors (IFP ~ 1.9 kPa), they found that the simple two compartment model described reasonable values of transvascular exchange and porosity, while less reasonable values were produced for higher IFP tumors (IFP ~ 3.7 kPa) suggesting the need to extend image based physiologic models to include phenomena such as convection and diffusion for tumors with higher IFPs. Chapter 2 aimed to present an extension by testing such a model by investigating small tu mors (IFP = 0.50 1.4 kPa) with high leakiness. Consequently, MRI estimated transvascular exchange was the dominant term in the tracer spatial and transient behavior. While this term was dominant, there was a subtle difference in experimental and simulated clearance rate, which aside from mismatches in AIF could be caused by an underprediction of convection or lymphatic function causing an overprediction of Gd DTPA in the tumor tissue. This chapter aims to further investigate and expand upon the image based porous media tumor model by: Presenting a validation study of the model for young tumors by investigating an additional three tumors. Simplifying the computational approach by using a voxelized (Cartesian) mesh. Presenting histology that shows extent of t umor infiltration and vascularization to inform model assumptions (e.g., tumor size) and inputs (e.g., leakiness map trends, cellular density). The results in Chapter 2 showed the importance of both K trans and maps on the Gd DTPA concentration distributi on in time. The inclusion of both of these maps improved previously presented results  Namely, better spatial and transient behavior of tracer concentration in tissue was observed. The addition of the K trans and
63 maps correspo nded to mathematical representations of the physiological environment in the patient specific tumor model. The K trans maps were important to provide the spatially varying leakiness due to non uniformly distributed leaky vessels, which created non uniformly distributed interstitial fluid and tracer sources. The maps were important to provide non uniformly distributed extracellular volume fraction due to varying cell density in in vivo tissue, which regulates the ability of interstitial fluid to dilute ext ravasated tracer in each voxel. With the incorporation of these maps, tracer concentration in tissue was found to be dominated by transvascular exchange for these particular small tumors (< 37 mm 3 ) with high leakiness. With that said, there were two curiou s findings: 1) later time point comparisons of simulated and experimental data yielded slight inconsistencies of Gd DTPA concentration in tissue and 2) the DCE MRI method of determining porosity values may have underestimated the actual value of in vivo po rosity. This suggested that physiological phenomena of the 3D porous media model were underestimated. Furthermore, physiological phenomena such as convection and diffusion can have effects on the perceived K trans and values. The effects of reducing these values are likely to be exacerbated in maturing tumors where vessel remodeling can drastically reduce vessel leakiness. A validation study is the next step forward with the model and allows for these and other concepts to be investigated. For example, it allows the accuracy and consistency of model predictions to be tested against experimental data. In this chapter, that prediction is tracer distribution, magnitude, and transient behavior among a larger sample size. Additionally, this validation study not only allows for the examination of the utility of the 3D porous media model, but it also is a means of investigating a broader sample of tumor characteristics
64 (e.g., size and leakiness) that might affect the relative importance of terms in the tracer trans port equation (i.e., convection, diffusion, and transvascular exchange). For example, the validation study can lend insight into a possible threshold tumor size, pressure, or leakiness, to name a few examples, at which diffusion and/or convection may play a more important role for the distribution of tracer over time. Conversely, tracer concentration in tumor tissue in tumor sizes under that threshold may be able to be approximated sufficiently using a two compartment model accounting for transvascular exch ange. Tumor size and boundary can affect the estimated location of leakiness with respect to the tumor. This, in turn, can affect the perceived location of tracer or drug delivery in a porous media with a tumor tissue interface. This study aims to discrimi nate tumor boundary based on MR measurements of the combination of leakiness pattern and T 2 values, or apparent diffusion coefficient (ADC) maps with a quantitative comparison with literature values of other studies. MR imaging methods inherently store sp atial data on a voxelwise basis. Because of this, MR image based computational models lend themselves to a numerical solution that is computed using voxelized (Cartesian) meshes. This approach has been used to simulate infusion into the rat spinal cord  and rat brain  For solid tumors, interstitial fluid and tracer transport simulations using an unstru ctured, like the one used in chapter 2, and voxelized mesh were shown to be comparable  Additionally tracer transport simulations based on chapter 2 data were shown to be consistent with the tracer measurements from the experimental data 
65 For the validation study in this chapter, heterogeneous tissue transport p roperties have been accounted for by incorporating vascular permeability and porosity maps for an additional three small KHT sarcomas. Permeability maps were created based on uptake of Gd DTPA in tissue at early time points using dynamic contrast enhanced magnetic resonance imaging (DCE MRI). A fixed porosity ( = 0.2) was used and a K trans scale factor (> 1) was optimized to account for possible reductions in their perceived values due to the interstitial flow. Also, a fixed porosity was used in this study to independently investigate the role of interstitial fluid flow on transient Gd DTPA behavior in these tumors, so that spatial differences in behaviors would be due to interstitial flow and not porosity. Segmentation of tumor boundaries was conducted bas ed on a combination of early DCE MRI patterns and T 2 relaxation times (n = 2), and diffusion weighted imaging (DWI) (n = 1). Assumptions made in the model about the tumor microenvironment were assessed via histology. Specifically, H&E and MECA 32 staining was used to examine the extent of tumor infiltration of the hind limb muscle, cell density, and extent of vascularization. A voxelized modeling approach was used to solve interstitial fluid and tracer transport for each of these specific tumors. As a part of the validation study, simulated tracer transport was compared to experimental tracer transport. Tracer transport was compared by quantifying peak tumor concentration, spatial distribution across tumor, and average uptake and washout behavior within the tumor. The porous media model was compared to the two compartment model because the two compartment model is the most widely used MR image based model to obtain tracer concentration curves in tumor tissue space and time. Because each tumor has unique chara cteristics such as size and leakiness, the validation study also has the
66 potential to elucidate the role that such characteristics might have on the relative importance of convection, diffusion, and transvascular exchange within the tumor. Garrett Astary ( Department of Biomedical Engineering, UF) collected MR data. Jingya Zhang (Department of Biomedical Engineering, UF) sectioned tumors. Marda Jorgensen (MBI Cell & Tissue Analysis Core, UF) provided assistance with staining. Dr. Lori Rice and Sharon Lepler (Department of Radiation Oncology, UF) provided assistance and materials for tumor passage. 3.2 Materials and Methods 3.2.1 Animal Preparation Three C3H /HeJ female mic e (Jackson Laboratories, Bar Harbor, ME ) were inoculated with between 1 10 5 to 2 10 5 murine KHT sarcoma cells in the gastrocnemius muscle. Murine KHT sarcoma cells were provided by Dr. Dietmar The tumor was grown for 5 to 7 days During animal preparation and imaging, the mouse was anesthetized using a gas mixture of 98% oxygen, 2% isoflurane The lateral tail vein was catheterized using a Terumo SurFlo 24 Gauge NJ) and given a 0.2 5 mmol/kg of body weight (bw) bolus tail vein injection of Gd DTPA (Omniscan, GE Healthcare Inc., Princeton, NJ) CA at a constant rate (~0.4 ml/min, 0.23 ml). The animal experiment was governed by the principles of the Guide for the Care and Use of Laboratory Animals and approved by the Universit y of Florida Institutional Animal Care and Use Committee (IACUC). 3.2.2 Magnetic Resonance Imaging (MRI) The MRI experiment was performed using a Bruker Avance imaging console (Bruker NMR Instruments, Billerica, MA) connected to Magnex Scientific 11.1 T
67 ho rizontal bore magnet system (Varian, Inc., Magnex Scientific Products, Walnut Creek California). Tumor bearing hind limb s of three mice were placed in a 1.5 cm inside diameter transmit and receive, volume coil The animal body was oriented in the prone pos ition with the leg secured to reduce motion Figure 3 1 shows the approximate anatomical location of the tumors in the hind limb as well as the orientation of the MR slice geometry. Multiple T 2 weighted spin echo (SE) scan s (TR = 2000 ms, TE = 15, 30, 45, 60 and 75 ms; subjects 1 and 2: field of view (FOV) = 2 cm 1 cm 1 cm, 10 slices, matrix size = 192 96, 2 averages; subject 3: FOV = 2 cm 1 cm 1.2 cm, 12 slices, matrix size = 192 96, 1 average), were acquired for the ca lculation of T 2 values. A variable time for recovery ( TR ) SE sequence (TR = 5000, 2000, 1000, 500 and 250 ms, TE = 15 ms), was acquired for calculation of native tissue T 1 values used in CA concentration calculations Serial D CE MR images, consisting of T 1 weighted SE sequence (su bjects 1 and 2: TR/TE = 330 ms/9.4 ms, FOV = 2 cm 1 cm 1 cm, 10 slices, matrix size = 192 96, 2 averages, total acquisition time = 1 min 4 s; subject 3: TR/TE = 368 ms/9.4 ms, FOV = 2 cm 1 cm 1.2 cm, 12 slices, matrix size = 192 96, 2 averages, total acquisition time = 1 min 11 s) were collected before and after CA administration For DWI, a diffusion weighted SE sequence was used with a total acquisition time = 33.8 min, FOV = 2.0 cm 1.0 cm 1.2 cm, TR = 1500 ms, TE = 21.5 ms, and one avera ge. Low diffusion weighted data (70 s mm 2 ) were acquired in 6 directions, defined by the tessellation of an icosahedron on a unit hemisphere, and high diffusion weighted data (520 s mm 2 ) were acquired in 21 directions. MR data was obtained at the
68 Advanc ed Magnetic Resonance Imaging and Spectroscopy (AMRIS) facility in the McKnight Brain Institute (MBI) of the University of Florida. 3.2.3 Histology After imaging sessions were completed, mice were euthanized and the tissue on the lateral side of the tibia and fibula was resected. This tissue primarily included tumor tissue as well as lateral gastrocnemius and soleus muscles. Tissue was fixed in 10% phosphate buffered formalin between 24 and 48 h, then moved into a histological cassette and placed in 70% eth anol before being embedded in paraffin wax. Tissue embedded paraffin blocks were sectioned with a thickness of 5 m using a microtome (HM 325, Microm, Germany). Four consecutive 5 m sections were cut every 50 m into the paraffin block. Two sections were h ema toxylin and e osin (H&E) s tain ed, processed with an automatic stainer (Leica Auto Stainer XL, Germany). The other two sections were stained with MECA 32 (BD Biosciences, Bedford, MA), a monoclonal antibody with high specificity for mouse endothelium A Leica microscope (DM 2500, Germany) was used to view slides and digital images were taken using a digital microscope camera (Optronics, Goleta, CA). Tumor area was estimated in the H&E stained slides ( 1.25 optical magnification). Vessel density was meas ured in peripheral and central regions of the tumor in the MECA 32 stained slides ( 20 optical magnification). Tumor infiltration of the hind limb muscle was investigated with the H&E s tain ed sections by measuring tumor area. Tumor area was estimated usin g an in house image analysis code (MATLAB v.7, Natick, MA) that counted tumor (purple) pixels from muscle and the background. Vessel density in the tumor was investigated with the MECA 32
69 stained sections. Vessel density was determined in the tumor rim and core by counting stained structures in up to 6 fields of 38 mm 2 at a magnification of 20. The number of counted fields was limited by the size of the tumor. 3.2.4 Image Segmentation Segmentation of tumor tissue for the computational model was manually co nducted (MATLAB v.7, Natick, MA) based on the combination of contrast differences in T 2 maps from the variable TE SE sequence and peripheral enhancement in DCE MRI (n = 2) or ADC maps (n = 1) from DWI. T 2 m easurements were guided by that of normal hind lim b mouse muscle (range = 18 22 ms ) at 11.7 T  and with the assumption that T 2 values for tumor are typically higher than normal muscle due to the increased mobility of extracellular water that occurs from apoptosis  Tumor boundary was demarcated in ADC maps based on the observed ADC differences between tumor and muscle  All of the tissue regions outside of the tumor were designated has host muscle in the model. 3.2 .5 Arterial Input Function, Vascular Permeability Maps, and Porosity Maps Chapter 2.2.3 showed the link between vascular leakiness ( K trans ) maps, porosity ( ) maps, and arterial input function (AIF). The shape (rate of clearance) of the biexponential AIF impacted the outcome of the K trans and maps. The impact of these maps on tracer uptake and wash out was borne out in Figure 2 3. The slow AIF produced the best fit for average concentration within the tumor volume and also showed a similar trend to the m easured biexponential in a vessel using DCE MRI (Figure 2 4B). An imaging protocol was established to obtain the enhancement patterns in a major vessel of the hind limb by setting up the imaging geometry such that a vessel
70 was contained in a single voxel. The advantage of this was to gain information on the AIF behavior during the same DCE MR imaging session as the one used to create permeability maps. Post imaging analysis revealed that no major vessels were captured in imaging sessions. While a signal en hancement curve like that of Figure 2 4B could not be established for individual data sets in this study, it does indicate the likely nature of the AIF for this experimental setup. Additionally, the results of using the slow AIF established similar uptake a washout over the early time points of average Gd DTPA concentration in tumor (Figure 2 3). For these reasons, a slow AIF [26, 32] was used where a 1 = 4.0 kg bw L 1 a 2 = 4.8 kg bw L 1 m 1 = 0.002 s 1 and m 2 = 0.0 002 s 1 in Equation 2 3; a 1 and m 1 represent the amplitude and rate constant, respectively, of the fast equilibrium between plasma and extracellular space; a 2 and m 2 represent the amplitude and rate constant of the slow component of the clearance. DCE MRI data was converted into concentration measurements based on Equation 2 1. K trans maps were created based on early time points (first 15 min) of concentration data using the same two compartment analysis as described in Equations 2 2 through 2 4 with a fixe d porosity of 0.2. Values of K trans were set to zero in voxels where the fit was poor (R 2 K trans values in pixels that showed no Gd DTPA uptake (< 0.005 mM) during the first two post scans were filtered out as well. 3.2.6 3D Porous Media Mathematic al and Computational Model The 3D porous media mathematical model presented in chapter 2.2.4 was used to predict the steady state interstitial fluid flow and transient Gd DTPA distribution assuming the tissue as porous media. The computational modeling tim es were reduced by using a voxelized (Cartesian) mesh to store the spatially discretized porous media transport solutions instead of an
71 unstructured mesh (Figure 3 2). Rectangular volumes were created (Gambit, Fluent, Lebanon, NH) that corresponded to the exact dimensions of DCE MRI data (field of view = 2 1 1 cm 3 for KHT 1 & KHT 2; field of view = 2 1 1.2 cm 3 for KHT 3). Eight node brick elements were used to create meshes consisting of 205,931 nodal points (KHT 1 & KHT 2) and 243,373 nodal points (KHT 3). The CFD software package (FLUENT 6.3, Fluent, Lebanon, NH) was used to solve porous media transport Tissue and transport properties in tumor and host muscle were assigned according to baseline values (Table 2 1) in FLUENT. Prediction of tumor tr acer distribution was a two step process. First, steady state interstitial plasma fluid flow was solved (Equation 2 6). Then, tracer transport (Equation 2 8) was solved at later time points The only difference between boundary conditions of unstructured m esh (chapter 2) and the Cartesian mesh was at t he skin boundary In order to maintain a no flux condition for interstitial fluid at the skin, brick elements outside of the muscle regions (exterior cells, Figure 3 2C) were assigned hydraulic conductivity va lues two orders of magnitude lower than muscle. This low hydraulic conductivity value was assigned in the exterior cells to effectively reduce the interstitial flow in this region to zero. There should be no interstitial flow here because there is no tissu e in this region. The convergence problem associated with the porous media solver in FLUENT for low hydraulic conductivity materials was avoided by reducing the under relaxation factor and step wise reductions of hydraulic conductivity to reach the final b aseline value. Effective diffusion was set to zero in the exterior cells so that tracer flux was zero at the skin boundary.
72 3.3 Results 3.3.1 Segmentation, Leakiness, and Histology Segmentation of tumor tissue resulted in increased (13 33%) T 2 values (KHT 1 & KHT 2) and decreased ( 44%) ADC value (KHT 3) in tumor regions of interest (ROIs) when compared with normal muscle ROIs (Figure 3 4). Contrast between assumed tumor and muscle was greater in ADC maps than T 2 maps. KHT 3 was found to be the largest (mo st mature tumor) of the three based on total segmented volume (120 mm 3 ) and most tumor infiltration observed in the H&E stained tumor cross section (area = 4.4 0.9 mm 2 ; Figure 3 5E). KHT 1 and KHT 2 showed less tumor infiltration in both the total segmen ted tumor volume (KHT 1: 43 mm 3 ; KHT 2: 63 mm 3 ) and area of tumor in the H&E stained sections (KHT 1: 1.8 0.3 mm 2 ; KHT 2: 1.5 0.7 mm 2 ). K trans maps showed leakiness heterogeneity in the tumor and surrounding tissue (Figure 3 6). Increased leakiness wa s observed around the perimeters for all tumors when compared to the tumor mass (> 100%). Average K trans value within the tumor decreased with tumor size. Within the tumor, leakiness variations were not as pronounced, which was corroborated by MECA 32 stai ning for endothelial cells. For the largest of the three tumors (KHT 3), MECA 32 staining showed the tumor to be well vascularized at both the rim and center (Figure 3 7) based on the mean number of vessels per mm 2 at the tumor rim (147 100) and tumor ce nter (158 49). Histological sections of the two smaller tumors were not large enough to distinguish a tumor rim and center.
73 3.3.2 Interstitial Fluid Transport Interstitial plasma fluid pressure was predicted to be higher within the tumor than in normal t issue for all three subjects (Figure 3 8). Intratumoral pressures increased with tumor size: KHT 1 (0.62 kPa; Figure 3 8A), KHT 2 (0.85 kPa; Figure 3 8B), and KHT 3 (1.79 kPa; Figure 3 8C). These peak pressures within the tumor fell slightly towards the tu mor boundary with the pressure gradient greater than 0.38 kPa mm 1 Subsequently, the largest boundary pressure gradient was observed in KHT 3. The increased pressure resulted in flow of interstitial fluid directed outward from the tumor boundary. Interst itial velocity of plasma fluid was predicted to be highest at or near the boundary of th e tumor (Figure 3 9). Average fluid velocity at the tumor boundary increased with tumor size: KHT 1 (0.10 m s 1 ; Figure 3 9A), KHT 2 (0.13 m s 1 ; Figure 3 9B), and KHT 3 (0.40 m s 1 ; Figure 3 9C). These areas of high velocity correspond to the larger pressure gradients. 3.3.3 Tracer Transport Experimental Gd DTPA spatial and transient concentration behavio r exemplified the non uniformity of tracer delivery. Furthermore, differences in behavior between tumors highlighted the patient specific nature of systemic delivery. Spatially, all three subjects showed increased uptake (> 80% at peak concentration) at th e tumor periphery when compared with the central tumor region. Temporally, uptake within and around tumor tissue was slow. At t = 8 min, the average concentration of Gd DTPA in the central tumor region was minimal (0.02 0.05 mM), but slowly increased (> 90 %) over the next 15 min in this region. Peak concentration was reached at t 30 min. Peak and washout characteristics varied between tumors, with slower washout corresponding to
74 increased tumor size. KHT 1 and KHT 2 showed signs of washout at 30 min and 6 5 min, respectively. The washout rate in the tumor was quicker for KHT 1 (0.11 mM hr 1 ) than KHT 2 (0.08 mM hr 1 ). The tumor of KHT 3 did not show Gd DTPA washout over the imaging time course ( t 1.7 hr). Peak Gd DTPA concentration occurred at t 30 min for all porous media simulations, with the average peak occurring later in time with increased tumor size. Also, washout rates decreased with increasing tumor size. While transient behavior of the experimental data was similar to the baseline porous media simulations, the magnitude of the concentration within the tissue was underpredicted (Figure 3 10). At peak Gd DTPA concentration, baseline simulations were underpredicted by less than 35%. Tracer transport simulations were optimized to provide optimal c oncentration magnitudes and transient behavior in the tumor with the assumption of constant porosity ( 0.2). K trans values were scaled by a constant to account for the perceived reduction in permeability in the case of low transvascular exchange. The tr ansient concentration behavior in the tumor using the porous media model compared better than the two compartment model over the course of the simulation based on the Pearson product moment correlation coefficient (PMCC), r where r > 0.9 for the porous me dia model (Figure 3 10). The two compartment model performed better as washout was less significant in the tumor. PMCC was lowest for KHT 1 (most washout) with r = 0.53 and highest for KHT 3 r = 0.98 (least washout). Increasing the K trans scale factor did not change the overall shape of the concentration curves; hence it did not appreciably
75 change PMCC (< 1%). However, peak Gd DTPA concentration values increased from the baseline peak value in response to an increase in the K trans scale factor. By compari ng the results of the simulations and experiment through quantifying error versus time, revealed that scaling K trans by a value greater than one resulted in reduced error over the time of the simulation compared with the baseline (Figure 3 11). Root mean s quared (RMS) errors, of the data in Figure 3 11 shows a decreasing trend of error with increasing K trans scale factor. At early time points (first three), the two compartment model resulted in lower absolute error than the porous media model, but the er ror increased with time. This was most evident in KHT 1 (Figure 3 10A) and KHT 2 (Figure 3 10B); at the final time points, the two compartment model resulted in absolute error > 0.05 mM, which was > 100% of the average concentration in the tumor. The porou s media model predictions resulted in smaller absolute error (< 0.02 mM) than the two compartment model at the final time point. Optimized porous media simulations showed similar spatial trends to the experimental data (Figure 3 12 and 3 13). Both simulate d and experimental Gd DTPA concentration profiles depicted the trend of reduced uptake within the center of tumor; the concentration of Gd DTPA increased at the boundaries. For the tumor with the highest washout rate (KHT 1), concentration simulations show ed reduced uptake (50% less) within the tumor at early time points when compared with periphery (Figure 3 12A). The higher washout rate lead to the largest decrease (50 %) in Gd DTPA concentration in tumor over the experimental and simulated time courses w hen compared with the other tumors. At later time points (Figure 3 12B), the concentration difference across the tumor boundary was less pronounced (42% less). All of the tumor
76 simulations resulted in more flattening of the concentration curve within the t umor over time than the experimental data. The greatest flattening of the concentration curve within the tumor occurred in KHT 3 (Figure 3 13) where the relative difference in peak and base concentration in the tumor was < 10% for the simulation and 85% fo r the experiment. 3.4 Discussion and Conclusions This chapter presents the application of the image based model to multiple small non necrotic tumors ( 120 mm 3 ) for the prediction of interstitial fluid and solute transport with heterogeneous leakiness. Th ese models predict interstitial hypertension and increased velocities at the host tissue tumor interface like the model presented in chapter 2. While chapter 2 showed the importance of porosity upon the distribution of Gd DTPA concentration in tissue over time, this chapter presented models using a fixed a porosity. A fixed porosity was necessary for two reasons. Most importantly, the uptake rate within tumor tissue was too slow to obtain a physiologically relevant value by using the two compartment model t o fit early time points of concentration data. Secondly, porosity values obtained in chapter 2 ranged (0.05 0.15) lower than values measured by DCE MRI independent methods (0.2). Fixing the porosity also allowed Gd DTPA clearance in the tumor due to inters titial fluid motion and not a low extracellular volume fraction. The simulations with fixed porosity showed the ability of interstitial fluid to transport extravasated solute out of the tumor. Transient behavior of average Gd DTPA in the tumor showed agree ment between the simulation and experiment. The establishment of a model that accounts for convection, diffusion, and transvascular
77 exchange underscores these phenomena role in solute transient behavior and distribution in tissue. The importance of the po rous media model for small molecular weight solutes as well as the potential effects of interstitial fluid flow on perceived MRI measured values of K trans and was demonstrated in this chapter. This is best illustrated by assuming pure transvascular excha nge over the entire course of the experiment to obtain these parameters, which is often assumed in literature [30, 31, 83] In this limited case, results of the model lead to K trans (inflow of tracer from the blood into the interstitial space) that are much smaller than outflow (tracer entering back into blood pool from the interstitial space). A common consequence is small, non physiologically relevant values (< 5%) within the tumor. Similar average values for h ave been reported before for this experimental tumor type (KHT sarcoma) using the standard kinetic model  While the H&E stains of the KHT sarcoma in the work of Nielsen et al. qualitatively revealed densely pac ked cells throughout most of the tumors, 5% is on the extreme low end of previously measured cell densities in grafted tumors  A porosity of 5% was measured in specific locations within the tumor and did not represent the average porosity of the whole tumor in the work of Jakobs en et al. The H&E staining of the KHT sarcomas presented in this chapter showed cell packing similar to that of Nielsen et al; hence it was reasonable to use a porosity of 0.2 which represented the low end of average porosity in tumors  Th e large difference in inflow and outflow, which can result in unrealistic implies other physiological phenomena aid clearance of Gd DTPA in tumors. This observation has been made in the DCE MRI pressure measurement work of Y. Hassid et al. (2006 and 2008 ). Hassid and cohorts suggested that the
78 apparently low values were not physiologically representative of the extracellular volume fraction itself, but that interstitial fluid flow is likely responsible for the clearance behavior within the tumor. The application of the porous media model to determine average Gd DTPA concentration in the tumor reduced the absolute error when compared with the two compartment model using a fixed porosity. This concept was apparent in both Figure 3 10 and Figure 3 11. Fu rthermore, the importance of interstitial fluid flow on the overall transient behavior in the tumor can be illustrated by using the two compartment model to simultaneously fit K trans and over the course of the simulations presented in this chapter. Even though the porosity was fixed at 0.20, KHT 1 and KHT 2 resulted in porosity values less than 0.06 while KHT 3 resulted in average porosity values less than 0.10 in the optimized, scaled K trans cases. These low values misrepresent the extracellular volume f raction used in the simulations and underscore the importance of clearance in tumor. Models with low leakiness ( K trans ) could benefit from DCE MRI independent imagi ng methods of obtaining porosity. Current imaging methods do not allow for the porosity measurements, but T 1 or T 2 weighted images may be a way to approximate porosity pattern. The interstitial fluid flow pattern resulted in an average Peclet number of a pproximately 2 5 revealing that Gd DTPA transport is slightly more convective than diffusive. This range of Peclet numbers has been presented in previous tumor models  Peclet number was defined by, Pe = vL / D eff where v is the velocity magnitude and L is characteristic length of 1 mm. The outward direction of the convective transport
79 aided in movement of extravasated solute to the boundaries of the tumor of t he entire time course of the simulation and experiment. Not only does this phenomenon contribute to a reduced apparent porosity as described earlier, but also a reduced K trans The measured K trans value in a given voxel could be lower than the actual K tran s value because of extravasated solute transported out of the voxel due to convective interstitial flow. The convective transport in the tumor may be responsible for the slower uptake seen in the experimental tumors in this study. This was the motivation f or scaling K trans values within the tumor to determine optimal transient behavior. MECA 32 staining confirmed prevalence of vessels throughout the tumor, but these vessels could have reduced leakiness due to vessel remodeling and thickening of vessel walls that can occur in maturing tumors  This remodeling may have also contributed to differences in early uptake rates seen in the tumor in chapter 2 and those in this chapter. This study showed that there is not a particular set of parameters that can be applied to all tumors. In other words, there is no one size fits all approach to modeling interstitial an d solute transport in the presented KHT sarcomas. In the case of vessels with low leakiness, the porous media model can be used to account for the interstitial fluid flow and the potential misrepresentation of K trans and that are important physiological inputs into the model. Fitting K trans values with a fixed, reasonable at early time points, then using those values to predict concentration in tumor at later time points using a two compartment model grossly overestimates concentration. The use of a bas eline porous media model to make that same prediction underestimates the concentration of solute in tumor at later time points, but allows for more accurate transient behavior than the two compartment model because it accounts for the effects
80 of the inters titial fluid flow. Scaling K trans values to account for reduction in measured leakiness due to convection does provide a path to more accurate predictions of concentration in tumor with low leakiness.
81 Figure 3 1 Orien tation of MR slices with respect to the mouse anatomy (top view). The first slice of MR data is labeled 1. Point of view of the MR slices is shown. Syringe shows the location of the inoculation. The dotted line shows the approximate location of the grafted tumors. 1 Z syringe
82 Figure 3 2 CFD compatible meshes A) Unstructured mesh of reconstructed hind limb from Figure 2 2B simplified to B) a voxelized (Cartesian) mesh and C) detailed view of a segmented slice.
83 Figure 3 4 Tumor segmentation of M RI data for small KHT sarcomas. Segmentation for KHT 1 based on A) early DCE MRI enhancement and B ) T 2 values. Segmented tumor boundary ( black ) and host muscle region of interest ( red ) shown with corresponding T 2 values for these regions. Segmentation for KHT 2 based on C) early DCE MRI enhancement and D) T 2 values with measured values for tumor and host muscle ROI. Segmentation for KHT 3 based on E ) early DCE MRI enhancement and F ) ADC values. ADC values for tumor ( black ) and normal host muscle ROI ( red ) a re shown. A C E B D F Tumor Host muscle ROI
84 Figure 3 5 H&E staining and example of estimating tumor area. Dense cellular regions in the tumor of A) KHT 1, B) KHT 2, and C) KHT 3. D ) KHT 3 H&E stained with corresponding E ) segmented H&E stain. D E Tumor area = 4.4 0.9 mm 2 Host muscle Tumor B A C 100 m
85 Figure 3 6. L eakiness ( K trans ) map s of small KHT sarcoma s. Maps for A) KHT 1, B) KHT 2, and C) KHT 3 Leakiness outside the tumor boundary ( black line ) is higher than inside the tumor boundary.
86 Figure 3 7 Example of MECA 32 staining for KHT 3. A cont inuous region of tissue is shown that shows the extent of vascularization at the tumor rim and tumor center at 20 magnification. The fluorescently green labeled areas ( arrows ) are the endothelial cells that line the vasculature. 100 m Tumor center Host muscle Tumor rim Vasculature
87 Figure 3 8 Interstiti al fluid pressure for small KHT sarcoma s. Pressure contours near tumor boundary ( black line ) of A) KHT 1, B ) KHT 2, and C) KHT 3.
88 Figure 3 9 Interstitial fluid velocity for small KHT sarcoma s. Velocity contours near tumor boundary ( black line ) o f A) KHT 1, B ) KHT 2, and C) KHT 3.
89 Fig ure 3 10 Comparison of simulated and experimental average Gd DTPA in tumor. Time course of average concentration curves shown for A) KHT 1, B) KHT 2, and C) KHT 3. Porous media model and two compartment mode l computed with = 0.2.
90 Fig ure 3 11 Comparison of error versus time quantified for simulations and experiment for KHT sarcomas. Root mean square (RMS) error is quantified. Porous media simulations resulted in lower RMS than the two compartment model f or KHT 1 and KHT 2. Absolute error versus time for KHT 3 is not shown (RMS = 0.1 10 3 for 1.3* K trans case; RMS = 0.3 10 3 for two compartment case).
91 Fig ure 3 12 Simulated tracer transport for small KHT sarcoma (KHT 1). A) Gd DTPA concentra tion distribution contours in KHT 1 ( black ) at an early time point ( t = 16 min) and B) later time point ( t = 90 min ) C) Gd DTPA concentration profile ( along dashed arrow ) comparing simulated and experimental tracer at an early and later time point across tumor boundary ( dotted line ).
92 Fig ure 3 13 Comparison of experimental and simulated concentration profiles across small KHT sarcomas. Gd DTPA distribution at early time points ( t = 16 min) for A) KHT 2 and B) KHT 3 with the tumor boundary labeled ( do tted line ) Gd DTPA distribution at later time points ( t = 100 min ) C ) KHT 2 and D ) KHT 3.
93 CHAPTER 4 PATIENT SPECIFIC COMPUTATION AL MODELS OF LARGE S OLID TUMORS: AN EXPERIMENTAL COMPARI SON 4.1 Introduction The previous chapter presented a validation st udy of a patient specific tumor model for predicting interstitial fluid and tracer transport within small, non necrotic tumors with low leakiness. This result revealed the importance of the 3D porous media model by accounting for the effects of convection and diffusion. Convection played an important role on the transient Gd DTPA concentration behavior within tumors with low leakiness and constant porosities. The porous media model provided better peak and transient concentration behavior than the two compa rtment model when the modeled role on reducing measured values of K trans and porosity was elucidated. This chapter aims to investigate the application of the 3D po rous media model in larger KHT sarcoma with greater tissue heterogeneity ( 160 mm 3 ). Larger tumors can complicate drug delivery due to the evolving nature of the tissue structure. As a tumor matures and grows larger, the vasculature at the tumor boundary has a more difficult time supplying distant cells in less vascularized regions with the proper concentration of oxygen and nutrients. The lack of oxygen and nutrients compromises the cell viability. An early study on this topic demonstrated the distance b etween vessels and cells to be a limiting factor for cell viability due to the increase diffusional distance required for oxygen and nutrients  The lack of proper cell maintenance can often lead to necrosis. Necrosis is a slow cell death in which groups of cells swell to the point of bursting the cellular membrane leading to the discharge of cytotoxic intracellular content into the tumor en vironment causing an inflammatory
94 response in nearby tissue [ 88] With healthy cells, phagocytes (white blood cells that ingest foreign particles) are signaled to clean up such harmful products, but necrotic cell death lacks these signaling capabilities leading to the accumulation of cellular debris. This cascade o f events due to a growing tumor exacerbates the non uniformity of the tumor environment. The drug delivery difficulties that arise due to necrosis have been studied using mathematical models. Baxter and Jain predicted the effects that a necrotic core would have on interstitial and solute transport in spherical tumor geometry  Because necrotic regions in tumor tissue can be described by lack of functioning blood vessels and lymph vessels, there is no transvascular exchange in t hese regions. This requires that necrotic tissue regions gain access to drug molecules from nearby vascularized necrosis did not significantly reduce the central interstitial p ressure of the tumor. Second, concentration of distributed solute was reduced in the tumor as a whole due to the necrotic region. Third, there existed a reservoir effect. At early times, there was little solute in the necrotic core. At later times, the out er tumor region had less solute concentration than the central necrotic region; solute in the outer tumor region was cleared by functional transvascular exchange and convection. Experimentally, the reservoir effect has been seen in lower molecular weight d rugs (methotrexate, MW ~ 450) [89, 90] which have a shorter time scale for u ptake, distribution, and washout than larger macromolecules. The importance of necrosis upon drug delivery was studied using a non systemic delivery scenario by Smith and Humphrey  They modeled infusion into a brain tumor with a necrotic core. The necrotic core lacked functioning
95 vasculature; hence, there was no transvascular exchange. Their model suggested that the presence of a necrotic core substantially increased the convective fluid velocity in its vicinity. More intuitive drug delivery difficulties in larger tumors have been shown by sparser solute sourc es that results in less dense K trans maps [66, 91] In many experimental cases, these sparser solute sources are indicative of reduced solute distribution to the tumor. Magnetic resonance imaging (MRI) has been used to detect necrosis and associated phenomena. Necrotic regions have been characterized by T 2 relaxation time  as well as by using diffusion weighted imaging (DWI) to calculate the apparent diffusion coefficient (ADC) [21, 67, 92, 93] DWI allows for structural tissue information to be probed with the self diffusion of water molecules. Expressly, DWI is an MRI technique that exploits the reduction in MR signal in biological tissues as water molecules move along a field gradient. DWI has been implemented for a number of tumor related applications: (1) the detection of tumors  (2) ability to distinguish tumor from sur roundings  and (3) ability to differentiate viable and necrotic regions of tumors [67 69] Non MR methods of investigating tissue has shown reduced integrity of cells in necrotic regions  The degradation of the cell matrix may increase the abili ty for water molecules to diffuse. Dynamic contrast enhanced (DCE) MRI has been used to show the delayed and increased presence of MR visible tracer in necrotic regions  This chapter aims to create an image based model that accounts for patient specific necrotic regions in conjunction with vascular permeability maps of the tumor rim. For the study in this chapter, heterogeneous tissue transport properties have been
96 accounted for by incorporating vascular permeability maps for an additional four large KHT sarcomas. Permeability maps and porosity maps were created based o n uptake of Gd DTPA in tissue at early time points using DCE MRI. Segmentation of tumor boundaries was conducted based on ADC maps. ADC maps will be fluid filled necrotic regions by which solute reduction and reservoir effects may be assessed for individua l tumors with regions of necroses. MR determined tumor infiltration of the hind limb muscle and leakiness were assessed via histology, H&E and MECA 32 staining, respectively. A voxelized modeling approach was used to solve interstitial fluid and tracer tra nsport for each of these specific tumors. As a part of th is chapter simulated tracer transport was compared to experimental tracer transport. Tracer transport was compared by quantifying peak tumor concentration, spatial distribution, and average uptake a nd washout behavior within the tumor. Larger tumors are susceptible to a wider variety of tissue heterogeneities than smaller tumors. Because of each tumor has unique characteristics such as size and leakiness, the re is the potential to elucidate the role that such characteristics such might have on the relative importance of convection, diffusion, and transvascular exchange within the tumor. 4.2 Materials and Methods 4.2.1 Animal Preparation Four C3H /HeJ female mic e (Jackson Laboratories, Bar Harbor, ME ) w ere inoculated with between 1 10 5 to 2 10 5 murine KHT sarcoma cells in the gastrocnemius muscle. Murine KHT sarcoma cells were provided by Dr. Dietmar The tumor was grown for 8 to 9 days D uring animal preparation and imaging, the mouse was anesthetized using a gas mixture of 98% oxygen, 2% isoflurane The lateral tail vein was catheterized using a
97 Terumo SurFlo 24 Gauge NJ) and given a 0.2 5 mmol/kg of body weight (bw) bolus tail vein injection of Gd DTPA (Omniscan, GE Healthcare Inc., Princeton, NJ) CA at a constant rate (~0.4 ml/min, 0.23 ml). The animal experiment was governed by the principles of the Guide for the Care and Use of L aboratory Animals and approved by the University of Florida Institutional Animal Care and Use Committee (IACUC). 4.2.2 Magnetic Resonance Imaging (MRI) The MRI experiment was performed using a Bruker Avance imaging console (Bruker NMR Instruments, Billeric a, MA) connected to Magnex Scientific 11.1 T horizontal bore magnet system (Varian, Inc., Magnex Scientific Products, Walnut Creek California). Tumor bearing hind limb s of four mice were placed in a 1.5 cm inside diameter transmit and receive, volume coil A variable time for recovery ( TR ) SE sequence (TR = 5000, 2000, 1000, 500 and 350 ms, TE = 15 ms), was acquired for calculation of native tissue T 1 values used in CA concentration calculations Serial D CE MR images, consisting of T 1 weighted SE sequence ( 2 averages, TR/TE, field of view (FOV), slices, matrix size, and, total acquisition time in table 4 1) were collected before and after CA administration DWI allows for structural tissue information to be probed with the self diffusion of water molecules. For DWI, a diffusion weighted SE sequence was used with the parameters in table 4 2 and two averages. Low diffusion weighted data were acquired in 6 directions, defined by the tessellation of an icosahedron on a unit hemisphere, and high diffusion weighte d data were acquired in 21 directions. MR data was obtained at the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS) facility in the McKnight Brain Institute (MBI) of the University of Florida.
98 Multiple linear regression  was us ed to fit multi slice diffusion weighted images to obtain the water translational diffusion tensor, D e A single component of the diffusion tensor, D eij was determined by, (4 1) where b ij is the diffusion weight factor; S ( b ) is the b dependent signal intensity; S (0) is T 2 weighted, proton de nsity dependent signal intensity in the absence of diffusion weighting. ADC was solved in each voxel by solving, (4 2) 4.2.3 Histol ogy After imaging sessions were completed, mice were euthanized and the tumor tissue on the lateral side of the tibia and fibula was resected. The histology and image acquisition protocol was followed as described in the histology section 3.2.3. Histology was conducted at the MBI Cell & Tissue Analysis Core (CTAC) with the aid of Marda Jorgensen and Jingya Zhang. 4.2.4 Image Segmentation Segmentation of tumor tissue for the computational model was manually conducted (MATLAB v.7, Natick, MA) based on contras t differences in ADC maps (n = 4) from DWI. Tumor boundary was demarcated in ADC maps based on the observed ADC differences between tumor and muscle  All of the tissue regions outside of the tumor were designated as host muscle in the model. High ADC regions (~1.59 10 3 mm 2 s 1 ) within the tumor boundary w ere segmented 
99 4.2.5 Arterial Input Function, Vascular Permeability Maps, and Porosity Maps Arterial input function, v ascular leakiness ( K trans ) and porosity was described based upon method in section 3.2.5. 4 .2.6 3D Porous Media Mathematical and Computational Model The 3D porous media mathematical model presented in chapter 2.2.4 was used to predict the steady state interstitial fluid flow and transient Gd DTPA distribution assuming the tissue as porous media. Voxelized (Cartesian) meshes were used to store the spatially discretized porous media transport solutions (Figure 3 2). Rectangular volumes were created (Gambit, Fluent, Lebanon, NH) that corresponded to the exact dimensions of DCE MRI data (field of vie w correspond to those in Table 4 1). Eight node brick elements were used to create meshes between 205,931 (10 slices of MR data) and 383,910 (12 slices of MR data) nodal points. CFD software package (FLUENT 6.3, Fluent, Lebanon, NH) was used to solve porou s media transport Tissue and transport properties in tumor and host muscle were assigned according to baseline values (Table 2 1) in FLUENT. Prediction of tumor tracer distribution was a two step process. First, steady state interstitial plasma fluid flo w was solved (Equation 2 6). Then, tracer transport (Equation 2 8) was solved at later time points No flux of interstitial fluid at the skin boundary was maintained by low hydraulic conductivity (two orders lower than muscle) and zero diffusivity, respect ively, in the exterior cells (Figure 3 2C). Segmented regions of high ADC were assumed to be fluid filled necrotic regions. They were considered to have no functioning blood or lymphatic vessels  High ADC regions were defined with an increased diffusion coefficient (1
100 10 9 m 2 s 1 ) based on MRI diffusion experiments of Gd DTPA in necrotic cyst fluid  and increased porosity (0.5). 4.3 Results 4.3.1 Segmentation, Leakiness, and Histology The heterogeneous tissue structure of large KHT sarcomas was illustrated by the range of ADC values measu red in regions of interest. ADC values in the tumor rim regions of interest (ROIs) for all subjects were lower than normal muscle and high ADC ROIs (Figure 4 1). High ADC regions (ADC 1.59 10 3 mm 2 s 1 ) were present in all tumors but one tumor (KHT 5). Two large KHT sarcomas (KHT 4 and KHT 6) developed high ADC regions at the center of the tumor while one (KHT 7) developed an additional high ADC regions at the tumor rim. High ADC regions accounted for < 6% the total tumor volume. KHT 7 was found to be t he largest (most mature tumor) of the four based on total segmented volume (895 mm 3 ) and most tumor infiltration observed in the H&E stain (area = 19.2 2.3 mm 2 ) in Figure 4 2 KHT 4, KHT 5, and KHT 6 showed less tumor infiltration in both the total segme nted tumor volume (KHT 4: 557 mm 3 ; KHT 5: 160 mm 3 ; KHT 6: 392 mm 3 ) and area of tumor in the H&E stained sections (KHT 4: 17.0 2.0 mm 2 ; KHT 5: 6.8 1.2 mm 2 ; KHT 6: 18.9 2.1 mm 2 ) K trans maps showed increased leakiness around the perimeter for all tum ors, but leakiness was patchy within the tumor. Percentage of fitted voxel within tumors was (< 15%) for all tumors. Increased leakiness density (46% voxels were fitted) was observed near fluid filled, high ADC regions subjects KHT 6 and KHT 7 (the high AD C region near the rim). MECA 32 staining revealed KHT 6 to have the most extensive vascularization, which showed higher vascular density towards the center (132 91
101 vessels per mm 2 ) of the tumor than at the rim (18 18 vessels per mm 2 ) (Figure 4 3). KHT 4 showed the opposite trend with higher vascularization observed at the tumor rim (23 6 vessels per mm 2 ) than the tumor center (5 8 vessels per mm 2 ). Labeled endothelial cells were sparse in tumors KHT 5 and KHT 7. 4.3.2 Interstitial Fluid Transport Hi ghest regions of interstitial fluid pressure were predicted to be within the tumor all four subjects (Figure 4 5). Peak pressures ranged from 1.9kPa (KHT 5) to 2.6 kPa (KHT 7). Broader regions of higher intratumoral pressure (> 2.0kPa) were predicted in th ree of the tumors (Figure 4 5A, 4 5C, and 4 5D). Additional localized tumor peaks were predicted within the tumors (Figure 4 5B and 4 5D) as well. These peaks ranged from 50 100% the magnitude of the broader pressure peaks. Pressure decreased from peak pre ssure regions with a gradient 0.10 0.15 kPa mm 1 for all tumors. Unique interstitial fluid flow patterns were observed in each tumor (Figure 4 6). All tumors showed increased fluid velocity (0.7 0.8 m s 1 ) and outward convection along the tumor boundary. One large sarcoma (KHT 6) showed a single central region (Figure 4 6C) of lower i nterstitial fluid velocity (0.1 m s 1 ) than the higher boundary velocities. The three other tumors (Figure 4 6A, 4 6B and 4 6D) showed more complex interstitial flow patterns. Namely, these tumors had multiple low velocity regions within the tumor. where interstitial fluid radially e manated. 4.3.3 Tracer Transport Experimental Gd DTPA concentration behavior demonstrated the unique patterns of tracer distribution as well as some of the difficulties of systemic delivery to larger
102 tumors. All four subjects showed early uptake ( t 5 min) in the tumor periphery when compared with the tumor region. However, all tumors shared similar patterns of uptake in the tumor periphery, which was slow with the peak concentration (0.10 0.25 mM) being reached at t 40 min. Within the tumor, peak concent ration behavior showed more variation for a given tumor and between tumor experiments (Figure 4 7). All experimental tumors showed regions where a peak concentration was reached followed by washout ( t > 50 min). Peak concentrations were reached at t ~ 50 m in for KHT 4 (Figure 4 7A) and KHT 5 (Figure 4 7B). A small region of KHT 4 (4% of tumor volume) showed slow and low uptake ( C t 0.01 mM) with no washout over the course of the experiment. The highest and most homogeneous Gd DTPA coverage (0.089 0.051 m M) was observed in the smallest tumor of this animal group (KHT 5; Figure 4 7B). Gd DTPA concentration within tumor tissue was lower and not as homogeneous for KHT 6 (0.031 0.020 mM; Figure 4 7C) and KHT 7 (0.014 0.038 mM; Figure 4 7D). Accumulation of Gd DTPA was observed in the high ADC, necrotic regions within these lower uptake tumors (KHT 6 and KHT 7). Accumulation of tracer in the peripheral high ADC region of KHT 7 occurred immediately after the bolus injection where as accumulation occurred late r ( t = 9 min) in the central high ADC region of KHT 6. Peak concentrations were achieved at later time points ( t > 65 min) than surrounding tissue in these high ADC regions for both KHT 6 (0.24 0.17 mM at peak) and KHT 7(0.22 0.10 mM at peak). Like t he experimental tumor data, the porous media simulations showed noticeable uptake at the tumor periphery at early time points ( t = 5 min). This was particularly true for KHT 4, KHT 5, and KHT 7 (Figure 4 8A, 4 8B, and 4 8D,
103 respectively). These tumors were predicted to contain large regions where the concentration was very low (< 0.005 mM) while peripheral concentration was higher (> 0.01 mM). Peripheral tissue concentration was high (> 0.01 mM) at early time points in KHT 6, but concentrations were also hi gh in nearly all locations within the tumor (Figure 4 8C). Peak concentration at the tumor periphery was reached later ( t 50 min) than in experiment for all cases except one, KHT 7, at t < 5 min Simulations of the four tumors showed distinctive distribu tion characteristics. Because of the low concentrations in the center of the tumor at early time points, three of the tumors (KHT 4, KHT 5 and KHT 7) showed inward diffusion from the tumor rim towards the center. This type of inward diffusion was not visib le in the experimental data. Over the course of 90 min, tissue concentration in the low concentration central regions increased to ~ 0.02 mM. KHT 5 had the most homogeneous coverage in the tumor with a 65% relative difference in peak (0.033 mM) and minimum concentration; however, the peak concentration in the tumor was underpredicted by more than twofold. Coverage was less complete for KHT 4. Even at the last simulated time point ( t = 90 min), concentration in more than half the tumor was < 0.005 mM). No wa shout within the tumor of KHT 4 or KHT 5 was predicted over 90 min; however the washout outside the tumor boundary occurred at a rate of 0.07 mM min 1 Inward diffusion was less significant for KHT 7. At t = 90 min, concentrations above 0.01 mM were predic ted 3 mm from the tumor boundary. A growing circular concentration front was predicted within this tumor as well. This was an anomaly related to the high velocity point source predicted in the interstitial fluid flow (Figure 4 6D).
104 Accumulation of Gd DTPA was underpredicted in the high ADC regions of KHT 6. Over the course of the 90 min simulation, concentration reached 0.04 mM. This tissue concentration was about 200% higher than the predicted tissue concentration in the surrounding tissue, but was 83% lo wer than the experimental measurement. Washout (~0.3 mM min 1 ) occurred t > 50 min in the tissue surrounding the high ADC region but like the experiment, there was no washout. Unlike the experimental data, accumulation in the high ADC regions of KHT 4 and KHT 7 was not observed 4.4 Discussion and Conclusions Chapter 4 presents the application of the MR image based model for multiple (n = 4) large necrotic tumors ( 160 mm 3 ). Drug delivery modeling challenges are elucidated by the heterogeneous nature of th e tissue structure as well as decreased density of solute sources (sparser K trans maps). The tissue heterogeneity present in this tumor series is observed by ADC maps, which is indicative of heterogeneous water content within the tumor. Higher ADC regions were assumed to have a higher effective diffusivity and porosity due to the increased water content. While intratumoral K trans maps were sparser than those in chapters 2 and 3, these models account for heterogeneous leakiness to predict interstitial hypert ension and interstitial fluid motion. Interstitial fluid motion was directed outwards from the tumor boundary, but inward diffusion was observed in the porous media models. The porous media models underpredicted the Gd DTPA concentrations within the tumors due i n part to the sparseness of K trans which reduced the number of leaky sources within the tumor. The model predicted inward diffusion in three of the tumors. In the model, regions of non zero K trans were Gd DTPA sources within the tumor boundary, at the tumor rim.
105 Over time, the Gd DTPA diffused inwards, in the opposite direction of outward flux of interstitial fluid motion. The interstitial velocity in the central region was low (<0.1 m s 1 ) such that transport of solute was diffusion dominant. It was difficult to confirm this phenomenon experimentally with these particular KHT sarcomas because of the noise present in the data, which made complicated the delineation of diffusion front. However, this phenomenon was observed in other DCE MRI experimen ts with CAKI 1 renal cell carcinomas conducted by our group and in experimental tumors of other researchers  The difference in magnitudes between predicted and experimental concentrations was a roadblock for validation. The large errors in concentration magnitude were likely due to the sparseness of K trans mapping. Th e sparseness of the maps resulted in an underprediction of tracer sources, which lead to a reduced Gd DTPA concentration within the tumor. Leakiness maps for larger KHT sarcomas were expected to be less dense towards the center of the tumo r Additionally, MECA 32 staining revealed low microvascular vessel density in these tumors; however, experimental Gd DTPA distribution suggested the possibility of more tracer sources than the K trans maps predicted. This factor could be compounded by underpredictions K tr ans which would lead to less uptake and slower transvascular washout. Noise in the concentration data was the likely culprit of the reduced properly fitted K trans values. These model i nputs are affected by the number of averages as well as the temporal an d spatial resolution of the DCE MR I sequence. Spatial and temporal resolution should be tailored to the expected behavior of the contrast agent. In the case of these low leakiness tumors, the number of averages of each scan can be increased to decrease noi se.
106 Table 4 1 Dynamic contrast enhanced magnetic resonance imaging parameters Subject Field of view (cm 3 ) Slices Matrix size TR/TE (ms ms 1 ) 4 2.0 1.4 1.4 14 190 133 430/9.4 5 2.0 1.0 1.0 10 192 96 330/9.4 6 2.0 1.4 1.1 11 190 133 33 7/9.4 7 2.0 1.4 1.4 14 190 133 430/9.4 Table 4 2 Diffusion weighted imaging parameters Subject Field of View (cm 3 ) Slices Matrix Size TE (ms) Total acquisition time (min) Low b (s mm 2 ) High b (s mm 2 ) 4 2.0 1.4 1.4 14 190 133 31.7 89.8 77 583 5 2.0 1.0 1.0 10 100 50 21.5 33.8 70 520 6 2.0 1.4 1.1 11 100 70 21.5 47.3 70 520 7 2.0 1.4 1.4 14 100 70 21.5 47.3 70 520
107 Figure 4 1 Tumor segmentation with corresponding H&E staining of small KHT sarcomas. Segmentation an d H&E staining for A) KHT 4, B) KHT 5, C) KHT 6 with an example of segmented tissue zones, and D) KHT 7 with ADC values for tumor rim ROI ( blue ), high ADC necrotic ROI ( green ), and normal muscle ( red ). Tumors outlined in black. For H&E stains the purple in dicates tumor cells, while the pink indicates host muscle. Tumor rim ROI Host muscle ROI Tumor rim ROI High ADC ROI A B C D
108 Figure 4 2 H&E staining of large KHT sarcomas for A) KHT 4, B) KHT 5, C) KHT 6, and D) KHT 7. Showing stained sections of the dense cellular tumor rim ( left panel ) and less dense regions toward the center ( right panel ). 100 m A B C D
109 Figure 4 3 Example of MECA 32 staining for KHT 6. Two regions (near necrotic core and tumor rim) is shown that depicts the extent of vascularization in these regions at 20 magnification. The fluorescently green la beled areas ( arrows ) are the endothelial cells that line the vasculature. Region near the necrosis is more vascularized than the tumor rim. 100 m
110 Figure 4 4. Leakiness ( K trans ) map s of large KHT sarcoma s. Maps for A) KHT 4, B) KHT 5, C) KHT 6, and D) KHT 7 L eakiness outside the tumor boundary ( black line ) is higher than inside the tumor boundary.
111 Figure 4 5 Interstitial fluid pressure for large KHT sarcoma s. Pressure contours near tumor boundary ( black line ) of A) KHT 4, B ) KHT 5, C) KHT 6, and D) KHT 7.
112 Figure 4 6 Interstitial fluid velocity for large KHT sarcoma s. Velocity contours near tumor boundary ( black line ) of A) KHT 4, B ) KHT 5, C) KHT 6, and D) KHT 7. A), B), and D) show complex pattern of interstitial fluid flow that occu r due to localized point sources in the K trans maps. C) shows a more conventional interstitial fluid flow patter with a region of low velocity near the central region and increasing velocity toward the tumor boundary.
113 Fig ure 4 7 Experimental G d DTPA concentration, C t distribution in large KHT sarcomas at time equals 30 min. Peak times and locations shown within the tumor ( black line ) for A) KHT 4, B) KHT 5, C) KHT 6, and D) KHT 7. A) A tumor region of KHT 4 (4% of tumor volume) showed slow and low uptake ( C t 0.01 mM) with no washout over the course of the experiment. B) The highest and most homogeneous coverage of Gd DTPA concentration in the tumor (0.089 0.051 mM) at peak concentration was observed in KHT 5. C) The high ADC region of KHT 6 accumulated tracer over the course of the experiment and reached a peak C t (0.24 0.17 mM). D) Significant region of no uptake (average C t < 0.008 mM) was observed in tumor KHT 7 (20% of tumor volume).
114 Fig ure 4 8 Simulated tracer transport for l arge KHT sarcomas. Distribution of Gd DTPA within tumor boundary ( black ) at early ( t = 16 min), intermediate ( t = 50 min), and later ( t = 90 min) time points for A) KHT 4, B) KHT 5 C) KHT 6, and D) KHT 7.
115 CHAPTER 5 CONCLUSION S AND FUTURE WORK The inten t of this chapter is to recapitulate and unite the key concepts and contributions of the image based modeling approach for solid tumors. An image based solid tumor model may be used to optimiz e and evaluat e treatment strategies for patient specific therapi es. An MRI based porous media model has the ability to mathematically incorporate physiologically relevant transport characteristics such as spatially varying transvascular exchange, diffusion, and convection, which affect the therapeutic levels of systemi cally drugs in both time and space. Because both the model and reality are the confluence of many physiological behaviors, it was the goal of the dissertation to investigate and validate positive predictive qualities of the image based model as well as to delineate areas of refinement. The use of experimental data is quite a stride for the development of tumor models because it allows for the assessment of tracer distribution and transient behavior. These are the first necessary steps toward building more a ccurate models for clinical application. The dissertation first presented a methodology for the first MRI based tumor model that accounts for non uniform distribution of solute by incorporating heterogeneous leakiness and extravascular volume fraction in a small KHT sarcoma with high leakiness. An appropriate arterial input function (AIF) was selected based on tumor volume averaged Gd DTPA uptake and washout behavior. The selection process showed that a slow AIF provided consistency between experimental and simulated uptake and washout in the tumor. The slow washout of Gd DTPA in the plasma compartment was observed in this experimental setup using dynamic contrast enhanced magnetic resonance imaging (DCE MRI). Sensitivity analysis was also
116 conducted on tissu e and vascular transport parameters. The sensitivity analysis revealed that interstitial fluid pressure and velocity was more sensitive to changes in parameters than tracer transport for this KHT sarcoma, which implied the dominance of transvascular exchan ge. The model that was developed using one animal was used to simulate interstitial fluid and tracer transport for two additional KHT sarcomas. By comparing simulated and experimental Gd DTPA distribution, concentration profiles were validated over the cou rse of the experiment; however, there were concentration magnitude discrepancies at later time points, which was likely due to incorrect later time point AIF behavior or inaccurate predictions of interstitial flow. Even though errors in concentration magni tude existed, the overall transient behavior and concentration profiles were superior to previously presented image based tumor models from our group. This promising result was contributed to the inclusion of porosity maps. It should be noted that DCE MRI derived porosity maps are susceptible to errors because the underlying interstitial fluid flow can affect the distribution and transient behavior of the MRI calculated concentration curves leading to a possible underestimation of porosity. This concern is likely to be more significant with more mature tumors. Small KHT sarcomas with more mature vasculature were observed to have slower leakiness. Slower leakiness made it difficult to obtain porosity maps by simultaneously fitting K trans and with early tim e points ( t < 19 min) of MR derived in vivo concentration data. In order to have leakiness maps based on early time points, a approach to obtaining K trans and A s such, it was appropriate to understand and adapt the porous media model for this low leakiness scenario. Additionally, it allowed for the
117 model to be examined independent of spatial porosity changes or potentially erroneous porosities. In other words, it allowed for a more explicit investigation of the effects of convection and diffusion on tracer transient behavior and distribution. This study showed a potential strength of the porous media model: the ability to account for clearance of tracer or drug d ue to convection and diffusion. The two compartment model overpredicted average Gd DTPA in the tumors at later time points. The baseline porous media model underpredicted this quantity, which was an important revelation. The underprediction showed that the interstitial fluid flow can affect the perceived values of K trans and when tumors have low leakiness. This fact, unfortunately, adds more complexity to modeling this scenario, as it may be necessary to account for the decrease in K trans value with a gre ater than unity scale factor. This highlights the point that there is not a particular set of parameters that can be applied to all tumors. In other words, there is no one size fits all approach to modeling interstitial and solute transport in the presente d KHT sarcomas. This study provided a model by which the underpredictions of K trans and in DCE MRI can be better understood. Furthermore, the methods employed in the study allow the comparison of experimental data to potentially quantify the underpredict ions. Large KHT sarcomas were observed to have heterogeneous vasculature and tissue. The MRI based model was further developed to account for regions of high fluid filled regions. Apparent diffusion coefficient (ADC) maps were used to detect and account fo r these fluid filled regions into the model. Lower leakiness in these tumors made it difficult to obtain porosity maps. Furthermore, the decreased K trans density complicated predictions of tracer distribution over time. Future models of larger tumors
118 that employ K trans would be likely aided by reducing the noise in the data by either decreasing temporal resolutions or number of averages of the DCE MRI scans. While K trans density is lower in larger tumors than smaller tumors, these changes in MR parameters w ill lead to more accurate K trans mapping and tumor interstitial fluid and tracer sources within the tumor. This dissertation showed the development and application of an image based modeling approach that incorporates more reality than previously present ed models. The model was applied to three distinctly unique tumor scenarios: 1) small tumors with high leakiness, 2) small tumors with low leakiness, and 3) larger tumors with greater tissue heterogeneity. Positive steps have been made towards better under standing and predicting tracer transport in smaller tumors. A t this point, models of larger tumors require greater refinement. Despite some modeling shortcomings, this is the first attempt to test the porous media approach to modeling tumors and comparing predictions with experimental data. The study contained in this dissertation stands at a place where short intermediate and long term goals should be established. For the short term, animal numbers for the small low leakiness tumors should be increase d. Currently, a method for this tumor scenario has been developed. Increased animal numbers would help to determine the ability of the model to account for the potential underestimation of K trans and at that particular stage of tumor growth. Currently, t he model was validated for distribution of Gd DTPA along a line through the tumor for young (6 7 days) KHT sarcomas. Transient response was validated for slightly more mature tumors (7 8 days). It is necessary to validate both spatial and transient behavio r within a tumor. Two things will be necessary
119 to accomplish this. First, it is likely that animal specific AIFs will be required. It may be sufficient to use the method of choosing AIF as presented in this dissertation. That is to vary the AIF, simulate t ransport for early time points, and select the best one based on average uptake and washout characteristics. It may be easier to obtain animal specific AIFs using MRI, though this too has its pitfalls. Obtaining animal specific AIFs was attempted several t imes during experimental MR data acquisition for this dissertation. Major vessels in the tail and leg were imaged to little success. The MR spatial resolution of the mouse tail was too low to discern a tail vein region of interest for AIF. Only one of many imaging sessions revealed vessel signal enhancement curves in the leg. Second, DCE MRI independent methods for determining porosity distribution will be useful. Assuming a constant porosity throughout the tumor, due to the slow leakiness limitation, in ch apter 3 decreased the detail of the Gd DTPA concentration distribution curve along a bisecting tumor line. This made comparison of experimental and simulated distribution difficult. Employing ADC maps or T 2 maps will not provide quantitative porosity valu es but may provide the relative distribution of extravascular volume fraction. These porosity distribution patterns may be scaled to a realistic value (0.2 0.4) to provide more detail of the distribution curve. The developed model utilized a low molecular weight tracer. An appropriate intermediate goal would be to employ the methods presented in this dissertation for large molecular weight tracer (e.g., Gd Albumin). Larger molecular weight tracers will give clues into the transport characteristics of larger molecular weight drugs (e.g., monoclonal antibodies). Substituting a bolus injection of Gd DTPA with Gd Albumin could identically follow the methodology presented in this dissertation. Several
120 preliminary experiments using Gd Albumin were conducted. Minim al extravasation of Gd Albumin was observed with the KHT sarcomas on a thirty minute time scale. It is recommended to use a different tumor model (e.g., MCF 7 breast cancer cell line) or to increase the time scale of observation (~2 h). While the model re quires further validation for all tumors sizes a road map for a patient specific model has been established. This road map gives promise to a long term ambition: a clinical model that would customize chemotherapeutic dose based on vironment. This requires an addition of a binding degradation term in solute transport. The benefit is that a predictive model would allow for the reduction of imaging time, which can be expensive and uncomfortable. Additionally, cardiotoxicity and other n egative systemic consequences could be limited and dose efficacy maximized.
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130 BIOGRAPHICAL SKETCH Greg Pi shko spent most of his life in Florida. His formative years were spent at Atlantic Community High School where he discovered a predilection for science. In 2006, he obtained a B.S. in aerospace engineerin g from the University of Florida free from debt thank s to his great parents and the friendly lottery players. In graduate school, he took a turn for the bioengineering world in order to avoid entanglement with the military industrial complex mission accomplished.