Computational Models for Localized Drug Delivery in Tumors

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Computational Models for Localized Drug Delivery in Tumors
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1 online resource (82 p.)
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english
Creator:
Kulam Najmudeen,Magdoom M
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University of Florida
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Gainesville, Fla.
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Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Sarntinoranont, Malisa
Committee Members:
Tran-Son-Tay, Roger
Sorg, Brian

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Subjects / Keywords:
cfd -- convection -- dce -- drug -- image -- intratumoral -- murine -- porous
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Systemic drug delivery to malignant tumors involving macromolecular therapeutic agents is challenging for many reasons. Amongst them is their chaotic microvasculature which often leads to inadequate and uneven uptake in solid tumors. Tumors are known to have highly tortuous, fenestrated, discontinuous vessels and large avascular areas. Such an abnormal microvasculature is thought to cause heterogeneous extravasation of drugs and elevated interstitial fluid pressures inside the tumor. Localized drug delivery is increasingly being used to circumvent such obstacles and convection-enhanced delivery (CED) which utilizes convection in addition to diffusion for distributing macromolecules has emerged as a promising local drug delivery technique. The focus of this thesis was to develop a three dimensional computational porous media transport model for solid tumors based on voxelized modeling methodology, which incorporates the actual tumor microvasculature from the data obtained through dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI). The model was used to predict interstitial fluid flow and tracer transport in tumors. First portion of the project was focused on the development and evaluation of the voxelized model for tumor transport. The model was developed for predicting the interstitial flow field and distribution of MR visible tracer (Gd-DTPA) in tumor following bolus tail vein injection. The results of the voxelized model were compared with that obtained from a previously developed CFD modeling approach using unstructured meshes. Furthermore, simulated Gd-DTPA distribution within the tumor was compared to MR-measured Gd-DTPA concentration data. The voxelized model was tested on three tumors with its predictions compared against the non-voxelized model and experimental results. Benefits of a voxel approach include less labor and less computational time. Sensitivity of the model to changes in arterial input function (AIF) parameters was also investigated. For comparison, statistical analysis and qualitative representation of both model results were presented. The analysis indicated similarity in both the model results with low root mean square error and high correlation coefficient. The voxelized model captured features of the flow field and tracer distribution such as the high interstitial fluid pressure (IFP) inside tumor and the heterogeneous distribution of tracer. Predictions of tracer distribution by the voxelized approach resulted in low error when compared with the MR-measured data over a 1 hr time course. The accuracy of the voxelized model results with experiment and non-voxelized model predictions were maintained across the tumors. The sensitivity of the model to changes in AIF parameters was found to be similar to that of the previous model approach. Secondly, the developed voxelized model was slightly modified for predicting the interstitial flow field and distribution of albumin tracer following CED at the hind-limb tumor in mice. The spatially varying transport properties were obtained via DCE-MRI experiments following systemic delivery of MR visible tracer, as mentioned in the previous paragraph. A point source was introduced in the governing equations to model the local infusion. The model was able to capture the heterogeneous/asymmetric tracer distribution and the linear variation of distribution volume with the infusion volume. Sensitivity of the model to changes in hydraulic conductivity and catheter placement were investigated. The albumin distribution was found to be sensitive to both the parameters under study. Increasing the values of the hydraulic conductivity map lowered the tumor IFP and raised the distribution volume within the whole leg. However within the tumor, the distribution volume decreased with increasing value of hydraulic conductivity, at later time points. The infusion at the tumor-host tissue interface resulted in larger distribution volume compared to that at the center and anterior end of the tumor, under baseline conditions. Within the tumor, the distribution volume was almost identical for infusions at the interface and center of the tumor. This image-based model thus serves as a potential tool for optimizing patient-specific cancer treatments and exploring the effects of heterogeneous vasculature on tumor transport.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Magdoom M Kulam Najmudeen.
Thesis:
Thesis (M.S.)--University of Florida, 2011.
Local:
Adviser: Sarntinoranont, Malisa.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-02-29

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COMPUTATIONALMODELSFORLOCALIZEDDRUGDELIVERYINTUMORSByMAGDOOMMOHAMEDKULAMNAJMUDEENATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2011

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c2011MagdoomMohamedKulamNajmudeen 2

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Tomymom,dadandfamily 3

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ACKNOWLEDGMENTS Itakethisopportunitytothankeveryonewhohashelpedmethroughoutthisjourney.IwouldliketoextendmygratitudetomyadvisorDr.MalisaSarntinoranontforgivingmeanopportunitytodoresearchwithnancialassistance,andhelpingmesinceIjoinedUF.Ithankallthemembersofmyresearchgroupwhohavehelpedmewiththisprojectinvaryingdegrees.AspecialthanksgoestoDr.GregoryPishkowithoutwhomthisprojectwouldnothavebeenaccomplished,Iamtrulyindebtedtohim.Ialsowanttoextendmythankstomycommitteemembers:Dr.BrianSorgandDr.Tran-Son-Tayforkindlyacceptingtobeinmycommittee.Formyresearch,IwouldliketothankDr.DietmarSiemann,Dr.LoriRiceandChrisPampoforprovidingmurineKHTsarcomacellsandtumorinoculation.IalsothankDr.ThomasMareciforprovidingMRIexpertiseandGarrettW.AstaryforhelpingwiththeDCE-MRexperiments.IappreciatethehelpDr.GregoryPishkohasofferedinthisproject,byprovidingmewithtissuetransportpropertymapsandsimulationresultsforthenon-voxelizedmodel,alongwithsegmentedMRimages.IalsowanttothankDr.JungHwanKimforsharinghisvaluableexperienceonvoxelizedmodeling.Moreimportantly,Ithankmymomanddadforprovidingmewithnancialandmoralsupport.Ialsothankmyfriendsandfamilymembersespeciallymyuncles,cousins,brothersandsistersforadvisingandhelpingmeindifculttimes.IwouldalsoliketothankmyfriendsatUFespeciallythosefrommyundergraduateschoolinIndiaforprovidingmewithvaluablesupportwhenIcametothiscountryforthersttime.Finally,IwouldliketothankalltheprofessorsIhaveinteractedwithforprovidingmewithbenecialknowledge,whichhashelpedmebecomethepersonIamtoday. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 13 1.1Background ................................... 13 1.2Objectives .................................... 14 1.2.1SpecicAim1 .............................. 15 1.2.2SpecicAim2 .............................. 15 2DEVELOPMENTOFVOXELIZEDMODELFORSYSTEMICDELIVERYINSOLIDTUMORS ................................... 16 2.1Overview .................................... 16 2.2Methods ..................................... 18 2.2.1Estimationofspatialvariationmapsofvascularleakiness ..... 18 2.2.2MathematicalModel .......................... 20 2.2.3ComputationalMethod ......................... 22 2.2.4StatisticalAnalysis ........................... 23 2.2.4.1RootMeanSquareError .................. 24 2.2.4.2PearsonProductMomentCorrelationCoefcient ..... 24 2.2.4.3ErrorHistogram ....................... 25 2.3Results ..................................... 25 2.3.1SensitivityAnalysis ........................... 27 2.3.2ValidationStudy ............................. 27 2.4Discussion ................................... 28 3APPLICATIONOFVOXELIZEDMODELFORCONVECTION-ENHANCEDDELIVERYINAHINDLIMBTUMOR ....................... 48 3.1Overview .................................... 48 3.2Methods ..................................... 51 3.2.1MathematicalModel .......................... 51 3.2.2ComputationalMethod ......................... 53 3.3Results ..................................... 55 3.3.1SensitivityAnalysis ........................... 57 3.4Discussion ................................... 58 5

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4CONCLUSIONSANDFUTUREWORK ...................... 73 REFERENCES ....................................... 75 BIOGRAPHICALSKETCH ................................ 82 6

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LISTOFTABLES Table page 2-1TissueandvascularparametersusedforsimulatingdistributionofGd-DTPAfollowingbolustailveininjectionatthehindlimbtumorinamice. ........ 33 2-2Statisticalparametersobtainedwhilecomparingvoxelizedandnon-voxelizedmodelresultsforthebaselinesimulationinthreeanimals. ............ 34 2-3Statisticalparametersobtainedwhilecomparingvoxelizedandnon-voxelizedmodelresultsforintermediateandfastarterialinputfunctioninanimalI. .... 35 2-4Comparisonoftracerwashoutratesandrootmeansquareerrorintracerconcentrationwithinthetumorvolumebetweenvoxelizedandnon-voxelizedmodelresultswithexperimentinthreeanimals. .................. 36 3-1TissueandvascularparametersusedforsimulatingdistributionofalbuminfollowingCEDatthehindlimbtumorinamice. .................. 62 7

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LISTOFFIGURES Figure page 2-1NormalizedconcentrationoftracerinbloodplasmaapproximatedbydifferentAIFsusedforsensitivityanalysis .......................... 37 2-2CFDcompatiblemeshes .............................. 38 2-3Horizontalandverticallinesusedforplottingtheoweldandtracertransportinvoxelizedandnon-voxelizedmodels ....................... 39 2-4ContoursofIFPpredictedbyvoxelizedandnon-voxelizedmodels,alongwithlineplotsalongthehorizontalandverticalbisectorsinthemid-slice. ...... 40 2-5ContoursofIFVpredictedbyvoxelizedandnon-voxelizedmodelsalongwithlineplotsalongthehorizontalandverticalbisectorsinthemid-slice. ...... 41 2-6Comparisonoftracerconcentrationcontours.Voxelizedandnon-voxelizedmodelcomparedwithMR-derivedtissueconcentrationatt=5,30,and60min. .......................................... 42 2-7Lineplotscomparingthepredictedtracerconcentrationinthetissuebyboththemodelswithexperiment,alongthehorizontalandverticalbisectorsofmid-sliceatt=5,30and60min. ........................ 43 2-8ErrorHistogramsforowandtransportinbaselinesimulationforvoxelizedmodelwithrespecttonon-voxelizedmodel. .................... 44 2-9LineplotscomparingtheIFPandIFVpredictedbyboththemodelsfortwodifferentAIFparametersets(intermediateandfast)alongtheverticalbisectorofmid-slice. ...................................... 45 2-10LineplotscomparingthetracerconcentrationinthetissuepredictedbyboththemodelsfortwodifferentAIFparametersets(intermediateandfast)alongtheverticalbisectorofmid-sliceatt=5and20min. ............... 46 2-11ErrorHistogramsfortracerconcentrationwithinthetumorforvoxelandnon-voxelmodelresultswithrespecttotheexperimentaldataatt=5,30and60min. ........................................ 47 3-1DepictionofbaselineCEDsimulation. ....................... 63 3-2Variationofscaledhydraulicconductivitywithporosityfordifferentvaluesofm. 64 3-3IFPandEFVcontoursatthetumormid-sliceforsystemicandlocalinfusion,alongwithaEFVconeplotcoloredbyitsmagnitudeforlocalinfusion. ..... 65 8

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3-4Normalizedtracerconcentrationcontoursattumormid-sliceatt=30,60,and120min.Alsoincludedpredictedevolutionofdistributedvolumeovertimeshownbyaniso-surfaceatthedistributionvolumethreshold. .......... 66 3-5VariationoftissuedistributionvolumeswithinfusionvolumeforthewholelegandtumorfollowingCEDofalbumin(0.3L/min)atthecenterofthetumor .. 67 3-6ComparisonofIFPandEFVcontoursatthetumormid-slicealongwithEFVconeplotcoloredbyitsmagnitude,forinfusionsatm=5&9. .......... 68 3-7Predictedevolutionofdistributedvolumeforinfusionsatthecenterofthetumorform=5and9att=30,60,and120min. .................... 69 3-8VariationoftissuedistributionvolumeswithinfusionvolumeforthewholelegandtumorfollowingCEDofalbumin(0.3L/min)atthecenterofthetumorform=0,5&9. ................................... 70 3-9Comparisonofnormalizedtracerconcentrationcontoursattumormid-sliceforinfusionsatthetumor-hostinterfaceandanteriorendofthetumoratt=30,60,and120min. ................................. 71 3-10VariationoftissuedistributionvolumeswithinfusionvolumeforthewholelegandtumorfollowingCEDofalbumin(0.3L/min)atthetumor-hosttissueinterfaceandanteriorendofthetumorwithm=0 ................ 72 9

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceCOMPUTATIONALMODELSFORLOCALIZEDDRUGDELIVERYINTUMORSByMagdoomMohamedKulamNajmudeenAugust2011Chair:MalisaSarntinoranontMajor:MechanicalEngineeringSystemicdrugdeliverytomalignanttumorsinvolvingmacromoleculartherapeuticagentsischallengingformanyreasons.Amongstthemistheirchaoticmicrovasculaturewhichoftenleadstoinadequateandunevenuptakeinsolidtumors.Tumorsareknowntohavehighlytortuous,fenestrated,discontinuousvesselsandlargeavascularareas.Suchanabnormalmicrovasculatureisthoughttocauseheterogeneousextravasationofdrugsandelevatedinterstitialuidpressuresinsidethetumor.Localizeddrugdeliveryisincreasinglybeingusedtocircumventsuchobstaclesandconvection-enhanceddelivery(CED)whichutilizesconvectioninadditiontodiffusionfordistributingmacromoleculeshasemergedasapromisinglocaldrugdeliverytechnique.Thefocusofthisthesiswastodevelopathreedimensionalcomputationalporousmediatransportmodelforsolidtumorsbasedonvoxelizedmodelingmethodology,whichincorporatestheactualtumormicrovasculaturefromthedataobtainedthroughdynamiccontrast-enhancedmagneticresonanceimaging(DCE-MRI).Themodelwasusedtopredictinterstitialuidowandtracertransportintumors.Firstportionoftheprojectwasfocusedonthedevelopmentandevaluationofthevoxelizedmodelfortumortransport.ThemodelwasdevelopedforpredictingtheinterstitialoweldanddistributionofMRvisibletracer(Gd-DTPA)intumorfollowingbolustailveininjection.TheresultsofthevoxelizedmodelwerecomparedwiththatobtainedfromapreviouslydevelopedCFDmodelingapproachusingunstructured 10

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meshes.Furthermore,simulatedGd-DTPAdistributionwithinthetumorwascomparedtoMR-measuredGd-DTPAconcentrationdata.Thevoxelizedmodelwastestedonthreetumorswithitspredictionscomparedagainstthenon-voxelizedmodelandexperimentalresults.Benetsofavoxelapproachincludelesslaborandlesscomputationaltime.Sensitivityofthemodeltochangesinarterialinputfunction(AIF)parameterswasalsoinvestigated.Forcomparison,statisticalanalysisandqualitativerepresentationofbothmodelresultswerepresented.Theanalysisindicatedsimilarityinboththemodelresultswithlowrootmeansquareerrorandhighcorrelationcoefcient.Thevoxelizedmodelcapturedfeaturesoftheoweldandtracerdistributionsuchasthehighinterstitialuidpressure(IFP)insidetumorandtheheterogeneousdistributionoftracer.PredictionsoftracerdistributionbythevoxelizedapproachresultedinlowerrorwhencomparedwiththeMR-measureddataovera1hrtimecourse.Theaccuracyofthevoxelizedmodelresultswithexperimentandnon-voxelizedmodelpredictionsweremaintainedacrossthetumors.ThesensitivityofthemodeltochangesinAIFparameterswasfoundtobesimilartothatofthepreviousmodelapproach.Secondly,thedevelopedvoxelizedmodelwasslightlymodiedforpredictingtheinterstitialoweldanddistributionofalbumintracerfollowingCEDatthehind-limbtumorinmice.ThespatiallyvaryingtransportpropertieswereobtainedviaDCE-MRIexperimentsfollowingsystemicdeliveryofMRvisibletracer,asmentionedinthepreviousparagraph.Apointsourcewasintroducedinthegoverningequationstomodelthelocalinfusion.Themodelwasabletocapturetheheterogeneous/asymmetrictracerdistributionandthelinearvariationofdistributionvolumewiththeinfusionvolume.Sensitivityofthemodeltochangesinhydraulicconductivityandcatheterplacementwereinvestigated.Thealbumindistributionwasfoundtobesensitivetoboththeparametersunderstudy.IncreasingthevaluesofthehydraulicconductivitymaploweredthetumorIFPandraisedthedistributionvolumewithinthewholeleg.Howeverwithinthetumor,thedistributionvolumedecreasedwithincreasingvalueofhydraulic 11

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conductivity,atlatertimepoints.Theinfusionatthetumor-hosttissueinterfaceresultedinlargerdistributionvolumecomparedtothatatthecenterandanteriorendofthetumor,underbaselineconditions.Withinthetumor,thedistributionvolumewasalmostidenticalforinfusionsattheinterfaceandcenterofthetumor.Thisimage-basedmodelthusservesasapotentialtoolforoptimizingpatient-speciccancertreatmentsandexploringtheeffectsofheterogeneousvasculatureontumortransport. 12

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CHAPTER1INTRODUCTION 1.1BackgroundTumororneoplasmisanabnormalmassoftissueusuallycausedduetogeneticmutations.Theycanbeclassiedintobenignandmaligntumorsdependingontheirabilitytoinvadeadjacenttissues.Maligntumorsinvadeanddestroyadjacenttissueswhilebenigntumorslackstheabilitytometastasize.Theonsetoftumorsareoftencharacterizedbyrapidformationofnewbloodvesselstosupplynutrientstothetumorcells.Angiogenesisintumortissuesisdifferentfromthatinthenormaltissue,tumorvasculatureisirregularandoftencharacterizedbyhighlytortuous,fenestrated,discontinuousvesselsandlargeavascularareas[ 1 25 33 40 42 ].Tumorsarealsoknowntoexhibitelevatedinterstitialuidpressure(IFP),whichisattributedtoitslackoflymphatics[ 60 ]alongwiththechaoticvasculature[ 7 14 ].ThereissignicantevidenceforelevatedIFPintumorsfromtheexperimentsperformedbyseveralresearchers[ 11 13 23 48 77 ].Theseabnormalitiesformvascularandinterstitialbarrierstothedeliveryofmacromoleculartherapeuticagentstotumors[ 6 29 ].Systemicdrugdeliverytotumorsisoftenknowntoresultininadequateandunevenuptake,therebypreventingthedrugfromreachingtherapeuticconcentrationsatthetargetsite.Thechaotictumormicrovasculatureleadstoheterogeneousextravasationofdrugs[ 20 ],therebyreducingitstherapeuticefciency.ThehighIFPincreasesthedrugtransportawayfromthetumorintonormaltissuesandreducesthetranscapillarytransport,causingundesirableside-effectsandlowerdruguptakeinthetumor.Overall,thesecharacteristicsofthetumormicroenvironmenthinderthesystemicdeliveryoftherapeuticagentstotumorcells.Localizeddrugdeliveryhasemergedasaplausiblealternativetosystemicdeliveryfortransportingmacromoleculartherapeuticagentstothetumors[ 18 22 56 72 74 ].Bydirectlyinjectingintothetumor,thiscircumventsthepreviouslymentionedvascularand 13

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interstitialbarriersandalsoreducestheside-effectsassociatedwithsystemicexposure.Amongsttheavailabletechniques,convection-enhanceddelivery(CED)appearspromisingbecauseatagiventimeitcanachievelargerdistributionvolumesthanbydiffusionalone[ 2 10 ].InCED,aninfusionpumpdeliversthedrugatconstantowrateorpressuretherebyutilizingthebulkowduetotheinfusionpressuredifference,todeliveranddistributemacromoleculestolargervolumesinthetissue.Sinceitsadvent,CEDhasbeenwidelyusedforinsitudeliveryofawiderangeofsubstancesincludingnanoparticles[ 50 ],liposomes[ 44 56 ],cytotoxins[ 57 ]andviruses[ 24 62 ].HoweverheterogeneousdistributionremainsasanobstacleforCEDtotumors. 1.2ObjectivesThefocusofthisthesiswastodevelopacomputationalmodelforpredictingdrugdistributionsfollowingCEDtotumors.Computationalmodelinghasgainedattentionpartlybecauseitcouldhelpinplanningandoptimizingpatient-specictreatments.Previousmathematicalmodelsoftransportintumorsassumetheoreticalvasculatureandsimplergeometries[ 7 59 64 66 ]neglectingthevascularheterogeneity.Giventhecriticalnatureofthemicrovasculatureintumordrugdelivery,ourgroupdevelopedaframeworkaccountingfortherealistictumormicrovasculatureusingthedataobtainedfromdynamiccontrastenhanced-magneticresonanceimaging(DCE-MRI)[ 53 ].Themodelaccountingfortheheterogeneoustumormicrovasculaturecouldpotentiallyhelpoptimizepatient-specictreatmentswithitsrealisticpredictions,andunderstandthebiophysicalIFPandinterstitialuidvelocity(IFV)changesduetoCED,whichareotherwisedifculttomeasureexperimentally.Themodelpreviouslydevelopedbyourgroupforthispurpose,involvedcomplexgeometricre-constructionwhichistimeconsumingandlaborintensive.Oneoftheobjectivesofthisprojectwastodevelopasimplermodelfortumortransportbasedonavoxelizedmodelingmethodology.Inthisapproach,tissuepropertiesandanatomicalboundariesareassignedonavoxel-by-voxelbasisusingMRIdata.Theseproperties 14

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arethenincorporatedintoaporousmediatransportmodeltopredictIFP,IFV,andtracertransport,therebyallowingforquickerbuildingofcomputationaltransportmodelsandrapidestimationoftracerdistribution.ThismodelavoidsthecomplexgeometricreconstructionastheMRdataisdirectlyimportedintothemesh. 1.2.1SpecicAim1Therstportionoftheprojectwasaimedatdevelopingandstudyingtheapplicabilityofthevoxelizedmodelfortumortransport.ThevoxelizedmodelwasdevelopedforpredictingthedistributionofsystemicallydeliveredMRvisibletracer(Gd-DTPA)inthehindlimbofmicethroughbolustailveininjection.Theresultsofthemodelwhichincludesthepredictedoweldandtracertransportwerecomparedwiththoseobtainedfromanon-voxelizedone[ 53 ].AvalidationstudyforthisapproachwasalsoconductedbycalculatingtheerrorbetweenGd-DTPAtissueconcentrationswithinthetumor,predictedusingavoxelizedmodelandthosemeasuredusingMRI.Sensitivityofthemodeltoarterialinputfunction(AIF)wasalsoinvestigated.Themodelwastestedwiththreesetsofanimaldata. 1.2.2SpecicAim2ThesecondportionoftheprojectwasfocusedonapplyingthedevelopedvoxelizedmodelforpredictingthedistributionofalbumintracerinthesametumorfollowingCEDasopposedtosystemicdelivery.Thegoverningowandtransportequationswereslightlymodiedtoaccountforthepointsourceandthevoxelizedmethodologywasusedtosolvethem.Forsensitivityanalysis,theeffectsofvaryinghydraulicconductivitymapsandcatheterplacement,onuidowandalbumintransportwereinvestigated.Infusionswerecarriedoutseparatelyattwodifferentsitesinthetumornamelyatthetumor-hosttissueinterfaceandanteriorendofthetumor,inadditiontothebaselinesimulationatthecenterofthetumor.Themodelcouldserveasapotentialtoolforoptimizingpatient-specictreatmentandstudyingtheeffectofheterogeneousvasculatureontumortransport. 15

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CHAPTER2DEVELOPMENTOFVOXELIZEDMODELFORSYSTEMICDELIVERYINSOLIDTUMORS 2.1OverviewAlthoughenormousadvancementshavebeenmadeinthediagnosisandtreatmentofcancers,targeteddrugdeliverytomalignanttumorsstillremainsachallenge.Transportofmacromoleculartherapeuticagentsinthetumormicrovasculatureplaysavitalroleinthetreatmentofsolidtumors[ 34 35 ].However,amajorobstacletosystemictransportintumorsisinadequateandunevenuptake,whichiswidelyattributedtotheheterogeneousarchitectureofthetumormicrovasculature[ 6 ].Tumorsareknowntocontainhighlytortuous,fenestrated,discontinuousvesselsandlargeavascularareas[ 1 25 33 40 42 ].Theresultingheterogeneousvasculatureleadstoirregularperfusion[ 9 32 ]whichcausesheterogeneousextravasationoftherapeuticagentsacrossthebloodvesselwall,dependingonthepressuredifferenceacrossthewallandspatiallyvaryingvascularpermeability[ 7 32 ].Anotherprofoundeffectofabnormalvasculargeometry,combinedwithalackoflymphatics[ 60 ]intumorsisthoughttobetheelevationofinterstitialuidpressure(IFP)[ 7 14 ].ExperimentsperformedbyseveralresearchershaverevealedincreasedIFPintumors[ 11 13 23 48 77 ].IthasbeenalsoobservedthatIFPisuniformthroughoutthecenterofthetumoranddropssharplyatitsperiphery[ 11 16 ].However,recentevidencealsosuggestsalesseruniformIFPinsidethetumors[ 26 ].AstudyconductedbyHassidandhiscolleaguesshowedthattheIFPinsideectopichumannon-small-celllungcancerincreasedfromtheperipheryinward,withahighplateauinsidethetumors.Withtheabsenceofpressuregradientsinthecenteroftumorineithercase,convectivetransportofdrugsisexpectedtobelessthanattheperipherywherepressuregradientsexist,resultinginahetergenousextravasation.ItisalsoexpectedthattheinterstitialuidowdrivenbytheIFPgradientisaffected.Interstitialuidvelocity(IFV)withinahumanneuroblastomawasexperimentallyfound 16

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toincreasefromthecentertowardstheperipheryofthetumor[ 16 ].Fromamodelingstudy,elevatedIFPisalsothoughttocausevascularconstrictionwhichmayleadtoreductionintumorbloodow[ 47 ],alsothepresenceofanecroticcorewasfoundtohaveanadverseaffectonthedistributionoflarge,slow-diffusingmolecules[ 8 ].Onthewhole,thesecharacteristicsofthetumormicroenvironmenthinderthesystemicdeliveryoftherapeuticagentstotumorcells.Hence,quanticationofextravasationanddrugdistributionisparamounttodevelopingsuccessfultreatmentstrategies.Previousmathematicalmodelsoftransportintumorsassumeeitheruniformlydistributedorregularpatternsofparallelandseriesbloodvessels[ 7 64 66 ]neglectingthevascularheterogeneity.JainandhiscolleaguesmodeledtheeffectsofuniformlydistributedleakybloodvesselsandminimallyfunctioninglymphaticsforthecaseofasphericalsolidtumorandshowedhowelevatedIFPleadstoheterogeneousextravasation[ 36 ].Pozrikidisdevelopedatheoreticalmodeltodescribethebloodowinwhich,tumormicrovasculaturewasgeneratedbybranchingcapillariesusingdeterministicandrandomparametersthusresultinginacapillarytree[ 54 ].Itshouldbenotedthattumorangiogenesispatternsinthesepreviousbloodvesselmodelsaretheoreticalandsimulatedbasedonrulestogeneratenetworkstructures.Recently,computationaluiddynamics(CFD)approacheswereusedbyourgroupandotherstostudytheextracellulartransportintumors[ 51 53 63 78 ].Inparticular,studiesconductedbyPishkoet.al.[ 53 ]accountedforrealistictumorvasculaturebyusingdynamiccontrastenhanced-magneticresonanceimaging(DCE-MRI)datatoestimatethespatialvariationoftransportproperties(ratetransferconstantbetweenplasmaandextracellularspace,Ktransandporosity,),whichweremappedintoaunstructuredmeshofaCFDmodelthatsolvesforIFP,IFVandtracertransport.Theresultsofthesestudiesareencouraging;however,thetime-intensivelaborinvolvedintheapproachmotivatedustodevelopasimplermodelfortumortransportbasedonavoxelizedmodelingmethodology.Earlier,thismethodologyhasbeenusedbyour 17

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grouptomodelinterstitialtransportintheratspinalcordandbrainduringtissueinfusion[ 38 39 ].Inthisapproach,tissuepropertiesandanatomicalboundariesareassignedonavoxel-by-voxelbasisusingMRIdata.ThesepropertiesarethenincorporatedintoaporousmediatransportmodeltopredictIFP,IFV,andtracertransport,therebyallowingforquickerbuildingofcomputationaltransportmodelsandrapidestimationoftracerdistribution.Thisvoxelmethodcircumventsthelaboriousgeometricreconstructioninvolvedinitsnon-voxelizedcounterpartbydirectlyimportingMRIdata.Inthisstudy,avoxelizedmodelforsystemictransportintumorswasdevelopedanditsresultswerecomparedwiththoseobtainedfromanon-voxelizedone[ 53 ].AvalidationstudyforthisapproachwasconductedbycalculatingtheerrorbetweenGd-DTPAtissueconcentrationspredictedusingavoxelizedmodelandthosemeasuredusingMRI.Themodelwasappliedtothreetumorsanditspredictionswerecomparedasdescribedpreviously.Sensitivityofthemodeltoarterialinputfunction(AIF)wasalsoinvestigated.TheshapeoftheAIFdeterminesthetimevariationoftheconcentrationofMRvisibletracerinbloodplasma.ThechoiceofAIFiscriticalinthepharmacokineticmodelingoftissuetranportproperties[ 28 ].AfasterAIFsignieshigherwash-outrateofthetracerandvice-versa. 2.2Methods 2.2.1EstimationofspatialvariationmapsofvascularleakinessDCE-MRIwasusedtoobtainvascularleakinessmaps.Thelowerhindlimbofananesthetizedmouse(C3H),inoculatedwithmurinesarcomacells(KHT)wasusedintheMRexperiment.SerialDCE-MRimages,consistingofaT1-weightedspin-echosequencewereacquiredbeforeandaftercontrastagent(tracer)administration.ThesameMRIdataaspresentedinPishkoet.al,[ 53 ]wasused.Thedataconsistedof9sliceswithamatrixof19296voxelsperslice.Thesizeofeachvoxelwas0.1040.1041mm3. 18

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DCE-MRImeasuresthetissueuptakeofaMRvisibletracer,whichinthiscaseisgadolinium-diethylene-triaminepenta-aceticacid(Gd-DTPA,MW590Da),afterasystemicbolustailveininjection.ThetracerconcentrationintissueandthemethodtocalculateKtransandwereidenticaltothatpresentedinPishkoet.al,[ 53 ].Thetracerdepositionintissuewasmeasuredbysignalenhancementwhichisdenedastheratioofthesignalintensitiesafterandbeforeinjectionofthetracer.Thisisthenmappedtotheactualtracerconcentrationinthetissue(Ct)byassumingalinearrelationshipbetweenCtandrelaxationtimes(T1&T2),andsubstitutingitintothestandardspin-echoequation[ 49 55 69 ].Afteralgebraicmanipulations,thefollowingexpressionforCtwasobtainedwithanaddedassumptionthattransverse-relaxationcontributiontosignalisunity, CMRI,t=1 R11 TRlnS(0) S(0))]TJ /F3 11.955 Tf 11.95 0 Td[(S(CMRI,t).(1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(TR=T10))]TJ /F5 11.955 Tf 19.03 8.09 Td[(1 T10(2)whereCMRI,tisthetissueconcentrationofGd-DTPAdeterminedbyMRI,R1isthelongitudinalrelaxivityofthetracerinwater,TRisthetimeforrecovery,S(CMRI,t)andS(0)arethesignalintensitiesattracerconcentrationsCMRI,tandzerorespectivelyandT10istheT1relaxtiontimewithouttracer.VascularleakinesscharacterisedbyKtransandwereestimatedusingatwo-compartmentkineticmodel[ 67 ].Thismodeldescribestheexchangeoftracerbetweentheplasmaandtissuecompartmentsineachvoxel.Thetwocompartmentmodelcanbedescribedby, dCt dt=KtransCp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Ktrans Ct(2)whereCtandCparetheconcentrationsofGd-DTPAintissueandbloodplasmarespectively. 19

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Thetracerconcentrationinthebloodplasma,CpfollowingabolusinjectioncanbedescribedbyanAIFofbiexponentialdecay: Cp(t)=da1e)]TJ /F9 7.97 Tf 6.58 0 Td[(m1t+a2e)]TJ /F9 7.97 Tf 6.59 0 Td[(m2t(2)wherea1,m1referstotheamplitudeandrateconstantofthefastequilibriumbetweenplasmaandextracellularspacerespectively,a2,m2referstotheamplitudeandrateconstantoftheslowcomponentofkidneyclearancerespectivelyanddisthedoseofthebolusinjection.Thebaselinewashoutparametersusedwereasfollows:a1=3.99kg=L,m1=0.114min)]TJ /F8 7.97 Tf 6.59 0 Td[(1,a2=4.78kg=L,andm2=0.0111min)]TJ /F8 7.97 Tf 6.59 0 Td[(1[ 67 76 ].InordertostudytheeffectsofAIFparametersinthemodel,twodifferentsetsofAIFparameterswerealsoused,whichareasfollows:a1=9.2kg=L,m1=0.23min)]TJ /F8 7.97 Tf 6.59 0 Td[(1,a2=4.2kg=L,andm2=0.05min)]TJ /F8 7.97 Tf 6.58 0 Td[(1describedthefastAIF[ 27 ];a1=13kg=L,m1=0.30min)]TJ /F8 7.97 Tf 6.59 0 Td[(1,a2=16kg=L,andm2=0.026min)]TJ /F8 7.97 Tf 6.59 0 Td[(1describedtheintermediateAIF[ 4 ].AqualitativerepresentationofthedifferentAIFsareprovidedinFigure 2-1 .Inthisgure,CpwasnormalizedsuchthattheinitialconcentrationisthesameforalltheAIFs.KnowingtheCpvaluesfromEquation( 2 ),Equation( 2 )wasthensolvedanalyticallytondanexpressionforCt(t)whichwasthenttedwiththeexperimentalvalues(CMRI,t)atearlytimepoints(20min)toobtaintheKtransandmaps.Thesemapswereincorporatedintotheporousmediatransportmodeltopredictthetracerdistributionatlatertimepoints. 2.2.2MathematicalModelThetissuecontinuumwasmodeledasaporousmediawithcontinuity[ 53 ]andmomentum(Darcy'slaw)equationsgivenby,r.v=Ktrans KtransJV V)]TJ /F3 11.955 Tf 11.95 0 Td[(Lp,lySL V(p)]TJ /F3 11.955 Tf 11.96 0 Td[(pL) (2)v=)]TJ /F3 11.955 Tf 9.3 0 Td[(Krp (2) 20

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wherevistheIFV, KtransistheaveragevalueofKtransovertumorandhosttissuevoxels,Lp,lyisthelymphaticvesselpermeability,SL=Visthelymphaticvesselsurfaceareaperunitvolumewhichwassettozerointumortissue,pistheIFP,pLispressureinthelymphaticvesselswhichwassettozeroandKisthetissuehydraulicconductivity.JV=VistheltrationrateofplasmaperunitvolumeoftissueintotheinterstitialspacewhichisgivenbyStarling'slawasfollows[ 68 ], JV V=LpS V(pv)]TJ /F3 11.955 Tf 11.96 0 Td[(p)]TJ /F4 11.955 Tf 11.96 0 Td[(T(v)]TJ /F4 11.955 Tf 11.96 0 Td[(i))(2)whereLpisthehydraulicconductivityofthemicrovascularwall,S=Visthebloodvesselsurfaceareaperunitvolume,pvisthevascularuidpressure,Tistheosmoticreectioncoefcientforplasmaproteins,v,iaretheosmoticpressuresoftheplasmaandinterstitialuid,respectively.Thersttermontherightsideofthecontinuityequation(Equation( 2 ))representstheuiduxacrossthemicrovascularwallperunitvolumeofthetissue.Thesecondtermaccountsforthelymphaticdrainagefrominterstitialspaceperunitvolumeoftissue.TransportofinterstitialGd-DTPAwassolvedusingtheconvectionanddiffusionequationforporousmedia[ 53 ], @Ct @t+v .rCt)]TJ /F3 11.955 Tf 11.96 0 Td[(Dr2Ct=KtransCp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Ct )]TJ /F3 11.955 Tf 11.96 0 Td[(Lp,lySL V(p)]TJ /F3 11.955 Tf 11.96 0 Td[(pL)Ct (2)whereDisthediffusioncoefcientforGd-DTPA.Thefollowingassumptionsaremadeintheaboveequation:thediffusioncoefcientisisotropicanduniformandthatthedispersioncoefcientismuchsmallerthanDandtherearenobindinginteractionsbetweenthemolecules.Thetermsontheleftsideoftheaboveequationreferstothetransient,convectionanddiffusionuxesrespectively.Thersttermontherighthandsideoftheequationdenotesthetransvascularsoluteexchangeandthesecondterm 21

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denotesthetraceroutuxduetothelymphatics.ThevaluesoftheparametersintheaboveequationsarelistedinTable 2-1 .TheMRimageconsistsofvoxelswhichnietherbelongtotumororhosttissue,i.e.exteriorvoxelswhichcorrespondtosurroundingair.Inthesevoxels,thesourcetermsincontinuityandtransportequations(Equations( 2 )and( 2 ))anddiffusivitywassettozero. 2.2.3ComputationalMethodThecontinuity,momentumandtracertransportequationsweresolvedusingtheCFDsoftwarepackage,FLUENT(version6.3,Fluent,Lebanon,NH).Forthe3Dcomputationaltissuemodel,arectangularvolume(20109mm3)enclosingthetumorwascreatedandmeshedwithquadrilateralelements(voxels)ofsizeequaltotheMRIresolution(0.1040.1041mm3)usingthemeshingsoftware(GAMBIT,Fluent,Lebanon,NH),withone-to-onemappingbetweentheCFDmeshandMRdata.Inthenon-voxelizedmodel[ 53 ],thegeometrywasmeshedusinganunstructuredgridwithapproximately2.7,2.5and2.3milliontetrahedralelementsforanimalsI,IIandIIIrespectively(Figures 2-2A and 2-2B ).Governingequationswerediscretizedwithacontrol-volumebasedtechniqueusingFLUENTasdonewiththenon-voxelapproach.WithinFLUENT,anuserdenedfunctionwasusedtoassignKtransandforeachvoxelinthemesh.Forcontinuityandtracertransportequations,auserdeneduxmacrowasusedtoaccountforthesourceterms.Standardpressureinterpolationschemewasusedtosolveforpressureandrstorderupwindmethodwasusedtosolveforvelocityandthetransportequations.TheSIMPLE(Semi-ImplicitMethodforPressure-LinkedEquations[ 3 ])pressure-velocitycouplingmethodwaschosenandconvergencecriterionwassetto1E-5.Initialconditionsfortracertransportassumednoinitialtracerinthetissue,Ct=0.Azerouidpressurecondition,p=0,wasappliedalongthecutendsandtheremainingouterboundariesofthegeometrywereassignedaswall. 22

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Thereisonedifferenceinthemodelingstrategybetweenvoxelizedandnon-voxelizedmodels,theimpermeabilityconditionalongtheskinboundaryinthevoxelizedmodelwasachievedbyassigninghydraulicconductivitytwoordersofmagnitudelowerthanthenormaltissueintheexteriorvoxels,whilethenon-voxelizedmodelimplementsitbydirectlyassigningthemwithawallboundaryconditionwithzeronormalux.Theassignmentoflowhydraulicconductivityintheexteriorvoxelscreatesamaterialthatisresistanttouidmotion.Forthechosenvalueofhydraulicconductivityattheexteriorvoxelsthemeanvelocityattheskinboundarywascalculatedtobeclosetozero(0.001m/s).Theeffectofchangingthebi-exponentialarterialinputfunction(AIF)parametersonthesolutionswasstudiedtounderstandthesensitivityofthevoxelizedmodelcomparedtoitscounterpart.ThesensitivityanalysiswasperformedonlyforanimalI.Apartfromthebaselinevalue,owandtransportfortwodifferentsetsofAIF(Figure 2-1 )parameterswassimulated.Fortheanalysis,tracerconcentrationwassimulatedfort20minandthedatawascomparedatdiscretetimepoints,t=5,10and20min. 2.2.4StatisticalAnalysisQuantitativemethodscomparedIFP,IFV,andtracerconcentrationintissuepredictedbyboththemodels.Suchanevaluationrequiredaone-to-onemappingbetweenboththemeshes(unstructuredandcartesian)whichwasmathematicallycumbersometoderive,henceasetofelementscommontoboththemesheswereidentiedbasedonthelocationoftheircellcentersandusedfortheanalysis.Assumingthevariationsindependentvariablesacrossdifferentnon-voxelelementswithinagivenvoxeltobesmall,thecriteriaformatchingwasthatthenon-voxelelementshouldliewithinthevoxelcomparedwith.Theabovecriteriaresultedinapproximately97%matchforanimalI,98%foranimalIIand98%foranimalIII.Afterndingthematchingelementsinboththemeshes,thevaluesofIFP,IFV,andtracerconcentrationintissue,intheseelementswereusedfortheanalysis. 23

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Additionally,aquantitativecomparisonwasconductedbetweentracerconcentrationintumortissuepredictedbybothmodelsandMRI-obtainedexperimentaldataatlatertimepoints.Gd-DTPAconcentrationsfromexperimentaldataandvoxelizedmodelweremappedtopointswithinthetumorboundaryofthenon-voxelizedmodeltocomparedistributionoftraceratagiventimepoint.Variousstatisticalmeasureswereusedtoascertainthesimilaritybetweenmodelpredictions.Theseincluderootmeansquareerror,correlationcoefcientanderrorhistogram. 2.2.4.1RootMeanSquare(RMS)Error,"Theerrorinthemagnitudeofdependentvariablesweremeasuredusingtherootmeansquareerrorwhichwasdenedasthesquarerootoftheaverageofthesquaresoftheerror.TheRMSerrorforIFPandIFVwerecomputedasshownbelow.ForIFV,inadditiontothemagnitude,theRMSerroroftheanglebetweenthetwovelocityvectorswerealsocalculated, "x=vuuut NPj=1xjvox)]TJ /F3 11.955 Tf 11.95 0 Td[(xjnvox2 N(2)WherexwasreplacedwithIFP,IFVmagnitude,andtracerconcentration,Nisthetotalnumberofmatchingelements,voxreferstovoxelvalueandnvoxreferstonon-voxelvalue. 2.2.4.2PearsonProductMomentCorrelationCoefcient(PMCC),rCorrelationcoefcientwasusedtomeasurethestatisticalrelationshipsbetweenboththeresults.PMCCisameasureoflineardependencebetweentwovariables.Itassumesthattherelationshipbetweenboththevariablescanbebestdescribedbyalinearfunctionanditisdenedastheratioofcovarianceofthetwovariablesandtheproductoftheirstandarddeviations.Thevalueofthecoefcientrangesfrom-1to1.Apositivesignindicatesthatthevariablesincreaseanddecreasetogether.Alargemagnitude(closeto1)impliesthatthereisastronglinearrelationshipbetweenboththevariables.Thisiscansummarizedasfollows, 24

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r=8>>>><>>>>:)]TJ /F5 11.955 Tf 9.3 0 Td[(1indicatesperfectnegativecorrelation0indicatesnocorrelation1indicatesperfectpositivecorrelation 2.2.4.3ErrorHistogramErrorhistogramsweregeneratedtoprovideagraphicalrepresentationoffrequencydistributionoferrorsinthedependentvariables,whichinthiscasewastheabsolutevalueofthedifferencebetweenthecomputedvaluesofvoxelizedandnon-voxelizedmodel.Asuitablerangefortheerrorwaschosenanddividedintoequalsizedintervals(orbins).Thenumberofoccurrencesoftheerrorwasthencalculatedforeachbinandrepresentedasabarplot. 2.3ResultsTumoroweldsandtracertransportobtainedusingbothcomputationalapproacheswerecomparedusingstatisticalanalysisforallthethreeanimaldatasets,andqualitativepresentationofthedependentvariables(IFP,IFVandtracerconcentrationintissue)foranimalIusingcontourplotsatthemid-slicesupplementedwithlineplotsalongthehorizontalandverticalbisectorsatthemid-slice(Figure 2-3 ).Foradetaileddescriptionofpredicteduidow,tracertransportandsensitivityanalysisinthenon-voxelizedtumormodel,thereaderisreferredtoPishkoet.al.,[ 53 ].TheIFPcontourandlineplotsforthetumorpredictedbythevoxelizedandnon-voxelizedmodelareshowninFigures 2-4A to 2-4D .ThevoxelizedmodelpredictedelevatedIFPinsidethetumor,pressurereachedpeakvalue(0.73-1.62kPainanimalI,0.32-0.71kPainanimalIIand0.39-0.87kPainanimalIII)atthetumorcoreandrapidlydecreasedatthetumorboundary.Asexpectedwithinthetumor,predictedpressuregradientswerelowestclosetothetumorcenter(14.2,38.5and11.2Pa/mminanimalI,IIandIIIrespectively)andhighest(1136.5,579.6and763.6Pa/mminanimalI,IIandIIIrespectively)nearitsperiphery.Thepressurepatternwascapturedbythevoxelizedmodel.However,thelineplotsclearlyindicatedadifferenceinthe 25

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predictedpressuresbetweenboththemodels.ThemagnitudeofpeakIFPpredictedbythevoxelizedmodelwasfoundtobe15%higherthanthatofitscounterpartinanimalI,29%inanimalIIand18%inanimalIII.DespitechangesinpredictedIFPbetweenboththemodels,theIFVcontoursandlineplots(Figures 2-5A to 2-5D )indicatedthatthedistributionspredictedarequalitativelysimilarwithhighestvelocities(0.75,0.40and0.42m/sinanimalI,IIandIIIrespectively)occurringalongthetumorboundarynearthecutends.ThecomputedIFVvalueswerefoundtobelowerinsidethetumor(0.03,0.03and0.01m/sinanimalI,IIandIIIrespectively).Thelowvelocityregionswerealsoobservedfarawayfromthetumorboundary.InterstitialdistributionofGd-DTPAtracerwassimulatedatvarioustimes(t=5,30and60min)afterinfusion.Thepredictedtracerdistributionofboththemodelsandtheactualexperimentaldata,washeterogeneouswithhighconcentrationregions(0.4,0.18and0.19mMinanimalI,IIandIIIrespectivelyatt=5min)outsidethetumor(Figure 2-6 ).Itcanalsobeobservedthatlowesttracerconcentration(0.03,0.05and0.03mMinanimalI,IIandIIIrespectivelyatt=5min)occurswithinthetumor.Thelineplots(Figure 2-7 )showsthatthetracerextravasationappearstobelessaffectedbythedifferencesintheoweldpredictedbyboththemodels.Conformingwiththestatisticalndings,theaccuracyofvoxelizedmodel'spredictionwithrespecttoitsnon-voxelcounterpartwasmaintainedforallthetimessimulated.Astimeproceeds,tracerconcentrationwasreducedandthedistributionbecamemoreuniform.ThestatisticalparameterscomparingboththemodelpredictionsforallthethreeanimalsarelistedinTable 2-2 .Thestatisticsofthemodelresultsappearedsimilaracrosstheanimals.ThePearsoncoefcientforIFPwashigh(r>0.7)indicatingsimilarpatternsinboththemodelpredictions.ThevalueofitsRMSerrorreectedthedifferenceinthepeakpressurespredictedbyboththemodels.ThelowRMSerrorinIFVandthehighcorrelationcoefcient(r>0.7)showedareasonabledegreeof 26

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similaritybetweenboththemodelpredictions.TheRMSerrorintracerconcentrationwasmaximumatinitialtimepointsanddecreasesthereafterwithtime.However,correlationcoefcientsdidnotchangemuchwithtime.Errorhistogramsforowandtracertransport(Figure 2-8 )followedanexponentialdistributionwithpeakaroundzero. 2.3.1SensitivityAnalysisResultsofsensitivityanalysisarepresentedintermsoflineplotsalongtheverticalbisectorofthemid-sliceforoweldandtracerconcentration.Similartothebaselineresults,theIFPpatternpredictedbythevoxelizedmodelmatchedwiththatofthenon-voxelmodelforallAIFs,althoughtherearedifferencesinthepredictedmagnitude(Figures 2-9A and 2-9B ).ThepredictedpressuresforintermediateAIFwerefoundtobecloselymatchingforboththemodels.TheIFVpredictedbynon-voxelizedmodelmatchedwellwiththatofitscounterpart(Figures 2-9C and 2-9D ).ConcentrationsofGd-DTPApredictedbythevoxelizedmodelroughlyfollowedthenon-voxelizedone(Figure 2-10 ).TheaccuracyofthepredictedconcentrationdidnotseemtochangewithAIFsandtime.StatisticsofthesensitivityanalysisareprovidedinTable 2-3 .ThecorrelationcoefcientsforIFPacrosstheAIF'swerealmostidenticalalthoughtheRMSerrorsweredifferentreectingthedifferencesinpredictedpressures.HighestandlowestpressureswereobservedforthebaselineandintermediateAIFrespectively.ItwasobservedthatPMCCsandRMSerrorsinIFVweresimilarfortheintermediateandfastAIFs.TracerconcentrationstatisticsalsoexhibitedasimilarbehaviourwithalmostidenticalRMSerrorvaluesandPMCCsacrosstheAIFs.Withincreasingtime,theRMSerrordecayedforallthecasesalthoughPMCCsremainsimilar. 2.3.2ValidationStudyQualitatively,asimilarpatternofGd-DTPAdistributionandwashoutwasobservedforthevoxelizedmodel,non-voxelizedmodel,andexperimentaldataoverthecourseof1hr(Figure 2-6 ).Highconcentrationregionswereobservedoutsidethetumorandat 27

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theedgeofthetumorjustwithinthetumorboundary.Washoutratewascomparedbycalculatingvolume-averagedGd-DTPAconcentrationwithinthetumorforvarioustimepointsandttingthedataintoamono-exponentialfunction(Table 2-4 ).Thevoxelizedandnon-voxelizedmodelbothcomparedwellwiththeexperimentaldata.RMSerrorwascalculatedforbothmodelsthroughouttheentiretumorvolumeaswellaserrorfrequencyhistograms(Figure 2-11 )toillustratethecomparisonofthemodelswiththeexperimentaldatainspaceandtime.Boththevoxelizedandnon-voxelizedmodelsshowedlowRMSerrorandhigherrorfrequencyclosetozero.However,foranimalIIItherewasslightlyhigherRMSerrorinthevoxelizedandnon-voxelizedmodelpredictionswiththeexperiment,eventhoughthewashoutratewasveryclosewiththeexperimentaldata. 2.4DiscussionAvoxelizedmodelingapproachwasusedtostudythetransportofGd-DTPAfollowingsystemicinjectionintumors.BenetsofthismethodologyincludeeasierandmorerapidbuildingofcomputationalporousmediatransportmodelscomparedtotraditionalCFDapproacheswhichinvolvescomplexgeometricreconstruction.Thusthevoxelmodelislesslaborintensiveandpotentiallysimplertoimplement.Spatially-varyingtissuetransportpropertiesandrealisticanatomicaltissuegeometrieswereincorporatedintoathree-dimensional,image-basedcomputationalmodel.Theporousmediasimulationpredictedinterstitialuidpressure,interstitialuidvelocity,andtracertransportthroughthetissueinterstitium.Theseresultswerecomparedwiththatobtainedusinganon-voxelapproach[ 53 ].ThesensitivityofthevoxelizedmodelfordifferentAIFswasinvestigatedandcomparedwiththenon-voxelmodel.Thevoxelizedandnon-voxelizedmodel'spredictionsoftracerdistributionwithinthetumorwerecomparedtoMRI-determinedtracerdistributionandthevoxelizedmodelwasfurtherevaluatedwithadditionalanimaldata. 28

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ThevoxelizedmodelpredictedelevatedIFPconformingwiththeexperimentalobservations[ 11 13 23 48 77 ]andpreviousmodelingresults[ 7 8 53 78 ].However,itcanbefoundthattheIFPpredictedbythevoxelizedmodelwashigherthanthatpredictedbythenon-voxelizedmodel.ThevalueofRMSerrorreectsthisdifferenceasitcanbeinterpretedasthestandarddeviationbetweentwovariables.Thisdiscrepancycanbeexplainedbythedifferencesinthetumorvolumeinboththemodels.Thetumorvolumeapproximatedbythevoxelizedmodelwasfoundtobehigherthanthatofthenon-voxelmodel(39%higherinanimalI,83%inanimalIIand36%inanimalIII).Thisisduetothedifferencesinthemeshingstrategyofthemodels.Thenon-voxelizedmodelusedvariablesizedelements(unstructuredmesh)whichlikelyapproximatesthetumorvolumeslightlybetterthanitsvoxelcounterpartwhichreliesonlyonxedsizeelements(cuboids).ThiseffectwasparticularlymorepronouncedinanimalIIwhichhasthesmallesttumorvolumeofall.SinceIFPvaluesarefoundtobecorrelatedwiththetumorvolume,withhigherIFPforlargetumors[ 23 ],thevoxelizedmodelwithhighertumorvolumeisexpectedtohaveIFPhigherthanthenon-voxelizedone.ThelowerdifferencesobservedinthepredictedIFPbyboththemodelsforanimalsIIandIIIcomparedtoanimalI,couldbeattributedtotheiractualdifferencesinthetumorvolumeapproximatedbyboththemodels.TheadditionaltumorvolumeforanimalsIIandIIIinthevoxelizedmodelwasanorderofmagnitudelowerthanthatinanimalI,thusresultinginsmallerchange.ThesedifferencesinthepredictedIFPbyboththemodelsdoesnothavemucheffectonthepredictedextracellularowwhichisdrivenbytheIFPgradientwhichwassimilarinboththecases.Itshouldbenotedthatthecorrelationcoefcientcanalsobeinterpretedasthedegreeofsimilaritybetweentheslopesoftwovariables,inotherwordsasimilarityindexforthegradientsofthevariables.FromTable 2-2 ,itisclearthattheyarehighforallthethreeanimals,therebyindicatingthehighdegreeofsimilarityintheIFPgradientcomputedbyboththemodelsthussupportingthisprevious 29

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argument.TheexponentialerrordistributionforIFPwithhighfrequencynearzeroerrorimpliesthedecayingnatureofthenumberofvoxelswithhighererrors.TheIFVpredictedbythevoxelizedmodelreectspreviousexperimentalnding[ 16 ].IntheexperimentalstudyconductedbyDiRestaandhiscolleagues,IFVinhumanneuroblastomawasfoundtoincreasefromthecentertowardstheperipheryofthetumor.ThisisduetothehighpressuregradientatthetumorboundarywhichincreasestheIFV.Ontheotherhandamoreuniformpressuredistributioninthetumorcoreleadstolowvelocitiesinthoseregions.StatisticalparametersobtainedforIFVindicateahigherdegreeofsimilaritybetweenthepredictionsofboththemodels.Asmentionedpreviously,theIFVdrivenbytheIFPgradientislessaffectedbythechangesinthepredictedpressure.Thisisalsoreectedintheirerrorhistograms,itcanbeobservedthattheerrordecaysmorerapidlythanthatofIFP,thusindicatingthehighaccuracyofthevoxelizedmodelinpredictingIFV.DistributionofGd-DTPAwasheterogeneousduetospatiallyvaryingdepositionandlimitedinterstitialtransportbydiffusionandconvection.ThelowtracerconcentrationinsidethetumorisconsistentwiththereduceduidltrationandhighIFP.Astheconcentrationisadvectedthroughthevelocityeld,itscorrelationcoefcientissimilartothatobtainedforIFV.Errorhistogramsalsoreectthisbehaviour,astrongpeakaroundzeroclearlyshowsthereliabilityofvoxelizedmodelinpredictingthetracerconcentrationdespitesomechangesinthepredictedoweld.Itshouldbenotedthat,althoughtherearedifferencesbetweentheresultsobtainedthroughboththemodels,thevoxelizedmodelfaithfullycapturesthetracerextravasationwhichisessentialforanydrugdeliverymodel.Insensitivityanalysis,theeffectsofvaryingAIFparameterswerealsoinvestigated.Ithasbeenfoundthattheowandtransportaresensitivetotheseparameters[ 53 ].ChangesintheowandtransportcanbemainlyattributedtothedifferencesintheKtransandmaps.Thesensitivityofvoxelizedmodelcomparedwiththatofits 30

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counterpartseemstobethesamefordifferentAIFsasindicatedbysimilarcorrelationcoefcientsandRMSerrors.Thisanalysisalsodemonstratestheapplicabilityofthemodelinadiversesetofconditions.Overall,thesensitivityofvoxelizedmodelwassimilartoitsnon-voxelizedcounterpart.ThevalidationstudyrevealedthattheGd-DTPAdistributionresultsobtainedvianon-voxelizedandvoxelizedmodelswereconsistentwiththeexperimentalobservations.Thisbecamemoreclearwhenthevoxelizedmodelpredictionsweresimilartonon-voxelizedpredictionsandexperiment,acrossthetumors.TheslightlyhighRMSerrorinanimalIIIcouldbeduetoerrorsinAIFparameterswhichvariesacrossthetumorsandtowhichtheresultsaresensitive.However,thewashoutratewasaccuratelypredictedbythevoxelizedmodel.Thisoutcomelendscredencetotheusageofvoxelizedporousmediatumormodelsforpredictionsoflowmolecularweighttracersanddrugdistribution.However,matchingthemodelingresultswiththeactualexperimentalvaluesisdifcultduetothepresenceofalargenumberofmodelparameterswhichneedtobedeterminedexperimentally.Alsothedifferencesinthegridsizesbetweenthenon-voxelizedmodel(approximately2.3-2.7millionelements)andvoxelizedmodel(approximately165,000elementswhichisjust6-7%ofthatinthenon-voxelmesh)mayalsoaccountfordiscrepanciesbetweenthem.ThelowresolutionofthevoxelizedmodelisduetolimitationsofMRIresolutionasitsdataaredirectlymappedintothemodel.Non-voxelizedmodelsontheotherhandaremoreexibleinthisaspectastheydonotdirectlymaptheMRdata,thusallowingforvariableresolution.Thenon-voxelmodel[ 53 ]wasalsousedforextensivesensitivityanalysisrequiringittocapturesteeppressuregradientsatthetumorboundary,hencethemeshsizewasincreasedforattainingconvergenceinFLUENT.Thecurrentvoxelizedmodelwasaimedatgaininganoverallunderstandingoftheuidowandtransportintumors,andprovidingareliablealternativetothenon-voxelizedapproach. 31

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Highcorrelationcoefcientsbetweenthevoxelizedandnon-voxelizedmodelresultsindicatesthatboththeresultsareinagreementwitheachother.However,thereissomedisparityintheresultsespeciallytheIFPwhichcanbeattributedtooneofthebasicdifferencesbetweenboththeapproaches,meshstructure.Thevoxelizedmodelusesanuniformandrectangularmeshwhilethenon-voxelizedmodelusesanunstructuredmesh.Thetypeofmeshusedcanaffectthesolutionintwoways:(1)tumor/hosttissuevolumeapproximation,(2)resolution.Usingthecartesianmesh,thevoxelizedmodelapproximatesthetumorandhosttissuevolumeswithrectangularelementstherebyneglectingcurvatureattissueboundarieswhilethenon-voxelizedmodelwithitsvariablesizeelementscanaccountforthis.Thisresultsinslightdifferencesintumorandhosttissuevolumeswhichinturnaffectsthesolutionasthetumorshapeandsizeareimportantfactorsdeterminingtheinterstitialuidow[ 17 58 ].Themeshdensityalsoaffectsthesolutionasitdeterminesthediscretizationofthedomainwithbetterresolutioninnermeshesandvice-versa.Inthisaspect,thevoxelizedmodelhasmuchlessermeshdensitycomparedwithitsnon-voxelizedcouterpartresultinginalowerresolutionasmentionedearlier.Itshouldhoweverbenotedthattheusageofverynemeshesiscomputationallyintensiveandtimeconsuming.Despitethesedifferences,voxelizedmodelwasstillabletocapturekeyfeaturesintheowandtransportthusmakingitaattractivealternatecandidatefortumormodeling. 32

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Table2-1. TissueandvascularparametersusedforsimulatingdistributionofGd-DTPAfollowingbolustailveininjectionatthehindlimbtumorinamice. VariableDescriptionValueReferences Lp(m=Pa.s)Vesselpermeability210)]TJ /F8 7.97 Tf 6.59 0 Td[(11t;310)]TJ /F8 7.97 Tf 6.58 0 Td[(12n;[ 53 ]S=V(m)]TJ /F8 7.97 Tf 6.59 0 Td[(1)Microvascularsurfaceareaperunitvolume20000t;7000n[ 7 ]Lp,lySL=V(m)]TJ /F8 7.97 Tf 6.58 0 Td[(1)Lymphaticltrationcoefcient110)]TJ /F8 7.97 Tf 6.59 0 Td[(7[ 53 ]K(m2=Pa.s)Hydraulicconductivity1.910)]TJ /F8 7.97 Tf 6.59 0 Td[(12t;3.810)]TJ /F8 7.97 Tf 6.59 0 Td[(13n[ 53 ]7.710)]TJ /F8 7.97 Tf 6.59 0 Td[(15epv(Pa)Microvascularpressure2300[ 53 ]i(Pa)Osmoticpressureininterstitialspace3230t;1330n[ 53 ]v(Pa)Osmoticpressureinmicrovasculature2670[ 53 ]T(Pa)Averageosmoticreectioncoefcientforplasma0.82t;0.91n[ 53 ]D(m2=s)DiffusioncoefcientofGd-DTPA110)]TJ /F8 7.97 Tf 6.59 0 Td[(9[ 53 ] t-tumor,n-normaltissue,e-exterior. 33

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Table2-2. Statisticalparametersobtainedwhilecomparingvoxelizedandnon-voxelizedmodelresultsforthebaselinesimulationinthreeanimals. VariableQuantityAnimalI(Baseline)AnimalIIAnimalIII "r"r"r IFPMagnitude167.09Pa0.9544.45Pa0.9738.72Pa0.97IFVMagnitude0.07m/s0.810.01m/s0.890.02m/s0.92Direction23.100.7225.640.5816.440.74CtAtt=5min0.10mM0.790.10mM0.710.08mM0.74Att=30min0.05mM0.790.06mM0.750.04mM0.77Att=60min0.04mM0.780.03mM0.770.03mM0.77 34

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Table2-3. Statisticalparametersobtainedwhilecomparingvoxelizedandnon-voxelizedmodelresultsforintermediateandfastarterialinputfunctioninanimalI. VariableQuantityIntermediateAIFFastAIF "r"r IFPMagnitude47.48Pa0.99119.88Pa0.99IFVMagnitude0.12m/s0.780.14m/s0.78Direction18.220.7018.450.70CtAtt=5min0.10mM0.720.07mM0.68Att=10min0.07mM0.750.07mM0.72Att=20min0.06mM0.750.03mM0.74 35

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Table2-4. Comparisonoftracerwashoutratesandrootmeansquareerrorintracerconcentrationwithinthetumorvolumebetweenvoxelizedandnon-voxelizedmodelresultswithexperimentinthreeanimals. Washoutrateofvolume-averagedGd-DTPAconcentrationwithintumorvolume(min)]TJ /F8 7.97 Tf 6.58 0 Td[(1)RMSerrorforconcentrationwithintumorvolume(mM) AnimalCaset=5mint=30mint=60min IExperimental-0.031VoxelizedModel-0.0220.0720.0390.031Non-voxelizedModel-0.0200.1200.0640.048IIExperimental-0.020VoxelizedModel-0.0210.0700.0600.036Non-voxelizedModel-0.0220.0890.0620.040IIIExperimental-0.025VoxelizedModel-0.0220.2950.2580.034Non-voxelizedModel-0.0220.2970.2590.037 36

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Figure2-1. NormalizedconcentrationoftracerinbloodplasmaapproximatedbydifferentAIFsusedforsensitivityanalysis 37

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A BFigure2-2. CFDcompatiblemeshes.(A)Schematicofvoxelized(cartesian)mesh(B)Unstructuredmeshofreconstructedhindlimb.Includestumor(lightgreen),skin(green),cutends(yellow),andrepresentationofmid-slice(darkblue). 38

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A BFigure2-3. Horizontalandverticallinesusedforplottingtheoweldandtracertransportinvoxelized(A)andnon-voxelized(B)models 39

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A B C DFigure2-4. ContoursofIFPpredictedby(A)voxelizedmodel(B)non-voxelizedmodel.Tumorandskinboundariesareoverlaidonthecontours.Alsoincluded,lineplotscomparingthepredictedIFP(C&D)byboththemodelsalongthehorizontalandverticalbisectorsinthemid-slicerespectively.Thetumorandskinboundariesarerepresentedbydashedanddash-dotlinesrespectively. 40

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A B C DFigure2-5. ContoursofIFVpredictedby(A)voxelizedmodel(B)non-voxelizedmodel.Tumorandskinboundariesareoverlaidonthecontours.Alsoincluded,lineplotscomparingthepredictedIFV(C&D)byboththemodelsalongthehorizontalandverticalbisectorsinthemid-slicerespectively.Thetumorandskinboundariesarerepresentedbydashedanddash-dotlinesrespectively. 41

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Experimental t=5min t=30min t=60min Voxelized Non-voxelized Figure2-6. Comparisonoftracerconcentrationcontours.Voxelizedandnon-voxelizedmodelcomparedwithMR-derivedtissueconcentrationatt=5,30,and60min.Tumorandskinboundariesareoverlaidonthecontours. 42

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A B C D E FFigure2-7. Lineplotscomparingthepredictedtracerconcentrationinthetissuebyboththemodelswithexperiment,alongthehorizontalandverticalbisectorsofmid-sliceatt=5(A&B),30(C&D),60(E&F)minrespectively.Thetumorandskinboundariesarerepresentedbydashedanddash-dotlinesrespectively. 43

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A B C D E FFigure2-8. ErrorHistogramsforow(A,B&C)andtransportatt=5(D),30(E)and60(F)minsinbaselinesimulationforvoxelizedmodelwithrespecttonon-voxelizedmodel 44

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A B C DFigure2-9. LineplotscomparingtheIFP(A&B)andIFV(C&D)predictedbyboththemodelsfortwodifferentAIFparametersets(intermediateandfast)alongtheverticalbisectorofmid-slicerespectively.Thetumorandskinboundariesarerepresentedbydashedanddash-dotlinesrespectively. 45

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A B C DFigure2-10. LineplotscomparingthetracerconcentrationinthetissuepredictedbyboththemodelsfortwodifferentAIFparametersets(intermediateandfast)alongtheverticalbisectorofmid-sliceatt=5(A&B)and20(C&D)minrespectively.Thetumorandskinboundariesarerepresentedbydashedanddash-dotlinesrespectively. 46

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Voxelized t=5min t=30min t=60min Non-voxelized Figure2-11. ErrorHistogramsfortracerconcentrationwithinthetumorforvoxelandnon-voxelmodelresultswithrespecttotheexperimentaldataatt=5,30and60min. 47

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CHAPTER3APPLICATIONOFVOXELIZEDMODELFORCONVECTION-ENHANCEDDELIVERYINAHINDLIMBTUMOR 3.1OverviewCancertreatmentsbasedonsystemicdeliveryoftherapeuticagentsareoftenhinderedduetothepoorandunevenuptakeofthedrugswithinthetumor.Theuniquecharacteristicsofthetumormicroenvironmentareknowntobeanimportantfactoraffectingtheefcacyoftheanti-cancertreatmentssuchaschemotherapy.Tumorsareknowntoexhibitelevatedinterstitialuidpressure(IFP)[ 11 13 23 48 77 ]andirregularmicrovasculature[ 19 31 33 ]whichleadstoinadequateuptakeandheterogeneousextravasationofdrugs[ 20 ]respectively,consequentlyloweringtheirtherapeuticindex.Intherecentyears,localizeddrugdeliveryhasemergedasaplausiblealternativetosystemicdeliveryfortransportingmacromoleculartherapeuticagentstothetumors[ 18 22 56 72 74 ].Bydirectlyinjectingintothetumor,thiscircumventsthepreviouslymentionedvascularandinterstitialbarriersandalsoreducestheside-effectsassociatedwithsystemicexposure.Amongsttheavailabletechniques,convection-enhanceddelivery(CED)appearspromisingbecauseatagiventimeitcanachievelargerdistributionvolumesthanbydiffusionalone[ 2 10 ].InCED,aninfusionpumpdeliversthedrugatconstantowrateorpressuretherebyutilizingthebulkowduetotheinfusionpressuredifference,todeliveranddistributemacromoleculestolargervolumesinthetissue.Sinceitsadvent,CEDhasbeenusedforinsitudeliveryofawiderangeofsubstancesincludingnanoparticles[ 50 ],liposomes[ 44 56 ],cytotoxins[ 57 ]andviruses[ 24 62 ].ExperimentalstudiesonCEDofliposomesintobraintumor(glioma)inratswereencouraging,itwasfoundthatthetechniqueeffectivelydistributedtheliposomesinthetumorandthesurroundingnormaltissue[ 56 ].AbroadanisotropicdistributionwasreportedtohaveresultedfromCEDofcytotoxinsintohumangliomas[ 57 ].SuchanasymmetricdistributionwasalsoreportedbyBoucherandhiscolleaguesin 48

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theirstudyinvolvingintratumoralinfusionofEvansblue-albumininsalineintosarcomaHSTS26T[ 12 ].ItshouldhoweverbenotedthatsphericallysymmetricdistributionsforcolonadenocarcinomaLS174Twerealsoreportedintheirstudy.ComputationalmodelingofCEDhasgainedattentionrecentlypartlybecauseitcouldhelpinplanningandoptimizingpatient-specictreatments.EarliertheoreticalmodelswerefocusedonpredictingdrugdistributionsfollowingCEDinmediumslikeagarosegels/braintissue[ 15 38 39 43 ].Fortumors,SmithandHumphreydevelopedatheoreticalmodelinwhichinfusionsinasphericaltumorwithanecroticcorewassimulated[ 59 ].Amainobjectiveoftheirstudywastoanalyzetheeffectoftransvascularuidexchangeontheoweldduringtheinfusion.Theyfoundthattheoweldwasverysensitivetotheratioofvascularconductivityandhydraulicconductivity,andinfusionclosetothetumorwasretardedbytheoutwardow.Weinbergandhiscolleaguesdevelopedaniteelementmodeltopredictthedistributionofdoxirubicinfollowingintratumoraldelivery[ 75 ].However,theconvectiveeffectsinthemodelwerereplacedwithaeliminationcoefcientinsteadoftheactualinterstitialuidvelocity(IFV).Itshouldbenotedthatthesemodelsutilizedtheoreticaltumormicrovasculatureandsimpliedtumorgeometries.Patient-speciccomputationalporousmediamodels,incorporatingrealisticgeometriesandspatiallyvaryingtransportpropertiesobtainedthroughMRI,forpredictingdrugdistributionshavebeendevelopedbyourgroup[ 38 39 45 51 53 78 ].Fortumorsinparticular,ourgroupdevelopedaframeworkwhichaccountsfortheactualtumormicrovasculaturebyusingDCE-MRIdatatoestimatethespatialvariationoftransportproperties(ratetransferconstantbetweenplasmaandextracellularspace,Ktransandporosity,)whichwereincludedinaporousmediamodeltosolveforowandtransportusingcomputationaluiddynamics(CFD)techniques[ 45 53 78 ].Inthisstudy,thismethodwasusedtopredictthedistributionofalbumininamurinesarcomafollowingCEDasopposedtosystemicdeliveryintheaforementionedpapers. 49

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Inparticular,CFDsimulationswerecarriedoutbasedonavoxelizedmodelingapproachdescribedinthepreviouschapter,whereitwasshownthatthepredictedoweldandtransportusingthisapproachwassimilartothatofamoretraditionalapproachbasedonunstructuredmeshes.Earlier,thismethodologyhasalsobeenusedbyourgrouptomodelCEDinratspinalcordandbraintissue[ 38 39 ].Inthisapproach,anisotropictissuepropertiesandanatomicalboundariesareassignedonavoxel-by-voxelbasisusingMRIdata.ThesepropertiesarethenincorporatedintoaporousmediatransportmodeltopredictIFP,IFVandtracerconcentrations.Thesevoxelizedmodelsallowforquickerbuildingofcomputationaltransportmodelsandrapidestimationofconcentrationproles.Inthisstudy,aDCE-MRIbasedvoxelizedmodelwasdevelopedforpredictingalbumintracerdistributionfollowingCEDinthelowerlimbofamouse(C3H)inoculatedwithmurinesarcomacells(KHT).Themodelaccountingfortheheterogeneoustumormicrovasculaturecouldpotentiallyhelpoptimizepatient-specictreatmentswithitsrealisticpredictions,andunderstandthebiophysicalIFPandIFVchangesduetoCED,whichareotherwisedifculttomeasureexperimentally.Asensitivityanalysiswasperformedtostudytheeffectsofvaryinghydraulicconductivitymapsandcatheterplacementonuidowandalbumintracertransport.ThiswasdonetounderstandthesensitivityofthemodelandrelatethemtokeyfactorscontributingtoCED.ThechoiceofvaryinghydraulicconductivityisbecauseofitsdirectinuenceonthetumorIFPandconvectiveoweldinintratumoralinfusions.ThehighervaluesofhydraulicconductivitywerethoughttoreduceIFPtherebyincreasingtheltrationofuidsandextravasationofmacromolecules[ 12 ].TheeffectofcatheterplacementwasknowntobeveryimportantinCED[ 2 ].Studiesinvolvinginfusionsindifferentsitesinthebrainhaverevealedthepresenceofaoptimalsiteforachievingmaximumdistributionvolumeatthetargettedarea[ 39 43 ].Inthecurrentstudy,infusionswerecarriedoutseparatelyattwodifferentsitesinthetumornamelyatthetumor-hosttissueinterfaceandanteriorendofthe 50

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tumor,inadditiontothetumorcenter.Thereasonforchoosingasiteatthetumor-hosttissueinterfacewasbecauseofthepresenceofhigherconvectiveeffectsinthatregionduetothesuddendecreaseinIFPwhichcouldresultinhigherIFV.Thechoiceofaninfusionsiteattheanteriorendandcenterofthetumorwastostudythedistributionatvariouspositionsinsidethetumor. 3.2Methods 3.2.1MathematicalModelThestudywasdividedintotwoparts:Firstthespatially-varyingtransportpropertiesoftheKHTmurinesarcomawerefoundthroughDCE-MRIfollowingbolustailveininjectionofMRvisibletracergadolinium-diethylene-triaminepenta-aceticacid(Gd-DTPA,MW590Da).ThemethodsforobtainingDCE-MRIderiveddatasuchasGd-DTPAconcentrationintissue,ratetransferconstant(Ktrans)maps,andporosity()mapsareidenticaltothoseinPishkoet.al.,[ 53 ].ThesecondpartinvolvesincorporatingtheabovecalculatedvariabletransportpropertiesintothecomputationalporousmediamodelforowandtransportbyCED.ThetissuecontinuumwasmodeledasaporousmediaandthegoverningequationsweresolvedateachvoxelafterassigningtheirrespectiveKtransandvalues.Thecontinuityequationisgivenby,r.v=Q V infAttheinfusionvoxel (3)=Ktrans KtransJV V)]TJ /F3 11.955 Tf 11.95 0 Td[(Lp,lySL V(p)]TJ /F3 11.955 Tf 11.96 0 Td[(pL)Atallothervoxelsintumorandhosttissue (3)wherevistheIFV,Qistheinfusionowrateofalbumin,V infisthevolumeoftheinfusedvoxel, KtransistheaveragevalueofKtransinhostandtumortissuevoxels,Lp,lyislymphaticvesselpermeability,SL=Visthelymphaticvesselsurfaceareaperunitvolumewhichwassettozerointumortissue,pistheIFPandpLispressureinthelymphaticvesselswhichwassettozero.JV=Vistheltrationrateofplasmaperunitvolumeof 51

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tissueintotheinterstitialspacewhichisgivenbyStarling'slawasfollows[ 68 ], JV V=LpS V(pv)]TJ /F3 11.955 Tf 11.96 0 Td[(p)]TJ /F4 11.955 Tf 11.96 0 Td[(T(v)]TJ /F4 11.955 Tf 11.96 0 Td[(i))(3)hereLpisthehydraulicconductivityofthemicrovascularwall,S=Visthebloodvesselsurfaceareaperunitvolume,pvisthevascularuidpressure,Tistheosmoticreectioncoefcientforplasmaproteins,v,iaretheosmoticpressuresoftheplasmaandinterstitialuid,respectively.Thersttermontherightsideofthecontinuityequationforvoxelsthatarenotinfusedwithalbumin(Equation( 3 ))representsthetransvascularuiduxacrossthemicrovascularwallperunitvolumeofthetissue,scaledbythenormalizedKtranstoaccountfortheheterogeneityinthemodel.Thesecondtermaccountsforthelymphaticdrainagefrominterstitialspaceperunitvolumeoftissue.Foraporousmedium,themomentumequationisgivenbyDarcy'slaw, v=)]TJ /F5 11.955 Tf 9.3 0 Td[(Krp(3)whereKisthehydraulicconductivitywhichislikelytobeheterogeneousintumorsandcanvarywiththelocalchangesinporosityofthetissues[ 30 41 61 70 ].InparticularLaiandMow[ 41 ],proposedanexponentialvariationofhydraulicconductivitywithdeformationwhichinturnwasrelatedtoporosity.Byusingasimilarrelation,theexponentialtermwasnormalizedwithitsmeanvaluetoensurethatthemeanhydraulicconductivitycalculatedoverthetumor/hosttissuevoxelsequalstheirbaselinevalues.Theresultingexpressionisgivenasfollows, K=8>>>>>>><>>>>>>>:K0tem(+0.1)Nt NtPi=1em(i+0.1)FortumorK0hem(+0.1)Nh NhPi=1em(i+0.1)Forhost(3) 52

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whereNt,Nharethenumberoftumorandhosttissuevoxelsrespectively,K0t,K0harethebaselinehydraulicconductivitiesoftumorandhosttissuesrespectivelyandmisanempiricalexponent.Albumin(MW66,776Da)isanon-bindingandnon-reactingmacromoleculewhichiswidelyusedastracerinCEDstudies.Assumingnotissuesourcesandsinkssincelargemolecularweightalbuminisnotexpectedtogobackintothecapillaries,transportinthetissuewasgivenbytheconvectionanddiffusionequation, @Ct @t+v .rCt)]TJ /F3 11.955 Tf 11.95 0 Td[(Der2Ct=0(3)whereCtistheconcentrationoftracerinthetissue,Deistheeffectivediffusivityofalbuminintheporousmediumgivenbythefollowingempiricalrelationbasedondiffusioninporousmedia[ 21 ], De=Dfreen(3)whereDfreeistheself-diffusioncoefcientofalbumininwaterandnisanempiricalexponentsetto4.Theconcentrationintheequationwasnormalizedusingthefollowingrelation, ^C=Ct C(t,i)i(3)whereC(t,i)andiaretheinfusateconcentrationandporosityoftheinfusedvoxelrespectively.ThevaluesoftheparametersinthegoverningequationsarelistedinTable 3-1 .TheMRimagealsoconsistedofvoxelspresentoutsidethemousewhichbelongneithertotumororhosttissue,i.e.exteriorvoxels.Inthesevoxels,thewholesourcetermforthecontinuityequationandthediffusivityweresettozero. 3.2.2ComputationalMethodThecontinuity,momentumandalbumintransportequationsweresolvedusingtheCFDsoftwarepackage,FLUENT(version12.0.16,ANSYS,Inc.,Canonsburg,PA).For 53

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the3Dcomputationaltissuemodel,arectangularvolume(20109mm3)enclosingthetumorwascreatedandmeshedwithquadrilateralelements(voxels)ofsizeequaltotheMRIresolution(0.1040.1041mm3)usingthemeshingsoftware(GAMBIT,Fluent,Lebanon,NH)withone-to-onemappingbetweentheCFDmeshandMRdata.Theamountoftumorandhost-tissuecontainedintheresultingvolumewerecalculatedtobe51.49and741.16mm3respectivelywiththeexteriorvoxelsoccupyingtherest.Governingequationswerediscretizedwithacontrol-volumebasedtechniqueusingFLUENT.Darcy'slawwassubstitutedfortheconservationofmomentumequation.WithinFLUENT,auser-denedfunctionwasusedtoassignKtrans,porosity,hydraulicconductivityanddiffusivityforeachvoxelinthemesh.Forthecontinuityequation,auser-deneduxmacrowasusedtoaccountforthesourceterms.Astandardpressureinterpolationschemewasusedtosolveforpressureandasecond-orderupwindmethodwasusedtosolvefortheowequations.TheSIMPLEC(Semi-ImplicitMethodforPressure-LinkedEquationsConsistent[ 71 ])pressure-velocitycouplingmethodwaschosen.Thetransportequationwasset-upusingtheuserdenedscalar(UDS)equationinFLUENTandsolvedusingrstorderupwindmethod.Theconvergencecriterionforallthethreeequationswassetto0.001.Infusionsimulationswerecarriedoutuptot=2hrsandtheinterstitialdistributionofalbuminwassimulatedatintermittenttimepoints,t=5,30,60and120mins.Initialconditionsfortracertransportassumednotracerinthetissue,^C=0exceptattheinfusionsitewhichisonevoxel(0.1040.1041mm3),whereitwassettoanormalizedvalueof1atallthetimesthroughanuser-denedfunctionwhichwasfedinduringthetransportsimulation.Thedistributionvolumewascalculatedasthevolumeoccupiedbyvoxelshavinganalbuminconcentrationgreaterthan1%oftheinfusionconcentration[ 10 ].Azerouidpressurecondition,p=0,wasappliedalongthecutendsandtheremainingouterboundariesofthegeometrywereassignedaswall.Theimpermeabilityconditionalongtheskinboundarywasachievedbyassigninghydraulic 54

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conductivitytwoordersofmagnitudelowerthanthenormaltissue,intheexteriorvoxels.Theassignmentoflowhydraulicconductivityintheexteriorvoxelscreatesamaterialthatisresistanttouidmotion.Forthechosenvalueofhydraulicconductivityattheexteriorvoxelsthemeanvelocityattheskinboundarywascalculatedtobeclosetozero(0.001m/s).Theinfusionatthecenterofthetumorwithlocallyconstanthydraulicconductivity(m=0)wastakenasthebaselinecase(Figure 3-1 ).Forcomparison,theoweldwasalsosimulatedforthesystemicdeliveryofalbuminbyneglectingtheinfusatesourceterm(Equation( 3 )),inthecontinuityequation.Theeffectofchangingthehydraulicconductivitywasachievedbyvaryingtheempiricalexponent(m)intheexpressionforhydraulicconductivity.Apartfromthebaselinevalue(m=0),owandtransportfortwodifferentvaluesofm=5and9(Figure 3-2 )werealsosimulated.Theeffectofcatheterplacementonthedistributionwasalsostudiedthroughinfusionsatthetumor-hosttissueinterfaceandanteriorendofthetumorwithm=0,inadditiontothebaselinesimulationatthetumorcenter.Thevesselpermeabilityanddiffusivitywasnotincludedinthesensitivityanalysisbasedontheresultsofourpreviousstudyontransportintumors[ 53 ],wheretheseparameterswerefoundtobeinsensitivetotracertransport.Moreover,diffusionbeingaslowprocess,changesindiffusivityisnotexpectedtoaffectthetracerdistributioninthesmalltimewindow(2hrs)understudy.Thechangesinowrateisalsonotexpectedtoaffectthetransportasthemodeldoesnothaveanymechanismforbackowandotherassociatedeffects. 3.3ResultsThebaselineresultsalongwiththesensitivityanalysisforthemodelareprovided.ThepredictedIFPforsystemicandlocalinfusionarerepresentedbycontourplotsatthemid-sliceofthetumorasshowninFigures 3-3A and 3-3B .Thelocalinfusionat0.3L/minincreasedthepressureattheinfusionsitebyapproximately1.27kPa.ThevoxelizedmodelpredictedelevatedIFPinsidethetumorthanthehosttissue.The 55

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contourplotsrevealalocalincreaseinIFPattheinfusionsitewhichmaskedthehighpressureinsidethetumorcomparedtothehosttissue.Atthetumormid-slice,themagnitudeofthepressuregradientwasmaximumattheinfusionsite(4.85kPa/mm)althoughsignicantvalueswerealsoobservedalongthetumor-hosttissueinterface(0.41-1.23kPa/mm).Theextracellularuidvelocity(EFV,v )forsystemicandlocalinfusion,isshownbyacontouralongthetumormid-slice(Figures 3-3C and 3-3D ).Thisisfurthersupplementedbyaconeplotdepictingthevelocityvectorscoloredbyitsmagnitudeforthewholelegwithlocalinfusion(Figure 3-3E ).Highervelocityregionswereobservedneartheinfusionsiteforlocalinfusion.Atthetumormid-sliceforlocalinfusion,peakvelocitieswereobservedatthepointofinfusion(36m/s)followedbysignicantvelocitiesatthetumor-hosttissueinterface(0.25-6.15m/s).Therewasalsoside-waysowoftheuidalongtheskinboundaryclosertothetumor.Thecontoursofthenormalizedalbuminconcentrationatvarioustimepoints,atthetumormid-sliceareshowninFigures 3-4A to 3-4C .Thepredicteddistributionofalbuminovertimewasasymmetricreectingtheanisotropicoweld.Theeffectoftheskinboundaryconditionnearthetumoronthedistributionpatternwasevidentatlatertimepointswithagradualoutwarduxofalbuminalongtheskinboundaryclosertothetumor.Aniso-surfaceatthedistributionvolumethreshold(0.01)fortimest=30,60and120minsshowninFigures 3-4D to 3-4F ,depictstheevolutionoftheconcentrationprolewithtime.Theiso-surfacesconrmstheasymmetricnatureofthedistributionandtheside-wiseuxofalbuminalongtheskinboundarynearthetumor.Aftertwohoursofinfusionat0.3L/min,albuminwasdistributedtoapproximately58%ofthetumorvolume.Thevariationofdistributionvolume(Vd)withinfusionvolume(Vi)withinthewholelegandtumorinparticular,isshowninFigure 3-5 .Theresultsdataindicatethatthedistributionvolumevarieslinearlywiththeinfusionvolumeforthewholeleg. 56

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Howeverthevariationwasslightlynon-linearwithinthetumor.TheratioVd=Viobtainedthroughlineartwasfoundtobe2.9forthewholelegand0.71forthetumor. 3.3.1SensitivityAnalysisSimilartothebaselineresults,themodelpredictedhigherIFPform=5and9althoughthepeakpressurevaluesweredifferent(Figure 3-6 ).Thesimulationresultsindicatedan48and75%reductioninthepeakIFPfromitsbaselinevalueform=5and9respectivelyinthetumormidslice.IncreasingthevalueofmloweredthepeakIFPinsidethetumorandtheconvectionvelocitybecamemoreheterogeneouswithincreasingm.Theincreaseinmappearedtoreinforceuidpathwayswithhigherporosities.Thevelocityvectorplotrevealstheincreaseinowinthecoronalplaneatm=5comparedtothebaselinevalue.Thisphenomenonbecamemorevisibleatm=9wheretherewasalargeoutowfromthetumor.Theuidleakageacrosstheskinboundaryclosertothetumorwaspresentatbothvaluesofm.ThepredictedevolutionofthedistributionvolumeovertimefordifferentvaluesofmisshowninFigure 3-7 .Theconvectiveeffectswereapparentontheshapesofthedistributionvolume,atm=5thedistributionpatterntendstogetmoreskewedintothetumorthanthebaselinevalue.Howeverastimeproceeds,thealbumintracertendstogoawayfromthetumor.Asimilarpatternwasobservedatm=9forinitialtimepointsbutthedistributiongotmoreheterogeneousandoutwardfromthetumorastimeprogressed.Thedistributionvolumeinthewholelegvariedlinearlywithinfusionvolumeform=5and9withslopesequalto3.8and4.7respectively(Figure 3-8A ).Howeverthevariationwithinthetumor,tendstobecomenon-linearatlatertimepoints(Figure 3-8B ).Atlatertimepoints,increasingthemdecreasedthedistributionvolumewithinthetumor.Thiseffectbecamemoreapparentforlargervaluesofm.Form=5,twohoursofinfusionat0.3L/minresultedincoveringapproximately55%ofthetumorvolume.Whereasform=9,approximately43%ofthetumorvolumewascoveredbythetracer. 57

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Theeffectofcatheterplacementonalbumindistributionisshownascontoursatthetumormid-sliceinFigure 3-9 .Anasymmetricdistributionwasobservedforinfusionsatboththelocations,tumor-hosttissueinterfaceandanteriorendofthetumor.Infusionattheinterfacetendstodistributealbuminmorealongthedorsalsidewhereasattheanteriorenditwasmoreskewedtowardstheanteriorsideoftheleg.Forthewholeleg,theresultsdataindicatedalinearvariationofVdwithVi,withhigherdistributionvolumeforinfusionattheinterfacethanattheanteriorendofthetumor(Figure 3-10A ).Withinthetumor,infusionattheinterfaceresultedincoveringapproximately58%ofthetumorwhileinfusionattheanteriorendresultedinapproximately18%(Figure 3-10B ). 3.4DiscussionAcomputationalmodelforpredictingdistributionofamacromolecularproteintracerfollowingCEDinthehindlimbtumorofamiceusingvoxelizedmodelingapproachwasdeveloped.Thisapproachaccountedforrealistictumormicrovasculatureandgeometry,andallowedformoreeasierandrapidbuildingofcomputationalporousmediatransportmodelcomparedtotraditionalapproachesutilizingunstructuredmeshesinvolvingcomplexgeometricreconstruction.Thismakesthemodellesslaborintensiveandeasiertoimplement.Spatially-varyingtissuetransportpropertiesbasedontheactualheterogeneoustumormicrovasculature,tissuestructureandnaturalanatomicaltissuegeometrieswereincorporatedintoathree-dimensional,image-basedcomputationalporousmediamodel.Themodelsolvesforinterstitialuidpressure,interstitialuidvelocity,andalbuminconcentrationthroughthetissueinterstitium,followingCED.Thesensitivityofthemodelfordifferenthydraulicconductivitymapsandcatheterplacementswereinvestigated.ThepredictedIFPreectedthepreviousexperimentalndingswhichsuggestedelevatedpressuresinsidethetumor[ 11 13 23 48 77 ].However,theinfusioninducedalocalpressuregradienttherebyexhibitingtheadvantageconvectiongivesindistributingalbumintolargertissuevolumesfollowingCED.Exceptattheinfusion 58

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site,thepressurewasuniforminsidethetumoranddroppedsteeplyinitsperipheryinagreementwiththepreviousndings[ 29 ].Outsidethetumor,theboundaryconditionplayedacriticalroleindeterminingIFP.Thecloseproximityofthetumortooneportionoftheimpermeableskinresultedinapressuregradientnearthatportionoftheskin,approximatelyfourtimeshigherthanattheskinfartherfromthetumor.TheconvectionvelocityeldpredictedbythemodelreectedthecomputedIFPastheowisdrivenbythepressuregradient.Thehighpressuregradientattheinfusionsiteandtumor-hosttissueinterface,resultedinhighervelocitiesinthoseregions.Inaddition,thepresenceofskinclosertothetumorwaspredictedtoaffecttheoweldcausingside-waysowoftheuidalongtheskin.ThelinearvariationofdistributionvolumewiththeinfusionvolumecapturedbythemodelisinaccordancewiththepreviousexperimentalndingbySaitoandhiscolleagues[ 56 ].TheyreportedsuchatrendforCEDofliposomesinratgliomas.Thevalueoftheratiowasknowntodependonmanyfactorsbutnotlimitedtoinfusateproperties,extracellularmatrix(ECM)amongothers,andawiderangeofvaluesfrom1to8.7hasbeenreportedintheliterature[ 2 ].Furthermore,thedistributionofalbuminwasasymmetricandheterogeneousconformingwiththepreviousexperimentalndings[ 12 46 57 ].Suchadistributionistheresultoftheoweldwhichadvectsthealbumininpathwaysofleastresistance(higherporosity).Thedistributionpatternwascloselyinterlinkedwiththeoweldwithhighconcentrationattheinfusionsiteandgradualleakageofalbuminalongtheskinboundaryclosertothetumor.Attheendoftwohours,CEDwasabletocoverapproximately58%ofthetumorvolumetherebyexhibitingtheeffectivenessofthemethodindeliveringmacromoleculardrugs.Inthisstudy,weinvestigatedthepossibilityofreducingthetumorIFPbyincreasingthesensitivityoftissuehydraulicconductivitytotissueporosity.Thiswasalsodonetoincreasetheheterogeneoustransport.Mathematicallythiswasimplementedbyvaryingtheempiricalparametermintheexpressionforhydraulicconductivity.Increasingthe 59

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hydraulicconductivityhasbeenpreviouslythoughttoreduceIFPandthusincreaseextravasationofmacromolecules[ 12 ].Theresultsofthesensitivityanalysisatm=9indicatedthatthepeaktumorIFPgotreducedbyapproximatelyfourtimesandtheresultingdistributionvolumeincreasedbyapproximately60%fromthebaselineaftertwohoursofinfusion.Howeverthiseffectwasnotreectedinthecomputeddistributionvolumewithinthetumor,whichgraduallyreducedwithtimeforhighervaluesofm.ThisisbecauseoftheveryhighreductionintheresultingtumorIFP,whichdirectstheinterstitialuidandalbumintracerawayfromthetumor.Thisdemonstratestheimportanceofmeasuringtheparametermforagiventumor,forachievingaccuratetracerdistributionwithinthetumor.Thesensitivityanalysiswasextendedtostudytracerdistributionatdifferentcatheterpositions,inanattempttondanoptimalplacementwhichcouldmaximizedistributionvolumeinthetargetsite.Theinfusionswerecarriedoutattwoothersitesinadditiontothebaselineposition:tumor-hosttissueinterfaceandanteriorendofthetumor.Forthegivensetofbaselineparameters,wefoundthattheinfusionatthetumor-hosttissueinterfaceproducedthemaximumdistributionvolumeforthewholeleg.Thisisduetothepresenceoflargerconvectiveeffectsaroundtheperipheryasopposingtojustoneattheinfusionsite.Itshouldhoweverbenotedthatthedistributionvolumewithinthetumorwasalmostidenticaltothebaselinevalue.Theincreasedconvectiveeffectapparentlydidnothavesignicanteffectonthedistributionvolumewithinthetumor.Theoutwardowofalbuminfromthetumorforinfusionsattheanteriorendofthetumorisduetoproximityofsitetothecutendsofthetumorwherezeropressureboundaryconditionwasspecied.Similarpatterncanbeexpectedforinfusionsattheposteriorendofthetumor.Byincreasingtheowrateorinfusiontimeand/ortestingadditionalinfusionsites,themodelcanbeusedtondanoptimalcatheterplacementforagiventumortoachievetotalcoverage.Inthisway,themodelcanhelpinsurgicalplanningbyprovidingeffectivetreatmentstrategiesonacase-by-casebasis. 60

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Toourknowledge,thisistherstimage-basedtumormodelthatincorporatestheactualtumormicrovasculatureandpredictsheterogeneous/asymmetricdrugdistributionfollowingCED.Ourmodelservesasapotentialtoolforconductingandoptimizingpatient-specictreatments.Alteringtheextracellularmatrix(ECM)ofthetumorisbeingexploredbyresearchersasapossibletechniquetoacheivebetterdistribution[ 29 ].SeveralcompoundssuchasVEGFinhibitors,hyaluronidase,mannitolamongotherswereusedtodisrupttheheterogeneoustumormicrovasculatureandnormalizeit,therebyimprovingdrugdeliveryandefcacy[ 29 37 ].Thismodelcouldbeusedtostudytheseeffectsoncethetransportproperties(Ktransand)oftheresultingECMwerefoundusingthemethodsdescribedin[ 53 ].Althoughtheresultsdiscussedinthisstudywererestrictedtothehindlimbtumorunderstudy,itshouldbenotedthattheapplicabilityofvoxelizedmodeltoawiderangeoftumorsispossible.Therelativeeaseinimplementingthemodelanditsreasonablepredictions,makesitapromisingcandidateforpredictingdrugdistributionsfollowingCEDintumors. 61

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Table3-1. Tissueandvascularparametersusedforsimulatingdistributionofalbuminfollowingconvection-enhanceddeliveryatthehindlimbtumorinamice. VariableDescriptionValueReferences Lp(m/Pa.s)Vesselpermeability210)]TJ /F8 7.97 Tf 6.58 0 Td[(11t;310)]TJ /F8 7.97 Tf 6.59 0 Td[(12n;[ 53 ]S=V(m)]TJ /F8 7.97 Tf 6.58 0 Td[(1)Microvascularsurfaceareaperunitvolume20000t;7000n[ 7 ]Lp,lySL=V(m)]TJ /F8 7.97 Tf 6.59 0 Td[(1)Lymphaticltrationcoefcient110)]TJ /F8 7.97 Tf 6.58 0 Td[(7[ 53 ]K0(m2/Pa.s)Baselinehydraulicconductivity1.910)]TJ /F8 7.97 Tf 6.59 0 Td[(12t;3.810)]TJ /F8 7.97 Tf 6.59 0 Td[(13n[ 53 ]7.710)]TJ /F8 7.97 Tf 6.59 0 Td[(15epv(Pa)Microvascularpressure2300[ 53 ]i(Pa)Osmoticpressureininterstitialspace3230t;1330n[ 53 ]v(Pa)Osmoticpressureinmicrovasculature2670[ 53 ]T(Pa)Averageosmoticreectioncoefcientforplasma0.82t;0.91n[ 53 ]Dfree(m2=s)Selfdiffusioncoefcientofalbumin5.810)]TJ /F8 7.97 Tf 6.59 0 Td[(11[ 5 ]Q(L/min)Infusionowrate0.3[ 39 ] t-tumor,n-normaltissue,e-exterior. 62

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Figure3-1. DepictionofbaselineCEDsimulation.Thesizeoftheinfusionneedlewasexaggeratedforclarity.Includesskin(green),tumor(blue),tumormid-slice(red)andinfusionneedle(magenta) 63

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Figure3-2. Variationofscaledhydraulicconductivitywithporosityfordifferentvaluesofm. 64

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A B C D EFigure3-3. Interstitialuidpressure(IFP)andextracellularuidvelocity(EFV)withsystemic(A&C)andlocal(B&D)infusiondescribedbyitscontoursatthetumormidslice.Tumorandskinboundariesareoverlaidonthecontoursandtheinfusionsiteisshownbyaplussign.Inthebottom,aEFVconeplot(E)coloredbyitsmagnitudeforlocalinfusion.Includespointsource(blacksphere),tumor(blue)andskin(green) 65

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At=30min Bt=1hr Ct=2hr Dt=30min Et=1hr Ft=2hrFigure3-4. Onthetop,normalizedtracerconcentrationcontoursattumormid-sliceatt=30,60,and120min.Tumorandskinboundariesareoverlaidonthecontoursandtheinfusionsiteisshownbyaplussign.Onthebottom,predictedevolutionofdistributedvolumeovertimeshownbyaniso-surfaceatthedistributionvolumethreshold.Includestumor(blue),skin(green)anddistributedvolume(red) 66

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Figure3-5. VariationoftissuedistributionvolumeswithinfusionvolumeforthewholelegandtumorfollowingCEDofalbumin(0.3L/min)atthecenterofthetumor.Includesequationforthelineartonthedata. 67

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m=5m=9 A B C D E FFigure3-6. Comparisonofinterstitialuidpressure(IFP,A&B),extracellularuidvelocity(EFV,C&D)contoursatthetumormid-sliceforinfusionsatm=5&9respectively.Tumorandskinboundariesareoverlaidonthecontoursandtheinfusionsiteisshownbyaplussign.TheEFVconeplots(E&F)coloredbyitsmagnitudeform=5and9isalsoshown.Includespointsource(blacksphere),tumor(blue)andskin(green) 68

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t=30mint=60mint=120min m=5 m=9 Figure3-7. Predictedevolutionofdistributedvolumeforinfusionsatthecenterofthetumorform=5and9att=30,60,and120min.Includestumor(blue),skin(green)anddistributedvolume(red) 69

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A BFigure3-8. Variationoftissuedistributionvolumeswithinfusionvolumeforthewholeleg(A)andtumor(B)followingCEDofalbumin(0.3L/min)atthecenterofthetumorform=0,5&9.Includesequationforthelineartonthedata. 70

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t=30mint=60mint=120min Interface Anterior Figure3-9. Comparisonofnormalizedtracerconcentrationcontoursattumormid-sliceforinfusionsatthetumor-hostinterfaceandanteriorendofthetumoratt=30,60,and120min.Tumorandskinboundariesareoverlaidonthecontoursandtheinfusionsiteisshownbyaplussigninacircle. 71

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A BFigure3-10. Variationoftissuedistributionvolumeswithinfusionvolumeforthewholeleg(A)andtumor(B)followingCEDofalbumin(0.3L/min)atthetumor-hosttissueinterfaceandanteriorendofthetumorwithm=0.Includesequationforthelineartonthedata. 72

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CHAPTER4CONCLUSIONSANDFUTUREWORKAcomputationalmodelforpredictingthedistributionofmacromoleculardrugfollowingconvection-enhanceddelivery(CED)inthehindlimbtumorofamiceusingvoxelizedmodelingapproachwasdeveloped.Theapproachaccountedforrealistictumormicrovasculatureandgeometry,andallowedformoreeasierandrapidbuildingofcomputationalporousmediatransportmodelcomparedtotraditionalapproachesutilizingunstructuredmeshesinvolvingcomplexgeometricreconstruction.Thismakesthemodellesslaborintensiveandeasiertoimplement.Spatially-varyingtissuetransportpropertiesbasedontheactualheterogeneoustumormicrovasculature,tissuestructureandnaturalanatomicaltissuegeometrieswereincorporatedintoathree-dimensional,image-basedcomputationalporousmediamodel.Themodelsolvesforinterstitialuidpressure,interstitialuidvelocity,anddrugconcentrationthroughthetissueinterstitium,followinginjection.ThisframeworkwaspreviouslyevaluatedwithexperimentandpredictionsfromatraditionalCFDapproachforsystemicdeliveryofMRvisibletracer.Therstportionofthisthesisdemonstratedtheapplicabilityofthevoxelizedmodelforpredictingtumortransport.ThiswasdonebycomparingitsresultswiththatobtainedfromapreviouslydevelopedCFDmodelingapproachusingunstructuredmeshesforsystemicdeliveryofthetracer,usingstatisticalmethodsandqualitativepresentation.Theresultinganalysisindicatedsimilarityinboththemodelresultswithlowrootmeansquareerrorandhighcorrelationcoefcient.Thevoxelizedmodelalsocapturedtypicalfeaturesoftheoweldandtracerdistributioninthetumorinterstitiumsuchasthehighinterstitialuidpressure(IFP)insidetumorandtheheterogeneousdistributionoftracer.TheobtainedtracerdistributionwithinthetumorwasalsosimilartoMR-measuredtracerconcentrationdata.Furthermore,theaccuracyofthevoxelizedmodelresultswithexperimentandnon-voxelizedmodelpredictionsweremaintainedacrossthethree 73

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tumorsunderstudy.Sensitivityofthevoxelizedandnon-voxelizedmodeltochangingarterialinputfunction(AIF)parameterswasalsofoundtobesimilar.Inthesecondportionofthethesis,thismodelwasslightlymodiedtopredictthetracerdistributionfollowingCED.Themodelwasabletocapturetheasymmetrictracerdistributionandthelinearvariationofdistributionvolumewiththeinfusionvolume.Sensitivityofthemodeltochangesinhydraulicconductivityandcatheterplacementwereinvestigated.Thetracerdistributionwasfoundtobesensitivetoboththeparametersunderstudy.IncreasingthevaluesofthehydraulicconductivitymaploweredthetumorIFPandraisedthedistributionvolumewithinthewholeleg.Howeverwithinthetumor,thedistributionvolumedecreasedwithincreasingvalueoftheempiricalparameter(m)usedtoincreasethehydraulicconductivity,atlatertimepoints.Theinfusionatthetumor-hosttissueinterfaceresultedinlargerdistributionvolumecomparedtothatatthecenterandanteriorendofthetumor,underbaselineconditions.Withinthetumor,thedistributionvolumewasalmostidenticalforinfusionsattheinterfaceandcenterofthetumor.Theaccuracyofthemodel'sCEDpredictionscouldbeimprovedwiththefollowingmodicationswhichwouldbethesubjectoffuturework.Firstly,transvascularsoluteexchangecanbeaccountedinthetransportequationforsmallermolecularweightcompounds.Itcanbeexpectedtoaffectthetracerdistributionbecauseoftheleakinessofthetumors.Secondly,binding,metabolismanduptakeofthemacromoleculardrugcanbeaccountedforusingreactionandbindingkinetics.Finally,themodelcanbevalidatedwithexperimentsacrossawiderrangeoftumors.Inadditiontotheaboveextensions,thevoxelizedmodelsolutionalsoneedstobetestedforgridandtimestepindependency,toensurethatdiscretizationandtruncationerrorsaresmall. 74

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BIOGRAPHICALSKETCH MagdoomMohamedreceivedhisBachelorofTechnologydegreeinMechanicalEngineeringfromNationalInstituteofTechnology(NIT),Tiruchirappalli,Indiain2008.Between2008-09,heworkedatIndianInstituteofTechnology(IIT),MadrasasprojectassistantintheDepartmentofBio-technology.InSpring2010,hewasadmittedtothegraduateprogramintheDepartmentofMechanicalandAerospaceEngineeringattheUniversityofFlorida.InSummer2011,hereceivedhisMSinMechanicalEngineeringfromtheUniversityofFlorida. 82