COMPUTATIONAL MODELS FOR LOCALIZED DRUG DELIVERY IN TUMORS

Transcription

1 COMPUTATIONAL MODELS FOR LOCALIZED DRUG DELIVERY IN TUMORS By MAGDOOM MOHAMED KULAM NAJMUDEEN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 211

2 c 211 Magdoom Mohamed Kulam Najmudeen 2

3 To my mom, dad and family 3

4 ACKNOWLEDGMENTS I take this opportunity to thank everyone who has helped me throughout this journey. I would like to extend my gratitude to my advisor Dr. Malisa Sarntinoranont for giving me an opportunity to do research with financial assistance, and helping me since I joined UF. I thank all the members of my research group who have helped me with this project in varying degrees. A special thanks goes to Dr. Gregory Pishko without whom this project would not have been accomplished, I am truly indebted to him. I also want to extend my thanks to my committee members : Dr. Brian Sorg and Dr.Tran-Son-Tay for kindly accepting to be in my committee. For my research, I would like to thank Dr. Dietmar Siemann, Dr. Lori Rice and Chris Pampo for providing murine KHT sarcoma cells and tumor inoculation. I also thank Dr. Thomas Mareci for providing MRI expertise and Garrett W. Astary for helping with the DCE-MR experiments. I appreciate the help Dr. Gregory Pishko has offered in this project, by providing me with tissue transport property maps and simulation results for the non-voxelized model, along with segmented MR images. I also want to thank Dr. Jung Hwan Kim for sharing his valuable experience on voxelized modeling. More importantly, I thank my mom and dad for providing me with financial and moral support. I also thank my friends and family members especially my uncles, cousins, brothers and sisters for advising and helping me in difficult times. I would also like to thank my friends at UF especially those from my undergraduate school in India for providing me with valuable support when I came to this country for the first time. Finally, I would like to thank all the professors I have interacted with for providing me with beneficial knowledge, which has helped me become the person I am today. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Background Objectives Specific Aim Specific Aim DEVELOPMENT OF VOXELIZED MODEL FOR SYSTEMIC DELIVERY IN SOLID TUMORS Overview Methods Estimation of spatial variation maps of vascular leakiness Mathematical Model Computational Method Statistical Analysis Root Mean Square Error Pearson Product Moment Correlation Coefficient Error Histogram Results Sensitivity Analysis Validation Study Discussion APPLICATION OF VOXELIZED MODEL FOR CONVECTION-ENHANCED DELIVERY IN A HIND LIMB TUMOR Overview Methods Mathematical Model Computational Method Results Sensitivity Analysis Discussion

6 4 CONCLUSIONS AND FUTURE WORK REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 2-1 Tissue and vascular parameters used for simulating distribution of Gd-DTPA following bolus tail vein injection at the hind limb tumor in a mice Statistical parameters obtained while comparing voxelized and non-voxelized model results for the baseline simulation in three animals Statistical parameters obtained while comparing voxelized and non-voxelized model results for intermediate and fast arterial input function in animal I Comparison of tracer washout rates and root mean square error in tracer concentration within the tumor volume between voxelized and non-voxelized model results with experiment in three animals Tissue and vascular parameters used for simulating distribution of albumin following CED at the hind limb tumor in a mice

8 Figure LIST OF FIGURES page 2-1 Normalized concentration of tracer in blood plasma approximated by different AIFs used for sensitivity analysis CFD compatible meshes Horizontal and vertical lines used for plotting the flow field and tracer transport in voxelized and non-voxelized models Contours of IFP predicted by voxelized and non-voxelized models, along with line plots along the horizontal and vertical bisectors in the mid-slice Contours of IFV predicted by voxelized and non-voxelized models along with line plots along the horizontal and vertical bisectors in the mid-slice Comparison of tracer concentration contours. Voxelized and non-voxelized model compared with MR-derived tissue concentration at t = 5, 3, and 6 min Line plots comparing the predicted tracer concentration in the tissue by both the models with experiment, along the horizontal and vertical bisectors of mid-slice at t = 5, 3 and 6 min Error Histograms for flow and transport in baseline simulation for voxelized model with respect to non-voxelized model Line plots comparing the IFP and IFV predicted by both the models for two different AIF parameter sets (intermediate and fast) along the vertical bisector of mid-slice Line plots comparing the tracer concentration in the tissue predicted by both the models for two different AIF parameter sets (intermediate and fast) along the vertical bisector of mid-slice at t = 5 and 2 min Error Histograms for tracer concentration within the tumor for voxel and non-voxel model results with respect to the experimental data at t = 5, 3 and 6 min Depiction of baseline CED simulation Variation of scaled hydraulic conductivity with porosity for different values of m IFP and EFV contours at the tumor mid-slice for systemic and local infusion, along with a EFV cone plot colored by its magnitude for local infusion

9 3-4 Normalized tracer concentration contours at tumor mid-slice at t = 3, 6, and 12 min. Also included predicted evolution of distributed volume over time shown by an iso-surface at the distribution volume threshold Variation of tissue distribution volumes with infusion volume for the whole leg and tumor following CED of albumin (.3 µl/min) at the center of the tumor Comparison of IFP and EFV contours at the tumor mid-slice along with EFV cone plot colored by its magnitude, for infusions at m = 5 & Predicted evolution of distributed volume for infusions at the center of the tumor for m = 5 and 9 at t = 3, 6, and 12 min Variation of tissue distribution volumes with infusion volume for the whole leg and tumor following CED of albumin (.3 µl/min) at the center of the tumor for m =, 5 & Comparison of normalized tracer concentration contours at tumor mid-slice for infusions at the tumor-host interface and anterior end of the tumor at t = 3, 6, and 12 min Variation of tissue distribution volumes with infusion volume for the whole leg and tumor following CED of albumin (.3 µl/min) at the tumor-host tissue interface and anterior end of the tumor with m =

10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COMPUTATIONAL MODELS FOR LOCALIZED DRUG DELIVERY IN TUMORS Chair: Malisa Sarntinoranont Major: Mechanical Engineering By Magdoom Mohamed Kulam Najmudeen August 211 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 1

11 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 11

12 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. 12

13 CHAPTER 1 INTRODUCTION 1.1 Background Tumor or neoplasm is an abnormal mass of tissue usually caused due to genetic mutations. They can be classified into benign and malign tumors depending on their ability to invade adjacent tissues. Malign tumors invade and destroy adjacent tissues while benign tumors lacks the ability to metastasize. The onset of tumors are often characterized by rapid formation of new blood vessels to supply nutrients to the tumor cells. Angiogenesis in tumor tissues is different from that in the normal tissue, tumor vasculature is irregular and often characterized by highly tortuous, fenestrated, discontinuous vessels and large avascular areas [1, 25, 33, 4, 42]. Tumors are also known to exhibit elevated interstitial fluid pressure (IFP), which is attributed to its lack of lymphatics [6] along with the chaotic vasculature [7, 14]. There is significant evidence for elevated IFP in tumors from the experiments performed by several researchers [11, 13, 23, 48, 77]. These abnormalities form vascular and interstitial barriers to the delivery of macromolecular therapeutic agents to tumors [6, 29]. Systemic drug delivery to tumors is often known to result in inadequate and uneven uptake, thereby preventing the drug from reaching therapeutic concentrations at the target site. The chaotic tumor microvasculature leads to heterogeneous extravasation of drugs [2], thereby reducing its therapeutic efficiency. The high IFP increases the drug transport away from the tumor into normal tissues and reduces the transcapillary transport, causing undesirable side-effects and lower drug uptake in the tumor. Overall, these characteristics of the tumor microenvironment hinder the systemic delivery of therapeutic agents to tumor cells. Localized drug delivery has emerged as a plausible alternative to systemic delivery for transporting macromolecular therapeutic agents to the tumors [18, 22, 56, 72 74]. By directly injecting into the tumor, this circumvents the previously mentioned vascular and 13

14 interstitial barriers and also reduces the side-effects associated with systemic exposure. Amongst the available techniques, convection-enhanced delivery (CED) appears promising because at a given time it can achieve larger distribution volumes than by diffusion alone [2, 1]. In CED, an infusion pump delivers the drug at constant flow rate or pressure thereby utilizing the bulk flow due to the infusion pressure difference, to deliver and distribute macromolecules to larger volumes in the tissue. Since its advent, CED has been widely used for in situ delivery of a wide range of substances including nanoparticles [5], liposomes [44, 56], cytotoxins [57] and viruses [24, 62]. However heterogeneous distribution remains as an obstacle for CED to tumors. 1.2 Objectives The focus of this thesis was to develop a computational model for predicting drug distributions following CED to tumors. Computational modeling has gained attention partly because it could help in planning and optimizing patient-specific treatments. Previous mathematical models of transport in tumors assume theoretical vasculature and simpler geometries [7, 59, 64 66] neglecting the vascular heterogeneity. Given the critical nature of the microvasculature in tumor drug delivery, our group developed a framework accounting for the realistic tumor microvasculature using the data obtained from dynamic contrast enhanced-magnetic resonance imaging (DCE-MRI) [53]. The model accounting for the heterogeneous tumor microvasculature could potentially help optimize patient-specific treatments with its realistic predictions, and understand the biophysical IFP and interstitial fluid velocity (IFV) changes due to CED, which are otherwise difficult to measure experimentally. The model previously developed by our group for this purpose, involved complex geometric re-construction which is time consuming and labor intensive. One of the objectives of this project was to develop a simpler model for tumor transport based on a voxelized modeling methodology. In this approach, tissue properties and anatomical boundaries are assigned on a voxel-by-voxel basis using MRI data. These properties 14

15 are then incorporated into a porous media transport model to predict IFP, IFV, and tracer transport, thereby allowing for quicker building of computational transport models and rapid estimation of tracer distribution. This model avoids the complex geometric reconstruction as the MR data is directly imported into the mesh Specific Aim 1 The first portion of the project was aimed at developing and studying the applicability of the voxelized model for tumor transport. The voxelized model was developed for predicting the distribution of systemically delivered MR visible tracer (Gd-DTPA) in the hind limb of mice through bolus tail vein injection. The results of the model which includes the predicted flow field and tracer transport were compared with those obtained from a non-voxelized one [53]. A validation study for this approach was also conducted by calculating the error between Gd-DTPA tissue concentrations within the tumor, predicted using a voxelized model and those measured using MRI. Sensitivity of the model to arterial input function (AIF) was also investigated. The model was tested with three sets of animal data Specific Aim 2 The second portion of the project was focused on applying the developed voxelized model for predicting the distribution of albumin tracer in the same tumor following CED as opposed to systemic delivery. The governing flow and transport equations were slightly modified to account for the point source and the voxelized methodology was used to solve them. For sensitivity analysis, the effects of varying hydraulic conductivity maps and catheter placement, on fluid flow and albumin transport were investigated. Infusions were carried out separately at two different sites in the tumor namely at the tumor-host tissue interface and anterior end of the tumor, in addition to the baseline simulation at the center of the tumor. The model could serve as a potential tool for optimizing patient-specific treatment and studying the effect of heterogeneous vasculature on tumor transport. 15

16 CHAPTER 2 DEVELOPMENT OF VOXELIZED MODEL FOR SYSTEMIC DELIVERY IN SOLID TUMORS 2.1 Overview Although enormous advancements have been made in the diagnosis and treatment of cancers, targeted drug delivery to malignant tumors still remains a challenge. Transport of macromolecular therapeutic agents in the tumor microvasculature plays a vital role in the treatment of solid tumors [34, 35]. However, a major obstacle to systemic transport in tumors is inadequate and uneven uptake, which is widely attributed to the heterogeneous architecture of the tumor microvasculature [6]. Tumors are known to contain highly tortuous, fenestrated, discontinuous vessels and large avascular areas [1, 25, 33, 4, 42]. The resulting heterogeneous vasculature leads to irregular perfusion [9, 32] which causes heterogeneous extravasation of therapeutic agents across the blood vessel wall, depending on the pressure difference across the wall and spatially varying vascular permeability [7, 32]. Another profound effect of abnormal vascular geometry, combined with a lack of lymphatics [6] in tumors is thought to be the elevation of interstitial fluid pressure (IFP) [7, 14]. Experiments performed by several researchers have revealed increased IFP in tumors [11, 13, 23, 48, 77]. It has been also observed that IFP is uniform throughout the center of the tumor and drops sharply at its periphery [11, 16]. However, recent evidence also suggests a lesser uniform IFP inside the tumors [26]. A study conducted by Hassid and his colleagues showed that the IFP inside ectopic human non-small-cell lung cancer increased from the periphery inward, with a high plateau inside the tumors. With the absence of pressure gradients in the center of tumor in either case, convective transport of drugs is expected to be less than at the periphery where pressure gradients exist, resulting in a hetergenous extravasation. It is also expected that the interstitial fluid flow driven by the IFP gradient is affected. Interstitial fluid velocity (IFV) within a human neuroblastoma was experimentally found 16

17 to increase from the center towards the periphery of the tumor [16]. From a modeling study, elevated IFP is also thought to cause vascular constriction which may lead to reduction in tumor blood flow [47], also the presence of a necrotic core was found to have an adverse affect on the distribution of large, slow-diffusing molecules [8]. On the whole, these characteristics of the tumor microenvironment hinder the systemic delivery of therapeutic agents to tumor cells. Hence, quantification of extravasation and drug distribution is paramount to developing successful treatment strategies. Previous mathematical models of transport in tumors assume either uniformly distributed or regular patterns of parallel and series blood vessels [7, 64 66] neglecting the vascular heterogeneity. Jain and his colleagues modeled the effects of uniformly distributed leaky blood vessels and minimally functioning lymphatics for the case of a spherical solid tumor and showed how elevated IFP leads to heterogeneous extravasation [36]. Pozrikidis developed a theoretical model to describe the blood flow in which, tumor microvasculature was generated by branching capillaries using deterministic and random parameters thus resulting in a capillary tree [54]. It should be noted that tumor angiogenesis patterns in these previous blood vessel models are theoretical and simulated based on rules to generate network structures. Recently, computational fluid dynamics (CFD) approaches were used by our group and others to study the extracellular transport in tumors [51 53, 63, 78]. In particular, studies conducted by Pishko et.al. [53] accounted for realistic tumor vasculature by using dynamic contrast enhanced-magnetic resonance imaging (DCE-MRI) data to estimate the spatial variation of transport properties (rate transfer constant between plasma and extracellular space, K trans and porosity, ϕ), which were mapped into a unstructured mesh of a CFD model that solves for IFP, IFV and tracer transport. The results of these studies are encouraging; however, the time-intensive labor involved in the approach motivated us to develop a simpler model for tumor transport based on a voxelized modeling methodology. Earlier, this methodology has been used by our 17

18 group to model interstitial transport in the rat spinal cord and brain during tissue infusion [38, 39]. In this approach, tissue properties and anatomical boundaries are assigned on a voxel-by-voxel basis using MRI data. These properties are then incorporated into a porous media transport model to predict IFP, IFV, and tracer transport, thereby allowing for quicker building of computational transport models and rapid estimation of tracer distribution. This voxel method circumvents the laborious geometric reconstruction involved in its non-voxelized counterpart by directly importing MRI data. In this study, a voxelized model for systemic transport in tumors was developed and its results were compared with those obtained from a non-voxelized one [53]. A validation study for this approach was conducted by calculating the error between Gd-DTPA tissue concentrations predicted using a voxelized model and those measured using MRI. The model was applied to three tumors and its predictions were compared as described previously. Sensitivity of the model to arterial input function (AIF) was also investigated. The shape of the AIF determines the time variation of the concentration of MR visible tracer in blood plasma. The choice of AIF is critical in the pharmacokinetic modeling of tissue tranport properties [28]. A faster AIF signifies higher wash-out rate of the tracer and vice-versa. 2.2 Methods Estimation of spatial variation maps of vascular leakiness DCE-MRI was used to obtain vascular leakiness maps. The lower hind limb of an anesthetized mouse (C3H), inoculated with murine sarcoma cells (KHT) was used in the MR experiment. Serial DCE-MR images, consisting of a T1-weighted spin-echo sequence were acquired before and after contrast agent (tracer) administration. The same MRI data as presented in Pishko et.al, [53] was used. The data consisted of 9 slices with a matrix of voxels per slice. The size of each voxel was mm 3. 18

19 DCE-MRI measures the tissue uptake of a MR visible tracer, which in this case is gadolinium-diethylene-triamine penta-acetic acid (Gd-DTPA, MW 59 Da), after a systemic bolus tail vein injection. The tracer concentration in tissue and the method to calculate K trans and ϕ were identical to that presented in Pishko et.al, [53]. The tracer deposition in tissue was measured by signal enhancement which is defined as the ratio of the signal intensities after and before injection of the tracer. This is then mapped to the actual tracer concentration in the tissue (C t ) by assuming a linear relationship between C t and relaxation times (T 1 & T 2 ), and substituting it into the standard spin-echo equation [49, 55, 69]. After algebraic manipulations, the following expression for C t was obtained with an added assumption that transverse-relaxation contribution to signal is unity, C MRI,t = 1 [ 1 R 1 TR ln S() 1 ] S() S(C MRI,t ).(1 e TR/T 1 ) T 1 where C MRI,t is the tissue concentration of Gd-DTPA determined by MRI, R 1 is the (2 1) longitudinal relaxivity of the tracer in water, TR is the time for recovery, S(C MRI,t ) and S() are the signal intensities at tracer concentrations C MRI,t and zero respectively and T 1 is the T 1 relaxtion time without tracer. Vascular leakiness characterised by K trans and ϕ were estimated using a two-compartment kinetic model [67]. This model describes the exchange of tracer between the plasma and tissue compartments in each voxel. The two compartment model can be described by, dc t dt = K trans C p K trans ϕ C t (2 2) where C t and C p are the concentrations of Gd-DTPA in tissue and blood plasma respectively. 19

20 The tracer concentration in the blood plasma, C p following a bolus injection can be described by an AIF of biexponential decay: C p (t) = d [ a 1 e m 1t + a 2 e m 2t ] (2 3) where a 1, m 1 refers to the amplitude and rate constant of the fast equilibrium between plasma and extracellular space respectively, a 2, m 2 refers to the amplitude and rate constant of the slow component of kidney clearance respectively and d is the dose of the bolus injection. The baseline washout parameters used were as follows: a 1 = 3.99 kg/l, m 1 =.114 min 1, a 2 = 4.78 kg/l, and m 2 =.111 min 1 [67, 76]. In order to study the effects of AIF parameters in the model, two different sets of AIF parameters were also used, which are as follows : a 1 = 9.2 kg/l, m 1 =.23 min 1, a 2 = 4.2 kg/l, and m 2 =.5 min 1 described the fast AIF [27]; a 1 = 13 kg/l, m 1 =.3 min 1, a 2 = 16 kg/l, and m 2 =.26 min 1 described the intermediate AIF [4]. A qualitative representation of the different AIFs are provided in Figure 2-1. In this figure, C p was normalized such that the initial concentration is the same for all the AIFs. Knowing the C p values from Equation (2 3), Equation (2 2) was then solved analytically to find an expression for C t (t) which was then fitted with the experimental values (C MRI,t ) at early time points ( 2 min) to obtain the K trans and ϕ maps. These maps were incorporated into the porous media transport model to predict the tracer distribution at later time points Mathematical Model The tissue continuum was modeled as a porous media with continuity [53] and momentum (Darcy s law) equations given by,.v = K trans J V K trans V L S L p,ly V (p p ) (2 4) L v = K p (2 5) 2

21 where v is the IFV, K trans is the average value of K trans over tumor and host tissue voxels, L p,ly is the lymphatic vessel permeability, S L /V is the lymphatic vessel surface area per unit volume which was set to zero in tumor tissue, p is the IFP, p L is pressure in the lymphatic vessels which was set to zero and K is the tissue hydraulic conductivity. J V /V is the filtration rate of plasma per unit volume of tissue into the interstitial space which is given by Starling s law as follows [68], J V V = L S p V (p v p σ T (π v π i )) (2 6) where L p is the hydraulic conductivity of the microvascular wall, S/V is the blood vessel surface area per unit volume, p v is the vascular fluid pressure, σ T is the osmotic reflection coefficient for plasma proteins, π v, π i are the osmotic pressures of the plasma and interstitial fluid, respectively. The first term on the right side of the continuity equation (Equation (2 4)) represents the fluid flux across the microvascular wall per unit volume of the tissue. The second term accounts for the lymphatic drainage from interstitial space per unit volume of tissue. Transport of interstitial Gd-DTPA was solved using the convection and diffusion equation for porous media [53], C t t + v ( ϕ. C t D 2 C t = K trans C p C ) t S L L p,ly ϕ V (p p )C t L ϕ (2 7) where D is the diffusion coefficient for Gd-DTPA. The following assumptions are made in the above equation : the diffusion coefficient is isotropic and uniform and that the dispersion coefficient is much smaller than D and there are no binding interactions between the molecules. The terms on the left side of the above equation refers to the transient, convection and diffusion fluxes respectively. The first term on the right hand side of the equation denotes the transvascular solute exchange and the second term 21

22 denotes the tracer outflux due to the lymphatics. The values of the parameters in the above equations are listed in Table 2-1. The MR image consists of voxels which niether belong to tumor or host tissue, i.e. exterior voxels which correspond to surrounding air. In these voxels, the source terms in continuity and transport equations (Equations (2 4) and (2 7)) and diffusivity was set to zero Computational Method The continuity, momentum and tracer transport equations were solved using the CFD software package, FLUENT (version 6.3, Fluent, Lebanon, NH). For the 3D computational tissue model, a rectangular volume (2 1 9 mm 3 ) enclosing the tumor was created and meshed with quadrilateral elements (voxels) of size equal to the MRI resolution ( mm 3 ) using the meshing software (GAMBIT, Fluent, Lebanon, NH), with one-to-one mapping between the CFD mesh and MR data. In the non-voxelized model [53], the geometry was meshed using an unstructured grid with approximately 2.7, 2.5 and 2.3 million tetrahedral elements for animals I, II and III respectively (Figures 2-2A and 2-2B). Governing equations were discretized with a control-volume based technique using FLUENT as done with the non-voxel approach. Within FLUENT, an user defined function was used to assign K trans and ϕ for each voxel in the mesh. For continuity and tracer transport equations, a user defined flux macro was used to account for the source terms. Standard pressure interpolation scheme was used to solve for pressure and first order upwind method was used to solve for velocity and the transport equations. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations [3]) pressure-velocity coupling method was chosen and convergence criterion was set to 1E-5. Initial conditions for tracer transport assumed no initial tracer in the tissue, C t =. A zero fluid pressure condition, p =, was applied along the cut ends and the remaining outer boundaries of the geometry were assigned as wall. 22

23 There is one difference in the modeling strategy between voxelized and non-voxelized models, the impermeability condition along the skin boundary in the voxelized model was achieved by assigning hydraulic conductivity two orders of magnitude lower than the normal tissue in the exterior voxels, while the non-voxelized model implements it by directly assigning them with a wall boundary condition with zero normal flux. The assignment of low hydraulic conductivity in the exterior voxels creates a material that is resistant to fluid motion. For the chosen value of hydraulic conductivity at the exterior voxels the mean velocity at the skin boundary was calculated to be close to zero (.1 µm/s). The effect of changing the bi-exponential arterial input function (AIF) parameters on the solutions was studied to understand the sensitivity of the voxelized model compared to its counterpart. The sensitivity analysis was performed only for animal I. Apart from the baseline value, flow and transport for two different sets of AIF (Figure 2-1) parameters was simulated. For the analysis, tracer concentration was simulated for t 2 min and the data was compared at discrete time points, t = 5, 1 and 2 min Statistical Analysis Quantitative methods compared IFP, IFV, and tracer concentration in tissue predicted by both the models. Such an evaluation required a one-to-one mapping between both the meshes (unstructured and cartesian) which was mathematically cumbersome to derive, hence a set of elements common to both the meshes were identified based on the location of their cell centers and used for the analysis. Assuming the variations in dependent variables across different non-voxel elements within a given voxel to be small, the criteria for matching was that the non-voxel element should lie within the voxel compared with. The above criteria resulted in approximately 97% match for animal I, 98 % for animal II and 98 % for animal III. After finding the matching elements in both the meshes, the values of IFP, IFV, and tracer concentration in tissue, in these elements were used for the analysis. 23

24 Additionally, a quantitative comparison was conducted between tracer concentration in tumor tissue predicted by both models and MRI-obtained experimental data at later time points. Gd-DTPA concentrations from experimental data and voxelized model were mapped to points within the tumor boundary of the non-voxelized model to compare distribution of tracer at a given time point. Various statistical measures were used to ascertain the similarity between model predictions. These include root mean square error, correlation coefficient and error histogram Root Mean Square (RMS) Error, ε The error in the magnitude of dependent variables were measured using the root mean square error which was defined as the square root of the average of the squares of the error. The RMS error for IFP and IFV were computed as shown below. For IFV, in addition to the magnitude, the RMS error of the angle between the two velocity vectors were also calculated, N ( ) 2 xvox j xnvox j j=1 ε x = N Where x was replaced with IFP, IFV magnitude, and tracer concentration, N is (2 8) the total number of matching elements, vox refers to voxel value and nvox refers to non-voxel value Pearson Product Moment Correlation Coefficient (PMCC), r Correlation coefficient was used to measure the statistical relationships between both the results. PMCC is a measure of linear dependence between two variables. It assumes that the relationship between both the variables can be best described by a linear function and it is defined as the ratio of covariance of the two variables and the product of their standard deviations. The value of the coefficient ranges from -1 to 1. A positive sign indicates that the variables increase and decrease together. A large magnitude (close to 1) implies that there is a strong linear relationship between both the variables. This is can summarized as follows, 24

25 r = 1 indicates perfect negative correlation indicates no correlation 1 indicates perfect positive correlation Error Histogram Error histograms were generated to provide a graphical representation of frequency distribution of errors in the dependent variables, which in this case was the absolute value of the difference between the computed values of voxelized and non-voxelized model. A suitable range for the error was chosen and divided into equal sized intervals (or bins). The number of occurrences of the error was then calculated for each bin and represented as a bar plot. 2.3 Results Tumor flow fields and tracer transport obtained using both computational approaches were compared using statistical analysis for all the three animal data sets, and qualitative presentation of the dependent variables (IFP, IFV and tracer concentration in tissue) for animal I using contour plots at the mid-slice supplemented with line plots along the horizontal and vertical bisectors at the mid-slice (Figure 2-3). For a detailed description of predicted fluid flow, tracer transport and sensitivity analysis in the non-voxelized tumor model, the reader is referred to Pishko et.al.,[53]. The IFP contour and line plots for the tumor predicted by the voxelized and non-voxelized model are shown in Figures 2-4A to 2-4D. The voxelized model predicted elevated IFP inside the tumor, pressure reached peak value ( kpa in animal I, kpa in animal II and kpa in animal III) at the tumor core and rapidly decreased at the tumor boundary. As expected within the tumor, predicted pressure gradients were lowest close to the tumor center ( 14.2, 38.5 and 11.2 Pa/mm in animal I, II and III respectively) and highest ( , and Pa/mm in animal I, II and III respectively) near its periphery. The pressure pattern was captured by the voxelized model. However, the line plots clearly indicated a difference in the 25

26 predicted pressures between both the models. The magnitude of peak IFP predicted by the voxelized model was found to be 15 % higher than that of its counterpart in animal I, 29 % in animal II and 18 % in animal III. Despite changes in predicted IFP between both the models, the IFV contours and line plots (Figures 2-5A to 2-5D) indicated that the distributions predicted are qualitatively similar with highest velocities (.75,.4 and.42 µm/s in animal I, II and III respectively) occurring along the tumor boundary near the cut ends. The computed IFV values were found to be lower inside the tumor (.3,.3 and.1 µm/s in animal I, II and III respectively). The low velocity regions were also observed far away from the tumor boundary. Interstitial distribution of Gd-DTPA tracer was simulated at various times (t = 5, 3 and 6 min) after infusion. The predicted tracer distribution of both the models and the actual experimental data, was heterogeneous with high concentration regions (.4,.18 and.19 mm in animal I, II and III respectively at t = 5 min) outside the tumor (Figure 2-6). It can also be observed that lowest tracer concentration (.3,.5 and.3 mm in animal I, II and III respectively at t = 5 min) occurs within the tumor. The line plots (Figure 2-7) shows that the tracer extravasation appears to be less affected by the differences in the flow field predicted by both the models. Conforming with the statistical findings, the accuracy of voxelized model s prediction with respect to its non-voxel counterpart was maintained for all the times simulated. As time proceeds, tracer concentration was reduced and the distribution became more uniform. The statistical parameters comparing both the model predictions for all the three animals are listed in Table 2-2. The statistics of the model results appeared similar across the animals. The Pearson coefficient for IFP was high (r >.7) indicating similar patterns in both the model predictions. The value of its RMS error reflected the difference in the peak pressures predicted by both the models. The low RMS error in IFV and the high correlation coefficient (r >.7) showed a reasonable degree of 26

27 similarity between both the model predictions. The RMS error in tracer concentration was maximum at initial time points and decreases thereafter with time. However, correlation coefficients did not change much with time. Error histograms for flow and tracer transport (Figure 2-8) followed an exponential distribution with peak around zero Sensitivity Analysis Results of sensitivity analysis are presented in terms of line plots along the vertical bisector of the mid-slice for flow field and tracer concentration. Similar to the baseline results, the IFP pattern predicted by the voxelized model matched with that of the non-voxel model for all AIFs, although there are differences in the predicted magnitude (Figures 2-9A and 2-9B). The predicted pressures for intermediate AIF were found to be closely matching for both the models. The IFV predicted by non-voxelized model matched well with that of its counterpart (Figures 2-9C and 2-9D). Concentrations of Gd-DTPA predicted by the voxelized model roughly followed the non-voxelized one (Figure 2-1). The accuracy of the predicted concentration did not seem to change with AIFs and time. Statistics of the sensitivity analysis are provided in Table 2-3. The correlation coefficients for IFP across the AIF s were almost identical although the RMS errors were different reflecting the differences in predicted pressures. Highest and lowest pressures were observed for the baseline and intermediate AIF respectively. It was observed that PMCCs and RMS errors in IFV were similar for the intermediate and fast AIFs. Tracer concentration statistics also exhibited a similar behaviour with almost identical RMS error values and PMCCs across the AIFs. With increasing time, the RMS error decayed for all the cases although PMCCs remain similar Validation Study Qualitatively, a similar pattern of Gd-DTPA distribution and washout was observed for the voxelized model, non-voxelized model, and experimental data over the course of 1 hr (Figure 2-6). High concentration regions were observed outside the tumor and at 27

28 the edge of the tumor just within the tumor boundary. Washout rate was compared by calculating volume-averaged Gd-DTPA concentration within the tumor for various time points and fitting the data into a mono-exponential function (Table 2-4). The voxelized and non-voxelized model both compared well with the experimental data. RMS error was calculated for both models throughout the entire tumor volume as well as error frequency histograms (Figure 2-11) to illustrate the comparison of the models with the experimental data in space and time. Both the voxelized and non-voxelized models showed low RMS error and high error frequency close to zero. However, for animal III there was slightly higher RMS error in the voxelized and non-voxelized model predictions with the experiment, eventhough the washout rate was very close with the experimental data. 2.4 Discussion A voxelized modeling approach was used to study the transport of Gd-DTPA following systemic injection in tumors. Benefits of this methodology include easier and more rapid building of computational porous media transport models compared to traditional CFD approaches which involves complex geometric reconstruction. Thus the voxel model is less labor intensive and potentially simpler to implement. Spatially-varying tissue transport properties and realistic anatomical tissue geometries were incorporated into a three-dimensional, image-based computational model. The porous media simulation predicted interstitial fluid pressure, interstitial fluid velocity, and tracer transport through the tissue interstitium. These results were compared with that obtained using a non-voxel approach [53]. The sensitivity of the voxelized model for different AIFs was investigated and compared with the non-voxel model. The voxelized and non-voxelized model s predictions of tracer distribution within the tumor were compared to MRI-determined tracer distribution and the voxelized model was further evaluated with additional animal data. 28

29 The voxelized model predicted elevated IFP conforming with the experimental observations [11, 13, 23, 48, 77] and previous modeling results [7, 8, 53, 78]. However, it can be found that the IFP predicted by the voxelized model was higher than that predicted by the non-voxelized model. The value of RMS error reflects this difference as it can be interpreted as the standard deviation between two variables. This discrepancy can be explained by the differences in the tumor volume in both the models. The tumor volume approximated by the voxelized model was found to be higher than that of the non-voxel model (39 % higher in animal I, 83 % in animal II and 36 % in animal III). This is due to the differences in the meshing strategy of the models. The non-voxelized model used variable sized elements (unstructured mesh) which likely approximates the tumor volume slightly better than its voxel counterpart which relies only on fixed size elements (cuboids). This effect was particularly more pronounced in animal II which has the smallest tumor volume of all. Since IFP values are found to be correlated with the tumor volume, with higher IFP for large tumors [23], the voxelized model with higher tumor volume is expected to have IFP higher than the non-voxelized one. The lower differences observed in the predicted IFP by both the models for animals II and III compared to animal I, could be attributed to their actual differences in the tumor volume approximated by both the models. The additional tumor volume for animals II and III in the voxelized model was an order of magnitude lower than that in animal I, thus resulting in smaller change. These differences in the predicted IFP by both the models does not have much effect on the predicted extracellular flow which is driven by the IFP gradient which was similar in both the cases. It should be noted that the correlation coefficient can also be interpreted as the degree of similarity between the slopes of two variables, in other words a similarity index for the gradients of the variables. From Table 2-2, it is clear that they are high for all the three animals, thereby indicating the high degree of similarity in the IFP gradient computed by both the models thus supporting this previous 29

30 argument. The exponential error distribution for IFP with high frequency near zero error implies the decaying nature of the number of voxels with higher errors. The IFV predicted by the voxelized model reflects previous experimental finding [16]. In the experimental study conducted by DiResta and his colleagues, IFV in human neuroblastoma was found to increase from the center towards the periphery of the tumor. This is due to the high pressure gradient at the tumor boundary which increases the IFV. On the other hand a more uniform pressure distribution in the tumor core leads to low velocities in those regions. Statistical parameters obtained for IFV indicate a higher degree of similarity between the predictions of both the models. As mentioned previously, the IFV driven by the IFP gradient is less affected by the changes in the predicted pressure. This is also reflected in their error histograms, it can be observed that the error decays more rapidly than that of IFP, thus indicating the high accuracy of the voxelized model in predicting IFV. Distribution of Gd-DTPA was heterogeneous due to spatially varying deposition and limited interstitial transport by diffusion and convection. The low tracer concentration inside the tumor is consistent with the reduced fluid filtration and high IFP. As the concentration is advected through the velocity field, its correlation coefficient is similar to that obtained for IFV. Error histograms also reflect this behaviour, a strong peak around zero clearly shows the reliability of voxelized model in predicting the tracer concentration despite some changes in the predicted flow field. It should be noted that, although there are differences between the results obtained through both the models, the voxelized model faithfully captures the tracer extravasation which is essential for any drug delivery model. In sensitivity analysis, the effects of varying AIF parameters were also investigated. It has been found that the flow and transport are sensitive to these parameters [53]. Changes in the flow and transport can be mainly attributed to the differences in the K trans and ϕ maps. The sensitivity of voxelized model compared with that of its 3

31 counterpart seems to be the same for different AIFs as indicated by similar correlation coefficients and RMS errors. This analysis also demonstrates the applicability of the model in a diverse set of conditions. Overall, the sensitivity of voxelized model was similar to its non-voxelized counterpart. The validation study revealed that the Gd-DTPA distribution results obtained via non-voxelized and voxelized models were consistent with the experimental observations. This became more clear when the voxelized model predictions were similar to non-voxelized predictions and experiment, across the tumors. The slightly high RMS error in animal III could be due to errors in AIF parameters which varies across the tumors and to which the results are sensitive. However, the washout rate was accurately predicted by the voxelized model. This outcome lends credence to the usage of voxelized porous media tumor models for predictions of low molecular weight tracers and drug distribution. However, matching the modeling results with the actual experimental values is difficult due to the presence of a large number of model parameters which need to be determined experimentally. Also the differences in the grid sizes between the non-voxelized model (approximately million elements) and voxelized model (approximately 165, elements which is just 6-7% of that in the non-voxel mesh) may also account for discrepancies between them. The low resolution of the voxelized model is due to limitations of MRI resolution as its data are directly mapped into the model. Non-voxelized models on the other hand are more flexible in this aspect as they do not directly map the MR data, thus allowing for variable resolution. The non-voxel model [53] was also used for extensive sensitivity analysis requiring it to capture steep pressure gradients at the tumor boundary, hence the mesh size was increased for attaining convergence in FLUENT. The current voxelized model was aimed at gaining an overall understanding of the fluid flow and transport in tumors, and providing a reliable alternative to the non-voxelized approach. 31

32 High correlation coefficients between the voxelized and non-voxelized model results indicates that both the results are in agreement with each other. However, there is some disparity in the results especially the IFP which can be attributed to one of the basic differences between both the approaches, mesh structure. The voxelized model uses an uniform and rectangular mesh while the non-voxelized model uses an unstructured mesh. The type of mesh used can affect the solution in two ways : (1) tumor/host tissue volume approximation, (2) resolution. Using the cartesian mesh, the voxelized model approximates the tumor and host tissue volumes with rectangular elements thereby neglecting curvature at tissue boundaries while the non-voxelized model with its variable size elements can account for this. This results in slight differences in tumor and host tissue volumes which in turn affects the solution as the tumor shape and size are important factors determining the interstitial fluid flow [17, 58]. The mesh density also affects the solution as it determines the discretization of the domain with better resolution in finer meshes and vice-versa. In this aspect, the voxelized model has much lesser mesh density compared with its non-voxelized couterpart resulting in a lower resolution as mentioned earlier. It should however be noted that the usage of very fine meshes is computationally intensive and time consuming. Despite these differences, voxelized model was still able to capture key features in the flow and transport thus making it a attractive alternate candidate for tumor modeling. 32

33 Table 2-1. Tissue and vascular parameters used for simulating distribution of Gd-DTPA following bolus tail vein injection at the hind limb tumor in a mice. Variable Description Value References L p (m/pa.s) Vessel permeability t ; n ; [53] S/V (m 1 ) Microvascular surface area per unit volume 2 t ; 7 n [7] L p,ly S L /V (m 1 ) Lymphatic filtration coefficient [53] K(m 2 /Pa.s) Hydraulic conductivity t ; n [53] e p v (Pa) Microvascular pressure 23 [53] π i (Pa) Osmotic pressure in interstitial space 323 t ; 133 n [53] π v (Pa) Osmotic pressure in microvasculature 267 [53] σ T (Pa) Average osmotic reflection coefficient for plasma.82 t ;.91 n [53] D(m 2 /s) Diffusion coefficient of Gd-DTPA [53] t - tumor, n - normal tissue, e - exterior. 33

34 Table 2-2. Statistical parameters obtained while comparing voxelized and non-voxelized model results for the baseline simulation in three animals. Variable Quantity Animal I (Baseline) Animal II Animal III ε r ε r ε r IFP Magnitude Pa Pa Pa.97 IFV Magnitude.7 µm/s.81.1 µm/s.89.2 µm/s.92 Direction C t At t = 5 min.1 mm.79.1 mm.71.8 mm.74 At t = 3 min.5 mm.79.6 mm.75.4 mm.77 At t = 6 min.4 mm.78.3 mm.77.3 mm.77 34

35 Table 2-3. Statistical parameters obtained while comparing voxelized and non-voxelized model results for intermediate and fast arterial input function in animal I. Variable Quantity Intermediate AIF Fast AIF ε r ε r IFP Magnitude Pa Pa.99 IFV Magnitude.12 µm/s µm/s.78 Direction C t At t = 5 min.1 mm.72.7 mm.68 At t = 1 min.7 mm.75.7 mm.72 At t = 2 min.6 mm.75.3 mm.74 35

36 Table 2-4. Comparison of tracer washout rates and root mean square error in tracer concentration within the tumor volume between voxelized and non-voxelized model results with experiment in three animals. Washout rate of volume-averaged Gd-DTPA concentration within tumor volume (min 1 ) RMS error for concentration within tumor volume (mm) Animal Case t = 5 min t = 3 min t = 6 min I Experimental -.31 Voxelized Model Non-voxelized Model II Experimental -.2 Voxelized Model Non-voxelized Model III Experimental -.25 Voxelized Model Non-voxelized Model

37 Normalized plasma concentration Baseline Intermediate Fast Time (minutes) Figure 2-1. Normalized concentration of tracer in blood plasma approximated by different AIFs used for sensitivity analysis 37

38 A B Figure 2-2. CFD compatible meshes. (A) Schematic of voxelized (cartesian) mesh (B) Unstructured mesh of reconstructed hind limb. Includes tumor (light green), skin (green), cut ends (yellow), and representation of mid-slice (dark blue). 38

39 A B Figure 2-3. Horizontal and vertical lines used for plotting the flow field and tracer transport in voxelized (A) and non-voxelized (B) models 39

40 A B Non Voxel Voxel Non Voxel Voxel IFP (Pa) IFP (Pa) Distance (mm) C Distance (mm) D Figure 2-4. Contours of IFP predicted by (A) voxelized model (B) non-voxelized model. Tumor and skin boundaries are overlaid on the contours. Also included, line plots comparing the predicted IFP (C & D) by both the models along the horizontal and vertical bisectors in the mid-slice respectively. The tumor and skin boundaries are represented by dashed and dash-dot lines respectively. 4

41 A B.6 Non Voxel Voxel.6 Non Voxel Voxel.5.5 IFV (µm/s).4.3 IFV (µm/s) Distance (mm) Distance (mm) C D Figure 2-5. Contours of IFV predicted by (A) voxelized model (B) non-voxelized model. Tumor and skin boundaries are overlaid on the contours. Also included, line plots comparing the predicted IFV (C & D) by both the models along the horizontal and vertical bisectors in the mid-slice respectively. The tumor and skin boundaries are represented by dashed and dash-dot lines respectively. 41

42 t = 3 min t = 6 min Non-voxelized Voxelized Experimental t = 5 min Figure 2-6. Comparison of tracer concentration contours. Voxelized and non-voxelized model compared with MR-derived tissue concentration at t = 5, 3, and 6 min. Tumor and skin boundaries are overlaid on the contours. 42

43 Non Voxel Voxel Experimental Non Voxel Voxel Experimental C t (mm) C t (mm) Distance (mm) Distance (mm) A B Non Voxel Voxel Experimental Non Voxel Voxel Experimental C t (mm) C t (mm) Distance (mm) Distance (mm) C D Non Voxel Voxel Experimental Non Voxel Voxel Experimental C t (mm) C t (mm) Distance (mm) Distance (mm) E F Figure 2-7. Line plots comparing the predicted tracer concentration in the tissue by both the models with experiment, along the horizontal and vertical bisectors of mid-slice at t = 5 (A & B), 3 (C & D), 6 (E & F) min respectively. The tumor and skin boundaries are represented by dashed and dash-dot lines respectively. 43

44 Frequency Frequency 15 1 Frequency Pressure (Pa) Velocity magnitude (µm/s) Velocity direction (degrees) A B C Frequency Concentration (mm) D Frequency Concentration (mm) E Frequency Concentration (mm) F Figure 2-8. Error Histograms for flow (A, B & C) and transport at t = 5 (D), 3 (E) and 6 (F) mins in baseline simulation for voxelized model with respect to non-voxelized model 44

45 7 65 Non Voxel Voxel 7 65 Non Voxel Voxel 6 6 IFP (Pa) 55 5 IFP (Pa) Distance (mm) A Distance (mm) B.8.7 Non Voxel Voxel.8.7 Non Voxel Voxel.6.6 IFV (µm/s) IFV (µm/s) Distance (mm) Distance (mm) C D Figure 2-9. Line plots comparing the IFP (A & B) and IFV (C & D) predicted by both the models for two different AIF parameter sets (intermediate and fast) along the vertical bisector of mid-slice respectively. The tumor and skin boundaries are represented by dashed and dash-dot lines respectively. 45

46 .5.4 Non Voxel Voxel.5.4 Non Voxel Voxel C t (mm).3.2 C t (mm) Distance (mm) Distance (mm) A B.5.4 Non Voxel Voxel.5.4 Non Voxel Voxel C t (mm).3.2 C t (mm) Distance (mm) Distance (mm) C D Figure 2-1. Line plots comparing the tracer concentration in the tissue predicted by both the models for two different AIF parameter sets (intermediate and fast) along the vertical bisector of mid-slice at t = 5 (A & B) and 2 (C & D) min respectively. The tumor and skin boundaries are represented by dashed and dash-dot lines respectively. 46

47 t = 5 min t = 3 min t = 6 min Voxelized Frequency Frequency Frequency Concentration (mm) Concentration (mm) Concentration (mm) Non-voxelized Frequency Frequency Frequency Concentration (mm) Concentration (mm) Concentration (mm) Figure Error Histograms for tracer concentration within the tumor for voxel and non-voxel model results with respect to the experimental data at t = 5, 3 and 6 min. 47

48 CHAPTER 3 APPLICATION OF VOXELIZED MODEL FOR CONVECTION-ENHANCED DELIVERY IN A HIND LIMB TUMOR 3.1 Overview Cancer treatments based on systemic delivery of therapeutic agents are often hindered due to the poor and uneven uptake of the drugs within the tumor. The unique characteristics of the tumor microenvironment are known to be an important factor affecting the efficacy of the anti-cancer treatments such as chemotherapy. Tumors are known to exhibit elevated interstitial fluid pressure (IFP) [11, 13, 23, 48, 77] and irregular microvasculature [19, 31, 33] which leads to inadequate uptake and heterogeneous extravasation of drugs [2] respectively, consequently lowering their therapeutic index. In the recent years, localized drug delivery has emerged as a plausible alternative to systemic delivery for transporting macromolecular therapeutic agents to the tumors [18, 22, 56, 72 74]. By directly injecting into the tumor, this circumvents the previously mentioned vascular and interstitial barriers and also reduces the side-effects associated with systemic exposure. Amongst the available techniques, convection-enhanced delivery (CED) appears promising because at a given time it can achieve larger distribution volumes than by diffusion alone [2, 1]. In CED, an infusion pump delivers the drug at constant flow rate or pressure thereby utilizing the bulk flow due to the infusion pressure difference, to deliver and distribute macromolecules to larger volumes in the tissue. Since its advent, CED has been used for in situ delivery of a wide range of substances including nanoparticles [5], liposomes [44, 56], cytotoxins [57] and viruses [24, 62]. Experimental studies on CED of liposomes into brain tumor (glioma) in rats were encouraging, it was found that the technique effectively distributed the liposomes in the tumor and the surrounding normal tissue [56]. A broad anisotropic distribution was reported to have resulted from CED of cytotoxins into human gliomas [57]. Such an asymmetric distribution was also reported by Boucher and his colleagues in 48

49 their study involving intratumoral infusion of Evans blue-albumin in saline into sarcoma HSTS 26T [12]. It should however be noted that spherically symmetric distributions for colon adenocarcinoma LS174T were also reported in their study. Computational modeling of CED has gained attention recently partly because it could help in planning and optimizing patient-specific treatments. Earlier theoretical models were focused on predicting drug distributions following CED in mediums like agarose gels/ brain tissue [15, 38, 39, 43]. For tumors, Smith and Humphrey developed a theoretical model in which infusions in a spherical tumor with a necrotic core was simulated [59]. A main objective of their study was to analyze the effect of transvascular fluid exchange on the flow field during the infusion. They found that the flow field was very sensitive to the ratio of vascular conductivity and hydraulic conductivity, and infusion close to the tumor was retarded by the outward flow. Weinberg and his colleagues developed a finite element model to predict the distribution of doxirubicin following intratumoral delivery [75]. However, the convective effects in the model were replaced with a elimination coefficient instead of the actual interstitial fluid velocity (IFV). It should be noted that these models utilized theoretical tumor microvasculature and simplified tumor geometries. Patient-specific computational porous media models, incorporating realistic geometries and spatially varying transport properties obtained through MRI, for predicting drug distributions have been developed by our group [38, 39, 45, 51 53, 78]. For tumors in particular, our group developed a framework which accounts for the actual tumor microvasculature by using DCE-MRI data to estimate the spatial variation of transport properties (rate transfer constant between plasma and extracellular space, K trans and porosity, ϕ) which were included in a porous media model to solve for flow and transport using computational fluid dynamics (CFD) techniques [45, 53, 78]. In this study, this method was used to predict the distribution of albumin in a murine sarcoma following CED as opposed to systemic delivery in the aforementioned papers. 49

50 In particular, CFD simulations were carried out based on a voxelized modeling approach described in the previous chapter, where it was shown that the predicted flow field and transport using this approach was similar to that of a more traditional approach based on unstructured meshes. Earlier, this methodology has also been used by our group to model CED in rat spinal cord and brain tissue [38, 39]. In this approach, anisotropic tissue properties and anatomical boundaries are assigned on a voxel-by-voxel basis using MRI data. These properties are then incorporated into a porous media transport model to predict IFP, IFV and tracer concentrations. These voxelized models allow for quicker building of computational transport models and rapid estimation of concentration profiles. In this study, a DCE-MRI based voxelized model was developed for predicting albumin tracer distribution following CED in the lower limb of a mouse (C3H) inoculated with murine sarcoma cells (KHT). The model accounting for the heterogeneous tumor microvasculature could potentially help optimize patient-specific treatments with its realistic predictions, and understand the biophysical IFP and IFV changes due to CED, which are otherwise difficult to measure experimentally. A sensitivity analysis was performed to study the effects of varying hydraulic conductivity maps and catheter placement on fluid flow and albumin tracer transport. This was done to understand the sensitivity of the model and relate them to key factors contributing to CED. The choice of varying hydraulic conductivity is because of its direct influence on the tumor IFP and convective flow field in intratumoral infusions. The higher values of hydraulic conductivity were thought to reduce IFP thereby increasing the filtration of fluids and extravasation of macromolecules [12]. The effect of catheter placement was known to be very important in CED [2]. Studies involving infusions in different sites in the brain have revealed the presence of a optimal site for achieving maximum distribution volume at the targetted area [39, 43]. In the current study, infusions were carried out separately at two different sites in the tumor namely at the tumor-host tissue interface and anterior end of the 5

51 tumor, in addition to the tumor center. The reason for choosing a site at the tumor-host tissue interface was because of the presence of higher convective effects in that region due to the sudden decrease in IFP which could result in higher IFV. The choice of an infusion site at the anterior end and center of the tumor was to study the distribution at various positions inside the tumor Mathematical Model 3.2 Methods The study was divided into two parts : First the spatially-varying transport properties of the KHT murine sarcoma were found through DCE-MRI following bolus tail vein injection of MR visible tracer gadolinium-diethylene-triamine penta-acetic acid (Gd-DTPA, MW 59 Da). The methods for obtaining DCE-MRI derived data such as Gd-DTPA concentration in tissue, rate transfer constant (K trans ) maps, and porosity (ϕ) maps are identical to those in Pishko et.al., [53]. The second part involves incorporating the above calculated variable transport properties into the computational porous media model for flow and transport by CED. The tissue continuum was modeled as a porous media and the governing equations were solved at each voxel after assigning their respective K trans and ϕ values. The continuity equation is given by,.v = Q V inf At the infusion voxel (3 1) = K trans J V K trans V L S L p,ly V (p p ) L At all other voxels in tumor and host tissue (3 2) where v is the IFV, Q is the infusion flow rate of albumin, V inf is the volume of the infused voxel, K trans is the average value of K trans in host and tumor tissue voxels, L p,ly is lymphatic vessel permeability, S L /V is the lymphatic vessel surface area per unit volume which was set to zero in tumor tissue, p is the IFP and p L is pressure in the lymphatic vessels which was set to zero. J V /V is the filtration rate of plasma per unit volume of 51

52 tissue into the interstitial space which is given by Starling s law as follows [68], J V V = L S p V (p v p σ T (π v π i )) (3 3) here L p is the hydraulic conductivity of the microvascular wall, S/V is the blood vessel surface area per unit volume, p v is the vascular fluid pressure, σ T is the osmotic reflection coefficient for plasma proteins, π v, π i are the osmotic pressures of the plasma and interstitial fluid, respectively. The first term on the right side of the continuity equation for voxels that are not infused with albumin (Equation (3 2)) represents the transvascular fluid flux across the microvascular wall per unit volume of the tissue, scaled by the normalized K trans to account for the heterogeneity in the model. The second term accounts for the lymphatic drainage from interstitial space per unit volume of tissue. For a porous medium, the momentum equation is given by Darcy s law, v = K p (3 4) where K is the hydraulic conductivity which is likely to be heterogeneous in tumors and can vary with the local changes in porosity of the tissues [3, 41, 61, 7]. In particular Lai and Mow [41], proposed an exponential variation of hydraulic conductivity with deformation which in turn was related to porosity. By using a similar relation, the exponential term was normalized with its mean value to ensure that the mean hydraulic conductivity calculated over the tumor/host tissue voxels equals their baseline values. The resulting expression is given as follows, Kt e m(ϕ+.1) N t N t e m(ϕ i +.1) i=1 K = Kh em(ϕ+.1) N h N h e m(ϕ i +.1) i=1 For tumor For host (3 5) 52

53 where N t, N h are the number of tumor and host tissue voxels respectively, Kt, Kh are the baseline hydraulic conductivities of tumor and host tissues respectively and m is an empirical exponent. Albumin (MW 66, 776 Da) is a non-binding and non-reacting macromolecule which is widely used as tracer in CED studies. Assuming no tissue sources and sinks since large molecular weight albumin is not expected to go back into the capillaries, transport in the tissue was given by the convection and diffusion equation, C t t + v ϕ. C t D eff 2 C t = (3 6) where C t is the concentration of tracer in the tissue, D eff is the effective diffusivity of albumin in the porous medium given by the following empirical relation based on diffusion in porous media [21], D eff = D free ϕ n (3 7) where D free is the self-diffusion coefficient of albumin in water and n is an empirical exponent set to 4. The concentration in the equation was normalized using the following relation, Ĉ = C t C (t,i) ϕ i (3 8) where C (t,i) and ϕ i are the infusate concentration and porosity of the infused voxel respectively. The values of the parameters in the governing equations are listed in Table 3-1. The MR image also consisted of voxels present outside the mouse which belong neither to tumor or host tissue, i.e. exterior voxels. In these voxels, the whole source term for the continuity equation and the diffusivity were set to zero Computational Method The continuity, momentum and albumin transport equations were solved using the CFD software package, FLUENT (version , ANSYS, Inc., Canonsburg, PA). For 53

54 the 3D computational tissue model, a rectangular volume (2 1 9 mm 3 ) enclosing the tumor was created and meshed with quadrilateral elements (voxels) of size equal to the MRI resolution ( mm 3 ) using the meshing software (GAMBIT, Fluent, Lebanon, NH) with one-to-one mapping between the CFD mesh and MR data. The amount of tumor and host-tissue contained in the resulting volume were calculated to be and mm 3 respectively with the exterior voxels occupying the rest. Governing equations were discretized with a control-volume based technique using FLUENT. Darcy s law was substituted for the conservation of momentum equation. Within FLUENT, a user-defined function was used to assign K trans, porosity, hydraulic conductivity and diffusivity for each voxel in the mesh. For the continuity equation, a user-defined flux macro was used to account for the source terms. A standard pressure interpolation scheme was used to solve for pressure and a second-order upwind method was used to solve for the flow equations. The SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent [71]) pressure-velocity coupling method was chosen. The transport equation was set-up using the user defined scalar (UDS) equation in FLUENT and solved using first order upwind method. The convergence criterion for all the three equations was set to.1. Infusion simulations were carried out upto t = 2 hrs and the interstitial distribution of albumin was simulated at intermittent time points, t = 5, 3, 6 and 12 mins. Initial conditions for tracer transport assumed no tracer in the tissue, Ĉ = except at the infusion site which is one voxel ( mm 3 ), where it was set to a normalized value of 1 at all the times through an user-defined function which was fed in during the transport simulation. The distribution volume was calculated as the volume occupied by voxels having an albumin concentration greater than 1 % of the infusion concentration [1]. A zero fluid pressure condition, p =, was applied along the cut ends and the remaining outer boundaries of the geometry were assigned as wall. The impermeability condition along the skin boundary was achieved by assigning hydraulic 54

55 conductivity two orders of magnitude lower than the normal tissue, in the exterior voxels. The assignment of low hydraulic conductivity in the exterior voxels creates a material that is resistant to fluid motion. For the chosen value of hydraulic conductivity at the exterior voxels the mean velocity at the skin boundary was calculated to be close to zero (.1 µm/s). The infusion at the center of the tumor with locally constant hydraulic conductivity (m = ) was taken as the baseline case (Figure 3-1). For comparison, the flow field was also simulated for the systemic delivery of albumin by neglecting the infusate source term (Equation (3 1)), in the continuity equation. The effect of changing the hydraulic conductivity was achieved by varying the empirical exponent (m) in the expression for hydraulic conductivity. Apart from the baseline value (m = ), flow and transport for two different values of m = 5 and 9 (Figure 3-2) were also simulated. The effect of catheter placement on the distribution was also studied through infusions at the tumor-host tissue interface and anterior end of the tumor with m =, in addition to the baseline simulation at the tumor center. The vessel permeability and diffusivity was not included in the sensitivity analysis based on the results of our previous study on transport in tumors [53], where these parameters were found to be insensitive to tracer transport. Moreover, diffusion being a slow process, changes in diffusivity is not expected to affect the tracer distribution in the small time window (2 hrs) under study. The changes in flow rate is also not expected to affect the transport as the model does not have any mechanism for back flow and other associated effects. 3.3 Results The baseline results along with the sensitivity analysis for the model are provided. The predicted IFP for systemic and local infusion are represented by contour plots at the mid-slice of the tumor as shown in Figures 3-3A and 3-3B. The local infusion at.3 µl/min increased the pressure at the infusion site by approximately 1.27 kpa. The voxelized model predicted elevated IFP inside the tumor than the host tissue. The 55

56 contour plots reveal a local increase in IFP at the infusion site which masked the high pressure inside the tumor compared to the host tissue. At the tumor mid-slice, the magnitude of the pressure gradient was maximum at the infusion site ( 4.85 kpa/mm) although significant values were also observed along the tumor-host tissue interface ( kpa/mm). The extracellular fluid velocity (EFV, v ) for systemic and local infusion, is shown ϕ by a contour along the tumor mid-slice (Figures 3-3C and 3-3D). This is further supplemented by a cone plot depicting the velocity vectors colored by its magnitude for the whole leg with local infusion (Figure 3-3E). Higher velocity regions were observed near the infusion site for local infusion. At the tumor mid-slice for local infusion, peak velocities were observed at the point of infusion ( 36 µm/s) followed by significant velocities at the tumor-host tissue interface ( µm/s). There was also side-ways flow of the fluid along the skin boundary closer to the tumor. The contours of the normalized albumin concentration at various time points, at the tumor mid-slice are shown in Figures 3-4A to 3-4C. The predicted distribution of albumin over time was asymmetric reflecting the anisotropic flow field. The effect of the skin boundary condition near the tumor on the distribution pattern was evident at later time points with a gradual outward flux of albumin along the skin boundary closer to the tumor. An iso-surface at the distribution volume threshold (.1) for times t = 3, 6 and 12 mins shown in Figures 3-4D to 3-4F, depicts the evolution of the concentration profile with time. The iso-surfaces confirms the asymmetric nature of the distribution and the side-wise flux of albumin along the skin boundary near the tumor. After two hours of infusion at.3 µl/min, albumin was distributed to approximately 58 % of the tumor volume. The variation of distribution volume (V d ) with infusion volume (V i ) within the whole leg and tumor in particular, is shown in Figure 3-5. The results data indicate that the distribution volume varies linearly with the infusion volume for the whole leg. 56

57 However the variation was slightly non-linear within the tumor. The ratio V d /V i obtained through linear fit was found to be 2.9 for the whole leg and.71 for the tumor Sensitivity Analysis Similar to the baseline results, the model predicted higher IFP for m = 5 and 9 although the peak pressure values were different (Figure 3-6). The simulation results indicated an 48 and 75 % reduction in the peak IFP from its baseline value for m = 5 and 9 respectively in the tumor mid slice. Increasing the value of m lowered the peak IFP inside the tumor and the convection velocity became more heterogeneous with increasing m. The increase in m appeared to reinforce fluid pathways with higher porosities. The velocity vector plot reveals the increase in flow in the coronal plane at m = 5 compared to the baseline value. This phenomenon became more visible at m = 9 where there was a large outflow from the tumor. The fluid leakage across the skin boundary closer to the tumor was present at both values of m. The predicted evolution of the distribution volume over time for different values of m is shown in Figure 3-7. The convective effects were apparent on the shapes of the distribution volume, at m = 5 the distribution pattern tends to get more skewed into the tumor than the baseline value. However as time proceeds, the albumin tracer tends to go away from the tumor. A similar pattern was observed at m = 9 for initial time points but the distribution got more heterogeneous and outward from the tumor as time progressed. The distribution volume in the whole leg varied linearly with infusion volume for m = 5 and 9 with slopes equal to 3.8 and 4.7 respectively (Figure 3-8A). However the variation within the tumor, tends to become non-linear at later time points (Figure 3-8B). At later time points, increasing the m decreased the distribution volume within the tumor. This effect became more apparent for larger values of m. For m = 5, two hours of infusion at.3 µl/min resulted in covering approximately 55 % of the tumor volume. Whereas for m = 9, approximately 43 % of the tumor volume was covered by the tracer. 57

58 The effect of catheter placement on albumin distribution is shown as contours at the tumor mid-slice in Figure 3-9. An asymmetric distribution was observed for infusions at both the locations, tumor-host tissue interface and anterior end of the tumor. Infusion at the interface tends to distribute albumin more along the dorsal side whereas at the anterior end it was more skewed towards the anterior side of the leg. For the whole leg, the results data indicated a linear variation of V d with V i, with higher distribution volume for infusion at the interface than at the anterior end of the tumor (Figure 3-1A). Within the tumor, infusion at the interface resulted in covering approximately 58 % of the tumor while infusion at the anterior end resulted in approximately 18 % (Figure 3-1B). 3.4 Discussion A computational model for predicting distribution of a macromolecular protein tracer following CED in the hind limb tumor of a mice using voxelized modeling approach was developed. This approach accounted for realistic tumor microvasculature and geometry, and allowed for more easier and rapid building of computational porous media transport model compared to traditional approaches utilizing unstructured meshes involving complex geometric reconstruction. This makes the model less labor intensive and easier to implement. Spatially-varying tissue transport properties based on the actual heterogeneous tumor microvasculature, tissue structure and natural anatomical tissue geometries were incorporated into a three-dimensional, image-based computational porous media model. The model solves for interstitial fluid pressure, interstitial fluid velocity, and albumin concentration through the tissue interstitium, following CED. The sensitivity of the model for different hydraulic conductivity maps and catheter placements were investigated. The predicted IFP reflected the previous experimental findings which suggested elevated pressures inside the tumor [11, 13, 23, 48, 77]. However, the infusion induced a local pressure gradient thereby exhibiting the advantage convection gives in distributing albumin to larger tissue volumes following CED. Except at the infusion 58

59 site, the pressure was uniform inside the tumor and dropped steeply in its periphery in agreement with the previous findings [29]. Outside the tumor, the boundary condition played a critical role in determining IFP. The close proximity of the tumor to one portion of the impermeable skin resulted in a pressure gradient near that portion of the skin, approximately four times higher than at the skin farther from the tumor. The convection velocity field predicted by the model reflected the computed IFP as the flow is driven by the pressure gradient. The high pressure gradient at the infusion site and tumor-host tissue interface, resulted in higher velocities in those regions. In addition, the presence of skin closer to the tumor was predicted to affect the flow field causing side-ways flow of the fluid along the skin. The linear variation of distribution volume with the infusion volume captured by the model is in accordance with the previous experimental finding by Saito and his colleagues [56]. They reported such a trend for CED of liposomes in rat gliomas. The value of the ratio was known to depend on many factors but not limited to infusate properties, extracellular matrix (ECM) among others, and a wide range of values from 1 to 8.7 has been reported in the literature [2]. Furthermore, the distribution of albumin was asymmetric and heterogeneous conforming with the previous experimental findings [12, 46, 57]. Such a distribution is the result of the flow field which advects the albumin in pathways of least resistance (higher porosity). The distribution pattern was closely interlinked with the flow field with high concentration at the infusion site and gradual leakage of albumin along the skin boundary closer to the tumor. At the end of two hours, CED was able to cover approximately 58 % of the tumor volume thereby exhibiting the effectiveness of the method in delivering macromolecular drugs. In this study, we investigated the possibility of reducing the tumor IFP by increasing the sensitivity of tissue hydraulic conductivity to tissue porosity. This was also done to increase the heterogeneous transport. Mathematically this was implemented by varying the empirical parameter m in the expression for hydraulic conductivity. Increasing the 59

60 hydraulic conductivity has been previously thought to reduce IFP and thus increase extravasation of macromolecules [12]. The results of the sensitivity analysis at m = 9 indicated that the peak tumor IFP got reduced by approximately four times and the resulting distribution volume increased by approximately 6 % from the baseline after two hours of infusion. However this effect was not reflected in the computed distribution volume within the tumor, which gradually reduced with time for higher values of m. This is because of the very high reduction in the resulting tumor IFP, which directs the interstitial fluid and albumin tracer away from the tumor. This demonstrates the importance of measuring the parameter m for a given tumor, for achieving accurate tracer distribution within the tumor. The sensitivity analysis was extended to study tracer distribution at different catheter positions, in an attempt to find an optimal placement which could maximize distribution volume in the target site. The infusions were carried out at two other sites in addition to the baseline position: tumor-host tissue interface and anterior end of the tumor. For the given set of baseline parameters, we found that the infusion at the tumor-host tissue interface produced the maximum distribution volume for the whole leg. This is due to the presence of larger convective effects around the periphery as opposing to just one at the infusion site. It should however be noted that the distribution volume within the tumor was almost identical to the baseline value. The increased convective effect apparently did not have significant effect on the distribution volume within the tumor. The outward flow of albumin from the tumor for infusions at the anterior end of the tumor is due to proximity of site to the cut ends of the tumor where zero pressure boundary condition was specified. Similar pattern can be expected for infusions at the posterior end of the tumor. By increasing the flow rate or infusion time and/or testing additional infusion sites, the model can be used to find an optimal catheter placement for a given tumor to achieve total coverage. In this way, the model can help in surgical planning by providing effective treatment strategies on a case-by-case basis. 6

61 To our knowledge, this is the first image-based tumor model that incorporates the actual tumor microvasculature and predicts heterogeneous/asymmetric drug distribution following CED. Our model serves as a potential tool for conducting and optimizing patient-specific treatments. Altering the extracellular matrix (ECM) of the tumor is being explored by researchers as a possible technique to acheive better distribution [29]. Several compounds such as VEGF inhibitors, hyaluronidase, mannitol among others were used to disrupt the heterogeneous tumor microvasculature and normalize it, thereby improving drug delivery and efficacy [29, 37]. This model could be used to study these effects once the transport properties (K trans and ϕ) of the resulting ECM were found using the methods described in [53]. Although the results discussed in this study were restricted to the hind limb tumor under study, it should be noted that the applicability of voxelized model to a wide range of tumors is possible. The relative ease in implementing the model and its reasonable predictions, makes it a promising candidate for predicting drug distributions following CED in tumors. 61

62 Table 3-1. Tissue and vascular parameters used for simulating distribution of albumin following convection-enhanced delivery at the hind limb tumor in a mice. Variable Description Value References L p (m/pa.s) Vessel permeability t ; n ; [53] S/V (m 1 ) Microvascular surface area per unit volume 2 t ; 7 n [7] L p,ly S L /V (m 1 ) Lymphatic filtration coefficient [53] K (m 2 /Pa.s) Baseline hydraulic conductivity t ; n [53] e p v (Pa) Microvascular pressure 23 [53] π i (Pa) Osmotic pressure in interstitial space 323 t ; 133 n [53] π v (Pa) Osmotic pressure in microvasculature 267 [53] σ T (Pa) Average osmotic reflection coefficient for plasma.82 t ;.91 n [53] D free (m 2 /s) Self diffusion coefficient of albumin [5] Q (µl/min) Infusion flow rate.3 [39] t - tumor, n - normal tissue, e - exterior. 62

63 Figure 3-1. Depiction of baseline CED simulation. The size of the infusion needle was exaggerated for clarity. Includes skin (green), tumor (blue), tumor mid-slice (red) and infusion needle (magenta) 63

64 1 9 8 m= m=5 m=9 7 6 K/K φ Figure 3-2. Variation of scaled hydraulic conductivity with porosity for different values of m. 64

65 1 3 Pa A C B D Pa E Figure 3-3. Interstitial fluid pressure (IFP) and extracellular fluid velocity (EFV) with systemic (A & C) and local (B & D) infusion described by its contours at the tumor mid slice. Tumor and skin boundaries are overlaid on the contours and the infusion site is shown by a plus sign. In the bottom, a EFV cone plot (E) colored by its magnitude for local infusion. Includes point source (black sphere), tumor (blue) and skin (green) 65

66 A t = 3 min B t = 1 hr C t = 2 hr D t = 3 min E t = 1 hr F t = 2 hr Figure 3-4. On the top, normalized tracer concentration contours at tumor mid-slice at t = 3, 6, and 12 min. Tumor and skin boundaries are overlaid on the contours and the infusion site is shown by a plus sign. On the bottom, predicted evolution of distributed volume over time shown by an iso-surface at the distribution volume threshold. Includes tumor (blue), skin (green) and distributed volume (red) 66

67 Distribution volume in mm Whole leg Tumor V d,leg = 2.9V i +.4 V d,tum =.71V i Infusion volume in µl Figure 3-5. Variation of tissue distribution volumes with infusion volume for the whole leg and tumor following CED of albumin (.3 µl/min) at the center of the tumor. Includes equation for the linear fit on the data. 67

68 m=5 m=9 1 Pa A 2 B C D E F Figure 3-6. Comparison of interstitial fluid pressure (IFP, A & B), extracellular fluid velocity (EFV, C & D) contours at the tumor mid-slice for infusions at m = 5 & 9 respectively. Tumor and skin boundaries are overlaid on the contours and the infusion site is shown by a plus sign. The EFV cone plots (E & F) colored by its magnitude for m = 5 and 9 is also shown. Includes point source (black sphere), tumor (blue) and skin (green) 68 Pa