COMPUTATIONAL MODELS FOR LOCALIZED DRUG DELIVERY IN TUMORS

By

MAGDOOM MOHAMED KULAM NAJMUDEEN

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2011

c⃝ 2011 Magdoom Mohamed Kulam Najmudeen

2

To my mom, dad and family

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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.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Specific Aim 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Specific Aim 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 DEVELOPMENT OF VOXELIZED MODEL FOR SYSTEMIC DELIVERY INSOLID TUMORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Estimation of spatial variation maps of vascular leakiness . . . . . 182.2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4.1 Root Mean Square Error . . . . . . . . . . . . . . . . . . 242.2.4.2 Pearson Product Moment Correlation Coefficient . . . . . 242.2.4.3 Error Histogram . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Validation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 APPLICATION OF VOXELIZED MODEL FOR CONVECTION-ENHANCEDDELIVERY IN A HIND LIMB TUMOR . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.2 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5

4 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . 73

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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LIST OF TABLES

Table page

2-1 Tissue and vascular parameters used for simulating distribution of Gd-DTPAfollowing bolus tail vein injection at the hind limb tumor in a mice. . . . . . . . . 33

2-2 Statistical parameters obtained while comparing voxelized and non-voxelizedmodel results for the baseline simulation in three animals. . . . . . . . . . . . . 34

2-3 Statistical parameters obtained while comparing voxelized and non-voxelizedmodel results for intermediate and fast arterial input function in animal I. . . . . 35

2-4 Comparison of tracer washout rates and root mean square error in tracerconcentration within the tumor volume between voxelized and non-voxelizedmodel results with experiment in three animals. . . . . . . . . . . . . . . . . . . 36

3-1 Tissue and vascular parameters used for simulating distribution of albuminfollowing CED at the hind limb tumor in a mice. . . . . . . . . . . . . . . . . . . 62

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LIST OF FIGURES

Figure page

2-1 Normalized concentration of tracer in blood plasma approximated by differentAIFs used for sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2-2 CFD compatible meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2-3 Horizontal and vertical lines used for plotting the flow field and tracer transportin voxelized and non-voxelized models . . . . . . . . . . . . . . . . . . . . . . . 39

2-4 Contours of IFP predicted by voxelized and non-voxelized models, along withline plots along the horizontal and vertical bisectors in the mid-slice. . . . . . . 40

2-5 Contours of IFV predicted by voxelized and non-voxelized models along withline plots along the horizontal and vertical bisectors in the mid-slice. . . . . . . 41

2-6 Comparison of tracer concentration contours. Voxelized and non-voxelizedmodel compared with MR-derived tissue concentration at t = 5, 30, and 60min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2-7 Line plots comparing the predicted tracer concentration in the tissue by boththe models with experiment, along the horizontal and vertical bisectorsof mid-slice at t = 5, 30 and 60 min. . . . . . . . . . . . . . . . . . . . . . . . . 43

2-8 Error Histograms for flow and transport in baseline simulation for voxelizedmodel with respect to non-voxelized model. . . . . . . . . . . . . . . . . . . . . 44

2-9 Line plots comparing the IFP and IFV predicted by both the models for twodifferent AIF parameter sets (intermediate and fast) along the vertical bisectorof mid-slice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2-10 Line plots comparing the tracer concentration in the tissue predicted by boththe models for two different AIF parameter sets (intermediate and fast) alongthe vertical bisector of mid-slice at t = 5 and 20 min. . . . . . . . . . . . . . . . 46

2-11 Error Histograms for tracer concentration within the tumor for voxel andnon-voxel model results with respect to the experimental data at t = 5, 30 and60 min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3-1 Depiction of baseline CED simulation. . . . . . . . . . . . . . . . . . . . . . . . 63

3-2 Variation of scaled hydraulic conductivity with porosity for different values of m. 64

3-3 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. . . . . . 65

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3-4 Normalized tracer concentration contours at tumor mid-slice at t = 30, 60,and 120 min. Also included predicted evolution of distributed volume over timeshown by an iso-surface at the distribution volume threshold. . . . . . . . . . . 66

3-5 Variation of tissue distribution volumes with infusion volume for the whole legand tumor following CED of albumin (0.3 µL/min) at the center of the tumor . . 67

3-6 Comparison of IFP and EFV contours at the tumor mid-slice along with EFVcone plot colored by its magnitude, for infusions at m = 5 & 9. . . . . . . . . . . 68

3-7 Predicted evolution of distributed volume for infusions at the center of the tumorfor m = 5 and 9 at t = 30, 60, and 120 min. . . . . . . . . . . . . . . . . . . . . 69

3-8 Variation of tissue distribution volumes with infusion volume for the whole legand tumor following CED of albumin (0.3 µL/min) at the center of the tumorfor m = 0, 5 & 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3-9 Comparison of normalized tracer concentration contours at tumor mid-slicefor infusions at the tumor-host interface and anterior end of the tumor at t =30, 60, and 120 min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3-10 Variation of tissue distribution volumes with infusion volume for the whole legand tumor following CED of albumin (0.3 µL/min) at the tumor-host tissueinterface and anterior end of the tumor with m = 0 . . . . . . . . . . . . . . . . 72

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Abstract of Thesis Presented to the Graduate Schoolof 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

By

Magdoom Mohamed Kulam Najmudeen

August 2011

Chair: Malisa SarntinoranontMajor: Mechanical Engineering

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

10

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

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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CHAPTER 1INTRODUCTION

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, 40, 42]. Tumors are also

known to exhibit elevated interstitial fluid pressure (IFP), which is attributed to its

lack of lymphatics [60] 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 [20], 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

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, 10]. 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 [50], 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

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.

1.2.1 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.

1.2.2 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.

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CHAPTER 2DEVELOPMENT 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, 40, 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 [60] 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

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

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

2.2.1 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 192 × 96 voxels per slice. The size of each voxel was 0.104 ×

0.104× 1 mm3.

18

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 ∼ 590 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 (Ct) by assuming a linear

relationship between Ct and relaxation times (T1 & T2), and substituting it into the

standard spin-echo equation [49, 55, 69]. After algebraic manipulations, the following

expression for Ct was obtained with an added assumption that transverse-relaxation

contribution to signal is unity,

CMRI,t =1

R1

[1

TRln

S(0)

S(0)− S(CMRI,t).(1− e−TR/T10)− 1

T10

](2–1)

where CMRI,t is the tissue concentration of Gd-DTPA determined by MRI, R1 is the

longitudinal relaxivity of the tracer in water, TR is the time for recovery, S(CMRI,t) and

S(0) are the signal intensities at tracer concentrations CMRI,t and zero respectively and

T10 is the T1 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,

dCtdt= K transCp −

K trans

ϕCt (2–2)

where Ct and Cp are the concentrations of Gd-DTPA in tissue and blood plasma

respectively.

19

The tracer concentration in the blood plasma, Cp following a bolus injection can be

described by an AIF of biexponential decay:

Cp(t) = d[a1e

−m1t + a2e−m2t

](2–3)

where a1,m1 refers to the amplitude and rate constant of the fast equilibrium between

plasma and extracellular space respectively, a2,m2 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: a1 =

3.99 kg/L, m1 = 0.114 min−1, a2 = 4.78 kg/L, and m2 = 0.0111 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 : a1 = 9.2 kg/L, m1 = 0.23 min−1, a2 =

4.2 kg/L, and m2 = 0.05 min−1 described the fast AIF [27]; a1 = 13 kg/L, m1 =

0.30 min−1, a2 = 16 kg/L, and m2 = 0.026 min−1 described the intermediate AIF [4]. A

qualitative representation of the different AIFs are provided in Figure 2-1. In this figure,

Cp was normalized such that the initial concentration is the same for all the AIFs.

Knowing the Cp values from Equation (2–3), Equation (2–2) was then solved

analytically to find an expression for Ct(t) which was then fitted with the experimental

values (CMRI,t) at early time points (∼ 20 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.

2.2.2 Mathematical Model

The tissue continuum was modeled as a porous media with continuity [53] and

momentum (Darcy’s law) equations given by,

∇.v = Ktrans

K transJVV

− Lp,lySLV(p − p

L) (2–4)

v = −K∇p (2–5)

20

where v is the IFV, K trans is the average value of K trans over tumor and host tissue voxels,

Lp,ly is the lymphatic vessel permeability, SL/V is the lymphatic vessel surface area

per unit volume which was set to zero in tumor tissue, p is the IFP, pL

is pressure in the

lymphatic vessels which was set to zero and K is the tissue hydraulic conductivity. JV /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],

JVV= Lp

S

V(pv − p − σ

T(πv − πi)) (2–6)

where Lp is the hydraulic conductivity of the microvascular wall, S/V is the blood

vessel surface area per unit volume, pv is the vascular fluid pressure, σT

is the osmotic

reflection coefficient for plasma proteins, πv ,πi 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],

∂Ct∂t+

vϕ.∇Ct −D∇2Ct = K trans

(Cp −

Ctϕ

)− Lp,ly

SLV(p − p

L)Ctϕ

(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

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.

2.2.3 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 (20 × 10 × 9 mm3) enclosing the

tumor was created and meshed with quadrilateral elements (voxels) of size equal to

the MRI resolution (0.104 × 0.104 × 1 mm3) 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, Ct = 0.

A zero fluid pressure condition, p = 0, was applied along the cut ends and the remaining

outer boundaries of the geometry were assigned as wall.

22

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 (0.001

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 ≤ 20 min and the data was compared at discrete time points, t = 5, 10 and 20 min.

2.2.4 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

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.

2.2.4.1 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,

εx =

√√√√√ N∑j=1

(x jvox − x jnvox

)2N

(2–8)

Where x was replaced with IFP, IFV magnitude, and tracer concentration, N is

the total number of matching elements, vox refers to voxel value and nvox refers to

non-voxel value.

2.2.4.2 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

r =

−1 indicates perfect negative correlation

0 indicates no correlation

1 indicates perfect positive correlation

2.2.4.3 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 (0.73 - 1.62 kPa in animal

I, 0.32 - 0.71 kPa in animal II and 0.39 - 0.87 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 (∼ 1136.5, 579.6 and 763.6 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

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 (∼ 0.75, 0.40 and 0.42 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 (∼ 0.03, 0.03 and 0.01 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,

30 and 60 min) after infusion. The predicted tracer distribution of both the models and

the actual experimental data, was heterogeneous with high concentration regions (∼

0.4, 0.18 and 0.19 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 (∼ 0.03, 0.05 and

0.03 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 > 0.7) 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 > 0.7) showed a reasonable degree of

26

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.

2.3.1 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 2-10). 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.

2.3.2 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

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

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

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

30

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 2.3-2.7 million elements)

and voxelized model (approximately 165,000 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

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

Table 2-1. Tissue and vascular parameters used for simulating distribution of Gd-DTPAfollowing bolus tail vein injection at the hind limb tumor in a mice.

Variable Description Value ReferencesLp(m/Pa.s) Vessel permeability 2× 10−11t ; 3× 10−12n; [53]S/V (m−1) Microvascular surface area per unit volume 20000t ; 7000n [7]Lp,lySL/V (m

−1) Lymphatic filtration coefficient 1× 10−7 [53]K(m2/Pa.s) Hydraulic conductivity 1.9× 10−12t ; 3.8× 10−13n [53]

7.7× 10−15epv(Pa) Microvascular pressure 2300 [53]πi(Pa) Osmotic pressure in interstitial space 3230t ; 1330n [53]πv(Pa) Osmotic pressure in microvasculature 2670 [53]σT(Pa) Average osmotic reflection coefficient for plasma 0.82t ; 0.91n [53]

D(m2/s) Diffusion coefficient of Gd-DTPA 1× 10−9 [53]

t - tumor, n - normal tissue, e - exterior.

33

Table 2-2. Statistical parameters obtained while comparing voxelized and non-voxelizedmodel results for the baseline simulation in three animals.

Variable Quantity Animal I (Baseline) Animal II Animal IIIε r ε r ε r

IFP Magnitude 167.09 Pa 0.95 44.45 Pa 0.97 38.72 Pa 0.97IFV Magnitude 0.07 µm/s 0.81 0.01 µm/s 0.89 0.02 µm/s 0.92

Direction 23.10◦ 0.72 25.64◦ 0.58 16.44◦ 0.74Ct At t = 5 min 0.10 mM 0.79 0.10 mM 0.71 0.08 mM 0.74

At t = 30 min 0.05 mM 0.79 0.06 mM 0.75 0.04 mM 0.77At t = 60 min 0.04 mM 0.78 0.03 mM 0.77 0.03 mM 0.77

34

Table 2-3. Statistical parameters obtained while comparing voxelized and non-voxelizedmodel results for intermediate and fast arterial input function in animal I.

Variable Quantity Intermediate AIF Fast AIFε r ε r

IFP Magnitude 47.48 Pa 0.99 119.88 Pa 0.99IFV Magnitude 0.12 µm/s 0.78 0.14 µm/s 0.78

Direction 18.22◦ 0.70 18.45◦ 0.70Ct At t = 5 min 0.10 mM 0.72 0.07 mM 0.68

At t = 10 min 0.07 mM 0.75 0.07 mM 0.72At t = 20 min 0.06 mM 0.75 0.03 mM 0.74

35

Table 2-4. Comparison of tracer washout rates and root mean square error in tracerconcentration within the tumor volume between voxelized and non-voxelizedmodel results with experiment in three animals.

Washout rate ofvolume-averaged

Gd-DTPA concentrationwithin tumor volume

(min−1)

RMS error for concentration withintumor volume (mM)

Animal Case t = 5 min t = 30 min t = 60 minI Experimental -0.031 – – –

Voxelized Model -0.022 0.072 0.039 0.031Non-voxelized Model -0.020 0.120 0.064 0.048

II Experimental -0.020 – – –Voxelized Model -0.021 0.070 0.060 0.036Non-voxelized Model -0.022 0.089 0.062 0.040

III Experimental -0.025 – – –Voxelized Model -0.022 0.295 0.258 0.034Non-voxelized Model -0.022 0.297 0.259 0.037

36

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (minutes)

Nor

mal

ized

pla

sma

conc

entr

atio

n

BaselineIntermediateFast

Figure 2-1. Normalized concentration of tracer in blood plasma approximated bydifferent AIFs used for sensitivity analysis

37

0

5

10

15

20

0

2

4

6

8

10

0

2

4

6

8

10

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

0 5 10 15 200

2

4

6

8

10

A B

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

39

A B

0 5 10 15 200

200

400

600

800

1000

1200

1400

1600

Distance (mm)

IFP

(P

a)

Non VoxelVoxel

C

0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

Distance (mm)

IFP

(P

a)

Non VoxelVoxel

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, lineplots comparing the predicted IFP (C & D) by both the models along thehorizontal and vertical bisectors in the mid-slice respectively. The tumor andskin boundaries are represented by dashed and dash-dot lines respectively.

40

A B

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

Distance (mm)

IFV

(µm

/s)

Non VoxelVoxel

C

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

Distance (mm)

IFV

(µm

/s)

Non VoxelVoxel

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, lineplots comparing the predicted IFV (C & D) by both the models along thehorizontal and vertical bisectors in the mid-slice respectively. The tumor andskin boundaries are represented by dashed and dash-dot lines respectively.

41

Exp

erim

enta

lt = 5 min t = 30 min t = 60 min

Voxe

lized

Non

-vox

eliz

ed

Figure 2-6. Comparison of tracer concentration contours. Voxelized and non-voxelizedmodel compared with MR-derived tissue concentration at t = 5, 30, and 60min. Tumor and skin boundaries are overlaid on the contours.

42

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

A

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

B

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

C

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

D

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

E

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Distance (mm)

Ct (

mM

)

Non VoxelVoxelExperimental

F

Figure 2-7. Line plots comparing the predicted tracer concentration in the tissue by boththe models with experiment, along the horizontal and vertical bisectors ofmid-slice at t = 5 (A & B), 30 (C & D), 60 (E & F) min respectively. The tumorand skin boundaries are represented by dashed and dash-dot linesrespectively.

43

0 200 400 600 800 10000

1000

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6000

Pressure (Pa)

Fre

quen

cy

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0 0.2 0.4 0.6 0.80

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Fre

quen

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E

0 0.1 0.2 0.3 0.4 0.5 0.60

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15000

Concentration (mM)

Fre

quen

cy

F

Figure 2-8. Error Histograms for flow (A, B & C) and transport at t = 5 (D), 30 (E) and 60(F) mins in baseline simulation for voxelized model with respect tonon-voxelized model

44

0 2 4 6 8 10350

400

450

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IFP

(P

a)

Non VoxelVoxel

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P (

Pa)

Non VoxelVoxel

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(µm

/s)

Non VoxelVoxel

C

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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IFV

(µm

/s)

Non VoxelVoxel

D

Figure 2-9. Line plots comparing the IFP (A & B) and IFV (C & D) predicted by both themodels for two different AIF parameter sets (intermediate and fast) along thevertical bisector of mid-slice respectively. The tumor and skin boundaries arerepresented by dashed and dash-dot lines respectively.

45

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

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Ct (

mM

)

Non VoxelVoxel

A

0 2 4 6 8 100

0.1

0.2

0.3

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0.5

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t (m

M)

Non VoxelVoxel

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0 2 4 6 8 100

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Ct (

mM

)

Non VoxelVoxel

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

Distance (mm)

Ct (

mM

)

Non VoxelVoxel

D

Figure 2-10. Line plots comparing the tracer concentration in the tissue predicted byboth the models for two different AIF parameter sets (intermediate and fast)along the vertical bisector of mid-slice at t = 5 (A & B) and 20 (C & D) minrespectively. The tumor and skin boundaries are represented by dashedand dash-dot lines respectively.

46

0 0.1 0.2 0.3 0.4 0.5 0.60

50

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Concentration (mM)

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quen

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Voxe

lized

t = 5 min

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Concentration (mM)

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quen

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t = 30 min

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Concentration (mM)

Fre

quen

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t = 60 min

0 0.1 0.2 0.3 0.4 0.5 0.60

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Concentration (mM)

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Non

-vox

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ed

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Concentration (mM)

Fre

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Figure 2-11. Error Histograms for tracer concentration within the tumor for voxel andnon-voxel model results with respect to the experimental data at t = 5, 30and 60 min.

47

CHAPTER 3APPLICATION 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 [20] 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, 10]. 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 [50], 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

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

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

50

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.

3.2 Methods

3.2.1 Mathematical Model

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 ∼ 590 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

VinfAt the infusion voxel (3–1)

=K trans

K transJVV

− Lp,lySLV(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, Vinf is the volume of the

infused voxel, K trans is the average value of K trans in host and tumor tissue voxels, Lp,ly is

lymphatic vessel permeability, SL/V is the lymphatic vessel surface area per unit volume

which was set to zero in tumor tissue, p is the IFP and pL

is pressure in the lymphatic

vessels which was set to zero. JV /V is the filtration rate of plasma per unit volume of

51

tissue into the interstitial space which is given by Starling’s law as follows [68],

JVV= Lp

S

V(pv − p − σ

T(πv − πi)) (3–3)

here Lp is the hydraulic conductivity of the microvascular wall, S/V is the blood vessel

surface area per unit volume, pv is the vascular fluid pressure, σT

is the osmotic

reflection coefficient for plasma proteins, πv ,πi 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 [30, 41, 61, 70]. 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,

K =

K 0t em(ϕ+0.1)Nt

Nt∑i=1

em(ϕi+0.1)For tumor

K 0h em(ϕ+0.1)Nh

Nh∑i=1

em(ϕi+0.1)For host

(3–5)

52

where Nt ,Nh are the number of tumor and host tissue voxels respectively, K 0t ,K 0h 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,

∂Ct∂t+

vϕ.∇Ct −Deff∇2Ct = 0 (3–6)

where Ct is the concentration of tracer in the tissue, Deff is the effective diffusivity

of albumin in the porous medium given by the following empirical relation based on

diffusion in porous media [21],

Deff = Dfreeϕn (3–7)

where Dfree 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,

C =CtC(t,i)ϕi

(3–8)

where C(t,i) and ϕi 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.

3.2.2 Computational Method

The continuity, momentum and albumin transport equations were solved using the

CFD software package, FLUENT (version 12.0.16, ANSYS, Inc., Canonsburg, PA). For

53

the 3D computational tissue model, a rectangular volume (20 × 10 × 9 mm3) enclosing

the tumor was created and meshed with quadrilateral elements (voxels) of size equal

to the MRI resolution (0.104 × 0.104 × 1 mm3) 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 51.49 and 741.16 mm3 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 0.001.

Infusion simulations were carried out upto t = 2 hrs and the interstitial distribution

of albumin was simulated at intermittent time points, t = 5, 30, 60 and 120 mins.

Initial conditions for tracer transport assumed no tracer in the tissue, C = 0 except

at the infusion site which is one voxel (0.104 × 0.104 × 1 mm3), 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 [10]. A zero fluid pressure condition, p = 0, 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

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

(0.001 µm/s).

The infusion at the center of the tumor with locally constant hydraulic conductivity

(m = 0) 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 = 0), 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 = 0, 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 0.3 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

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 (∼

0.41 - 1.23 kPa/mm).

The extracellular fluid velocity (EFV,v

ϕ) for systemic and local infusion, is shown

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 (∼ 0.25 - 6.15 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 (0.01) for times t = 30, 60 and

120 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 0.3 µL/min, albumin was distributed to approximately 58 % of the

tumor volume. The variation of distribution volume (Vd ) with infusion volume (Vi ) 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

However the variation was slightly non-linear within the tumor. The ratio Vd/Vi obtained

through linear fit was found to be 2.9 for the whole leg and 0.71 for the tumor.

3.3.1 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 0.3 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

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 Vd with Vi , with higher distribution volume

for infusion at the interface than at the anterior end of the tumor (Figure 3-10A). 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 3-10B).

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

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

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 60 % 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.

60

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

Table 3-1. Tissue and vascular parameters used for simulating distribution of albuminfollowing convection-enhanced delivery at the hind limb tumor in a mice.

Variable Description Value ReferencesLp (m/Pa.s) Vessel permeability 2× 10−11t ; 3× 10−12n; [53]S/V (m−1) Microvascular surface area per unit volume 20000t ; 7000n [7]Lp,lySL/V (m−1) Lymphatic filtration coefficient 1× 10−7 [53]K 0 (m2/Pa.s) Baseline hydraulic conductivity 1.9× 10−12t ; 3.8× 10−13n [53]

7.7× 10−15epv (Pa) Microvascular pressure 2300 [53]πi (Pa) Osmotic pressure in interstitial space 3230t ; 1330n [53]πv (Pa) Osmotic pressure in microvasculature 2670 [53]σT

(Pa) Average osmotic reflection coefficient for plasma 0.82t ; 0.91n [53]Dfree (m2/s) Self diffusion coefficient of albumin 5.8× 10−11 [5]Q (µL/min) Infusion flow rate 0.3 [39]

t - tumor, n - normal tissue, e - exterior.

62

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

63

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

9

10

φ

K/K

0

m=0m=5m=9

Figure 3-2. Variation of scaled hydraulic conductivity with porosity for different values ofm.

64

0 5 10 15 200

2

4

6

8

10

A

0 5 10 15 200 Pa

3000 Pa

B

0 5 10 15 200

2

4

6

8

10

C D

E

Figure 3-3. Interstitial fluid pressure (IFP) and extracellular fluid velocity (EFV) withsystemic (A & C) and local (B & D) infusion described by its contours at thetumor mid slice. Tumor and skin boundaries are overlaid on the contours andthe 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 (blacksphere), tumor (blue) and skin (green)

65

0 5 10 15 200

2

4

6

8

10

A t = 30 min0 5 10 15 20

B t = 1 hr

0 5 10 15 200

0.3

C t = 2 hr

D t = 30 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= 30, 60, and 120 min. Tumor and skin boundaries are overlaid on thecontours and the infusion site is shown by a plus sign. On the bottom,predicted evolution of distributed volume over time shown by an iso-surfaceat the distribution volume threshold. Includes tumor (blue), skin (green) anddistributed volume (red)

66

0 10 20 30 400

20

40

60

80

100

120

Vd,leg

= 2.9Vi+0.4

Vd,tum

= 0.71Vi+6.2

Infusion volume in µL

Dis

trib

utio

n vo

lum

e in

mm

3

Whole legTumor

Figure 3-5. Variation of tissue distribution volumes with infusion volume for the whole legand tumor following CED of albumin (0.3 µL/min) at the center of the tumor.Includes equation for the linear fit on the data.

67

m = 5 m = 9

0 5 10 15 200

2

4

6

8

10

A

0 5 10 15 200 Pa

1000 Pa

B

0 5 10 15 200

2

4

6

8

10

C D

E F

Figure 3-6. Comparison of interstitial fluid pressure (IFP, A & B), extracellular fluidvelocity (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 andthe infusion site is shown by a plus sign. The EFV cone plots (E & F) coloredby its magnitude for m = 5 and 9 is also shown. Includes point source (blacksphere), tumor (blue) and skin (green)

68

t = 30 min t = 60 min t = 120 min

m=

5m

=9

Figure 3-7. Predicted evolution of distributed volume for infusions at the center of thetumor for m = 5 and 9 at t = 30, 60, and 120 min. Includes tumor (blue), skin(green) and distributed volume (red)

69

0 10 20 30 400

20

40

60

80

100

120

140

160

180

Vd,m=0

= 2.9Vi+0.4

Vd,m=5

= 3.8Vi−2.7

Vd,m=9

= 4.7Vi−5.1

Infusion volume in µL

Dis

trib

utio

n vo

lum

e in

mm

3

m = 0m = 5m = 9

A

0 10 20 30 400

5

10

15

20

25

30

Vd,m=0

= 0.71Vi+6.2

Vd,m=5

= 0.66Vi+7.1

Vd,m=9

= 0.47Vi+7.8

Infusion volume in µL

Dis

trib

utio

n vo

lum

e in

mm

3

m = 0m = 5m = 9

B

Figure 3-8. Variation of tissue distribution volumes with infusion volume for the whole leg(A) and tumor (B) following CED of albumin (0.3 µL/min) at the center of thetumor for m = 0, 5 & 9. Includes equation for the linear fit on the data.

70

t = 30 min t = 60 min t = 120 min

0 5 10 15 200

2

4

6

8

10

Inte

rface

0 5 10 15 20

0 5 10 15 200

0.3

0 5 10 15 200

2

4

6

8

10

Ant

erio

r

0 5 10 15 20

0 5 10 15 200

0.3

Figure 3-9. Comparison of normalized tracer concentration contours at tumor mid-slicefor infusions at the tumor-host interface and anterior end of the tumor at t =30, 60, and 120 min. Tumor and skin boundaries are overlaid on thecontours and the infusion site is shown by a plus sign in a circle.

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0 10 20 30 400

20

40

60

80

100

120

Vd,center

= 2.9Vi+0.4

Vd,interface

= 3Vi−0.88

Vd,anterior end

= 2.1Vi+4.4

Infusion volume in µL

Dis

trib

utio

n vo

lum

e in

mm

3

CenterInterfaceAnterior end

A

0 10 20 30 400

5

10

15

20

25

30

Vd,center

= 0.71Vi+6.2

Vd,interface

= 0.72Vi+6.1

Vd,anterior end

= 0.18Vi+3.8

Infusion volume in µL

Dis

trib

utio

n vo

lum

e in

mm

3

CenterInterfaceAnterior end

B

Figure 3-10. Variation of tissue distribution volumes with infusion volume for the wholeleg (A) and tumor (B) following CED of albumin (0.3 µL/min) at thetumor-host tissue interface and anterior end of the tumor with m = 0.Includes equation for the linear fit on the data.

72

CHAPTER 4CONCLUSIONS AND FUTURE WORK

A computational model for predicting the distribution of macromolecular drug

following convection-enhanced delivery (CED) in the hind limb tumor of a mice using

voxelized modeling approach was developed. The 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 drug concentration through the

tissue interstitium, following injection. This framework was previously evaluated with

experiment and predictions from a traditional CFD approach for systemic delivery of MR

visible tracer.

The first portion of this thesis demonstrated the applicability of the voxelized model

for predicting tumor transport. This was done by comparing its results with that obtained

from a previously developed CFD modeling approach using unstructured meshes for

systemic delivery of the tracer, using statistical methods and qualitative presentation.

The resulting analysis indicated similarity in both the model results with low root mean

square error and high correlation coefficient. The voxelized model also captured typical

features of the flow field and tracer distribution in the tumor interstitium such as the high

interstitial fluid pressure (IFP) inside tumor and the heterogeneous distribution of tracer.

The obtained tracer distribution within the tumor was also similar to MR-measured

tracer concentration data. Furthermore, the accuracy of the voxelized model results

with experiment and non-voxelized model predictions were maintained across the three

73

tumors under study. Sensitivity of the voxelized and non-voxelized model to changing

arterial input function (AIF) parameters was also found to be similar.

In the second portion of the thesis, this model was slightly modified to predict

the tracer distribution following CED. The model was able to capture the 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 tracer 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

the empirical parameter (m) used to increase the 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.

The accuracy of the model’s CED predictions could be improved with the following

modifications which would be the subject of future work. Firstly, transvascular solute

exchange can be accounted in the transport equation for smaller molecular weight

compounds. It can be expected to affect the tracer distribution because of the leakiness

of the tumors. Secondly, binding, metabolism and uptake of the macromolecular drug

can be accounted for using reaction and binding kinetics. Finally, the model can be

validated with experiments across a wider range of tumors. In addition to the above

extensions, the voxelized model solution also needs to be tested for grid and time step

independency, to ensure that discretization and truncation errors are small.

74

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81

BIOGRAPHICAL SKETCH

Magdoom Mohamed received his Bachelor of Technology degree in Mechanical

Engineering from National Institute of Technology (NIT), Tiruchirappalli, India in 2008.

Between 2008-09, he worked at Indian Institute of Technology (IIT), Madras as project

assistant in the Department of Bio-technology. In Spring 2010, he was admitted to the

graduate program in the Department of Mechanical and Aerospace Engineering at the

University of Florida. In Summer 2011, he received his MS in Mechanical Engineering

from the University of Florida.

82