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Research Papers

# Numerical Study of Transport of Anticancer Drugs in Heterogeneous Vasculature of Human Brain Tumors Using Dynamic Contrast Enhanced-Magnetic Resonance ImagingOPEN ACCESS

[+] Author and Article Information
Ajay Bhandari

Department of Mechanical Engineering,
Indian Institute of Technology,
Kanpur 208016, India
e-mail: ajayb@iitk.ac.in

Ankit Bansal

Department of Mechanical and
Industrial Engineering,
Indian Institute of Technology,
Roorkee 247677, India
e-mail: abansfme@iitr.ac.in

Anup Singh

Centre for Biomedical Engineering,
Indian Institute of Technology,
Delhi 110016, India;
Department of Biomedical Engineering,
All India Institute of Medical Sciences,
Delhi 110016, India
e-mail: anupsm@cbme.iitd.ac.in

Niraj Sinha

Department of Mechanical Engineering,
Indian Institute of Technology,
Kanpur 208016, India
e-mail: nsinha@iitk.ac.in

1Corresponding author.

Manuscript received August 15, 2017; final manuscript received November 7, 2017; published online March 16, 2018. Assoc. Editor: Ram Devireddy.

J Biomech Eng 140(5), 051010 (Mar 16, 2018) (10 pages) Paper No: BIO-17-1366; doi: 10.1115/1.4038746 History: Received August 15, 2017; Revised November 07, 2017

## Abstract

Systemic administration of drugs in tumors is a challenging task due to unorganized microvasculature and nonuniform extravasation. There is an imperative need to understand the transport behavior of drugs when administered intravenously. In this study, a transport model is developed to understand the therapeutic efficacy of a free drug and liposome-encapsulated drugs (LED), in heterogeneous vasculature of human brain tumors. Dynamic contrast enhanced-magnetic resonance imaging (DCE-MRI) data is employed to model the heterogeneity in tumor vasculature that is directly mapped onto the computational fluid dynamics (CFD) model. Results indicate that heterogeneous vasculature leads to preferential accumulation of drugs at the tumor position. Higher drug accumulation was found at location of higher interstitial volume, thereby facilitating more tumor cell killing at those areas. Liposome-released drug (LRD) remains inside the tumor for longer time as compared to free drug, which together with higher concentration enhances therapeutic efficacy. The interstitial as well as intracellular concentration of LRD is found to be 2–20 fold higher as compared to free drug, which are in line with experimental data reported in literature. Close agreement between the predicted and experimental data demonstrates the potential of the developed model in modeling the transport of LED and free drugs in heterogeneous vasculature of human tumors.

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## Introduction

Cancer is one of the leading causes of deaths worldwide. A major obstacle in treatment of cancer lies in insufficient drug accumulation and poor efficacy due to the abnormal physiology of tumors, which give rise to a set of transport barriers that limit the rate and extent of drug delivery to tumors. The four major unusual properties that constitute the pathophysiological state of tumors are accumulated solid stress, abnormal and heterogeneous blood vessel networks, elevated interstitial fluid pressure (IFP), and dense interstitial structure. These abnormalities lead to low and nonuniform perfusion rates in tumors. With the advent of nanotechnology, there has been widespread interest in the development and exploitation of nanomedicines, including liposome-based formulations, to increase drug uptake in tumors [1]. It has been shown that nanosized drug delivery systems result in more drug accumulation at tumor site as compared to free drug [2].

Since the drug delivery process involves many variables related to chemotherapeutic drugs and tissues, a mathematical model accounting for drug properties, tumor abnormalities, and barriers in drug transport could provide better insight into the complex transport process of nanomedicines. It will also aid in examining the roles of interacting components in a systematic manner via computational simulations. In this context, a number of numerical studies related to the free drug and liposome mediated drug delivery have been reported [37]. Baxter and Jain used a continuum porous media model of solid tumor to solve for interstitial fluid flow and solute transport parameters [3,4]. Harashima et al. developed a mathematical model for the delivery of nonthermo sensitive liposomes to the tumor tissue and optimized their release rate for maximum antitumor effect [5]. El-Kareh and Secomb demonstrated through simulations that continuous infusion for optimal duration is superior to other infusion methods in liposomal delivery [6]. Stapleton et al. developed a theoretical framework to predict the intratumoral accumulation of liposomes and highlighted the relationship between interstitial flow and liposome accumulation [7].

In recent years, computational fluid dynamics (CFD) simulations are increasingly being used to predict transport of macromolecules within tumor tissues [810]. Zhan et al. modeled human liver tumor vasculature and concluded that not only drug accumulates faster in well vascularized regions but also clears out from these regions quickly, facilitating less tumor cell killing [11]. Attempts have also been made to model the heterogeneous vasculature of tumor tissues [12]. However, most of the studies had certain assumptions such as use of homogeneous model of tumors, use of global intravascular concentration or arterial input function (AIF), and heterogeneous models developed for animals with use of simple tofts kinetic model [13].

In pursuit of overcoming these limitations, a computational model has been developed previously for contrast agent as well as liposomes transport in human brain tumors [14,15]. Taking the work forward, the aim of the present study is to predict the accumulation and efficacy of liposome encapsulated drug (LED) and free drug in heterogeneous vasculature of human brain tumors in addition to their transport. Heterogeneous vasculature is determined using dynamic contrast enhanced-magnetic resonance imaging (DCE-MRI) data that allow for incorporation of volume fraction of interstitial space (also called porosity) and cell densities into the computational model. The developed computational model accounts for transport of drug in vascular, interstitial or extracellular and intracellular compartments. Binding of the drug with the proteins present in vascular and interstitial compartment has also been taken into account. Effect of heterogeneous vasculature on interstitial and intracellular drug concentration has been investigated. Two different drug delivery modes have been examined and compared: intravenous administration of free drug and intravenous administration of LED, which releases the drug at an optimum release rate. Therapeutic efficacy of both was compared in terms of tumor cell density and results were validated with literature. Effect of heterogeneous vasculature on drug delivery in tumors has been studied earlier by Zhan et al. [11]. However, current study utilizes DCE-MRI to explicitly model the heterogeneous vasculature of human brain tumor by calculating vascular and interstitial volume fraction voxel-wise. This helps in studying their effects on drug transport. Second, this study measures the time-dependent and patient-specific AIF of contrast agent to accurately calculate interstitial and vascular volume fraction within the tumor. This, in turn, provides true picture of accumulation of drug in heterogeneous tumor areas. Finally, this study models heterogeneous and nonuniform tumor cell density at each voxel to accurately predict the drug efficacy in heterogeneous tumor region. To the extent of our knowledge, CFD analysis of transport of LED and free drug in heterogeneous vasculature of human brain tumors based on DCE-MRI data has not been reported till date.

## Methodology

In the current study, the DCE-MRI data was acquired and analyzed to obtain the porosity and plasma volume fraction maps for tumor and normal tissue by general tracer kinetic model (GTKM) also called extended tofts model [16].

###### Magnetic Resonance Imaging Protocol.

Magnetic resonance imaging (MRI) of four brain tumor patients was performed according to the protocol mentioned in our previous work [15]. Briefly, DCE-MRI data were acquired using a three-dimensional fast field echo (T1-FFE) sequence (TR/TE = 4.38 ms/2.3 ms, flip angle ($θ$) = 10 deg, field of view = 240 × 240 mm2, slice thickness = 6 mm, matrix size = 256 × 256). A dose of 0.1 mmol/kg body weight of Gd-BOPTA (GadobenateDimeglumine) (Multihance, Bracco, Italy) was administered intravenously with the help of a power injector. From DCE-MRI images, concentration of contrast agent $Ct$ on each voxel was calculated from signal intensity by using equation spoiled gradient recalled echo (SPGR)/FFE sequence, which is expressed as Display Formula

(1)$S(t)S(0)=k0 exp(−TER2C(t))1−exp(−TR(T10−1+R1C(t)))1−cos(θ)exp(−TR(T10−1+R1C(t)))$

where

$k0=1−cos(θ)exp(−TRT10−1)1−exp(−TRT10−1)$

For derivation of Eq. (1), readers are referred to Singh et al. [17]. Longitudinal ($R1$) and transverse ($R2$) relaxivity of contrast agent in body were taken as 6.3 $mmol−1s−1L$ and 17.5 $mmol−1s−1L$, respectively [18]. Pre-contrast T1 (T10) estimation was done using three fast spin echo image (T1—weighted, T2—weighted, and proton density—weighted) as mentioned in Singh et al. [17]. Local AIF or patient-specific AIF was estimated using a method described by Singh et al. [19]. Plasma ($vp$) and interstitial volume fraction maps ($ve$) were obtained at each voxel by solving GTKM (Eq. (2)) using the concentration values obtained from Eq. (1)Display Formula

(2)$C(t)=vpCp(t)+Ktrans∫0tCp(τ)eKtransve(τ−t)dτ$

General tracer kinetic model has been used in this study, since it incorporates the intravascular term, which cannot be ignored during analysis of human tumors [16].

## Tissue Transport Model

Tissue transport model used in this study consists of interstitial fluid flow and drug transport equations. First, the interstitial fluid flow equations are solved to have a basic understanding of the biomechanical environment of drug transport.

###### Mass Balance Equation.

Mass balance equation for interstitial fluid in biological tissues (normal and tumor) is expressed as [3] Display Formula

(3)$∇.v=∅B−∅L$
where
$∅B=LpSV[Peff−Pi]; Peff=Pv−σT(πv−πi) and ∅L=Lp,LySLV[Pi−PL]$

For details of the aforementioned equation, readers are referred to Ref. [15]. Since lymphatic system does not play a role in interstitial fluid removal from human brain, its value has been assumed to be negligible in this study. Values of all the aforementioned transport parameters for normal and tumor tissue of human brain were taken from literature and are mentioned in our previous work [15].

###### Momentum Balance Equation.

Since tissue interstitium is considered as porous media and the interstitial fluid as Newtonian fluid, the interstitial fluid flow through the interstitium is modeled with the help of Darcy's law that is expressed as [3] Display Formula

(4)$v=−K∇Pi$

###### Drug Transport Equation.

Once administered into human body, the drug is distributed in three different compartments: vascular (plasma), interstitial, and intracellular. Accordingly, models of drug transport in these three compartments are described in this section.

###### Drug Concentration in Blood Plasma.

The drug transport in the vascular (plasma) compartment has been modeled by using a bi-exponential decay function [20,21] that is represented as Display Formula

(5)$Cv(t)=a1e−m1t+a2e−m2t$

where $a1$ and $a2$ are plasma and extracellular space compartment parameters and $m1$ and $m2$ are compartment clearance rates. These parameters were found by fitting the previous equation to the blood plasma concentration available in literature for free drug (doxorubicin) as well as LED (DOXIL) for dosage of 50 mg/m2 [21]. Besides, it has been experimentally observed by Greene et al. [22] that drug binds extensively to proteins present in the plasma compartment. They found that 75% of the drug is present in bound form and the remaining in free form. Therefore, free and bound drug concentration in blood plasma is expressed as Display Formula

(6)$Cfp=0.25Cv; Cbp=0.75Cv$

The drug transport equations for free and bound drug in remaining two compartments are described next.

###### Free Drug Concentration in Interstitial Fluid.

The governing equation for transport of free drug in interstitial space is expressed as [23] Display Formula

(7)$∂Cfi∂t+v.∇Cfi−Dfi∇2Cfi=SV+SB+SI$

In the previous equation, $SV$ is the source term that represents net free drug transport rate per unit volume from blood vessels and lymphatic vessels into the interstitial space and is expressed as [3] Display Formula

(8)$SV=∅B(1−σfl)Cfp+PfiSV(Cfp−Cfi)PefePef−1−∅LCfi$

$Pef$ is the trans-capillary Peclet number for free drug, which is the ratio of free drug convection to diffusion across the capillary wall. It is expressed as [3] Display Formula

(9)$Pef=∅B(1−σfl)Pfi(SV)$

$SB$ is the source term that represents association and dissociation of drug bounded with the proteins and is expressed as [23] Display Formula

(10)$SB=kdCbi−kbCfi$

$SI$ is the source term that represents influx and efflux from tumor cells and is expressed as [6] Display Formula

(11)$SI=Cs(ω−δ)$

where $Cs$ is the cell density. The heterogeneity in cell density is taken into account by calculating it at each voxel. Assuming the cell volume to be $10−9mlor10−15m3$ [6], it is expressed as Display Formula

(12)$Cs=(1−ve−vp)1015cells/m3$

###### Bound Drug Concentration in Interstitial Fluid.

The governing equation for transport of bound drug in interstitial space is expressed as [23] Display Formula

(13)$∂Cbi∂t+v.∇Cbi−Dbi∇2Cbi=SVb−SB$

where $SVb$ is the source term that represents bound drug transport rate per unit volume from blood vessels into the interstitial space. It is expressed as [3] Display Formula

(14)$SVb=∅B(1−σbl)Cbp+PbiSV(Cbp−Cbi)PebePeb−1$

$Peb$ is the trans-capillary Peclet number for bound drug and is expressed as [3] Display Formula

(15)$Peb=∅B(1−σbl)Pbi(SV)$

###### Intracellular Drug Concentration.

Once the drug is available in free form in interstitial space, it enters the intracellular space for cell killing and its concentration is expressed as [6] Display Formula

(16)$∂Ci∂t=δ−ω$
$δ$ and $ω$ are expressed as [6] Display Formula
(17)$δ=Vmax(CfiCfi+keve);ω=Vmax(CiCi+kive)$

Since heterogeneous vasculature of tissue was considered in this study, free drug transport in interstitial space is expressed as Display Formula

(18)$∂Cfi∂t+vve.∇Cfi−Dfi∇2Cfi=∅B(1−σfl)Cfp+PfiSV(Cfp−Cfive)PefePef−1−∅LCfive+kdCbive−kbCfive+Cs(Vmax(CiCi+kive)−Vmax(CfiCfi+keve))$

Similarly, equation used for bound drug transport in interstitial fluid is expressed as Display Formula

(19)$∂Cbi∂t+vve.∇Cbi−Dbi∇2Cbi=∅B(1−σbl)Cbp+PbiSV(Cbp−Cbive)PebePeb−1−kdCbive+kbCfive$

###### Drug Transport Equations for Liposome Release.

Liposome-encapsulated drugs administered into human body circulate in blood plasma. They extravasate to interstitial space and release their drug payload so that it can be taken up by cellular compartment since liposomes are not directly taken up by cells. LED drug concentration in blood plasma ($Clp$) also follows a bi-exponential decay function [11] and was modeled in similar way as described in Sec. 3.2.1. Equations describing liposome-released drug (LRD) transport are expressed as follows.

###### Liposome-Encapsulated Drug Concentration in Interstitial Fluid.

The governing equation for transport of liposome encapsulated drug in interstitial space is expressed as [6] Display Formula

(20)$∂Cli∂t+v.∇Cli−Dli∇2Cli=SVl−SR$

where $SVl$ is the source term that represents net liposome encapsulated drug transport rate per unit volume from blood vessels and lymphatic vessels into the interstitial space and is expressed as [3] Display Formula

(21)$SVl=∅B(1−σll)Clp+PliSV(Clp−Cli)PelePel−1−∅LCli$

$Pel$ is the trans-capillary Peclet number for liposome encapsulated drug and is expressed as [3] Display Formula

(22)$Pel=∅B(1−σll)Pli(SV)$

$SR$ represents amount of drug released from liposome as it gets extravasated into interstitial space and is expressed as [6] Display Formula

(23)$SR=krelCli$

For heterogeneous vasculature, Eq. (20) is expressed as Display Formula

(24)$∂Cli∂t+vve.∇Cli−Dli∇2Cli=∅B(1−σll)Clp+PliSV(Clp−Clive)PelePel−1−∅LClive−krelClive$

Equation for transport of liposome released free drug in interstitial space after its release from liposome is expressed as Display Formula

(25)$∂Cfi∂t+vve.∇Cfi−Dfi∇2Cfi=∅B(1−σfl)Cfp+PfiSV(Cfp−Cfive)PefePef−1−∅LCfive+kdCbive−kbCfive+Cs(Vmax(CiCi+kive)−Vmax(CfiCfi+keve))+krelClive$

Equations for transport of bound drug concentration in interstitial space and intracellular concentration are same as Eqs. (16), (17), and (19).

###### Pharmacodynamics Model.

As the drug enters the intracellular compartment and is taken up by the tumor cells, it starts killing the cells. As a result, cell density keeps on changing during treatment and can be described by the pharmacodynamics model as [24] Display Formula

(26)$dCsdt=−fmaxCiEC50+CiCs+kcCs$

The free drug and LED used in this study are doxorubicin and DOXIL, respectively. Both of them are increasingly being used for treatment of human brain tumors [25,26]. Value of cell proliferation rate has been taken for human brain tumors (glioblastoma) [27]. Pharmacokinetic parameters of LED and free drug and their AIF were taken from literature and are listed in Table 1.

## Computational Method

The computational study was done using a 64-bit Intel (R) Xeon (R) processor (3.40 GHz) with 16 GB RAM. The computational domain consisted of a rectangular volume of size 67 × 65 × 72 $mm3$ enclosing the tumor and normal tissue. The domain was created and meshed in OpenFOAM with mesh element size taken in accordance with the MRI resolution (0.9375 × 0.9375 × 6 $mm3$) to ensure one-to-one mapping between MRI data and CFD mesh. Further, no potential offset existed in one-to-one mapping between MRI voxels and computational elements. The voxel-wise assignment of perfusion parameters obtained from analysis of DCE-MR images and transport properties was done in OpenFOAM by declaring the variables as nonuniform field. Governing equations for interstitial fluid flow and drug transport were discretized with finite volume method. For discretization of gradient and Laplacian terms, Gauss linear scheme was used. The semi-implicit method for pressure linked equations algorithm [28] was used to solve pressure and velocity, which were further used to solve drug transport equations. Since the interstitial fluid flow is not a convection dominated flow therefore, in light of weak convection coupling between fluid flow and drug transport, the solution methodology adopted in this paper is adequate. First-order Euler scheme was adopted to solve for time derivative. A fixed time-step of 60 s was used. The convergence criteria for residual tolerances were 1 × 10−6 for momentum equations and 1 × 10−8 for drug transport equations.

###### Boundary Conditions.

A zero fluid pressure boundary condition was applied at all the boundaries. This is because IFP in normal tissue, which is located at distance from tumor tissue will be close to zero. Further, zero gradient boundary condition was implemented at all boundaries for calculation of interstitial fluid velocity (IFV) and drug concentration. Initial condition for drug transport in the tissue was set to zero (C = 0) assuming that there is no drug present in the tissue initially.

###### Grid Independence Test.

Grid independence study was done to see the effect of change in number of mesh elements on the simulated solution. To perform it, 58,788, 235,152 and 940,608 mesh elements were used in the simulation. With 235,152 mesh elements (that is, four times the initial number of mesh elements), less than 2% change in simulated drug concentration and fluid flow parameters was observed. On further increasing the mesh elements to 940,608 (that is 16 times the initial number of mesh elements), no change in the drug concentration and fluid flow parameters was found. Therefore, 235,152 mesh elements were used for the entire simulation. The computational time involved in solving the fluid flow and drug transport equation using original mesh was approximately 1.5 h, which increased to 5 h when four times finer mesh was used.

## Results and Discussion

Figures 1(a) and 1(b) show the precontrast and postcontrast MRI images, respectively, of one slice of brain (slice 8) along with zoomed portion of tumor (please see Fig. 1(c)). Figure 1(d) shows CFD mesh of the same slice enclosing tumor. Porosity, plasma volume fraction, and cell density maps obtained at each voxel after analysis of DCE-MR images are shown in Figs. 2(a)2(c), respectively. It should be noted that values of porosity and plasma volume fraction highly depend on the grade of the tumor [17,29]. The range of porosity and plasma volume fraction values obtained in this study were in accordance with the grade of the tumor simulated (Grade 1, Meningioma). First, the fluid flow parameters IFP and IFV were simulated. A higher and uniform IFP of 1530 Pa was observed inside the tumor region, which rapidly decreased at the tumor boundary. IFP values were found to be independent of heterogeneous vasculature of tumor, which is consistent with the finding of Baxter and Jain [4]. The simulated IFP value becomes close to effective pressure ($Peff$) in tumor region due to which the convective transport of drug becomes negligible within the tumor region. Convective transport of drug is significant only at the tumor periphery due to steep pressure gradient. A higher IFV value of 0.023 μm/s was observed at the tumor periphery. These higher IFV values at tumor periphery also contribute to outward convective transport of drug at the tumor periphery. The simulated IFP and IFV values are in agreement with the values reported previously in the literature for humans via experiments [30,31]. Additionally, these simulated results of IFP and IFV are in good agreement with the analytical results generated by Baxter and Jain [3].

###### Systemic Administration of Bolus Injection of Free Drug.

Free drug extravasates from the leaky vasculature in the interstitial space. The free drug concentration was simulated for 48 h by assuming negligible tumor growth during this time [23]. Figure 3 shows spatial distribution of free doxorubicin in interstitial space of tumor at four different times. Contour plots show heterogeneity as well as nonuniform accumulation of free doxorubicin in tumor area. Drug concentration increases in tumor area for some time (up to 1 h) and then starts getting washed out due to decrease in plasma concentration of drug than interstitial concentration. The drug perfusion is uniform within the whole tumor area up to initial 1 h due to uniform permeability of the drug. However, after that drug starts getting washed out from the low-porosity regions or high-vascular regions quickly within 1 h. Another interesting observation was that tumor areas having higher interstitial volume fraction retained higher drug concentration for longer amount of time. This is due to large interstitial volume fraction or porosity, implying more accessible space for the free drug to diffuse in the interstitial space of tumor. As the time progresses, the drug starts getting diffused to higher porosity areas and gets accumulated there in larger quantities. However, diffusion of drug takes place slowly as compared to extravasation of drug. As the drug diffuses in interstitial space, it is absorbed by the cells present in the cellular compartment. The intracellular concentration accumulation pattern was found to be similar to that of interstitial concentration, with higher concentration present in higher interstitial volume fraction areas. One can conclude through this observation that intracellular concentration is directly dependent on interstitial concentration.

Figure 4 shows line plots of interstitial and intracellular concentration of free drug along a horizontal line bisecting the tumor at different times. Line plots quantitatively show that the concentration in both the compartments increases in tumor area initially for some time due to higher concentration in blood plasma and increased permeability in tumor area. However, it decreases rapidly after 12 h due to rapid decrease in plasma levels of drug, with almost negligible concentration at 48 h.

Upon absorption by cells, anticancer effect of drug starts and is predicted by change in the cell density in tumor area through pharmacodynamics model. It was observed that cell density decreased in the tumor areas having higher interstitial volume fraction because of higher interstitial and intracellular concentration of drug in those areas.

###### Systemic Administration of LED and Release of Drug.

A nonuniform interstitial concentration of LED was found that increased rapidly in the tumor area with more accumulation in the areas of higher interstitial volume fraction. The concentration gradually decreased with time because of decrease in plasma concentration. However, the decrease was relatively slower at longer times. This is because DOXIL is a stealth liposomes and circulates in blood plasma for longer period of time without being trapped in reticulo-endothelial system [21,32].

Once the LED gets extravasated to interstitial space, it starts releasing its drug payload at an optimum release rate [33,34]. Some portion of the drug is also released in blood plasma. Upon release from liposome, the drug (doxorubicin) follows the same transport phenomenon as free drug. It was observed that interstitial concentration of LRD was higher within the tumor region due to enhanced permeability within the tumor area. Also, drug accumulation was more in regions of higher interstitial volume fraction areas due to heterogeneous vasculature. This pattern is similar to the one obtained in case of free drug. Additionally, the accumulation of LRD in higher porosity areas was found to be more as compared to free drug at all times qualitatively. Figure 5 shows line plots of interstitial as well as intracellular concentration of LRD along horizontal bisector of slice. These line plots show that the LRD concentration in interstitial space increases rapidly for some time and then gradually decreases at a very slow rate due to slow decrease in plasma concentration of DOXIL, as mentioned earlier. Same pattern was observed for intracellular concentration as well.

Anticancer effect of the LRD was calculated by pharmacodynamics model. It was observed that tumor cell density achieved through LRD decreased at a higher rate in higher interstitial volume fraction areas at all the times as compared to free drug qualitatively. To compare the efficacy of LRD and free drug, line plots were plotted for interstitial concentration at various times along a horizontal line bisecting the tumor slice (please see Fig. 6). Interstitial concentration of LRD was found to be higher than free drug initially. A 2–20 fold enhancement was found in the concentration of LRD as compared to free drug in the tumor region as time progressed, implying enhanced retention of LRD in tumors. Similar trend was observed for intracellular concentration. Also, same enhancement in concentration is reported in literature for different types of human tumors by Gabizon et al. [35] and for brain tumors in rats by Siegel et al. [36]. Presence of LRD at later times in large quantities indicates the increased efficacy of LRD towards tumor. This analysis was done for three different human brain tumors and similar fold enhancement was found in all three tumors simulated. This demonstrates the robustness of the developed computational model in predicting the accumulation of LED and free drug.

In Fig. 7, a comparison was made between tumor cell density obtained by LRD and free drug. No significant difference was found between tumor cell densities achieved by both the drugs initially. However, after 1 h, tumor cell density changed drastically, with more number of cells being killed by LRD at longer times. Number of tumor cells being killed by LRD at higher times was found to be approximately 30–40% more than the free drug. This is because the intracellular concentration of free drug decreases after 12 h to a level where cell kill rate falls below cell proliferation rate. Consequently, the cell density in tumor area starts to increase. On the contrary, the tumor cell density continues to decrease for 48 h in case of LRD due to higher intracellular concentration as compared to free drug.

###### Importance of Heterogeneous Vasculature of Tumor.

Adopting constant value of interstitial volume fraction (as in homogeneous model) may result in significant overestimation of drug accumulation in tumors since in reality tumor vasculature is highly heterogeneous, disorganized, and chaotic. Therefore, heterogeneous vasculature of tumor in this study was determined by DCE-MRI that helped in calculating voxel-wise vascular and interstitial volume fraction. Finally, to investigate the implication of considering heterogeneous vasculature over homogeneous vasculature in drug distribution, simulations of drug concentration were performed for a case with constant value of porosity throughout the tumor region. For the homogeneous case, the drug distribution was observed to be uniform throughout the interstitial space, with same drug concentration at each voxel. On the other hand, as can be seen in Figs. 3 and 4, nonhomogeneous distribution was observed with heterogeneous vasculature.

###### Study Limitations and Future Work.

The developed computational model helps us in understanding the drug transport mechanism in human brain tumors. However, there are following assumptions used in this model. First, to simulate the systemic administration of free drug as well as LED in human brain tumors, AIF of drugs were taken from literature. Second, the release rate of the LED has been assumed to be constant throughout the simulation. Third, uniform permeability and transport properties of the drug as well as LED were assumed for the whole tumor region. Fourth, it has been assumed in this study that cell kill rate and cell proliferation rate will not vary a lot for different tumor cell lines [11,23,27]. Finally, the governing equations used for drug prediction in the current study consist of lot of parameters which are specific to each patient such as interstitial volume fraction, vascular volume fraction, patient-specific AIF, and volume of tumor. Therefore, there is a need to perform a sensitivity analysis of aforementioned parameters. This analysis will help in assessing the effect of each parameter on the interstitial fluid flow and drug concentration to determine the most influential parameter. The aforementioned sensitivity analysis was not performed in the current study due to unavailability of experimental data of large number of patients; however, it can be carried out in future. Additionally, nonuniform permeability of tumor to drug and drug carriers can be taken into account in future by correlating it with tumor-specific permeability of contrast agent. This will help in modeling the selective leakage of chemotherapeutic drugs in tumor areas of a specific patient. This will further help in improving the accuracy of the developed computational model, while predicting drug transport and its efficacy in realistic human tumors.

## Conclusions

The present study helps us to understand the effect of heterogeneous vasculature on the distribution of free drugs as well as LED in human brain tumors. The current study tells us the spatial variation of tumor vasculature, tumor interstitial space, and tumor cell density and their effect on nonuniform distribution of drug within the tumor region with the help of DCE-MRI data. The computational results have important implications because they demonstrate that (i) heterogeneous vasculature leads to preferential accumulation of drugs at the tumor position, and (ii) more drug accumulation takes place at higher interstitial volume fraction areas at longer times, thereby killing larger number of tumor cells at those areas. Due to longer circulation time and higher concentration, LRD has better therapeutic efficacy. A 2–20 fold concentration enhancement in tumor tissue was found with LRD as compared to free drug that is similar to the previously published studies [35,36]. The computational model developed in this study can be applied to any type of tumor for planning patient-specific treatment strategy. Additionally, most appropriate chemotherapeutic drug or a combination of suitable drugs for a specific patient can be found through in silico studies by using this model.

## Acknowledgements

The authors would like to thank Dr. R.K. Gupta and Dr. R.K.S. Rathore for providing clinical data and technical support in DCE-MRI data analysis, respectively.

## Funding Data

• Indian Institute of Technology Kanpur (Grant No. IITK/ME/2013370).

• Science and Engineering Research Board (Grant No. YSS/2014/000092).

## Nomenclature

• $Cbi$ =

bound drug concentration in interstitial space (kg/m3)

• $Cbp$ =

time-dependent bound drug concentration in blood plasma (kg/m3)

• $Cfi$ =

free drug concentration in interstitial space (kg/m3)

• $Cfp$ =

time-dependent free drug concentration in blood plasma (kg/m3)

• $Cv$ =

drug concentration in blood plasma (kg/m3)

• $Ci$ =

intracellular drug concentration (kg/105 cells)

• $Cli$ =

liposome-encapsulated drug concentration in interstitial space (kg/m3)

• $Clp$ =

time-dependent liposome-encapsulated drug concentration in blood plasma (kg/m3)

• $C(t)$ =

time-dependent concentration of contrast agent in interstitial space (mmol/Lt)

• $Cp(t)$ =

time dependent concentration of contrast agent in blood plasma (AIF) (mmol/Lt)

• $Dbi$ =

diffusion coefficient of bound drug molecules in interstitial space ($m2 s−1$)

• $Dfi$ =

diffusion coefficient of free drug molecules in interstitial space ($m2 s−1$)

• $Dli$ =

diffusion coefficient of liposome encapsulated drug in interstitial space ($m2 s−1$)

• $EC50$ =

drug concentration producing 50% of $fmax$ respectively (kg/105 cells)

• $fmax$ =

cell kill rate constant (s−1)

• $kb$ =

drug–protein binding rate (s−1)

• $kc$ =

cell proliferation rate constant (s−1)

• $kd$ =

dissociation rate (s−1)

• $ke$ =

parameter for cellular transmembrane transport (kg/m3)

• $ki$ =

parameter for cellular transmembrane transport (ng/105 cells)

• $krel$ =

rate at which drug is released from liposomes (s−1)

• K =

hydraulic conductivity of tissue ($m2Pa−1 s−1)$

• $Ktrans$ =

rate transfer constant of contrast agent from plasma to interstitial space (s−1)

• $Lp$ =

hydraulic conductivity of micro vascular wall ($mPa−1 s−1$)

• $Lp,Ly(SL/V)$ =

lymphatic filtration coefficient ($Pa−1 s−1$)

• $pL$ =

pressure in lymphatic vessels (Pa)

• $pi$ =

interstitial fluid pressure (IFP) (Pa)

• $pv$ =

vascular fluid pressure (Pa)

• $Pbi$ =

micro vascular permeability coefficient to bound drug (m s−1)

• $Pfi$ =

micro vascular permeability coefficient to free drug (m s−1)

• $Pli$ =

micro vascular permeability coefficient to liposome-encapsulated drug (m s−1)

• $Peff$ =

effective pressure (Pa)

• S(t) =

signal intensity at a particular time point after administration of contrast agent

• S(0) =

signal intensity when no contrast agent is given

• $(S/V)$ =

blood vessel surface area per unit volume ($m−1$)

• $TR$ =

repetition time (ms)

• $TE$ =

echo time (ms)

• $T1 and T2$ =

spin lattice and spin–spin relaxation time, respectively (ms)

• $T10$ =

Relaxation time without contrast agent administration (ms)

• $ve$ =

volume fraction of interstitial space or porosity

• $vp$ =

volume fraction of plasma space

• v =

interstitial fluid velocity (IFV) (m/s)

• $Vmax$ =

transmembrane transport rate (kg/105 cells s)

• $δ$ =

cell uptake of drug from interstitial space

• $πv and πi$ =

osmotic pressures of plasma and interstitial fluid, respectively (Pa)

• $σbl$ =

osmotic reflection coefficient for bound drug molecules

• $σfl$ =

• $σll$ =

osmotic reflection coefficient for liposome encapsulated drug molecules

• $σT$ =

osmotic reflection coefficient for plasma proteins

• $ω$ =

efflux from cells to interstitial space

## References

Pattni, B. S. , Chupin, V. V. , and Torchilin, V. P. , 2015, “New Developments in Liposomal Drug Delivery,” Chem. Rev., 115(19), pp. 10938–10966. [PubMed]
Allen, T. M. , and Torchilin, P. R. , 2013, “Liposomal Drug Delivery Systems: From Concept to Clinical Applications,” Adv. Drug Delivery Rev., 65(1), pp. 36–48.
Baxter, L. T. , and Jain, R. K. , 1989, “Transport of Fluid and Macromolecules in Tumors—I: Role of Interstitial Pressure and Convection,” Microvasc. Res., 37(1), pp. 77–104. [PubMed]
Baxter, L. T. , and Jain, R. K. , 1990, “Transport of Fluid and Macromolecules in Tumors II. Role of Heterogeneous Perfusion and Lymphatics,” Microvasc. Res., 40(2), pp. 246–263. [PubMed]
Harashima, H. , Iida, S. , Urakami, Y. , Tsuchihashi, M. , and Kiwada, H. , 1999, “Optimization of Antitumor Effect of Liposomally Encapsulated Doxorubicin Based on Simulation by Pharmacokinetic/Pharmacodynamics Modeling,” J. Controlled Release, 61(1–2), pp. 93–106.
Elkareh, A. W. , and Secomb, T. W. , 2000, “A Mathematical Model for Comparison of Bolus Injection, Continuous Infusion and Liposomal Delivery of Doxorubicin to Tumor Cells,” Neoplasia, 2(4), pp. 325–338. [PubMed]
Stapleton, S. , Milosevic, M. , Allen, C. , Zheng, J. , Dunne, M. , Yeung, I. , and Jaffray, D. A. , 2013, “A Mathematical Model of the Enhanced Permeability and Retention Effect for Liposome Transport in Solid Tumors,” PLoS ONE, 8(2), p. e81157. [PubMed]
Goh, Y. M. F. , Kong, H. L. , and Wang, C. H. , 2001, “Simulation of Delivery of Doxorubicin to Hepatoma,” Pharm. Res., 18(6), pp. 761–770. [PubMed]
Arifin, D. Y. , Lee, K. Y. T. , Wang, C. H. , and Smith, K. A. , 2009, “Role of Convective Flow in Carmustine Delivery to a Brain Tumor,” Pharm. Res., 26(10), pp. 2289–2302. [PubMed]
Soltani, M. , and Chen, P. , 2011, “Numerical Modeling of Fluid Flow in Solid Tumors,” PLoS One, 6(6), p. e20344. [PubMed]
Zhan, W. , Gedroyc, W. , and Xu, X. Y. , 2014, “Effect of Heterogeneous Microvasculature Distribution on Drug Delivery to Solid Tumour,” J. Phys. D: Appl. Phys., 47(47), p. 475401.
Magdoom, K. N. , Pishko, G. L. , Kim, J. H. , and Sarntinoranont, M. , 2012, “Evaluation of a Voxelized Model Based on DCE-MRI for Tracer Transport in Tumor,” ASME J. Biomech. Eng., 134(9), p. 091004.
Tofts, P.S., Brix, G., Buckley, D. L., Evelhoch, J. L., Henderson, E ., Knopp, M. V., Larsson, H. B. W., Lee, T. Y., Mayr, N. A., Parker, G. J. M., Port, R. E., Taylor, J., and Weisskoff, R. M., 1999, “Estimating Kinetic Parameters From Dynamic Contrast-Enhanced T1-Weighted MRI of a Diffusable Tracer: Standardized Quantities and Symbols,” J. Magn. Reson. Imaging, 26(4), pp. 871–880.
Bhandari, A. , Bansal, A. , Singh, A. , and Sinha, N. , 2017, “Perfusion Kinetics in Human Brain Tumor With DCE-MRI Derived Model and CFD Analysis,” J. Biomech., 59, pp. 80–89. [PubMed]
Bhandari, A. , Bansal, A. , Singh, A. , and Sinha, N. , 2017, “Transport of Liposomes Encapsulated Drugs in Voxelized Computational Model of Human Brain Tumors,” IEEE Trans. Nanobiosci., 16(7), pp. 634–644.
Tofts, P. S., 1997, “Modeling Tracer Kinetics in Dynamic Gd-DTPA MR Imaging,” J. Magn. Reson. Imaging, 3, pp. 91–101.
Singh, A. , Haris, M. , Purwar, A. , Sharma, M. , Husain, N. , Rathore, R. K. S. , and Gupta, R. K. , 2007, “Quantification of Physiological and Hemodynamic Indices Using T1 DCE-MRI in Intracranial Mass Lesions,” J. Magn. Reson. Imaging, 26(4), pp. 871–880. [PubMed]
Pintaske, J. , Martirosian, P. , Graf, H. , Erb, G. , Lodemann, K. P. , Claussen, C. D. , and Schick, F. , 2006, “Relaxivity of Gadopentetate Dimeglumine (Magnevist), Gadobutrol (Gadovist), and Gadobenate Dimeglumine (MultiHance) in Human Blood Plasma at 0.2, 1.5 and 3 Tesla,” Invest. Radiol., 41(3), pp. 213–221. [PubMed]
Singh, A. , Rathore, R. K. S. , Haris, M. , Verma, S. K. , Husain, N. , and Gupta, R. K. , 2009, “Improved Bolus Arrival Time and Arterial Input Function Estimation for Tracer Kinetic Analysis in DCE-MRI,” J. Magn. Reson. Imaging, 29(1), pp. 166–176. [PubMed]
Tofts, P. S. , and Kermode, A. G. , 1991, “Measurement of the Blood-Brain Barrier Permeability and Leakage Space Using Dynamic MR Imaging,” Magn. Reson. Med., 17(2), pp. 357–367. [PubMed]
Gabizon, A. , Isacson, R. , Libson, E. , Kaufman, B. , Uziely, B. , Catane, R. , Rabello, E. , Cass, Y. , Peretz, T. , Sulkes, A. , Chisin, R. , and Barenholz, Y. , 1994, “Clinical Studies of Liposome Encapsulated Doxorubicin,” Acta Oncol., 33(7), pp. 779–786. [PubMed]
Greene, R. F. , Collins, J. M. , Jenkins, J. F. , Speyer, J. L. , and Myers, C. E. , 1983, “Plasma Pharmacokinetics of Adriamycin and Adriamycinol: Implications for the Design of In Vitro Experiments and Treatment Protocols,” Cancer Res., 43(7), pp. 3417–3421. [PubMed]
Eikenberry, S. , 2009, “A Tumor Cord Model for Doxorubicin Delivery and Dose Optimization in Solid Tumors,” Theor. Bio. Med. Model., 6, p. 16.
Eliaz, R. E. , Nir, S. , Marty, C. , and Szoka, F. C. , 2004, “Determination and Modeling of Kinetics of Cancer Cell Killing by Doxorubicin and Doxorubicin Encapsulated in Targeted Liposomes,” Cancer Res., 64(2), pp. 711–718. [PubMed]
Khalifa, A. , Dodds, D. , Rampling, R. , Paterson, J. , and Murray, T. , 1997, “Liposomal Distribution in Malignant Gliomas: Possibilities for Theraphy,” Nucl. Med. Commun., 18, pp. 17–23. [PubMed]
Hau, P. , Fabel, K. , Baumgart, U. , Rummele, P. , Grauer, O. , Bock, A. , Dietmaier, C. , Dietmaier, W. , Dietrich, J. , Dudel, C. , Hubner, F. , Jauch, T. , Drechsel, E. , Kleiter, I. , Wismeth, C. , Zellner, A. , Brawanski, A. , Stienbrecher, A. , Marienhagen, J. , and Bogdahn, U. , 2005, “Pegylated Liposomal Doxorubicin-Efficacy in Patients With Recurrent High-Grade Gliomas,” Am. Can. Soc., 100(6), pp. 1199–1207.
Murray, J. D. , 2012, “Glioblastoma Brain Tumours: Estimating the Time From Brain Tumour Initiation and Resolution of a Patient Survival Anomaly After Similar Treatment Protocols,” J. Bio. Dyn., 6(2), pp. 118–127.
Anderson, D. A. , Tannehill, J. C. , and Pletcher, R. H. , 1984, Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, pp. 671–674.
Abe, T. , Mizobuchi, Y. , Nakajima, K. , Otomi, Y. , Irahara, S. , Obama, Y. , Majigsuren, M. , Khashbat, D. , Kagezi, T. , Nagahiro, S. , and Harada, M. , 2015, “Diagnosis of Brain Tumors Using Dynamic Contrast-Enhanced Perfusion Imaging With a Short Acquisition Time,” Springer Plus, 4, p. 88. [PubMed]
Boucher, Y. , Salehi, H. , Witwer, B. , Harsh , G. R., IV ., and Jain, R. K. , 1997, “Interstitial Fluid Pressure in Intracranial Tumors in Patients and in Rodents,” Br. J. Cancer, 75, pp. 829–836. [PubMed]
Guttman, R. , Leunig, M. , Feyh, J. , Goetz, A. E. , Messmer, K. , and Kastenbauer, E. , 1992, “Interstitial Hypertension in Head and Neck Tumors in Patients: Correlation With Tumor Size,” Cancer Res., 52(7), pp. 1993–1995. [PubMed]
Wu, Z. N. , Da, D. , Rudoll, T. L. , Needham, D. , Whorton, A. R. , and Dewhirst, M. W. , 1993, “Increased Microvascular Permeability Contributes to Preferential Accumulation of Stealth Liposomes in Tumor Tissue,” Cancer Res., 53(16), pp. 3765–3770. [PubMed]
Gabizon, A. , Goren, D. , Horowitz, A. T. , Tzemach, D. , Lossos, A. , and Siegal, T. , 1997, “Long-Circulating Liposomes for Drug Delivery in Cancer Therapy: A Review of Bio-Distribution Studies in Tumor-Bearing Animals,” Adv. Drug Delivery Rev., 24(2–3), pp. 337–344.
Gabizon, A. , Shmeeda, H. , and Barenholz, Y. , 2003, “Pharmacokinetics of Pegylated Liposomal Doxorubicin, Review of Animal and Human Studies,” Clin. Pharm., 42(5), pp. 419–436.
Gabizon, A. , Catane, R. , Uziely, B. , Kaufman, B. , Safra, T. , Cohen, R. , Martin, F. , Huang, A. , and Barenholz, Y. , 1994, “Prolonged Circulation Time and Enhanced Accumulation in Malignant Exudates of Doxorubicin Encapsulated in Polyethylene-Glycol Coated Liposomes,” Cancer Res., 54(4), pp. 987–992. [PubMed]
Siegel, T. , Horowitz, A. , and Gabizon, A. , 1995, “Doxorubicin Encapsulated in Sterically Stabilized Liposomes for the Treatment of a Brain Tumor Model: Bio Distribution and Therapeutic Efficacy,” J. Neurosurg., 83(6), pp. 1029–1037. [PubMed]
Topics: Drugs , Tumors
View article in PDF format.

## References

Pattni, B. S. , Chupin, V. V. , and Torchilin, V. P. , 2015, “New Developments in Liposomal Drug Delivery,” Chem. Rev., 115(19), pp. 10938–10966. [PubMed]
Allen, T. M. , and Torchilin, P. R. , 2013, “Liposomal Drug Delivery Systems: From Concept to Clinical Applications,” Adv. Drug Delivery Rev., 65(1), pp. 36–48.
Baxter, L. T. , and Jain, R. K. , 1989, “Transport of Fluid and Macromolecules in Tumors—I: Role of Interstitial Pressure and Convection,” Microvasc. Res., 37(1), pp. 77–104. [PubMed]
Baxter, L. T. , and Jain, R. K. , 1990, “Transport of Fluid and Macromolecules in Tumors II. Role of Heterogeneous Perfusion and Lymphatics,” Microvasc. Res., 40(2), pp. 246–263. [PubMed]
Harashima, H. , Iida, S. , Urakami, Y. , Tsuchihashi, M. , and Kiwada, H. , 1999, “Optimization of Antitumor Effect of Liposomally Encapsulated Doxorubicin Based on Simulation by Pharmacokinetic/Pharmacodynamics Modeling,” J. Controlled Release, 61(1–2), pp. 93–106.
Elkareh, A. W. , and Secomb, T. W. , 2000, “A Mathematical Model for Comparison of Bolus Injection, Continuous Infusion and Liposomal Delivery of Doxorubicin to Tumor Cells,” Neoplasia, 2(4), pp. 325–338. [PubMed]
Stapleton, S. , Milosevic, M. , Allen, C. , Zheng, J. , Dunne, M. , Yeung, I. , and Jaffray, D. A. , 2013, “A Mathematical Model of the Enhanced Permeability and Retention Effect for Liposome Transport in Solid Tumors,” PLoS ONE, 8(2), p. e81157. [PubMed]
Goh, Y. M. F. , Kong, H. L. , and Wang, C. H. , 2001, “Simulation of Delivery of Doxorubicin to Hepatoma,” Pharm. Res., 18(6), pp. 761–770. [PubMed]
Arifin, D. Y. , Lee, K. Y. T. , Wang, C. H. , and Smith, K. A. , 2009, “Role of Convective Flow in Carmustine Delivery to a Brain Tumor,” Pharm. Res., 26(10), pp. 2289–2302. [PubMed]
Soltani, M. , and Chen, P. , 2011, “Numerical Modeling of Fluid Flow in Solid Tumors,” PLoS One, 6(6), p. e20344. [PubMed]
Zhan, W. , Gedroyc, W. , and Xu, X. Y. , 2014, “Effect of Heterogeneous Microvasculature Distribution on Drug Delivery to Solid Tumour,” J. Phys. D: Appl. Phys., 47(47), p. 475401.
Magdoom, K. N. , Pishko, G. L. , Kim, J. H. , and Sarntinoranont, M. , 2012, “Evaluation of a Voxelized Model Based on DCE-MRI for Tracer Transport in Tumor,” ASME J. Biomech. Eng., 134(9), p. 091004.
Tofts, P.S., Brix, G., Buckley, D. L., Evelhoch, J. L., Henderson, E ., Knopp, M. V., Larsson, H. B. W., Lee, T. Y., Mayr, N. A., Parker, G. J. M., Port, R. E., Taylor, J., and Weisskoff, R. M., 1999, “Estimating Kinetic Parameters From Dynamic Contrast-Enhanced T1-Weighted MRI of a Diffusable Tracer: Standardized Quantities and Symbols,” J. Magn. Reson. Imaging, 26(4), pp. 871–880.
Bhandari, A. , Bansal, A. , Singh, A. , and Sinha, N. , 2017, “Perfusion Kinetics in Human Brain Tumor With DCE-MRI Derived Model and CFD Analysis,” J. Biomech., 59, pp. 80–89. [PubMed]
Bhandari, A. , Bansal, A. , Singh, A. , and Sinha, N. , 2017, “Transport of Liposomes Encapsulated Drugs in Voxelized Computational Model of Human Brain Tumors,” IEEE Trans. Nanobiosci., 16(7), pp. 634–644.
Tofts, P. S., 1997, “Modeling Tracer Kinetics in Dynamic Gd-DTPA MR Imaging,” J. Magn. Reson. Imaging, 3, pp. 91–101.
Singh, A. , Haris, M. , Purwar, A. , Sharma, M. , Husain, N. , Rathore, R. K. S. , and Gupta, R. K. , 2007, “Quantification of Physiological and Hemodynamic Indices Using T1 DCE-MRI in Intracranial Mass Lesions,” J. Magn. Reson. Imaging, 26(4), pp. 871–880. [PubMed]
Pintaske, J. , Martirosian, P. , Graf, H. , Erb, G. , Lodemann, K. P. , Claussen, C. D. , and Schick, F. , 2006, “Relaxivity of Gadopentetate Dimeglumine (Magnevist), Gadobutrol (Gadovist), and Gadobenate Dimeglumine (MultiHance) in Human Blood Plasma at 0.2, 1.5 and 3 Tesla,” Invest. Radiol., 41(3), pp. 213–221. [PubMed]
Singh, A. , Rathore, R. K. S. , Haris, M. , Verma, S. K. , Husain, N. , and Gupta, R. K. , 2009, “Improved Bolus Arrival Time and Arterial Input Function Estimation for Tracer Kinetic Analysis in DCE-MRI,” J. Magn. Reson. Imaging, 29(1), pp. 166–176. [PubMed]
Tofts, P. S. , and Kermode, A. G. , 1991, “Measurement of the Blood-Brain Barrier Permeability and Leakage Space Using Dynamic MR Imaging,” Magn. Reson. Med., 17(2), pp. 357–367. [PubMed]
Gabizon, A. , Isacson, R. , Libson, E. , Kaufman, B. , Uziely, B. , Catane, R. , Rabello, E. , Cass, Y. , Peretz, T. , Sulkes, A. , Chisin, R. , and Barenholz, Y. , 1994, “Clinical Studies of Liposome Encapsulated Doxorubicin,” Acta Oncol., 33(7), pp. 779–786. [PubMed]
Greene, R. F. , Collins, J. M. , Jenkins, J. F. , Speyer, J. L. , and Myers, C. E. , 1983, “Plasma Pharmacokinetics of Adriamycin and Adriamycinol: Implications for the Design of In Vitro Experiments and Treatment Protocols,” Cancer Res., 43(7), pp. 3417–3421. [PubMed]
Eikenberry, S. , 2009, “A Tumor Cord Model for Doxorubicin Delivery and Dose Optimization in Solid Tumors,” Theor. Bio. Med. Model., 6, p. 16.
Eliaz, R. E. , Nir, S. , Marty, C. , and Szoka, F. C. , 2004, “Determination and Modeling of Kinetics of Cancer Cell Killing by Doxorubicin and Doxorubicin Encapsulated in Targeted Liposomes,” Cancer Res., 64(2), pp. 711–718. [PubMed]
Khalifa, A. , Dodds, D. , Rampling, R. , Paterson, J. , and Murray, T. , 1997, “Liposomal Distribution in Malignant Gliomas: Possibilities for Theraphy,” Nucl. Med. Commun., 18, pp. 17–23. [PubMed]
Hau, P. , Fabel, K. , Baumgart, U. , Rummele, P. , Grauer, O. , Bock, A. , Dietmaier, C. , Dietmaier, W. , Dietrich, J. , Dudel, C. , Hubner, F. , Jauch, T. , Drechsel, E. , Kleiter, I. , Wismeth, C. , Zellner, A. , Brawanski, A. , Stienbrecher, A. , Marienhagen, J. , and Bogdahn, U. , 2005, “Pegylated Liposomal Doxorubicin-Efficacy in Patients With Recurrent High-Grade Gliomas,” Am. Can. Soc., 100(6), pp. 1199–1207.
Murray, J. D. , 2012, “Glioblastoma Brain Tumours: Estimating the Time From Brain Tumour Initiation and Resolution of a Patient Survival Anomaly After Similar Treatment Protocols,” J. Bio. Dyn., 6(2), pp. 118–127.
Anderson, D. A. , Tannehill, J. C. , and Pletcher, R. H. , 1984, Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, pp. 671–674.
Abe, T. , Mizobuchi, Y. , Nakajima, K. , Otomi, Y. , Irahara, S. , Obama, Y. , Majigsuren, M. , Khashbat, D. , Kagezi, T. , Nagahiro, S. , and Harada, M. , 2015, “Diagnosis of Brain Tumors Using Dynamic Contrast-Enhanced Perfusion Imaging With a Short Acquisition Time,” Springer Plus, 4, p. 88. [PubMed]
Boucher, Y. , Salehi, H. , Witwer, B. , Harsh , G. R., IV ., and Jain, R. K. , 1997, “Interstitial Fluid Pressure in Intracranial Tumors in Patients and in Rodents,” Br. J. Cancer, 75, pp. 829–836. [PubMed]
Guttman, R. , Leunig, M. , Feyh, J. , Goetz, A. E. , Messmer, K. , and Kastenbauer, E. , 1992, “Interstitial Hypertension in Head and Neck Tumors in Patients: Correlation With Tumor Size,” Cancer Res., 52(7), pp. 1993–1995. [PubMed]
Wu, Z. N. , Da, D. , Rudoll, T. L. , Needham, D. , Whorton, A. R. , and Dewhirst, M. W. , 1993, “Increased Microvascular Permeability Contributes to Preferential Accumulation of Stealth Liposomes in Tumor Tissue,” Cancer Res., 53(16), pp. 3765–3770. [PubMed]
Gabizon, A. , Goren, D. , Horowitz, A. T. , Tzemach, D. , Lossos, A. , and Siegal, T. , 1997, “Long-Circulating Liposomes for Drug Delivery in Cancer Therapy: A Review of Bio-Distribution Studies in Tumor-Bearing Animals,” Adv. Drug Delivery Rev., 24(2–3), pp. 337–344.
Gabizon, A. , Shmeeda, H. , and Barenholz, Y. , 2003, “Pharmacokinetics of Pegylated Liposomal Doxorubicin, Review of Animal and Human Studies,” Clin. Pharm., 42(5), pp. 419–436.
Gabizon, A. , Catane, R. , Uziely, B. , Kaufman, B. , Safra, T. , Cohen, R. , Martin, F. , Huang, A. , and Barenholz, Y. , 1994, “Prolonged Circulation Time and Enhanced Accumulation in Malignant Exudates of Doxorubicin Encapsulated in Polyethylene-Glycol Coated Liposomes,” Cancer Res., 54(4), pp. 987–992. [PubMed]
Siegel, T. , Horowitz, A. , and Gabizon, A. , 1995, “Doxorubicin Encapsulated in Sterically Stabilized Liposomes for the Treatment of a Brain Tumor Model: Bio Distribution and Therapeutic Efficacy,” J. Neurosurg., 83(6), pp. 1029–1037. [PubMed]

## Figures

Fig. 1

(a) Precontrast T1 weighted MR image of one slice of human brain, (b) postcontrast T1 weighted image, (c) zoomed view of tumor portion, and (d) segmented CFD single slice including tumor (dark blue) and remaining normal tissue

Fig. 2

Contour maps of (a) interstitial volume fraction (porosity), (b) plasma volume fraction, and (c) cell density of slice 8 of MR data set

Fig. 3

Contour maps representing distribution of free drug at different times: (a) 1 h, (b) 12 h, (c) 24 h, and (d) 48 h. Scale bar is same as in Fig. 2.

Fig. 4

Line plots along horizontal bisector through tumor region of slice at different times of free drug (a) interstitial concentration and (b) intracellular concentration (dashed lines indicate tumor boundary)

Fig. 5

Line plots along horizontal bisector through tumor region of slice at different times of LRD (a) interstitial concentration and (b) intracellular concentration (dashed lines indicate tumor boundary)

Fig. 6

Line plots showing comparison of interstitial concentration by LRD and free drug along horizontal bisector of slice at different times: (a) 1 h, (b) 12 h, (c) 24 h, and (d) 48 h (dashed lines indicate tumor boundary)

Fig. 7

Line plots showing comparison of cell density by LRD and free drug along horizontal bisector of slice at different times: (a) 1 h, (b) 12 h, (c) 24 h, and (d) 48 h (dashed lines indicate tumor boundary)

## Tables

Table 1 Normal and tumor tissue pharmacokinetic values for LED, free drug, and bound drug
NT: normal tissue, TT: tumor tissue.

## Discussions

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