Three-dimensional simulation of paclitaxel delivery to a brain tumor. Yingying Huang, Davis Y. Arifin, Chi-Hwa Wang

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1 Three-dimensional simulation of paclitaxel delivery to a brain tumor Yingying Huang, Davis Y. Arifin, Chi-Hwa Wang Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore ABSTRACT Purpose. In this report, a better understanding of the mechanism of paclitaxel delivery to a brain tumor is presented through mathematical model and computer simulation. Both one-dimensional simulation and three-dimensional simulations are performed and compared to investigate the crucial factors affecting the drug delivery to human brain tumor, especially the role of convective flow. Methods. The geometry model from a brain tumor patient with tumor removal surgery and Gliadel wafer implantation is obtained from a reliable literature [8]. The simulation is performed by solving a simple diffusion-convection reaction model with finite volume techniques. Results. The release profile of paclitaxel presented by simulations shows the force for the penetration into the tumor is a combination of diffusion and convection, though the former appears to be more dominant. The role of convective flow is shown by a non-uniform drug penetration into the tumor: the remnant tumor fraction towards the ventricle is less exposed to the drug than that of away from ventricle. Conclusions. There is a reasonably good penetration of paclitaxel for wafer implanted in brain, i.e. it can penetrate through the whole remnant tissues. However, as a consequence the paclitaxel is also present in normal tissues during the simulation;hence, managing the dosage of paclitaxel is crucial as to minimize the toxic exposure to the normal tissues.. In addition, the local convective flow greatly affects the distribution of drug in the remnant tumor and nearby tissues by 3 5 orders of magnitude. Therefore, the incision of tumor and the implantation site of wafer should be the main concern. Alternatively, a modified microsphere to be implanted at the interface, i.e. embedded in a polymeric film, might help. INTRODUCTION 1

2 Paclitaxel, an anti-microtubule, anti-neoplastic agent, has been clinically effective against a variety of human cancers, including ovarian and breast cancer. It is also one of the most widely chemotherapeutic agents investigated for brain tumor treatment. While the in-vitro and in-vivo analysis has been widely performed, there is still lack of understanding on how the drug is transported in the brain tissue. The complex characteristic of interstitial fluid flow in human brain even more complicates the performance of the drug delivery system. Hence, the aim of this study is to further understand the mechanism of paclitaxel delivery in the brain tumor and how various factors influence the drug penetration to a brain tumor. This report also includes the development of mathematical model for paclitaxel as well as a simple onedimensional study as a proof of concept, suggesting on how fluid flow might cause variability in the drug penetration and a subsequent three-dimensional simulation to address the overall effect of the brain interstitial fluid flow. MATHEMATICAL TRANSPORT MODEL To evaluate the dynamics of local interstitial, mathematical models have been developed to describe the transport of drugs into the tissue near the implant. When a drug-loaded polymer is implanted, the drug is released at a localized site. The released drug migrates away from the polymer/tissue interface, principally via the extracellular space (ECS). Drug molecules move in the ECS by diffusion and fluid convection; they may also be internalized by cells to react with subcellular components; the drug molecules can also enter the vascular system, be transported to the rest of the body, or be eliminated systemically. If the drug is sufficiently lipophilic, it may penetrate cell membranes rapidly enough to move by a transcellular path. Therefore, the fate of drug molecules delivered to the tissue depends on the rates of transport (via diffusion and fluid convection), elimination (by degradation, metabolism and transcapillary permeation) and internalization. [4] Fluid flow through tissue in the brain has been simplified as the following with substitution of appropriate scales and parameters from Table I with the interstitial fluid assumed to be water with p p K 2 2 ρ v s L b ra in physical properties at 37 ºC. [8]: ( ve n tric le ou te r ) v = R e p The equation of drug conservation is simplified as: t C 2 = D C * v C kc i 2

3 Detailed derivation is presented in appendix A. Knowledge of the physicochemical and biological characteristics of the drug paclitaxel is required to estimate the parameters in the equations, and they are presented in table II. Table I. Interstitial fluid-related Parameter Values [8] Parameter Cavity Remnant Tumor ρ Density of the interstitial fluid (kg/m 3 ) 1,000 µ Viscosity of the interstitial fluid (Pa-s) p v Vascular pressure (Pa) N/A 4,610 Normal Tissue S/V Blood vessel exchange area (m -1 ) N/A 20,000 7,000 π v Vascular osmotic pressure (Pa) N/A 3,440 π i Interstitial osmotic pressure (Pa) N/A 1, σ Osmotic reflection coefficient of plasma N/A L p Hydraulic conductivity (m/pa/s) N/A K Darcy s permeability (m 2 ) Fv Rate of fluid gain from the capillary blood flow per unit vol of tissue Table II. Paclitaxel-related Parameter Values α Parameter Volume fraction of interstitial/extracellular space N/A Cavity Remnant Tumor Normal Tissue [3] β Volume fraction of intracellular space [3] P c/i P m/i K i, K c D i Partition coefficient between cellular and interstitial phase Partition coefficient between membrane(octane) and interstitial phase(water) Binding constant between free and bound drugs (Assume Ki=Kc) Diffusion coefficient in interstitial phase (m 2 /s) N/A 1 N/A 3162 [2] N/A 5 [1] 9.00x10-10[2] k bbb Drug elimination to blood capillaries (1/s) N/A 1.39x10-4[2] k e Drug elimination due to enzymatic/nonenzymatic reactions (1/s) 6.79x x10-7[2] α* retardation constant N/A D Lumped diffusion coefficient (m 2 /s) 9.0x x x10-13 k Lumped first-order elimination constant (1/s) 6.8x x x10-8 3

4 Φ (predicted) Thiele modulus Pe Peclet number: LV/D measures the relative importance of drug convection over diffusion in tissue Thiele modulus φ = Ld k D is the ratio of drug elimination rate to diffusion transport, where L d is the diffusion/reaction length scale, i.e.3.7mm. Φ is predicted to be 1, meaning that penetration is mainly due to diffusion which is against the rapid transcapillary elimination. This value is compared with experimental data obtained from published literatures [6] using best line fitting method, and is found out to be fairly accurate. During the experiment, paclitaxel was incorporated into polilactofate microspheres (Paclimer) at 10% loading and compressed into 10-mg discs containing 50% PEG-1000 by weight for implantation in the rat parietal lobe. Brains were removed at 7 and 30 days after implantation and divided into implant and contralateral hemispheres. Each hemisphere was sectioned at 2-mm intervals in the coronal plane, and the amount of paclitaxel present in the section was quantitated by liquid scintillation counting. Fig.1(a). Coronal plane showing both contralateral hemisphere and implant hemisphere. In this case, the effect of convection can be neglected as the convection pattern is comparable in each coronal plane. The concentration profile of paclitaxel in the implant hemisphere is presented below. 4

5 Relative Paclitaxel Concentration Concentration Profiles of free paclitaxel Distance from implantation site (m) Experiment - Day 7 Experiment - Day 30 Model Fig.1(b). The penetration profile of paclitaxel to the rate brain in implant hemisphere. Through best line fitting, Φ is evaluated to be 1, which agrees well with predicted value. RESULTS AND DISCUSSION A one-dimensional simulation is firstly performed using COMSOL to solve a simple diffusion-convection-reaction problem. Subsequently, the brain geometry is constructed using a preprocessor grid generator GAMBIT (Fluent Inc., 2006), which is then exported to commercial computational fluid dynamics (CFD) software FLUENT (Fluent Inc., 2006) for a user-defined threedimensional simulation. One-Dimensional Simulation To suggest on how fluid convection might cause variability in the drug penetration, a onedimensional study was initially performed to simplify the mechanisms. The one-dimensional model has been designed to consist of three rectangular blocks as shown below, which represent cavity, remnant tumor and nearby tissue respectively. The mean thickness of remnant tumor was taken as 2.3mm, and nearby tissue was represented by a thin 1.5mm layer, whereas the thickness of cavity is non-uniform due to the irregular shape of the section and arbitrary wafer placement, and was taken to be 0mm, 2mm and 5mm to suggest on how different thickness of this layer might affect the distribution of drug in cavity. [8] The cavity domain exists because the wafers implanted are often not big enough to fill the cavity created by the tumor removal surgery. 5

6 Fig.2. The schematic of one dimensional approach. Therefore, this model is simplified into a one-dimensional diffusion-convection-reaction problem through three different layers. Taking into consideration the possibility of brain edema in which the interstitial fluid increases in correspondence to the trauma, three interstitial fluid velocities: 1 x 10-7 m/s, 2 x 10-7 m/s and 5 x10-7 m/s were applied respectively to study the effect of convection. In addition, the convection is assumed to occur in two directions only: (1) towards the treatment domain which brings about enhanced convection, and (2) against the treatment domain which brings about repressed convection. The boundary conditions are constant concentrations of 1 mol/m 3 at the polymer surface [5,10,11] and negligible concentration at the periphery of nearby normal tissue considering high Φ value in remnant tumor and nearby tissue due to rapid capillary elimination. Fig.3. The drug penetration profile in the remnant tumor for different Vx / Vs values. Vs is the characteristic interstitial fluid velocity of 1 x 10-7 m/s obtained from scaling analysis, and velocity vector V is replaced with scalar quantity of Vx as the flow investigated is uni-directional. Three different magnitudes of interstitial fluid velocity are discussed: (1) the interstitial fluid velocity is the same as characteristics velocity (Vx / Vs = 1); (2) the interstitial fluid velocity is two times of characteristics velocity (Vx / Vs = 2); (1) the interstitial fluid velocity is five times of characteristics velocity (Vx / Vs = 5).The cavity thickness is taken to be 0 mm; i.e. the polymer surface is in intact with the remnant tumor. As seen from the graph, the penetration of paclitaxel is rather sensitive to the effect of convection, especially at high interstitial fluid velocity, because the graphs with enhanced convection and repressed convection at high interstitial fluid velocity deviate far from the pure diffusion curve, 6

7 and at the end of remnant tumor, i.e. penetration distance=2.3mm, they differ by a magnitude of about 15, indicating that convection plays an important role in comparable magnitude with diffusion - in the drug penetration. Fig.4. The penetration profiles in the remnant tumor when polymer wafer interface is not in tact with the remnant tumor, separated by a cavity layer of 2mm or 5mm, with comparison with the case when polymer surface is intact with the remnant tumor (i.e. without cavity) for (a) enhanced convection with the interstitial fluid velocity Vx / Vs = 1; (b) repressed convection with the interstitial fluid velocity Vx / Vs = 1;(c) enhanced convection with the interstitial fluid velocity Vx / Vs = 2; (d) repressed convection with the interstitial fluid velocity Vx / Vs = 2. Penetration distance is always measured from the remnant tumor interface. These graphs have shown a variability of drug distribution due to the presence of cavity of different thickness. When drug travels along the cavity, the convective flow is dominant due to less resistance than the tissue. As we can see from the graphs, when cavity is present, the initial concentration at the remnant tumor interface is reduced by both enhanced and repressed convection and the decrease in initial concentration is proportional to the thickness of cavity layer as we can see 7

8 from the comparison between cavity of thickness 2mm and 5mm,eg. for enhanced convection with an interstitial fluid velocity Vx / Vs = 1, the initial concentration for 2mm cavity is reduced by 1/5 of the initial concentration without cavity, whereas for 5mm cavity it is reduced by 7/20. In the case of repressed convection, the reduction of initial concentration is more prominent (almost doubles). In addition, when the interstitial fluid velocity increases, decrease in initial concentration in the case of enhanced convection is reduced, eg. the initial concentration for enhanced convection for 2mm cavity is reduced by 0.2 mol/m 3 with an interstitial fluid velocity Vx / Vs = 1 and 0.13 mol/m 3 with an interstitial fluid velocity Vx / Vs = 2; whereas the decrease in initial concentration in the case of repressed convection is aggravated, eg. the initial concentration for repressed convection for 2mm cavity is reduced by 0.35 mol/m 3 with an interstitial fluid velocity Vx / Vs = 1 and 0.45 mol/m 3 with an interstitial fluid velocity Vx / Vs = 2. Hence, tapping drug delivery devices at the cavity/remnant tumor interface could reduce the variability, and therefore enhance the amount of drug available to the tumor in all directions. Therefore, a film-based drug delivery device has been suggested to be useful. [8] Three-Dimensional Simulation In reality, the fluid flow in the brain is rather complicated as it involves fluid permeation from the ventricle and from the blood vasculature. Hence, a three-dimensional simulation with accurate brain geometry is performed and analyzed using FLUENT with the available literatures data (the concentration boundary conditions and the magnitudes of the interstitial fluid flow). Fig.5 (a). Concentration contours obtained from FLUENT showing the penetration of paclitaxel to remnant tumor around the cavity domain. Shown in concentration contours are the 3D cavity surface and the 2D coronal surface of remnant tumor and nearby tissue domain. From the contour plots, the maximum intracellular paclitaxel concentration C i = x 1000 = M at the cavity domain. It can be drawn from the contour plots that paclitaxel can penetrate well through the remnant tumor as there is a significant amount of paclitaxel even outside the remnant tumor. The mean concentration in remnant tumor is about 5x10-5 mol/m 3 whereas the mean concentration in the nearby 8

9 tissues is about 3x10-7 mol/m 3. Hence smaller dosage of paclitaxel should be used so as to prevent the presence of the drug in the nearby tissues. The contour plots also show that the penetration of paclitaxel is very much affected by the convection as we can see from the distorted contour lines, the surfaces towards the ventricle provide rapid decrease of paclitaxel concentration and hence lower drug concentration at the remnant tumor located towards the ventricle, due to repressed convection, whereas those directed away from the ventricle show slow deterioration of paclitaxel concentration and hence higher drug concentration at the remnant tumor away from the ventricle, due to enhanced convection. Hence, effect of convection should also be taken into consideration during the designing of the drug and plantation of the drug into the brain tumor. REFERENCES 1. Lawrence K. Fund, Matthew G. Ewend, Allen Sills, Eric P. Sipos, Reid Thompson, Mark Watts, O. Michael Colvin, Henry Brem, and W. Mark Saltzman. Pharmacokinetics of Interstitial Delivery of Carmustine, 4-Hydroperoxycyclophosphamide, and Paclitaxel from a Biodegradable Polymer Implant in the Monkey Brain. Cancer Research 58: , February 15, Hyo-Jeong Kuh, Seong Hoon Jang, M.Guillaume Wientjes, Jessie L.-S. AU. Computational Model of Intracellular Pharmacokinetics of Paclitaxel. The Journal of Pharmacology and Experimental Therapeutics by the American Society for Pharmacology and Experimental Therapeutics. JPET 293: , 2000/2252/ C.Nicholson. Interaction between diffusion and Michaelis-Menten uptake of dopamine after iontophoresis in striatum, Biophysical Journal 68: (1995). 4. Lawrence K. Fung and W. Mark Saltzman.Polymeric implants for cancer chemotherapy. Advanced Drug Delivery Reviews.Volume 26, Issues 2-3, 7 July 1997, Pages W.M. Saltzman and M.L. Radomsky, Drugs released from polymers: diffusion and elimination in brain tissue. Chem. Eng. Sci. 46 (1991), pp Khan W. Li, Wenbin Dang, Betty M. Tyler,Greg Troiano, Tarik Tihan, Henry Brem, and Kevin A. Walter. Polilactofate Microspheres for Paclitaxel Delivery to Central Nervous System Malignancies. Clinical Cancer Research 3441.Vol. 9, , August 15, R.Byron Bird, Warren E. Stewart. Edwin N. Lightfoot. Transport Phenomena - second edition. John Wiley and Sons, Inc

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