Modeling Blood Flow in a Brain Tumor Treated Concurrently with Radiotherapy and Chemotherapy

Ranadhir Roy^1, and Daniel N Riahi¹

¹Mathematics Department, University of Texas-Pan American, Edinburg, Texas

Cite this paper:
Ranadhir Roy and Daniel N Riahi. Modeling Blood Flow in a Brain Tumor Treated Concurrently with Radiotherapy and Chemotherapy. Applied Mathematics and Physics. 2013; 1(3):67-77. doi: 10.12691/amp-1-3-4

Abstract

Blood flow through a tumor plays a critical role in tumor growth and cancer therapies. Hence, fluid dynamics is an appropriate method to study blood flow through a tumor. Drug transport in the tumor interstitial depends on convection and diffusion. To investigate characteristics of blood flow through a spherical tumor, a coupled convection-diffusion models for simulating the interactions between two anti cancer drugs has been developed in this paper. This model provides the computational transportation to evaluate systematically and quantitatively the effects of interactions on the concentration within the tumor. Mathematical expressions for the spatial variations of the interstitial velocity and interstitial pressure are developed and calculated analytically, while variations of drug concentrations within a tumor are determined numerically in this paper. We determined the way interstitial pressure and velocity vary in the radial direction, which agreed with the experiments, as well as we studied the way one drug concentration changes in the presence or absence of a second drug within the tumor. We found that the concentration of a drug in the tumor could be improved in the presence of another drug in the tumor.

Keywords:
brain tumor cancer therapy convection and diffusion blood flow drug concentration fluid dynamics

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References:

[1]	G. Powathi, M. Kohandel, S. Sivaloganathan, A. Oza and M. Milosevic, Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy, Phys. Med. Biol. 52 3291-3306 (2007).
[2]	L. Barazzuol, NG. Burnet, R Jena, B. Jones, SJ. Jefferies, NF. Kirkby, A mathematical model of brain tumor response to radiotherapy and chemotherapy considering radiobiological aspects, Journal of Theoretical Biology, 262, 553-565, (2010).
[3]	R. K. Jain, Barriers to drug delivery in solid tumors, Sci Am 271, 58-65, (1994).
[4]	R. K. Jain, Normalization of tumor vasculature: An emerging concept in antiangiogenic therapy. Science 307, 58-62, (2005).
[5]	L. T. Baxter, R. K. Jain, Transport of fluid and macromolecules in tumors. (I) role of interstitial pressure and convection. Microvasc. Res. 37, 77-104, (1989).
[6]	L. T. Baxter, T., R. K. Jain, Transport of fluid and macromolecules in tumors. (II) role of heterogeneous perfusion and lymphatics. Microvasc. Res. 40, 246-263, (1990).
[7]	L. T. Baxter, R. K. Jain, Transport of fluid and macromolecules in tumors. (III) role of binding and metabolism. Microvasc. Res. 41, 5-23, (1991).
[8]	C. H. Wang, J. Li, Three dimensional iggs delivery to tumors. Chemical Eng. Science 53, 3579-3600, (1998).
[9]	J. Zhao, H. Salmon, M. Sarntinoranont, Effect of heterogeneous vasculature on interstitial transport within a solid tumor. Macrovasc. Res. 73, 224-236, (2007).
[10]	M. Soltani, P. Chen, Numerical modeling of fluid flow in solid tumors. Plos One 6, issue 6, e20344, (2011).
[11]	W. M. Saltzman, M. L. Radomsky, Drugs released from polymers: diffusion and elimination in brain tissue. Chemical Eng. Science 46, 2429-2444, (1991).
[12]	W. H. K. Tan, F. J. Wang, T. Lee, C. H. Wang, Computer simulation of the delivery of etanidazole to brain tumor from PLGA wafers: Comparison between linear and double burst release systems. Biotechnology and Bioengineering 82, 278-288, (2003).
[13]	C. S. Teo, W. Tan, H. K., Lee, T., Wang, C. H, Transient interstitial fluid flow in brain tumors: Effects on drug delivery. Chemical Eng. Science 60, 4803-4821, (2005).
[14]	D. Y, Arifin, K. Y, Lee, C. H, Wang, Chemotherapeutic drug transport to brain tumor. J. Control Release, 137, 203-210, (2009).
[15]	J. Sinek, H. Frieboes, X. Zheng, V. Cristini, Two-dimensional chemotherapy simulations demonstrates fundamental transport and tumor response limitations involving nanoparticles. Biomed. Microdevices. 6(4), 297-309 (2004).
[16]	X. Zheng, SM, Wise, V. Cristini, Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Biol. 67, 211–259, (2005).
[17]	K. R. Swanson, , E. C. Alvord, J. D, Murray, Virtual brain tumors (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy, British J. Cancer 86, 14-18, (2002).
[18]	S. E. Eikenberry, T. Sankar, MC. Preul; EJ; Kostelich, CJ Thalhauser, Y. Kuang,, Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model, Cell Proliferation, 42, 511-528, (2009).
[19]	D. N Riahi and R. Roy, Mathematical modeling of fluid flow in brain tumor, under review, (2012).
[20]	R. K. Jain, L. T. Baxter, Mechanisms of heterogeneous of monoclonal anti-bodies and other macromolecules in tumors: significance of elevated interstitial pressure. Cancer Res. 48, 7022-7032, (1988).
[21]	R. K. Jain, Transport of molecules in the tumor interstitium: A review. Cancer Res. 47, 3039-3051, (1987).
[22]	J. W, Baish, P. A. Netti, R. K. Jain, Transmural coupling of fluid flow in microcirculatory network and interstitium in tumors , Microvascular Research, 53, 128-141, (1997).
[23]	Y. Boucher, Baxter, L. T, Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: Implication for therapy. Cancer Res. 50, 4478-4484, (1990).
[24]	R. K. Jain, R. T. Tong, L. L, Munn, Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis. Cancer Res. 67, 2729-2735, (2007).
[25]	L. D. Landau, E. M, Lifshitz, Fluid Mechanics, 2nd edition. (1987), Pergamon Press, Oxford.