Mathematical modeling and optimization of intratumor drug transport
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MATHEMATICAL MODELING AND OPTIMIZATION OF INTRATUMOR DRUG TRANSPORT D. A. Klyushin,a N. I. Lyashko,b and Yu. N. Onopchukb
UDC 519.8.612.007
A pseudohyperbolic problem of optimal control of intratumoral drug distribution is formulated. It takes into account the heterogeneity of tumor tissues and effects of convection diffusion in a fissured porous medium. A mathematical model constructed and the corresponding optimal control problem are shown to be correct. Keywords: mathematical model, pharmacological drug distribution, tumor, optimal control, pseudohyperbolic problem. TRADITIONAL MODELS In medical practice of leading oncological clinics, methods of preoperative chemotherapy of tumors are increasingly used and make it possible to decrease tumor sizes and to facilitate the subsequent treatment. These methods are divided into intravenous intraarterial and intratumoral ones. The efficiency of any selected treatment method depends on the chemical drug resistance. The problem is complicated by the fact that many drugs lose their efficiency during their transport within the circulatory system. As is shown by investigations of R. Jain [1], S. Jang, et al. [2], a high density of tumor cells that complicates blood flow, a high pressure of the intersticial liquid within tumors, and also a permeability increase in their microcapillaries form three physiological barriers that prevent the delivery of a drug to its target. In this connection, high hopes are laid on preoperative intratumoral chemotherapy. Mathematical models of pharmacodynamics, transport, and distribution of intratumoral drugs within tumors that arise in this case were investigated by L. Baxter and R. Jain in [3–5], J. Lankelma et al. in [6], and E. Goldberg and et al. in [7]. Sources of the drugs intended for intratumoral distribution are microspheres filled with polymeric gels. The questions connected with mathematical modeling of transport of drug substances diffusing through the microspheres implanted into a tumor are considered by J. Ward and J. King in [8]. In the latter work, they investigated the transport of substances whose diffusive properties are similar to those of glucose. Taking into account that concrete drugs (paclitaxel and doxorubicine) are used in clinical practice, the problems connected with their diffusion within a tumor are of special interest. Mathematical models of transport of paclitaxel and doxorubicine in tumor tissues of a mammary gland are described by A. Tzafriri et al. in [9] and W. Kaowumpai et al. in [10], respectively. The models proposed in the mentioned works are initial-boundary diffusion-convection problems taking into account pharmacokinetic properties of drugs in microscopic scale. We dwell on the mathematical model proposed by A. Tzafriri et al. in [9] and extend it to include the assumption that the coefficient of diffusion and speed of convective transport depends on spatial variables. In this model, drug transport is considered in two regions, namely, in the intersticial W 1 and endocellular W 2 ones. The distribut
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