# American Institute of Mathematical Sciences

December  2019, 39(12): 6825-6842. doi: 10.3934/dcds.2019233

## Free boundary problems associated with cancer treatment by combination therapy

 1 Mathematical Bioscience Institute & Department of Mathematics, Ohio State University, Columbus, OH, USA 2 Institute for Mathematical Sciences, Renmin University of China, Beijing, China

* Corresponding author: Avner Friedman

Received  June 2018 Revised  October 2018 Published  June 2019

Fund Project: The first author is supported by NSF grant DMS 0931642

Many mathematical models of biological processes can be represented as free boundary problems for systems of PDEs. In the radially symmetric case, the free boundary is a function of $r = R(t)$, and one can generally prove the existence of global in-time solutions. However, the asymptotic behavior of the solution and, in particular, of $R(t)$, has not been explored except in very special cases. In the present paper we consider two such models which arise in cancer treatment by combination therapy with two drugs. We study the asymptotic behavior of the solution and its dependence on the dose levels of the two drugs.

Citation: Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233
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The profiles of functions $h(C)$ and $f(C)$ given by (25) and (27) respectively
Illustration of the situation where $R\to0$ or $R\to \infty$ in terms of $\gamma_A$ and $\gamma_V$. Dashed line represents $\lambda+\gamma_A\delta = \gamma_V$. Dotted line is defined by $\gamma_A = \frac{\lambda}{\sqrt{4\frac{\lambda}{K}(1+\delta)-\delta}} = \gamma_A^*$. The solid curve represents $C^* = C^{**}$ which is given by $\gamma_V^2+[2(1+\delta)-(\lambda+\gamma_A\delta)]\gamma_V+(1+\delta)\left[1+\delta+\frac{\lambda}{K}-(\lambda+\gamma_A\delta)\right] = 0$. The pairs $(C^*,C^{**})$ exists in the region bounded by the three curves. Here $K = 2$, $\lambda = 2$, $\delta = 1$
The shape of functions $f(C)$ and $h(C)$. (a) The case (40), $\gamma_A-\lambda<\mu-\gamma_V$. (b) The case (42), $\gamma_A-\lambda>\mu-\gamma_V$
Illustration of the situation where $R\to0$ or $R\to \infty$ in terms of $\gamma_A$ and $\gamma_V$. Dashed line represents $\gamma_A+\gamma_V = \lambda+\mu$. Dotted line denotes $\gamma_A\gamma_V = \frac{1}{4}(\lambda+\mu)^2$. Dash-doted line represents $\gamma_V = \mu$. Solid curve represents either $C^* = C^{**}$ or $C^*_+ = C^{**}$. The pairs $(C^*,C^{**})$ exists in the region below the dashed line, while the pairs $(C^*_+,C^{**})$ exists in the region bounded by the dashed line and doted curve. Here $\lambda = 0.5$, $\mu = 2$
The comparison between $C^{**}$ and $C^{*}$ (Fig. 3(a)), $C^*_\pm$ (Fig. 3(b)). Note that $C^*$ exists if (40) holds; $C^*_\pm$ exists if (42) holds; $C^{**}$ exists if $\gamma_V<\mu$
 $\gamma_A^2+4\gamma_A\gamma_V>(\lambda+\mu)^2$ $\gamma_A^2+4\gamma_A\gamma_V\le(\lambda+\mu)^2$ $\lambda> \mu$ $\lambda \le \mu$ $\lambda< \mu$ $\lambda \ge \mu$ $\lambda\mu>\gamma_A \gamma_V$ $C^*_-C^{**} $$C^*_+>C^{**} C^*>C^{**}$$ C^*_+>C^{**}$ $\lambda\mu<\gamma_A \gamma_V$ $C^*_->C^{**}$ $C^* $ \gamma_A^2+4\gamma_A\gamma_V>(\lambda+\mu)^2  \gamma_A^2+4\gamma_A\gamma_V\le(\lambda+\mu)^2  \lambda> \mu  \lambda \le \mu  \lambda< \mu  \lambda \ge \mu  \lambda\mu>\gamma_A \gamma_V  C^*_-C^{**} $$C^*_+>C^{**}   C^*>C^{**}$$ C^*_+>C^{**}  \lambda\mu<\gamma_A \gamma_V  C^*_->C^{**}  C^*
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