# American Institute of Mathematical Sciences

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, doi: 10.3934/dcds.2019233
##### References:
 [1] J.-F. Cao, W.-T. Li and M. Zhao, On a free boundary problem for a nonlocal reaction-diffusion model, Discrete & Continuous Dynamical Systems - B, 23 (2018), 4117-4139. [2] J. A. Carrillo and J. L. Vazquez, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Philos. Trans. A Math. Phys. Eng. Sci., 373 (2015), 20140275. [3] H. A. Chang-Lara, N. Guillen and R. W. Schwab, Non local branching Brownians with annihilation and free boundary problems, arXiv: 1807.02714, (2015). [4] X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Transactions of the American Mathematical Society, 357 (2005), 4771-4804. doi: 10.1090/S0002-9947-05-03784-0. [5] S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth, Transactions of the American Mathematical Society, 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4. [6] A. De Massi, P. A. Ferrari, E. Presutti and N. Soprano-Loto, Non local branching Brownians with annihilation and free boundary problems, arXiv: 1711.06390, (2017). [7] A. Friedman, Mathematical Biology Modeling and Analysis, Conference Board of the Mathematical Sciences Reginal Conference Series in Mathematics, 127, American Mathematical Society, Singapore, 2018. [8] A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM Journal on Mathematical Analysis, 42 (2010), 2013-2040. doi: 10.1137/090772630. [9] A. Friedmen, C.-Y. Kao and R. Leander, On the dynamics of radially symmetric granulomas, Journal of Mathematical Analysis and Applications, 412 (2014), 776-791. doi: 10.1016/j.jmaa.2013.11.017. [10] A. Friedman and X. Lai, Combination therapy for cancer with oncolytic virus and checkpoint inhibitor: A mathematical model, PLoS ONE, 13 (2018), e0192449. [11] A. Friedman and K.-Y. Lam, On the stability of steady states in a granuloma model, Journal of Differential Equations, 256 (2014), 3743-3769. doi: 10.1016/j.jde.2014.02.019. [12] X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitor: A mathematical model, PLoS ONE, 12 (2017), e0178479. doi: 10.3934/mbe.2017020. [13] J. Lee, A free boundary problem with non local interaction, Mathematical Physics, Analysis and Geometry, 21 (2018), 1-22. doi: 10.1007/s11040-018-9282-4. [14] J. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, Journal of Differential Equations, 260 (2016), 5875-5893. doi: 10.1016/j.jde.2015.12.023. [15] J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, Journal of Differential Equations, 262 (2017), 4907-4930. doi: 10.1016/j.jde.2017.01.012.

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##### References:
 [1] J.-F. Cao, W.-T. Li and M. Zhao, On a free boundary problem for a nonlocal reaction-diffusion model, Discrete & Continuous Dynamical Systems - B, 23 (2018), 4117-4139. [2] J. A. Carrillo and J. L. Vazquez, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Philos. Trans. A Math. Phys. Eng. Sci., 373 (2015), 20140275. [3] H. A. Chang-Lara, N. Guillen and R. W. Schwab, Non local branching Brownians with annihilation and free boundary problems, arXiv: 1807.02714, (2015). [4] X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Transactions of the American Mathematical Society, 357 (2005), 4771-4804. doi: 10.1090/S0002-9947-05-03784-0. [5] S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth, Transactions of the American Mathematical Society, 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4. [6] A. De Massi, P. A. Ferrari, E. Presutti and N. Soprano-Loto, Non local branching Brownians with annihilation and free boundary problems, arXiv: 1711.06390, (2017). [7] A. Friedman, Mathematical Biology Modeling and Analysis, Conference Board of the Mathematical Sciences Reginal Conference Series in Mathematics, 127, American Mathematical Society, Singapore, 2018. [8] A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM Journal on Mathematical Analysis, 42 (2010), 2013-2040. doi: 10.1137/090772630. [9] A. Friedmen, C.-Y. Kao and R. Leander, On the dynamics of radially symmetric granulomas, Journal of Mathematical Analysis and Applications, 412 (2014), 776-791. doi: 10.1016/j.jmaa.2013.11.017. [10] A. Friedman and X. Lai, Combination therapy for cancer with oncolytic virus and checkpoint inhibitor: A mathematical model, PLoS ONE, 13 (2018), e0192449. [11] A. Friedman and K.-Y. Lam, On the stability of steady states in a granuloma model, Journal of Differential Equations, 256 (2014), 3743-3769. doi: 10.1016/j.jde.2014.02.019. [12] X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitor: A mathematical model, PLoS ONE, 12 (2017), e0178479. doi: 10.3934/mbe.2017020. [13] J. Lee, A free boundary problem with non local interaction, Mathematical Physics, Analysis and Geometry, 21 (2018), 1-22. doi: 10.1007/s11040-018-9282-4. [14] J. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, Journal of Differential Equations, 260 (2016), 5875-5893. doi: 10.1016/j.jde.2015.12.023. [15] J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, Journal of Differential Equations, 262 (2017), 4907-4930. doi: 10.1016/j.jde.2017.01.012.
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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