# 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
##### References:

show all references

##### References:
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^*
 [1] Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093 [2] Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129 [3] Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253 [4] Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099 [5] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625 [6] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [7] Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019207 [8] Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989 [9] Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 [10] Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565 [11] Jianjun Paul Tian. The replicability of oncolytic virus: Defining conditions in tumor virotherapy. Mathematical Biosciences & Engineering, 2011, 8 (3) : 841-860. doi: 10.3934/mbe.2011.8.841 [12] Joanna R. Wares, Joseph J. Crivelli, Chae-Ok Yun, Il-Kyu Choi, Jana L. Gevertz, Peter S. Kim. Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1237-1256. doi: 10.3934/mbe.2015.12.1237 [13] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [14] Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 [15] Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041 [16] Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431 [17] Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689 [18] Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843 [19] Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 [20] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

2018 Impact Factor: 1.143

## Metrics

• HTML views (247)
• Cited by (0)

• on AIMS