# American Institute of Mathematical Sciences

May  2019, 24(5): 2237-2250. doi: 10.3934/dcdsb.2019093

## Antagonism and negative side-effects in combination therapy for cancer

 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: xiulanlai@ruc.edu.cn

Received  January 2018 Revised  January 2019 Published  March 2019

Fund Project: The first author is supported by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS 0931642.

Most clinical trials with combination therapy fail. One of the reasons is that not enough forethought is given to the interaction between the different agents, as well as the potential negative side-effects that may arise in the combined therapy. In the present paper we consider a generic cancer model with combination therapy consisting of chemotherapy agent $X$ and checkpoint inhibitor $A$. We use a mathematical model to investigate the results of injecting different amounts $\gamma_X$ of $X$ and $\gamma_A$ of $A$. We show that there are some regions in the $(\gamma_A,\gamma_X)$-plane where as increase in $\gamma_X$ or $\gamma_A$ actually decreases the tumor volume; such 'regions of antagonism' should be avoided in clinical trials. We also show how to achieve the same level of tumor volume reduction with least negative-side effects, where the side-effects are represented by the level of inflammation of the tumor microenvironment.

Citation: 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
##### References:

show all references

##### References:
Interaction of immune cells with cancer cells. Sharp arrows indicate proliferation/activation, blocked arrow indicates killing/blocking, inverted sharp arrow indicates recruitment/chemoattraction. $C$: cancer cells, $T$: effector T cells
and 3, for a mouse model">Figure 2.  Average densities/concentrations, in ${\rm g}/{\rm cm}^3$, of all the variables of the model in control case (no drugs). All parameter values are the same as in Tables 2 and 3, for a mouse model
and 3, for a mouse model">Figure 3.  Growth of tumor volume under treatment with $\gamma_X$ or $\gamma_A$, or combination ($\gamma_X,\gamma_A$). The chemotherapy or/and anti-PD-1 treatments. (a) $\gamma_X = 5\times 10^{-13}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$, $\gamma_A = 2\times 10^{-11}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$; (b) $\gamma_X = 3\times 10^{-13}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$, $\gamma_A = 5\times 10^{-11}$ ${\rm g}/{\rm cm}^3\cdot {\rm day}$. All other parameter values are the same as in Tables 2 and 3, for a mouse model
and 3">Figure 4.  Efficacy of combination therapy at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here (a) $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. (b) $\gamma_X = 0.9\times 10^{-12} - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 1\times 10^{-10} - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
and 3">Figure 5.  Average T cell density at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
and 3">Figure 6.  Average concentration of TNF-$\alpha$ at day 30 for different pair of $(\gamma_X, \gamma_A)$. Here $\gamma_X = 0 - 2\times 10^{-12}$ ${\rm g}/{\rm cm}^3$ and $\gamma_A = 0 - 1.8\times 10^{-10}$ ${\rm g}/{\rm cm}^3$. All other parameter values are the same as in Tables 2 and 3
List of variables (in units of g/${\rm cm}^3$)
 Notation Description Notation Description $D$ density of DCs $I_{12}$ IL-12 concentration $T$ density of effector T cells $M_P$ MCP-1 (CCL2) concentration $M_1$ density of proinflammatory macrophages M1 $I_{10}$ IL-10 concentration $M_2$ density of anti-proinflammatory macrophages M2 $T_\beta$ TGF- $\beta$ : concentration $C$ density of cancer cells $P$ PD-1 concentration $A$ concentration of anti-PD-1 $L$ PD-L1 concentration $X$ concentration of a chemotherapy agent $Q$ PD-1-PD-L1 concentration
 Notation Description Notation Description $D$ density of DCs $I_{12}$ IL-12 concentration $T$ density of effector T cells $M_P$ MCP-1 (CCL2) concentration $M_1$ density of proinflammatory macrophages M1 $I_{10}$ IL-10 concentration $M_2$ density of anti-proinflammatory macrophages M2 $T_\beta$ TGF- $\beta$ : concentration $C$ density of cancer cells $P$ PD-1 concentration $A$ concentration of anti-PD-1 $L$ PD-L1 concentration $X$ concentration of a chemotherapy agent $Q$ PD-1-PD-L1 concentration
Summary of parameter values
 Notation Description Value used References $\delta_D$ diffusion coefficient of DCs $8.64 \times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_T$ diffusion coefficient of T cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_M$ diffusion coefficient of macrophages $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_C$ diffusion coefficient of tumor cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_{I_{12}}$ diffusion coefficient of IL-12 $6.05\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_{\beta}}$ diffusion coefficient of TGF-$\beta$ $8.52\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{I_{10}}$ diffusion coefficient of IL-10 $9.11\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_\alpha}$ diffusion coefficient of TNF-$\alpha$ $8.46\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{M_P}$ diffusion coefficient of MCP-1 $1.12\times 10^{-1}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_A$ diffusion coefficient of anti-PD-1 $4.73\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\delta_X$ diffusion coefficient of $X$ $0.27$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\sigma_0$ flux rate of $T$ cells at the boundary 1 ${\rm cm}^{-1}$ [8] $\chi_M$ chemoattraction coefficient of MCP-1 $10$ ${\rm cm}^5/{\rm g}\cdot {\rm day}$ [13,14] $\lambda_{DC}$ activation rate of DCs by tumor cells $10$ ${\rm g}/{\rm cm}^3\cdot{\rm day}$ [16] $\lambda_{TI_{12}}$ activation rate of T cells by IL-12 $16.2$ ${\rm day}^{-1}$ estimated $\lambda_{M_1}$ activation rate of M1 macrophages $1.35$ ${\rm day}^{-1}$ [16] $\lambda_{M_2}$ activation rate of M2 macrophages $1.01$ ${\rm day}^{-1}$ [16] $\beta_{M_1}$ phenotype change rate of M1 to M2 macrophages $0.3$ ${\rm day}^{-1}$ estimated $\beta_{M_2}$ phenotype change rate of M2 to M1 macrophages $4.68\times 10^{-3}$ ${\rm day}^{-1}$ estimated $\lambda_{C}$ growth rate of cancer cells $1.92$ ${\rm day}^{-1}$ estimated $\lambda_{I_{12}D}$ production rate of IL-12 by DCs $1.38\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{12}M_1}$ production rate of IL-12 by M1 macrophages $5.52\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\beta C}$ production rate of TGF-$\beta$ by cancer cells $2.79\times 10^{-10}$ ${\rm day}^{-1}$ estimated $\lambda_{T_\beta M_2}$ production rate of TGF-$\beta$ by M2 macrophages $6.97 \times 10^{-9}$ ${\rm day}^{-1}$ estimated $\lambda_{I_{10} C}$ production rate of IL-10 by cancer cells $2.07\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{10} M_2}$ production rate of IL-10 by M2 macrophages $1.65\times 10^{-9}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha M_1}$ production rate of TNF $\alpha$ by M1 macrophages $1.36\times 10^{-5}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha T}$ production rate of TNF $\alpha$ by Th1 cells $9.06\times 10^{-8}$ ${\rm day}^{-1}$ estimated $\lambda_{M_PM_2}$ production rate of MCP-1 by M2 macrophages $1.2\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{M_PC}$ production rate of MCP-1 by cancer cells $8.24\times 10^{-7}$ ${\rm day}^{-1}$ [16] $d_{D}$ death rate of DCs 0.1 ${\rm day}^{-1}$ [8] $d_{T}$ death rate of T cells $0.18$ ${\rm day}^{-1}$ [8] $d_{M_1}$ death rate of M1 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{M_2}$ death rate of M2 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{C}$ death rate of tumor cells $0.17$ ${\rm day}^{-1}$ [8] $d_{I_{12}}$ degradation rate of IL-12 $1.38$ ${\rm day}^{-1}$ [8] $d_{T_\beta}$ degradation rate of TGF-$\beta$ $499.066$ ${\rm day}^{-1}$ [16] $d_{I_{10}}$ degradation rate of IL-10 $8.3178$ ${\rm day}^{-1}$ [16] $d_{T_\alpha}$ degradation rate of TGF-$\alpha$ $55.01$ ${\rm day}^{-1}$ [16] $d_{M_P}$ degradation rate of MCP-1 $55.01$ ${\rm day}^{-1}$ [10] $d_{A}$ degradation rate of anti-PD-1 $0.047$ ${\rm day}^{-1}$ [16] $d_X$ degradation rate of docetexel $1.11$ ${\rm day}^{-1}$ estimated
 Notation Description Value used References $\delta_D$ diffusion coefficient of DCs $8.64 \times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_T$ diffusion coefficient of T cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_M$ diffusion coefficient of macrophages $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_C$ diffusion coefficient of tumor cells $8.64\times 10^{-7}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [8] $\delta_{I_{12}}$ diffusion coefficient of IL-12 $6.05\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_{\beta}}$ diffusion coefficient of TGF-$\beta$ $8.52\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{I_{10}}$ diffusion coefficient of IL-10 $9.11\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{T_\alpha}$ diffusion coefficient of TNF-$\alpha$ $8.46\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_{M_P}$ diffusion coefficient of MCP-1 $1.12\times 10^{-1}$ ${\rm cm}^2$ ${\rm day}^{-1}$ [16] $\delta_A$ diffusion coefficient of anti-PD-1 $4.73\times 10^{-2}$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\delta_X$ diffusion coefficient of $X$ $0.27$ ${\rm cm}^2$ ${\rm day}^{-1}$ estimated $\sigma_0$ flux rate of $T$ cells at the boundary 1 ${\rm cm}^{-1}$ [8] $\chi_M$ chemoattraction coefficient of MCP-1 $10$ ${\rm cm}^5/{\rm g}\cdot {\rm day}$ [13,14] $\lambda_{DC}$ activation rate of DCs by tumor cells $10$ ${\rm g}/{\rm cm}^3\cdot{\rm day}$ [16] $\lambda_{TI_{12}}$ activation rate of T cells by IL-12 $16.2$ ${\rm day}^{-1}$ estimated $\lambda_{M_1}$ activation rate of M1 macrophages $1.35$ ${\rm day}^{-1}$ [16] $\lambda_{M_2}$ activation rate of M2 macrophages $1.01$ ${\rm day}^{-1}$ [16] $\beta_{M_1}$ phenotype change rate of M1 to M2 macrophages $0.3$ ${\rm day}^{-1}$ estimated $\beta_{M_2}$ phenotype change rate of M2 to M1 macrophages $4.68\times 10^{-3}$ ${\rm day}^{-1}$ estimated $\lambda_{C}$ growth rate of cancer cells $1.92$ ${\rm day}^{-1}$ estimated $\lambda_{I_{12}D}$ production rate of IL-12 by DCs $1.38\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{12}M_1}$ production rate of IL-12 by M1 macrophages $5.52\times 10^{-6}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\beta C}$ production rate of TGF-$\beta$ by cancer cells $2.79\times 10^{-10}$ ${\rm day}^{-1}$ estimated $\lambda_{T_\beta M_2}$ production rate of TGF-$\beta$ by M2 macrophages $6.97 \times 10^{-9}$ ${\rm day}^{-1}$ estimated $\lambda_{I_{10} C}$ production rate of IL-10 by cancer cells $2.07\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{I_{10} M_2}$ production rate of IL-10 by M2 macrophages $1.65\times 10^{-9}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha M_1}$ production rate of TNF $\alpha$ by M1 macrophages $1.36\times 10^{-5}$ ${\rm day}^{-1}$ [16] $\lambda_{T_\alpha T}$ production rate of TNF $\alpha$ by Th1 cells $9.06\times 10^{-8}$ ${\rm day}^{-1}$ estimated $\lambda_{M_PM_2}$ production rate of MCP-1 by M2 macrophages $1.2\times 10^{-8}$ ${\rm day}^{-1}$ [16] $\lambda_{M_PC}$ production rate of MCP-1 by cancer cells $8.24\times 10^{-7}$ ${\rm day}^{-1}$ [16] $d_{D}$ death rate of DCs 0.1 ${\rm day}^{-1}$ [8] $d_{T}$ death rate of T cells $0.18$ ${\rm day}^{-1}$ [8] $d_{M_1}$ death rate of M1 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{M_2}$ death rate of M2 macrophage $0.015$ ${\rm day}^{-1}$ [9] $d_{C}$ death rate of tumor cells $0.17$ ${\rm day}^{-1}$ [8] $d_{I_{12}}$ degradation rate of IL-12 $1.38$ ${\rm day}^{-1}$ [8] $d_{T_\beta}$ degradation rate of TGF-$\beta$ $499.066$ ${\rm day}^{-1}$ [16] $d_{I_{10}}$ degradation rate of IL-10 $8.3178$ ${\rm day}^{-1}$ [16] $d_{T_\alpha}$ degradation rate of TGF-$\alpha$ $55.01$ ${\rm day}^{-1}$ [16] $d_{M_P}$ degradation rate of MCP-1 $55.01$ ${\rm day}^{-1}$ [10] $d_{A}$ degradation rate of anti-PD-1 $0.047$ ${\rm day}^{-1}$ [16] $d_X$ degradation rate of docetexel $1.11$ ${\rm day}^{-1}$ estimated
Summary of parameter values
 Notation Description Value used References $K_D$ half-saturation of dendritic cells $4\times 10^{-4}$ g/${\rm cm}^3$ [16] $K_{T}$ half-saturation of T cells $3\times 10^{-3}$ g/${\rm cm}^3$ [16] $K_{M_1}$ half-saturation of M1 macrophages $10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $K_{M_2}$ half-saturation of M2 macrophages $3.2\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $K_{C}$ half-saturation of tumor cells $0.4$ g/${\rm cm}^3$ [8] $K_{I_{12}}$ half-saturation of IL-12 $8\times 10^{-10}$ g/${\rm cm}^3$ [16] $K_{T_\beta}$ half-saturation of TGF-$\beta$ $2.68\times 10^{-13}$ ${\rm g}/{\rm cm}^3$ [16] $K_{I_{10}}$ half-saturation of IL-10 $8.75\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{T_\alpha}$ half-saturation of TNF-$\alpha$ $3\times 10^{-11}$ g/${\rm cm}^3$ [11] $K_{M_P}$ half-saturation of MCP-1 $2\times 10^{-7}$ ${\rm g}/{\rm cm}^3$ [10] $K_{X}$ half-saturation of $X$ $8.02\times 10^{-11}$ ${\rm g}/{\rm cm}^3$ estimated $K_{TI_{10}}$ inhibition of function of T cells by IL-10 $4.375\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{TQ}$ inhibition of function of T cells by PD-1-PD-L1 $4.86\times 10^{-20}$ ${\rm g}^2/{\rm cm}^6$ estimated $K_{CX}$ inhibition of proliferation of cancer cells by docetexel $8.02\times 10^{-10}$ ${\rm g}/{\rm cm}^3$ estimated $D_0$ density of inactive DCs $2\times 10^{-5}$ g/${\rm cm}^3$ [8] $T_{0}$ density of naive T cells in tumor $6\times 10^{-4}$ g/${\rm cm}^3$ estimated $M_{10}$ density of monocytes $1.2\times 10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $M_{20}$ density of monocytes $3.84\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $C_M$ carrying capacity of cancer cells $0.8$ g/${\rm cm}^3$ [8] $\hat T$ density of T cells from lymph node $6\times 10^{-3}$ g/${\rm cm}^3$ [16] $\eta$ killing rate of tumor cells by T cells $210$ ${\rm cm}^3/{\rm g}\cdot {\rm day}$ [16] $\mu_{XC}$ absorbtion rate of $X$ by cancer cells estimated $\rho_P$ expression of PD-1 in T cells $2.49\times 10^{-7}$ [15] $\rho_L$ expression of PD-L1 in T cells $3.25\times 10^{-7}$ [15]
 Notation Description Value used References $K_D$ half-saturation of dendritic cells $4\times 10^{-4}$ g/${\rm cm}^3$ [16] $K_{T}$ half-saturation of T cells $3\times 10^{-3}$ g/${\rm cm}^3$ [16] $K_{M_1}$ half-saturation of M1 macrophages $10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $K_{M_2}$ half-saturation of M2 macrophages $3.2\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $K_{C}$ half-saturation of tumor cells $0.4$ g/${\rm cm}^3$ [8] $K_{I_{12}}$ half-saturation of IL-12 $8\times 10^{-10}$ g/${\rm cm}^3$ [16] $K_{T_\beta}$ half-saturation of TGF-$\beta$ $2.68\times 10^{-13}$ ${\rm g}/{\rm cm}^3$ [16] $K_{I_{10}}$ half-saturation of IL-10 $8.75\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{T_\alpha}$ half-saturation of TNF-$\alpha$ $3\times 10^{-11}$ g/${\rm cm}^3$ [11] $K_{M_P}$ half-saturation of MCP-1 $2\times 10^{-7}$ ${\rm g}/{\rm cm}^3$ [10] $K_{X}$ half-saturation of $X$ $8.02\times 10^{-11}$ ${\rm g}/{\rm cm}^3$ estimated $K_{TI_{10}}$ inhibition of function of T cells by IL-10 $4.375\times 10^{-11}$ g/${\rm cm}^3$ [16] $K_{TQ}$ inhibition of function of T cells by PD-1-PD-L1 $4.86\times 10^{-20}$ ${\rm g}^2/{\rm cm}^6$ estimated $K_{CX}$ inhibition of proliferation of cancer cells by docetexel $8.02\times 10^{-10}$ ${\rm g}/{\rm cm}^3$ estimated $D_0$ density of inactive DCs $2\times 10^{-5}$ g/${\rm cm}^3$ [8] $T_{0}$ density of naive T cells in tumor $6\times 10^{-4}$ g/${\rm cm}^3$ estimated $M_{10}$ density of monocytes $1.2\times 10^{-4}$ ${\rm g}/{\rm cm}^3$ [16] $M_{20}$ density of monocytes $3.84\times 10^{-3}$ ${\rm g}/{\rm cm}^3$ [16] $C_M$ carrying capacity of cancer cells $0.8$ g/${\rm cm}^3$ [8] $\hat T$ density of T cells from lymph node $6\times 10^{-3}$ g/${\rm cm}^3$ [16] $\eta$ killing rate of tumor cells by T cells $210$ ${\rm cm}^3/{\rm g}\cdot {\rm day}$ [16] $\mu_{XC}$ absorbtion rate of $X$ by cancer cells estimated $\rho_P$ expression of PD-1 in T cells $2.49\times 10^{-7}$ [15] $\rho_L$ expression of PD-L1 in T cells $3.25\times 10^{-7}$ [15]
 [1] Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307 [2] Shengqiang Liu, Lin Wang. Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Mathematical Biosciences & Engineering, 2010, 7 (3) : 675-685. doi: 10.3934/mbe.2010.7.675 [3] Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021140 [4] Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871 [5] Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561 [6] Tinevimbo Shiri, Winston Garira, Senelani D. Musekwa. A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters. Mathematical Biosciences & Engineering, 2005, 2 (4) : 811-832. doi: 10.3934/mbe.2005.2.811 [7] Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233 [8] Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020343 [9] Luis A. Fernández, Cecilia Pola. Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1563-1588. doi: 10.3934/dcdsb.2014.19.1563 [10] 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 [11] Reihaneh Mostolizadeh, Zahra Afsharnezhad, Anna Marciniak-Czochra. Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004 [12] Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229 [13] Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075 [14] Xian Zhang, Vinesh Nishawala, Martin Ostoja-Starzewski. Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects. Evolution Equations & Control Theory, 2019, 8 (1) : 231-246. doi: 10.3934/eect.2019013 [15] Gesham Magombedze, Winston Garira, Eddie Mwenje. Modelling the immunopathogenesis of HIV-1 infection and the effect of multidrug therapy: The role of fusion inhibitors in HAART. Mathematical Biosciences & Engineering, 2008, 5 (3) : 485-504. doi: 10.3934/mbe.2008.5.485 [16] Nikolay Pertsev, Konstantin Loginov, Gennady Bocharov. Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2365-2384. doi: 10.3934/dcdss.2020141 [17] Sarra Delladji, Mohammed Belloufi, Badreddine Sellami. Behavior of the combination of PRP and HZ methods for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 377-389. doi: 10.3934/naco.2020032 [18] Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185 [19] Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065 [20] Valeria Artale, Cristina L. R. Milazzo, Calogero Orlando, Angela Ricciardello. Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2067-2091. doi: 10.3934/jimo.2017032

2019 Impact Factor: 1.27