American Institute of Mathematical Sciences

January  2017, 14(1): 305-319. doi: 10.3934/mbe.2017020

The role of TNF-α inhibitor in glioma virotherapy: A mathematical model

 1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 2 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Illinois, 62026-1653, USA 3 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

Received  June 22, 2016 Accepted  June 30, 2016 Published  October 2016

Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

Citation: Elzbieta Ratajczyk, Urszula Ledzewicz, Maciej Leszczynski, Avner Friedman. The role of TNF-α inhibitor in glioma virotherapy: A mathematical model. Mathematical Biosciences & Engineering, 2017, 14 (1) : 305-319. doi: 10.3934/mbe.2017020
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References:
Flow chart of the model for virotherapy
Graphs of model variables for $C=0,\; D=0.$
Graphs of model variables for $C=5\cdot10^{-7},\; D=0.$
Graph of $R(50)$ for $b\in[70,110]$ and different values od C
Graph of $R(50)$ for $b\in [110,150]$ and different values od C
Graphs of model variables for $C=5\cdot10^{-7},\; D=15.$
Graph of R(50) for $b \in [70,90]$ with different D and fixed $C=5\cdot10^{-7}$
Graph of R(50) for $b \in [110,130]$ with different D and fixed $C=3\cdot10^{-7}$
Efficacy map for b = 90
Efficacy map for b = 150
Parameters of the model
 Parameter Description Num. values Dimension $\alpha$ Proliferation rate of uninfected tumor cells $0.2$ 1/day $\beta$ Infection rate of tumor cells by viruses $2\cdot10^{4}$ $\frac{cm^{3}}{g\cdot day}$ $\rho$ Rate of loss of viruses during infection $4\cdot10^{-2}$ $\frac{cm^{3}}{g\cdot day}$ $k$ Effectiveness of the inhibitory action of TNF-$\alpha$ $0.4$ 1/day $\delta_{y}$ Infected tumor cell death rate $0.2$ 1/day $\lambda$ TNF-$\alpha$ production rate $2.86\cdot10^{-3}$ 1/day $\delta_{T}$ TNF -$\alpha$ cell degradation rate $55.45$ 1/day $\delta_{M}$ Macrophages death rate $0.015$ 1/day $b$ Burst size of infected cells during apoptosis $(50-150) \cdot10^{-6}$ $b_{1}$ Burst size of infected cells during necrosis $0,\,\ll b$ $K$ Carrying capacity of the TNF-$\alpha$ $5\cdot10^{-7}$ $\frac{g}{cm^{3}}$ $\kappa$ Degradation of TNF-$\alpha$ $4\cdot10^{-10}$ 1/day due to its action on infected cells $\delta_{v}$ Virus lysis rate $0.5$ 1/day $A$ Constant source of macrophages $9\cdot10^{-7}$ $\frac{g}{cm^{3}\cdot day}$ $s$ Stimulation rate of macrophages by infected cells $0.15$ $\frac{cm^{3}}{g\cdot day}$ without stimulus $\delta_{x}$ death rate of uninfected cancer cells $0.1$ 1/day $\mu$ removal rate of dead cells $0.25$ 1/day $\theta_{0}$ average of total densitiy of cells $0.9$ $g/cm^{3}$ $C$ constant infusion of the virus $(0-5) \cdot10^{-7}$ $\frac{g}{cm^{3}\cdot day}$ $D$ constant infusion of the TNF-$\alpha$ inhibitor $0-30$
 Parameter Description Num. values Dimension $\alpha$ Proliferation rate of uninfected tumor cells $0.2$ 1/day $\beta$ Infection rate of tumor cells by viruses $2\cdot10^{4}$ $\frac{cm^{3}}{g\cdot day}$ $\rho$ Rate of loss of viruses during infection $4\cdot10^{-2}$ $\frac{cm^{3}}{g\cdot day}$ $k$ Effectiveness of the inhibitory action of TNF-$\alpha$ $0.4$ 1/day $\delta_{y}$ Infected tumor cell death rate $0.2$ 1/day $\lambda$ TNF-$\alpha$ production rate $2.86\cdot10^{-3}$ 1/day $\delta_{T}$ TNF -$\alpha$ cell degradation rate $55.45$ 1/day $\delta_{M}$ Macrophages death rate $0.015$ 1/day $b$ Burst size of infected cells during apoptosis $(50-150) \cdot10^{-6}$ $b_{1}$ Burst size of infected cells during necrosis $0,\,\ll b$ $K$ Carrying capacity of the TNF-$\alpha$ $5\cdot10^{-7}$ $\frac{g}{cm^{3}}$ $\kappa$ Degradation of TNF-$\alpha$ $4\cdot10^{-10}$ 1/day due to its action on infected cells $\delta_{v}$ Virus lysis rate $0.5$ 1/day $A$ Constant source of macrophages $9\cdot10^{-7}$ $\frac{g}{cm^{3}\cdot day}$ $s$ Stimulation rate of macrophages by infected cells $0.15$ $\frac{cm^{3}}{g\cdot day}$ without stimulus $\delta_{x}$ death rate of uninfected cancer cells $0.1$ 1/day $\mu$ removal rate of dead cells $0.25$ 1/day $\theta_{0}$ average of total densitiy of cells $0.9$ $g/cm^{3}$ $C$ constant infusion of the virus $(0-5) \cdot10^{-7}$ $\frac{g}{cm^{3}\cdot day}$ $D$ constant infusion of the TNF-$\alpha$ inhibitor $0-30$
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