# 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
##### References:

show all references

##### 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 $0-30$
 [1] Elzbieta Ratajczyk, Urszula Ledzewicz, Maciej Leszczyński, Heinz Schättler. Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 425-441. doi: 10.3934/dcdsb.2018029 [2] 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 [3] 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 [4] Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019233 [5] 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 [6] 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 [7] 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 [8] Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058 [9] Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469 [10] Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial & Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453 [11] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [12] Daniel Vasiliu, Jianjun Paul Tian. Periodic solutions of a model for tumor virotherapy. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1587-1597. doi: 10.3934/dcdss.2011.4.1587 [13] Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004 [14] Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701 [15] Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249 [16] 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 [17] Mika Yoshida, Kinji Fuchikami, Tatsuya Uezu. Realization of immune response features by dynamical system models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 531-552. doi: 10.3934/mbe.2007.4.531 [18] Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091 [19] Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507 [20] Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415

2018 Impact Factor: 1.313