Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system

Elzbieta Ratajczyk; Urszula Ledzewicz; Maciej Leszczyński; Heinz Schättler

doi:10.3934/dcdsb.2018029

Article Contents

2018, Volume 23, Issue 1: 425-441. Doi: 10.3934/dcdsb.2018029

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Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system

1.
Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland
2.
Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA
3.
Dept. of Electrical and Systems Engr., Washington University, St. Louis, Mo, 63130, USA

^* Corresponding author
^* Corresponding author

Received: July 31, 2016

Published: November 2017

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Abstract

Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-α inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.

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Mathematics Subject Classification: Primary: 92C50, 70K50; Secondary: 93C15.

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Figure 1. Illustration of Proposition 3 for $c=0$

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Figure 2. 'Graph' of polynomial $P$ if $w^0 < 1$ (top) and in the two subcases that arise for $w^0>1$ (bottom)

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Figure 3. Illustration of Propositions 3 and 4

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Figure 4. Transcritical bifurcation

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Figure 5. Values of the tumor free and positive equilibrium points for $B=90$ (left), $B=125$ (middle) and $B=140$ (right) plotted as function of $C$. In each case there is a saddle-node bifurcation which generates an upper ($z_u$) and lower branch $(z_\ell)$ of positive solutions. The upper branch only exists for the short range and terminates as $\bar{y}=0.1=\frac{\delta_M}{s}$ which causes $\bar{M}\rightarrow \infty$. The lower branch exists until $C=2.5$ and terminates as $\bar{y}$ becomes zero

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Figure 6. Periodic orbit for $C=0.3$ and $B=140$

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Table 1. States and parameters of the model

States	Description	Dimension	Num. Value(s)
x	density of uninfected cancer stem cells	$\frac{g}{{c{m^3}}}$
y	density of infected cancer stem cells	$\frac{g}{{c{m^3}}}$
M	density of macrophages	$\frac{g}{{c{m^3}}}$
T	density of TNF-α	$\frac{g}{{c{m^3}}}$
v	density of virus	$\frac{g}{{c{m^3}}}$
z	z = (x; y; M; T; v^#8224;
Parameter	Description	Dimension	Num. Value(s)
α	proliferation rate of uninfected tumor cells	1/day	0.2
β	infection rate of tumor cells by viruses	$\frac{{c{m^3}}}{{g \cdot day}}$	2·10⁴
ρ	rate of loss of viruses during infection	$ρ$	0.04
k	effectiveness of the inhibitory action of TNF-α	1/day	0.4
δ_y	infected tumor cell death rate	1/day	0.2
λ	TNF-α production rate	1/day	2.86·10^-3
δ_T	TNF-α cell degradation rate	1/day	55.45
δ_M	macrophages death rate	1/day	0.015
b	burst size of infected cells during apoptosis	×10^-6	50 - 150
K_T	carrying capacity of TNF-α	$\frac{g}{{c{m^3}}}$	5·10^-7
κ	degradation of TNF-α due to its action on infected cells	1/day	4·10^-10
δ_v	virus lysis rate	1/day	0.5
A	constant source of macrophages	$\frac{g}{{c{m^3} \cdot day}}$	0.9·10^-6
s	stimulation rate of macrophages by infected cells	$\frac{{c{m^3}}}{{g \cdot day}}$	0.15
δ_x	death rate of uninfected cancer cells	1/day	0.1
c	constant infusion of the virus	$\frac{{g \times {{10}^{-6}}}}{{c{m^3} \cdot day}}$	0 - 150
d	constant infusion of the TNF-α inhibitor

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