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2016 A 5D tumour-immune-virus system

Christophe LETELLIER
12/04/2017
GIF - 21.3 ko
Raluca Eftimie

There are some evidences that oncolytic viruses can be effective in treating cancer. in order to better understand the interactions between tumour cells, oncolytic viruses and immune cells that could lead to tumor control or tumor escape, mathematical modelling of cancer oncolytic therapies appears as one possible ways for has been used to investigate the biological mechanisms behind the observed patterns of tumour growth. The mathematical model is here presented to investigate the chaotic tumour-immune-virus dynamics. The describe the response to a vaccine virus and an oncolytic virus, which carry the same tumour antigen and are administered consecutively into tumour-bearing mice. The vaccine induces an immune response against the tumour antigens. The oncolytic virus, Vesicular Stomatitis Virus, which is , induces a much stronger anti-tumour immune response that could theoretically eliminate tumours. However, the Vesicular Stomatitis Virus does not always eliminate tumours [1]. The model - proposed by Raluca Eftimie and her coworkers - describes a memory response induced by the injection of the oncolytic virus following the vaccine virus [2]. It is the set of five ordinary differential equations


\left\{
  \begin{array}{l}
     \displaystyle
     \dot{x}_{\rm u} =R x_{\rm u} \left( \displaystyle - \frac{x_{\rm u} + x_{\rm i}}{\kappa-C (y_{\rm m} +y_{\rm e})}  \right)
         - \frac{d_{\rm v} x_{\rm u} v}{h_{\rm u} + x_{\rm u}} 
         - \frac{d_{\rm x} x_{\rm u} y_{\rm e}}{h_{\rm e}+y_{\rm e}} \\[0.3cm]
      \displaystyle
      \dot{x}_{\rm i} =d_{\rm v} x_{\rm u} v}{h_{\rm u}+ x_{\rm u}} - \delta x_{\rm i}
     - d_{\rm x} \frac{x_{\rm i} y_{\rm e}}{h_{\rm e} +y_{\rm e}} \\[0.3cm]
     \displaystyle
      \dot{y}_{\rm m} =p_{\rm m} \left( \displaystyle 1 - \frac{y_{\rm m}}{M} \right) 
      \frac{v y_{\rm m}}{h_{\rm v}+v} \\[0.3cm]
      \displaystyle
      \dot{y}_{\rm e} =p_{\rm e} \frac{v y_{\rm m}}{h_{\rm v} + v} - d_{\rm e} y_{\m e} 
      - d_{\rm t} x_{\rm u} y_{\rm e} \\[0.3cm]
      \dot{v} = \delta b x_{\rm i} - \omega v \\
  \end{array}
\right.

where xu and xi are the densities of infected and uninfected tumour cells, ym and ye the densities of effector immune (that kill tumour cells) and memory cells (that proliferate in the presence of viral or tumour antigens), and v is the density of oncolytic virus particles.

With appropriate parameter values, this system produces a chaotic attractor (Fig. 1) characterized by a unimodal smooth first-return map to the Poincarť section


 {\cal P} \equiv \{ (x_{{\rm i},n}, x_{{\rm u},n}, y_{{\rm m},n}, v_n) ~|~
 y_{{\rm e},n} =1100, \dot{y}_{{\rm e},n} >0 \}

This attractor is thus observed after a period-doubling cascade as a route to chaos.

JPG - 19.5 ko
Fig. 1. Chaotic attractor produced by the tumor-immune-virus system.

Parameter values used for producing the attractor shown in Fig. 1 are


\left\{
  \begin{array}{ll}
  R=0.927  & \mbox{days}^{-1} \\
  \kappa= \frac{1}{5.5 \cdot 10^{-9}} & \mbox{cells.vol}^{-1} \\
  d_\mbox{v} =0.0026 & \\
  d_\mbox{x} =2.6 & \mbox{days}^{-1} \\
   h_\mbox{u}=1  & \mbox{cells.vol}^{-1} \\
      h_{\Ám e} =2 \cdot 10^{+3} & \mbox{cells.vol}^{-1} \\
      h_\mbox{v}= 10^4 &  \mbox{PFU.vol}^{-1} \\
      \delta=1 & \mbox{days}^{-1} \\
      p_\mbox{m}=2.5 & \mbox{days}^{-1} \\
      M=10^{4} & \mbox{cells.vol}^{-1} \\
      p_\mbox{e} =0.4 & \mbox{days}^{-1} \\
      d_\mbox{e}=0.1 & \mbox{days}^{-1} \\
      d_\mbox{t} =5 \cdot 10^{-9} & \mbox{days}^{-1} \\
      \omega=1.257    & \\ 
      b=1000 & \mbox{PFU/vol.cell/vol} \\
      C=0

\end{array}
\right.

[1] B. W. Bridle, K. B. Stephenson, J. . Boudreau, S. Koshy, N. Kazdhan, E. Pullenayegum, J. BrunelliŤre, J. L. Bramson, B. D. Lichty, Y. Wan, Potentiating cancer immunotherapy using an oncolytic virus, Molecular Therapy, 18 (8), 1430-1439, 2010.

[2] R. Eftimie, C. K. MacNamara, J. Dusho, J. D. Bramson & D. J. D. Earn, Bifurcations and chaotic dynamics in a tumour-immune-virus system, Mathematical Models in Natural Phenomena, 11 (5), 65-85, 2016.

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