| Citation: |
Mohamed Ch-Chaoui, Karima Mokni. ASYMPTOTIC ANALYSIS OF AN INTEGRO-DIFFERENTIAL SYSTEM MODELING THE BLOW UP OF CANCER CELLS UNDER THE IMMUNE RESPONSE[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1763-1785. doi: 10.11948/20210296
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ASYMPTOTIC ANALYSIS OF AN INTEGRO-DIFFERENTIAL SYSTEM MODELING THE BLOW UP OF CANCER CELLS UNDER THE IMMUNE RESPONSE
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Abstract
In this paper, we derive and analyze a phenomenological model at the cellular level of the immune response to cancer evolution based on the kinetic theory of active particles. The model consists of a system of nonlinear integro-differential equations describing the binary interactions between epithelial, tumor, naive immune cells, and activated immune cells. It also takes into account the phenotypic mutations in the epithelial and immune cells, which are known to result in the uncontrolled growth of tumor cells. We prove the well-posedness of the related Cauchy problem and the non-negativity of the solution. We give sufficient conditions for which the solution may exist globally in time. A detailed asymptotic analysis has been developed with the aim of predicting the effect of mutation events on the tumor-immune dynamics. The analysis shows that under some critical values of the model's parameters and initial conditions, we can specify some biological states of the blow up of tumor cells. Indeed, the analysis gives useful indications to be properly explored toward the design of therapeutical actions.
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References
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Mohamed Ch-Chaoui, Karima Mokni. ASYMPTOTIC ANALYSIS OF AN INTEGRO-DIFFERENTIAL SYSTEM MODELING THE BLOW UP OF CANCER CELLS UNDER THE IMMUNE RESPONSE[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1763-1785. doi: 10.11948/20210296
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Figure 1. Numerical simulation of the model showing the evolution of the number density of cancer cells versus time for different value of their proliferation rate. The mutation events are absent, that is,
$ \varepsilon_1=0 $ and$ \varepsilon_2 = 0 $ -
Figure 2. Numerical simulation of the model (2.9)-(2.10) showing the evolution the evolution of tumor cells (
$ n_2 $ ) and activated immune cells ($ n_4 $ ) versus time$ t $ . -
Figure 3. Numerical simulation of the model (2.9)-(2.10) showing the evolution of the distribution functions
$ f_2(t,u) $ for$ \varepsilon_2=0 $ (left panel) and$ \varepsilon_2=0.6 $ (right panel). These two figures are performed with an intermediate value of$ \beta_2=0.2 $ and$ \varepsilon_1=0 $ -
Figure 4. Numerical simulation of the model (2.9)-(2.10) showing the evolution of
$ n_2 $ versus time, for different values of$ \varepsilon_1 $ . The mutation events in activated immune cells are absents$ \varepsilon_2=0 $ -
Figure 5. Numerical simulation of the model (2.9)-(2.10) showing the evolution the evolution of tumor cells (
$ n_2 $ ) versus time$ t $ in the case$ \varepsilon_2 <\varepsilon_2^*=0.03 $ et$ n_{40}< n_{40}^{*}=0.1 $