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On a coupled SDE-PDE system modeling acid-mediated tumor invasion

  • * Corresponding author: Sandesh Athni Hiremath

    * Corresponding author: Sandesh Athni Hiremath 
This research was supported by the DFG, grant SU807/1-1.
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  • We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

    Mathematics Subject Classification: Primary: 34F05, 35R60, 92C17; Secondary: 35Q92, 60H10, 92C50.

    Citation:

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  • Figure 1.  Initial conditions in 1D and 2D.

    Figure 2.  Time snapshots of a sample solution in the case of a 1D domain. Blue line: cancer cell density $c$; green line: extracellular proton concentration $p$, red line: intracellular proton concentration $h$. Choice of functions and coefficients as in (47).

    Figure 3.  Time snapshots of the numerical mean in the case of a 1D domain. Blue line: cancer cell density $c$; green line: extracellular proton concentration $p$, red line: intracellular proton concentration $h$. Choice of functions and coefficients as in (47).

    Figure 4.  Qualitative behavior of $J$ as a function of $c$.

    Figure 5.  Time snapshots of three different sample solutions (out of 1000 simulations) in a 1D domain. Blue: cancer cell density, green: extracellular proton concentration, red: intracellular proton concentration. Choice of functions and coefficients as in (48).

    Figure 6.  Time snapshots of the numerical mean in a 1D domain. Choice of functions and coefficients as in (48).

    Figure 7.  Time snapshots of three different sample solutions to (3)-(4) in a 2D domain. Functions and coefficients as in (48).

    Figure 8.  Time snapshots of the numerical mean in a 2D domain. Functions and coefficients as in (48).

    Figure 9.  Time snapshots of the sample solution 335 in the case of nonlocal coupling and for a 1D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).

    Figure 10.  Time snapshots of the numerical mean in the case of nonlocal coupling and for a 1D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).

    Figure 11.  Time snapshots of the numerical mean in the case of nonlocal coupling and for a 2D domain. Functions $f_3$ and $g$ as in (48), $f_1$ and $f_2$ as in (49).

    Table 1.  Numerical parameters

    Numerical parameters (48), (49) (47)
    Parameter 1D 2D 1D
    N (# time steps) 8000 1500 5000
    M (# Monte Carlo simulations) 1000 1000 1000
    $\tau$ (temporal step size) 0.1 0.1 0.1
    $\delta_x$ (spatial step size along $x$) 0.01 0.01 0.01
    $M_x$ (grid resolution along $x$) 301 41 301
    $\delta_y$ (spatial step size along $y$) - 0.01 -
    $M_y$ (grid resolution along $y$) - 41 -
     | Show Table
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    Table 2.  Simulation parameters (1D and 2D)

    Growth and decay parameters
    phenomenological relevance value in (48), (49) value in (47)
    $\gamma_{_{f_1}}$ rate const. for cancer proliferation 0.009 0.09
    $\gamma_{_{f_2}}$ rate const. for extracellular protons 0.4 36.8
    $\rho$ const. within the logistic term of $p$ - $\frac{1}{36.8}$
    $\gamma_{_{f_3}}$ rate const. for intracell. protons 1 0.08
    $ \gamma_{_g}$ noise intensity intracell. proton dyn. 3 0.03
    Migration parameters
    phenomenological relevance value in (48), (49) value in (47)
    $\gamma_{_{_D}}$ diffusion coefficient for protons 0.0001 0.0001
    $\gamma_{_{\Phi}}$ diffusion coefficient for cancer cells 0.00005 0.00005
    $\gamma_{_{\Psi}}$ pH-taxis coefficient 0.02 0.002
    $k_1$ conversion rate from $h$ to $p$ 0.07 0.06
    $k_2$ conversion rate from $p$ to $h$ 0.01 0.07
    $k_3$ decay rate $h$ due to $c$ - 0.06
    $k_4$ decay rate $c$ due to interaction with $p$ - 0.01
    $\alpha_1$ const. in diffusion coefficient $\Phi$ (47) 1 1
    $\alpha_2$ const. in diffusion coefficient $\Phi$ (47) 4 4
     | Show Table
    DownLoad: CSV
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