# American Institute of Mathematical Sciences

August  2018, 23(6): 2339-2369. doi: 10.3934/dcdsb.2018071

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

 1 Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany 2 Karl-Franzens-Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author: Sandesh Athni Hiremath

Received  July 2017 Revised  September 2017 Published  January 2018

Fund Project: This research was supported by the DFG, grant SU807/1-1

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.

Citation: Sandesh Athni Hiremath, Christina Surulescu, Anna Zhigun, Stefanie Sonner. On a coupled SDE-PDE system modeling acid-mediated tumor invasion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2339-2369. doi: 10.3934/dcdsb.2018071
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##### References:
Initial conditions in 1D and 2D.
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).
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).
Qualitative behavior of $J$ as a function of $c$.
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).
Time snapshots of the numerical mean in a 1D domain. Choice of functions and coefficients as in (48).
Time snapshots of three different sample solutions to (3)-(4) in a 2D domain. Functions and coefficients as in (48).
Time snapshots of the numerical mean in a 2D domain. Functions and coefficients as in (48).
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).
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).
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).
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 -
 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 -
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
 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
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