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

Sandesh Athni Hiremath; Christina Surulescu; Anna Zhigun; Stefanie Sonner

doi:10.3934/dcdsb.2018071

Article Contents

2018, Volume 23, Issue 6: 2339-2369. Doi: 10.3934/dcdsb.2018071

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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
* Corresponding author: Sandesh Athni Hiremath

Received: June 30, 2017

Revised: August 31, 2017

Early access: January 2018

Published: June 2018

This research was supported by the DFG, grant SU807/1-1.

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Abstract

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.

Keywords:

Multiscale model,
intra-and extracellular proton dynamics,
tumor acidity,
stochastic differential equations,
partial differential equations,
pH-taxis.

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

Citation:

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

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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).

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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).

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

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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).

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

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

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

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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).

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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).

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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).

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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	-

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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

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