Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix

Rebecca M. Crossley, Jan-Frederik Pietschmann, Markus Schmidtchen

TL;DR

The paper addresses a volume-filling, cross-diffusion invasion model where cell density $u$ interacts with ECM density $m$ via a degenerate parabolic PDE for $u$ coupled to an ODE for $m$. It develops a rigorous existence theory for weak solutions by exploiting an entropy structure and a regularized, time-discrete scheme that yields uniform a priori estimates, enabling compactness arguments to pass to the limit. The main contributions include a well-posedness result for a PDE-ODE coupled system with carrying capacity constraints, and a numerical exploration that uncovers traveling wave behavior and the impact of ECM degradation on wave speed, including the asymptotic regime as $\lambda\to\infty$. The work provides mathematical grounding for cell-ECM invasion dynamics and informs the qualitative and quantitative understanding of wave propagation in tissue invasion contexts.

Abstract

We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the density of cells, coupled with an ordinary differential equation (ODE) describing the ECM density. The model contains cross-species density-dependent diffusion and proliferation terms that capture the role of the ECM in providing structural support for the cells during invasion while also preventing growth via volume-filling effects. Furthermore, the model includes ECM degradation by the cells. We present an existence result for weak solutions which is based on carefully exploiting the partial gradient flow structure of the problem which allows us to overcome the non-regularising nature of the ODE involved. In addition, we present simulations based on a finite difference scheme that illustrate that the system exhibits travelling wave solutions, and we investigate numerically the asymptotic behaviour as the ECM degradation rate tends to infinity.

Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix

TL;DR

The paper addresses a volume-filling, cross-diffusion invasion model where cell density interacts with ECM density via a degenerate parabolic PDE for coupled to an ODE for . It develops a rigorous existence theory for weak solutions by exploiting an entropy structure and a regularized, time-discrete scheme that yields uniform a priori estimates, enabling compactness arguments to pass to the limit. The main contributions include a well-posedness result for a PDE-ODE coupled system with carrying capacity constraints, and a numerical exploration that uncovers traveling wave behavior and the impact of ECM degradation on wave speed, including the asymptotic regime as . The work provides mathematical grounding for cell-ECM invasion dynamics and informs the qualitative and quantitative understanding of wave propagation in tissue invasion contexts.

Abstract

We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the density of cells, coupled with an ordinary differential equation (ODE) describing the ECM density. The model contains cross-species density-dependent diffusion and proliferation terms that capture the role of the ECM in providing structural support for the cells during invasion while also preventing growth via volume-filling effects. Furthermore, the model includes ECM degradation by the cells. We present an existence result for weak solutions which is based on carefully exploiting the partial gradient flow structure of the problem which allows us to overcome the non-regularising nature of the ODE involved. In addition, we present simulations based on a finite difference scheme that illustrate that the system exhibits travelling wave solutions, and we investigate numerically the asymptotic behaviour as the ECM degradation rate tends to infinity.
Paper Structure (7 sections, 12 theorems, 105 equations, 4 figures)

This paper contains 7 sections, 12 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main theorem and structure of the paper
  3. Proof of the main theorem
  4. Time discretisation and regularisation
  5. A priori estimates
  6. Existence of weak solutions
  7. A numerical exploration of large ECM degradation rates

Key Result

Theorem 1.2

Given $(u_0,m_0) \in L^2(\Omega) \times H^1(\Omega)$ such that $0 \le m_0, \,u_0$ and $u_0 + m_0 \le 1$ a.e. in $\Omega$, there exists a weak solution $(u,m) \in (L^2(0,T;H^1(\Omega)))^2$ to system eq:main in the sense of Definition def:weak-soln, which also satisfies the box constraints In addition, there exist non-negative constants $C_\mathrm{u}$ and $C_\mathrm{m}$, depending on $\Omega$, $T$

Figures (4)

  • Figure 1: On the left, plot of the relationship between the ECM degradation rate, $\lambda$, and the numerically estimated travelling wave speed, $c$, (solid lines) of solutions of the system \ref{['eq:main']} subject to zero flux boundary conditions and initial conditions \ref{['IC_u']} and \ref{['IC_m']} for various initial ECM densities, $m_0$. The numerically estimated travelling wave speed is obtained by interpolation to find the point $X(t)$ such that $u(X(t), t) = 0.1$ at all times $t\geq0$. The dashed lines indicate the value of the analytically predicted minimum travelling wave speed, $2(1-m_0)$crossley2023travelling. On the right, plots of solutions to system \ref{['eq:main']} subject to zero flux boundary conditions and initial conditions \ref{['IC_u']} and \ref{['IC_m']} in one dimension for the cells (blue) and for the ECM (red), where $m=0.5$. Plots are shown subject to ECM degradation rates $\lambda=10^0,\, 10^2, \, 10^4, \, 10^6$ at times $t=25, \, 50, \, 75\, \text{and}\, 100,$ from left to right.
  • Figure 2: Plots of solutions to system \ref{['eq:main']} subject to zero flux boundary conditions and initial conditions \ref{['IC_u']} and \ref{['IC_m']} for the cells (blue, plotted only on the right half of the plots) and for the ECM (red, plotted only on the left half of the plots), where $m_0=0.5$. Results are symmetric, hence plotted in this way. Plots are shown subject to ECM degradation rates $\lambda=1,\, 10$ at times $t=0,\,2,\,4,\,6,\,8.$
  • Figure 3: Plots of solutions to system \ref{['eq:main']} subject to zero flux boundary conditions and initial conditions for the cells (blue, plotted only on the right half of the plots) as in Eq. \ref{['IC_u']} and random initial data for the ECM, smoothed using a Gaussian filter with $\sigma=5$ (red, plotted only on the left half of the plots). Results are symmetric, hence plotted in this way. Plots are shown subject to ECM degradation rates $\lambda=1,\, 10$ at times $t=0,\,2,\,4,\,6,\,8.$
  • Figure 4: Plots of solutions to system \ref{['eq:main']} subject to zero flux boundary conditions and initial conditions for the cells given by Eq. \ref{['IC_u']}, but for the ECM as $m(x,0)=0.5+0.25\,\text{sin}(x/10)$. Solutions are shown in one dimension for the cells (blue) and for the ECM (red) at times $t=25, \, 50, \, 75\, \text{and}\, 100,$ from left to right.

Theorems & Definitions (24)

  • Theorem 1.2: Existence
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4: Uniform upper and lower bounds on $u_k,\,m_k$
  • proof
  • Corollary 2.5
  • ...and 14 more