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Multiphase cross-diffusion models for tissue structures: modeling, analysis, numerics

Ansgar Jüngel, Cordula Reisch, Sara Xhahysa

Abstract

Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell-Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak-strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

Multiphase cross-diffusion models for tissue structures: modeling, analysis, numerics

Abstract

Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell-Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak-strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

Paper Structure

This paper contains 18 sections, 11 theorems, 94 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model setting
  3. Equal drag coefficients
  4. Varying drag coefficients
  5. Multiphase modeling
  6. Derivation of the cross-diffusion equations
  7. Entropy structure
  8. Mathematical analysis: equal drag coefficients
  9. Proof of Theorem \ref{['thm.ex']}
  10. Proof of Theorem \ref{['thm.decay']}
  11. Proof of Theorem \ref{['thm.wsu']}
  12. Tumor-growth model of Jackson and Byrne
  13. Mathematical analysis: different drag coefficients
  14. Numerical experiments
  15. Implementation of the scheme
  16. ...and 3 more sections

Key Result

Theorem 1

Let Assumptions (A1)--(A4) hold, the matrix $(q_{ij}+r_{ij})_{i,j=1}^n$ be positive definite with smallest eigenvalue $\alpha>0$, and $q_{ij}\ge r_{ij}$ for $i,j=1,\ldots,n$. Then there exists a global bounded weak solution $u$ to 1.qr--1.samek satisfying $u(x,t)\in\overline{\mathcal{D}}$ for a.e. $ Moreover, the entropy inequality holds for $0<t<T$: where $h_B(u)=\sum_{i=0}^n u_i(\log u_i-1)$ an

Figures (6)

  • Figure 1: Convergence rate in the discrete $L^1$ norm for the tumor-growth model ($\beta=0.0015$, $\theta=100$).
  • Figure 2: Volume fractions of the tumor cells (left) and the ECM (right) using $\theta=30$ (top) and $\theta=1000$ (bottom). The tumor cell front and the ECM peaks move from left to right as time increases.
  • Figure 3: Volume fractions of the tumor cells (left) and the ECM (right) using the symmetric values $\beta_c=\beta_m=1$ and the supercritical parameter $\theta=100$.
  • Figure 4: Relative entropy for the tumor-growth model versus time using $\beta_c=\beta_m=1$ and $\theta=100$.
  • Figure 5: Volume fractions of the multiphase model \ref{['1.vfSKT']} using $\theta_1=1$ and $\theta_2=10$ (top) as well as $\theta_2=100$ (bottom).
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 1: Global existence of solutions
  • Theorem 2: Exponential decay
  • Theorem 3: Weak--strong uniqueness
  • Lemma 4: Boltzmann entropy equality
  • proof
  • Lemma 5: Rao entropy inequality
  • proof
  • Lemma 6: Combined Boltzmann and Rao entropies
  • proof
  • Remark 7
  • ...and 11 more