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On a mathematical model for tissue regeneration

Shimi Chettiparambil Mohanan, Nishith Mohan, Christina Surulescu

TL;DR

The paper develops a mesoscopic-to-macroscopic modeling pipeline for tissue regeneration in a hyaluron-impregnated scaffold, starting from kinetic transport equations for densities $p_1$ and $p_2$ and deriving a macroscopic PDE-ODE system for $c_1$, $c_2$, and $h$ via parabolic scaling. It proves global existence of classical solutions in a 2D bounded domain and employs a change of variable $z = c_1 e^{- rac{b}{a_1} h}$ to obtain robust a priori estimates, enabling the analysis. Linear stability and bifurcation analysis reveal that taxis with strength $b$ can drive Hopf bifurcations, producing spatial-temporal patterns, with a critical value $b_c$ determined by Routh-Hurwitz-type conditions and Neumann Laplacian modes. Numerical simulations in 1D and 2D corroborate the theory, showing how larger $b$ induces oscillatory, pattern-forming regimes and how initial tissue/distribution of hyaluron shapes regeneration outcomes, offering scaffold-design insights for enhanced tissue repair.

Abstract

We propose a PDE-ODE model for tissue regeneration, obtained by parabolic upscaling from kinetic transport equations written for the mesoscopic densities of mesenchymal stem cells and chondrocytes which evolve in an artificial scaffold impregnated with hyaluron. Due to the simple chosen turning kernels, the effective equations obtained on the macroscopic level are of the usual reaction-diffusion-taxis type. We prove global existence of solutions to the coupled macroscopic system and perform a stability and bifurcation analysis, which shows that the observed patterns are driven by taxis. Numerical simulations illustrate the model behavior for various tactic sensitivities and initial conditions.

On a mathematical model for tissue regeneration

TL;DR

The paper develops a mesoscopic-to-macroscopic modeling pipeline for tissue regeneration in a hyaluron-impregnated scaffold, starting from kinetic transport equations for densities and and deriving a macroscopic PDE-ODE system for , , and via parabolic scaling. It proves global existence of classical solutions in a 2D bounded domain and employs a change of variable to obtain robust a priori estimates, enabling the analysis. Linear stability and bifurcation analysis reveal that taxis with strength can drive Hopf bifurcations, producing spatial-temporal patterns, with a critical value determined by Routh-Hurwitz-type conditions and Neumann Laplacian modes. Numerical simulations in 1D and 2D corroborate the theory, showing how larger induces oscillatory, pattern-forming regimes and how initial tissue/distribution of hyaluron shapes regeneration outcomes, offering scaffold-design insights for enhanced tissue repair.

Abstract

We propose a PDE-ODE model for tissue regeneration, obtained by parabolic upscaling from kinetic transport equations written for the mesoscopic densities of mesenchymal stem cells and chondrocytes which evolve in an artificial scaffold impregnated with hyaluron. Due to the simple chosen turning kernels, the effective equations obtained on the macroscopic level are of the usual reaction-diffusion-taxis type. We prove global existence of solutions to the coupled macroscopic system and perform a stability and bifurcation analysis, which shows that the observed patterns are driven by taxis. Numerical simulations illustrate the model behavior for various tactic sensitivities and initial conditions.
Paper Structure (14 sections, 14 theorems, 136 equations, 9 figures)

This paper contains 14 sections, 14 theorems, 136 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model deduction
  3. Mesoscopic level
  4. Parabolic upscaling
  5. Analysis of the macroscopic model
  6. Global existence
  7. Change of variable
  8. Local existence and extensibility criterion
  9. A priori estimates
  10. Linear stability and bifurcations
  11. Numerical simulations and patterns
  12. 1D simulations
  13. 2D simulations
  14. Appendix

Key Result

Theorem 3.1

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with smooth boundary, assume that $a_1, a_2, b, \alpha, \delta, \beta, \gamma_1$ and $\gamma_2$ are positive. Then for any $(c_1^0, c_2^0, h_0)$ satisfying assu1 with some $\omega \in (0, 1)$, the problem modeladmits a unique global classical sol

Figures (9)

  • Figure 1: ADSC, chondrocyte, and hyaluron/ECM density for Scenario 1. Upper row: $b>b_c$, lower row $b<b_c$.
  • Figure 2: ADSC, chondrocyte, and hyaluron/ECM density for Scenario 2. Upper row: $b>b_c$, lower row $b<b_c$.
  • Figure 3: ADSC, chondrocyte, and hyaluron/ECM density for Scenario 3, $b>b_c$.
  • Figure 4: ADSC, chondrocyte, and hyaluron/ECM density for Scenario 4. Upper row: $b>b_c$, lower row $b<b_c$.
  • Figure 5: Initial conditions \ref{['eq:ICs-2D']} for ADSC, chondrocyte, and hyaluron/ECM density.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 17 more