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Global existence of weak solutions to a cell migration and (de)differentiation model with double haptotaxis in the context of tissue regeneration

Nishith Mohan, Christina Surulescu

TL;DR

The paper addresses the global existence of weak solutions for a cross-diffusion model of mesenchymal stem cell migration and (de)differentiation with double haptotaxis in a hyaluron-coated scaffold. It introduces a smooth regularization with $F_\varepsilon$ and diffusion terms, derives an entropy-type functional to obtain uniform estimates, and employs compactness arguments to pass to the limit $\varepsilon\to 0$, constructing a global weak solution. The analysis extends prior work on single haptotaxis to a two-taxis setting, providing a rigorous mathematical foundation for tissue-regeneration models and informing numerics. Overall, the results deliver a robust existence theory for the model's weak solutions under biologically meaningful assumptions on the transition functions and parameters.

Abstract

We study a model for the spread and (de)differentiation of mesenchymal stem cells and chondrocytes in a scaffold whose fibers are coated with hyaluron. The chondrocytes produce new extracellular matrix, which, together with hyaluron, serves as haptotactic cue for the stem cell migration. We prove global existence of weak solutions of the corresponding cross-diffusion system with double haptotaxis.

Global existence of weak solutions to a cell migration and (de)differentiation model with double haptotaxis in the context of tissue regeneration

TL;DR

The paper addresses the global existence of weak solutions for a cross-diffusion model of mesenchymal stem cell migration and (de)differentiation with double haptotaxis in a hyaluron-coated scaffold. It introduces a smooth regularization with and diffusion terms, derives an entropy-type functional to obtain uniform estimates, and employs compactness arguments to pass to the limit , constructing a global weak solution. The analysis extends prior work on single haptotaxis to a two-taxis setting, providing a rigorous mathematical foundation for tissue-regeneration models and informing numerics. Overall, the results deliver a robust existence theory for the model's weak solutions under biologically meaningful assumptions on the transition functions and parameters.

Abstract

We study a model for the spread and (de)differentiation of mesenchymal stem cells and chondrocytes in a scaffold whose fibers are coated with hyaluron. The chondrocytes produce new extracellular matrix, which, together with hyaluron, serves as haptotactic cue for the stem cell migration. We prove global existence of weak solutions of the corresponding cross-diffusion system with double haptotaxis.

Paper Structure

This paper contains 5 sections, 14 theorems, 139 equations.

Table of Contents

  1. Introduction and model set-up
  2. A family of approximate problems
  3. Global existence for approximate problems
  4. An entropy-type functional
  5. Construction of weak solutions

Key Result

Theorem 1.2

Assume that oic2 holds true and the transition functions $\alpha_i$, where $i \in \{1, 2\}$, satisfy alpha_def. Then, the problem nondim1-oic has at least one global weak solution in the sense of Definition solution_definition.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Global existence for \ref{['model1']}
  • proof
  • ...and 18 more