Global existence of weak solutions to a tissue regeneration model
Nishith Mohan, Christina Surulescu
TL;DR
The paper analyzes a cross-diffusion tissue-regeneration model with MSCs and chondrocytes interacting through haptotaxis and chemotaxis, regulated by a differentiation medium. It constructs global weak solutions via a regularization scheme that adds $\varepsilon$-dependent terms and mollified sources, and employs an entropy-type functional to derive uniform, $\varepsilon$-independent a priori bounds. Through compactness arguments (Aubin–Lions, weak/strong convergence), it passes to the limit $\varepsilon\to 0$ and demonstrates the existence of a global weak solution to the original system for spatial dimension $n\le 3$. The work provides a rigorous mathematical foundation for a simplified yet biologically meaningful tissue regeneration model and contributes to the analytical understanding of reaction-diffusion-taxis systems with indirect signal production.
Abstract
We study a cross-diffusion model for tissue regeneration which involves the dynamics of human mesenchymal stem cells interacting with chondrocytes in a medium containing a differentiation factor. The latter acts as a chemoattractant for the chondrocytes and influences the (de)differentiation of both cell phenotypes. The stem cells perform haptotaxis towards extracellular matrix expressed by the chondrocytes and degraded by themselves. Cartilage production as part of the extracellular matrix is ensured by condrocytes. The growth factor is provided periodically, to maintain the cell dynamics. We provide a proof for the global existence of weak solutions to this model, which is a simplified version of a more complex setting deduced in \cite{surulescu_AMM}.