Geometric stability and optimization in reaction–diffusion models for tissue engineering: A case study with hydrogel scaffolds

This paper presents a thorough qualitative analysis of coupled reaction–diffusion equations, with an emphasis on the careful selection of boundary conditions that govern the interactions between stem cell and nanozyme concentrations within hydrogel scaffolds. These scaffolds, widely applied in cartilage tissue repair, possess unique attributes such as tunable mechanical strength and high water content, which are captured within the model through carefully chosen parametric constraints. The precision in defining boundary conditions is particularly crucial, as stem cell proliferation within these scaffolds is highly sensitive to external parameters, directly influencing their behavior. Our approach employs a geometric framework to analyze the stability of the coupled system, with a focus on discerning the delicate balance between stem cell populations and nanozyme concentrations. The goal is to ensure that the scaffold sustains optimal mechanical integrity, achieves uniform cell distribution, and maintains controlled degradation rates. Through a rigorous stability and bifurcation analysis of the reaction–diffusion equations, we explore the system’s equilibrium points and establish conditions under which these equilibria are stable or unstable. This analysis serves as a theoretical foundation for optimizing the design of hydrogel scaffolds, particularly in the context of ensuring consistent mechanical properties and efficient cellular integration. In addition to the qualitative geometric methods, we integrate Optuna, an advanced optimization algorithm, to fine-tune the model parameters. This computational tool enhances the accuracy of the parameter selection process by efficiently navigating the solution space, thereby refining the predictions of scaffold performance. The combination of geometric stability analysis with the state-of-the-art optimization provides a novel framework for addressing challenges in tissue engineering, offering new insights into the dynamics of scaffold design and the development of experimental protocols. These findings not only elucidate the stability conditions of the system but also contribute in establishing rigorous guidelines for the control of stem cell populations and nanozyme delivery rates in practical applications.