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Stability of Solutions of the Porous Medium Equation with Growth with Respect to the Diffusion Exponent


Core Concepts
The core message of this article is to establish Hölder continuous dependence of the solutions of the porous medium equation with a growth term on the diffusion exponent.
Abstract
The article considers a macroscopic model for the growth of living tissues that incorporates pressure-driven dispersal and pressure-modulated proliferation. Assuming a power-law relation between the mechanical pressure and the cell density, the model can be expressed as the porous medium equation with a growth term. The key highlights and insights are: The authors prove Hölder continuous dependence of the solutions of the model on the diffusion exponent. This is an important result for validating and calibrating the model using experimental data. The main difficulty lies in the degeneracy of the porous medium equations at vacuum. To address this, the authors first regularize the equation by shifting the initial data away from zero and then optimize the stability estimate derived in the regular setting. The authors discuss the importance of this stability result for applying inverse problem methodologies, both in deterministic approaches involving parameter estimation and in stochastic Bayesian approaches for model calibration. The authors point out that without the stability of solutions with respect to parameters, solving the inverse problem becomes unfeasible, as both stochastic and deterministic approaches rely on numerical approximation, which involves addressing finite-dimensional problems. The authors also discuss how the stability of the solution with respect to the diffusion exponent is required to design efficient Monte Carlo algorithms for Bayesian inference.
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Deeper Inquiries

How can the stability analysis be extended to more complex models that incorporate additional biological processes, such as cell-cell adhesion or angiogenesis

To extend the stability analysis to more complex models incorporating additional biological processes like cell-cell adhesion or angiogenesis, we need to consider the impact of these processes on the overall dynamics of the system. Cell-Cell Adhesion: Cell-cell adhesion plays a crucial role in tissue growth and can be incorporated into the model by introducing terms that represent the adhesion forces between cells. This would involve modifying the growth term in the porous medium equation to include interactions between neighboring cells. The stability analysis would then need to account for the effects of adhesion on the overall behavior of the system. Angiogenesis: Angiogenesis, the formation of new blood vessels, can be included in the model by introducing additional equations that describe the dynamics of blood vessel growth and their interaction with the cell density field. This would lead to a more complex system of equations that would require a comprehensive stability analysis to understand how angiogenesis affects the overall behavior of the tissue growth model. By extending the stability analysis to incorporate these additional biological processes, we can gain a more comprehensive understanding of how different factors influence the growth and behavior of living tissues.

What are the potential limitations of the power-law relation between pressure and cell density, and how could alternative constitutive relations be incorporated into the model while preserving the stability properties

The power-law relation between pressure and cell density, while commonly used in models of tissue growth, may have limitations in capturing the full complexity of biological systems. Some potential limitations include: Biological Realism: The power-law relation may oversimplify the actual biological processes governing pressure-driven dispersal and proliferation. Alternative constitutive relations could be more biologically realistic and capture the nuances of cell behavior more accurately. Model Flexibility: Alternative constitutive relations can provide more flexibility in modeling different biological scenarios. For example, incorporating nonlinear or time-dependent relations between pressure and cell density could better represent the dynamic nature of tissue growth. To incorporate alternative constitutive relations while preserving stability properties, it is essential to ensure that the new relations satisfy key mathematical properties required for stability analysis. This may involve conducting additional mathematical analysis to verify the stability of the model with the updated constitutive relations.

The article focuses on the porous medium equation with a growth term, but the authors mention that similar stability studies have been undertaken for other types of hyperbolic conservation laws and quasilinear parabolic equations. What insights from these other contexts could be leveraged to further develop the stability analysis for the porous medium equation with growth

Insights from stability studies on hyperbolic conservation laws and quasilinear parabolic equations can be leveraged to further develop the stability analysis for the porous medium equation with growth. Entropy Solutions: Drawing from the stability studies on entropy solutions for hyperbolic conservation laws, we can explore the concept of entropy in the context of the porous medium equation with growth. Understanding the entropy properties of the system can provide valuable insights into its stability behavior. Numerical Methods: Techniques used in stability studies for other types of equations, such as finite element methods or spectral methods, can be adapted to analyze the stability of the porous medium equation with growth. These numerical methods can help in simulating the behavior of the system under different conditions and parameter values. By integrating insights and methodologies from related stability studies, we can enhance the robustness of the stability analysis for the porous medium equation with growth and potentially uncover new avenues for research and application in mathematical modeling of tissue growth.
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