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Derivation of PS-PDEs from mesoscale kinetic internal-state models

Derive, via rigorous scaling limits, phenotype-structured partial differential equations of the general form (equation (general)) starting from mesoscale kinetic equations that incorporate internal state variables, thereby obtaining explicit PS-PDE formulations from kinetic-level descriptions.

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Background

Beyond agent-based to continuum derivations, the literature includes mesoscale kinetic formulations with internal state variables that can be scaled to macroscopic PDEs describing movement and sensing. However, the authors note the absence of explicit derivations that yield the PS-PDE structure they review, indicating a gap between kinetic internal-state models and PS-PDEs.

Bridging this gap would provide a principled route to link intracellular dynamics encoded at the kinetic level to population-level phenotype structuring in space and trait, enhancing the mechanistic foundations of PS-PDE models.

References

Moreover, structuring variables have also been incorporated into ‘mesoscale’ kinetic equations, which can also be scaled to a PDE form (see e.g. \citealt{erban2004individual,engwer2015glioma,lorenzi2024phenotype}); we are, however, unaware of any such derivations that lead explicitly to the PS-PDE form (\ref{general}).

— Phenotype structuring in collective cell migration:a tutorial of mathematical models and methods (2410.13629 - Lorenzi et al., 17 Oct 2024) in Section 4.1 (Tools and techniques to derive PS-PDE models from agent-based models)