Rahimabadi, A. | Search bioRxiv

bioRxiv2024

Parameter Dependence in Identifiability Applied to FP-Fisher-KPP Reaction-Diffusion Equations Parameterized for Tauopathy Network Modeling

Parameterization and a priori identifiability analysis are two interconnected steps that should be carried out in advance of model calibration. In the first place, we propose a framework for parameterizing a recently introduced and analytically studied generalization of the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) reaction-diffusion (Re-Di) equation with fractional polynomial (FP) terms to model heterogeneous nonlinear diffusion in the propagation of a given species through directed networks, exemplified by the tauopathy progression in Alzheimers disease (AD). Next, we present our results on identifiability in a generic sense for the parameterized FP-Fisher-KPP Re-Di equations with regular multi-experimental designs, seamlessly applicable to meromorphic systems, encompassing analytic systems. In particular, the concept of generic local minimal dependence of unknown parameters and regularly parameterized initial conditions will be formalized through the use of one-parameter Lie groups of transformations, and a decomposition method to explore this new concept will be devised. Finally, the Allen Mouse Brain Connectivity Atlas (AMBCA) dataset is utilized to develop a model for tauopathy progression in the mouse brain, which will subsequently be employed to implement the proposed methodology for analyzing a priori identifiability.

systems biology↗

bioRxiv2023

Extended fractional-polynomial generalizations of diffusion and Fisher-KPP equations on directed networks: Modeling neurodegenerative progression

In a variety of practical applications, there is a need to investigate diffusion or reaction-diffusion processes on complex structures, including brain networks, that can be modeled as weighted undirected and directed graphs. As an instance, the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) reaction-diffusion equation are becoming increasingly popular for use in graph frameworks by substituting the standard graph Laplacian operator for the continuous one to study the progression of neurodegenerative diseases such as tauopathies including Alzheimers disease (AD). However, due to the porous structure of neuronal fibers, the spreading of toxic species can be governed by an anomalous diffusion process rather than a normal one, and if this is the case, the standard graph Laplacian cannot adequately describe the dynamics of the spreading process. To capture such more complicated dynamics, we propose a diffusion equation with a nonlinear Laplacian operator and a generalization of the Fisher-KPP reaction-diffusion equation on undirected and directed networks using extensions of fractional polynomial (FP) functions. A complete analysis is also provided for the extended FP diffusion equation, including existence, uniqueness, and convergence of solutions, as well as stability of equilibria. Moreover, for the extended FP Fisher-KPP reaction-diffusion equation, we derive a family of positively invariant sets allowing us to establish existence, uniqueness, and boundedness of solutions. Finally, we conclude by investigating nonlinear diffusion on a directed one-dimensional lattice and then modeling tauopathy progression in the mouse brain to gain a deeper understanding of the potential applications of the proposed extended FP equations.

systems biology↗