Exploring tau protein and amyloid-beta propagation: a sensitivity analysis of mathematical models based on biological data

TL;DR

A sensitivity analysis of heterodimer and Fisher-Kolmogorov models is performed to evaluate the impact of the equilibrium values of protein concentration on the solution patterns of Alzheimer's disease patients and controls.

Abstract

Alzheimer's disease is the most common dementia worldwide. Its pathological development is well known to be connected with the accumulation of two toxic proteins: tau protein and amyloid-$β$. Mathematical models and numerical simulations can predict the spreading patterns of misfolded proteins in this context. However, the calibration of the model parameters plays a crucial role in the final solution. In this work, we perform a sensitivity analysis of heterodimer and Fisher-Kolmogorov models to evaluate the impact of the equilibrium values of protein concentration on the solution patterns. We adopt advanced numerical methods such as the IMEX-DG method to accurately describe the propagating fronts in the propagation phenomena in a polygonal mesh of sagittal patient-specific brain geometry derived from magnetic resonance images. We calibrate the model parameters using biological measurements in the brain cortex for the tau protein and the amyloid-$β$ in Alzheimer's patients and controls. Finally, using the sensitivity analysis results, we discuss the applicability of both models in the correct simulation of the spreading of the two proteins.

Figures (8)

• Figure 1: Bifurcation surface for the heterodimer model with respect to the parameters $p_\Delta$, $p_\mathrm{min}$, and $q_\mathrm{max}$ (left), and bifurcation lines intersected with the significant planes in the following simulations.
• Figure 2: Comparison between empirical CDFs and theoretical CDFs for the model parameters in the tau protein simulations.
• Figure 3: Comparison between empirical CDFs and theoretical CDFs for the model parameters in the amyloid-$\beta$ simulations.
• Figure 4: Brain section with the specification of the polygonal mesh (a), the axonal directions with the distinction between white (red) and grey (blue) matters (b), and the initial conditions of the simulation of tau protein (c) and amyloid-$\beta$, respectively.
• Figure 5: Sensitivity analysis of heterodimer model in tau protein for the variation of the parameter $q_\mathrm{max}$ (a), $p_\mathrm{min}$ (b), $p_\Delta$ (c). For each parameter, in the first row, we report the space average of the healthy protein in time (left), of the misfolded protein in time (center), and of both in the phase space (right). In the second row, we report some snapshots of the solutions for three selected parameter values.