  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-$\beta$. 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-$\beta$ 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.
