Biomechanics of the brain: A theoretical and numerical study of Biot's equations of consolidation theory with deformation-dependent permeability

Various mathematical models of hydrocephalus and other brain abnormalities have appeared in the literature over the past 2 decades. In this paper, we study a class of models based on Biot's theory of consolidation with boundary forcing. By simplifying the geometry, we derive a single non-linear parabolic equation for the unidirectional deformation of brain tissue and focus on the effects of variable permeability. Using the theory of semigroups, we prove the existence and uniqueness of weak solutions to a class of problems which includes our particular case under consideration. Numerical solutions of our problem are used to motivate a discussion as to whether it is reasonable to pursue the development and implementation of models that incorporate deformation dependent permeability for more complex geometric configurations that are of relevance for models of the clinical condition of hydrocephalus.