A hybridizable discontinuous Galerkin method for nonlinear porous media viscoelasticity with applications in ophthalmology

Abstract

The interplay between biomechanics and blood perfusion in the optic nerve head (ONH) has a critical role in ocular pathologies, especially glaucomatous optic neuropathy. Elucidating the complex interactions of ONH perfusion and tissue structure in health and disease using current imaging methodologies is difficult, and mathematical modeling provides an approach to address these limitations. The biophysical phenomena governing the ONH physiology occur at different scales in time and space and porous media theory provides an ideal framework to model them. We critically review fundamentals of porous media theory, paying particular attention to the assumptions leading to a continuum biphasic model for the phenomenological description of fluid flow through biological tissues exhibiting viscoelastic behavior. The resulting system of equations is solved via a numerical method based on a novel hybridizable discontinuous Galerkin finite element discretization that allows accurate approximations of stresses and discharge velocities, in addition to solid displacement and fluid pressure. The model is used to theoretically investigate the influence of tissue viscoelasticity on the blood perfusion of the lamina cribrosa in the ONH. Our results suggest that changes in viscoelastic properties of the lamina may compromise tissue perfusion in response to sudden variations of intraocular pressure, possibly leading to optic disc hemorrhages.