European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS49: Mathematical modeling of biological fluids: theoretical and numerical aspects
Chair: Marcela Szopos
25 mins
Stabilizing DG methods on polygonal meshes via computable dual norms
Silvia Silvia Bertoluzza, Ilaria Perugia, Daniele Prada
Abstract: Discontinuous Galerkin Methods based on polygonal an polyhedral meshes are quite attractive when dealing with complex geometries, as it often happens in the simulation of biophysical phaenomena such as, for instance, blood circulation in a complex vascular system. Relaxing the requirement that the elements of the tessellation are triangles/quadrangles or tetrahedra/hexahedra, allows to better capture the geometrical features without the need for unnecessary refinement. The choice of the stabilization terms (or, equivalently, of the definition of the numerical fluxes and/or traces) is a key issue in the design of DG methods. Usually, some form of penalization is somehow added, where some residual term is measured in a mesh dependent norm, designed to somehow mimic the norm of the dual space where the residual naturally exists. The treatment, in the analysis, of such mesh dependent norms calls for the combination of direct and inverse inequalities, which results in a loss of optimality whenever these do not compensate each other, as it happens, when considering the dependence on the mesh size parameter h, if the mesh quality deteriorates, or, also for strongly shape regular tessellations, when studying the dependence on the polynomial degree k. With the final aim of overcoming such limitations, we will present, in this talk, a way to design “cheap” computable norms (and scalar products) for the dual spaces, and of using them in designing the stabilization term for DG type methods.
25 mins
Parameter robust preconditioning for multi-compartmental Darcy equations
Kent-Andre Mardal, Marie Rognes, Eleonora Piersanti
Abstract: In this paper, we propose a new finite element solution approach to the multi-compartmental Darcy equations describing flow and interactions in a porous medium with multiple fluid compartments. We introduce a new numerical formula- tion and a block-diagonal preconditioner. The robustness with respect to variations in material parameters is demonstrated by theoretical considerations and numerical examples.
25 mins
A Continuum-Based Model for the Theoretical Study of Aqueous Humor Secretion Coupling Electrochemical and Fluid-Dynamical Transmembrane Mechanism
Riccardo Sacco, Lorenzo Sala, Aurelio Giancarlo Mauri, Dario Messenio, Giovanna Guidoboni, Alon Harris
Abstract: Intraocular pressure, resulting from the balance of aqueous humor (AH) production and drainage, is the only approved treatable risk factor in glaucoma~\cite{kiel1998}. AH production is determined by the concurrent function of ion pumps and aquaporins in the ciliary processes but their individual contribution is difficult to characterize experimentally. In this work, we propose a novel unified modeling and computational framework for the finite element simulation of the role of the main ion pumps and exchangers involved in AH secretion, namely, the sodium-potassium pump, the calcium-sodium exchanger, the chloride-bicarbonate exchanger and the sodium-proton exchanger. The theoretical model is developed at the cellular scale and is based on the coupling between electrochemical and fluid-dynamical transmembrane mechanisms characterized by a novel description of the electric pressure exerted by the ions on the intrapore fluid that includes electrochemical and osmotic corrections~\cite{camcos2019}. Considering a realistic geometry of the ion pumps, the proposed model is demonstrated to correctly predict their functionality as a function of (1) the permanent electric charge density over the pore surface; (2) the osmotic gradient coefficient; and (3) the stoichiometric ratio between the ion pump currents enforced at the inlet and outlet sections of the pore. In particular, theoretical predictions of the transepithelial membrane potential for each simulated pump/exchanger allow us to perform a first significant model comparison with experimental data for monkeys. This is a significant step for future multidisciplinary studies on the action of molecules on AH production and may help provide a quantitative method to relate phenomena at the cellular scale with the whole eye level that can be accessed clinically~\cite{sala2018}.
25 mins
Mathematical assessment of the role of three factors entangled in the development of glaucoma by means of the Ocular Mathematical Virtual Simulator
Lorenzo Sala, Christophe Prud'homme, Giovanna Guidoboni, Marcela Szopos, Alon Harris
Abstract: Glaucoma is a multifactorial neurodegenerative disease that involves the optic nerve head and the death of the retinal ganglion cells. The main challenge in medicine is to understand the origin of this degeneration. In this paper we present a virtual clinical study employing the Ocular Mathematical Virtual Simulator (OMVS), a mathematical model, which is able to disentangle the hidden mechanisms and to investigate the causes of this ocular neurodegeneration. In particular, we focus our attention on the influence that intraocular pressure, intracranial pressure and arterial blood pressure set on the ocular hemodynamics.