A Framework for Multiscale/Multiphysics Mathematical Modeling of Cerebral Aneurysm Rupture

Lehigh University, Bethlehem PA

Investigators

Abstract

Cerebral aneurysm (CA) is a diseased dilatation of an intracranial artery, and its rupture is the leading cause of subarachnoid bleeding. However, the mechanisms behind aneurysm formation, growth and rupture remain an enigma. While the current clinical technology cannot yet provide a lot of mechanistic details of these processes in vivo, many efforts have been devoted in modeling the biomechanics of the cerebral aneurysms. Specifically, numerical simulations have elucidated some of the physics associated with the arterial wall damage and aneurysm rupture. The proposed work aims to provide a multiphysics/multiscale mathematical model along with a numerical framework to understand the mechanism of cerebral aneurysm rupture. The new knowledge will be introduced into both graduate and undergraduate level courses. The resultant software will be ready for classroom use as friendly and free opensource routines for instructors and students. This project aims to develop a new methodology for addressing fundamental open questions in multiscale and multiphysics modeling of brain aneurysms, and to study the interactions between the arterial wall and the blood flow with an emphasis on simulating the rupture phenomena. To be specific, the computational domain is composed of three regions: the fluid (blood) simulated as incompressible Newtonian flow, the fracture solid (aneurysm fundus wall) modeled by the nonlocal peridynamic theory, and the solid (arterial wall) described by a viscoelastic model. These three subregions will be numerically coupled to each other with proper interface boundary conditions. In preliminary work, the PI has: (1) developed new schemes for fluid-structure interaction (FSI) to stabilize and accelerate the coupling between the fluid solver and the classical solid solver; (2) designed an efficient long-term integration method for fractional-order PDEs (FPDEs) and found that the fractional order might serve as an indicator for the aneurysm wall strength; (3) investigated the Dirichlet-Dirichlet boundary condition for the peridynamic-classical theory coupling. In the next three years, the PI will work on both theoretical and numerical aspects. For the theoretical part, new models will be addressed to describe the viscoelastic behavior of the arterial walls and to capture the material failure near the aneurysm fundus. Regarding the numerical effort, high-performance computational tools based on high-order continuous/discontinuous Galerkin methods will be developed, which could accurately simulate the new models as well as provide a coupling framework for problems composed of heterogeneous domains with multiscale/multiphysics dynamics. Technically, the PI will: (1) further validate the fractional-order PDE models that better describe the viscoelastic behavior of cerebral aneurysm walls; (2) for the first time develop two-component peridynamic theory for modeling the aneurysm rupture, and develop high-order numerical solvers for this model based on the discontinuous Galerkin method; (3) design partitioned approaches for coupling the 3D continuum formulations of peridynamics and classical theory, by investigating proper mathematical interface conditions directly derived from conservation laws. The coupling techniques the PI has investigated in preliminary work (FSI coupling) would also be adopted into the multiscale coupling problem here. This project is co-funded by the Computational Mathematics Program of the Division of Mathematical Sciences, the BioMAPS Initiative and the Biomedical Engineering program of the Division of Chemical, Bioengineering, Environmental and Transport Systems Division (CBET).