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

脑动脉瘤（CA）是颅内动脉的病变扩张，其破裂是蛛网膜下腔出血的主要原因。然而，动脉瘤形成、生长和破裂的机制仍然是一个谜。虽然目前的临床技术还不能提供这些过程在体内的机械细节很多，许多努力已经致力于建模的脑动脉瘤的生物力学。具体而言，数值模拟已经阐明了与动脉壁损伤和动脉瘤破裂相关的一些物理学。本研究的目的是提供一个多物理场/多尺度的数学模型沿着数值框架来理解脑动脉瘤破裂的机制。新知识将被引入研究生和本科生课程。由此产生的软件将作为教师和学生的友好和免费的开源例程准备在课堂上使用。本项目旨在开发一种新的方法来解决脑动脉瘤的多尺度和多物理场建模中的基本开放问题，并研究动脉壁和血流之间的相互作用，重点是模拟破裂现象。具体而言，计算域由三个区域组成：流体（血液）模拟为不可压缩的牛顿流，断裂固体（动脉瘤基底壁）由非局部周壁理论建模，和固体（动脉壁）描述的粘弹性模型。这三个子区域将在适当的界面边界条件下相互耦合。在初步工作中，PI已经：（1）开发了新的流固耦合（FSI）方案，以稳定和加速流体求解器和经典固体求解器之间的耦合;（2）设计了一种有效的分数阶偏微分方程（FPDE）长期积分方法，并发现分数阶可以作为动脉瘤壁强度的指标;（3）研究了周波-经典理论耦合的Dirichlet-Dirichlet边界条件。在接下来的三年里，PI将在理论和数值方面开展工作。对于理论部分，将提出新的模型来描述动脉壁的粘弹性行为，并捕获动脉瘤底部附近的材料失效。关于数值工作，将开发基于高阶连续/不连续Galerkin方法的高性能计算工具，这些工具可以精确地模拟新模型，并为具有多尺度/多物理场动力学的异质域组成的问题提供耦合框架。从技术上讲，PI将：（1）进一步验证了分数阶PDE模型能更好地描述脑动脉瘤壁的粘弹性行为：（2）首次提出了用于模拟动脉瘤破裂的双组分周波理论，并基于间断Galerkin方法开发了该模型的高阶数值求解器;（3）通过研究由守恒律直接导出的适当的数学界面条件，设计耦合三维连续介质理论和经典理论的分区方法。PI在前期工作中研究的耦合技术（FSI耦合）也将被采用到多尺度耦合问题中。该项目由数学科学部的计算数学计划，BioMAPS计划和化学，生物工程，环境和运输系统部（CBET）的生物医学工程计划共同资助。

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).