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

脑动脉瘤（CA）是颅内动脉病变扩张，其破裂是导致蛛网膜下腔出血的主要原因。然而，动脉瘤形成、生长和破裂背后的机制仍然是一个谜。虽然目前的临床技术还不能提供体内这些过程的许多机制细节，但许多努力已经投入到脑动脉瘤的生物力学建模中。具体来说，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现象。提出的工作旨在提供一个多物理场/多尺度的数学模型以及一个数值框架来理解脑动脉瘤破裂的机制。这些新知识将被引入研究生和本科生的课程中。由此产生的软件将准备在课堂上使用，作为教师和学生友好和免费的开源例程。本项目旨在开发一种新的方法来解决脑动脉瘤多尺度和多物理场建模中的基本开放性问题，并研究动脉壁与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：模拟为不可压缩牛顿流的流体（血液）、非局部周动力理论模拟的破裂实体（动脉瘤底壁）和粘弹性模型描述的实体（动脉壁）。这三个子区域将在适当的界面边界条件下相互数值耦合。在前期工作中，PI开发了新的流固耦合（FSI）方案，以稳定和加速流体求解器与经典固体求解器之间的耦合；(2)设计了一种有效的分数阶PDEs （FPDEs）长期积分方法，发现分数阶可以作为动脉瘤壁强度的指标；(3)研究了周动力-经典理论耦合的Dirichlet-Dirichlet边界条件。未来三年，PI将在理论和数值方面开展工作。在理论部分，将建立新的模型来描述动脉壁的粘弹性行为，并捕捉动脉瘤眼底附近的材料破坏。在数值方面，将开发基于高阶连续/不连续伽辽金方法的高性能计算工具，以准确模拟新模型，并为具有多尺度/多物理场动力学的异构域问题提供耦合框架。从技术上讲，PI将：(1)进一步验证分数阶PDE模型，该模型能更好地描述脑动脉瘤壁的粘弹性行为；(2)首次建立了动脉瘤破裂双分量动力学模型，并基于不连续Galerkin方法建立了该模型的高阶数值解；(3)通过研究直接由守恒定律推导出的合适的数学界面条件，设计耦合三维连续体周动力公式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）也可以应用于这里的多尺度耦合问题。该项目由数学科学部的计算数学项目、生物生物aps计划和化学、生物工程、环境和运输系统部（CBET）的生物医学工程项目共同资助。

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).