Collaborative Research: Analysis and Control in Multi-Scale Interface Coupling between Deformable Porous Media and Lumped Hydraulic Circuits

该项目将发展理论和计算技术，以促进对心血管系统血液循环的理解，从大动脉到组织内的小血管。这对数百万患有青光眼的人尤其重要，青光眼是一种进行性眼睛退行性疾病，会导致不可逆转的视力丧失；了解血流动力学因素的作用可能会给25%以上的青光眼患者带来新的希望，尽管他们接受了药物治疗，但仍继续发展为失明。该项目将培养广大受众的科学素养，并使下一代STEM学生多样化。本科生和研究生将由研究人员在这个项目的理论和应用方面进行指导。调查人员将举办关于“眼睛：我们身体的一扇窗户”的虚拟峰会，以培养北卡罗莱纳州和密苏里州高中女性和少数族裔对数学研究和数学相关职业的兴趣。该研究项目的重点是组织灌注模型的分析和控制，这是一种与周围血液循环的整体特征相关的局部现象。该模型涉及偏微分方程（具有不可压缩成分和渗透率非线性依赖于膨胀的拟线性Biot模型）和常微分方程（描述流体通过血管网络的非线性系统）之间的多尺度界面耦合。本研究将应用数学与计算数学相结合，运用分析与控制理论、数值分析、非凸优化和灵敏度分析等方法，研究局部因素和全局特征对系统解的影响。在青光眼的应用中，该研究将阐明眼压（局部因素）和血压（全局或全体性特征）在确保足够的血液流向眼睛方面所起的作用，从而促进新治疗方法的发展，共同调节眼压和血压，可能有助于许多青光眼患者。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will develop theoretical and computational techniques to advance understanding of the circulation of blood in the cardiovascular system, from large arteries to small vessels embedded within the tissues. This is particularly relevant for the millions of people who suffer from glaucoma, a progressive degenerative disease of the eye that leads to irreversible vision loss; understanding the role of hemodynamic factors could bring new hope to the more than 25% of glaucoma patients who continue to progress to blindness despite medical treatment. The project will cultivate science literacy among a broad audience and diversify the next generation of STEM students. Undergraduate and graduate students will be mentored by the investigators on both the theoretical and applied aspects of this project. The investigators will run Virtual Summits on “The Eye: A Window into Our Body” to build interest in mathematics research and mathematics-related careers among women and underrepresented minorities at the high school level in North Carolina and Missouri.The research program is focused on analysis and control in models of the perfusion of a tissue, a local phenomenon studied in correlation with the global features of the surrounding blood circulation. The models involve multi-scale interface coupling between partial differential equations (quasi-linear Biot model with incompressible constituents and permeability nonlinearly depending on dilation) and ordinary differential equations (nonlinear system describing fluid flow through a vascular network). The research will bridge applied and computational mathematics and apply methods from analysis and control theory, numerical analysis, non-convex optimization, and sensitivity analysis to study how local factors and global features affect the solution of the system. In the application to glaucoma, the study will elucidate the roles played by intraocular pressure (local factor) and blood pressure (global, or systemic, feature) in securing adequate blood flow to the eye, thereby facilitating the development of novel therapeutic approaches coregulating intraocular pressure and blood pressure that could potentially help many glaucoma patients.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.