Modelling and Scientific Computing (CMCS), Mathematics Institute of Computational Science and Engineering (MATHICSE), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 8, CH-1015 Lausanne, Switzerland.
Int J Numer Method Biomed Eng. 2013 Jul;29(7):741-76. doi: 10.1002/cnm.2559. Epub 2013 Jun 25.
The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.
心血管数学中的反问题求解计算代价高昂。在本文中，我们应用域参数化技术来降低正向问题的几何和计算复杂性，并使用计算成本较低的降阶基近似代替不可压缩纳维-斯托克斯方程的有限元解。这大大降低了正向问题模拟的成本。然后，我们分别从确定性和统计两个角度来考虑反问题的求解：通过求解最小二乘问题进行确定性求解，以及使用贝叶斯框架来量化不确定性进行统计求解。考虑了血流动力学建模中的两个反问题：（i）在狭窄动脉的一部分中简化的流固耦合模型问题，通过根据压力测量结果来确定动脉壁的材料参数，从而量化动脉粥样硬化的风险；（ii）基于压力测量结果识别主动脉分支中不确定的残余流的股浅动脉旁路移植模型的稳健形状设计。
