Nonlinear Dynamics and Analysis of Intracranial Saccular Aneurysms with Growth and Remodeling

Abstract

A new mathematical model for the interaction of blood flow with the arterial wall surrounded by cerebral spinal fluid is developed with applications to intracranial saccular aneurysms. The blood pressure acting on the inner arterial wall is modeled via a Fourier series, the arterial wall is modeled as a spring-mass system incorporating growth and remodeling, and the surrounding cerebral spinal fluid is modeled via a simplified one-dimensional compressible Euler equation with inviscid flow and negligible nonlinear effects. The resulting nonlinear coupled fluid-structure interaction problem is analyzed and a perturbation technique is employed to derive the first-order approximation solution to the system. An analytical solution is also derived for the linearized version of the problem using Laplace transforms. The solutions are validated against related work from the literature and the results suggest the biological significance of the inclusion of the growth and remodeling effects on the rupture of intracranial aneurysms.

Mathematical Analysis and Simulation of a Coupled Nonlinear Fluid Structure Interaction Model With Applications to Aneurysms

“…In [24], the blood pressure was modeled via a Fourier series assuming blood flow is pulsatile whereas the CSF was modeled using a simplified one-dimensional compressible Euler equation for inviscid flow and negligible nonlinear effects. Recently, this model was extended to incorporate the complex biological structure of the arterial wall by including the growth and remodeling effects of two main constituents, elastin and collagen [2].The elastin is a protein that functions as a resistance to the formation of an aneurysm, whereas the collagen prevents rupture after the formulation of aneurysm occurs. Although this model showed the significance of collagen and elastin in the development and rupture of aneurysms which is very consistent with experimental observations, some assumptions were made in this paper, in order to improve the model of the CSF and perform an analysis of the resulting new model.…”

“…It must also be pointed out that modeling such biological multi-physics applications often require solution to coupled nonlinear partial differential equations describing the fluid-structure interaction (FSI). While it is possible to find exact solutions for simplified FSI problems either through linear approximations [24] or well-chosen boundary conditions [2], often these problems do not often admit exact solutions. One has to therefore resort to numerical methods such as finite difference methods [24] or finite element methods [23].…”