Journal of Nonlinear Dynamics

Volume 2016 (2016), Article ID 2869083, 12 pages

http://dx.doi.org/10.1155/2016/2869083

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

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

Received 30 March 2016; Accepted 5 May 2016

Academic Editor: Giovanni P. Galdi

Copyright © 2016 Manal Badgaish et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Intracranial saccular aneurysm is a focal dilatation of the arterial wall that can be found in the circle of Willis which is a network of vessels at the base of the brain. The aneurysm which is a soft tissue interacts with a variety of flows including the blood as well as the cerebral spinal fluid. Based on the influence of various biomechanical factors, the growing aneurysm can potentially rupture that leads to either a neurological disorder or death. About 80% to 90% of ruptured aneurysms lead to death [1]. One of the invasive treatments for intracranial saccular aneurysms is to employ surgical procedures which may be associated with risk and death [2].

Over the last two decades, there have been several efforts to investigate the genesis of this disease and to develop a way for prediction of rupture through mathematical modeling [3–6]. Different groups of researchers had identified the elastodynamics of the arterial wall interaction with the blood flow to be the main reason for the rupture of an aneurysm [7–9]. A coupled fluid-structure model to understand the elastodynamics better was later studied more extensively [10–13]. These models introduced mathematical models of increasing complexity for intracranial saccular aneurysms that described the coupled interaction between blood, arterial wall, and cerebral spinal fluid (CSF). In [11], the CSF was modeled using simplified Navier-Stokes equations, whereas the arterial wall structure was modeled using a spring-mass system. A Fourier series was used to model the interaction between blood pressure and inner wall. While the model developed yielded good insight into understanding rupture, this model did not accommodate growth and remodeling effects of the soft-tissue wall.

There are three main constituents of the artery wall, namely, the elastin, the collagen, and the smooth muscle [14, 15]. The elastin is a stable protein and is considered the most load bearing element that functions as resistance to the formation of an aneurysm, whereas the collagen is the protein that is responsible for preventing rupture after formation of an aneurysm. The growth of the aneurysm is associated with deficiency of elastin and weakening of the artery wall [16]. Hence, elastin and collagen should be incorporated into the modeling of arterial wall in order to obtain an accurate biological model of the aneurysm that maybe can lead to better interpretation and prediction for this disease. This is one of the main contributions of this work. In the model we will consider in this paper we will assume elastin and collagen to be one-dimensional passive fibers and also, for simplicity, we will not consider the effects of the smooth muscle for this work.

In Section 2, we will describe the mathematical model that we will consider to solve a coupled fluid-structure problem. Section 3 discusses the nonlinear analysis and perturbation solution for the model. In Section 4, the linearization of the growth and remodeling model is considered and solved analytically using Laplace transformation. After that, some important computational results and comparisons are discussed in Section 5. Finally, we conclude by some biological interpretation of the results and suggestion for future work.

#### 2. Models and Background

The mathematical modeling of intracranial saccular aneurysm builds on models developed in [11, 12] which included a simple mathematical model of a thin-walled, spherical intracranial aneurysm surrounded by cerebral spinal fluid which is referred to as CSF (see Figure 1).