A nonlinear mathematical model for the development and rupture of intracranial fusiform aneurysms

Laplace's law, which describes a linear relation between the tension and the radius, is often used to characterize the mechanical response of the aneurysm wall to distending pressures. However, histopathological studies have confirmed that the wall of the fully developed aneurysm consists primarily of collagen and is subject to large increases in tension for small increases in the radius, i.e., a nonlinear relationship exists between the tension within the aneurysm wall and the radius. Thus, a nonlinear version of Laplace's law is proposed to accurately describe the development and rupture of a fusiform saccular aneurysm. The fusiform aneurysm was modelled as a thin-walled ellipsoidal shell with a major axis radius, Ra, minor axis radius, Rb, circumferential tension, S0, and meridional tension, S phi, with phi defining the angle from the surface normal. Using both linear and nonlinear models, differential expressions of the volume distensibility evaluated at 90 degrees were used to determine the critical radius of the aneurysm along the minor axis from S0 and S phi in terms of the following geometric and biophysical variables; A, elastic modulus of collagen; E, elastic modulus of the aneurysm (elastin and collagen); t, wall thickness; P, systolic pressure; and Ra. For typical physiological values of A = 2.8 MPa, E = 1.0 MPa, T = 40 microns, P = 150 mmHg, and Ra = 4Rb, the linear model yielded critical radii of 4.0 mm from S phi and 2.2 mm from S0. The resultant critical radius was 4.56 mm. Using the same values, the critical radii from the tension components of the nonlinear model were 3.5 mm from S phi and 1.9 mm from S0.(ABSTRACT TRUNCATED AT 250 WORDS)