A NONLINEAR MATHEMATICAL-MODEL FOR THE DEVELOPMENT AND RUPTURE OF INTRACRANIAL FUSIFORM ANEURYSMS

Laplace's law, which describes a linear relation between the tension a nd the radius, is often used to characterize the mechanical response o f the aneurysm wall to distending pressures. However, histopathologica l studies have confirmed that the wall of the fully developed aneurysm consists primarily of collagen and is subject to large increases in t ension for small increases in the radius, i.e., a nonlinear relationsh ip 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, R(a), minor axis radius, R(b) circumferentia l tension, S-theta 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 9 0 degrees were used to determine the critical radius of the aneurysm a long the minor axis from S-theta and S-phi in terms of the following g eometric and biophysical variables: A, elastic modulus of collagen; E, elastic modulus of the aneurysm (elastin and collagen); t, wall thick ness; P, systolic pressure; and R(a). For typical physiological values of A = 2.8 MPa, E = 1.0 MPa, T = 40 mu m, P = 150 mmHg, and R(a) = 4R (b), the linear model yielded critical radii of 4.0 mm from S-phi and 2.2 mm from S-theta. The resultant critical radius was 456 mm. Using t he 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 S-theta. The r esultant critical radius was 3.98 mm. In conclusion, a nonlinear model has been developed for the fusiform aneurysm which accounts for the c ontribution of collagen through all stages of development and rupture and compared to the standard linear model. In comparison to the linear model, the nonlinear model provides for a slightly wider range of cri tical radii.