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

Abstract

Mathematical models of aneurysms are typically based on Laplace's law which defines a linear relation between the circumferential tension and the radius. However, since the aneurysm wall is viscoelastic, a nonlinear model was developed to characterize the development and rupture of intracranial spherical aneurysms within an arterial bifurcation and describes the aneurysm in terms of biophysical and geometric variables at static equilibrium. A comparison is made between mathematical models of a spherical aneurysm based on linear and nonlinear forms of Laplace's law. The first form is the standard Laplace's law which states that a linear relation exists between the circumferential tension, T, and the radius, R, of the aneurysm given by T = PR/2t where P is the systolic pressure. The second is a 'modified' Laplace's law which describes a nonlinear power relation between the tension and the radius defined by T = ARP/2At where A is the elastic modulus for collagen and t is the wall thickness. Differential expressions of these two relations were used to describe the critical radius or the radius prior to aneurysm rupture. Using the standard Laplace's law, the critical radius was derived to be Rc = 2Et/P where E is the elastic modulus of the aneurysm. The critical radius from the modified Laplace's law was R = [2Et/P]2At/P. Substituting typical values of E = 1.0 MPa, t = 40 microns, P = 150 mmHg, and A = 2.8 MPa, the critical radius is 4.0 mm using the standard Laplace's law and 4.8 mm for the modified Laplace's law.(ABSTRACT TRUNCATED AT 250 WORDS)