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

Mathematical models of aneurysms are typically based on Laplace's law which defines a linear relation between the circumferential tension an d the radius. However, since the aneurysm wall is viscoelastic, a nonl inear 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 variable s at static equilibrium. A comparison is made between mathematical mod els of a spherical aneurysm based on linear and nonlinear forms of Lap lace'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 s ystolic pressure. The second is a 'modified' laplace's law which descr ibes a nonlinear power relation between the tension and the radius def ined by T = AR(P/2At) where A is the elastic modulus for collagen and t is the wall thickness. Differential expressions of these two relatio ns were used to describe the critical radius or the radius prior to an eurysm rupture. Using the standard Laplace's law, the critical radius was derived to be R(c) = 2Et/P where E is the elastic modulus of the a neurysm. 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 mu m , P = 150 mmHg, and A = 2.8 MPa, the critical radius is 4.0 mm using s he standard Laplace's law and 4.8 mm for the modified Laplace's law. I n conclusion, a biomathematical model has been developed based on a no nlinear expression of Laplace's law which integrates the quantitative influence of collagen in the tension of the aneurysm wall. The nonline ar model better describes the influence of biophysical variables on th e critical radius in comparison to the model based on the standard Lap lace's law. The critical radius from the modified Laplace's law more a ccurately predicts aneurysm rupture based on previously published clin ical observations.