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

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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 = i2Et/P]2At/p. Substituting typical values of E = 1.0 MPa, t = 40 ¼m, 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. In conclusion, a biomathematical model has been developed based on a nonlinear expression of Laplace’s law which integrates the quantitative influence of collagen in the tension of the aneurysm wall. The nonlinear model better describes the influence of biophysical variables on the critical radius in comparison to the model based on the standard Laplace’s law. The critical radius from the modified Laplace’s law more accurately predicts aneurysm rupture based on previously published clinical observations. [Neurol Res 1994; 16: 376-384]

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