Biomathematical models of intracranial aneurysms can provide qualitative and quantitative information on stages of aneurysm development through elucidation of biophysical interactions and phenomena. However, most current aneurysm models, based on Laplace's law, are renditions of static, linearly elastic spheres. The primary goal of this study is to: 1. develop a nonlinear constitutive quasi-static model and 2. derive an expression for the critical size/pressure of an aneurysm, with subsequent applications to clinical data. A constitutive model of an aneurysm, based on experimental data of tissue specimens available in the literature, was incorporated into a time-dependent set of equations describing the dynamic behavior of a saccular aneurysm in response to pulsatile blood flow. The set of differential equations was solved numerically, yielding mathematical expressions for aneurysm radius and pressure. This model was applied to clinical data obtained from 24 patients presenting with ruptured aneurysms. Aneurysm development and eventual rupture exhibited an inverse relationship between aneurysm size and blood pressure. In general, the model revealed that rupture becomes highly probable for an aneurysm diameter greater than 2.0 mm and a systemic blood pressure greater than 125 mmHg. However, an interesting observation was that the critical pressure demonstrated a minimal sensitivity to the critical radius, substantiating similar clinical and experimental observations that blood pressure was not correlated, to any degree, with aneurysm rupture. Undulations in the aneurysm wall, presented by irregular multilobulated morphologies, could play an important role in aneurysm rupture. However, due to the large variation in results, more extensive studies will be necessary for further evaluation and validation of this model.