A mathematical model for the mechanics of saccular aneurysms.

We constructed and discussed a mathematical model of intracranial saccular aneurysms based on the static mechanics of hollow vessels and were able to focus on three variables that are fundamental to the process of enlargement and rupture of these lesions. They are blood pressure (P), wall strength (sigma), and total wall substance (VT), which, if assigned values of 150 mm Hg, 10 MPa, and 1.0 mm3, lead to model-predicted values of 8 mm for the diameter and 40 micron for the wall thickness for the critical geometry of aneurysmal rupture. These are quantitatively similar to published measurements. The model is based on the assumption of a uniform thin spherical shell for the saccular aneurysm. The interrelationship of the variables, expressed in the equation for critical size at rupture (dc) (i.e., dc = [4 sigma VT/(pi P)]1/3), draws attention to the need for quantitative studies on aneurysmal geometry and on the stereology of the structural fraction of the aneurysmal wall. We concluded that tissue recruitment from around the initial site or hypertrophy of the wall tissue is commonly involved in the aneurysmal process. We identify the paradox of elastic stiffness and stability, which are characteristic of autopsy specimens in the laboratory, in contrast to plastic behavior and irreversible strain, which are essential to the natural process of enlargement of saccular aneurysms.