Mathematical model of the rupture mechanism of intracranial saccular aneurysms through daughter aneurysm formation and growth

Abstract

Objectives: Daughter aneurysms have been strongly associated with saccular aneurysm rupture. We constructed a mathematical model to help explain this association as a possible hemodynamic mechanism for intracranial saccular aneurysm rupture.

Methods: Our model is based on the assumption that when an aneurysm reaches a state of imminent rupture, the weakest area of the aneurysm wall responds passively to a surge of intra-aneurysmal pressure by forming a daughter aneurysm that will be the site of the eventual rupture. The daughter and parent aneurysms were assumed to be spherical. Using mathematical modeling, the growth of the daughter aneurysm was observed. To obtain the change in tensile stress in the daughter aneurysm wall under constant pressure and changing geometry, the Law of Laplace was applied to the parent and the daughter aneurysms.

Results: The model reveals that the stress factor, i.e. tensile stress in the daughter aneurysm wall relative to the wall strength (rupture point), is dependent on two geometric parameters: the orifice factor (mu), which represents the relative size of the daughter aneurysm orifice radius to the parent aneurysm radius; and the aspect ratio (lambda), which represents the height-to-orifice ratio of the daughter aneurysm. As the daughter aneurysm develops, the stress factor first decreases to protect against rupture. Minimal stress is attained at an aspect ratio (lambda) of 0.577 regardless of the orifice factor. This is a relatively stable state. Further growth of the daughter aneurysm results in an increase of stress above the minimum, eventually leading to rupture at a stress factor of 1. A smaller orifice factor mu allows this aneurysm to grow to a higher aspect ratio lambda before rupture.

Discussion: Daughter aneurysm formation is a likely path to aneurysm rupture. The formation of a daughter aneurysm temporarily decreases the tensile stress within a parent aneurysm in which rupture is imminent, indicating a temporary protective role of daughter aneurysm development. Aneurysms harboring daughter aneurysms are at a more advanced stage of development, hence at a greater risk for rupture. The severity of the rupture risk can be estimated on the basis of daughter aneurysm geometry; aspect ratio lambda > 0.577 indicates a greater risk of rupture. Furthermore, daughter aneurysms with larger orifices are associated with a greater risk of rupture.