Vascular tortuosity: a mathematical modeling perspective

Although vascular tortuosity is a ubiquitous phenomenon, almost no mathematical models exist to describe its shape. Given that the shape of tortuous vessel curves seems fairly uniform across orders of magnitude of vessel size and across vast differences in anatomic substrata, it is hypothesized that the shape of tortuosity is not purely random but rather is governed by physical principles. We present a mathematical model of tortuosity based on optimality principles, and show how this model can potentially be used to distinguish physiologic tortuosity from abnormal tortuosity which may exist in disease states. Using the calculus of variations, a model of tortuosity has been developed which minimizes average curvature per unit length. The model is tested against curves in normal vessels and in diseased vessels in a case of Fabry's disease. It is found that the theoretical model provides a good fit for normal vessel tortuosity. This suggests that blood vessels obey optimality principles, and curve in such a way as to minimize average curvature. The model may also be able to distinguish physiologic tortuosity from abnormal tortuosity found in disease states.