Vascular tortuosity: modeling using optimality analysis

Abstract

Although vascular tortuosity is a ubiquitous phenomenon, very few mathematical models exist to describe its shape. Given that the shape of tortuous vessel curves seems fairly uniform across great variations in anatomic substrata, it is hypothesized that tortuosity is is governed by physical principles. We present a mathematical model of vascular tortuosity based on optimality principles. The model minimizes average curvature per unit length; it produces a “sine‐generated” curve. Thirty‐four tortuous vessel segments are analyzed and compared to the results of the model. Curve shapes are characterized by the parameters L/R and L/λ, where L is curve length, λ is wavelength, and R is radius of curvature. These parameters are measured for each vessel segment and calculated for the corresponding theoretical curves. Comparison between measured and predicted values produces average percent errors of about 6%. This suggests that blood vessels obey optimality principles, and bend in such a way as to minimize average curvature. Vessels from a case of Fabry's disease were also analyzed, and deviated significantly from theoretical results, suggesting that the model can potentially be used to distinguish physiologic curvature from the abnormal tortuosity which occurs in disease states.