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Abstract
Introduction/Purpose Although vascular tortuosity is a ubiquitous phenomenon, no mathematical models exist to describe its shape. Given that the shape of tortuous vessel curves seems fairly uniform across great variations in vessel size and anatomic substrata, it is hypothesized that tortuosity is not purely random, but is governed by physical principles. We develop a mathematical model based on optimality principles, and show how it can potentially be used to distinguish physiologic from abnormal tortuosity.
Materials/Methods A model is developed which minimizes average curvature per unit length; it produces a “sine-generated” curve. Thirty-four tortuous vessel segments (from retinal, superficial temporal, coronary and splanchnic vessels) are analyzed and compared to the results of the model. Curve shapes are characterized by the parameters L/R and L/l, where L is curve length, l is wavelength, and R is radius of curvature. These parameters are measured for each vessel segment and calculated for the corresponding theoretical curves. The accuracy of the model is compared with that of a standard sine curve in trying to model vessel tortuosity. Also, six retinal vessel segments from a case of Fabry's disease (which produces abnormal tortuosity) are analyzed and compared to the model.
Results Comparison between measured and predicted values produces average percent errors of 5.8% in L/R and 5.5% in L/l. Also, sine-generated curves produce a significantly better fit than standard sine curves (p=0.001). For the case of Fabry's disease, there is a significant deviation from the theoretical model, with average percent errors of 21% in L/l and 33% in L/R.
Conclusions 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. This makes physical sense, as it would minimize shear stress against vessel walls and the energy needed to accelerate blood through the vessel. Although anecdotal, the case of Fabry's disease differed significantly from theoretical results, suggesting that these vessels deviate from the normal pattern of tortuosity. The model may be useful in distinguishing physiologic tortuosity from the abnormal tortuosity which may occur in disease states.
