Robust measures of three-dimensional vascular tortuosity based on the minimum curvature of approximating polynomial spline fits to the vessel mid-line

The clinical recognition of abnormal vascular tortuosity is important in the diagnosis of many diseases. This paper presents a novel approach to the quantification of vascular tortuosity, using robust metrics based on unit speed parameterizations of three-dimensional (3D) curvature. The use of approximating polynomial spline-fitting obviates the need for arbitrary filtering of mid-line data which is necessary with other tortuosity indices. The metrics were tested using both two-dimensional and three-dimensional synthesized images that mimicked clinically significant pathologies: two of the three metrics were scale invariant, additive, and produced tortuosity values tailored to be independent of the resolution of the imaging system. Our methodology is designed to explicitly handle the challenge of noisy data, and is largely tolerant of the inaccuracies in the mid-line extraction. While all the proposed metrics are sensitive to gently curved vessels, the rms curvature of the smoothest path was more effective in recognizing abnormalities involving high-frequency coiling such as occurs in malignant tumors. We have also indicated how values from two projection images, such as acquired in biplane angiography, can be combined to give an accurate approximation of the 3D value of this metric. Our proposed methodology is well-suited to automated detection and measurement, which are a prerequisite for clinical implementation.