  Measures of vascular tortuosity--how curved and twisted a vessel is--are associated with a variety of vascular diseases. Consequently, measurements of vessel tortuosity that are accurate and comparable across modality, resolution, and size are greatly needed. Yet in practice, precise and consistent measurements are problematic--mismeasurements, inability to calculate, or contradictory and inconsistent measurements occur within and across studies. Here, we present a new method of measuring vessel tortuosity that ensures improved accuracy. Our method relies on numerical integration of the Frenet-Serret equations. By reconstructing the three-dimensional vessel coordinates from tortuosity measurements, we explain how to identify and use a minimally-sufficient sampling rate based on vessel radius while avoiding errors associated with oversampling and overfitting. Our work identifies a key failing in current practices of filtering asymptotic measurements and highlights inconsistencies and redundancies between existing tortuosity metrics. We demonstrate our method by applying it to manually constructed vessel phantoms with known measures of tortuousity, and 9,000 vessels from medical image data spanning human cerebral, coronary, and pulmonary vascular trees, and the carotid, abdominal, renal, and iliac arteries. 
Improving blood vessel tortuosity measurements via highly sampled   numerical integration of the Frenet-Serret equations