Table 1 Overview of tortuosity measures.

From: Accuracy of vascular tortuosity measures using computational modelling

Measure

Symbol

Formula

Previous work which used these tortuosity measures

Tortuosity index

\({\varvec{\tau}}\)

\(\frac{{\varvec{L}}}{{\varvec{C}}}\)

1,2,30

Total absolute-curvature*

\({{\varvec{\kappa}}}_{{\varvec{t}}{\varvec{a}}}\)

\({\int }_{{{\varvec{t}}}_{1}}^{{{\varvec{t}}}_{2}}{\varvec{\kappa}}({\varvec{t}})\boldsymbol{ }{\varvec{d}}{\varvec{t}}\)

31,39,40

Total squared-curvature*

\({{\varvec{\kappa}}}_{tr}\)

\({\int }_{{{\varvec{t}}}_{1}}^{{{\varvec{t}}}_{2}}{{\varvec{\kappa}}}^{2}({\varvec{t}})\boldsymbol{ }{\varvec{d}}{\varvec{t}}\)

31,41

Average absolute-curvature

\({{\varvec{\kappa}}}_{{\varvec{a}}}\)

\(\frac{{\int }_{{{\varvec{t}}}_{1}}^{{{\varvec{t}}}_{2}}{\varvec{\kappa}}({\varvec{t}})\boldsymbol{ }{\varvec{d}}{\varvec{t}}}{L}\)

31

RMS-curvature

\({{\varvec{\kappa}}}_{r}\)

\(\sqrt{\frac{{\int }_{{{\varvec{t}}}_{1}}^{{{\varvec{t}}}_{2}}{{\varvec{\kappa}}}^{2}({\varvec{t}})\boldsymbol{ }{\varvec{d}}{\varvec{t}}}{L}}\)

31

Average squared-derivative-curvature

\({{\varvec{\kappa}}}_{{\varvec{d}}}\)

\(\frac{{\int }_{{{\varvec{t}}}_{1}}^{{{\varvec{t}}}_{2}}{(\frac{{\varvec{d}}{\varvec{\kappa}}({\varvec{t}})}{{\varvec{d}}{\varvec{t}}})}^{2}\boldsymbol{ }{\varvec{d}}{\varvec{t}}}{{\varvec{L}}}\)

35

  1. L = length of vessel, C = length of chord between vessel ends, \(\kappa\) = curvature.
  2. *The total absolute-curvature and the total squared curvature are not considered in this study, since these metrics are not scale invariant and dependent on the arc length of the vessels.
Back to article page