Tortuosity

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Tortuosity is a property of curve being tortuous (twisted; having many turns). There have been several attempts to quantify this property.

1 Tortuosity in 2-D
2 Tortuosity in 3-D
3 Applications of tortuosity
4 References

[edit] Tortuosity in 2-D

Subjective estimation (sometimes aided by optometric grading scales^[1]) is often used.

The most simple mathematic method to estimate tortuosity is arc-chord ratio: ratio of the length of the curve (L) to the distance between the ends of it (C):

$\tau = \frac{L}{C}$

Arc-chord ratio equals 1 for a straight line and is infinite for a circle.

Another method, proposed in 1999^[2], is to estimate the tortuosity as integral of square (or module) of curvature. Dividing the result by length of curve or chord has also been tried.

In 2002 several Italian scientists^[3] proposed one more method. At first, the curve is divided into several (N) parts with constant sign of curvature (using hysteresis to decrease sensitivity to noise). Then the arc-chord ratio for each part is found and the tortuosity is estimated by:

$\tau = \frac{{N - 1}}{L} \cdot \sum\limits_{i = 1}^N {\left( {\frac{{L_i }}{{S_i }} - 1} \right)}$

In this case tortuosity of both straight line and circle is estimated to be 0.

In 1993^[4] Swiss mathematician Martin Mächler proposed an analogy: it’s relatively easy to drive a bicycle or a car in a trajectory with a constant curvature (an arc of a circle), but it’s much harder to drive where curvature changes. This would imply that roughness (or tortuosity) could be measured by relative change of curvature. In this case the proposed "local" measure was derivative of logarithm of curvature:

$\frac{d}{{dx}}\log \left( \kappa \right) = \frac{{\kappa'}}{\kappa}$

However, in this case tortuosity of a straight line is left undefined.

In 2005 it was proposed to measure tortuosity by an integral of square of derivative of curvature, divided by the length of a curve^[5]:

$\tau = \frac{{\int\limits_{t_1 }^{t_2 } {\left( {\kappa'\left( t \right)} \right)^2 } dt}}{L}$

In this case tortuosity of both straight line and circle is estimated to be 0.

In most of these methods digital filters and approximation by splines can be used to decrease sensitivity to noise.

[edit] Tortuosity in 3-D

Usually subjective estimation is used. However, several ways to adapt methods estimating tortuosity in 2-D have also been tried. The methods include arc-chord ratio, arc-chord ratio divided by number of inflection points and integral of square of curvature, divided by length of the curve (curvature is estimated assuming that small segments of curve are planar) ^[6].

[edit] Applications of tortuosity

Tortuosity of blood vessels (for example, retinal and cerebral blood vessels) is known to be used as a medical sign.

In mathematics, cubic splines minimize the functional, equivalent to integral of square of curvature (approximating the curvature as the second derivative).

In hydrogeology, the tortuosity refers to the ratio of the diffusivity in the free space to the diffusivity in the porous medium^[7] (analogous to arc-chord ratio of path).

In acoustics and following initial works by Maurice Anthony Biot in 1956, the tortuosity is used to describe sound propagation in fluid-saturated porous media. In such media, when frequency of the sound wave is high enough, the effect of viscous drag force between the solid and the fluid can be ignored. In this case, velocity of sound propagation in the fluid in the pores is non-dispersive and compared with the value of the velocity of sound in the free fluid is reduced by a ratio equal to the square root of the tortuosity. This has been used for a number of applications including the study of materials for acoustic isolation, and for oil prospection using acoustics means.

[edit] References

^ Richard M. Pearson. Optometric Grading Scales for use in everyday practice. Optometry Today, Vol. 43, No. 20, 2003, ISSN 0268-5485
^ William E. Hart, Michael Goldbaum, Brad Cote, Paul Kube, Mark R. Nelson. Automated measurement of retinal vascular tortuosity. International Journal of Medical Informatics, Vol. 53, No. 2-3, p. 239-252, 1999
^ Enrico Grisan, Marco Foracchia, Alfredo Ruggeri. A novel method for automatic evaluation of retinal vessel tortuosity. Proceedings of the 25th Annual International Conference of the IEEE EMBS, Cancun, Mexico, 2003
^ M. Mächler, Very smooth nonparametric curve estimation by penalizing change of curvature, Technical Report 71, ETH Zurich, May 1993
^ Patasius, M.; Marozas, V.; Lukosevicius, A.; Jegelevicius, D.. Evaluation of tortuosity of eye blood vessels using the integral of square of derivative of curvature // EMBEC'05: proceedings of the 3rd IFMBE European Medical and Biological Engineering Conference, November 20 - 25, 2005, Prague. - ISSN 1727-1983. - Prague. - 2005, Vol. 11, p. [1-4]
^ E. Bullitt, G. Gerig, S. M. Pizer, Weili Lin, S. R. Aylward. Measuring tortuosity of the intracerebral vasculature from MRA images. IEEE Transactions on Medical Imaging, Volume 22, Issue 9, Sept. 2003, p. 1163 - 1171
^ Watanabe, Y. and Nakashima, Y. (2001) Two-dimensional random walk program for the calculation of the tortuosity of porous media. Journal of Groundwater Hydrology, 43, 13-22