|
|
| Line 1: |
Line 1: |
| {{SI}}
| |
| {{WikiDoc Cardiology Network Infobox}}
| |
| {{CMG}}
| |
|
| |
|
| [[Image:Nowitna river.jpg|thumb|A tortuous river ([[meander]] of [[Nowitna River]], [[Alaska]])]]
| |
| '''Tortuosity''' is a property of [[curve]] being [[wiktionary:tortuous|tortuous]] (twisted; having many turns). There have been several attempts to quantify this property. Tortuosity is commonly used to describe diffusion in porous media <ref>Epstein, N. (1989), On tortuosity and the tortuosity factor in flow and diffusion through porous media, Chem. Eng. Sci., 44(3), 777– 779. [http://dx.doi.org/10.1016/0009-2509(89)85053-5]</ref>. This concept is also used for porous media as soils and snow <ref>Kaempfer, T. U., M. Schneebeli, and S. A. Sokratov (2005), A microstructural approach to model heat transfer in snow, Geophys. Res. Lett., 32, L21503,[http://dx.doi.org/10.1029/2005GL023873]</ref>
| |
| ==Tortuosity in 2-D==
| |
|
| |
|
| Subjective estimation (sometimes aided by optometric grading scales<ref>Richard M. Pearson. Optometric Grading Scales for use in everyday practice. Optometry Today, Vol. 43, No. 20, 2003, ISSN 0268-5485 [http://www.optometry.co.uk/articles/docs/e022e939c23ddd43951691b84a1efa90_pearson20031017.pdf]</ref>) is often used.
| | #redirect:[[Angulation and tortuosity]] |
| | |
| The most simple mathematic method to estimate tortuosity is arc-chord ratio: ratio of the [[Arc length|length]] of the curve (''L'') to the distance between the ends of it (''C''):
| |
| | |
| :<math>\tau = \frac{L}{C}</math>
| |
| | |
| Arc-chord ratio equals 1 for a straight line and is infinite for a circle.
| |
| | |
| Another method, proposed in 1999<ref>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 [http://www.cs.sandia.gov/~wehart/Papers/1997/HarGolCotKubNel97-amia.ps.gz]</ref>, 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<ref>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 [http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=28608&arnumber=1279902&count=267&index=227]</ref> 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:
| |
| | |
| :<math>\tau = \frac{{N - 1}}{L} \cdot \sum\limits_{i = 1}^N {\left( {\frac{{L_i }}{{S_i }} - 1} \right)}</math>
| |
| | |
| In this case tortuosity of both straight line and circle is estimated to be 0.
| |
| | |
| In 1993<ref>M. Mächler, Very smooth nonparametric curve estimation by penalizing change of curvature, Technical Report 71, ETH Zurich, May 1993 [ftp://ftp.stat.math.ethz.ch/Research-Reports/71.ps.gz]</ref> 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:
| |
| | |
| :<math>\frac{d}{{dx}}\log \left( \kappa \right) = \frac{{\kappa'}}{\kappa}</math>
| |
| | |
| 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<ref>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]</ref>:
| |
| | |
| :<math>\tau = \frac{{\int\limits_{t_1 }^{t_2 } {\left( {\kappa'\left( t \right)} \right)^2 } dt}}{L}</math>
| |
| | |
| 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.
| |
| | |
| ==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 point]]s and integral of square of curvature, divided by length of the curve (curvature is estimated assuming that small segments of curve are planar) <ref>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 [http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1225850]</ref>. Another method used for quantifying tortuosity in 3D has been applied in 3D reconstructions of solid oxide fuel cell cathodes where the Euclidean distance sums of the centroids of a pore were divided by the length of the pore.<ref>Gostovic, D., et al., Three-dimensional reconstruction of porous LSCF cathodes. Electrochemical and Solid State Letters, 2007. 10(12): p. B214-B217. [http://scitation.aip.org/journals/doc/ESLEF6-ft/vol_10/iss_12/B214_1.html]</ref>
| |
| | |
| ==Applications of tortuosity==
| |
| | |
| Tortuosity of [[blood vessels]] (for example, [[Retina|retinal]] and [[Brain|cerebral]] blood vessels) is known to be used as a [[medical sign]].
| |
| | |
| In mathematics, [[cubic spline]]s minimize the [[functional]], equivalent to integral of square of curvature (approximating the curvature as the second derivative).
| |
| | |
| In many engineering domains dealing with mass transfer in porous materials, such as [[hydrogeology]] or [[heterogeneous catalysis]], the tortuosity refers to the ratio of the diffusivity in the free space to the diffusivity in the [[porous medium]]<ref>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 [http://staff.aist.go.jp/nakashima.yoshito/programs/rw2d.txt]</ref> (analogous to arc-chord ratio of path). Strictly speaking, however, the effective diffusivity is proportional to the reciprocal of the square of the geometrical tortuosity <ref> Gommes, C.J., Bons, A.-J., Blacher, S. Dunsmuir, J. and Tsou, A. (2009) Practical methods for measuring the tortuosity of porous materials from binary or gray-tone [[tomographic reconstruction]]s. American Institute of Chemical Engineering Journal, 55, 2000-2012 [http://www3.interscience.wiley.com/journal/122465353/abstract?CRETRY=1&SRETRY=0]</ref>
| |
| | |
| 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.
| |
| | |
| In [[analytical chemistry]] applied to [[polymer]]s and sometimes small molecules tortuosity is applied in [[Gel permeation chromatography]] (GPC) also known as Size Exclusion Chromatography (SEC). As with any [[chromatography]] it is used to separate [[mixture]]s. In the case of GPC the separation is based on [[molecular size]] and it works by the use of stationary media with appropriately dimensioned pores. The separation occurs because larger molecules take a shorter, less tortuous path and elute more quickly and smaller molecules can pass into the pores and take a longer, more tortuous path and elute later.
| |
| | |
| ==References==
| |
| <references />
| |
| | |
| [[Category:Differential geometry]]
| |
| [[Category:Porous media]]
| |
| [[Category:Riemannian geometry]]
| |
| [[Category:Multivariable calculus]]
| |
| [[Category:Curves]]
| |
| | |
| [[de:Tortuosität]]
| |
| [[es:Tortuosidad]]
| |
| [[ja:迂回]]
| |
| [[pl:Krętość kanałów porowych]]
| |
| [[ru:Извилистость]]
| |
| [[sv:Tortuositet]]
| |
