View source for Gyrification/Addendum

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==Quantitative measures of gyrification==
Commonly used measures of the exent of cortical folding include<ref name=Rodriguez-carranza2008>{{citation
 | last1 = Rodriguez-Carranza | first1 = C.E.
 | last2 = Mukherjee | first2 = P.
 | last3 = Vigneron | first3 = D.
 | last4 = Barkovich | first4 = J.
 | last5 = Studholme | first5 = C.
 | year = 2008
 | title = A framework for in vivo quantification of regional brain folding in premature neonates
 | journal = Neuroimage
 | volume = 41
 | pages = 462
 | doi = 10.1016/j.neuroimage.2008.01.008
 | url = http://linkinghub.elsevier.com/retrieve/pii/S1053811908000311
}}</ref><ref name=Pienaar2008>{{citation
 | last1 = Pienaar | first1 = R.
 | last2 = Fischl | first2 = B.
 | last3 = Caviness | first3 = V.
 | last4 = Makris | first4 = N.
 | last5 = Grant | first5 = P.E.
 | year = 2008
 | title = A methodology for analyzing curvature in the developing brain from preterm to adult
 | journal = International Journal of Imaging Systems and Technology
 | volume = 18
 | issue = 1
 | pages = 42–68
 | doi = 10.1002/ima.20138
 | url = http://www3.interscience.wiley.com/journal/119877321/abstract
}}</ref>:
*[[L2 norm|<math> L^2 norms</math>]]:
**<math>LN_G = \tfrac{1}{4\pi} \textstyle \sqrt{\sum_A K^2}</math>, with <math>K = k_1 k_2</math> being the [[Gaussian curvature]], computed from the two [[principal curvature]]s <math>k_1</math> and <math>k_2</math>
**<math>LN_M =\tfrac{1}{4\pi} \textstyle \sum_A H^2</math>, with <math>H=\tfrac{1}{2}(k_1 + k_2)</math> being the [[Mean curvature]] and <math>A</math> the area of the surface in question
*Folding index
**<math>FI =\tfrac{1}{4\pi} \textstyle \sum_A k^{\ddagger}</math>, with <math>k^{\ddagger}=|k_1|(|k_1|-|k_2|)</math> 
*Intrinsic curvature index
**<math>ICI =\tfrac{1}{4\pi} \textstyle \sum_A K^+</math>, with <math>K^+</math> being the positive Gaussian curvature
*Curvedness
**<math>C =\sqrt{\tfrac{k_1^2+k_2^2}{2}}</math>
*Sharpness of folding
**<math>S =(k_1-k_2)^2</math>
*Bending energy
**<math>E_b =\int_A{(k_1+k_2)}^2dA</math>
*Willmore energy
**<math>E_W =\int_A{(k_1-k_2)}^2dA =\int_A{H}^2dA - \int_A{K}dA </math>
*Gyrification index
**<math>GI_{slice} (n) =\tfrac{A(n)_{outer}}{A(n)_{inner}}</math>, with <math>n</math> indicating the number of the slice, and <math>A(n)_{outer}</math> and <math>A(n)_{inner}</math> being the outer and inner cortical [[contour]] in that slice. Anatomically, the inner contour can be thought of as representing the [[pia mater]], the outer one the [[arachnoid mater]]. The latter correspondence is rough, since the arachnoid also encloses [[venous sinuse]]s.
**<math>GI_{mesh} (n) =\tfrac{A(n)_{outer}}{A(n)_{inner}}</math>, with <math>n</math> indicating the number of the region, and <math>A(n)_{outer}</math> and <math>A(n)_{inner}</math> being the outer and inner cortical [[surface area]] in that region. The anatomical correspondences apply equally to the slice-based and regional definitions.
*Gyrification-White index
**<math>GWI =\tfrac{A_{gw}}{A_{gc}}</math>, with <math>A_{gw}</math> being the surface area of the boundary between gray matter and white matter 
*White matter folding
**<math>WMF =\tfrac{A_{gw}}{{V_w}^{2/3}}</math>, with <math>V_w</math> being the volume of the white matter
*Cortical complexity
*Fractal dimension
*Shape index
*Roundness
**<math>RN =\tfrac{A}{^3\sqrt{36 \pi V^2}}</math>

==References==
{{reflist}}