Gyrification/Addendum

Quantitative measures of gyrification

Commonly used measures of the exent of cortical folding include:

• $L^{2}norms$:
  • $LN_{G}={\tfrac {1}{4\pi }}\textstyle {\sqrt {\sum _{A}K^{2}}}$, with $K=k_{1}k_{2}$ being the Gaussian curvature, computed from the two principal curvatures $k_{1}$ and $k_{2}$
  • $LN_{M}={\tfrac {1}{4\pi }}\textstyle \sum _{A}H^{2}$, with $H={\tfrac {1}{2}}(k_{1}+k_{2})$ being the Mean curvature and $A$ the area of the surface in question
• Folding index
  • $FI={\tfrac {1}{4\pi }}\textstyle \sum _{A}k^{\ddagger }$, with $k^{\ddagger }=|k_{1}|(|k_{1}|-|k_{2}|)$
• Intrinsic curvature index
  • $ICI={\tfrac {1}{4\pi }}\textstyle \sum _{A}K^{+}$, with $K^{+}$ being the positive Gaussian curvature
• Curvedness
  • $C={\sqrt {\tfrac {k_{1}^{2}+k_{2}^{2}}{2}}}$
• Sharpness of folding
  • $S=(k_{1}-k_{2})^{2}$
• Bending energy
  • $E_{b}=\int _{A}{(k_{1}+k_{2})}^{2}dA$
• Willmore energy
  • $E_{W}=\int _{A}{(k_{1}-k_{2})}^{2}dA=\int _{A}{H}^{2}dA-\int _{A}{K}dA$
• Gyrification index
  • $GI_{slice}(n)={\tfrac {A(n)_{outer}}{A(n)_{inner}}}$, with $n$ indicating the number of the slice, and $A(n)_{outer}$ and $A(n)_{inner}$ 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 sinuses.
  • $GI_{mesh}(n)={\tfrac {A(n)_{outer}}{A(n)_{inner}}}$, with $n$ indicating the number of the region, and $A(n)_{outer}$ and $A(n)_{inner}$ 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
  • $GWI={\tfrac {A_{gw}}{A_{gc}}}$, with $A_{gw}$ being the surface area of the boundary between gray matter and white matter
• White matter folding
  • $WMF={\tfrac {A_{gw}}{{V_{w}}^{2/3}}}$, with $V_{w}$ being the volume of the white matter
• Cortical complexity
• Fractal dimension
• Shape index
• Roundness
  • $RN={\tfrac {A}{^{3}{\sqrt {36\pi V^{2}}}}}$