. 2019 Mar 13;9:4329. doi: 10.1038/s41598-019-40437-5

Table 5.

Radiomic shape feature formulas.

Feature	Description	Formula
Volume	Compute the enclosed volume of the object of interest. The enclosed volume is evaluated by triangulation (i.e. dividing the surface into connected triangles)	Green-Ostrogradski formula: $∭_{V} \vec{\nabla} \vec{F} .dV = \iint_{S} \vec{F} .dS$ where F corresponds to the vector field deduced from the triangulation
Surface area¹⁷	Area of the surface encompassing the volume of interest, calculated by triangulation	$A = \sum_{i = 1}^{N} \frac{1}{2} \| a_{i} b_{i} \times a_{i} c_{i} \|$
Surface-to-volume ratio¹⁷	Ratio of surface to volume	$S u r f a c e t o v o l u m e r a t i o = \frac{A}{V}$
Compactness1¹⁷	Describes how much the shape of a tumor resembles that of a sphere/can be encompassed by a sphere Compactness of a sphere = 1	$c o m p a c t n e s s 1 = \frac{V}{\sqrt{π .} A^{2 / 3}}$
Compactness2¹⁷		$c o m p a c t n e s s 2 = 36. π . \frac{V^{2}}{A^{3}}$
Compactness3⁸		$c o m p a c t n e s s 3 = \frac{V^{1 / 3} . {(36 π)}^{1 / 6}}{\sqrt{A}}$
Spherical disproportion¹⁷	The ratio of the surface area of the tumor to the surface area of a sphere with the same volume as the tumor	$S p h e r i c a l d i s p r o p o r t i o n = \frac{A}{4 π R^{2}}$
Sphericity¹⁷	Measure of the roundness or spherical nature of the tumor, where the sphericity of a sphere is the maximum value of 1	$S p h e r i c i t y = \frac{{(π)}^{1 / 3} {(6 V)}^{2 / 3}}{A}$
Fractional concavity³⁴	The ratio between the surface of the convex hull encompassing the tumor, and the actual surface of the tumor.	$F c c_{-} 3 D = \frac{S u r f a c e o f t h e c o n v e x h u l l}{A}$

A: area, V: volume, R: radius.