Before our program, there are several algorithms to calculate the geometric volumes of protein. They fall into three categories. The first is 3D grid-based calculations which highly sensitive to orientation of the protein and include VOIDOO (Kleywegt, 1994), AVP [8], 3V [9], Voronoia [10]. The second category uses analytical methods and includes MSROLL [11] , VORLUME [12] and ALPHAVOL [13]. The third category includes calculations based on Delaunay triangulation such as VADAR [14] or Monte Carlo method such as MCVOL [15].
All the methods that rely on 3D grid are sensitive to position of protein. Compared to VOIDOO, AVP using separate probes to delineate accessible region and to detect voids, because AVP is more focus on analyzing void volumes in proteins.

VOIDOO: https://www.esrf.fr/exp_facilities/px_soft/usf/voidoo_man.html#H6 ;

AVP: despite of calculate protein volume, more focused on calculate the cavity volumes; "AVP unites the analysis of individual discrete voids and overall packing quality using a single algorithm. To achieve this, separate probes are used to delineate solvent accessible regions and to detect voids, thus resolving problems associated with using a single probe size."
3V: channel volume, discrete volume, 2 rolling probes to find internal volumes
Voronoia: Kristian Rother, Peter Werner Hildebrand, Andrean Goede, Bjoern Gruening, Robert Preissner, Voronoia: analyzing packing in protein structures, Nucleic Acids Research, Volume 37, Issue suppl_1, 1 January 2009, Pages D393–D395, https://doi.org/10.1093/nar/gkn769 ； construction of hyperboloid instead of planar interfaces, assigns individual radii for each atom; atomic-scale packing density analysis, from ; it can predit the protein structure

MSROLL: modeled as a static collection of hard spheres
VORLUME: Gauss’ divergence theorem
ALPHAVOL: within framework of alpha-shape method and relate discontinuities to combinatorial changes in the subcomplex of the Delaunay triangulation

VADAR: comprehensive web server for protein structure evaluation that both complements and adds to existing structure assessment programs. a compilation of >30 key structural parameters derived from 15 well-known algorithms or previously published techniques for quantitatively evaluating protein structures.
MCVOL: Monte Carlo method, The surface of the molecule is defined by rolling a probe sphere over the surface of the spheres

ProteinVolume : flood-fill algorithm:

some program are not providing molecular surface volume, but for our purpose Vmc is not important.
Most of the program are using probes to line the out外缘 of the protein, beside of the problem of size of probes, "the boundary separating protein and bulk solvent is highly dependent on the method used for the placement of the solvent molecules."

https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-015-0531-2#Sec1
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底层理论： https://www.annualreviews.org/doi/pdf/10.1146/annurev.bb.06.060177.001055

https://static-content.springer.com/esm/art%3A10.1186%2Fs12859-015-0531-2/MediaObjects/12859_2015_531_MOESM1_ESM.pdf