Numerical values

1. Number of tetrahedra

We recall that the total number of tetrahedra in the ART volume for a resolution n=4 is: (24 * 8^n) = 98304, and the total number of tetrahedra for n=5 is: (24 * 8^n) = 786432.

After the HEAD labelling for n=4, i.e. the process used to find the subset H of tetrahedra belonging to the head among the whole tetrahedra, we found 27044 tetrahedra composing H. For n=5, we found 203745 tetrahedra composing H.

During the process used to label the tetrahedra in H with the anatomical tissues labels, for n=4, we found 8210 tetrahedra for the brain, 5647 tetrahedra for the skull and 13187 tetrahedra for the scalp.
For n=5, we found 82399 tetrahedra for the brain, 26453 tetrahedra for the skull and 94893 tetrahedra for the scalp.

We can see in the following table the different numerical values:

Number of tetrahedra / Resolution	n = 4	n = 5
Number of tetrahedra in the ART volume	98 304	786 432
Number of tetrahedra in the HEAD	27 044	203 745
Number of tetrahedra in the BRAIN	8 210	82 399
Number of tetrahedra in the SKULL	5 647	26 453
Number of tetrahedra in the SCALP	13 187	94 893

For a given ART volume with a resolution n, the different numbers of tetrahedra labelled with the anatomical tissues and their repartitions also depend on the chosen threshold.
2. Sizes of the tetrahedra

To build an ART volume, we apply successive subdivisions on an initial set of 24 tetrahedra that share a common vertex. This intitial set is called Geometric Constructor and is denoted by GC. We saw in section I. Introduction that this initial set contained the MRI segmented volume.

Here we discribe more precisely the construction of the GC, since it has an influence on the size of the tetrahedra composing the head.

First, we choose the center C of the segmented volume. We consider an unitary subdivision invariant tetrahedra Tu with egdes close to 1 and we build an unitary Geometric Constructor, denoted by GCu, that is the polyhedra composed of 24 Tu that share the vertex C. Then, GC is the polyhedra obtained by scaling GCu with a factor S (in mm) such that GC contains the MRI segmented volume.

The scale factor S is one parameter of the ART volume construction. The choice of S directly influences the size and the number of tetrahedra composing H: the higher is the parameter S, the bigger and the less numerous are the tetrahedra composing H. We used the value S = 160 in this site. We can compare the different numbers of tetrahedra for S = 160 and S = 128 in the following table:

Number of tetrahedra / n and S	n = 4		n = 5
Number of tetrahedra / n and S	S = 160	S = 128	S = 160	S = 128
Number of tetrahedra in the ART volume	98 304	98 304	786 432	786 432
Number of tetrahedra in the HEAD	27 044	50 808	203 745	383 457
Number of tetrahedra in the BRAIN	8 210	17 916	82 399	160 637
Number of tetrahedra in the SKULL	5 647	9 208	26 453	50 332
Number of tetrahedra in the SCALP	13 187	23 684	94 893	172 488

As we can see for n=4, the numbers of tetrahedra for the different anatomical tissues is around half less for S=160 than for S=128. The number of tetrahedra in the ART volume is obviously always the same for the different values of S.
For a given n, the size of the edges of the tetrahedra depends on the scale factor. Indeed, if we choose for example S=160, then the length of the edges of a tetrahedron in GC is close to 16cm; for n=4, the length of the edges of a tetrahedron in the ART volume is close to 1cm (see next table for more values).

n / S	S = 160 mm	S = 128 mm
n = 0	160 mm	128 mm
n = 1	80 mm	64 mm
n = 2	40 mm	32 mm
n = 3	20 mm	16 mm
n = 4	10 mm	8 mm
n = 5	5 mm	4 mm

Approximate sizes of the edges of the tetrahedra in the ART volume, according to n and S.

I. Introduction II. Topological tools III. Method to label the head IV. First results and remarks V. Results with n=5 VI. Numerical values VII. Visible Human with CSF

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