# How do we measure thickness, area, and volume of the cerebral cortex?

There are various ways one could estimate morphometric parameters of the cortex, such as its thickness, area, and volume. For example, it is possible to use voxelwise partial volume effects using volume-based representations of the brain, such as in voxel-based morphometry (VBM), in which estimates per voxel become available. Volume-based representations also allow for estimates of thickness, as suggested, for example, by Hutton et al. (2004), or from a surface representation of the cortex, in which it can be measured as a form of distance between the mesh that represents the pia mater (the pial surface) and the mesh that represents the interface between gray and white matter (the white surface).

Here we focus on the surface-based representation as that offers advantages over volume-based representations (Van Essen et al., 1998). Software such as FreeSurfer uses magnetic resonance images to initially construct the white surface. Once that surface has been produced, a copy of it can be offset outwards until tissue contrast in the magnetic resonance image is maximal, which indicates the location of the pial surface. This procedure ensures that both white and pial surfaces have the same topology, with each face and each vertex of the white surface having their matching pair in the pial. This convenience facilitates the computations indicated below.

## Cortical surface area

For a triangular face $ABC$ of the surface representation, with vertex coordinates $\mathbf{a}=[x_A \; y_A \; z_A]'$, $\mathbf{b}=[x_B \; y_B \; z_B]'$, and $\mathbf{c}=[x_C \; y_C \; z_C]'$, the area is $|\mathbf{u} \times \mathbf{v}|/2$, where $\mathbf{u} = \mathbf{a}-\mathbf{c}$, $\mathbf{v} = \mathbf{b}-\mathbf{c}$, $\times$ represents the cross product, and the bars $|\;|$ represent the vector norm. Even though such area per face (i.e., facewise) can be used in subsequent steps, most software packages can only deal with values assigned to each vertex (i.e., vertexwise). Conversion from facewise to vertexwise is achieved by assigning to each vertex one-third of the sum of the areas of all faces that meet at that vertex (Winkler et al., 2012).

## Cortical thickness

The thickness at each vertex is computed as the average of two distances (Fischl and Dale, 2000; Greve and Fischl, 2018): the first is the distance from each white surface vertex to their corresponding closest point on the pial surface (not necessarily at a pial vertex); the second is the distance from the corresponding pial vertex to the closest point on the white surface (again, not necessarily at a vertex). Other methods are possible, however, see table below (adapted from Lerch and Evans, 2005):

Method Reference
Distance solved using the Laplace’s equation. Jones et al. (2000)
Distance between corresponding vertices. MacDonald et al. (2000)
Distance to the nearest point in the other surface. MacDonald et al. (2000)
Distance to the nearest point in the other surface, computed for both surfaces, then averaged. Fischl and Dale (2000)
Distance along the normal. MacDonald et al. (2000)
Distance along the iteratively computed normal. Lerch and Evans (2005)

## Cortical volume

### Product method

If the area of either of these surfaces is known, or if the area of a mid-surface, i.e., the surface running half-distance between pial and white surfaces is known, an estimate of the volume can be obtained by multiplying, at each vertex, area by thickness. This procedure is still problematic in that it underestimates the volume of tissue that is external to the convexity of the surface, and overestimates volume that is internal to it; both cases are undesirable, and cannot be solved by merely resorting to using an intermediate surface as the mid-surface. Figure 1: A diagram in two dimensions of the problem of measuring the cortical volume. If volume is computed using the product method (a), considerable amount of tissue is left unmeasured in the gyri, or measured repeatedly in sulci. The problem is minimised, but not solved, with the use of the mid-surface. In the analytic method (b), vertex coordinates are used to compute the volume of tissue between matching faces of white and pial surfaces, leaving no tissue under- or over-represented.

### Analytic method

In Winkler et al. (2018) we propose a different approach to measure volume. Instead of computing the product of thickness and area, we note that any pair of matching faces can be used to define an irregular polyhedron, of which all six coordinates are known from the surface geometry. This polyhedron is an oblique truncated triangular pyramid, which can be perfectly divided into three irregular tetrahedra, which do not overlap, nor leave gaps. Figure 2: A 3D diagram with the proposed solution to measure the cortical volume. In the surface representation, the cortex is limited internally by the white and externally by the pial surface (a). These two surfaces have matching vertices that can be used to delineate an oblique truncated triangular pyramid (b) and (c). The six vertices of this pyramid can be used to define three tetrahedra, the volumes of which are computed analytically (d).

From the coordinates of the vertices of these tetrahedra, their volumes can be computed analytically, then added together, viz.:

1. For a given face $A_w B_w C_w$ in the white surface, and its corresponding face $A_p B_p C_p$ in the pial surface, define an oblique truncated triangular pyramid.
2. Split this truncated pyramid into three tetrahedra, defined as: $\begin{array}{lcllllll} T_1 &=& (&A_w,&B_w,&C_w,&A_p&)\\ T_2 &=& (&A_p,&B_p,&C_p,&B_w&)\\ T_3 &=& (&A_p,&C_p,&C_w,&B_w&) \end{array}$
3. For each such tetrahedra, let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ represent its four vertices in terms of coordinates $[x\;y\;z]'$. Compute the volume as $|\mathbf{u}\cdot(\mathbf{v} \times \mathbf{w})|/6$, where $\mathbf{u} = \mathbf{a}-\mathbf{d}$, $\mathbf{v} = \mathbf{b}-\mathbf{d}$, $\mathbf{w} = \mathbf{c}-\mathbf{d}$, the symbol $\times$ represents the cross product, $\cdot$ represents the dot product, and the bars $|\;|$ represent the vector norm.

No error other than what is intrinsic to the placement of these surfaces is introduced. The resulting volume can be assigned to each vertex in a similar way as conversion from facewise area to vertexwise area. The above method is the default in FreeSurfer 6.0.0.

## Is volume at all useful?

Given that volume of the cortex is, ultimately, determined by area and thickness, and these are known to be influenced in general by different factors (Panizzon et al, 2009; Winkler et al, 2010), why would anyone bother in even measuring volume? The answer is that not all factors that can affect the cortex will affect exclusively thickness or area. For example, an infectious process, or the development of a tumor, have potential to affect both. Volume is a way to assess the effects of such non-specific factors on the cortex. However, even in that case there are better alternatives available, namely, the non-parametric combination (NPC) of thickness and area. This use of NPC will be discussed in a future post here in the blog.