Measuring and comparing brain cortical surface area and other areal quantities

Abstract

Structural analysis of MRI data on the cortical surface usually focuses on cortical thickness. Cortical surface area, when considered, has been measured only over gross regions or approached indirectly via comparisons with a standard brain. Here we demonstrate that direct measurement and comparison of the surface area of the cerebral cortex at a fine scale is possible using mass conservative interpolation methods. We present a framework for analyses of the cortical surface area, as well as for any other measurement distributed across the cortex that is areal by nature. The method consists of the construction of a mesh representation of the cortex, registration to a common coordinate system and, crucially, interpolation using a pycnophylactic method. Statistical analysis of surface area is done with power-transformed data to address lognormality, and inference is done with permutation methods. We introduce the concept of facewise analysis, discuss its interpretation and potential applications.

Fig. 1

An example demonstrating differences in the nature of measurements. In this analogy, the depth of the soil is similar to brain cortical thickness, whereas the number of trees is similar to areal quantities distributed across the cortex. These areal quantities can be the surface area itself (in this case, the area of the terrain), but can also be any other measurement that is areal by nature (such as the number of trees).

Fig. 2

Diagram of the steps to analyze the cortical surface area. For clarity, the colors represent the convexity of the surface, as measured in the native geometry.

Fig. 3

(a) Areal interpolation between a source and a target face uses the overlapping area as a weighting factor. (b) For a given target face, each overlapping source face contributes an amount of areal quantity. This amount is determined by the proportion between each overlapping area (represented in different colors) and the area of the respective source face. (c) The interpolation is performed at multiple faces of the target surface, so that the amount of areal quantity assigned to a given source face is conservatively redistributed across one or more target faces.

Fig. 4

A study-specific template (target for the registration) caused less systematic accumulation of areal quantities across the brain when compared with a non-specific template. Using default parameters, areal accumulation was less pronounced and unrelated to sulcal patterns using Spherical Demons in comparison with FreeSurfer registration. Gains and losses refer to the area per face that would be expected for areal quantities being redistributed with no bias, i.e. the zero corresponds to the average total surface area of all subjects, divided by the number of faces.

Fig. 5

(a) The area of the cortical surface is not normally distributed (upper panels). Instead, it is lognormally distributed throughout most of the brain (middle panels). A Box–Cox transformation can further improve normality (lower panels). The same pattern is present without (left) or with (right) smoothing (FWHM = 10 mm). (b) Spatial distribution of the parameter λ across the brain. When λ approaches zero, the distribution is more lognormal. See the Supplemental Material for the other views of the brain and histograms for λ.

Fig. 6

Distribution of the uncorrected p-values of the Shapiro–Wilk normality test. For normally distributed data, 5% of these tests are always expected to be declared as not normal with a significance level of α = 0.05. Without transformation or smoothing, near 80% are found as not normal. Logarithmic and Box–Cox transformations render the data more normally distributed. Observe that the frequencies are shown in logscale. The dashed line (blue) is at the frequency that would be observed for uniformly distributed p-values.

Fig. 7

After barycentric interpolation of the coordinates in the surface of the sphere, a new, subject-specific retessellated model is constructed. Areas can be computed directly from the retessellated model and, once divided by the areas of the homologous vertices or faces of the reference brain, constitute the measurement of expansion/contraction.

Fig. 8

Average area (left panels) or expansion/contraction (right panels) per vertex, without (upper panels) and with smoothing (lower panels). Areal analyses and expansion/contraction differ across space. Smoothing has little global impact.

Fig. 9

(a) The sum of the area per vertex correlates well with the average across space of the expansion/contraction at each vertex (i.e. equivalent to a weighed sum considering each vertex as having the same initial area) for the 168 hemispheres analyzed. For the expansion/contraction, this is not the same as computing the ratio between the global surface area in native geometry and of the template, in which case, the result would be a perfectly straight line. The high correlation implies that the regional differences in general compensate each other to produce a similar global effect. (b) The correlation between average spatial maps across the 84 subjects, both hemispheres, is very poor between the methods. [Note that, for (b), attempts to simultaneously plot all the > 300 thousand vertices would not produce meaningful plots in a small space; for this reason only 5% of the vertices were randomly selected for plotting. The R² were computed from all vertices and, for both (a) and (b), the value corresponds to the goodness of a linear fit.].

Fig. 10

For each isolated vertex, the linear relationship between areal analyses and expansion/contraction is very high across subjects, being above R² = 0.90 virtually across the whole cortex.

Fig. 11

(a) The common grid can be a geodesic sphere produced from recursive subdivision of a regular icosahedron. At each iteration, the number of faces is quadrupled. (b) After the first iteration, however, the faces no longer have regular sizes, with the largest face being approximately 1.3 times larger than the smallest as n increases.

Fig. 12

Differences between presentation of facewise and vertexwise data can be observed in this zoomed portion of the mesh representation of the cortex. Vertices are dimensionless and, to display vertexwise data, the faces have to be colored using linear interpolation. This is not necessary for facewise data, which can be shown directly in the uniform colors that represent the underlying data. In either case, the presentation can be improved by using a shading model, such as Gouraud in this example. Although the vertexwise presentation may be visually more appealing, it contains only half the resolution of the facewise image.