Fractal dimension in human cortical surface: multiple regression analysis with cortical thickness, sulcal depth, and folding area

Abstract

Fractal dimension (FD) has been widely used to provide a quantitative description of structural complexity in the cerebral cortex. FD is an extremely compact measure of shape complexity, condensing all details into a single numeric value. We interpreted the variation of the FD in the cortical surface of normal controls through multiple regression analysis with cortical thickness, sulcal depth, and folding area related to cortical complexity. We used a cortical surface showing a reliable representation of folded gyri and manually parcellated it into frontal, parietal, temporal, and occipital regions for regional analysis. In both hemispheres the mean cortical thickness and folding area showed significant combination effects on cortical complexity and accounted for about 50% of its variance. The folding area was significant in accounting for the FD of the cortical surface, with positive coefficients in both hemispheres and several lobe regions, while sulcal depth was significant only in the left temporal region. The results may suggest that human cortex develops a complex structure through the thinning of cortical thickness and by increasing the frequency of folds and the convolution of gyral shape rather than by deepening sulcal regions. Through correlation analysis of FD with IQ and the number of years of education, the results showed that a complex shape of the cortical surface has a significant relationship with intelligence and education. Our findings may indicate the structural characteristics that are revealed in the cerebral cortex when the FD in human brain is increased, and provide important information about brain development.

Figure 1

Estimation of FD using artificial data. The original model is folded in confined space like a brain (FD = 2.164) (a). The model with deeper folds than the original model, but representing the same frequency of folds and convolution, shows an increased FD (FD = 2.246) (b). The model of a more convoluted shape (c) and more frequency of folds (d) with the same depth compared to the original model also shows an increased FD, which is the same value of the deeper folding model (FD = 2.246). In the human brain, the depth, frequency, and convolution of gyral shape could affect the FD of the cortical surface integratedly.

Figure 2

Procedure for image preprocessing, cortical surface extraction, and manual lobe parcellation.

Figure 3

Scatterplot of logN(r) vs. log(1/r) for cortical surfaces of right hemisphere and temporal lobe of one subject. This figure shows data with artificial shifts in the vertical direction for visual assessment. Each numerical value in the graph indicates estimated fractal dimension that is the slope of the fitted line in the gray colored range showing linearity. Solid line is fitted for right hemisphere and dashed line for right temporal lobe. Solid line shows higher slope than dashed line, indicating higher FD value. Note that r is the length of box size in the grid and N(r) is the number of grid boxes occupied by one or more vertices of cortical surface.

Figure 4

Folding area measured from outer cortical surface (a) and smoothed surface (b). Folding area represents the area of folded regions in cortical surface, which consist of vertices whose mean curvature is negative. Folding area index represents the ratio of the folding area to the whole surface area. The bluish‐green color in the surface model represents the regions with a negative mean curvature, included to the measurement of folding area. The surfaces in the first column are outer cortical surface and the smoothed surface. The surfaces in the second column are inflated surfaces for visualization of mean curvature. The surfaces in the third column are inner cortical surfaces for observing folding patterns. Mean curvature in the 3‐D triangular model is sensitive to local geometrical small changes of the outer cortical surface (a). The cortical surface was smoothed geometrically to represent the folding patterns of the cortical surface with a mean curvature (b). Examples of brains exhibit high and low folding area in the left hemispheres. Folding area indices are shown below each cortical surface model (c). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 5

Scatterplots of the FD against each of the independent variables (cortical thickness, length‐weighted average sulcal depth, and folding area) for the significant results (y: FD, x₁: cortical thickness, x₂: length‐weighted average sulcal depth, x₃: folding area). Cortical thickness and folding area are significant variables for the regression model and shows a strong relationship with FD in both hemispheres (Left: y = –0.04 x₁ +0.923 x₃ +2.210, Right: y = –0.037 x₁ +0.767 x₃ +2.290) (a). In the left temporal lobe, folding area and length weighted average sulcal depth are selected as significant variables with positive correlation (Left: y = 0.003 x₂ +0.496 x₃ +1.954) (b). Folding area is also included in the regression model showing positive correlation in the left frontal lobe (Left: y = 1.848 x₃ +2.070) (c) and left/right parietal lobe (Left: y = 1.824 x₃ +1.922, Right: y = 3.028 x₃ +1.902) (d). Red marker and fitting line are for right hemisphere and blue for left (y: FD, x₁: cortical thickness, x₂: length‐weighted average sulcal depth, x₃: folding area). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 6

Scatterplot of FD vs. IQ (a) and the number of years of education (b) for both hemispheric cortical surfaces. The graphs show the positive correlation of FD with IQ in the right hemisphere and the number of years of education in the left and right hemispheres. The correlation with the FD and IQ in the left hemisphere is positive but not statistically significant. Red marker and fitting line are for right hemisphere and blue for left. (*P < 0.05, **P < 0.01, EDU: number of years of education). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]