2.3. Computing similarity matrices of cortical growth patterns

To define the growth patterns of the fetal cerebral cortex, ideally, we should use longitudinal fetal MRI data. This is, however, very difficult to obtain due to both ethical and practical issues. Therefore, as an alternative, we leveraged healthy fetuses in a cross‐sectional study, with each fetal subject only having one time point, to perform growth‐pattern‐based cortical parcellation. Specifically, we sorted all fetal cortical surfaces by incremental gestational age and then constructed the growth trajectories of cortical properties for each vertex. Herein, we adopted the surface area, as the convoluted cerebral cortex is achieved predominately by an increase in the surface area of a smooth sheet rather than its thickness, or other cortical properties (Rakic, 1988; Rakic et al., 2009). However, our method is generic and can also work on other cortical properties, for example, cortical thickness and local gyrification index, as long as their computation is reliable. Thus, our goal is to create a population‐level cortical parcellation map, which will be used to equip our constructed fetal cortical atlases, based on the growth patterns of surface area in the fetal brain. To this end, we first computed the similarities of growth patterns between each pair of vertices on the cortical surface. To comprehensively capture the similarities of growth patterns, we constructed two complementary similarity matrices S₁ and S₂, based on (a) growth trajectories of surface area and (b) growth correlation profiles of surface area, respectively.

Specifically, we defined the first similarity matrix S₁ by considering the growth trajectory of surface area at each surface vertex as a feature vector F₁. Between each pair of vertices i and j, we computed the Pearson's correlation coefficient p of their growth trajectories and obtained their similarity as follows.

Here, N is the total number of vertices on the cortical surface (N = 2562 × 2, considering both hemispheres in our case), and vectors F₁(i) and F₁(j) are the growth trajectories of surface area of vertex i and j, respectively. S₁ ranges from 0 to 1. Intuitively, high correlations between two vertices indicate high similarities of growth patterns. However, this similarity definition is inherently linear (low‐order), thus ignoring the complex and high‐order similarity of growth patterns.

To address this issue, we defined the second similarity matrix S₂ to capture the complex similarity of cortical growth patterns among vertices. First, we uniformly sampled 320 vertices from both cortical hemispheres as reference points, marked by the small yellow balls in Figure ​Figure2a.2a. Then, for each vertex, we calculated the Pearson's correlation coefficient between its growth trajectory of surface area and that of each reference point. In this way, for each vertex, we constructed a growth correlation profile as a new feature vector F₂(i), representing the correlation of growth trajectories between this vertex and each reference point. We then computed the similarity matrix S₂ based on the growth correlation profiles of vertices as:

Intuitively, two vertices with a high correlation of their growth correlation profiles indicate a high similarity of their growth patterns. Thus, S₂, based on “correlations of correlations”, captures more complex and high‐order nonlinear similarity in growth patterns.