Introduction

Although the cortical folding of mammalian brains varies greatly across species, it follows a scaling law, which describes the global relationship between cortical surface area and the square root of cortical thickness¹. This relationship is derived from a simple physical model, which minimizes the effective free energy associated with cortical shape¹. Specifically, the discovered relation is: A_tT^1/2 = kA_e^1.305 in which T is the average cortical thickness, A_t is the total surface area, A_e is the exposed cerebral hull surface area, and k is a free parameter¹. A recent study² further verified that the adult human brains at different ages largely conform to the theoretically predicted global relation: A_tT^1/2 = kA_e^5/4 by Mota¹. However, it is still unclear about the local relationship of these cortical properties, especially for the infant with rapid brain development during the first two years^3,4. To fill this knowledge gap, we explore the local relationship maps between surface area, cortical thickness, and cortical folding of 73 normal infants at 0, 1, and 2 years of age.

Methods

Totally 219 longitudinal MR images from 73 healthy infants at 0, 1, and 2 years of age were acquired on a Siemens head-only 3T scanner³. The cortical surfaces were reconstructed using the UNC Infant Pipeline⁵ and resampled as 163,842 vertices accordingly. Then, the cerebral hull surfaces were reconstructed and the vertex-to-vertex correspondences between cortical surfaces and cerebral hull surfaces were established. The vertex-wise cortical thickness was calculated as the average value of the minimum distance between inner and outer surfaces. The vertex-wise surface area was computed as the one third of the areas of all triangles connecting to this vertex. To study the local relationship between surface area, cortical thickness and cortical folding, for each cortical vertex we adopted the N-ring neighborhood on the resampled cortical surface and their corresponding regions on the cerebral hull surfaces, thus ensuring a meaningful comparison of anatomically corresponding regions across ages and subjects³. Then, in the local region of each vertex, the local features lT, lA_t, and lA_e were computed. Next, for each vertex, the relationship between lT, lA_t, and lA_e was established by linearly fitting the corresponding values from 73 subjects according to the following equation: log₁₀ (lA_t * lT^1/2) = s * log₁₀ lA_e + log₁₀ k. Here, the slope ‘s’ was calculated on each vertex at each age, thus the longitudinal maps of local relationship of surface area, cortical thickness, and cortical folding were created for infants. Meanwhile, for each subject, the mean lT, mean lA_t, and mean lA_e of all vertices were calculated, and then the 73 pairs of values were linearly fitted with the above equation.

Results

Fig. 1-a displays the average local relationship between x = log₁₀ lA_e and y = log₁₀ (lA_t * lT^1/2) of all 73 subjects and their fitting models at 0, 1, and 2 years of age. As we can see, an intrinsic linear fitting model exists at all three ages although there is obvious age-related difference. Specifically, the model is y = 1.5510 * x - 1.2662 (R² = 0.9491) at birth, y = 1.4408 * x - 0.8892 (R² = 0.9339) at year 1, and y = 1.3875 * x - 0.7010 (R² = 0.9464) at year 2. Both slope and offset decrease during the first two years. Fig. 1-b displays the distribution histogram of the slope ‘s’ of all the vertices at 0, 1, and 2 years of age. For each age, the slope values of more than 80% vertices locate in the range 0.84-1.36. Fig. 2 shows the longitudinal maps of local relationship of surface area, cortical thickness, and cortical folding in infants at 0, 1, and 2 years of age. The color of each vertex on the surface denotes the value of slope ‘s’. Clearly, the vertices in the frontal, inferior temporal and occipital cortices show large slope values, while those in the superior temporal and parietal cortices show small slope values. The mean slope values are 1.0523, 0.9929, 0.9915 at 0, 1, and 2 years, respectively. As we can see, the slope values significantly decrease in the first year and then slightly decrease in the second year.

Discussion and Conclusion

We have systematically explored the longitudinal local relationship between surface area, cortical thickness, and cortical folding in infants. We revealed that their local relationship is age-specific and region-specific. These findings are likely related to the underlying biological mechanisms and developmental trajectories of surface area, cortical thickness, and cortical folding. Future work will include interpreting the mechanisms of our findings and investigating whether neurodevelopmental disorders change this relationship.

Acknowledgements

This work was supported in part by NIH grants (MH100217, MH108914, MH107815, MH110274, MH070890, MH064065 and HD053000).