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Building 4D neonatal cortical surface atlases using Wasserstein barycenter1China Jiliang University, Hangzhou, China, 2University of North Carolina at Chapel Hill, Chapel Hill, NC, United States
Synopsis
Spatiotemporal (4D) neonatal cortical surface atlases are important tools for understanding the dynamic early brain development. To better preserve the sharpness and clarity of cortical folding patterns on surface atlases, we propose to compute the Wasserstein barycenter under the Wasserstein distance metric, for the construction of 4D neonatal surface atlases at each week from 39 to 44 postmenstrual weeks, based on a large-scale dataset with 764 neonates. Our atlases show sharper and more geometrically-faithful cortical folding patterns than the atlases built by the state-of-the-art method, thus leading to boosted accuracy for spatial normalization and facilitating early brain development studies.
Introduction
Spatiotemporal (4D) neonatal cortical surface atlases with densely sampled ages are important tools for understanding the dynamic early brain development. However, the current atlases cannot preserve sharp cortical folding pattern in some regions with large inter-subject variability, e.g., the middle frontal cortex. To obtain sharper folding patterns, we propose to build cortical surface atlases by computing the Wasserstein barycenter1, which represents a geometrically faithful population mean under the Wasserstein distance2 metric. Comparing to the direct vertex-wise Euclidean average, the Wasserstein distance takes into account the alignment of spatial distribution of cortical attributes, thus is more robust to potential registration errors during atlas building. By using this advanced method, 4D neonatal cortical surface atlases with multiple cortical attributes are constructed at each week, from 39 to 44 postmenstrual weeks, based on 764 neonates.Methods
T2-weighted brain MR images were acquired from 764 neonates from 39 to 44 postmenstrual weeks. All images were processed following the infant cortical surface pipeline3. For each hemisphere, the inner cortical surface was mapped onto a sphere by minimizing geometric distortion4. The cortical correspondences among subjects were established using the group-wise surface registration5 and then each surface was resampled using the same mesh tessellation.
As the Wasserstein barycenter is a geometrically and physically more meaningful average than the Euclidean average1, it is therefore particularly suitable for building cortical surface atlases, even with the presence of the potential registration errors. Specifically, atlases are built using Wasserstein distance following 5 steps. 1) For each vertex, the spherical patch centered at this vertex was localized from each subject, and its cortical folding attribute map x (with the size of d) was extracted. 2) Each extracted attribute map x was normalized, so that it is nonnegative and its L1 norm is one, thus following a probabilistic distribution. 3) The Wasserstein barycenter of the attribute maps of spherical patches from all subjects at this vertex was computed. Specifically, the Wasserstein barycenter is the optimal distribution that has the minimal distance to the distributions from each spherical patch centered at this vertex under the Wasserstein metric. To get a unique solution, we use an entropy regularized Wasserstein distance between two distributions x and y, which is defined as W(x,y)= min(tr(MT* N)+aH(M)|M*1d=x, MT*1d=y). The symbol 'tr()' stands for the trace of the product of two matrices, '*' represents the cross product and 1d represents the all-one vector with d dimension. The second term is the entropic penalty weighted by a>0, and H(M)= sum(M.* log(M)) with the convention of 0log0=0, '.*' denotes the element-wise multiplication and 'sum()' sums all elements of matrix. The cost matrix N is the adjacent matrix of each vertex within the patch according to the Euclidean distance of their spatial location. 4) The Wasserstein barycenter of the attribute maps was rescaled using the minimum value and scaling factor used in step 2. 5) The final cortical folding attribute at this vertex is computed as the average of its estimated values on all associated patches.
Results
Fig. 1 shows the constructed 4D neonatal cortical surface atlases with the color-coded average convexity and mean curvature at each week on both (a) spherical surfaces and (b) average inner surfaces. As can be seen, the cortex has a remarkable non-uniform development from 39 to 44 weeks, especially in the parietal region, zoomed for better inspection in (c). To show the effectiveness of our method, we compared our results with the atlases generated using two other methods in Fig. 2. The first one (denoted as ‘Average’) is the simple Euclidean average, and the second one is the spherical patch-based sparse representation (denoted as ‘Sparse’) with the state-of-the-art performance6. By visually comparing the curvature pattern on these three atlases at 41 postmenstrual weeks, we can see that our atlas preserves the sharpest folding patterns.Conclusion
In this paper, we proposed a spherical patch-based optimal transport averaging method to construct 4D neonatal cortical surface atlases from 39 to 44 postmenstrual weeks, based on a large cohort of neonates with 764 subjects, thus better characterizing the dynamic cortical development during early postnatal weeks. Benefit from the consideration of geometric information of cortical attribute distributions, our constructed 4D atlases better preserve sharp cortical folding than the state-of-the-art method, thus leading to better performance for spatial normalization of subjects. These 4D neonatal cortical surface atlases will be released to the public to facilitate early brain development studies.Acknowledgements
This work was supported in part by NIH grants (MH100217, MH108914, MH107815, MH110274, MH070890, MH064065, HD053000, MH109773, MH116225 and MH117943), and Zhejiang Natural Science Foundation Project under grant LQ18A010003.References
[1] Bonneel, N., et al., Wasserstein barycentric coordinates: histogram regression using optimal transport, ACM Trans. Graph., 2016; 35(4): 71:1-71:10.
[2]. Solomon, J., et al., Convolutional wasserstein distances: Efficient optimal transportation on geometric domains, ACM Transactions on Graphics, 2015; 34(4): 66:1-66:11.
[3]. Li, G., L. Wang, F. Shi, et al., Construction of 4D high-definition cortical surface atlases of infants: Methods and applications. Medical image analysis, 2015; 25(1):22-36.
[4]. Fischl, B., et al.: "Cortical surface-based analysis: II: inflation, flattening, and a surface-based coordinate system", Neuroimage, 1999; 9(2):195-207.
[5]. Yeo, B.T., et al.: "Spherical demons: fast diffeomorphic landmark-free surface registration", IEEE Trans. Med. Imaging, 2010; 29 (3): 650-668.
[6]. Wu, Z., et al.: "Construction of spatiotemporal neonatal cortical surface atlases using a large-scale Dataset". IEEE ISBI, 2018, 1056-1059.
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