## 2 Materials and Methods
### Quantification of Shape Deformation  
To assess the degree of skull correction after surgery, it is necessary to quantify the expansion and reduction of the regional cranial bones between the preoperative and  
Figure 4: Deformation process of the mean normal skull surface model associated with preoperative and postoperative skull surface models in severe sagittal craniosynostosis. (**a**) Preoperative skull surface model, (**b**) postoperative skull surface model, (**c**) mean normal skull surface model with regional bone partitioning, (**d**) deformed mean normal surface skull model into a preoperative skull surface model, (**e**) redeformed deformed mean normal skull surface model into a postoperative skull surface model.  
postoperative skulls. Therefore, we propose an expansion and reduction indices using the positive and negative regional top distance indices and the positive and negative regional area ratio indices.  
First, the regional distance is the distance value representing the degree of correction that has been extended or reduced from the bones of each part of the skull before surgery to the skull after surgery, and the regional area ratio is the proportion of areas where correction is expanded or reduced in the bones of each part of the skull before surgery. The regional distances and the regional area ratio indices are calculated using the local deformation calculated during the deformable registration process. The local deformation (LD) value is calculated according to the magnitude of the displacement vector of the deformable registration in Equation (1), and the sign of the local deformation is determined by the direction of movement of the mean normal skull surface model that is deformed toward the inside or outside of the postoperative skull surface model.  
\[LD(i)=\left\{\begin{array}{cc}|V_{i}|&\text{, toward inside}\\ -1\times|V_{i}|&\text{, toward outside}\end{array}\right. \tag{1}\]  
where \(i\) is the point of the mean normal skull surface model and \(V_{i}\) is the magnitude of the displacement vector of the deformable registration.  
Second, to represent the degree of major deformation in the expansion and reduction measures of the regional cranial bone in the deformed mean-normal skull surface model, the positive and negative regional top distance indices (PTD and NTD) are calculated as the average values of the positive and negative top 20% regional distances in Equations (2) and (3), respectively.  
\[\text{PTD}=\frac{1}{N_{p20}}\sum_{i=0}^{N_{p20}}LD(i) \tag{2}\]  
\[\text{NTD}=\frac{1}{N_{n20}}\sum_{i=0}^{N_{n20}}LD(i) \tag{3}\]  
where \(LD(i)\) is the signed local deformation distance at point \(i\) and \(N_{p20}\) and \(N_{n20}\) are the number of points with positive and negative top 20% local deformation distances, respectively.  
To represent the degree of distribution of the expanded and reduced areas of the regional cranial bones in the deformed mean normal skull surface model, the positive and negative regional area ratio indices (PAR and NAR) are calculated as the ratio of areas with corresponding positive and negative local deformation distances in the regional cranial bone area in Equations (4) and (5), respectively.  
\[\text{PAR}=\frac{N_{p}}{N_{all}} \tag{4}\]  
\[\text{NTD}=\frac{N_{n}}{N_{all}} \tag{5}\]  
where \(N_{all}\) is the number of points in each cranial bone and \(N_{p}\) and \(N_{n}\) are the number of points with the positive and negative local deformation distances, respectively.  
Finally, to consider the major degree of expanded and reduced correction in the regional cranial bone and the area distribution in which the correction occurred, the expansion and reduction indices are calculated by multiplying the positive and negative regional top distance indices by the positive and negative regional area ratio indices, respectively.