## Materials and Methods
### Geometrical Parameters and Morphology Characterization  
It has been clearly stated above that an accurate definition of the SCs morphology is important. One crucial point is that the cupulae are part of the vestibular cavity producing an increase in the diameter of the SCs and should be withdrawn from the calculations. Here, we explored a systematic way to restrict the geometrical analysis to the ducts themselves, reducing progressively the length of the canal considered and testing the effects on the geometrical parameters. The procedure is as follows: once the original central lines of the ducts have been identified (Figure 3a), we characterized the minimum set of geometrical parameters that can describe their morphology. From these recovered centerlines, our first step was to obtain a plane that fits best (least-squares of the normal distance to the plane) of the set of points that define the trajectory. From the normal vector of this plane, we recovered the 3D spatial orientation of each canal, and we could obtain the relative orientation between each pair (the angle between these vectors). Other parameters that we obtained are the non-planarity (normal distance from the canal to the plane) and the ellipse that fits best (least-squares algebraic distance) to the projection of the canal on the plane. Depending on the cupula section considered as part of the semicircular ducts, the planes that represent the canals show slight differences. For each point of the original trajectory, we computed the distance from the surface of the volume \(r(\vec{d})\). In Figure 4, we present these values for all the canals and volunteers. As the different canals have slightly different sizes, we normalized the position along each duct with its corresponding length. The variability of the canal radii between different volunteers (gray lines) and along the canal position is larger than the uncertainty due to the algorithm (i.e., the size of the binarized voxel). \begin{table}
\begin{tabular}{c c} \hline
1 & **Begin** \\
2 & **Input:** RAW MRI Image; \\
3 & Perform an \(\times 3\) upscale; \\
4 & Perform a binarization (Intensity-based, threshold 20\%) ; \\
5 & Perform a boundary/internal voxels labeling (Neighbourhood based); \\
6 & Define saddle-points/local minima (Potential field); \\
7 & Skeleton curves generation ([PERSON] iterative method); \\
8 & Reunify redundant points/curves; \\
9 & **Output:** centerline curves; \\
10 & **end** \\ \hline \end{tabular}
\end{table} 
Table 2: Pseudo-code of labyrinth skeletonization method based on potential field. These distances were retained as an indicator of the diameter of the duct along the centerline and we refer to them as radius, as each section can be assimilated to a cylinder. The cross-section of the canal defined by a plane perpendicular to the skeleton has an arbitrary shape, in most cases a convex shape different from a circle. Here, we used the generalization of the definition of diameter as the largest distance between two tangent straight lines at opposite sides of the shape boundary, and the radius is the shortest distance from the centerline to the surface. The local diameter \(T\) of the duct varies along the trajectory \(\vec{d}\) and corresponds to \(T(\vec{d})\gtrsim 2\,r(\vec{d})\). We defined a threshold for these radii to determine the ampulae region: all points above that threshold were assumed to be part of the copula of the channel and were removed from the calculation. We repeated the geometrical analysis described above determining the planes, ellipses, non-planarities and angles. Once this step was finished, we slightly decreased the threshold and checked its effect on the new parameters. The main advantage of this process is that we could sequentially determine the effects of the excluded area in the SCs definition without subjectivity due to the action of the operator. Table 3 shows a schematic description of the SCC morphological characterization code.