## Materials and Methods
### Labyrinth Skeletonization Method  
One of the possibilities for the analysis of the ducts' orientation is to assume that they follow a specific geometry, such as a circle, ellipse, or other 3D shape and then adjust the one that best describes the data. Here, we relaxed this restriction and obtained an arbitrary shape of the vestibular system,  
Figure 2: (**Top left**) original section from the MRI scan; (**Top right**) binarization using the same resolution; (**Bottom left**) binarization upscaling the original matrix; and (**Bottom right**) isovolume obtained using the upscaled matrix. and then analyzed how the characteristics of these reconstructed ducts depend on the specific details of the analysis. The shape of the ducts was obtained through a skeletonization [17] process of the binarized volume. This name encompasses a broad battery of different algorithms that generate a curve retaining the topology of the original 3D volume (Figure 3). The difference among the various algorithms depends on how this curve is defined. Skeletonization methods have been studied for decades and their results have been tested by many authors in different experimental and applied topics, such as virtual navigation, computer graphics animation, medical applications, data analysis, etc. [18; 19; 20; 21; 22]. Some methods are based on the definition of this line as the set of points that are equidistant to the surface (using a geometric distance that can be Euclidean or not). Other methods erode the original volume reducing the thickness until it collapses to a line (thinning methods). However, these two families are very sensitive to noise and to the roughness of the volume boundary. Here, we focused on a specific family called _Potential Field Methods_ that are less sensitive to these effects. The idea is that a fictitious repulsive and conservative force is associated to each one of the surface elements, and then a scalar potential field is created in the interior of the considered volume. This potential field will decrease when we move away from the surface and when we approach the central part of the volume. For any possible cross section of each canal, there will be at least one point inside where this potential reaches its minimum value. The skeleton defined through this algorithm is the line \(\vec{d}\) that connects the set of minima that will be recovered from any plane that chops a duct. From a practical point of view, we can define a vector force field from this potential. Any fictitious particle under the action of this force will be pushed to the skeleton and once it reaches this region it will restrict its dynamics to displacements along this line. We used an implemented MatLab (The MathWorks, Massachusetts, USA) toolbox named Volume Skeleton Toolbox (VSK) [23]. In this toolbox, the specifically used method is called _Generalized Potential Field method_. This script splits the volume into boundary elements (voxels with at least one empty neighbor voxel) and interior voxels. Then, a potential field \(V\) is generated (we used a spatial dependence with the fifth power \(V\sim 1/r^{5}\) of the distance to the surface \(r\) ). A dynamical system is defined where the acceleration of the fictitious particles depends on \(\vec{\Phi}=-\
abla V\). These particles will find (along the trajectory obtained connecting all these minima) some points where the dynamic stops (critical points). We can classify these critical points as local minima of the potential and saddle-points using the Jacobian of this function \(\vec{\Phi}\). The skeleton can be recovered simulating the evolution of points  
Figure 3: Transparent volume of volunteer S18 SCs. Inside the volume, each colored line represent the centerline of one duct: (**a**) the whole centerline is considered; and (**b**) the ampulla is excluded from the centerline. The sphere and diamonds correspond to the beginning and end of the retained trajectory (thick line) from the original centerline (thin line).