## 2 Materials and Methods
### VBQ Determination  
The \(VBQ\) score is determined from pixel intensity values of T1-weighted non-contrast MR images using the following expression:  
\[VBQ=\frac{SI_{L1-L5}}{SI_{CSF}}\enspace, \tag{9}\]  
where \(SI_{L1-L5}\) is the mean pixel signal intensity (SI) value of the \(L1\), \(L2\), \(L3\), \(L4\) and \(L5\) vertebrae, and \(SI_{CSF}\) is the mean signal intensity value of _CSF_. When drawing ROIs manually, a circular ROI was placed on the L1-L5 vertebral body, avoiding the lesion and posterior venous plexus. _CSF SI_ was measured near the L3 vertebra. If there was a complete obstruction of the posterior CSF region, the adjacent segment CSF was chosen [12]. Four different methods of automatic VBQ determination were investigated. For each method, the CSF region SI had to be extracted from the segmentation provided by the neural network using an additional algorithm, since the trained neural network model generates segmentation of the entire spinal canal and not just the CSF region. For this purpose, a connected threshold algorithm was chosen. In the first method (T1_VBQ), only T1-weighted data and the corresponding segmentation were used. The SI value of the CSF region was extracted in the following way: First, the point of local minimum was found in the segmentation of the spinal canal, with coordinates restricted to the vertical axis between L1 and L5. This local minimum was used as the starting point for the connected threshold algorithm. The lower limit of the threshold was chosen as the local minimum SI value and the upper limit was calculated using the following expression:  
\[Upper\ boundary=SI_{min}+\frac{SI_{max}-SI_{min}}{3}\enspace, \tag{10}\]  
where \(SI_{\text{min}}\) and \(SI_{\text{max}}\) are the minimum and maximum SI of the constrained spinal canal region, respectively. The connectivity value was set to 1. This was carried out for each slice of the segmented series where spinal canal segmentation was present. The second method (T1_VBQ_eroded) uses the same CSF extraction algorithm, but changes the volume of vertebral segmentation and thus the ROI used for VBQ determination. This was achieved using the erosion algorithm. In erosion, pixels that are non-zero and have a neighboring pixel with a value of 0 are removed from the segmentation masks, essentially removing the outer layer of pixels from each segmented region. This method of changing the vertebral ROI volume was chosen to avoid sharp changes in SI at the edge of the segmented vertebral body, which can negatively affect the VBQ value. In the third method (T2_VBQ), T2-weighted images and segmentation masks are used together with T1-weighted images. Since the T2-weighted MRI data of the spine have hyperintense SI in the CSF region, the CSF can be easily extracted using a segmented threshold algorithm. Similar to the first method, the local maximum point in the segmented region of the spinal canal in the T2-weighted image was used as the initial value for the connected threshold algorithm. The upper limit was set as the local maximum SI value and the lower limit as the maximum SI/1.5. The connectivity value was set to 1. The coordinates of this segmented region were then used to determine the SI in the CSF from the T1-weighted image of the same study (patient). T1- and T2-weighted images belonging to the same study do not necessarily match spatially, even if they were acquired sequentially as part of the same MRI protocol. This may be due to the movement of the patient during imaging or to the different parameter values of the T1 and T2 MR sequences, such as the number of slices or the spatial resolution. To address this, a co-registration algorithm implemented in the FMIRB Software Library (FSL) [25] was used together with a Python script to resize T1- and T2-weighted images together with their segmentation masks to the same size. The fourth method (T2_VBQ_eroded) uses the same CSF SI extraction algorithm as the third method, together with erosion of the vertebral segmentation region, as used in the second method.