## 3 Results
### Quantitative Evaluation  
Various evaluation metrics are used to assess the segmentation performance of DL architectures. Most commonly, the Dice Similarity Coefficient (DSC) and Jaccard index (JC) are used to quantify the performance of the predicted segmentation area with the ground truth area. Additionally, Hausdorff Distance (HD) and Average Surface Distance (ASD) are used to quantify boundary similarity. Desirable segmentation results are associated with higher DSC and JC scores and lower HD and MSD. The described metrics are defined as follows:  
\[DSC=\frac{2|A\cap B|}{|A|+|B|} \tag{1}\]  
\[JC(A,B)=\frac{|A\cap B|}{|A\cup B|} \tag{2}\]  
\[HD(A,B)=\max\{\sup_{a\in A}\mathop{\mathrm{inf}}_{b\in B}d(a,b),\sup_{b\in B }\mathop{\mathrm{inf}}_{a\in A}d(b,a)\} \tag{3}\]  
\[ASD(A,B)=\frac{1}{|A|+|B|}\left(\sum_{a\in A}\min_{b\in B}d(a,b)+\sum_{b\in B }\min_{a\in A}d(b,a)\right) \tag{4}\]  
where \(A\) and \(B\) represent the ground truth segmentation area and the predicted segmentation area, respectively. \(|A\cap B|\) and \(|A\cap B|\) are the number of pixels in the intersection and union of \(A\) and \(B\), respectively, while \(|A|\) is the number of pixels in \(A\) and \(|B|\) is the number of elements in \(B\). \(d(a,b)\) denotes the distance between points \(a\) and \(b\), sup represents the supremum (or the least upper bound), and \(\inf\) represents the infimum (or the greatest lower bound). Table 2 provides a comprehensive comparison of various deep learning (DL) architectures used for prostate segmentation. It includes details about the type of data (2D or 3D), the number of patients involved, the number of images used for training and testing, and key performance metrics. The DSC values in the table range from 0.87 to 0.95, indicating a generally high level of accuracy across different methods. [PERSON] et al. [20,21] reported consistently high DSC scores of 0.93, while [PERSON] et al. [24] achieved the highest DSC of 0.95, suggesting superior segmentation accuracy with their method. These high DSC values reflect the effectiveness of the various DL architectures in accurately segmenting the prostate. The JC, which measures the similarity between the predicted and ground truth segmentations, was not reported by several studies. For those that did report JC values, the scores range from 0.82 to 0.93. [PERSON] et al. (2022) [33] and [PERSON] et al. (2024) [34] reported the highest JC score of 0.93, indicating strong agreement between their predicted and actual segmentations. Average Surface Distance values were provided by fewer studies, but the reported values generally remained below 1 mm, indicating precise boundary predictions. [PERSON] et al. (2020) [27] reported the lowest ASD of 0.10 mm, reflecting highly accurate boundary delineation. However, [PERSON] et al. (2022) [36] reported a higher ASD of 3.41 mm, suggesting a possible improvement in boundary accuracy. Hausdorff Distance values varied widely, ranging from 1.12 mm to 8.37 mm. [PERSON] et al. (2022) [22] reported a relatively high HD of 5.48 mm, while [PERSON] et al. (2019) [37] and [PERSON] et al. (2022) [39] reported even higher HD values of 8.37 mm and 5.89 mm, respectively. These results highlight the challenges in achieving precise boundary segmentation, specifically in the case of more complex or less clear imaging data. Comparing 2D and 3D methods, most 2D methods demonstrated high DSC scores, with notable performance by [PERSON] et al. [20] and [PERSON] et al. [28]. Two-dimensional methods tend to be less computationally intensive than three-dimensional methods, making them suitable for real-time clinical applications. However, 3D methods, such as those by [PERSON] et al. [22] and [PERSON] et al. (2022) [31], generally reported high DSC scores but showed variation in HD values. The advantage of 3D methods lies in their ability to capture volumetric information, which can enhance the accuracy of PV estimation, although they require more computational resources.