## 5 Discussion
Our proposed method is superior to other methods in terms of DSC, 95 HD, and ASD. The DSC values for liver and tumor segmentation reached \(0.966\pm 0.022\) and \(0.865\pm 0.026\), 95 HD reached \(3.62\pm 1.91\) and \(4.12\pm 1.55\), and ASD reached \(0.92\pm 0.28\) and \(1.02\pm 0.32\). The visualization of the segmentation results shows that compared with the baseline model, the proposed model can detect small tumors better, and the segmentation results of the target edge region are smoother. Several works have explored the application of distance sign maps to medical images. [PERSON] et al. [58] converted the segmentation task to predicting SDM. They introduced an approximate Heaviside function to train the model by simultaneously predicting SDM and segmentation mask so that the segmentation results have better smoothness and shape continuity. [PERSON] et al. [63] embed the global geometric information of objects into the learning framework through the classic geodesic active contours (GAC) and propose a level set function (LSF) regression network, which not only predicts the segmentation mask but also minimize the GAC energy function. [PERSON] et al. [64] incorporated SDM into the semi-supervised segmentation task to enforce geometric shape constraints in the segmentation output. A multitask deep network is proposed to predict semantic segmentation jointly and SDM of object surfaces. Based on this, [PERSON] et al. [65] transformed the SDM into an approximate segmentation map via a differentiable task conversion layer. They introduced dual-task consistency regularization between the SDM map and the directly predicted segmentation map. However, it has yet to be found that the predicted distance sign map and segmentation results are applied to the liver tumor segmentation of CE-MRI images. This study is the first to explore this. Obtaining high-quality annotated images for HCC patients is a challenging endeavor, as deep-learning models need big datasets to realize their full potential. To alleviate the manual labeling burden for training data, we employ SSL methods by designing a pretraining task focused on liver and tumor segmentation. By collecting A significant volume of unlabeled data sourced from hospitals, the model can learn meaningful visual representations from this extensive dataset before transitioning to supervised learning. Experimental results show that our self-supervised pretraining task significantly boosts the accuracy of the final segmentation model. Specifically, without self-supervised learning, the DSC scores for liver and tumor segmentation are 96.15% and 86.18%, respectively, with 95 HD values of 3.76 mm and 4.21 mm and ASD values of 0.97 mm and 1.09 mm. Afterapplying self-supervised learning, the DSC scores remain unchanged at 96.62% and 86.47%, while the HD95 values improve to 3.62 mm and 4.12 mm, and the ASD values reduce to 0.92 mm and 1.02 mm. Despite the strong performance of our method on the test set, several limitations remain in this study. The most significant limitation is that the test set comprises only 32 patients, all from a single medical center. This relatively small and homogeneous dataset may not capture the variability in imaging protocols, scanner types, and patient populations that exist across different institutions. For example, in our study, all enrolled cases were patients diagnosed with hepatocellular carcinoma (HCC), which accounts for approximately 80-90% of all liver cancer cases. Other types of primary liver tumors, such as intrahepatic cholangiocarcinoma, combined hepatocellular-cholangiocarcinoma, and hepatic angiosarcoma, were not included in our dataset. As a result, there may be potential dataset biases, and the current results may not fully generalize to broader clinical settings. To address this issue, multicenter validation is essential to evaluate the robustness and applicability of the proposed method in real-world clinical practice.