## 5 Discussion and Conclusions  
In this study, we propose G-UNETR++, an effective hybrid segmentation network for accurate liver segmentation using CT scans. We introduce gradient-enhanced encoders (G-Encoders) for learning 3D geometric features to achieve better performance in liver segmentation. To evaluate the performance of the proposed method, comprehensive ablation studies on the LiTS dataset were conducted. The results of the ablation studies \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **DSC** & **VOE** & **RAVD** & **ASSD** & **MSSD** \\ \hline FCN + ACM [35] & 0.9620 & - & - & 1.40 & 23.80 \\ U-Net [48] & 0.9500 & 0.0932 & 0.0108 & 2.85 & 59.15 \\ DRAUNet [8] & 0.9685 & 0.0618 & 0.0105 & 1.18 & - \\ Proposed & 0.9732 & 0.0520 & 0.0141 & 0.76 & 18.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of different models on Sliver07 dataset. Figure 8: Segmentation results of the proposed method over the LiTS (**upper** row), Sliver07 (**middle** row), and 3D-IRCADb (**bottom** row) datasets with one example from each dataset in different views (left: axial, middle: coronal, and right: sagittal). The green regions represent the ground truth, surrounded by red lines representing the segmentation results of our proposed method. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **DSC** & **VOE** & **RAVD** & **ASSD** & **MSSD** \\ \hline mU-Net [49] & 0.9601 & 0.0973 & 0.0038 & 3.11 & 9.20 \\ EDCNN [50] & 0.9520 & 0.0910 & 0.0700 & 1.43 & 19.37 \\ S-Net + T-Net + SEL & 0.9650 & 0.0960 & 0.0030 & 3.21 & - \\ + PRL [36] & 0.9660 & 0.0565 & 0.0023 & 2.61 & - \\ DefED-Net [37] & 0.9618 & 0.0721 & 0.0002 & 1.20 & - \\ HFRU-Net [39] & 0.9720 & 0.0570 & 0.0110 & 1.15 & 36.24 \\ RMAU-Net [7] & 0.9703 & 0.0517 & 0.0013 & - & - \\ DRAUNet [8] & 0.9741 & 0.0540 & 0.0018 & 0.76 & - \\ Proposed & 0.9750 & 0.0488 & 0.0206 & 0.59 & 12.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of different models on 3D-IRCADb dataset. showed that our G-Encoder, the proposed loss function, and our post-processing method are effective for liver segmentation. To further evaluate the performance of the proposed method, we compared our method with other state-of-the-art methods on the LiTS dataset. The comparison results showed that our method was superior to the other methods, achieving a DSC of 0.9738 for liver segmentation. To evaluate the generalizability of the proposed method, we tested our method with the other two non-training datasets, Sliver07 and 3D-IRACDb. The results showed that our method was superior to other methods, achieving DSC scores of 0.9732 and 0.9750 on the two datasets, respectively. These results demonstrated that our method shows strong generalizability over the different datasets collected under different conditions for liver segmentation. Though the proposed G-UNETR++ achieved higher accuracy in liver segmentation than the baseline model UNETR++, the two proposed G-Encoders increased the complexity level of our model in terms of the number of parameters (97.73M) and FLOPs (73.12G), compared with the number of parameters (42.96M) and FLOPs (47.98G) of UNETR++. However, our model achieved a 1% higher dice score than UNETR++ for liver segmentation. With a similar complexity level to TransUNet and UNETR, our model achieved a 3% higher dice score. From this aspect, the increase in the complexity level is worthwhile. Still, future research is needed to reduce the complexity level of G-UNETR++. One potential direction is to employ a multi-resolution approach for the second-order partial derivatives computation. Instead of calculating the second-order partial derivatives for the entire input volume, this method focuses only on important regions of interest (ROI). Such a strategy will significantly reduce computational costs and memory usage, making the model more efficient without compromising segmentation accuracy. Another direction lies in adaptively activating gradient-enhanced encoders based on the complexity of the input data.