## 4 Discussion
The prediction performance of 0.65 as the area under the receiver operator characteristic curve (AUC) for disease-free survival by using only radiomics was improved to 0.70 by integrating clinical variables into radiomics. To investigate the impact of clinical variables on the prediction performance of our developed models, tumor size as the T-value in the tumor, lymph node, metastasis (TNM) classification and clinical stage were incorporated into the CT-\(R\), CT-\(L\), DD-\(L\), DD-\(T\), DD-\(RT\), DD-\(LT\), and CD-\(RLT\) models as the clinical variables. The balanced C-indices of seven models combined with and without clinical variables are shown in Table 4. The balanced C-indices decreased in all models with the addition of clinical variables. In addition, there were no statistically significant differences between the treatment failure-free probability curves in all models. Although an integrated analysis of clinical variables was expected to enhance prediction accuracy, this was not achieved. Onepossible explanation is that the clinical variables used in this investigation were limited to only two that were available at this time. Therefore, enhancing prediction performance by integrating additional clinical variables, such as performance status score, human papillomavirus status, histological subtype, and others should be studied in the future. Although a DL model achieved the relatively high balanced C-index of 0.743, as shown in Table 3, its predictive performance did not significantly outperform the handcrafted models when the C-indices in the training and test datasets are interpreted separately. In addition, there was no statistically significant difference in the treatment failure-free probability curves in the test dataset. Potential causes for the DL model's limitations may include suboptimal network architecture and insufficiently tailored input data. In future work, we plan to explore more advanced or optimal network architecture [40] and refine the input data design. The Betti numbers in this study provide a novel topological perspective on both tumor morphology and dose distribution. Betti numbers (b0 and b2), which quantify the number of connected components and cavities, respectively, may capture aspects of tumor spatial heterogeneity that are not reflected on conventional radiomics or dosiomics features. For instance, a high b2 value in a tumor may indicate internal necrosis or cavitation, which has been associated with poor prognosis in certain cancer types [41]. Similarly, Betti numbers can capture the complex patterns of dose distribution, such as hot and cold spots, which may correspond to dose conformity or coverage that may potentially affect the probability of tumor control [42]. The topological features could serve as surrogate markers for biological and clinical characteristics, such as tumor aggressiveness, hypoxia, or response to radiation. Although further biological validation is required, the integration of topological data may enhance our understanding of spatial patterns in radiation oncology and support the development of more personalized treatment strategies. Local recurrence, locoregional recurrence, distant metastasis, and residual tumors were defined as treatment failures in this study, owing to the insufficient number of cases for each outcome. This study aimed to predict treatment failure in patients with local recurrence, locoregional recurrence, distant metastasis, or residual tumors. Treatment failure is primarily caused by tumor resistance. Resistant tumor cells can survive treatment (residual tumor), contribute to local recurrence or locoregional recurrence, or lead to the development of distant metastasis. Thus, our approach may be able to extract resistant tumor cells from the phenotype in medical images through a more detailed analysis in the future.