## 1 Introduction  
Head-and-neck squamous cell carcinoma (HNSCC) affects approximately 136,000 people worldwide annually [1]. The 5-year overall survival rate (5y-OS) for HNSCC is estimated to be approximately 60%. Furthermore, advanced HNSCC (stages III and IV) has a poor prognosis, with a 5y-OS of \(<\)50%. Radiotherapy (RT), combined with chemoradiotherapy (CRT), plays an important role in the definitive treatment of HNSCC. However, the 5y-OS remains low despite advances in treatment, owing to treatment failure. Recurrence (local, locoregional, and regional recurrence), distant metastasis, and residual tumors were included as treatment failures. Recurrence occurs in 16-52% of patients with head-and-neck squamous cell carcinoma (HNSCC) treated with definitive CRT [2, 3, 4, 5, 6, 7]. Therefore, the prediction of treatment failure in each patient with HNSCC prior to treatment planning can lead to more optimal treatment plans or selection of a better treatment approach. In recent years, radiomics and dosionics have been widely applied for the recurrence prediction of HNSCC [8; 9; 10], preoperative prediction of the malignancy of parotid gland tumors [11; 12], and prognosis prediction in patients with HNSCC [13; 14; 15]. Radiomics is a rapidly developing field that analyzes the relationships between clinical outcomes and the features extracted from medical images. In addition, dosionics is a radiomics approach, and the dose distribution simulated in RT treatment planning is analyzed instead of using medical images. Dose distribution is closely related to tumor control at RT. Tumors consist of heterogeneous tumor cells as well as resistant and sensitive cells. Therefore, the inadequate distribution of an insufficient dose to resistant cells can cause treatment failure. We previously reported recurrence prediction after RT in patients with HNSCC using dosionics [10]. This performance is better than that of the conventional radiomics approaches. Local binary pattern (LBP) is a conventional radiomics feature. LBP can describe the textural characteristics of the images by comparing the intensity of a center pixel in a small neighborhood with the intensity of its surrounding pixels. Some studies have applied LBP to radiomics analysis to evaluate tumor heterogeneity [14; 15]. Moreover, we reported the first dosionics analysis applying LBP to predict recurrence in patients with HNSCC [10]. The dosionics model performance was higher than the conventional radiomics approach. Therefore, a mixed radiomics and dosionics approach (radiodoisomics) by using the LBP could characterize the tumor heterogeneity, which is associated with treatment failure. Topological radiomics is a novel approach in radiomics research [12; 13]. It was analyzed using topological features, particularly Betti numbers, extracted from medical images. Betti numbers are topological invariants used in algebraic topology to summarize the connectivity and geometry of spaces. Betti numbers provide information on the number of connected components, holes, and higher-dimensional cavities in the data [16; 17; 18]. Figure 1 shows an example of CT and binary images of non-treatment failure (upper panel) and treatment failure cases (lower panel). In radiomics, Betti numbers can be used to analyze and quantify the topological properties of medical images. Higher Betti numbers indicate a greater number of holes or voids in the image, which may be associated with different pathological conditions or anatomical structures. In dosionics, topological features may represent inhomogeneous dose distributions related to treatment failure. Therefore, topological features based on the Betti number may reveal tumor properties. They may improve the predictive performance of tumor characterization, prognosis, and treatment response. [PERSON] et al. reported two-dimensional topological radiomics for predicting the malignancy of parotid gland tumors [12]. The topology-based approach showed a better performance (accuracy: 0.975) than the conventional approach (accuracy: 0.795).