## 2 Classification of Breast Cancer by Histological, Molecular, and Morphological Characteristics
### Histopathological Classification of Breast Cancer  
Histologically, tumors are broadly classified as either invasive carcinomas or in situ carcinomas, depending on whether malignant cells have spread from breast lobules or ducts into the surrounding stroma at the time of diagnosis [12]. Figure 1: Classifications of breast cancer by distribution, histology, and machine learning methods: (**a**) techniques in machine learning for predicting various forms of breast cancer [13], reproduced with permission from [PERSON] et al., (2020), Creative Common Attribution 4.0; (**b**) classification of breast cancer subtypes [14], reproduced with permission from [PERSON] et al. (2023), Creative Common Attribution 4.0; (**c**) histological subtypes and distribution rates [15], reproduced with permission from [PERSON] et al., (2023), Creative Common Attribution 4.0; (**d**) classification of breast cancer types by immune staining of tumor tissue [16], reproduced with permission from [PERSON] et al., (2020), Creative Common Attribution 4.0. Ductal carcinoma in situ (DCIS) is the most prevalent form of preinvasive breast cancer, with about 10% to 30% of cases progressing to invasive cancer, as there are insufficient prognostic bioindicators for predicting the development of invasive or metastatic disease [10]. In the last 2 decades, there has been a notable rise in the incidence of ductal carcinoma in situ, constituting 20-25% of newly identified cases of breast cancer [11]. The widespread adoption of mammographic screening in numerous countries is a major factor contributing to this upsurge. The rising trend is evident across various age groups and all genders, underscoring the need for additional considerations. As the incidence of DCIS is increasing in developed nations alongside a general rise in life expectancy, a growing number of elderly women will face a DCIS diagnosis, prompting questions about the feasibility of reducing local-regional therapy. Common risk factors for male breast cancer, including DCIS, involve aging, hyperestrogenism, and a positive family history [12]. While the precise lifetime risk of transitioning from DCIS to invasive breast cancer remains uncertain, it is believed to be less than 50% [12]. This study systematically examines authors focused on two distinct subgroups: male patients and older women with concomitant conditions [12]. Invasive ductal carcinoma (IDC), which accounts for 60-75% of cases, and invasive lobular carcinoma (ILC), which makes up 10-15% of tumors [16], are the two most frequent types of invasive carcinomas. This category of carcinomas is characterized by heterogeneity and is further classified based on cell shape. An uncommon phenomenon, invasive carcinoma with neuroendocrine characteristics makes for 2-5% of all invasive carcinomas of the breast. With varying proportions of neuroendocrine markers, it shares several histological characteristics with other neuroendocrine tumors of the bronchopulmonary system, pancreas, and digestive tract. Immunohistochemistry characters, which are usually not employed to detect breast cancers, are utilized to validate the diagnosis of neuroendocrine tumors in the breast. As a result, the precise prevalence is yet unknown. Studies suggest that invasive carcinomas, including solid papillary carcinoma, invasive ductal carcinoma, and mucinous carcinoma, are frequently associated with neuroendocrine tumors, observed in up to 30% of breast cancer cases [17]. There are several subtypes of invasive ductal carcinoma, each with distinct characteristics. * Medullary carcinoma is characterized by its slow growth, featuring soft and fleshy tumors that bear a resemblance to the medulla of the brain. This subtype accounts for less than 1% of all breast cancers [18];
* Tubular carcinoma is a rare histological subtype of invasive ductal carcinoma, constituting 1% to 5% of all invasive breast cancers. These tumors exhibit slow growth and are characterized by cancer cells with tube-like structures [19].