## 1 Introduction
### MWR Method for Detecting Breast Cancer  
Mass-produced medical device RTM-01-RES with modifications (manufactured by RES, Ltd., Russia, Moscow [48], see images of device in Figure 1) is the basis of medical diagnostics of both diseases of the breast [12; 13] and other organs [4; 8; 56]. Our measurement data for temperature distributions for breast were obtained with this device, which provides an accuracy of approximately \(\pm 0.2\)\({}^{\circ}\)C [32]. The application of MWR is always based on a combined use with IR thermography, which gives surface skin temperature distribution. Analysis of these two spatial temperature distributions underlies breast cancer diagnosis. Cancer formations are an additional local source of heat inside tissue (Figure 4) and lead to characteristic changes in temperature distribution \(T_{IR}\) and \(T_{B}\). Figure 4: Dependence of tumor specific heat release \(Q^{(an)}\) on doubling time for 128 breasts, where tumor diameter is 0.4 cm \(\leq D\leq\) 4 cm [62]. Figure 3: The MWR method has been successfully applied to the following organs, tissues, and diseases: 1—brain [7; 25; 43; 56]; 2—breast [10; 12; 13; 16]; 3—carotid arteries [49]; 4—venous diseases of the lower extremities [8]; 5—coronavirus infection (COVID-19) [4]; 6—arthritis of the large joints (elbow (6a), knee (6b), ankle (6c) [46; 57], rheumatoid arthritis knee joints [11], sacroliliac joints [45]; 7—spine arthritis [58]; 8—acute inflammatory diseases of the prostate [42]; 9—kidney disease [42; 59]; 10—cardiovascular diseases [50]; 11—bladder [59]; 12—pathology of the female reproductive system [54]; 13—diabetic foot ulceration pathology [60]; 14—thyroid [61]. The traditionally used diagnostic methods (ultrasound, computer tomography, mammography) do not allow one to effectively detect a tumor at an early stage, when its size is small and the heat generated is low. One of the most promising methods for improving efficiency of mammographic screening and early differential diagnosis is the MWR method. A separate area of breast cancer diagnostics based on MWR data is analysis of feature spaces [2; 63; 64]. These are built from of various combinations of temperature differences measured at different points in the breast. Algorithms such as random forest, XGBoost, k-nearest neighbors, support vector machine, cascade-correlation neural network, deep neural network, convolutional neural network, decision trees, and naive Bayesian classifier are used to classify thermometric data [22; 63; 65]. These cause an increase in value sensitivity and specificity for cancer detection and a decrease in errors in diagnoses in early stages of tumor growth [16; 63; 66]. In [67], deep neural network showed the best result on a test set with a significant improvement over other machine learning algorithms. However, the disadvantage of neural networks is the \"black box\" type structure, which does not allow for interpretation of results [68]. Analysis of thermometric data should be part of expert diagnostic systems. Results of classification should be presented in terms that allow justification of the diagnosis, which is extremely important for medical practice. Medical examinations and analysis of MWR data revealed important features of breast cancer [63; 64; 69]. These include the following: increased value of thermal asymmetry between corresponding points of the breast; high temperature difference between separate points of the breast with a tumor; high difference between nipple temperatures; increased nipple temperature compared to average temperature of breast, etc. Analysis of feature spaces required development of special descriptive mathematical models of the patient's diagnostic state, on the basis of which it is possible to effectively classify models and justify the diagnostic result [70]. Classification algorithms are based on feature spaces such as temperature, thermometric features, and second, third, and fourth degree polynomial features [70].