## 4 Diagnosis of Indolent and Aggressive Forms of PCa before Treatment
### Modeling and Results  
The classification rule was sought using the algorithm of the method of monotonic functions. The software implementation of the algorithm includes a step-by-step feature selection technique and a cross-validation method to assess the quality of the classification. The best classification results were obtained according to the following three indicators: GrT, GrGl, and tPSA. Dependences \(U(V)\) of the probability of successful detection of patients with the indolent form of the disease \(U(\mathbf{x})\) on the probability of misclassification of patients with an aggressive form of the disease \(V(\mathbf{x})\) (probability of false alarm) are presented in Figure 3, where line 1 shows the dependence \(U(V)\) when tested on the training set, and line 2 shows the dependence \(U(V)\) averaged over 10 tests using \(k\)-fold cross-validation at \(k=10\). The vertical segments in Figure 1 indicate the values of the standard deviations. Notably, the classification results differed little when tPSA was replaced with PHI or fPSA/tPSA. Adding other features to the selected set of indicators significantly increased the discrepancy between the test results for the training set and the test results using cross-validation. In this case, the result of the test of the classification rule for the training set significantly improved, and the result of the cross-validation test slightly worsened. Therefore, such options for solving the problem were not considered. Figure 3 shows that the results of testing the classification rule based on the training sample and cross-validation are very close. Therefore, to explain the decision, it is permissible to use logical rules obtained from the training sample. The \(U(V)\) dependence determines the threshold that provides the same decision-making efficiency when classifying patients with indolent and aggressive PCa [30,31]. For this threshold, the decision on the training sample (line 1) corresponds to the probability of the successful classification of patients with indolent PCa \(U(\mathbf{x})=0.765\) and the probability of misclassification of patients with aggressive PCa \(V(\mathbf{x})=0.235\). Figure 3: Dependences of the probability of successful classification of patients with indolent prostate cancer (PCA) \(U\) on the probability of misclassification of patients with an aggressive form of the disease \(V\). 1—dependence \(U(V)\) when testing on a training sample; 2—dependence \(U(V)\) obtained by 10-fold cross-validation; vertical segments define the values of standard deviation. The threshold \(\alpha=0.235\) corresponds to the rule:  
\[ \begin{array}{ccccc}&\text{(tPSA $\leq$ 9.72)}&\text{\bf AND}&\text{(GrGl $=$ 1$)}\\ \text{\bf OR}&\text{(tPSA $\leq$ 8.24)}&\text{\bf AND}&\text{(GrGl $\leq$ 2$)}&\text{\bf AND}&\text{(GrT $=$ 1$)}\\ \text{\bf OR}&\text{(tPSA $\leq$ 11.39)}&\text{\bf AND}&\text{(GrGl $=$ 1$)}&\text{\bf AND}&\text{(GrT $=$ 1$)}\\ \end{array}  \tag{8}\]  
**THEN**  
_the patient has indolent PCa with a probability \(U(\mathbf{x})=0.765\) and with the probability of misclassification of a patient with aggressive PCa, \(V(\mathbf{x})=0.235\)._  
If this condition is not met, then the patient has aggressive PCa with a probability of \(1-V(\mathbf{x})=0.765\), whereas the probability of misclassification of a patient with indolent PCa is \(1-U(\mathbf{x})=0.235\). Below, we provide several examples of the found classification rule for the thresholds \(\alpha=0.10\), \(\alpha=0.15\), \(\beta=0.90\), and \(\beta=0.85\). The threshold \(\alpha=0.10\) corresponds to the rule:  
\[ \begin{array}{ccccc}&\text{(tPSA $\leq$ 8.21)}&\text{\bf AND}&\text{(GrGl $=$ 1$)}\\ \text{\bf OR}&\text{(tPSA $\leq$ 8.40)}&\text{\bf AND}&\text{(GrGl $=$ 1$)}&\text{\bf AND}&\text{(GrT $=$ 1$)}\\ \end{array}  \tag{9}\]  
**THEN**  
_the patient has indolent PCa with a probability \(U(\omega_{1})=0.548\) and with the probability of misclassification of a patient with aggressive PCa, \(V(\omega_{1})=0.097\)._  
The threshold \(\alpha=0.15\) corresponds to the rule:  
\[