## 2 Materials and Methods  
### Dataset Description  
This research utilized surrogate data in the absence of adequate prostate cancer data. More specifically, the surrogate data generation was based on the clinical study by [1], which was a pilot urological health promotion program that took place in Ireland. In that study, 660 subjects between 18 and 67 years of age were recruited, and no patient had clinical evidence of PCa. PSA values were evaluated and, in the entire cohort, the mean PSA level was 1.7 with a median of 0.9. The mean age of the patients was 58 (range 25-70). The reason for this cutoff in the age range was justified by the fact that beyond age 70, the PSA measurement may not be beneficial because of the fact that in a patient aged 75 or 80, PCa is rarely life-threatening (shorter life expectancy until the tumor progresses). Moreover, in patients aged 80 and above, there is a high probability that malignant PCa is indeed present. Of course, the above statement also depends on the overall physical condition and state of health of an individual. On the other hand, below 40 years of age, the PSA measurement mainly reflects benign conditions such as prostatitis. In order to produce surrogate PSA values, we utilized the fact the PSA data reported in [1] approximately followed the lognormal distribution. In Probability Theory, the lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if a random variable \(X\) follows the lognormal distribution, \(Y=ln(X)\) follows the normal distribution [41]. Equivalently, if \(Y\) follows the normal distribution, \(X=exp(Y)\) follows the lognormal distribution. A random variable that follows the lognormal distribution takes only positive real values. We sought to find the values of the \(\mu\) and \(\sigma^{2}\) parameters for the variable \(Y\), where \(Y=logX\sim Normal\) (\(\mu\), \(\sigma^{2}\)) with mean value \(\mu\) and variance \(\sigma^{2}\)[42]. So, the PSA values are represented by the variable \(X\) and, given that the mean value for the lognormal distribution is \(\Theta=E(X)=exp\) (\(\mu+\sigma^{2}/2\)) [42], we can derive that \(\mu=-0.105\). Moreover, the median value for the lognormal distribution is given by \(median=exp\) (\(\mu\)) = 0.9, where, in combination with \(\Theta=1.7\), we obtain \(\mu=-0.105\) and \(\sigma=1.12\). Furthermore, with the use of the _numPy library_ in Python 3.9 [43], we used the _lognormal_ (\(\mu\), \(\sigma^{2}\), 660) function to produce random numbers that follow the lognormal distribution with mean \(\theta\) = 1.7 and _median_ 0.9 for 660 subjects. With this procedure, we acquired surrogate PSA values in the desired range. The minimum PSA value was 0.12 ng/mL, and the maximum value was 9.99 ng/mL. In Table 1, we list the PSA values and the probability of PCa. The PSA index alone cannot rule out the possibility of PCa, and additional indexes should be evaluated. Additionally, we considered the age of the subjects, as there is a close relationship between age and PCa. According to epidemiological findings, 16% of men over 50 years of age in the USA will develop PCa, and the risk increases after 40 years of age. In our dataset, the minimum age was 35 and the maximum was 70 with a mean age of 54.7 years and a standard deviation of 7.9. The match between age and PSA values was random to reflect the fact that an older man could have a low PSA result and a younger man with PCa could have a higher PSA value. An additional very important variable is the PSA density, which was utilized as an additional feature of our data (PSAD). It was obtained by dividing the PSA value by the prostatic volume in mL, i.e., \(\text{PSAD}=\frac{PSA\left(\frac{\text{mL}}{\text{mL}}\right)}{Prostatic\ Volume\ (\text{mL})}\). In clinical procedures, the prostatic volume is determined with the help of Transrectal ultrasound (TRUS) and even with the simple, non-invasive procedure known as abdominal ultrasound.