## 2 Materials and Methods
### Dataset Description
In the absence of real data, we generated random values between 14 mL and 80 mL to make sure that we accounted for not only small, normal prostate glands but also hypertrophic glands [45]. So, the advantage of this index is that medical doctors may be able to distinguish between the situation called Prostate Gland Hypertrophy and PCa. A very important research study [46] analyzed a total of 5.291 men with PSA \(\geq\) 3 ng/mL along with measurement of the prostate volume. The results showed that omitting prostate biopsy in men with \(\text{PSAD}\leq\) 0.07 (ng/mL\({}^{2}\)) would ultimately result in 19.7% fewer biopsies, while 6.9% of clinically significant prostate cancers would be missed. Using PSAD values of 0.10 and 0.15 ng/mL\({}^{2}\) as a threshold resulted in the detection of 77% and 49%, respectively, of prostate cancers with Gleason Score \(\geq\) 7, in other words, clinically significant PCa [46]. Furthermore, in another major study [47] involving 2162 men, of which 56% were African American with PSA values from 4 to 10 ng/mL who eventually underwent into biopsy, it was found that a threshold of PSAD = 0.08 ng/mL\({}^{2}\) has a 95-96% NPV (Negative Predicted Value), i.e., it correctly predicts participants who do not have PCa when the biopsy is negative. For those reasons, in our system, we also chose a threshold limit of PSAD = 0.08 ng/mL\({}^{2}\), with some small tolerance, as other studies chose a PSAD limit of 0.1 ng/mL\({}^{2}\). Moreover, it is worth noting that the PSAD index is more significant for smaller-sized glands, as a larger-sized prostate usually produces more prostate antigen, and as a result, prostate cancer may be hidden in an apparently normal PSAD [47; 48]. On the other hand, in a prostate gland with a smaller volume, a value greater than 0.08 or \begin{table}
\begin{tabular}{c c} \hline
**PSA (ng/mL)** & **Probability of Prostate Cancer** \\ \hline
0–4 & 17\% \\ \hline
4–10 & 30\% \\ \hline \(>\)10 & \(>\)49\% \\ \hline \end{tabular}
\end{table} 
Table 1: The values should be evaluated as reference values because of the fact that different studies consider different probabilities [44]. 0.09 should alert the treating physician to carry out further investigation. In addition to PSAD, the PSA-TZ (Transition Zone PSA), which is defined as total PSA to the volume of the transition zone of the prostate, is sometimes estimated. In a study spanning 1997 to 1998 [48; 49] including 273 men with PSA values from 2.5 to 4 ng/mL, it was found that with threshold values of 0.1 ng/mL\({}^{2}\), a sensitivity of 93.181% and a specificity of 17.985% were acquired. In conclusion, the PSA-TZ is less effective. For the purposes of our study, we use only the PSAD with a minimum value of 0.007 ng/mL\({}^{2}\), a maximum value of 0.356 ng/mL\({}^{2}\), a mean value of 0.066 ng/mL\({}^{2}\), and a standard deviation of 0.047 ng/mL\({}^{2}\), as they were obtained by surrogate data. A crucial diagnostic indicator is the PSA velocity (PSAV), which is defined as the rate at which the PSA value changes between two consecutive measurements, with a relative distance of one year. Its measurement unit is ng/mL/year. It can highlight an incipient prostate cancer, as a significant increase from the previous measurement may hide cancer (of course, an increase may also be due to inflammation). A particularly important indicator is the so-called doubling time, i.e., the time in which the PSA value doubles from the previous measurement. Thus, certain threshold values were set, which should trigger the physician to complete further testing. The PSAV is mathematically defined as follows:  
\(\mathrm{PSAV}=\frac{PSA_{f}-PSA_{i}}{PSA_{i}}\pm\) 100%, where \(\mathrm{PSA_{f}}\) stands for the PSA final measurement and \(\mathrm{PSA_{i}}\) stands for the PSA initial measurement, within a distance of a year.