## 2 Materials and Methods  
The THz spectroscopy technique based on the effect of freely decaying polarization involves periodic induction and decay of macroscopic polarization in a sample of a gas mixture. Two types of instruments can be used: a phase manipulation spectrometer and a fast frequency sweep spectrometer [14]. The approach based on fast frequency sweep through the absorption line [15] is implemented in a THz spectrometer with a radiation source based on a backward wave tube (0.118-0.178 THz). The fast frequency sweep mode allows simultaneous measurement of all absorption lines and detection of several substances. The urine was studied with the fast frequency sweep spectrometer. The line shape obtained with such a spectrometer typically has a form [16] of beats between a freely decaying polarization signal and modulated radiation from the source. The advantage of terahertz gas spectroscopy is its high resolution because the absorption lines at working pressure are narrow and overlapped rarely. The line of the rotational spectrum is the fingerprint of a specific substance because absorption line parameters, such as central frequency and line strength, are determined by molecular structure. It has benefits over THz time-domain spectroscopy and IR spectroscopy (where frequency resolution is lower). Moreover, the detection of only one absorption line is, in principle, enough for revealing the substance in a multicomponent gas mixture, which is of greater benefit than the spectroscopic methods where spectral lines corresponding to the molecular parts or molecular bonds are detected. Some substances of urine samples are volatile and appear in a gas mixture over the sample without heating. The heating of samples in liquid or solid state allows the gaseous state of the sample to be obtained. Some large biological molecules, such as proteins, sugars, and fats, are decomposed on heating, but there are specific features in the content of the resulting gas mixture of the products of thermal decomposition that characterize the patient's state. Urine samples from 12 patients with PC were studied (patient information is available in Table 1). This study is approved by the Institutional Review Board of Nizhny Novgorod Regional Oncology Hospital protocol code 5, from 18 May 2015. The following abbreviations are used in this table. The TNM Staging System of Cancer includes the extent of the tumor (T), the spread to the lymph nodes (N), and the presence of metastasis (M). The ISUP system grades the cancer between 1 and 5 depending on the Gleason score. The lower the grade, the less likely the cancer is to spread. WBC--white blood cells; RBC--red blood cells; PSA--prostate-specific antigen; UAm--urinalysis; HGB--hemoglobin. In addition, the composition of thermal decomposition products of samples of cancer patients' urine was compared with the composition of urine samples of 4 conditionally healthy volunteers (male, 25-34 years old, without chronic diseases). The samples were prepared as follows: A 1-2 mL sample was poured into a flask where it was dehydrated to ensure accurate spectroscopic measurements aimed at identifying substances present in the sample in trace concentrations. A cryogenic trap with liquid nitrogen was used to dehydrate the liquid sample. The sample was frozen, most of the water was removed, and a thin film or crystallized residue remained in the flask. Volatile substances from the resulting residue were released, first without heating and then with heating to 200 \({}^{\circ}\)C, into the measuring cell, which had been evacuated to a pressure of \(10^{-3}\) mbar. The operating pressure in the cell was \(5\times 10^{-2}\)-\(1\times 10^{-1}\) mbar. Before lettingthe mixture of thermal decomposition products into the cell, the entire operating range of the spectrometer was registered, which was subsequently used as a background record for identifying absorption lines. The absorption spectra of thermal decomposition products of urine taken from cancer patients were measured.