Blood vessels biomechanical models in the form of multilayered anizothropic viscoelastic cylindrical shells with viscoelastic compressible biotissue from their one side and blood flow (taken as a compressible visco-elastic fluid) from the other side have been built. The vessels' acoustical resonant and antiresonant frequencies for middle-sized and large arteriole (veinlet), artery (vein), and aorta, both within the vacuum and within their biological medium, have been detected. Blood vessels' models that imitate such pathologies as arteriosclerosis and diabetes are elaborated. The informative frequency range for these vessels pathologies lays within 60 Hz--6 MHz. The lower spectrum of the range deals with the large vessels, the upper one deals with middle-sized and small vessels. The most probable detected vessels' acoustical resonances respond to the first three oscillation modes. Thanks to the fact that blood vessels have an elastic modulus less by the factor 10E3(centered dot)(centered dot)(centered dot)10E6, then the well-known ``metallic'' or plastic shells' elastic moduli, their (vessels') dispersion characteristic as well as their modus of interaction between the moving blood column oscillations and the vessel itself, has quite a different character. This is due to the low sound propagation velocity of the longitudinal and shear waves in the vessel when compared with the velocity of the same waves within the viscoelastic blood. A diabetes case is detected by blood viscosity changes, that cause the obvious (up to 30%) vessel's main resonance shift. Arteriosclerotic pathologies cause the appearance of an extra two or three resonances next to the main resonance of the vessel given, and they can be detected in the echosignal when compared to this vessel type anatomical standard that is filed in a computer database. Blood flow rate determination in the vessel is a trivial procedure. The suggested technique has been approved in vitro on the canina vessels' models and is confirmed in clinical conditions (with real patients).