The derivation of theoretical ultrasonic Doppler blood flow spectrum

This paper deals with theoretical aspects of blood flow and the ultrasonic Doppler spectrum. Blood flow in the vessel can be considered as a pipe flow with a circular cross-section. Blood flow is, in general, complex and pulsatile. Navier-Stokes and the continuity equations are the governing equations for the blood flow, and the Navier's equations for the blood vessel. If the flow is pulsatile with a large Womersely number, the velocity profile becomes blunt due to the inertia forces. Since the ultrasonic Doppler flowmeter detects the velocities of blood cells across the vessel, Doppler power spectrum is dependent on the velocity profile. If the velocity profile is blunt, the Doppler power spectrum will be narrow banded. If the flow is sinusoidal, the Womersely number is only a parameter describing the pulsatility. Blood flow waveforms, however, differ from the sinusoidal wave. They consist of a steady flow component and many harmonics with amplitudes and phases. In this study, the influence of the harmonic contents and the Womersely number on velocity profiles and the Doppler power spectrum are examined.