# Hemodynamic principles and calculations

## Calculation of hemodynamic parameters by ultrasound

Hemodynamics is the study of blood flow dynamics. The physical laws that govern blood flow are fundamental in echocardiography. Conventional two-dimensional (2D) echocardiography and Doppler studies are sufficient to study velocities, volumes, and pressure conditions in the heart. These techniques allow for the calculation of all clinically relevant hemodynamic parameters, with few exceptions. Prior to the Doppler era, hemodynamic assessments were performed by means of right heart catheterization (with *Swan-Ganz catheter* [*pulmonary artery catheter*]). However, hemodynamic measurements derived from Doppler studies are considered reliable and comparable to catheterization, and Doppler studies have largely replaced catheterization.

This section discusses hemodynamic principles and how these can be leveraged to calculate stroke volume, cardiac output, pressure conditions, severity of stenoses and regurgitations, *etc*. These calculations are based on simple mathematical equations, which are based on hemodynamic principles. It is fundamental to be familiar with these principles in order to fully understand echocardiography. In clinical practice, the investigator performs simple measurements and Doppler recordings, which the ultrasound machine uses to calculate various hemodynamic parameters.

## The Doppler effect

The central principle to all hemodynamic calculations is the Doppler effect, which was discussed previously (see Doppler Effect and Doppler Echocardiography). Only a brief summary of the Doppler effect is provided here.

The Doppler effect is used to assess the velocity and direction of blood flow. This is possible because sound waves that hit moving objects are reflected at an altered frequency. Sound waves that hit an object moving towards the sound source, will be reflected with higher frequency than the sound waves emitted from the sound source. Sound waves that hit an object moving away from the sound source, will instead be reflected with a lower frequency than the sound waves emitted from the sound source. The difference in frequency between the emitted and the reflected sound waves is called **Doppler shift**.

Erythrocytes in motion will alter the frequency of reflected sound waves. Erythrocytes flowing towards the transducer will reflect the sound waves with higher frequency, whereas erythrocytes flowing away from the transducer will reflect sound waves with reduced frequency (Figure 1).

### The Doppler shift

The Doppler shift depends on the velocity of blood flow (v), the frequency of the emitted ultrasound (f_{e}), the frequency of the reflected ultrasound (f_{r}), the ultrasound velocity in the tissue (c) and the cosine of the angle between the direction of blood flow and the reflected ultrasound wave (cos θ). The **Doppler equation** is as follows:

v = [c · (f_{r}-f_{e})] / [2 · f_{u} · cos ϴ]

The speed of sound (c) in the human body is constant (1540 m/s), and cos ϴ can be ignored unless there is significant angle error (cos 0° = 1), making this formula easier to handle.

Velocity and direction of blood flow can be calculated by using the Doppler equation.

## Blood flow in the heart and vessels

A liquid flowing in a straight cylinder exhibits **laminar flow**, which implies that the flow velocity is highest at the center of the cylinder and the lowest along the cylinder walls. The liquid flows in concentric layers, with gradually diminishing velocity with increasing distance from the center of the cylinder. This results in a parabolic shape of the flow (Figure 2A).

Figure 2B illustrates how the flow profile changes when the diameter of the cylinder decreases. As illustrated, the difference in velocity between the layers is reduced as the diameter becomes smaller. This is explained by the fact that the velocity in the outer layers increases as the diameter decreases. As the flow continues in the cylinder, the flow gradually reassumes a parabolic shape.

These principles are relevant when assessing valvular stenoses and regurgitations. Regardless of size and location, stenoses always have the same effect on blood flow; the velocity accelerates just before the stenosis (*pre-stenotic acceleration*) and the flow becomes turbulent after passing the stenosis (*post-stenotic turbulence*). As illustrated in Figure 3, the jet of the blood flow will be narrowest just after the stenosis and this part is called **vena contracta**. The diameter of vena contracta is slightly smaller than the diameter of the stenotic orifice. The more pronounced the stenosis (i.e the smaller the orifice), the greater the pre-stenotic acceleration.