# The Bernoulli principle and estimation of pressure gradients

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## The Bernoulli principle and pressure gradients using Doppler measurements

Continuous wave Doppler and pulsed wave Doppler can measure the velocity of erythrocytes as they travel through the heart and vessels. The velocity of erythrocytes (*i.e* blood) can be used to estimate pressure gradients (pressure differences) between the atria, ventricles, and connecting vessels. The estimation of pressure gradients is done using *the Bernoulli principle*. The Bernoulli principle is based on the *law of conservation of energy*, which states that the total energy of an isolated system remains constant over time; energy can neither be created nor destroyed, it can only be transformed or transferred from one form to another. Blood flowing through the heart and vessels obey the law of conservation of energy. It follows that the sum of kinetic energy (K) and pressure energy (P) of blood must be equal in two separate points in the system (Figure 1).

According to the Bernoulli principle, the sum of kinetic energy (K) and pressure energy (P) is constant as blood flows through the circulatory system. The equality of kinetic and pressure energy at two separate points can be formulated as follows:

*Formula 1*:

P_{1} + K_{1} = P_{2} + K_{2}

Kinetic energy (K) is a function of velocity (v) and density (D) of the liquid:

*Formula 2*:

K = 0.5 • D_{blood} • v^{2}

With regards to echocardiography and ultrasound imaging in general, v is the maximum velocity measured using Doppler. Moreover, the first part of the formula (0.5 • D_{blood}) can be approximated to 4, meaning that *Formula 2* can be rewritten as follows:

*Formula 3*:

K = 4v^{2}

*Formula 1* can be rewritten as follows:

*Formula 4:*

P_{1} + 4v_{1}^{2} = P_{2} + 4v_{2}^{2}

The pressure difference will then be:

*Formula 5:*

P_{1} – P_{2 }= 4v_{2}^{2 }– 4v_{1}^{2}

Which can be rewritten:

*Formula 6:*ΔP = 4(v

_{2}

^{2}– v

_{1}

^{2})

This formula is excellent for measuring pressure gradients across small openings, such as the valves. Importantly, in the setting of valvular stenosis or regurgitation, the proximal velocity (v_{1}) is very small compared to the distal velocity (v_{2}), and the difference becomes even greater after squaring the velocities. Thus, v_{1} can be ignored, which results in the **simplified Bernoulli equation**:

*Formula 7:*ΔP = 4v

_{2}

^{2}

This equation is also referred to as the* modified Bernoulli equation*. ΔP is the pressure gradient (mmHg) across a valve.

**Example 1**: A maximum velocity of 4 m/s is measured across the aortic valve. The pressure gradient equals:

4 · 4^{2} = 64 mmHg

The pressure gradient between the left ventricle and the aorta is 64 mmHg.

The Bernoulli principle can be used to calculate pressure gradients across valvular stenoses and regurgitations. The equation is agnostic to the direction of the blood flow; it merely measures the pressure gradient across a small orifice. According to the Bernoulli principle, the flow through the orifice will depend on the pressure gradient across it.

**Example 2**: A maximum velocity of 3 m/s is measured across the tricuspid valve. The pressure gradient equals:

4·3^{2} = 36 mmHg.

The pressure gradient between the right ventricle and the right atrium is 36 mmHg.

## Disadvantages of the Bernoulli equation

The Bernoulli equation is highly dependent on the precision of the Doppler measurement. The Doppler beam must be parallel to the direction of the blood flow (refer to The Doppler Equation). Any angle error between the Doppler beam and the blood flow will result in an underestimation of the velocity. In clinical practice, angle errors less than 15° are acceptable (cos 15° = 0.97). Velocity at v_{2} will be miscalculated by approximately 6% at an angle error of 15°

There are situations where v_{1} (proximal velocity) can not be ignored. The most common situation is when assessing aortic stenosis in the presence of a narrowing of the LVOT (left ventricular outflow tract). Such narrowings are due to septal hypertrophy or subaortic membrane (Figure 2).

Figure 2. (A) Septal hypertrophy and (B) subaortic membrane. LVOT is narrowed in (A) and (B).