Principles of hemodynamics

Chapter 1: 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).

Figure 1. The Doppler effect.

The Doppler shift

The Doppler shift depends on the velocity of blood flow (v), the frequency of the emitted ultrasound (fe), the frequency of the reflected ultrasound (fr), 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 · (fr-fe)] / [2 · fu · 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 2. Laminar flow in a cylinder results in a parabolic flow profile, with the highest velocity at the center, and the lowest velocity adjacent to the wall of the cylinder.

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.

Figure 3. Post-stenotic turbulence and location of vena contracta.

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Chapter 2: The Bernoulli principle and estimation of pressure gradients

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).

Figure 1. The Bernoulli principle.

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:P1 + K1 = P2  + K2

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

Formula 2:K = 0.5 • Dblood • v2

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 • Dblood) can be approximated to 4, meaning that Formula 2 can be rewritten as follows:

Formula 3:K = 4v2

Formula 1 can be rewritten as follows:

Formula 4:P1  + 4v12 = P2  + 4v22

The pressure difference will then be:

Formula 5:P1 – P2 = 4v22 – 4v12

Which can be rewritten:

Formula 6:ΔP = 4(v22 – v12)

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 (v1) is very small compared to the distal velocity (v2), and the difference becomes even greater after squaring the velocities. Thus, v1 can be ignored, which results in the simplified Bernoulli equation:

Formula 7:ΔP = 4v22

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 · 42 = 64 mmHgThe 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:4·32 = 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 v2 will be miscalculated by approximately 6% at an angle error of 15°

There are situations where v1 (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).

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Chapter 3: The Continuity Equation (The Principle of Continuity)

The Continuity Equation: What Goes In Must Come Out

As previously discussed, stroke volume is usually calculated by measuring area and VTI (velocity time integral) in the LVOT (left ventricular outflow tract). However, stroke volume can also be calculated by quantifying the blood volume flowing over the mitral valve or the tricuspid valve. This is explained by the principle of continuity (the continuity equation), which states that the volume of blood flowing into a chamber must be equal to the volume flowing out of the same chamber (Figure 1). Thus, the blood volume flowing through the mitral valve in diastole is equal to the volume flowing through the aortic valve during systole (Figure 2). The continuity equation is explained by the fact that the velocity of blood is inversely related to the area of the orifice; velocity increases with diminishing area of the orifice, and vice versa.

Figure 1. The principle of continuity (the continuity equation).

Figure 2. The principle of continuity (the continuity equation) states that the volume of blood passing the mitral valve must be equal to the volume passing the aortic valve.

Stroke volume, the amount of blood ejected into the aorta, is calculated by measuring the area and VTI in the LVOT:

SV = areaLVOT • VTILVOTSV = stroke volume; LVOT = left ventricular outflow tract; VTI = velocity time integral.

According to the formula, stroke volume is the product of area and VTI in LVOT. However, the continuity equation states that the stroke volume can be calculated by quantifying the volume flowing through the mitral valve, tricuspid valve or pulmonary valve. These volumes can be calculated using the same principle as for the aorta (i.e the product of the area and VTI). Although the continuity equation is correct, stroke volume is measured in the LVOT in the vast majority of cases, which is explained by the following:

The aortic valve is easy to visualize and image quality is typically high in multiple views. The diameter of the aortic valve is relatively constant during systole; this is important because a representative diameter is crucial for calculation of the area. There is usually no – or only a negligible – amount of regurgitation in the aortic valve. The prevalence of tricuspid valve regurgitation and pulmonic valve regurgitation is relatively high.

To calculate the flow (volume) over the mitral valve, the maximum diameter of the valve is measured in diastole in apical four-chamber view. VTI is obtained in the same view with the sample volume placed in the center of the mitral annulus.

SVmitral = areamitral • VTImitral

To calculate the flow over the tricuspid valve, the diameter and VTI are measured in the RVOT (Right Ventricular Outflow Tract). The measurements are made at the aortic valve plane in the parasternal short-axis view.

SVtricuspid = AreaRVOT • VTIRVOT

Continuity equation and valvular regurgitation

The continuity equation implies that the amount of blood flowing into the left ventricle (across the mitral valve) is equal to the amount of blood flowing out of the left ventricle (across the aortic valve). In other words, the stroke volume across the aortic valve is equivalent to the stroke volume across the mitral valve. This principle can be used to quantify regurgitations and stenoses. For example, in mitral regurgitation, the regurgitant volume can be quantified by calculating the difference in stroke volume across the aortic valve and the mitral valve, according to the formula:

RVmitral = SVmitral – SVaortaRV = regurgitant volume; mitral = mitral valve.

Similarly, the volume leaking back into the left ventricle in aortic regurgitation can be calculated using the following formula:

RVaorta = SVaorta – SVmitralis

Grading of the severity of stenoses using the continuity equation

All valvular stenoses must be graded in order to optimize management according to disease severity. The most important parameter, for grading of stenoses, is the opening area of the stenosis. The smaller the area, the more pronounced the stenosis and, accordingly, the greater the hemodynamic effect. The continuity equation can be used to calculate the area of the valve. The following formula indicates that flow over the mitral valve is equivalent to flow over the aortic valve:

areamitralis • VTImitralis = areaaorta • VTIaorta

The formula shows that either valve area can be calculated by measuring the other three variables.

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Chapter 4: Proximal Isovelocity Surface Area (PISA)

Hemodynamic calculations with PISA (Proximal Isovelocity Surface Area)

PISA (Proximal Isovelocity Surface Area) is a phenomenon that occurs when liquid flows through a circular orifice. The flow will converge and accelerate just proximal to the orifice. The change in flow profile results in the formation of a hemisphere with several layers. Flow velocity is equal within each layer (Figure 1).

Figure 1. (A) Flow velocity increases as a liquid approach a circular opening. The flow profile gradually assumes the shape of a hemisphere with multiple layers. Flow velocity is equal within each layer (depicted with different colors). (B) Schematic illustration of mitral regurgitation with PISA and the resulting regurgitant jet. MR jet = mitral regurgitation jet.

PISA is the hemisphere itself. It appears as a semicircle in 2D images (Figure 1). The radius of PISA can be used to calculate the diameter of the orifice. This has fundamental clinical implications as it enables the investigator to calculate the area of stenoses and regurgitations. Such area estimations are fundamental in the management of valvular conditions, such as aortic stenosis, aortic regurgitation, mitral valve stenosis, mitral valve regurgitation, etc. The radius of PISA is measured from the surface of the hemisphere to the narrowest segment of the Doppler beam, which is located within the orifice (Figure 2).

Figure 2. Measuring the radius of PISA.

Color Doppler is used to revealing PISA. As discussed earlier, aliasing occurs when using color Doppler to analyze velocities greater than the Nyquist limit. Aliasing implies that neither the direction nor velocity of flow can be determined. This results in the Doppler signal shifting color, such that blue turns red, and red turns blue. For color Doppler aliasing usually occurs when velocities exceed 0.5 m/s, which they generally do in the setting of significant stenoses and regurgitations.

Thus, aliasing is exploited to reveal PISA. An optimal assessment of PISA requires adjusting the Nyquist limit until PISA assumes the shape of a semicircle. The radius and area of PISA are calculated as follows:

areaPISA = 2 • π • rPISA2

The flow (Q) can be calculated using PISA, as follows:

QPISA = areaPISA • valiasingvaliasing = aliasing speed

According to the principle of continuity, the flow in PISA must be equivalent to the flow through the orifice itself. This implies that PISA can be used to quantify regurgitation volume. In the case of mitral regurgitation (MR), the regurgitant area can be calculated using the following formula:

areaMR = 2 • π • rPISA • (valiasing / VmaxMR)MR = mitral regurgitation; VmaxMR = maximum velocity of mitral regurgitation; valiasing = aliasing speed.

This formula actually calculates the area of vena contracta (Figure 3), which is approximately equal to the area of the orifice. The areaMR is also called EROA (Effective Regurgitant Orifice Area).

Figure 3. Vena contracta.

The regurgitant volume (RV) can be calculated by the following formula:

RV = areaMR • VTIMRRV = regurgitant volume; VTI = velocity time integral.

These formulas for PISA performs best when the surface surrounding the orifice is flat, which is often not the case for the valves. For example, a closed aortic valve assumes the shape of a cone. Fortunately, this can be accounted for by including a correction for the angle, as follows:

areaPISA = 2 • π • rPISA2 • (Ø / 180)Ø = angle.

Figure 4 shows the angle to be measured.

Figure 3. Angle correction for measurement of PISA.

The width of vena contracta can also be used to estimate the severity of a regurgitation.

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Chapter 5: Stroke Volume, VTI (Velocity Time Integral) & Cardiac Output

Principles of flows and volumes in the heart

If the flow in a cylinder is constant, then flow (Q) is the product of the cylinder area (a) and flow velocity (v):

Q = a • v

This principle can be used to estimate blood flow across the valves. As illustrated in Figure 1, the orifice of the aortic valve and the ascending aorta can be regarded as a cylinder, and the same assumption can be made for the other valves. The area is calculated by measuring the diameter of the valve (area = π × radius2, where radius = diameter/2), and velocity is measured by means of Doppler (Figure 1).

Figure 1. Calculating flow through a cylinder.

Velocity Time Integral (VTI, stroke distance)

The formula Q = a · v states that flow (Q) is the product of area (a) of the cylinder and the velocity (v) of the fluid (i.e blood). The volume (V) that passes a specific segment is the product of the flow (Q) and time (t):

V = Q · tV = volume; t = time (seconds).

However, this equation can only be used if the flow (Q) is constant, which is not the case in the heart. Blood flow is pulsatile during the cardiac cycle; flow is high during systole and ceases during diastole. Moreover, there are pronounced variations in flow during each phase, with rapidly accelerating flow in early systole, decelerating flow in late systole, and no flow during diastole. Doppler is capable of recording these flow variations with high precision.

To calculate flow across a valve, the Doppler line is placed in the valve orifice. The ultrasound machine displays the recorded flows as the spectral curve (Pulsed wave Doppler or Continuous wave Doppler). The area within the spectral curve is calculated automatically by the machine. This area is referred to as the VTI (Velocity Time Integral), and it measures how far blood travels during the time period. VTI is also called stroke distance. Figure 2 illustrates the recording of VTI using pulsed wave doppler in the aortic valve.

Figure 2.

VTI (Velocity Time Integral) is the area within the spectral curve and indicates how far blood travels during the flow period. VTI can be used for various volume calculations, such as calculation of stroke volume.

Stroke volume (SV), cardiac output (CO) and cardiac index (CI)

Stroke volume is the amount of blood ejected into the aorta during systole. Stroke volume is calculated by measuring the Doppler flow in the aortic valve. In the left ventricular outflow tract (LVOT), the following two measurements are performed:

Diameter of the aortic annulus: This measurement is made in the parasternal long-axis view during systole, when the diameter is greatest (usually halfway through systole). Zoom in LVOT to improve the accuracy of the measurement.Flow velocity in LVOT: Velocity is measured in apical four-chamber view (4C) or five-chamber view (5C) using pulsed wave doppler with sample volume located in the valve orifice. The ultrasound machine calculates VTI (Velocity Time Integral) automatically.

This approach assumes that the valve orifice is approximately circular, such that the orifice area can be calculated using the diameter (area = π × radius2, where radius = diameter/2). The formula for stroke volume (SV) will then be:

SV = area • VTISV = stroke volume, VTI = velocity time integral.

Corresponding formula for measurements in the aortic valve:

SVaorta = areaLVOT • VTILVOT

The area is measured in cm2. VTI has the unit cm/contraction and stroke volume has the unit cm3/contraction (i.e ml/contraction). Figure 3 illustrates schematically how area and VTI are used to calculate stroke volume.

Figure 3. Registration of VTI with pulsed wave doppler in the aortic opening.

Cardiac output (CO)

Cardiac Output (L/min) is the product of stroke volume and heart rate:

CO = SV • HRCO = cardiac output; SV = stroke volume; HR = heart rate.

Cardiac Index (CI)

If cardiac output (CO) is divided by body surface area (BSA), then the Cardiac Index (L/min/m2) is obtained:

CI = CO / BSACI = cardiac index; CO = cardiac output; BSA = body surface area.

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