# Basic principles of non-invasive hemodynamics, volume and flow measurement

Echocardiography allows a noninvasive evaluation of haemodynamics. This was initially done using M-mode and later 2D imaging, which allowed measurement of dimensions that could be translated into volumetric data. The development of Doppler echocardiography now provides a more direct and quantitative technique to derive hemodynamic information. Currently 2D imaging combined with a selection of Doppler measurements is the preferred method to assess haemodynamics and has in some cases replaced cardiac catheterization. Accordingly, a quantitative Doppler assessment of blood flow, estimation of intracardiac pressures, valvular disease, pulmonary vascular resistance (PVR), ventricular systolic and diastolic function, and anatomic defects is feasible and should be part of the echocardiographic examination. The accuracy of the Doppler technique for measuring blood velocity has been validated in many studies.

## Blood flow measurement

Accurate flow measurement requires care to as much as possible minimize interference from neighboring blood flows and align the ultrasound beam parallel to the blood flow of interest. Blood flow can be measured in an orifice as the product of flow velocity and cross-sectional area (CSA), where the CSA can be measured by two-dimensional imaging and flow velocity can be determined directly with Doppler. If flow had a constant velocity, it would be easy to determine velocity at any point in time and solve the equation accordingly. In the cardiovascular system, however, flow is pulsatile and therefore individual velocities during the ejection phase must be sampled and then integrated to measure average volume flow. This sum of velocities is called the time velocity integral (TVI) and is equal to the area enclosed by the Doppler velocity profile during one ejection period. Integrating the area under the velocity curve is simply measuring the velocities at each point in time and summing all these velocities. When TVI and the corresponding cross-sectional area (in centimeters squared) are measured at the same point, their product equals stroke volume (in centimeters cubed or milliliters), which is the volume of blood ejected by the heart with each contraction (assuming no valvular regurgitation or cardiac shunt). It is essential to remember that cross-sectional area must be measured at the same point in space where the Doppler signal is sampled. For example, if blood flow is measured through the aortic valve, both the Doppler signal and the cross-sectional area must be measured at the same level.

Figure 1 demonstrates how these concepts can be applied to aortic flow to measure stroke volume (SV). SV is calculated as \[SV = CSA \cdot TVI\]

The CSA of the aortic annulus is nearly circular, with little variability during systole. Because the area of a circle = π · r^{2}, the area of the aortic annulus is derived from the annulus diameter (D) measured in the parasternal long-axis view as:
\[CSA = \pi D^2/4 \approx 0.785 D^2\]

Considering this equation, it is obvious that any error in the measurement of the diameter of the orifice is “squared” and thus contributes greatly to errors in the final determination. For this reason, particular care must be taken to ensure accurate determination of orifice diameter.

It is crucial to keep in mind also the Doppler equation (Figure 2), and especially the importance of the angle θ, that is the angle between the ultrasound beam and blood flow direction. Because the cosine function varies between 0 and 1 and appears in the numerator of the Doppler equation, errors in θ will have a predictable effect on measured velocities. For example, if θ is between 0 and 20 degrees, the cosine of θ will range between 1.0 and 0.92, leading to a slight underestimation of true velocity. As θ increases to more than 20 degrees, the cosine decreases rapidly and the degree of velocity underestimation increases quickly. Hence, aligning the ultrasound beam as close as possible to the direction of flow is critical if true velocity is to be measured. Equally important, misalignment between the ultrasound beam and flow can only result in underestimation of velocity, never overestimation.

Another factor that will affect the accuracy of the Doppler equation is the pattern of blood flow where velocity is being measured. Normal flow in the heart and great vessels is laminar, meaning that the fluid is traveling at approximately the same velocity and in the same direction. If a sample volume is placed within such a flow pattern, the Doppler will record a clean signal of uniform velocity. Flow becomes increasingly disturbed or turbulent (i.e., less laminar) as the velocity increases or the cross-sectional area changes. Viscosity also affects the flow profile. The highest velocities and most laminar flow generally occur at the center of the profile, whereas at the edge of the flow pattern, near the vessel wall, flow tends to be slower and more turbulent. This spatial distribution of velocities across the three-dimensional flow is called the flow velocity profile. In a large, straight vessel, with laminar flow, it tends to be flat, whereas in smaller curved vessels, the profile has a parabolic shape. Velocity will be higher at the center and lower at the margins (Figure 3). Fortunately, flow passing through a normal heart valve or the proximal great vessels tends to be laminar with a flat profile and is therefore suitable for quantitative analysis.

Physiologic blood flow is never perfectly uniform. That is, at any point in time, a distribution of velocities occurs, resulting in a broadening of the Doppler signal. The greater the range of velocities is at any point in time, the broader is the Doppler signal. The darker line through the center of the distribution represents the modal frequency, i.e., the velocity at which the largest number of blood cells are traveling. When tracing the velocity to derive a VTI, it is best to trace the outer edge of the most dense portion of the spectral tracing (i.e., the modal velocity) and ignore the dispersion that occurs near peak velocity. When measuring TVI, multiple cycles (usually three to five) should be traced and averaged to minimize error. In patients with atrial fibrillation, between five and 10 beats should be analyzed.

## Volume measurement

Echocardiographic measurement of cardiac chamber dimensions are widely used in clinical practice and research. Some formulas have been proposed to calculate left ventricular (LV) volumes from linear dimensions, such as the Teichholz or Quinones methods. Unfortunately, linear measures of LV function are problematic especially when the heart chambers are deformed such as in regional hypokinesia, valvular diseases, where geometric assumptions required to convert a linear measurement to a 3D volume may result in inaccuracies. Therefore measurements from 2D tracing are preferred. The most important views are the midpapillary short-axis view and the apical 4- and 2-chamber views, where volumetric measurements are done by manual tracing of the endocardial border. Accurate measurements require optimal visualization of the endocardial border to minimize the need for extrapolation.

The most commonly used 2D measurement for volume measurements is the biplane method of disks (modified Simpson’s rule), which is the currently recommended method of choice. The principle underlying this method is that the total LV volume is calculated from the summation of a stack of elliptical disks. The height of each disk is calculated as a fraction (usually 1/20) of the LV long axis based on the longer of the two lengths from the 2- and 4-chamber views. The cross-sectional area of the disk is based on the two diameters obtained from the 2- and 4-chamber views. When two adequate orthogonal views are not available, a single plane can be used and the area of the disk is then assumed to be circular. The limitations of using a single plane are greatest when extensive wall-motion abnormalities are present.

End-diastolic volume (EDV) and end-systolic volume (ESV) can be then assessed and the EF is calculated as follows (Figure 4): \[EF = (EDV - ESV) ⁄ EDV\]

The LA volume can also be determined by the Simpson’s rule or the ellipsoid method. The Simpson’s rule, similar to its application for LV measurements, states that the volume of a geometric figure can be calculated from the sum of the volumes of smaller figures of similar shape (Figure 5). Most commonly, Simpson’s algorithm divides the LA into a series of stacked oval disks whose height is h and whose orthogonal minor and major axes are d_{1} and d_{2} (method of disks). The volume of the entire LA can be derived from the sum of the volume of the individual disks.
\[ volume = \sum h \pi \frac{d_1}{2} \frac{d_2}{2} = \frac{h \pi}{4} \sum d_1 d_2\]

The formula is integrated with the aid of a computer and the calculated volume provided by the software packages. The use of the Simpson’s method in this way requires the input of biplane LA planimetry to derive the diameters. Optimal contours should be obtained orthogonally around the long axis of the LA using TTE apical views. Care should be taken to exclude the pulmonary veins from the LA tracing. The inferior border should be represented by the plane of the mitral annulus. The ellipsoid model assumes that the LA can be represented as a prolate ellipse by the following formula: \[ volume=\frac{4 \pi}{3} \frac{L}{2}\frac{D_1}{2}\frac{D_2}{2}= \frac{\pi}{6} L D_1 D_2 \]

, where L is the long axis (ellipsoid) and D_{1} and D_{2}2 are orthogonal short-axis dimensions.
Volume determined using linear dimensions is very dependent on careful selection of the location and direction of the minor-axis dimensions and has been shown to significantly underestimate LA volume.

Three-dimensional echocardiography should provide the most accurate evaluation of cardiac chamber volumes, since avoids geometric assumptions. Recent improvements in semiautomated volumetric analysis are changing the clinical applications of 3D echocardiography, evolving from a complicated and time-consuming research tool into a simple and fast imaging modality ready for everyday clinical use.

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