Continuity Equation

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The continuity equation

This is a simple equation based on the conservation of mass principle that typically is used to calculate aortic valve area in aortic valve stenosis. It was first published by the group of Hatle[1]. The principle works as follows: for any flow through a tube, the product of cross-sectional area and flow velocity (averaged over that area) must be equal, provided the fluid is incompressible, because fluid particles (mass) cannot disappear or be created anew. The principle holds both instantaneously and over time, e.g. over a heart cycle or an ejection period. Therefore, if we know the product at one cross-section of the tube (say, location 1), we can infer that it is the same at any other cross-section (location 2):

A1 × v1 = A2 × v2

For calculating stenotic aortic valve orifice area , the following data are needed:

  1. Pre-stenotic cross-sectional area ALVOT(e.g., left ventricular outflow tract area); this is calculated from the aortic annulus diameter (D), assuming that the cross-section of the outflow tract is circular and that the annulus diameter is representative for the outflow tract: ALVOT = π • (D/2)2 = π •D2/4
  2. The corresponding pre-stenotic (averaged) flow velocity vLVOT or velocity-time integral VTILVOT (pulsed-wave Doppler of velocity in the outflow tract).
  3. The flow velocity vsten or velocity-time integral VTIsten at the cross-section where we want to know the area, i.e., at the stenotic orifice area (AVA), derived from a continuous-wave Doppler recording across the aortic valve. The continuity equation then simply states that

ALVOT • vLVOT = AVA • vsten or ALVOT • VTILVOT = AVA • VTIsten which can be solved for AVA, e.g. AVA = ALVOT • VTILVOT / VTIsten

Application

The continuity principle is usually applied to calculate valve orifice area in stenotic aortic valves[2]. However, it can in principle also be applied to other stenotic valves, if stroke volume is known, or to vascular stenoses. The principle is largely independent of flow laminarity or turbulence, or of pressure recovery. Example (see Fig.1,2):

ContinEquatWiki1.jpg ContinEquatWiki2.jpg
Figure 1 Figure 2

Left ventricular outflow tract diameter is 2 cm, and systolic velocity-time integral in the outflow tract by pulsed-wave Doppler is 11 cm. Hence, stroke volume is π•(2.2 cm / 2)2 • 19 cm = 35 cm3 or 35 mL. Maximal velocity-time integral across the aortic valve by continuous-wave Doppler is 64 cm. The stenotic aortic valve orifice area (AVA) then can be calculated as

AVA = 35 mL / 64 cm = 0.5 cm2

representing a very tight aortic stenosis, in spite of low gradients (maximal and mean, 45 mmHg and 27 mmHg), because left ventricular systolic function and stroke volume are low.

Limitations of continuity equation

While the continuity principle is a fundamental principle of hydrodynamics, the available data for application in a given case (e.g., to characterize the pre-stenotic cross-section and flow velocity) have a number of error sources and limitations:

  1. Pre-stenotic cross-sectional area. For the left ventricular outflow tract, it is assumed that its shape is circular and its area can be calculated as π•r2, where r=D/2 (r radius, D diameter in the parasternal long-axis view). This is not strictly true, since the outflow tract is in fact elliptic in cross-section and the parasternal long-axis view shows neither the largest nor the smallest ellipse diameter. Furthermore, the true outflow tract diameter is often underestimated if a tangential long-axis cut of the outflow tract is recorded, leading to underestimation of the diameter, radius, and ultimately aortic valve are. This is generally regarded as the most important error source in the application of the continuity equation. A possible alternative to the full continuity equation therefore is the “velocity ratio”, which is the ratio of peak velocities or velocity-time integrals in the outflow tract and across the stenosis.
  2. Pre-stenotic flow velocity and velocity-time integral. The flow field in the outflow tract, though laminar, is not homogeneous and blood flows faster at the septal (anterior) side than at the posterior side of the outflow tract. Moreover, velocities by Doppler are often recorded at an angle between flow direction and echo beam, leading to underestimation of velocities. Furthermore, in atrial fibrillation, different heart cycle lengths and different preceding heart cycles make it difficult to find equivalent heart cycles for the acquisition of peak velocities or velocity-time integrals in the outflow tract and across the stenotic valve.
  3. If the maximal velocity across the stenotic orifice is missed by continuous-wave Doppler, the continuity equation will produce erroneous results. The continuity equation therefore by no means can correct for a suboptimal Doppler interrogation of the aortic valve.

Note that the continuity equation calculates an "effective" orifice area, which by definition is smaller than the "geometric" or "anatomic" orifice area[3]. The ratio between geometric orifice area and effective orifice area, the "coefficient of contraction", is somewhat variable, depending on geometrical features (e.g., smooth or abrupt inlet) and also on flow rate, at least if the latter is low. Furthermore, since blood is a viscous fluid, inaccuracies arise from ignoring the boundary layer of the fluid in motion[4][5][6].

Conclusion

In clinical practice, in spite of the aforementioned limitations, the continuity equation works remarkably well, probably because some of the mistakes cancel out. It is an indispensable part of the routine evaluation of stenotic aortic valves. It can further conceptually also be used to calculate other stenotic areas, because the conservation of mass principle is universal. For example, if one assumes no valvular regurgitation, one can calculate effective orifice areas of all other valves by calculating the left ventricular outflow tract stroke volume (ALVOT • VTILVOT) and dividing this by the velocity-time integral of the stenotic valve (e.g., the mitral or pulmonary valve). In practice, however, the assumption of no valvular regurgitation is rarely satisfied.

References

  1. Skjaerpe T, Hegrenaes L, Hatle L. Noninvasive estimation of valve area in patients with aortic stenosis by Doppler ultrasound and two-dimensional echocardiography. Circulation 1985;72:810-8
  2. Baumgartner H, Hung J, Bermejo J, Chambers JB, Evangelista A, Griffin BP, Iung B, Otto CM, Pellikka PA, Quiñones M; EAE/ASE. Echocardiographic assessment of valve stenosis: EAE/ASE recommendations for clinical practice. Eur J Echocardiogr. 2009 Jan;10(1):1-25
  3. Flachskampf FA, Weyman AE, Guerrero JL, Thomas JD. Influence of orifice shape, size, and flow rate on effective valve area: an in vitro study. J Am Coll Cardiol 1990;15:1173 80
  4. Burwash IG, Thomas DD, Sadahiro M, Pearlman AS, Verrier ED, Thomas R, Kraft CD, Otto CM. Dependence of Gorlin formula and continuity equation valve areas on transvalvular volume flow rate in valvular aortic stenosis. Circulation 1994,89:827-835
  5. Segal J, Lerner DJ, Miller DC, Mitchell RS, Alderman EA, Popp RL. When should Doppler-determined valve area be better than the Gorlin formula? J Am Coll Cardiol 1987;9:1294-305
  6. Kadem L, Rieu R, Dumesnil JG, Durand LG, Pibarot P. Flow-dependent changes in Doppler-derived aortic valve effective orifice area are real and not due to artifact. J Am Coll Cardiol. 2006;47:131-7
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