Author + information
- Wei Li, MD, PhD, FESC, FACC⁎ (, )
- Kim H. Parker, PhD and
- Michael Henein, MSc, PhD, FESC, FACC
- ↵⁎Royal Brompton Hospital, National Heart and Lung Institute, Imperial College, Echocardiography, Sydney Street, London, SW3 6NP, United Kingdom
The comment given in the Limitations section of our study (1) regarding the modified Bernoulli equation was aimed at highlighting the fact that the equation should not be used for low velocities owing to mathematical properties of the exponential curves.
If we make the simple assumption that the flow through the coarctation is dominated by resistance, then the flow is determined by that resistance and the pressure difference across it. Thus,where Ris the resistance and P1(t) and P2(t) are the proximal and distal pressures, respectively. If we also assume that the vessels proximal and distal to the coarctation have compliances C1and C2, then the conservation of mass in the proximal and distal vessels can be written in diastole when there is no flow in the proximal vessels from the left ventricle. Thus,where V1and V2are the volume of the proximal and distal vessels, respectively.
Defining the pressure difference across the coarctation, , the two differential equations for pressure can be combined into the single equationIf we define the net compliance, , then the equation for the pressure difference can be written as,which has the solutionwhere P0is the pressure difference at t= 0, the time of closure of the aortic valve. From the very first equation, the time-dependent flow through the coarctation followswhere Q0= P0/Ris the flow through the coarctation at t= 0.
This solution predicts an exponential fall-off in velocity through the coarctation starting at the time of closure of the aortic valve. The time constant of this exponential fall-off is given by RC, the product of the resistance to flow within the coarctation and the net compliance of the vessels proximal and distal to the coarctation. This prediction is in good agreement with the Doppler velocity measurements through the coarctations of the patients. The model does not take into account the distal flow through the microcirculation, and therefore the exponential fall in distal pressure due to the arterial Windkessel. It could be included, but would probably lead to the prediction of a double exponential fall-off of velocity, which would be much harder to fit to the experimental data.
As a result of this analysis, we would agree with the comments by Dr. DeGroff that the results should be dependent on aortic compliance. His argument, which is based on the extreme condition of zero compliance, is only partially correct in that behavior of flow through the coarctation depends upon both the proximal and distal compliance. Because Dr. DeGroff only mentions one compliance, “the aortic compliance,” his reasoning does not seem to us to be as helpful as is the above model.
- American College of Cardiology Foundation