In some points of view, diastolic function of the left ventricle is more important for the abnormality usually appears ahead of the abnormal systolic function. In 1976, Weiss
et al. [
1] found that left ventricular pressure was able to be plotted and fit to an exponential function:
where P is the left ventricular pressure, e=2.71828…, is the natural logarithmic base, t is time from –dP/dt max, T is left ventricular relaxation time constant, also called Tau, and B is a constant. From Eq. (1), the derivative of the left ventricular pressure is expressed by:
Since P and -dP/dt are variables that can be obtained in catheter lab, the important parameter, Tau, becomes available through a simple calculation as shown above and its accuracy can be ensured through the accessibility of P and -dP/dt in the catheter lab. Obviously this method is inconvenient and invasive. Since 1992 many methods of noninvasive measurement of Tau by continuous-wave Doppler in patients with mitral regurgitation [
2-
5] or aortic regurgitation [
6] have been reported. However, the derivation of the important parameters seems time-consuming and complicated
via the proposed traditional method in Echo lab. I have introduced a simple method [
7] to calculate Tau in patients in mitral regurgitation by pure mathematical derivative based on Weiss’ formula and simplified Bernoulli’s equation. Here a similar method is developed to calculate Tau by continuous-wave Doppler in patients with aortic regurgitation.
Before mathematical deduction, let’s refresh our knowledge about continuous-wave Doppler aortic regurgitation spectrum. Fig. (
1) is a schematic description of aortic regurgitation spectrum.
![Fig. (1) Fig. (1)](picrender.fcgi?artid=2570579&blobname=TOCMJ-2-28_F1.gif) | Fig. (1) Schematic description of aortic regurgitation spectrum. AVC is aortic valves closure, MVO is mitral valves opening, MVC is mitral valves closure and AVO is aortic valves opening. |
From Fig. (
1) we see the typical aortic regurgitation spectrum bordered by AVC-MVO-MVC-AVO, X axis is sweep time, the unit is ms. Y axis is aortic regurgitation spectrum velocity, the unit is m/s. Suppose there is no mitral opening, we can expect the ascending limb and the descending limb of the spectrum connected smoothly like the upper dash line. If we can add the mitral spectrum in the mean time, we will see the E spectrum and the A spectrum are within the MVO and the MVC. Time interval from AVC to MVO is isovolumic diastolic period. Time interval from MVC to AVO is isovolumic systolic period. For the ascending limb, the upper part could be bending more, making the (t3-t1)/(t2-t1)>>2. From MVO to MVC, the spectrum border is like a straight line, but actually it is not. According to simplified Bernoulli’s equation, the velocity is dictated by the pressure gradient between ADP and LVDP, ADP is aortic diastolic pressure, LVDP is left ventricular diastolic pressure:
For this LVDP is measured during mitral valves opening, we have LVDP<< ADP. Roughly, we have:
Whenever there is a severe aortic regurgitation, ADP decreases quickly, causing rapid decreasing of v. That’s why we classify the severity of aortic regurgitation with the help of pressure half-time and slope of MVO-MVC.
–dp/dt max happens shortly after AVC [
1]. Or there is a short time delay between AVC and –dp/dt max. [
8,
9] In addition, after the return of pressure to the level of end-diastolic pressure, passive viscoelastic properties may be of importance and its effect on the evaluation of Tau should be modeled. [
1] In another word, neither the beginning nor the end of this period fits the function P=E TOCMJ-2-28-img-01 very accurately, although the middle part of the isovolumic diastolic period does.