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# The determination of coronary artery wave speed using information theory

Abstract

A method is described for estimating the wave speed of a coronary artery from simultaneous measurements of pressure and velocity. It involves finding the wave speed that minimises the distance between the probability density functions of the measured pressure and velocity and the probability density functions of the pressure and velocity obtained from the separation of the waves into their forward and backward components using the assumed wave speed.

# Introduction

The local wave speed in an artery is an important physical parameter. It depends upon the distensibility of the artery and hence is a parameter of direct clinical interest. It is also involved in the characteristic impedance of the artery and hence plays an important role in the impedance analysis of the arterial system. From the water hammer equations we know that the product of the density $$\rho$$ (which we assume to be constant) and the local wave speed $$c$$ is the constant of proportionality between the change in pressure, $$dP$$, and the change in velocity, $$dU$$, across a wave front $dP_\pm = \pm \rho c dU_\pm$ where the subscript $$\pm$$ refers to forward and backward traveling waves.

In wave intensity analysis, knowledge of the local wave speed is required to separate the measured $$dP$$ and $$dU$$ into their forward and backward components $dP_\pm = (dP \pm \rho c dU)/2$ $dU_\pm = \pm (dP \pm \rho c dU)/2 \rho c$ The net wave intensity does not depend upon the wave speed $dI = dP dU$ but the forward and backward wave intensities require knowledge of the wave speed $dI_\pm = dP_\pm dU_\pm = \pm (dP \pm \rho c dU)^2/4\rho c$

This separation of pressure, velocity and wave intensity into their forward and backward components has proven useful in the aorta, the pulmonary artery and other large systemic arteries [refs]. In the coronary arteries, moreover, it is proving essential for the analysis of the complex influences introduced by the compression of the intra-myocardial blood vessels during systole. Unfortunately, these additional mechanical complexities make it much more difficult to estimate the local wave speed in the coronary arteries.

# Theory

## wave speed in non-coronary arteries

The direct measurement of local wave speed by the foot-to-foot method, i.e. measuring time of travel of the foot of a waveform between two measurement sites a known distance apart, is problematic in practice. The wave speed varies with axial distance along an artery because of anatomical and structural changes in the artery wall and so the ’local’ wave speed must be measured over relatively short distances. Small distances between measuring sites means that travel times are small and it is difficult to make measurements with sufficiently high temporal resolution with currently available transducers. Wave speed (i.e. pulse wave velocity) measured clinically by the transit time (e.g.foot-to-foot method) is actually a weighted average measure of many local wave speeds over the arterial path length, typically carotid-to-femoral, and is not a good estimate of the local wave speed at a particular location.

Local wave speed is usually measured by the simultaneous measurement of pressure and velocity (or volume flow rate). The measurement of the local characteristic impedance, which is closely related to the local wave speed, involves averaging the impedance $$Z = \widetilda{P}/\widetilde{Q}$$ over a certain range of harmonics, where $$\widetilde{P}$$ and $$\widetilde{Q}$$ are the Fourier transforms of the measured pressure, $$P$$, and flow rate, $$Q$$. [ref. Segers] This method relies upon the assumption that reflections of the higher frequency components will die away due to viscous effects. However, the amplitude of the Fourier components of $$P$$ and $$U$$ also decrease rapidly with frequency and their ratio becomes sensitive to noise. The success of this method of calculating the characteristic impedance relies upon the judicious choice of frequencies that are free from reflections and free from the effects of noise.

In wave intensity analysis, the most common method for determining the local wave speed in arteries is the PU-loop method where $$P$$ is plotted against $$U$$ and the slope of the straight line during early systole is taken as the product $$\rho c$$. This is based on the assumption that there are only forward waves during early systole and so the forward water hammer equation holds during this period $dP = dP_+ = \rho c dU_+ = \rho c dU \qquad \mbox{when} \qquad dP_- = dU_- = 0$ In practice this method of determining wave speed is somewhat subjective because of the choice of the segment of the loop that should be fitted as a straight line in never clear cut. Also, the separation of the waves into forward and backward waves, using the wave speed, generally indicate that there are small but significant backward waves at the end of diastole. This is a consequence of the falling diastolic pressure at a time when the velocity is zero. Since there is not mechanism to suddenly stop these backward waves at the start of arterial systole, it could be argued that early systole is not free of backward waves as assumed in the PU-loop method. The presence of equal and opposite forward and backward waves necessitated by a falling pressure at a time of zero velocity is obviated if we first separate the pressure into a reservoir and an excess pressure [refs] and then plot the excess PU-loop. This may be an important virtue of the reservoir/excess pressure hypothesis [ref Ashraf, Allesandro??].