# PulseCO System Autocorrelation Algorithm

#### Step 1 - Arterial Pressure Transformation into a Volume Time Waveform

An accurate way of determining the change in blood volume in the arterial tree from maximum dilatation to minimum dilatation would allow an estimate of the volume of blood flowing out of the arterial tree during a period slightly longer than diastole. Since the whole period of the cardiac cycle usually bears a (near) fixed relation to diastole simple scaling up would give the stroke volume. The relationship between the capacity of the arterial side of the circulation and the intravascular pressure can be expressed as the compliance: pressure change per unit volume change. All would be relatively simple if this was constant. However, arterial compliance has been shown to change as arterial pressure changes. A stiffening of the vasculature occurs as pressure and volume increase such that, at higher pressures, a given increase in pressure expands the arterial tree by a smaller volume. Nevertheless, the form of this curvilinear relationship (approximately exponential), though differing in its scaling, appears to be very similar in different subjects. Using a lookup table the pressure waveform can be used as the basis for calculating a continuous curve describing the general form of the arterial volume changes with every cardiac cycle for which arterial blood pressure is available.

#### Step 2 - Deriving Nominal Stroke Volume and the Heartbeat Duration

In order to obtain cardiac output (volume per unit time) we require stroke volume (or at least a value proportional to it ie nominal stroke volume) and also the duration of the cardiac cycle to calculate flow. The autocorrelation technique gives us both.

Nominal Stroke Volume: First of all the software subtracts the mean value of the derived arterial blood volume record, giving a description of how much the arterial blood volume changes around it. This is periodic like a sine wave but with different shaped areas above and below zero. The figure below shows how the method works by first using a pure sine wave and subjecting it to the procedure. We firstly obtain an estimate of the mean deviation from zero by multiplying all values of the waveform by themselves. This gives positive waves for both the positive and negative parts of the original sine wave - effectively a double waveform. The mean of the values of the new (double) waveform is otherwise known as the mean square. The square root of this value is a constant proportion of the amplitude of the original waveform known as the root mean square, or RMS value. It is approximately 0.7 of the amplitude. The original sine wave and the squared (double) waveform are shown for three cycles.

Estimation of the Heart Beat Duration: The precise period of the cycle can be obtained by moving one version of the volume waveform successively, step by step, relative to another. Cross multiplication and addition of the answers now gives values, which are both positive and negative. The sum for a given displacement (often referred to as a tau shift) becomes less, and with maximum opposition of the two versions of the record a negative mean square difference is obtained. This is not quite as large as the positive version originally obtained because of the asymmetrical nature of the waveform. The sum of the two magnitudes (maximum reinforcement at tau 0 and maximum opposition at some intermediate point in the cycle - not precisely half way) gives a shifted times unshifted value very close to the same squared value for zero tau shift.

Continuation of the step by step movement of one version of the waveform relative to the other eventually causes positive reinforcement again, when the moving waveform arrives one cycle further along relative to the other. This process does not give a very close value to the mean square volume: the two versions tend to vary sufficiently once they are one cycle shifted relative to each other. However, the second positive peak in the autocorrelogram occurs at a tau shift, which represents the duration of the cardiac cycle.

#### Step 3 - Nominal Stroke Volume & Calibration

With an estimate of the square of the volume draining from the arteries as they shrink from their maximum to their minimum size we have a volume linearly related to the stroke volume ie the nominal stroke volume. With the period of the heartbeat defined precisely by the interval to the first peak in the autocorrelogram we can therefore calculate a value linearly related to the cardiac output following calibration with an indicator dilution system.

The nominal stroke volume and cardiac output are initially uncalibrated. They are converted to calibrated data ie 'true stroke volume/cardiac output' by multiplying the nominal stroke volume by a calibration factor (patient specific correction factor [CF].) The patient specific CF is determined by inputting the cardiac output derived from an independent method of deriving cardiac output ie LiDCO System.