- Open Access
Investigations concerning the application of the cross-correlation method in cardiac output measurements
© Gawlikowski and Pustelny; licensee BioMed Central Ltd. 2012
Received: 6 October 2011
Accepted: 2 April 2012
Published: 20 May 2012
In spite of numerous non-invasive examinations the “gold clinical standard” of cardiac output measurements is the invasive pulmonary artery catheterization by means of the Swan-Ganz catheter and the application of the thermodilution method to estimate the blood flow. The results obtained by means of thermodilution are sensitive to many physical and biological disturbances. The unreliability of this method amounts to 20-45% and depends on the given variant of the method. Therefore some other method, more accurate and resistant to disturbances, was looked for. This paper presents a new approach to cardiac output measurements, based on cross-correlation signal analysis. The goal of investigations was to verify experimentally the application of the cross-correlation method of cardiac output measurements.
In 99.2% of the examined cases the extreme of the cross-correlation function was easy to be estimated by numerical algorithms. In 0,8% of the remaining cases (with a plateau region adjacent to the maximum point) numerical detection of the extreme was inaccurate. The typical unreliability of the investigated method amounted o 5.1% (9.8% in the worst case). Investigations performed on a physical model revealed that the unreliability of cardiac output measurements by means of the cross-correlation method is 3–5 times better than in the case of thermodilution.
The performed investigations and theoretical analysis have shown, that the cross-correlation method may be applied in cardiac output measurements. This kind of measurements seems to be more accurate and disturbance-resistant than clinically applied thermodilution.
The most fundamental hemodynamic parameter is the cardiac output (CO). It is defined as the average blood flow which is pumped through the cardiac muscle in the course of one minute [1, 2]. For over 40 years pulmonary artery catheterization (PAC) and CO measurements by means of thermodilution constituted the “gold standard” of hemodynamics diagnostics [2–6]. Nowadays, invasive examinations are slowly taken off the intensive care wards and the no-invasive one (e.g. echocardiography) or minimally invasive (e.g. PiCCO , LidCO ) have been put into practice. It should be emphasized that non-invasive methods of CO measurements are recognized as less accurate [2, 3, 6, 8] and minimally invasive methods must be calibrated before examination by means of thermodilution [7, 8]. Actually there are a few methods based on other physical principles [9, 10], but nowadays they are being developed. The PAC technique by means of the Swan-Ganz catheter allows to obtain or calculate other valuable hemodynamic parameters basing on intracardiac pressures, e.g. pulmonary artery wedge pressure, pulmonary resistance etc. [2, 6]. It has been evidently proved that invasive PAC examinations did not increase the mortality rate . Therefore, in spite of their invasiveness, PAC and CO measurements by thermodilution still constitutes the “gold standard” and they are frequently utilized in intensive care wards.
where: c b , c i – specific heat of the blood and indicator, ρ b , ρ i – mass density of the blood and indicator, V i – indicator volume, T b , T i – temperature of the blood and indicator.
In spite of certain imperfections, thermodilution is essentially a clinical method of CO measurements. Low accuracy and repeatability cause that prospecting other measurement methods, based on other physical principles, is justified.
One of the flow measurement methods (occasionally applied for atypical liquid or intricate objects) is the cross-correlation [13, 16, 17]. The goal of investigations was to verify experimentally the application of the cross-correlation method for cardiac output measurements.
Background of cross-correlation flow measurements
where: T – finite time of integration.
The relationship between the transit time of disturbance and its velocity is more complex and depends on the spatial distribution of the flow. At a laminar flow through the cylindrical vessel the spatial distribution of the flow velocity is parabolic . For this kind of flow the average velocity v AV equals half the maximum velocity v max .
where: Q – volumetric flow, v AV – average flow velocity, A – area of the cross section of the vessel.
The procedure of the experiment
Experiments were performed on the pulsating flow at the following conditions: heart rate 40–70 beats per minute, duty cycle 25-50%.
After the injection of the indicator the signals originated by the sensors (0)…(3) were registered (sampling frequency 1 kHz). Next, the registered signals were processed (LabVIEW, SignalProcessing Toolkit) in the following way: filtering (digital Bessel low-pass filter, cutoff frequency 0.25 Hz), normalization, cross-correlation calculation (Fourier’s method), searching for the cross-correlation function extreme, estimation of the transit time τ of the disturbance. The extreme of the cross-correlation function was determined by the differential method. The last step was the calculation of the volumetric flow value by means of the algorithm described below:
● step 1: calculation of v max value from (4) – signals from the detectors (0) and (3) were analyzed,
where: T SYS –ejection time of the liquid during its pulsatory flow, T – period of flow pulsation.
● step 3: from Re the value of the factor m in ( 5) is estimated, basing on Figure 2,
● step 4: the average flow velocity v AV is calculated making use of the equation ( 5),
● step 5: the volumetric flow is calculated by means of ( 6).
In the applied physical model of pulmonary circulation the pulmonary trunk diameter was D PT = 1.3 cm. As the liquid flows through elements with different diameters (right ventricle and pulmonary trunk), the equivalent diameter of the vessel was assumed to be A z = 4 cm2 .
The total amount of the measured flow set points was N = 31. For all set points n = 8 the transit time was measured. The measurements were performed for a flow in the range of 1.3 – 4.0 L/min (the corresponding average and maximum flow velocity was from 0.10 to 0.26 m/s and 0.20 – 0.55 m/s, respectively). Those values correspond to the maximum flow velocity through the pulmonary artery valve (0.6 m/s, refer to ). At these conditions the Reynolds number was in the range of 700 – 2200.
In 99.2% of the examined cases the extreme of the cross-correlation function was easy to be estimated by numerical algorithms. In 0,8% of the remaining cases (with a plateau region adjacent to the maximum point) numerical detection of the extreme was inaccurate.
Unreliability of the cross-correlation method
Standard deviation = ±3.5%
The worst case
For fuzzy maximum of the cross-correlation function
One of the assumptions of the cross-correlation method is the immutability of the disturbance signal form received by the detectors [16, 17]. In the performed investigations the disturbance was excited artificially by injecting the indicator. In this case the shape of the disturbance signal is modified mainly by a diffusive dilution of the indicator. This phenomenon was discussed theoretically  and defined in the form of the stochastic Local Diffusion Random Walk model of dilution [13, 15]. Actually, the effect of the modification of the shape of the disturbance signal can be observed on raw signals received by the detectors (see: Figure 6). In spite of that a sharp extreme is to be observed in the cross-correlation function (Figure 8). Only in 0.8% of the analyzed cases the maximum of the cross-correlation function was fuzzy, resulting in a poor accuracy of estimation of the transit time. It was proved  that the recognition of the function extreme by means of the weighted mean method allows to achieve a 30% lower systematic error than by simple max. or min. searching (e.g. by differentiation of the analysed function).
Transit times lower than 420 ms were immeasurable by means of the presented method. The reason of this effect has not been explained unambiguously (low sampling frequency and near-field of thermal disturbance were taken into account). Therefore, in order to estimate of Re, τ 23 was calculated as a τ 03 – τ 02 . The transit time τ 03 was well-measurable and correlated with the reference flow. The τ = f(Q) dependence is non-linear but it may be precisely approximated by the power function (R 2 = 0.961). The reason of nonlinearity has not been unambiguously explained and the best fitting by power function was found experimentally.
Unreliability of various cardiac output measurement methods comparison
The worst case
There are many other methods of CO measurements and some of them have been applied clinically. Among non-invasive methods the most significant one is echocardiography. In Doppler’s echocardiography the CO is calculated by means of the blood velocity [13, 22]. The spatial profile of the flow of blood through the specific vessel is unknown (in calculations a laminar flow is assumed), therefore the results of measurements are inaccurate. Another approach is based on the estimation of the left ventricle volume by means of its two-dimensional cross section [13, 23]. The estimation is realized by rough equations which influents the uncertainty of measurements. Another non-invasive method is rheocardiography [3, 5]. The method itself was designed for continuously monitoring of tthe hemodynamic parameters in healthy patients (aircraft pilots). It was clinically proved  that with reference to patients with cardiac a illness this method is suitable for monitoring long-term tendency rather than isolated measurement. Most of the modern minimally invasive methods based on the pulse contour analysis (e.g. PiCCO  and LidCO ) assume an immutable transfer function between excitation (blood flow generated by the heart) and response (pressure in peripheral artery). Generally this transfer function is unknown; therefore, its empirical estimation by means of other methods (most often by thermodiluion) is indispensable.
In fact, the accuracy of the methods mentioned above is unknown. In many papers only the correlation of the investigated method with the reference is estimated. Mostly, the reference constitutes thermodilution (or - hardly ever - the Fick method) with a typical accuracy in the range of 18-35% [2, 12, 13].
The application of the cross-correlation method is not confined to cardiac output measurements only. One of the most fundamental problems connected with the mechanical heart supporting therapy is the estimation of the flow generated by the heart prosthesis [9, 24–27]. The cross-correlation method with a spontaneously excited disturbance (e.g. the blood pressure variation produced by artificial valves) may be applied in ventricular assisting devices.
The performed investigations and theoretical analysis have shown, that the cross-correlation method may be applied in cardiac output measurements. This kind of measurements seems to be more accurate and disturbance-resistant than clinically applied thermodilution. The idea of cross-correlation cardiac output measurements has been patented  (patent pending).
The presented investigations were performed on one geometry of physical models of heart and vessels. Therefore the obtained results should be treated as preliminary ones. Investigations should be continued on other geometries and – in future – on animals.
This work was supported by the Polish Ministry of Science and Higher Education (grant No. N518 336135).
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