Moving-average processing enables accurate quantification of time delay and compares the trending ability of cardiac output monitors with different response times

doi:10.21203/rs.3.rs-7116969/v1

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Research Article

Moving-average processing enables accurate quantification of time delay and compares the trending ability of cardiac output monitors with different response times

https://doi.org/10.21203/rs.3.rs-7116969/v1

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published 06 Oct, 2025

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Background

Continuous cardiac output (CCO) monitoring using pulmonary artery (PA) thermodilution and newly introduced beat-to-beat cardiac output (CO) monitoring technologies exhibits different response time delays. These differences can hinder accurate comparisons of their trending abilities. To address this, we applied moving average processing to the beat-to-beat CO monitor data to evaluate its effect on trending assessment accuracy. This study aimed to confirm the effectiveness of moving average processing for such comparisons.

Results

This was a single-center, retrospective, observational study conducted at a 916-bed university hospital. A total of 20 patients undergoing kidney transplantation were included. We analyzed the trending ability of arterial pressure cardiac index (APCI) and estimated continuous cardiac index (esCCI) relative to continuous cardiac index (CCI) derived from PA thermodilution. Trending ability was assessed using a Polar plot and Bland-Altman analyses. A wide range of moving average windows (0–60 minutes) was applied to APCI and esCCI. The polar concordance rate at 30° exceeded 92% for moving average windows between 20 and 30 minutes, with APCI peaking between 21 and 27 minutes. These improvements reflected both time-shifting and filtering effects of the moving average process.

Conclusions

Moving average processing over 20 to 30 minutes significantly enhanced concordance between esCCI and reference CCI, with APCI demonstrating similarly high concordance in the same time window. This approach effectively compensates for differences in response time delays between CO monitoring modalities, enabling more accurate assessment of trending ability.

Cardiac Output

Pulse Wave Transit Time

Response Time

Trends

Continuous cardiac output (CCO) monitoring using pulmonary artery (PA) thermodilution is known to provide delayed information in clinical settings due to its reliance on historical data1,2. Moreover, several reports have suggested that hemodynamic management using the Swan-Ganz catheter does not improve patient outcomes, 3–5, although it continues to be used. 6 This highlights the need for caution in its application. 7–9

In contrast, various continuous cardiac output (CO) monitoring technologies that operate on a beat-to-beat basis—such as those using blood pressure waveforms10 or bioimpedance11—have been introduced over the past two decades. One such example is the estimated continuous cardiac output (esCCO) monitor, which calculates CO based on the time interval from the R-wave of the electrocardiogram to the arrival of the peripheral pulse oximetry waveform12,13.

Recent studies have compared beat-to-beat monitors, such as arterial pressure-based cardiac output (APCO) and esCCO, to traditional cardiac output (CCO) monitors. 14,15 However, the validity of these comparisons remains uncertain. As noted by Saugel et al.16, when evaluating the trending ability of two CO monitors, it is crucial to account for their respective response delays.

While the exact cause of delays in CCO monitoring remains unclear, previous research17 suggests that moving average signal processing may be a contributing factor. Moving averages can introduce two key effects on a signal: time shift and filtering. Therefore, to understand the nature of such delays, it is important to examine these two effects separately in addition to evaluating the moving average as a whole.

The moving average is a well-known low-pass filter and has been previously used to estimate central blood pressure waveforms from peripheral ones18 and to remove motion artifacts from distorted pulse waveforms19. The amplitude and phase characteristics of a signal processed by a moving average can be described analytically. If the original signal is a sinusoidal waveform:

$$\:\text{y}=\text{x}\left(t\right)=\text{sin}\left(2\pi\:ft\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)$$

And the moving average is calculated over a time window t, then the filtered signal becomes:

$$\:{y}^{\tau\:}=\frac{1}{\tau\:}\underset{t-\tau\:}{\overset{t}{\int\:}}\text{sin}\left(2\pi\:ft\right)dt=\frac{1}{2\pi\:f\tau\:}\sqrt{2\left(1-\text{cos}\left(2\pi\:f\tau\:\right)\right)}\times\:\text{sin}\left(2\pi\:f\left(t-\frac{\tau\:}{2}\right)\right)\:\:\:\left(2\right)$$

From Eq. (2), the amplitude of the filtered signal is:

$$\:\frac{1}{2\pi\:f\tau\:}\sqrt{2\left(1-\text{cos}\left(2\pi\:f\tau\:\right)\right)}$$

and the phase delay corresponds to half the averaging time:

$$\:\frac{\tau\:}{2}$$

If two signals have different moving average times t0 and t, then their amplitude ratio r and phase difference f are given by:

$$\:\text{r}=\raisebox{1ex}{$\tau\:$}\!\left/\:\!\raisebox{-1ex}{${\tau\:}_{0}$}\right.\times\:\raisebox{1ex}{$\text{sin}\left(\pi\:f{\tau\:}_{0}\right)$}\!\left/\:\!\raisebox{-1ex}{$\text{sin}\left(\pi\:f\tau\:\right)$}\right.$$

$$\:{\phi\:}=\raisebox{1ex}{${\tau\:}_{0}-\tau\:$}\!\left/\:\!\raisebox{-1ex}{$2$}\right.$$

According to Equations (3) and (4), when t = t0, the amplitude ratio is 1 and the phase difference is 0. Under these conditions, Polar plot analysis, as described by Critchley et al.2, yields optimal trending agreement, characterized by a mean polar angle of 0° and a minimal polar angle standard deviation (SD).

If the delay in CCO monitoring arises primarily from moving average processing, then applying appropriate moving average adjustments to CO monitor data should improve their concordance, particularly when evaluated using polar plot metrics.

Accordingly, this study applied moving average processing to compare the trending ability of two CO monitors with different response delays. In addition to standard moving averages, we independently assessed the individual effects of time shifts and signal filtering. To address the potential for information loss inherent in averaging20, we applied a wide range of moving average windows, exceeding those previously examined14,15, in search of the optimal range.

Besides Polar plot analysis—which evaluates trending ability—we also conducted Bland-Altman analysis to assess the standard deviation (SD) of differences and to investigate the presence of proportional bias. The rationale for including Bland-Altman analysis is that the total variance observed in the differences includes both between-subject and within-subject components. Since trending ability is inherently a within-subject property, a prominent within-subject variance would be reflected in the SD of differences. The degree of proportional bias is likewise indicative of trending performance21.

In summary, this study investigates the necessity and optimal conditions of moving average processing when comparing CCO monitors with other CO monitoring methods. The relationship between moving average time and trending performance, as revealed by Polar plot and Bland-Altman analyses, forms the basis for this approach using clinical data.

We conducted simulations of continuous cardiac index (CCI) data measured by pulmonary artery (PA) thermodilution, as well as arterial pressure cardiac index (APCI) and estimated continuous cardiac index (esCCI) data. These datasets were originally obtained in the study by Terada et al. [13]. CCI was measured using balloon-tipped, flow-directed thermodilution pulmonary artery catheters (Edwards Lifesciences, Irvine, CA, USA) and the Hemodynamic Monitor Vigilance II (Edwards Lifesciences, Irvine, CA, USA). APCI was measured using the FloTrac/Vigileo™ system, Version 3 (Edwards Lifesciences, Irvine, CA, USA). Pulse wave transit time for esCCI was obtained using a BSM-9101 bedside monitor (Nihon Kohden, Tokyo, Japan), and transmitted to a personal computer, where esCCI was calculated using a custom C-compiled program. APCI values were averaged over 20-second intervals, while esCCI was computed using a dynamic average of 64 consecutive heartbeats.

The dataset comprised 20 patients undergoing kidney transplantation. Data were collected at 1-minute intervals, yielding a total of 5,027 data points over 83.8 hours. The study was approved by the ethics committee of Toho University Omori Medical Center, and all procedures conformed to the ethical standards of the 1964 Declaration of Helsinki and its later amendments. Written informed consent was obtained from all participating patients.

Three types of signal processing were applied to the esCCI and APCI data, as illustrated in Fig. 1:

1. Moving Average: The signal was averaged over a past window of duration τ (minutes) up to the current time:

$$\:{a}_{mov\_avg}=\raisebox{1ex}{$\sum\:_{j=0}^{m}{a}_{i-j}$}\!\left/\:\!\raisebox{-1ex}{$m+1$}\right.$$

2. Time Shift: The signal was shifted in time by τ/2, using the past value as the

$\:{a}_{shift}={a}_{i-n}$ (6)

3. Filtering: A symmetric moving average was applied, centered on the current time, using values from τ/2 before to τ/2 after:

$$\:{a}_{filt}=\raisebox{1ex}{$\sum\:_{j=-n}^{n}{a}_{i-j}$}\!\left/\:\!\raisebox{-1ex}{$2n+1$}\right.$$

Here, if τ is even, m = τ and n = τ/2.

Polar Concordance Rate Analysis (esCCI vs. CCI and APCI vs. CCI)

CCI was used as the reference signal, and esCCI and APCI served as test signals after application of a moving average with varying durations τ. τ was varied from 0 minutes (no averaging) to 60 minutes in 1-minute increments, generating 61 different datasets each for esCCI and APCI. For each τ, we computed the differences in measurements over time intervals t1, ranging from 1 to 60 minutes. This resulted in 3,660 (61 × 60) combinations of τ and t1 for both esCCI and APCI, respectively. For each of the 20 patients, we computed the differential data pairs between CCI and the corresponding esCCI/APCI values and performed Polar plot analysis [2].

Data pairs with an average CCI change and test signal change (esCCI or APCI) of less than 0.6 L/min/m² were excluded. This threshold corresponded to 15% of the mean CCI value, consistent with the previous study [13].

Polar Plot and Bland-Altman Analyses (esCCI vs. CCI and APCI vs. CCI)

Polar plot analyses were conducted with CCI as the reference and esCCI and APCI (moving averaged at τ) as test signals. The time interval t1 was fixed at 10 minutes, based on prior studies [22], which suggested it as an appropriate duration for assessing fluid responsiveness. For each τ, we calculated the polar concordance rate at 30°, the mean polar angle, and the standard deviation (SD) of polar angles. The exclusion zone was defined as 0.6 L/min/m².

For Bland-Altman analysis, we computed the bias, the SD of the differences, and the Pearson correlation coefficient between the differences (Y-axis) and the means (X-axis). The correlation coefficient was interpreted as an index of proportional bias.

Comparison of Signal Processing Methods: Polar Plot and Bland-Altman Analyses

To evaluate the effects of different processing methods, esCCI signals were processed using three approaches: moving average, time shift, and filtering. The reference remained CCI. For all methods, t1 was fixed at 10 minutes, and τ was varied. Polar plot and Bland-Altman analyses were conducted as described above. The exclusion zone was set to 0.6 L/min/m² in all comparisons.

All statistical analyses were conducted using Microsoft Excel (Office 365; Microsoft Corp., Redmond, WA, USA) and MATLAB R2023a (MathWorks, Natick, MA, USA).

Polar Concordance Rate at 30° for esCCI vs. CCI and APCI vs. CCI

Figure 2 shows the results of the polar concordance rate at 30°, applying an exclusion zone of 0.6 L/min/m², across varying values of t and t1. For both esCCI and APCI, the concordance rate fluctuated more noticeably at lower t1 values, indicating a higher dependency on t. As t1 increased, these fluctuations diminished. Across the entire t1 range, the highest concordance rates were consistently observed at mid-range values of t.

Polar Plot and Bland-Altman Analyses for esCCI vs. CCI and APCI vs. CCI

Figure 3A illustrates the polar concordance rate at 30° with an exclusion zone of 0.6 L/min/m², analyzed while varying t and keeping t1 = 10 minutes fixed. For esCCI, the concordance rate was below 60% without any moving average processing (t = 0). However, when t increased to between 17 and 30 minutes, the concordance rate exceeded 90%, peaking between 20 and 30 minutes. In comparison, APCI reached approximately 90% concordance within a narrower window between t = 21 and 27 minutes.

Figure 3B shows that the mean polar angle decreased with increasing t for both APCI and esCCI. The mean polar angle crossed zero at t = 19 minutes for APCI and t = 12 minutes for esCCI. Similarly, the polar angle standard deviation (SD) also trended downward with increasing t (Fig. 3C).

In the Bland-Altman analysis (Fig. 3D), the bias shifted in the negative direction as t increased for both signals. The smallest bias (closest to zero) was observed at t = 0. Figure 3E presents the correlation coefficient between the difference (Y-axis) and the mean value (X-axis), which initially showed positive values at t = 0 and transitioned toward negative values with increasing t. The zero crossing of this coefficient occurred at t = 8 minutes for APCI and t = 12 minutes for esCCI.

As shown in Fig. 3F, the SD of the differences decreased with increasing t, reaching a minimum at τ = 24 minutes for APCI and τ = 21 minutes for esCCI, before increasing again at higher པ values.

Comparison of esCCI vs. CCI Using Three Signal Processing Methods: Moving Average, Time Shift, and Filtering

In addition to the moving average method described above, esCCI was also analyzed using time shift and filtering techniques, with t as the varying parameter and t1 = 10 minutes fixed.

As shown in Fig. 4A, the polar concordance rate at 30° peaked between t = 20 and 30 minutes in both the moving average and time shift methods. However, the maximum concordance rate using the time shift method did not exceed 85.5%. In contrast, the filtering method did not exhibit a similar τ-dependent trend in concordance rate.

In terms of mean polar angle (Fig. 4B), both the moving average and filtering methods showed a decreasing trend with increasing t. The time shift method, however, did not display a clear trend. Regarding polar angle SD (Fig. 4C), the filtering method showed a consistent decrease beyond τ = 10 (equivalent to a 5-minute filtering window). The time shift method exhibited a U-shaped pattern, with SD decreasing to a minimum between t = 20–30 (i.e., a shift of 10–15 minutes), followed by an increase with further t increments.

Figure 4D shows that bias decreased in the negative direction with increasing t for the time shift method, whereas the filtering method demonstrated an increase in bias that approached zero.

Figure 4E presents the correlation coefficients between the difference and mean values. Similar to the moving average method, both the time shift and filtering methods showed a shift from a positive value at t = 0 toward negative values as t increased.

Finally, as shown in Fig. 4F, SD decreased with increasing t in the time shift method, reaching a minimum at t = 20 (a shift of 10 minutes), before increasing again. In contrast, the filtering method showed a continuous decrease in SD with increasing t.

This study evaluated the effectiveness of applying moving average processing when comparing CCI with beat-to-beat cardiac indices exhibiting different response times, namely esCCI and APCI. We confirmed that esCCI achieved an acceptable concordance rate when a moving average of 20–30 minutes was applied, while APCI demonstrated a high concordance rate within the 21–27-minute range (Fig. 3(A)).

We further explored the underlying mechanisms of moving average effects, specifically decomposing them into time shift and filtering components. As shown in Fig. 4(B), the mean polar angle crossed zero at t = 8 minutes due to filtering, subsequently shifting further into the negative direction. Figure 4(C) demonstrated that the decrease in polar angle SD up to t = 24 minutes was primarily attributed to time shift effects; beyond this point, filtering effects predominated. These findings suggest that moving averages exert their influence through a combination of time shift and signal smoothing (filtering).

Both the mean polar angle and the correlation coefficient between the difference (Y-axis) and mean (X-axis) values in the Bland-Altman analysis shifted toward the negative direction as τ increased. In esCCI, zero crossings occurred at t = 12 minutes for both the mean polar angle and the correlation coefficient. In APCI, the zero crossing of the mean polar angle occurred at t = 19 minutes, while that of the correlation coefficient occurred earlier at t = 8 minutes. The alignment of zero crossings in esCCI suggests a correspondence between systematic (proportional) error in Bland-Altman analysis and the mean polar angle in polar plot analysis, as previously described [21].

Although polar angle SD decreased with increasing τ beyond 30 minutes in polar plot analysis (Fig. 3(C)), the SD of the Bland-Altman analysis increased (Fig. 3(E)). This discrepancy prompted further comparison between polar plot analyses conducted with and without an exclusion zone. Specifically, we compared results using an exclusion zone of 0.6 L/min/m² versus 0.0 L/min/m². As seen in Fig. 5(C), omitting the exclusion zone increased polar angle SD.

These results suggest that, at τ > 30 minutes, the amplitude of esCCI was significantly attenuated, and the time shift was further prolonged, resulting in greater divergence between CCI and esCCI. Consequently, the SD in the Bland-Altman analysis and the polar angle SD without exclusion increased. However, since the magnitude of changes in both CCI and moving-averaged esCCI decreased as τ increased, the number of excluded data points (i.e., exclusion rate) also rose. This increasing exclusion rate contributed to the observed reduction in polar angle SD when an exclusion zone of 0.6 L/min/m² was applied. Appendix 1 further illustrates the relationship between exclusion rate and polar angle SD, confirming that higher exclusion rates consistently yielded lower SD values.

Overall, this study applied moving averages to facilitate a more accurate comparison of trending ability between CCI and other cardiac output indices with differing time responses. The optimal moving average duration (t) for achieving a concordance rate above 92% between CCI and esCCI was found to be 20–30 minutes. Critchley et al. [2] proposed a threshold of 92% concordance when using a 15% exclusion zone based on mean CO; our findings support this criterion within the specified t range. Although APCI did not reach the 92% threshold, it approached 90% within 21–27 minutes (Fig. 3(A)).

The wide variability in effective moving average times suggests that multiple factors may contribute to the delayed response observed in CCI. These include technical aspects such as the thermal filament heating algorithm [1, 23], clinical conditions like mitral regurgitation [1, 24], and physiological changes associated with bleeding or resuscitation [3, 26].

Given the variability and uncertainty in delay times, CCO alone may be insufficient for clinical decision-making. As noted by Mihm et al. [26], the reliability of CCO is condition-dependent and should be interpreted alongside other continuous hemodynamic parameters.

Oh et al. [14] assessed the trending ability of APCO and CCO using R-squared-based time adjustments for APCO. However, their four-quadrant plot analysis yielded unacceptably low concordance. As shown in Fig. 4(A), simple time shifting did not achieve satisfactory results in our study either, emphasizing the advantage of moving averages, which incorporate both time shift and filtering.

Takakura et al. [15] compared esCCO and CCO values before and after extubation in ICU patients, with and without applying a 20-minute moving average to esCCO. They reported that moving averaging reduced the SD of the difference between the two indices, supporting our current findings, although they did not employ polar plot analysis.

Study Limitations (Rewritten)

A primary limitation of this study was the lack of detailed information regarding the specific averaging algorithm employed by the CCO monitor. We inferred that the observed response delay was attributable to a moving average process. Accordingly, the objective of this report was to evaluate the effectiveness of applying moving average processing when assessing trending ability between two monitors with differing response times.

Another limitation concerns the unequal number of data points analyzed across different t (moving average window) values. This discrepancy arose due to two factors:

Larger t values resulted in delayed start times and earlier end times for output data, thereby reducing the total number of available data points.
The moving average was not computed if the averaging window included even a single missing value. As τ increased, the likelihood of encountering such missing data also rose, further reducing the number of usable data points.

A key challenge for future research is to determine how best to mitigate the effects of data loss and delayed signal responsiveness introduced by large time averaging windows (τ), which can obscure physiologically relevant changes and affect method comparisons. Clear, enough?

Finally, although Critchley et al. [27] recommend that the polar mean angle remain within ± 5° and the radial limits of agreement within ± 30° when comparing CO monitors against thermodilution reference measurements, we focused on trending ability using a single index—namely, the polar concordance rate at 30°.

This study retrospectively analyzed clinical data using polar plots and Bland-Altman methods to investigate how moving average time affects trending ability among cardiac output monitors. Due to the combined effects of time shift and filtering inherent in moving average processing, an acceptable polar concordance rate at 30° was achieved between CCI and esCCI when esCCI was averaged over 20–30 minutes. Similarly, high concordance rates were observed between CCI and APCI with moving average windows in a comparable range.

These findings suggest that moving average processing is a practical and effective method for evaluating the trending ability of cardiac output monitors that differ in response time.

Acknowledgments

The authors have no acknowledgments.

Author contributions

YS analyzed the literature, created the model, analyzed the data, post-processed the findings and was a major contributor to writing the manuscript. RO analyzed the literature, designed the study, created the model, and was a major contributor to reviewing and editing the manuscript.

Funding

Open Access funding enabled and organized by Nihon Kohden Corp.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable requests.

Ethics approval and consent to participate

This study received approval from the ethics committee of Toho University Omori Medical Center. All procedures adhered to the principles of the 1964 Helsinki Declaration and its subsequent revisions or similar ethical standards. Written consent was obtained from all patients who had undergone kidney transplants.

Consent for publication

Not applicable.

Competing interests

Yoshihiro Sugo works for Nihon Kohden Corporation.

Ryoichi Ochiai is an advisor to Nihon Kohden Corporation.

Author details

¹Development Department, Ogino Memorial Laboratory, Nihon Kohden Corporation, Saitama, Japan.

²Faculty of Medicine, Toho University, Tokyo, Japan.

³Senior Counselor, Tokushukai Medical Corporation, Tokyo, Japan

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No competing interests reported.

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Moving-average processing enables accurate quantification of time delay and compares the trending ability of cardiac output monitors with different response times

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