Abstract
We study complex scaling properties of RR and QT intervals of electrocardiograms (ECGs) with their equivalences at the cellular level, that is, interbeat intervals (IBI) and field potential durations (FPD) of spontaneously beating humaninduced pluripotent stem cellderived cardiomyocyte (hiPSCCM) aggregates. Our detrended fluctuation analysis and Poincaré plots reveal remarkable similarities between the ECG and hiPSCCM data. In particular, no statistically significant difference was found in the short and longterm scaling exponents α_{1} and α_{2} of RR and QT intervals and their cellular equivalences. Previously unknown scaling properties of FPDs of hiPSCCM aggregates reveal that the increasing scaling exponent of QT intervals as a function of the time scale, is an intrinsic feature at the cellular level.
Introduction
In the last few decades, it has been shown that a healthy heart commonly exhibits fractal scaling (longrange correlations) in heart rate variability, i.e., variations in beattobeat RR intervals in electrocardiograms (ECGs)^{1}. Alterations in the scaling properties have been observed in the case of cardiac diseases, such as congestive heart failure^{2,3}, myocardial infarction^{4,5}, atrial fibrillation^{6}, and dilated cardiomyopathy^{7}. QT intervals also exhibit spontaneous beattobeat fluctuations^{8} and QT variability has been an important measure in cardiac safety and drug development, because prolongation of QT intervals increases the risk of ventricular arrhythmias, such as Torsade de Pointes^{9,10}, and repolarisation lability^{11}. On the other hand, relatively few studies have characterised the longrange scaling properties of QT intervals under different physical conditions^{12,13,14}.
Less is known about the beattobeat scaling properties at the cellular level. With the rise of humaninduced pluripotent stem cell (hiPSC) technology^{15}, it is now possible to study the complex nonlinear properties at the cellular level. Spontaneous contraction of hiPSCderived cardiomyocyte (hiPSCCM) aggregates produces a field potential comparable to an ECG waveform. In particular, peaktopeak intervals in the field potential denoted here as interbeat intervals (IBIs) correspond to RR intervals, and field potential durations (FPDs) are equivalent to QT intervals^{16,17,18,19,20}.
Recently intrinsic powerlaw behaviour of IBIs of the isolated clusters of cardiomyocytes (CMs) has been characterised with one or two scaling exponents over predefined scale ranges^{21,22,23}. In this study, we assess complex variabilities of IBI as well as FPD times series more in depth. To the best of our knowledge, scaling properties of FPDs of isolated CMs have never been investigated. Complex variabilities of RR and QT intervals of in vivo heart are also evaluated. Understanding the complex dynamics of IBIs and FPDs in comparison to the RR and QT variabilities is important in establishing hiPSCCM aggregates as an ideal in vitro model of the human heart. Moreover, these studies provide new insights into the intrinsic QTRR dynamics in the absence of the autonomic nervous system.
Methods
ECG recordings
Raw ECG recordings are obtained from the MITBIH Normal Sinus Rhythm database of PhysioNet^{24}. RR and QT intervals are extracted using the PhysioNet algorithm^{24} and other software^{25,26,27}. Low quality signals and ectopic beats have been discarded (see Preprocessing). The final set of ECG data contain 18 RR and QT interval time series of 24 hours from healthy individuals of 13 women (age from 20 to 50 years) and 5 men (age from 26 to 45 years). The average length of RR and QT intervals is around 9800 beats.
Cell culture and differentiation
Oral and written information of the study has been provided and a signed informed consent has been obtained from the participants. The study has been approved by the ethics committee of Pirkanmaa Hospital District to establish, culture, and differentiate the hiPSC lines (R08070). All experiments were carried out in accordance with following all relevant rules and regulations set by Tampere University. Healthy control hiPSCs were derived from skin fibroblasts of a 55yearold female and a 44yearold male (hiPSC lines UTA.04602.WT, UTA.04511.WT). Both subjects showed no detectable cardiac diseases when the skin biopsy was taken. The hiPSCs were generated and characterised as described by Takahashi et al.^{28}. The hiPSCs were cultured and differentiated as previously described^{29}. All the hiPSCs were genotyped to ensure that no major cardiac genetic disease mutations were present. A normality test on Graphpad Prism 8 software (GraphPad Software, Inc., USA) was conducted on the beating frequency and field potential duration of each sample to ensure the normally distributed population.
Multielectrode array (MEA) measurements
Spontaneously beating cardiomyocyte (CM) aggregates (day 30–70) were manually dissected and plated on 1% gelatincoated 6well MEAs (Multichannel Systems, Reutlingen, Germany). Field potential signals were recorded from the CM aggregates under serumfree EB medium (knockout DMEM, nonessential amino acids, GlutaMAX and penicilin/streptomycin) at 36 ± 1 °C at 10 kHz sampling frequency using MEA 1060InvBC and MC_Rack software (Multichannel Systems, Reutlingen, Germany). The field potentials were continuously recorded for 30 minutes at the baseline. The data obtained from MEA were analysed using a custommade analysis module in Origin2018 software (OriginLab Corporation, USA). The signals displaying the highest amplitudes, low signaltonoise ratios, and clear repolarisation phases were chosen for the analysis.
Two parameters that were extracted were IBIs and FPDs. An IBI is defined as the time period between two consecutive depolarisation peaks. A FPD is defined as the time period measured from the first upstroke of the depolarisation wave to the baseline of the repolarisation wave. FPDs have been shown to correlate with APD90 from action potential measurements^{16,20}. Both IBIs and FPDs were extracted using a customdeveloped analysis module in Origin 2017 (MicroCal Origin™, USA), in which each upstroke of the depolarisation wave is detected as the start of a field potential, and the end of the field potential is calculated semiautomatically by detecting where the repolarisation decay phase intersects with the 0 μV line abscissa. The lengths of IBI and FPD time series range from 900 to 3000 beats with the average length around 1600 beats. The IBI and FPD datasets are available from the corresponding author on reasonable request.
Preprocessing
Each time series was filtered before analysis in order to discard any artificial noises and ectopic beats. All the RR and QT intervals or their cellular equivalences below 200 and above 3000 ms were systematically discarded as nonphysical values. Then an appropriate envelope, or lower and upper limit, was selected around a global trend, so that the intervals outside the envelope were filtered out. The trend was calculated with a 5th order polynomial fit and twice the standard deviation of the time series was used for the envelope size, or limits.
Poincaré plot
A Poincaré plot is a standard method to measure and visualise the temporal correlation of a time series at the shortest time scale. For a given time series {x_{t}}_{t = 1, …, N}, each x_{t} is plotted against x_{t + 1}. To characterise the distribution of the data points on the plane, an ellipse fitting technique^{30} is employed. The standard deviation of the data points perpendicular to the line x_{t} = x_{t + 1}, denoted as SD1, represents shortterm variability of the data^{31}. The standard deviation along the line x_{t} = x_{t + 1}, denoted as SD2, reflects longterm variability, implied by the relation:
where σ is the standard deviation of the time series^{31}. Equation 1 is equivalent to the statement that the sum of shortterm and longterm variability is the total variability. SD1 and SD2 are computed from the eigenvalues of the covariance matrix between x_{t} and x_{t + 1}.
A complementary measure to SD1 and SD2 is the Pearson’s correlation coefficient r, which represents the linear correlation between time series x_{t} and x_{t + 1}, defined by
where cov stands for the covariance and σ is the standard deviation. The Pearson r ranges from −1 to 1, where values closer to 1 indicate positive linear correlation, and values closer to −1 indicate negative correlation.
Detrended fluctuation analysis
Detrended fluctuation analysis (DFA), originally introduced by Peng et al.^{32}, has been established as a reliable method to detect longrange correlation in a nonstationary time series. We follow the algorithms as described by Kantelhardt et al.^{33}. The implementation of the algorithm has been validated against the PhysioNet DFA software package^{24}; see section 1 of the supplementary information for more details. For a time series of length N with observations \({\{{x}_{t}\}}_{t=1,\ldots ,N}\), the DFA procedure can be summarised in four steps:

1.
The profile of the time series is defined by taking an integrated sum of the series:
$$y(k)=\sum _{t=1}^{k}({x}_{t}\langle x\rangle ).$$ 
2.
The profile is divided into N/s nonoverlapping windows of equal length s. In each window, an ndegree polynomial approximation y_{tr} representing a local trend is computed by a leastsquares fit. For our analysis we use the firstorder DFA, in which a linear trend is eliminated from each window. When the windows does not divide the profile evenly, a reverse ordering of the window is averaged with the original ordering, so that the windows cover the whole profile.

3.
The rootmeansquare of the average variance of the residuals (y − y_{tr}) over all 2N/s windows defines the fluctuation F for window size s.
$$F(s)=\sqrt{\frac{1}{2N/s}\,\sum _{m=1}^{2N/s}\,[\frac{1}{s}\sum _{i=1}^{s}{[{y}_{m}(i){y}_{tr,m}(i)]}^{2}]}$$(2)The window sizes range from s = 4 to s = N/4.

4.
In presence of powerlaw scaling, F(s) ~ s^{α}. The scaling exponent α is the slope of F(s) in loglog scale.
The scaling exponent α describes the nature of the correlation present in the data. The white noise with no correlation and Brown noise are characterised by α = 0.5 and α = 1.5, respectively. Values 0.5 < α < 1.5 indicate longrange correlation, and values α < 0.5 correspond to anticorrelations. The value α = 1 corresponds to 1/f or “pink” noise, often referred to as fractal.
More than one scaling exponents may be required to describe different correlations at different scales. It is a common practice to define two scaling exponents, α_{1} and α_{2}, to describe shortrange and longrange correlations, respectively. It is also possible to calculate the gradient of F(s) as a function of s and define a spectrum α(s), also known as continuous α or local α. Here we use the αβ filter^{34}, which is a simplified version of a Kalman filter, to recursively estimate a local leastsquares fit for tracking the evolution of the gradient of F(s) in loglog scale. The α spectrum provides a more complete description of complex correlation properties of a given time series than two scaling exponents with predefined scale ranges. Despite possible limitations of the αβ filter, such as oversmoothing or undersmoothing of the gradient depending on the choice of its parameters, the method is sufficient for our purpose to assess the general scaling patterns of our time series data. The limitations may be overcome by advanced filtering techniques using other types of smoother based on Kalman filters^{35}.
Statistical analyses
Normality of the measures obtained from Poincaré analysis and DFA was checked with ShapiroWilk test. When comparing the measures between ECG and hiPSCCM data Welch’s ttest was employed to determine the statistical differences. If a variable did not meet the normality requirement for ttest, nonparametric Wilcoxon ranksum test was employed. All the measures are presented in minmax, median, and interquartile range (Q1–Q3). Distributions of the measures are plotted in section 2 of the supplementary information.
Results
Quantitative analysis of Poincaré plots
Figure 1 shows the representative Poincaré plots of RR and QT intervals extracted from ECGs. Most of the RR intervals show a shape of an ellipse, as in Fig. 1(a), but 20% of the samples have a fanlike shape, which is spread out towards larger RR intervals, as shown in Fig. 1(b). Results are in line with the previous studies showing the Poincaré plots of RR intervals are spread along the line of identity^{36}.
The Poincaré plots of the corresponding QT intervals, also shown in Fig. 1 in the same time scale, have elliptical shapes along the line of identity, but with less eccentricity and variation, compared to those of the RR intervals. The standard quantitative measures to characterise the poincaré plots, averaged over 18 control ECG recordings, are listed in Table 1.
Figure 2 and Table 1 show the Poincaré plots and the related measures of IBI and FPD time series of the hiPSCCM aggregates, which correspond to RR and QT interval times series of ECGs. Each Poincaré plot Fig. 2(a) is plotted for the IBI time series, measured from a cell aggregate in each well, labelled as A, B, C, D, E, and F on a sixwell MEA. The Poincaré plots in Fig. 2(b) are plotted for the corresponding FPD time series.
The IBI and FPD values vary considerably among the aggregates. The hiPSCCM aggregates tend to behave more erratically as they contract spontaneously without any inputs from the autonomous nervous system. Their beattobeat variations are also extremely sensitive to small fluctuations in the environment, e.g., in the temperature, pressure, oxygen levels, and ion concentration. Consequently, the absolute measures of SD1 and SD2 have large variances. In order to make the visual comparison of the Poincaré plots among the aggregates easier, we have normalised the data by subtracting the mean offset and scaling by the standard deviation, so that each Poincaré plot in Fig. 2 is centred at zero and comparable with each other in magnitude. Therefore, instead of computing SD1 and SD2 separately, we examine the ratio SD1/SD2 and Pearson’s correlation coefficient r between the different groups, which are unaffected by the normalisation. There is a remarkable agreement between RR and IBI in SD1/SD2 and r, while a discrepancy exists between QT and FPD. The significantly larger SD1/SD2 (with statistical significance p = 0.023) and smaller r (p = 0.015) of the FPD Poincaré plots suggest that the FPD time series have significantly larger shortterm variability with respect to the longterm variability.
Detrended fluctuation analysis
We first quantify two scaling exponents α_{1} and α_{2}, which represent shortrange and longrange correlations, respectively. In particular, α_{1} is calculated in the time scale of less than 20 beats, α_{2} in the scale of more than 30 beats. The α_{1} and α_{2} values of RR and QT intervals of ECGs and IBIs and FPDs of the hiPSCCM measurements are visualised in Fig. 3. Each corner of the quadrilaterals corresponds to a scaling exponent. The mean α values are marked with bold lines and the spread of the values is shown in coloured bands. The means and standard deviations of the α_{1} and α_{2} values are also listed in Table 2.
Figure 3 allows us to compare the relative magnitudes of the α values from the shapes of the quadrilaterals. On the average, the quadrilateral for the hiPSCCM aggregates resembles that of ECGs. The top sites of the quadrilaterals connecting the α_{1} and α_{2} of RR intervals and IBIs are flat, i.e., α_{1} ≈ α_{2} ≈ 1, suggesting that the fractallike scaling property is invariant over the time scale. On the other hand, QT intervals have significantly different α_{1} and α_{2}. The mean α_{2} is close to one, indicating that QT intervals are longrange correlated as the RR intervals, while the mean α_{1} is much closer to 0.5, indicating that the correlation properties resemble those of white noise in the short time scale. The scaling behaviour of FPDs is in notable agreement with that of QT intervals, with a small discrepancy in α_{2} (p = 0.09).
In further analysis, we examine the spectra of the scaling exponent α defined over a continuous time scale. The estimation of local α using the αβ filter^{34} produces a smooth spectrum, which depicts the evolution of the scaling exponents over the scales from shortrange to longrange. The left panel of Fig. 4 shows the characteristic scaling patterns of the α spectra of RR and QT intervals. The α spectra of RR intervals approach a constant value with increasing scale resulting α_{1} ≈ α_{2} ≈ 1. In contrast, the α spectra of QT intervals start at lower values and increase with scale, hence α_{2} > α_{1}. Similar results have been shown in the case study of the heart rate variability during pregnancy^{12}.
Overall, Fig. 4 reveals notable similarities in the scaling patterns of ECGs and the hiPSCCM aggregates. The hiPSCCM aggregates tend to behave more erratically. Consequently, their α spectra are not as precise as those of ECGs. However, the scaling patterns observed in ECG, namely, the descending α spectra of RR intervals and the increasing α spectra of QT intervals, are clearly present in the spectra of hiPSCCM aggregates. Unfortunately, the field potential measurements of the hiPSCCM aggregates are limited in length to about 1600 intervals, so that the maximum window size in DFA is about 400 (see Methods). Therefore, we are not able to compare the spectra for larger scales than what is shown in Fig. 4. This result, however, already shows that the scaling properties in beat rates of the hiPSCCM aggregates are well comparable to those of ECGs.
Discussion
Our quantitative analysis reveals that RR intervals and their in vitro equivalent from hiPSCCM aggregates (IBIs) share a common geometry in Poincaré plots with a positive linear correlation between two consecutive beats. Quantitative measures according to the ellipse fitting technique^{30} show that the ratio of shortterm to longterm variabilities is consistent between the ECGs and the hiPSCCM data. Therefore, clusters of hiPSCCMs exhibit beat rate dynamics comparable to a human heart. This has also been confirmed in previous studies^{21,22}. On the other hand, there is a significant difference between the hiPSCCM aggregates and human hearts in variations of the field potential durations, i.e., QT intervals and FPDs. The ratio of shortterm to longterm variability is larger in FPDs. This may be due to the erratic and immature nature of the hiPSCCMs, causing random fluctuations between consecutive beats. However, as it did not affect the IBIs in the same way, the reason may be found in the nontrivial relationship between QT and RR intervals (see below).
The complexity of the RR and QT variabilities and their equivalences at the cellular level is further examined in their longerrange scaling properties. The conventional way to compute short and longterm scaling exponents α_{1} and α_{2} in two predefined scales shows a remarkable agreement between RR intervals and IBIs (Fig. 3 and Table 2). The results are in line with previous reports^{3,21,22}. Here, more complete descriptions of the scaling properties are given by the full spectra of the scaling exponents. The α spectra of RR and QT in Fig. 4 are in good agreement with results from previous studies^{12,34}.
The α spectra of IBI and FPD, reported here for the first time, show a notable similarity with the RR and QT intervals of the ECG data. The overall resemblance implies that the distinct scaling patterns of RR and QT intervals are independent of the autonomous regulation of the nervous system, as it is also present in the isolated hiPSCCM aggregates without any external stimulation. The α spectra of the FPDs stay at low scaling exponents, but the increasing trend towards larger scales–similar to that of the QT intervals–is present. The less prominent longrange correlation may be due to the different QTRR relationship at the cellular level.
In a normal heart, QT variability is often described in the context of heart rate variability, because the variability in RR intervals is the major physiological source of variability in QT intervals^{8}. There is a clear response of QT to the acceleration and deceleration of the heart rate^{37}. Moreover, recent studies have shown that there is a nontrivial dynamic relationship in terms of transfer entropy between RR and QT intervals^{38}. The relationship is not as clear at the cellular level; FPDs do not necessarily have a positive correlation with IBIs, contrary to the wellknown QTRR relationship (See Fig. 5). Therefore, the discrepancies that appear only for FPDs but not IBIs, such as lower scaling exponents and relatively large short to longterm variability ratio of FPDs, depicted by the Poincaré plots above, may be due to the less significant influence of IBIs on FPD variability compared to that of RR interval changes on QT variability.
The spontaneously beating hiPSCCMs are relatively immature cells, which better represent a fetal heart with an underdeveloped contraction machinery and organised structural filaments, when compared to adult cardiomyocytes^{39}. Even though the hiPSCCMs have an expression of similar panel of ion channels as adult cardiomyocytes, they may have divergent expression levels of atrial, nodal, and ventricular cardiomyocytes^{40}. The varying expression levels and other factors, such as differentiation techniques, age of the cells, variations in temperature and ion concentrations in the recording solutions, and recording protocols may introduce erratic variability in the beat rates of hiPSCCM aggregates. Therefore, the interpretation of the results must be practised with caution. However, our results show that the beating dynamics of the hiPSCCM aggregates resemble those of in vivo heart, despite their relative immaturity and the absence of the autonomic neural input.
Even though the hiPSCCM aggregates were derived from only two subjects, we assume they are independent, as there are many factors in culture and differentiation process of the hiPSCs that introduce variations among cells, such as passage number, age of the cells, and batch of cultures and differentiation. After the cardiac differentiation, different cell types (nodal, atrial, and ventricular), various sizes and number of beating cells in the culture further give rise to variations among the cell aggregates.
A recent study has suggested that in the presence of intercellular connections along with electromechanical interactions, intrinsic clocklike signalling of the pacemaker cells in sinoatrial node tissues, which are similar to hiPSCCM aggregates in vitro, adapts and modifies their beat dynamics, contributing to the overall fractallike behaviour of the heart^{41}. Therefore, our findings support the idea that hiPSCCM aggregates could be an ideal in vitro model of a heart, i.e., a suitable platform (i) to model cardiac diseases, (ii) to screen new treatment options, and (iii) to assess cardiac safety of new chemical entities of potential new drug candidates.
Fractality and powerlaw behaviour of IBIs have been attributed to ionchannel gating and intracellular mechanisms that are themselves nonlinear processes. In an adult CM, variation in intracellular Ca^{2+} is the main trigger for excitationcontraction coupling, which generates mechanical contraction. Due to immaturity, the Ca^{2+} transient is slower and smaller in amplitude for hiPSCCMs^{42}, but is clearly present and closely related to the beat rate variability (BRV) of hiPSCCM aggregates. In particular, the intracellular sarcoplasmic reticulum (SR) Ca^{2+} cycling and mitochondrial Ca^{2+} extrusion, and the crosstalk between SR and mitochondria^{43} exhibit fractal behaviour, hence contribute to the fractal BRV at the cellular level^{22,44}.
Intracellular Ca^{2+} cycling, among other cellular processes, also contributes to the beattobeat variation of the overall repolarisation, which causes variations in QT intervals at a stationary heart rate^{8}. Therefore, we may postulate that the fractal behaviour of the intracellular Ca^{2+} cycling is also accountable for the intrinsic fractal scaling of FPDs in hiPSCCM aggregates. There are also other factors, such as stochastic fluctuations in ion currents and intercellular interactions that cause variations in QT intervals. Similar effects may be present also in FPDs, which calls for further systematic investigations.
Other standard HRV time and frequencydomain measures are available in the section 3 of the supplementary information.
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Acknowledgements
This project would not have been possible without the kind help form Markus Haponen, Henna Lappi for their technical expertise with the stem cell culture and cardiac differentiation. The project is supported by the Finnish Academy of Science and Letters, Academy of Finland (key project 304458: Health tracking through fractal analysis of complex signals), Aarne Koskelo Foundation, Tampere University of Technology graduate school grant, and University of Tampere graduate school grant. The authors acknowledge the Tampere facility of electrophysiological measurements for their service.
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E.R. and K.A. conceived the project, D.S. performed the experiments, J.K., I.P., and J.L., analysed the results. J.K. wrote the paper and all authors reviewed the manuscript.
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Kim, J., Shah, D., Potapov, I. et al. Scaling and correlation properties of RR and QT intervals at the cellular level. Sci Rep 9, 3651 (2019). https://doi.org/10.1038/s41598019402479
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DOI: https://doi.org/10.1038/s41598019402479
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Acta Biotheoretica (2021)
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