Stefan F. J.: Autocorrelation Patterns in the Left-Ventricular Pressure Course of Isolated Working Hearts at Sinus Rhythm.

Introduction

Left-ventricular pressure (LVP) curves of isolated hearts working at steady state present with very uniform shape of successive beat intervals. The screen of a triggered oscilloscope may appear as almost frozen while writing a series of beats. Such observations suggest a pooling (overlaying) of LVP data from consecutive beats to facilitate enhanced methods, as curve fitting algorithms, to determine hemodynamic parameters. However, short and medium term pattern analyses of LVP curves, necessary to justify such data pooling, have been lacking until now.

Extensive observations about impaired contractility in the presence of considerable arrhythmic excitation are available [1]. Regularity and normal fluctuation of the electrical excitation generated in sino-atrial cell clusters is also known [2,3]. Autonomous neural regulation adds non-linear fluctuation [4,5], that has been noticed as an important diagnostic criterion to discriminate between normal and pathological arrhythmias [5,6,7]. However, our knowledge is trifold limited: first, electrical interbeat variability was assessed rather than mechanical; second, severe arrhythmias rather than normal variability in isolated hearts were investigated; third, the interest focused on contractility rather than the whole LVP course.

The present study uses autocorrelation in LVP curves from a large number of isolated working small animal hearts to clarify on a sound statistical basis:

  1. (a) Do LVP curves of isolated hearts contain short or medium term patterns of similarity?

  2. (b) Which simple hemodynamic beat-to-beat parameters become affected by such rhythms?

  3. (c) Is there a stochastic process describing the re-occurrence of similar LVP curve shapes in series of beat intervals?

Figure 1.

Left-ventricular pressure course of an isolated working Sprague-Dawley rat heart at normal and low temperature (LVP, in mmHg, intracaval subminiatur manometer). Intersections of the upstroke with the average LVP (upper panel) to determine beat interval length precisely; for comparison, electocardiogram (ECG, in arbitrary units, obtained between right atrium and aortic cannula) showing slower upstroke. Both curves sampled at 10 kHz.

icfj.2016.8.57-g001.jpg

Methods

The study is based on LVP curves, obtained from isolated working hearts of male Sprague-Dawley (SpD) and Wistar-Kyoto (WKy) rats, and of mixed sex guinea pigs and ferrets. Samples I contain measurements of four seconds duration, see Table 1 for number of specimen and apparent hemodynamics. Subsets of these samples undergo repetitive measurements, changing temperature or cardiac output, or electrical stimulation. Samples II consist of LVP curves of ten minutes duration from guinea pig and Sprague-Dawley rat hearts.

Experimental setup

The artificial circulation apparatus, filled with bicarbonate buffer (protein-free, 2.5 mM Ca2+, O2 saturated), and the mounting of the working left heart specimen are previously given in detail [8].

A high-fidelity subminiature manometer was introduced to the left ventricle via the aortic valve; LVP is digitised at 1 kHz rate, binned in 0.075 mmHg resolution. Cardiac output was set to about 40 mL min-1 in the guinea pig and rat, and to 60 mL min-1 in the (larger) ferret hearts, by adjusting the roller pump feeding the left atrium (pressure undulations damped by two windkessel and a resistor in between). Mean aortic pressure of 75 mmHg (rat, ferret) or 60 mmHg (guinea pig, due to poorer durability) was maintained by a variable aortic outflow resistor. Temperated water jackets kept 37°C.

Figure 2.

Coeffcients of autocorrelation from a left-ventricular pressure curve. Sprague-Dawley rat at 37°C, 75 mmHg mean aortic pressure, 40 mL min−1 cardiac output. Autocorrelation window width 2s, lagged in millisecond steps. Ninty-five per cent limits of confidence are given to each coeffcient. Upper panel; First second of the autocorrelation curve. Lower panel; First three maxima at lags 273, 546 and 819 ms (for comparison: mean beat interval length from 13 consecutive beats is 272.9 ± 1.13 ms standard deviation). Significantly best autocorrelation at the lag by two beats (duorhythm).

icfj.2016.8.57-g002.jpg

Temperature was lowered to 31°C, or flow or beat frequency was changed, keeping aortic pressure constant. Steady state was carefully established; the specimens of Sample II (10 min recordings) were additionally protected by a temperated windbreak or by immersing in buffer. Electrical pacing (2 ms rectangular pulses, voltage as necessary, typically below 0.7 V) was applied to 50 guinea pig and 66 SpD rat hearts between a spring electrode, clamped to the right atrium, and the aortic cannula. Pacing frequency was set to shorten the spontaneous beat interval by about ten per cent.

Table 1.

Hemodynamics and LVP autocorrelation from 4s–intervals (Samples I)

G. Pig SpD Rat WKy Rat Ferret
N: 494 705 109 29
mLV mg 9431280616 802950580 681846566 2828
Hemodynamics:
Ao mmHg 60 (c) 75 (c) 75 (c) 75 (c)
Fcard mL/min 38.443.926.9 38.243.927.1 38.243.323.5 56.1
nBI [1] 151813 182213 162013 14
tBI ms 242298209 209252163 225297189 261
“, SD % 0.41 0.99 0.53 1.16 0.51 1.40 0.33
tRise ms 536737 374429 357028 56
P[max]LV mmHg 839666 11112693 120142106 104
P[min]LV mmHg 2.48.2-1.9 1.38.7-1.8 1.18.0-3.0 0.16
PED mmHg 4.615-0.8 4.113-5.0 4.612-0.5 2.2
Ṗ[max]LV mmHg/s 260136681813 500963723613 621893015145 3056
[min]LV -mmHg/s 178824281343 239633681934 246240701871 1664
t1Rlx ms 8610855 597943 588319 82
tRlx ms 395527 365923 328817 55
P[Rlx]0 mmHg 38.65629 43.96225 48.58631 44.3
τ0 ms 23.64012 19.6359.6 22.7689.6 34.7
τ ms 18.9659.8 12.4335.5 11.5442.0 15.5
Autocorrelation (LVP, 2s-window):
r[max](3) [1] 0.99989 0.99953 0.99924 0.99993
“, L95 [1] 0.99850 0.99319 0.98614 (0.9922)
| ΔP | [Med]LV % 0.222 0.38 0.250 0.57 0.238 0.58 0.181
| ΔP | [max]LV % 2.44 5.24 5.57 13.3 7.49 17.7 2.28
Mono % 45.550.141.0 67.871.364.2 768467 76
Duo % 41.946.437.5 25.228.72.0 172510 21
Tri % 12.615.89.7 7.09.15.1 7143 3

Entries are sample medians with (in the large samples) the limits of the central 90%–mass of the sample; if appropriate, single sided limits of 90% sample values are given. Species: Guinea pig, Sprague–Dawley (SpD) and Wistar–Kyoto (WKy) rat, ferret. mLV, left ventricular mass; ¯Ao, mean aortic pressure, (c)ontrolled; FCard, cardiac output; nBI, number of beats within the 4s–interval of observation; tBI, interbeat interval, with standard deviation (SD) within 4s; tRise, time from begin of systole to PmaxLV, the peak left–ventricular pressure; P minLV, minimal pressure; PED, end–diastolic pressure; maxLV, peak pressure rise and, minLV , decrease velocity; t1Rlx, time from P maxLV to minLV ;tRlx, duration of isochoric relaxation; P Rlx0 , isochoric initial pressure (regression estimated); τ0, τ, initial and asymptotic time constant of the isochoric pressure decay. rmax(3), peak coeffcient of autocorrelation from 2s–windows moving by 1 up to 3 beats in the ventricular pressure curve, accompanied by L95, confidence limit to expect 95% (80% among ferrets) of future observations exceeding the given figure. |ΔP| Med,maxLV , median and maximum absolute difference between the 2s–window and the lagged ventricular pressure curve (at rmax(3)), expressed as percentage of P maxLV. Mono/Duo/Tri–rhythms: LV counted percentages, with 95% confidence range, of r max(3) occurring at window lag 1, 2, or 3 beats.

Samples I were utilised in further research unrelated to the present study. All animals recieved humane care according to the Tierschutzgesetz (German Animal Protection Act).

Data processing

Each LVP curve is preliminarily checked for irregular beat intervals (BI). BI lengths are defined and determined as the time between consecutive systolic upstroke through the average LVP, see Fig. 1. Average LVP is the mean of the individual 4s-interval (Samples I) or the gliding mean from a 20s window in the 10min-intervals (Samples II). Intervals from Samples I (37 and 31°C) with more than 6% standard deviation of BI always contain irregularities (premature, delayed, or extra contraction) which were overlooked at the time of data sampling; these measurements are discarded from the study. For their duration, Samples II are allowed to contain solitary irregular beats. These were identified, and the longest series of regular BI within the 10 min recording of LVP data is used for analyses which require undisturbed curves.

Figure 2 illustrates the autocorrelation analysis. The coefficients, r, of correlation are calculated in lag steps of one millisecond and accompanied with upper and lower 95% confidence limits. Calculation is protected from digit extinction by using double precision (15 decimals) numbers. Window widths of one and two seconds, and of the length of the first BI, are used for comparison, even wider windows are deployed in regular parts of Samples II. Only the positive r peaks, noted in sequence as r1 . . rn, are of further interest.

The LVP sample is said to be monorhythmic if r1 does not stay significantly below r2 or r3. It is called duo- or triplerhythmic if r2 or r3, respectively, becomes significantly (p < 0.05) higher. Let rmax(3) be the significant maximum among r1, r2, r3. Patterns of higher order (i.e, rmax(k) with k>3) are seperately dealt with among Samples II.

For stochastic purposes (among Samples II), the first BI of approximately median duration (“stencil BI”) is chosen as an autocorrelation window in the respective LVP recording. Following BIs are qualified as “very similar” if their respective rk excedes ninety per cent of the peak r values (disregarding significance, including irregular beats). The numbers of BIs to await the next very similar one are counted, and the cumulative frequencies of waiting times are summed up (thus, the first frequency becomes the total number of waiting periods itself) [9].

Statistics

Sample size in guinea pig and SpD rat is for effective statistics; WKy rat and ferret in Samples I are supplementary. Hemodynamic parameters are extracted beat-by-beat from the LVP curve or from other recordings. Intervals of measurement are characterised by the respective medians. Mean and standard deviation is used instead for BI duration to meet the accepted standard [6]. Logistic time constants of relaxation, τ0, τ, are obtained by regression on pooled LVP decay data [10]. To deal with the influence of beat patterns, the odd- and the even-numbered BIs within each LVP recording are pooled separately (last BI omitted if odd).

Statistics are non-parametric. Samples are described by median and central 90 per cent mass, i.e. the narrowest vicinity around the median that covers 90% of the sample; if appropriate, one-sided percentiles are given instead.

Ninety-five per cent fiducial limits (intervals of confidence) of medians are calculated by order-statistics, those of counted percentages by angular transformation [11].

Table 2.

Species differences in LVP autocorrelation patterns (Samples I)

Spec Mono Duo Tri ΣL
GP n 225 (298.5) 207 (151.1) 62 (44.3) 494
PH < 10-5 < 10-5 0.00370
SR n 478 (426.1) 178 (215.7) 49 (63.3) 705
PH < 10-5 < 10-5 0.00622
WR n 83 (65.9) 18 (33.3) 8 (9.8) 109
PH 0.00279 0.00525 0.53297
Fer n 22 (17.5) 6 (8.9) 1 (2.6) 29
PH 0.34332 0.72613 0.58486
ΣC 808 409 120 1337
Statistics: X2> 76.84; c> 0.1695; p< 10-5

Samples I (GP, guinea pig; SR, Sprague–Dawley rat; WR, Wistar–Kyoto rat; Fer, Ferret) and pattern of left–ventricular pressure curves (Mono/Duo/Tri) as in Table 1. 4×3 table of contingency; n, observed and (expected) numbers; ΣL,C, Line/Column sums; pH, error probability (rounded upwards) for assuming a cell number differing from expectation, adjusted for multiple (viz. 12) comparisons; c, normalised coeffcient of contingency; p, error probability for assuming heterogeneity

Counted frequencies are compared in cross tables by χ2 statistics, with error probabilities of the individual cell adjusted for multiple testing according to Hommel [11]. Hemodynamic differences between odd and even BIs within a recording are tested by matched-pair Wilcoxon tests [11]. Elevated mean absolute deviation at duorhythms is single-sided U-tested [11]. All testing but the latter is two-sided, significance always stated if error probability p < 0.05. Calculating p is omitted if given fiducial limits already prove for significance (by non-overlapping).

Stochastic of occurring similar beats in the LVP curve is analysed as a Poisson process [9], where time is expressed as number of BI. Poisson parameters and their univariate 95% fiducial limits are estimated by non-linear regression [12] from each LVP recording.

Results

Samples I issue short term patterns in LVP at normothermia, their stability and influence on the hemodynamics. Samples II attend to medium and long term patterns at normo- and hypothermia.

Samples I

Table 1 gives the results from autocorrelating 2s-windows (about four BIs). The maximum among the first three coefficients of correlation typically lies above 0.999, with narrow lower fiducial limits, L95. The lagged LVP window differs in median by less than half a per cent of maximum LVP; the maximum LVP difference is, roughly, tenfold and higher in rat hearts. Monorhythm is predominant, clearly in rat and ferrets, among guinea pig almost level-pegging with duorhythm. Applying a 1s-window reduces the predominance of mono- and duorhythms (see Samples II).

Single-beat autocorrelation

If the first BI is used as autocorrelation window, median rmax(10) excels 0.9999 in all samples. The ten possibilities, rmax(10)∈ {r1 . . r10}, become almost equidistributed, albeit the zero hypothesis, tested by the 2I statistic [11], can be scantly rejected in the large samples (guinea pig, p < 0.014; SpD rat, p < 0.039; WKy rat, p < 0.020; ferret, p > 0.47). However, the individual 95% fiducial limits of the frequencies of almost all patterns embrace the expected ten per cent in all samples

Species and sex differences

Species differences seen in the occurrence of autocorrelative patterns are significant but for the small ferret sample, Table 2. The according 2 × 3 cross table for sex differences within the guinea pig sample is significantly (p < 0.023) inhomogeneous; however, this is exclusively due to the frequency of triplerhythm being 7.3% in male against 16.7% in female. The ferret sample is almost perfectly homogeneous by sex, p > 0.7.

Pattern stability with time and flow

Isolated hearts (guinea pig, SpD rat) significantly tend to keep their rhythmic pattern over hours, if control conditions are maintained, Table 3. However, this tendency is not too strict, with coeffcient of contingency about 0.2.

If cardiac output is doubled from below 30 mL min−1 to approximately 60 mL min−1 (10 mL min−1 added in ferret), the heart tends to switch to monorhythm. All 2 × 3 cross tables, counting the three pattern types according to “preserved” or “changed” by the flow increase among 447 guinea pig, 580 SpD rat, 97 WKy rat, and 22 ferret hearts, respectively, are significantly contingent (coeffcients greater 0.58). Seventy-three to ninety per cent (each sample) of the hearts presenting with monorhythm at low flow are monorhythmic at high flow also. On the contrary, only 7 to 16 per cent of duo- and triplerhythm are preserved.

Electrical stimulation

Artificial pacing limits the standard deviation of BI length (obtained from LVP) to below 0.5%, median 0.3%. The observed autocorrelation patterns differ substantially from those (Table 1) seen at sinus rhythm. Monorhythm is significantly (p < 0.030) reduced to 304518% (median, 95% fiducial limits) in guinea pig, and heavily reduced to 263816% in SpD rat hearts. Duorhythm becomes predominant, 526637% in guinea pig, 506337% in SpD rat. Triplerhythms occur in 18318% of guinea pig and in 243615% of SpD rat specimen.

Hemodynamic effects

No hemodynamic differences to speak of exist between the mono and duorhythmic subsamples. U-testing the twelve tabulated parameters for differences between the respective subsamples of the three large guinea pig and rat groups yields six significances (out of 36 tests), four of whom in the greatest sample (SpD). Table 4 compares the interbeat variability of hemodynamics in the monorhythmic against the duorhythmic LVP recordings.

As expected, the mean absolute deviation odd-vs.-even beats always turns out to be higher at duorhythm, in the three large species samples. Amount and significance of the difference vary by parameter.

Table 3.

Stationarity of LVP autocorrelation patterns

Pattern Mono Duo Tri ΣL
Guinea Pig (n = 36; 3 h):
1mon n 169 (134.1) 54 (88.3) 36 (36.6) 259
PH < 10-5 < 10-5 0.89197
1duo n 121 (149.1) 119 (98.2) 48 (40.7) 288
PH 0.00004 0.00196 0.17888
1tri n 29 (35.7) 37 (23.5) 3 (9.7) 69
PH 0.25565 0.00141 0.04006
ΣC 319 210 87 616
Statistics: X2> 47.10; c> 0.1955; p< 10-5
Sprague-Dawley Rat (n = 67; 3.5 h):
1mon n 840 (788.0) 138 (174.8) 66 (81.3) 1044
PH < 10-5 < 10-5 0.00081
1duo n 158 (149.1) 75 (41.5) 15 (19.3) 248
PH 0.00002 < 10-5 0.51928
1tri n 30 (52.8) 15 (11.7) 25 (5.4) 70
PH < 10-5 0.28069 < 10-5
ΣC 1028 228 106 1362
Statistics: X2> 127.5 c> 0.2163; p< 10-5

Repetitive observations of the autocorrelation pattern in left–ventricular pressure curves among subsets of Samples I. The pattern found in the first 4s–interval of the specimen determines the line (1mono/1duo/1tri) the following observations, taken every ten minutes, are counted in. Control conditions maintained by adjusting pre– and afterload. Notation as in Table 2.

Testing the monorhythmic 4s LVP recordings individually for such differences mostly states significance by chance. For example, 225 such recordings appear in guinea pig, allowing to expect 11 (5%) false positive tests, because no p-adjustment for multiple testing is made. Most hemodynamic parameters do not exceed that number; the same holds true in the monorhythmic rat samples, see the latin squares given in Table 4. On the contrary, significant differences between odd and even beats are often revealed in duorhythmic LVP recordings. This greater portion of significant outcomes is confirmed by latin square tests, as marked in Table 4.

Figure 3 demonstrates the effect, related to rhythm pattern, of performing the logistic regression (P0Rlx, τ0, τ) on isochoric LVP decay phases (tRlx) separately on odd and even beats. This procedure does not reduce the regression variance in most recordings (F<1). Especially, duorhythmic recordings (R>1) present with improved regression fit (F>1) just negligibly more often than monorhythms.

Samples II

The percentage (median with upper 90% percentile) of irregular BIs in all obtained 10 min LVP recordings is in guinea pig: 0.045.35 (37°C) and 0.523.81 (31°C), in SpD rats: 0.122.94 (37°C) and 0.306.64 (31°C). In each of these four groups, more than ninety percent of recordings contain at least 44 seconds free from irregularities. Only these phases are subjected to medium term analyses (N shown in Table 5). The same material, in its full 10 min lengths, is used for long term analyses, but limited to recordings that contain at most 3% (37°C) or 6% (31°C) irregularities (N in legend to Fig. 4).

Medium term patterns by different windows of autocorrelation

Table 5 demonstrates how the position of found peak correlation depends on the width of the autocorrelated window. One-second windows yield many distant best correlations (n > 2, partly given in Table 5), especially among rat. As window width increases, peak autocorrelation more and more clusters together at one- or two-beat lags.

This observation becomes even more pertinent at hypothermia (not explicated in Table 5, but seen from the “missing” percentages after n = 1, 2). All window widths much less prefer one-or-two beat lags at hypothermia, as compared with normothermia.

Long term pattern in single-beat autocorrelation

Single-beat autocorrelation always finds highly correlating later BIs, with median peak r > 0.99997, in both species and temperatures. Such excellent match may occur quite in distance, as after 50 beats or more. However, the peak r usually degrades in the course of time, but about a quarter of specimens presents with increasing trend in r (linear regression over 10 min). Figure 4 (upper panels) shows the pertinent scattering of r, and also its decreasing trend in median.

Stochastic analyses in each 10 min recording reveal average BI-to-wait, λ, for the next “very similar” beat of (median with central 90% mass of sample): 3.38.01.0 (guinea pig, 37°C), 2.64.10.8 (guinea pig, 31°C), 5.59.50.6 (SpD rat, 37°C), 5.09.51.1 (SpD, 31°C). The individual full 95% fiducial ranges, given per cent of λ, of these estimates emerge as, respectively: 7.515.12.6 3.2, 11.218.05.1, and 6.817.63.2, 9.823.04.0%. Figure 4 (bottom panel) interprets pooled recordings as a single Poisson process. As seen from the too-steep decay, zero waiting (i.e., immediately consecutive beats being “very similar”) and one-BI waiting intervals occur more often than predicted by the model; the same applies to most individual recordings. Most “very similar” beats occur within the first 5 min, these portions are in median: 80% (37°C), 90% (31°C) among guinea pig, and 72% among SpD rats (both temperatures); in a fifth of each sample, “very similar” beats are even absent in the second half of recordings. However, λ estimates do not change substantially (except guinea pig, 31°C) if the analysis is restricted to the first five minutes: λ values turn to (in sequence as in Fig. 4): 4.1, 5.2, 5.7, 5.8. Removing the linear trend from each 10 min recording has no substantial effect.

Discussion

Autocorrelation, as a method, entered cardiology research sixty years ago [13]. It was used to analyse electrical fluctuations in acute atrial fibrillation [13], to auto-recognise tachycardia and fibrillation [14], and to access ventricular performance at random RR intervals [15] or in chronic atrial fibrillation [16]. Some standard heart rate variability indices are mathematically related to the autocorrelation within RR-interval time series [7]. As based upon electropathology, all these findings are not directly related to the present ones. This is the first study focused on autocorrelation in LV pressure data; the discussion pays special attention to discriminate between mechanical and electrical issues, therefore Laboratory research and standard clinical diagnostics may prefer autocorrelation windows that encompass a sequence of beat intervals; we touch upon this use in the Applications subsection. Physiologically, the single beat interval should be ecognised as a natural window length instead. We first make an issue of previous findings related to the present results.

Table 4.

Influence of LVP autocorrelation pattern on hemodynamic parameters

Giuinea Pig SpD Rat WKy Rat Ferret
n: 432 656 101 28
P Md|Δ%| ns sg Md|Δ%| ns sg Md|Δ%| ns sg Md|Δ%|
tBI M 2.59 0.729 215 10 2.29 0.739 434 44 0.431 2.30 79 4 0.567 1.52
D 3.57 0.936 180 †27 0.772 3.19 156 22 1.002 3.20 13 †5 0.320 2.37
tRise M 0.810 2.63 218 7 0.645 2.06 473 5 0.480 2.00 83 0 0.949 1.90
D 0.844 2.95 199 8 0.798 2.69 171 *7 0.466 2.48 18 0 0.346 0.74
t1Rlx M 0.440 1.73 216 9 0.568 10.4 469 9 0.930 9.06 81 2 0.717 1.93
D 0.752 3.84 173 ‡34 1.177 10.2 145 ‡33 1.867 55.2 14 †4 0.426 0.88
tRlx M 1.644 6.77 218 7 1.858 13.2 458 20 2.421 13.1 79 4 1.598 4.33
D 1.993 8.37 195 12 3.243 18.3 147 ‡31 4.110 27.6 14 *4 0.552 0.90
PED M 2.491 19.7 214 11 2.116 15.6 461 17 1.420 9.94 79 4 2.652 12.5
D 3.149 19.0 196 11 3.826 15.2 148 ‡30 2.199 23.6 15 3 1.032 5.77
P[max]LV M 0.052 0.35 197 28 0.082 0.36 447 31 0.064 0.31 76 7 0.095 0.34
D 0.157 1.01 147 ‡60 0.263 1.57 130 ‡48 0.186 1.67* 12 †6 0.068 0.23
P[min]LV M 0.902 9.40 214 11 0.923 9.16 461 17 1.455 6.97 77 6 1.902 36.7
D 1.554 15.9 173 ‡34 2.629 18.1 143 ‡35 5.335 31.7 14 4 1.873 20.2
P[dot/max]LV M 0.271 1.02 206 19 0.261 0.90 453 25 0.247 0.93 79 4 0.174 0.61
D 0.477 2.46 166 ‡41 0.525 2.91 136 ‡42 0.254 1.55 15 3 0.286 0.42
P[dot/min]LV M 0.495 1.85 211 14 0.471 2.92 451 27 0.748 3.42 75 8 0.663 1.93
D 0.644 3.11 178 †29 0.821 3.65 149 ‡29 1.252 14.5* 14 4 0.278 1.07
P[Rlx]0 M 0.852 3.01 216 9 1.111 9.59 448 30 1.641 10.3 77 6 1.129 5.22
D 1.206 4.97 192 15 1.624 9.73* 151 ‡27 3.067 21.8 14 4 0.786 3.01
τ0 M 1.103 6.81 209 16 1.292 35.9 449 29 3.917 42.2 79 4 0.858 6.62
D 1.528 8.46* 178 *29 2.875 51.5 142 ‡36 9.899 372* 16 2 0.625 1.77
τ M 4.161 38.0 213 12 2.844 55.9 448 30 6.513 57.9 75 8 1.564 9.75
D 6.212 35.0* 185 *22 5.106 78.4 150 ‡28 18.65 324* 15 3 1.532 33.7

Subsets from Samples I (Table 1), presenting with (pattern, P) either mono– (M) or duorhythm (D) in the left–ventricular pressure curve. Md |Δ%|: median of the absolute mean difference in the respective parameter (see legend to Table 1) between the odd– and the even–numbered beats within each 4s recording, expressed as percentage of the mean value in that recording; superset is the upper limit of 90% of the species sample, subset indicates significance (see bottom line) of assuming higher interbeat differences in the duorhythmic (D) specimens, obtained by single–sided U tests.

Each recording is checked (in each parameter) on difference between consecutive odd–even numbered beats by matched–paired Wilcoxon tests, and counted as significant (sg) if p < 0.05, double–sided; sg and non–significant (ns) cases are counted in the presented latin squares (M/D×ns/sg; omitted in Ferret due to insuffcient sample size). Significant dependence on P is tested by the χ2 statistic and indicated in the latin squares. Error probabilities: *, p < 0.05; †, p < 0.01; ‡, p < 0.001.

Table 5.

Medium term LVP autocorrelation patterns at normo– and hypothermia (Samples II)

Guinea Pi SpD Rat
37 °C N = 23 N = 79
nBI 82 (68-91) 94 (80-112)
Lag 1s 2s 4s 1s 2s 4s
1 n 6 6 7 14 20 41
% 26 4910 26 4910 30 5313 18 2810 25 3716 52 6440
2 n 5 6 9 7 9 20
% 22 447 26 4910 39 6219 9 183 11 215 25 3716
3 n 1 2 2 4
4 n 1 6
5 n 1 1 1
6 n 1 1 2
7 n 1 1 1 4
8 n 1 2
9 n 1 1
10 n 1 1
11 n 1 2
12 n 4 1
13 n 1 1 1 1 4 1
14 n 5 1
15 n 2 1 1 2 1
8s 16s 24s 8s 16s 24s
1 n 10 10 12 54 60 60
% 43 6623 43 6623 52 7430 68 7956 76 8565 80 8969
2 n 8 11 10 21 17 14
% 35 5816 48 7026 43 6623 27 3817 22 3313 19 3010
31 °C Nh = 10 Nh = 36
nBI 57 (50-59) 58 (44-72)
Lag 1s 2s 4s 1s 2s 4s
1 n 1 2 5 1 5 11
% 10 450 20 563 50 8218 3 150 14 304 31 5016
2 n 1 1 1 2
% 0 260 10 450 0 260 3 150 3 150 6 201
8s 16s 24s 8s 16s 24s
1 n 6 9 10 28 31
% 60 8826 90 10055 100 10074 80 9263 86 9670
2 n 3 4
% 0 260 0 260 0 260 9 242 11 327

Left–ventricular pressure curves (sinus rhythm) are lagged to autocorrelation windows of 1 to 24 seconds width (see Table) by 1 to 20,000 ms, covering nBI beats (median, with central 90% range). N, sizes of Samples II, Guinea pig and Sprague–Dawley rat (SpD); Nh, size of subsample at hypothermia. Lag, number of beat intervals to the maximum coeffcient of autocorrelation, Table cut off at Lag 15 or 2, respectively; n, counted observations (blank cells are zeroes); %, percentage with 95% limits of confidence, additionally given for Lag 1 and 2.

Physiological basics

Intrinsic sino-atrial excitation is utilised in all but the artificial pacing experiments; we discuss its regularity first, therefore. Eliminating electrical irregularities suggests heterometric autoregulation i.e., (Frank-Starling law) to explain present findings.

As previous investigations of interbeat variability rely on electrophysiology rather than ventricular mechanics, the basal regularity of the pressure-related BI definition, used in the present study, should be explicated first. Variability of BI in LVP is not higher than in electrical RR intervals but even smaller if analysed at 10 kHz [17] (compare LVP with ECG in Fig. 1). Spontaneous pulse generation in the sino-atrial node, and its further propagation, yield BI differing from each other by less than 1 ms (theoretically: Weibull distributed) at constant normothermia, whereas random oscillation (normal distribution in Poincaré plots [4,7] occurs at hypothermia [17]. Increasing resistance of the electrical intercell coupling [18] may explain for this effect, because BI variability of pacemaker cell clusters decreases as the number of (electrically) interconnected cells rises [2,3]. In consequence, we rule electrical fluctuations (sinus node, atrio-ventricular and His bundle conduction) out of explaining the autocorrelative patterns found in LVP recordings at normothermia. This approach is strikingly affrmed by the pacing experiments encountering more LVP duorhythms than present at sinus rhythm.

Figure 3.

Influence of the autocorrelative pattern on the goodness-of-fit of regression on the isochoric pressure decay. Isolated rat and guinea pig hearts, working at standard conditions (see text) and presenting mono- or duorhythm. R = r−22r21 is the squared ratio of the autocorrelation coeffcients at lags 2 and 1 beat, respectively. F = σ21 σ−22 is the ratio of the empirical variances left by the four-parametric logistic regression on the isochoric pressure decay, σ21 with pooling of all beats, σ22 with pooling odd and even beats seperately. Panel clipped around the central data mass. Symbols: circles, guinea pigs; diamonds, Sprague- Dawley rats; black filled symbols (appearing at F > 1 only!) indicate significantly (p < 0.05 in F test) improved regression fits by seperately pooling alternating beats.

icfj.2016.8.57-g003.jpg

Heterometric autoregulation is the most pertinent (if not only) mechanism providing an immediate, viz., end-diastole-to-systole regulation of LVP development [16,19,20]. It remains effective up to the highest possible ventricular stretching [19,21] and responds very sensitively (e.g., eightfold twitch force by 15% fiber length increase [19]). The response curve is adjusted (shifted) by other regulatory pathways and attributed to Ca2+ sensitivity of the myocardial filament which increases with precontraction sarcomere length (length dependent activation) [20,22,23,24]. As standing to reason, this effect has been thoroughly investigated by sudden preload changes [25,26] and in the presence of spontaneous [16] or artificially induced ventricular arrhythmia [27,28]. Physiological importance of Frank-Starling’s law is acknowledged in attuning right and left cardiac output [21]; apart of that, it is considered somewhat to compensate for lacking neuro-humoral regulation (elderly [19]), transplant recipients [5]) or arrhythmia [16,28]. It has been stated that Starling’s law cannot readily been demonstrated even by gross volume changes in healthy subjects, due to autonomic nervous counteraction [25]; subsequent studies relativised this notion [22,26].

Interpretation of the results

We are going to consider some autoregulative and some random components to explain the present result but would like to tackle two technical issues first.

The observed patterns depend, in a systematic way, on the arbitrary window size. However, the predominance of best correlation at lags of one or two beats is a consistent finding. Monorhythm does not seem to need further explanation, as it just shows that a series of beats resembles its LVP curve during some seconds. Significant duorhythm deserves attention. Rhythms of higher order become a scarse finding in multi-beat windows, consequently, a process analysis based on single-beat windows appears to be more appropriate.

Small pressure undulations may be effective in Starling’s law. Such are not forwarded to the left atrium by the roller pump, but emerge in the pre-atrial windkessel as a retroaction of ventricular contraction. Forced damped oscillation may impinge on the filling pressure of the next heart beat. This effect may play a rôle in initiating slight beat-to-beat imbalance in preload but does not explain stationarity of such rhythm. This is demonstrated by the pacing intervention: Any interference or resonance should have even chance to occur at intrinsic frequency (sinus rhythm) or at the slightly elevated one, but stable duorhythms emerge significantly more often at pacing.

Heterometric autoregulative component (Frank-Starling)

Significantly increasing number of duorhythms emerging at slightly elevated heart rate (Langer paradox) is the most unexpected finding of the present study and, as yet, unknown to the literature. The pacing experiments were originally done to exclude possible intrinsic RR fluctuations as a cause for mechanical BI fluctuations, but a contrariant effect emerges. Obviously, pacing at constant cardiac inflow immediately unloads ventricular end- diastolic filling and, consequently, diminishes ejection fraction [16] and stroke volume. Thus, ventricular contraction becomes seemingly downregulated to a highly sensitive part of the Frank- Starling response curve, with compensation at the next following beat, as mean cardiac inflow and BI length remain constant. The persistence of the emerged alternating pattern confirms, in terms of regulation theory [29], that heterometric autoregulation reacts lag-free and does not propagate regulatory information to consecutive beats. A Bowditch effect (development of higher inotropy at pacing, due to higher intramyocytal Ca2+ turnover [19,21,27]), if any present at this slight overpacing, may further stabilise this alternating beat-to-beat imbalance, because there is no hemodynamic need for an elevated excitation-contraction coupling, which, in turn, only provides more elbowroom to heterometric beat-to-beat regulation. It is known (and quite understood) that the first or a few beats after shortening the BI develop slower and lower contraction force [1,21]; enough time for a Bowditch phenomenon to settle was awaited before starting the LVP recordings. Later on, neither the change in heart rate nor in its variability cause contractile effects in the present frequency range [27,28].

Figure 4.

Ten-minutes autocorrelation in left-ventricular pressure curves of isolated guinea pig (GPig, N=20 at 37°C, 10 at 31°C) and Sprague-Dawley rat hearts (Rat, N=64 at 37°C, 34 at 31°C), working at standard conditions (see Table 1 for BI lengths). Upper panels: Time-course of the peak coeffcients (r) of autocorrelation by moving an initial 2s-window. Curves are medians from 1s each, at 37°C (white) with 95 per cent interval of confidence (gray, clipped downwards at GPig) of the median, and at 31°C (black ). Bottom panels: Occurrence of similar beats as a Poisson process. Window of autocorrelation is one beat interval (BI), those ten per cent of following BIs presenting with the highest r values are qualified as “very similar”; black bars, distribution of the periods free from similar beats (number of BIs to wait), given as percentage of the total number, n, of waiting periods; gray curves, best fitting exponentials with λ, expected number of BIs to wait. Black exponential curves (having λ=9) would be predicted by a Poisson process with neither microrhythms nor fade-out.

icfj.2016.8.57-g004.jpg

The given explanation of microrhythmic findings by heterometric autoregulation presumes that duorhythm is somewhat unrelated to a pathological condition known as pulsus alternans. Such condition can be provoked in normal isolated rat hearts by pacing with at least twice the normal heart rate, and it usually occurs after an abrupt jump in pacing rate [30]. The present pacing experiments do by far not reach this level. Alternating pulse is characterised by a substantial difference ∆PmaxLV changing its sign from beat to beat, and is related to insufficient recovery time for sarcoplasmatic Ca2+ reuptake [30]. This seems to discriminate alternans from duorhythm. Frequency-∆PmaxLV response curves may further clarify, whether or not a contiguous development from duorhythm to alternans exists.

Results from the preload tests obviously support this concept to explain microrhythmic changes. Reducing the left atrial inflow, at quite constant heart rate, provides (besides pacing) just an other way to alleviate the single beats’ preload. Vanishing of most duo- and triplerhythms at elevated cardiac inflow, as seen in the study, accords with the concept derived from the pacing tests, i.e., enhanced sensitivity of heterometric autoregulation at low preloads.

Afterload (mean aortic pressure) is not in the focus of the present study, but the observations on guinea pig specimens indicate its indirect relevance in causing microrhythmic LVP patterns. Rat and ferret specimens are set to control conditions resembling rest. End-diastolic pressure is known to be “near maximal at rest” [26]. Guinea pig hearts, different to the other species, are set to reduced aortic pressure, just for reasons of durability. They present significantly more often with duorhythms (Table 2) as the rat subspecies do. Reduced afterload immediately increases the beat volume, thus reducing residual volume and end-diastolic pressure. Homeometric autoregulation (Anrep effect) counteracts, re-adjusting the end-diastolic pressure within a series of beats [19,21,31]. Such adjustment settles during the mounted guinea pig hearts adapt to the artificial circulation. However, homeometric autoregulation may leave the steady-state end-diastolic pressure with some small amount of reducement into higher sensitivity to heterometric regulation, or other processes may eventually drive the heart towards such sensitivity, thus causing duorhythm to appear more often.

Generally, preload-contraction sensitivity of the isolated hearts varies from specimen to specimen and depends especially on a considerate mounting procedure. Control conditions are defined as an average load, independent from individual specimen’s per-formance. The similar distribution of microrhythmic patterns in the LVP of different species (with the particulars discussed about the guinea pig sample), and its long term persistence, argue for that fixed condition as enabling the specimens to express different types of microrhythms individually. Inspecting power spectra of individual peak r curves (and their derivatives) from the present material does not reveal consistent rhythms with periods above two BI lengths, according to previous negative findings in the spectra of BI length itself [17].

Random component (Poisson)

Aforesaid remarks, and the scattering in r time series (Fig. 4), suggest an adequate random component superimposing onto microrhythmic formation. Such random contribution takes no issue with previous assertions, because we reject the monorhythm assumption only if higher-lag r turn out significantly above r1 in the particular specimen. Consequently, randomness is better assessed by a stochastic approach, viz., by declaring an arbitrary portion (10%, presently) of beats, following a randomly selected one, as “very similar” to the referenced one, and by observing the occurrence of such beats in the course of time. If the probability to encounter a “very similar” beat within some time converges to a fixed value as that time is reduced toward zero, and if there is no mutual relation or “memory” [5,15], a Poisson process is expected [9]. An artificial Poisson generator has been used to stimulate isolated hearts, perceiving the arrival of ventricular excitation in case of right atrial fibrillation as a Poisson process [15].

Each Poisson process is fully characterised by its intensity, i.e. the constant in the exponent [9], presently noted as λ-1 to obtain its reciprocal, λ, as the “average time to wait for the next similar” (in numbers of BI). As Poisson’s distribution is indefinitely divisible [9], we are allowed to pool as well as discard data if such partial data can be considered as similarly intense Poisson processes. This is the proper averaging method; the arithmetic mean is biased by non-linearity.

The present recordings strongly support the notion of an apparent Poisson process by high goodness of the exponential fits and by estimating credible intensities, λ-1. Individual recordings present with distinctive λ values, but propagate their essentials un-mitigated into the pooled data (Fig. 4). The data pool reveals model’s appropriateness as well as indicative model violations, according to microrhythms: Short intermediate waiting times (λ = 0 or 1 BI) correspond to mono- and duorhythm, respectively, and appear as well over-represented in the Poisson analysis. Another surplus is counted among long waiting times (beyond abscissa scale in Fig. 4, bottom, as invisible on ordinate scale anyhow), caused just by spanning the stencil BI over more than five minutes. This is a trifle because the existence of a Poisson process is in question, not its duration with a particular stencil BI. Obviously, one should refresh the stencil from time to time if long term analysis is desired (or consider an inhomogeneous process [5,9]). Generally, a series of about thousand BIs suffices; Fig. 4 depicts the full 10 min data just to bring out this aspect clearly. Especially, the robustness of λ estimates against a shortage of observation time is noteworthy. The mean number of intercalary beats to await is nine, by definition. If very similar beats cluster together in the first 5 min, this number reduces towards 4 to 6 in that phase. Exactly this is revealed by the analysis, even with the full 10 min data. λ has barely changed in three of four samples by removing the almost void terminal half. This robust behaviour is a feature of least-squares minimisation, paying most attention to the high-count data (this also explains the small λ in Fig. 4, GPig 31°C). In conclusion, λ describes physiological reality rather than spurious effects of sampling time [5].

Hemodynamics of microrhythmic patterns

As correlation is a non-specific indicator of similarity, found non- uniformities in the beat-to-beat LVP course should be detailed out in terms of prevalent hemodyamics. Extensive research has been done especially for the (“chrono-”)inotropic effects of systematic or random changes in the rhythm of myocardial excitation [1,15,16,27,28]. Its most widespread result says that any remarkable variability in the heart rhythm (at given mean frequency) has a negative inotropic effect. Probably for the (clinical) importance of this finding, only few, if any, attention is paid to the beat-to-beat variability of hemodynamic parameters at normal sinus rhythm, except the RR interval variability [4,6] itself. Typical research papers either deal with apparent arrhythmia and give single-beat hemodynamics, or focus on steady state and present averaged data obtained from a number of “representative” beats. In-depth discussion on individual parameters goes beyond the present scope also, but some remarks only about the dispersion of hemodynamic parameters (Table 4) may be issued.

Contrary to what may be expected, duorhythm does not present with much higher dispersion of the mechanical BI length, tBI. Systolic pressure rising time, tRise, remains even more stationary. End-diastolic pressure shows significantly but small higher variability in duorhythm. This accords equally well with its proposed rôle in triggering heterometric autoregulation especially in duorhythm, and also with the known minuteness of pressure necessary to stretch myofibrilles considerably at this condition. The output parameters of Starling’s law, PmaxLV and (with one exception) maxLV show considerable and significant higher variability in duorhythms. All this supports (besides the pacing experiments) the notion of an effective rôle of heterometric autoregulation particularly at duorhythm.

Unrelated to the microrhythmic pattern, variability is highest in those parameters which depend, by their definition, on LVP points near zero (this holds, e.g., for tRlx, as depending on PED, and for t, in turn relying on tRlx). The percentages given in Table 4 are calculated with dividing the difference by the respective mean, thus an inter-parameter comparison is unfavourable.

Applications

The study originally aimed for an improvement in data acquisition from multi-beat LVP recordings. We discuss this topic and a potential clinical utilisation of the nouveau finding.

Basic research

In order to characterise a steady state in LVP recordings, simple statistics with single-beat data should be avoided in favour of sound data pooling, done by overlaying well adjusted LVP courses (or parts of them) from consecutive beats. This seems of minor importance in punctual parameters; however, it should be realised that even then not only the pressure but also the time, at which the respective pressure appears within distinct beats, is subjected to (biological) variability that should undergo statistical balancing. Anyhow, LVP overlaying becomes almost unavoidable to enhance the data basis if the parameter in question is based on regression analysis, and reliable fiducial limits are desired concomitantly. Time constants of isochoric (isovolumic) pressure decays [10,12] are by far the most prevalent parameters of this kind in cardiology. Performing such data pooling, the scientist would like to provide evidence of acceptability [5]. No pertinent (systematic or heavy) interbeat differences must be concealed.

Autocorrelation in the LVP recordings provides a straightforward assessment of interbeat similarity by a single index, avoiding multi-dimensional and fractal [4,5] descriptions. Window size and the range of correlation coefficients to expect may be a matter of recording length, environment (isolated or in vivo), and the purpose of the investigation. Determining appropriate standards and r thresholds may lead to a valuable pretest in order to trigger data sampling or to arouse the experimenters’ attention. Sensitive and specific auto-detection of certain conditions is not to be expected.

Clinical diagnostics

Safe, easy, and fast estimation of contractile reserve may be obtained by means of pacing induced microrhythms (Langer phenomenon). Well performing hearts do not exhaust their heterometric autoregulative capacity at resting conditions. Consequently, such condition may be indicated by changes in the microrhythm as arising from a slight deloading of the heart beat, induced by artificial electrical stimulation. Hearts in state of depressed contractility may be found unable to present with this phenomenon, due to an already exploited Frank-Starling mechanism. Furthermore, mutated troponin, depressing this mechanism, is found in pathological hearts [22]. Left heart catheters, provided with pacing electrodes and pressure sampling facility, and a few seconds of respiratory arrest (or compensatory data processing) should suffice to obtain the data. Due to the immediateness of the effect, it will hardly be necessary to block any other regulatory pathway.

Limitations

The study observed microrhythms at a uniform condition in isolated working small-animal left hearts with their intrinsic sinus rhythm. Other or less stationary conditions, as well as hearts from other species, will probably yield other percentiles of observable correlation coefficients. Influences apparent in situ, as pericardial pressure [26], ventricular right-left interference, respiratory cycle (undulation of mediastinal pressure and reflexive respiratory sinus arrhythmia), neuroregulative heart rate variability, and baroceptor reflexes, are not addressed. Selected control parameters (heart rate, preload, temperature) were changed dichotomously, without assessing quantitative extent-effect relationship in detail. Conclusions toward a pertinent rôle of the Frank-Starling law are drawn from the observed end-diastolic pressure; interlinked parameters, as end-diastolic ventricular strain or fiber length [24], were not directly monitored. Finally, the study observed steady state rather than emergence. Discussing microrhythmicity focuses on physiology and remains within the confines of elemental stochastic; specific process models, which account for such microrhythms, should become mathematically defined. Hinting at a probable diagnostic application is tentative as yet.

Conclusions

We conclude the introduced issues as follows:

  1. a) As assessed by autocorrelation, the left-ventricular pressure course of isolated hearts at sinus rhythm does contain, either, no pertinent short term rhythm (above the beat interval itself, “monorhythm”), or an alternating beat-to-beat pattern (“duorhythm”). Stationary undulations pitched at lengths of more than two beat intervals are absent. Both findings do not depend, in quality, on the number (up to a hundred) of beats encompassed with the autocorrelation window. A key rôle of heterometric autoregulation (Frank-Starling) in the formation of duorhythm is surmised. Particularly, variability of the electrical heart rate is of no account; quite the contrary, the pertaining observation, the present study encountered with, may be pointed out as the Langer paradox: Reducing (a quite normal) electrical heart rate variability may induce more beat-to-beat variability in the mechanical action of the ventricle. This phenomenon might provide diagnostic information about contractility conditions.

  2. b) Ventricular end-diastolic and maximum pressures contribute to duorhythmical alterations in the pressure course. Duration of the beat interval, and of systolic pressure rise, do not. Quantifying the variability should be improved with respect to those hemodynamic parameters, which depend on the fidelity of pressure scale’s zero calibration. Due to the small variability even at duorhythm, data pooling seems to be generally justified for most conventional hemodynamic parameters at eurhythmic steady states. If at issue, odd and even numbered beat intervals should be pooled separately.

  3. c) Mutually similar beats (of isolated hearts at sinus rhythm) occur in the left-ventricular pressure curve stochastically according to a Poisson process with fade-out over several minutes. The observation time (number of beats), necessary to obtain reliable estimates of the process intensity, is uncritical, despite the fade-out. The expected time (reciprocal intensity) to wait for the next similar beat obviously depends on the preset definition of similarity, but does not depend on temperature, if time is counted in units of beat intervals which last longer at hypothermia.

Acknowledgements

The author thanks the Sonnenfeld Foundation, Berlin, who partly financed the laboratory equipment. He is much obliged to fourteen research fellows of the Institute who made their experimental data available for this investigation. The author states that he adhere to the statement of ethical publishing of the International Cardiovascular Forum Journal [32].

Conflict of interest

The author has no conflict of interest to declare.

References

1. 

Koch-Weser J, Blinks JR The influence of the interval between beats on myocardial contractility [review]. Pharmacol. Rev. 1963; 15: 601–652http://pharmrev.aspetjournals.org/content/15/3/601.long

2. 

Clay JR, DeHaan RL Fluctuations in interbeat interval in rhythmic heart–cell clusters. Biophys. J. 1979; 28: 377–390 10.1016/s0006-3495(79)85187-5

3. 

Jongsma HJ, Tsjernina L, DeBruijne J The establishment of regular beating in populations of pacemaker heart cells. A study with tissue-cultured rat heart cells. J. Mol. Cell. Cardiol. 1983; 15: 123–133 10.1016/0022-2828(83)90288-2

4. 

Babloyantz A, Destexhe A Is the normal heart a periodic oscillator?. Biol. Cybern. 1988; 58: 203–211 10.1007/bf00364139

5. 

Meyer M, Stiedl O Self–affne fractal variability of human heartbeat interval dynamics in health and disease [review]. Eur. J. Appl. Physiol. 2003; 90: 305–316 10.1007/s00421-003-0915-2

6. 

Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology. Heart rate variability. Standards of measurement, physiological interpretation, and clinical use. Eur. Heart J. 1996; 17: 354–381 10.1093/oxfordjournals.eurheartj.a014868andCirculation 1996; 93: 1043–1065 10.1161/01.CIR.93.5.1043

7. 

Hwang JS, Hu TH, Chen LC An index related to the autocorrelation function of RR intervals for the analysis of heart rate variability. Physiol. Meas. 2006; 27: 339–352 10.1088/0967-3334/27/4/002

8. 

Langer SFJ, Schmidt HD Different left ventricular relaxation parameters in isolated working rat and guinea pig hearts. Influence of preload, afterload, temperature and isoprenaline. Int. J. Card. Imaging 1998; 14: 229–240 10.1023/a:1006083306901

9. 

Karlin S, Taylor HM A First Course in Stochastic Processes [2nd ed, reprint]. New York etc:Academic Press2011;

10. 

Langer SFJ Ransacking the curve of cardiac isovolumic pressure decay by logistic–and–oscillation regression. Jpn. J. Physiol. 2004; 54: 347–356 10.2170/jjphys-iol.54.347

11. 

Sachs L Angewandte Statistik [6th ed; engl. ed. as: Applied Statistics, 2nd ed]. Berlin etc:Springer1984; 10.1007/978-3-662-05750-6

12. 

Langer SFJ Effcient exponential regression with exact fiducial limits to fit cardiac pressure data. Comput. Methods Programs Biomed. 1997; 53: 57–64 10.1016/s0169-2607(97)01802-6

13. 

Angelakos ET, Shepherd GM Autocorrelation of electrocardiographic activity during ventricular fibrillation. Circ. Res. 1957; 5: 657–658 10.1161/01.RES.5.6.657

14. 

Aubert AE, Denys BG, Ector H, DeGeest H Automatic detection of ventricular tachycardia and fibrillation using ECG processing and intramyocardial intramyocardial pressure measurement. Comput. Biomed. Res. 1994; 27: 367–382 10.1006/cbmr.1994.1028

15. 

Meijler FL, Strackee J, van Capelle FJL, du Perron JC Computer analysis of the RR interval–contractility relationship during random stimulation of the isolated heart. Circ. Res. 1968; 22: 695–702 10.1161/01.RES.22.5.695

16. 

Gosselink ATM, Blanksma PK, Crijns HJGM, van Gelder IC, de Kam PJ, Hillege HL, Niemeijer MG, Lie KI, Meijler FL Left ventricular beat-to–beat performance in atrial fibrillation: contribution of Frank–Starling mechanism after short rather than long RR intervals. J. Am. Coll. Cardiol. 1995; 26: 1516–1521 10.1016/0735-1097(95)00340-1

17. 

Langer SFJ, Lambertz M, Langhorst P, Schmidt HD Interbeat interval variability in isolated working rat hearts at various dynamic conditions and temperatures. Res. Exp. Med. 1999; 199: 1–19 10.1007/s004330050128

18. 

Bukauskas FF, Weingart R Temperature dependence of gap junction properties in neonatal rat heart cells. Pflugers Arch. 1993; 423: 133–139 10.1007/bf00374970

19. 

Lakatta EG Length modulation of muscle performance: Frank-Starling law of the heart. In:. Fozzard HA et al [eds]The Heart and Cardiovascular System. Vol. 2: Chap. 40.New YorkRaven Press1986;

20. 

deTombe PP, Mateja RD, Tachampa K, Ait Mou Y, Farman GP, Irving TC Myofilament length dependent activation [review]. J. Mol. Cell. Cardiol. 2010; 48: 851–858 10.1016/j.yjmcc.2009.12.017

21. 

Levick JR An Introduction to Cardiovascular Physiology [3rd ed], Chap. 7. London etc:Edward Arnold Publ.2000;

22. 

Kobirumaki-Shimozawa F, Inoue T, Shintani SA, Oyama K, Terui T, Minamisawa S, Ishiwata S, Fukuda N Cardiac thin filament regulation and the Frank-Starling mechanism [review]. J. Physiol. Sci. 2014; 64: 221–232 10.1007/s12576-014-0314-y

23. 

Neves JS, Leite-Moreira AM, Neiva–Sousa M, Almeida–Coelho J, Castro–Ferreira R, Leite-Moreira AF Acute myocardial response to stretch: What we (don’t) know [review]. Front. Physiol. 2016; 6: Article 408 10.3389/fphys.2015.00408

24. 

Ait-Mou Y, Hsu K, Farman GP, Kumar M, Greaser ML, Irving TC, de Tombe PP Titin strain contributes to the Frank–Starling law of the heart by structural rearrangements of both thin– and thick–filament proteins. Proc. Natl. Acad. Sci. U S A. 2016; 113: 2306–2311 10.1073/pnas.1516732113

25. 

Frye RL, Braunwald E Studies on Starling’s law of the heart. I. The circulatory response to acute hypervolemia and its modification by ganglionic blockade. J. Clin. Invest. 1960; 39: 1043–1050 10.1172/JCI104119

26. 

Boettcher DH, Vatner SF, Heyndrickx GR, Braunwald E Extent of utilization of the Frank-Starling mechanism in conscious dogs. Am. J. Physiol. 1978; 234: H338–H345http://ajpheart.physiology.org/content/234/4/H338.long

27. 

Bershitskaya ON, Izakov VY, Lysenko LT, Protsenko JL, Trubetskoy AV Certain characteristics of myocardial contractility of isovolumic dog heart at randomly variable heart rhythm. Basic Res. Cardiol. 1985; 80: 156–164 10.1007/10.1007%2FBF01910463

28. 

Popović ZB, Yamada H, Mowrey KA, Zhang Y, Wallick DW, Grimm RA, Thomas JD, Mazgalev TN Frank-Starling mechanism contributes modestly to ventricular performance during atrial fibrillation. Heart Rhythm 2004; 1: 482–489 10.1016/j.hrthm.2004.06.016

29. 

MacDonald N Time Lags in Biological Models. Lecture Notes in Biomathematics, Vol. 27. Berlin etc:Springer1978; 10.1007/978-3-642-93107-9

30. 

Dumitrescu C, Narayan P, Efimov IR, Cheng Y, Radin MJ, McCune SA, Altschuld RA Mechanical alternans and restitution in failing SHHF rat left ventricles. Am. J. Physiol. Heart. Circ. Physiol. 2002; 282H1320–H1326 10.1152/ajp-heart.00466.2001

31. 

Schipke JD, Sunderdiek U, Arnold G Effect of changes in aortic pressure and in coronary arterial pressure on left ventricular geometry and function Anrep vs. gardenhose effect. Basic Res. Cardiol. 1993; 88: 621–637 10.1007/bf00788879

32. 

Shewan LG, Coats AJS, Henein M Requirements for ethical publishing in biomedical journals. International Cardiovascular Forum Journal 2015; 2: 2 10.17987/icfj.v2i1.4



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