- Email: [email protected]
Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics
Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29Q3 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Q4 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics Stephen D. McIntyre a,c, Virendra Kakade b,c, Yoichiro Mori a, Elena G. Tolkacheva c,n a
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Electrical and Computer Engineering Department, University of Minnesota, Minneapolis, MN 55455, USA c Biomedical Engineering Department, University of Minnesota, Minneapolis, MN 55455, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 May 2013 Received in revised form 28 January 2014 Accepted 13 February 2014
A beat-to-beat alternation in the action potential duration (APD) of myocytes, i.e. alternans, is believed to be a direct precursor of ventricular fibrillation in the whole heart. A common approach for the prediction of alternans is to construct the restitution curve, which is the nonlinear functional relationship between the APD and the preceding diastolic interval (DI). It was proposed that alternans appears when the magnitude of the slope of the restitution curve exceeds one, known as the restitution hypothesis. However, this restitution hypothesis was derived under the assumption of periodic stimulation, when there is a dependence of the DI on the immediate preceding APD (i.e. feedback). However, under physiological conditions, the heart rate exhibits substantial variations in time, known as heart rate variability (HRV), which introduces deviations from periodic stimulation in the system. In this manuscript, we investigated the role of HRV on alternans formation in isolated cardiac myocytes using numerical simulations of an ionic model of the cardiac action potential. We used this model with two different pacing protocols: a periodic pacing protocol with feedback and a protocol without feedback. We show that when HRV is incorporated in the periodic pacing protocol, it facilitated alternans formation in the isolated cell, but did not significantly change the magnitude of alternans. On the other hand, in the case of the pacing protocol without feedback, alternans formation was prevented, even in the presence of HRV. & 2014 Published by Elsevier Ltd.
Keywords: Action potential Periodic pacing Cardiac dynamics Numerical simulations
1. Introduction One of the most fundamental characteristics of cardiac cells is the shortening of the action potential duration (APD) as the heart rate increases, a phenomenon known as electrical restitution. Restitution plays a vital role in heart function: for a given heart rate, a shorter APD allows for a longer diastolic interval (DI), thereby giving adequate time for the heart to refill with blood. Although important for life at moderate heart rates, at higher rates, restitution may result in life-threatening cardiac rhythms and ventricular fibrillation (VF), in particular (Zipes and Wellens, 1998; Franz, 2003). It is generally believed that T-wave alternans, defined as an alternating change in the amplitude or shape of the T-wave in the ECG, is a precursor of cardiac electrical instability (Karma, 1994; Watanabe et al., 1995; Gilmour and Chialvo, 1999; Fox et al., 2002). T-wave alternans results from APD alternans at the cellular level.
n Correspondence to: University of Minnesota, 312 Church Street SE, 6-128 Nils Hasselmo Hall, Minneapolis, MN 55455, USA. Tel.: þ1 612 626 2719. E-mail address: [email protected] (E.G. Tolkacheva).
A common technique for studying the initiation and maintenance of alternans and other complex rhythms is to analyze the restitution curve, the nonlinear functional relationship between the APD and the preceding DI. While detailed ionic models were used extensively to study the response of cardiac myocytes to stimulation, mapping models were introduced to focus on restitution (Nolasco and Dahlen, 1968; Guevara et al., 1984). Specifically, it was proposed that the APD could be determined as a function of the preceding DI, essentially forming the one-dimensional mapping model. APDn þ 1 ¼ f ðDIn Þ:
ð1Þ
here, f is the restitution curve, APDn þ 1 is the APD generated by the (nþ1)st stimulus and DIn is the nth DI, i.e., the interval during which the tissue recovers to its resting state after the end of the previous (nth) action potential. In 1968, Nolasco and Dahlen (Nolasco and Dahlen, 1968) developed a graphical method to analyze and predict APD alternans in a mapping model (Eq. (1)) under the assumption that pacing occurs at a constant rate, i.e. when the APD and DI are related through the pacing relation. APDn þ DIn ¼ BCLn
ð2Þ
http://dx.doi.org/10.1016/j.jtbi.2014.02.015 0022-5193 & 2014 Published by Elsevier Ltd.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
2
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
where the basic cycle length BCLn ¼BCL (a constant) under periodic pacing. Using Eqs. (1) and (2), it was proposed that the normal cardiac rhythm, or so-called 1:1 response, becomes unstable and alternans occurs when the magnitude of the slope of the restitution curve exceeds one, 0 df Sr ¼ ¼ f j Z1; ð3Þ dDI DI ¼ DIn known as the restitution hypothesis. This restitution hypothesis has been confirmed in some experiments (Koller et al., 1998) and has led to the assumption that flattening the restitution curve will help prevent VF (Koller et al., 1998; Hall and Gauthier, 1999; Gilmour, 2002; Banville and Gray, 2002; Riccio et al., 1999). However, recent experimental results have shown that this hypothesis is incorrect in many situations, where the normal cardiac rhythm is observed when the restitution curve is very steep or in which the transition to alternans occurs in the presence of a flat restitution curve (Koller et al., 1998; Hall and Gauthier, 1999; Gilmour, 2002; Banville and Gray, 2002; Narayan et al., 2008). One of the reasons why the restitution condition fails experimentally is that Nolasco and Dahlen's approach is valid only for periodic pacing, i.e. when BCLn ¼ BCL is a constant in Eq. (2), and cannot be applied for complex stimulation regimes, such as those which result in physiological heart rates that exhibit heart rate variability (HRV) (Malik and Camm, 1995; Goldberger et al., 2000; Kleiger et al., 1987; La Rovere et al., 2003). It is known that the heart rate is regulated by the autonomic nervous system, baroreceptors, and other factors (Malik and Camm, 1995). The sympathetic components of the autonomic nervous system increase heart rate by releasing the neural hormones catecholamine, epinephrine, and norepinephrine; while the parasympathetic components decrease heart rate through the releasing of the neurohormone acetylcholine. HRV is affected by, but not limited to, respiration, thermoregulation, hormonal regulation, blood pressure, etc. (Malik and Camm, 1995). HRV is a temporal variation between sequences of consecutive heartbeats, which reflects the balance between sympathetic and parasympathetic mediators. HRV alters pacing relation Eq. (2), in which BCLn is no longer a constant, and therefore, might affect the overall dynamics of cardiac rhythm. Despite these important consequences, the effect of HRV on alternans formation in the heart has never been investigated. The purpose of this manuscript was to determine the role of HRV on alternans formation in isolated cardiac myocytes using numerical simulations of physiological ionic model of cardiac
Fig. 1. The correspondence between APD, DI, and BCL values and the RT, TR, and RR intervals from the ECG, respectively.
action potential. First, we analyzed ECG data from Healthy and Diseased patients, separately, to determine HRV, feedback, and several other important physiological parameters used to design pacing protocols for the numerical simulations. Specifically, we designed two pacing protocols: periodic pacing with and without feedback. We then used both protocols to investigate the influence of HRV on the formation of alternans in an ionic model of cardiac action potential.
2. Methods 2.1. HRV data analysis ECG data analyses from 14 patients taken from Physionet database (Goldberger et al., 2000) were performed. Based on the information provided in Physionet, the data sets were divided in two different categories: Healthy (n ¼8), and Diseased (n ¼6). All Diseased patients were diagnosed with myocardial infarction. Each ECG trace was approximately 120 s long. We mainly applied band-pass filtering to each data set, and calculated the following parameters: RR intervals, determined as a distance between RR peaks; standard deviation (SDRR) and average (AVGRR) of RR intervals; TR and RT intervals, determined as a time between T and R peaks, and R and T peaks within RR intervals. Fig. 1 shows a direct correspondence between APD, DI, and BCL values and the RT, TR, and RR intervals from the ECG, respectively. HRV for each ECG data set was determined as HRV ¼
SDRR n100%: AVGRR
ð4Þ
The sensitivity of each ECG data set, which is an indirect representation of feedback, was calculated using the following equation s¼
SDTR AVGRT ; AVGTR SDRT
ð5Þ
where AVGTR and AVGRT are the average TR and RT values from each ECG data set, respectively. All these parameters were calculated separately for Healthy and Diseased patients and the results are presented in Tables 1 and 2, respectively. 2.2. Numerical simulations To investigate the influences of HRV on alternans formation, we used a physiological ionic model of a canine cardiac action potential (Fox et al., 2002). This model exhibits APD alternans while being periodically paced at progressively decreased BCLs, and therefore, a distinct value of BCL for the onset of alternans, BCLstart, and for the end of alternans, BCLend, can be defined. The system of ordinary differential equations was solved using a two-step Runge–Kutta method with a time step of Δt¼0.05 ms. The APD was calculated at 80% repolarization. Two different pacing protocols were used to model feedback and HRV in numerical simulations, based on ECG analysis: Protocol 1, a periodic pacing protocol with feedback, and Protocol 2, a pacing protocol without feedback. Specifically, the RR data from ECG data analysis was used for Protocol 1, and TR data was used for Protocol 2. The periodic pacing protocol is described by Eq. (2), where BCLn ¼BCL if HRV is absent. This pacing protocol entails a strong connection between APDn and DIn, and thus possesses feedback associated with periodic pacing, as described in Section 1. We started pacing at BCL ¼400 ms, and then decreased BCL by increments of 10 ms down to 100 ms. 120 stimuli were applied at each BCL in order to reach steady state. HRV was modeled by
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
3
was defined as
modifying the BCLn in Eq. (2) to the following: BCLn BCLHRV ¼ BCL þ δðHRVÞ;
ð6Þ
where δ(HRV) is a random number with a mean of zero and a standard deviation HRVnBCL, and HRV is defined by Eq. (4). In the pacing protocol without feedback, the connection between APDn and DIn was eliminated by fixing the DIn values. The DI values were set to correspond to the steady state values of DI from Protocol 1 with HRV ¼ 0% when 1:1 responses were present. When alternans was present in Protocol 1, we chose three different DI values: the high DI value, the low DI value and the DI value corresponding to the average of two consecutive stimuli. However, since the results of numerical simulations were similar for all three DI values, for demonstration purpose we chose to show results only for average DI values. Similar to Protocol 1, HRV was modeled as the following, DIn DIHRV ¼ DI þ δðHRVÞ
ð7Þ
where δ(HRV) is a random number with a mean of zero and a standard deviation HRVnBCL, and HRV is defined by Eq. (4). For both protocols, the value of HRV was varied from 0% to 6% to cover physiological range of HRV calculated from the ECG data, both for Healthy and Diseased patients. For each value of HRV, both protocols were run 10 times to mimic a stochastic process. The last 20 APDs for each BCL (or DI) for each run, representing steady state responses, were divided into even and odd beats and the mean values of 〈APD〉even and 〈APD〉odd, and corresponding standard errors were calculated separately. For each run, alternans
ΔAPD ¼ j〈APD〉even 〈APD〉odd j Z 5 ms:
ð8Þ
When HRV was present, we ensured that the magnitude of alternans is larger than the standard deviation of the mean BCL, to account for noise. Eq. (8) was also used to determine the BCL at which alternans appears, BCLstart, and the BCL at which alternans ends, BCLend, at each run. The mean 〈BCL〉start and 〈BCL〉end with their respective standard errors were then determined from the 10 runs. Once BCLstart was determined at each run, a restitution curve was constructed from the last 20 APD and DI values taken from each BCL prior to the onset of alternans, BCLstart. The restitution curve was then fitted with an exponential function, and the slope of the restitution, Smax , was calculated at the mean DI value r corresponding to the BCL value immediately preceding BCLstart. Note that Smax represents the maximum slope of the restitution r curve just prior the onset of alternans, i.e. during 1:1 behavior. The mean 〈Smax 〉 and the standard error were then determined from r the 10 runs.
2.3. Statistical analysis Statistical significance of the ECG data, presented in Tables 1 and 2, was determined by one-way ANOVA analysis. Data was considered statistically significant if p o0.05. Statistical significance of numerical data was determined using Student's t-test. Numerical results were considered statistically significant if po 0.01.
Table 1 ECG analysis of Healthy patients representing average RR (AVGRR), TR (AVGTR) and RT (AVGRT) intervals, standard deviations of the RR (SDRR), TR (SDTR) and RT (SDRT) intervals, and corresponding HRVs. Patient ID
S0306 S0303 S0292 S0304 S0308 S0452 S0457 S0460 AVG n
# $
RR
TR
RT
AVGRR (ms)
SDRR (ms)
HRVRR (%)
AVGTR (ms)
SDTR (ms)
HRVTR (%)
AVGRT (ms)
SDRT (ms)
HRVRT (%)
975.8 771.5 874.3 888.9 839.2 1047.5 1044.0 1113.8 944.3 742.1
60.9 34.5 39.3 22.6 27.4 46.0 67.7 69.4 46.0 7 6.4
6.2 4.5 4.5 2.5 3.3 4.4 6.5 6.2 4.8 7 1.5
702.3 531.6 637.1 592.6 588.8 747.1 771.1 808.4 672.4 7 93.0
59.2 34.4 38.7 23.0 27.6 45.1 67.9 68.7 45.6 7 16.7
8.4 6.5 6.1 3.9 4.7 6.0 8.8 8.5 6.6 71.7
270.6 239.9 237.3 296.4 250.3 300.5 272.9 305.3 271.7 7 25.6
4.8 3.0 3.3 3.3 3.1 4.4 3.6 3.2 3.6 7 0.6
1.8 1.2 1.4 1.1 1.3 1.5 1.3 1.0 1.3 7 0.2
n
n
n
n,#,$
n,$
n,#,$
#,$
n,#,$
n,#,$
Denotes statistical significance (p o 0.05) between Healthy and Diseased patients. Denotes statistical significance (p o 0.05) between RR and TR (RT) parameters within Healthy patients. Denotes statistical significance (po 0.05) between TR and RT parameters within Healthy patients.
Table 2 ECG Analysis of Diseased Patients representing average RR (AVGRR), TR (AVGTR) and RT (AVGRT) intervals, standard deviations of the RR (SDRR), TR (SDTR) and RT (SDRT) intervals, and corresponding HRVs. Patient ID
TWA18 TWA38 TWA31 TWA11 TWA20 TWA42 AVG n
# $
RR
TR
RT
AVGRR (ms)
SDRR (ms)
HRVRR (%)
AVGTR (ms)
SDTR (ms)
HRVTR (%)
AVGRT (ms)
SDRT (ms)
HRVRT (%)
699.7 711.7 710.0 786.0 814.8 824.0 757.2 7 23.3
19.6 18.5 13.8 20.8 25.7 34.5 22.2 7 2.9
2.8 2.6 1.9 2.6 3.2 4.2 2.9 70.8
441.2 464.3 454.2 538.4 571.4 557.6 504.57 52.6
19.8 18.8 14.2 20.5 25.5 34.5 22.2 76.4
4.5 4.1 3.1 3.8 4.5 6.2 4.4 7 0.9
258.4 247.5 256.1 247.9 243.4 266.4 253.3 7 7.8
9.3 3.2 6.2 4.2 4.0 6.1 5.5 7 2.0
3.6 1.3 2.4 1.7 1.6 2.3 2.2 70.7
n
n
n
n,#,$
n,$
n,#,$
#,$
n,#,$
n,$
Denotes statistical significance (p o 0.05) between Healthy and Diseased patients. Denotes statistical significance (p o 0.05) between RR and TR (RT) parameters within Diseased patients. Denotes statistical significance (po 0.05) between TR and RT parameters within Diseased patients.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
3. Results 3.1. The role of feedback in alternans formation: analytical insight As mentioned in Section 1, Eqs. (1) and (2) have been widely used to describe dynamics of cardiac myocytes subject to periodic pacing. Let us look at these equations a little closer, since we want to determine the behavior of the system near the 1:1 steady state value of Eq. (1). Let APDn and DIn be the steady state values of Eq. (1), such that APDn ¼ f ðBCL APDn Þ ¼ f ðDIn Þ:
ð9Þ
Taking the difference between Eqs. (1) and (9), we can write APDn þ 1 APDn ¼ f ðDIn Þ f ðDIn Þ:
ð10Þ
We can approximate the right-hand side of Eq. (10) as the following: 0
f ðDIn ÞðDIn DIn Þ ¼ Sr ðDIn DIn Þ:
ð11Þ
Therefore, combining Eqs. (10) and (11) we obtain n
n
ðAPDn þ 1 APD Þ ¼ Sr ðDIn DI Þ:
ð12Þ
Eq. (12) describes the restitution properties of the heart, suggesting that small changes in APD are proportional to small changes in preceding DI. It is important to note that Eq. (12) is linear, since Sr is a constant for any given small changes, and therefore, cannot describe complex cardiac rhythms, such as alternans. The physiological basis for restitution properties of the heart, arising from, among other factors, incomplete recovery from
inactivation of delayed rectifier ionic channels, has been discussed previously in greater details (Rudy and Silva, 2006). In the case of periodic pacing with constant BCL (pacing Protocol 1), the APD and DI are related through the pacing relation (Eq. (2)), which can be rewritten as DIn ¼ BCL APDn DIn ¼ BCL APDn ;
ð13Þ
and therefore, the following relation can be derived: APDn APDn ¼ ðDIn DIn Þ
ð14Þ
Eq. (14) describes the feedback associated with periodic pacing protocol, and suggests that small changes in DI are proportional with opposite sign to small changes in APD. Combination of the restitution properties of the myocytes (Eq. (12)) with the relationship derived based on periodic pacing (Eq. (14)) leads to a recurrence relation APDn þ 1 APDn ¼ Sr ðAPDn APDn Þ
ð15Þ
that has been used to describe different dynamical behavior of cardiac myocytes: 1:1 responses and possible presence of alternans. Indeed, for |Sr| 4 1, any small changes in APD that appear after nth stimulus is applied, will enhance the difference APDn þ 1 APDn during further pacing, i.e. at the next (n þ1)st stimulus. If |Sr| o1, relation (Eq. (15)) describes 1:1 responses, since small changes in APD will eventually die out. In the case when the connection between APDn and DIn are eliminated by fixing the DIn value (pacing Protocol 2), Eq. (13) is no longer valid, so feedback is absent. Therefore, we cannot derive Eq. (15) and as a result are unable to mathematically explain the mechanism of alternans formation in isolated cells. Indeed, mapping model (Eq. (1)) will return a single APDn þ 1 value for a fixed DIn in the absence of feedback. The difference between pacing protocols with and without feedback is illustrated in Fig. 2, where BCLs (stars), APDs (open circles), and DIs (open triangles) are shown for Protocol 1 (Fig. 2A) and Protocol 2 (Fig. 2B) as a function of beat number for the case of HRV ¼0%. For Protocol 1, the BCL was changed from 270 ms to 170 ms, and corresponding values of DI were calculated using Eq. (13). Note that the slope of the restitution curve Sr 41 at BCL ¼170 ms, leading to the formation of alternans due to the presence of feedback (Eq. (14)), and therefore, recurrence relation (Eq. (15)). In contrast, no alternans is present in Fig. 2B, when DI is changed from 105 ms to 48 ms. Note, that the calculated values of BCLs are very similar to those from Fig. 2A. 3.2. ECG data analysis
Fig. 2. BCL (stars), APD (open circles), and DI (open triangles) as a function of beat number for (A) constant-BCL pacing protocol and (B) constant-DI pacing protocol for the case of HRV ¼0%. For (A), alternans formation is shown when BCL is changed from 270 ms to 170 ms is shown. For (B), no alternans is present when DI is changed from 105 ms to 85 ms (which correspond to roughly the same BCLs).
Fig. 3A shows a representative example of RR intervals as a function of beat number taken from an ECG of a Healthy patient's data, illustrating the presence of HRV. Fig. 3B shows the histogram of RR intervals from the data in Fig. 3A. Note the approximately normal distribution of RR intervals in Fig. 3B. Both Healthy and Diseased patients' RR distributions were qualitatively similar to the one shown in Fig. 1, and therefore we chose a normal distribution for the random numbers in Eqs. (6) and (7). Fig. 1 illustrates the relationship of the RR, RT, and TR intervals to the BCL, APD, and DI respectively. Comprehensive analyses of ECG traces from the two groups of patients, Healthy and Diseased, are presented in Tables 1 and 2, respectively. Note that AVGRR, (944.38 742.08 ms vs. 757.7 7 23.25 ms, po 0.05), SDRR, (45.98 76.42 ms vs. 22.15 72.92 ms, po 0.01), and HRV (4.767 1.45% vs. 2.897 0.75%, po 0.05) are significantly larger for Healthy patients in comparison with Diseased patients.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Fig. 3. (A) An ECG trace of a 32-year-old male with normal sinus rhythm taken from Goldberger et al. (2000). (B) The histogram of the RR intervals calculated from the ECG trace shown in A.
5
Fig. 4. (A) Representative plot of RT vs. TR intervals for two Healthy (circles) and two Diseased (triangles) patients. (B) Values of s from all Healthy and Diseased patients. n represent statistical significance (p o0.01).
In order to determine whether feedback is present in the Healthy and Diseased patients, we separately calculated the nth TR (TRn) and RT (RTn) intervals from the entire ECG data set, corresponding to DIn and APDn, respectively, as indicated in Fig. 1. Therefore, the relationship between TR and RT intervals correlates with the relationship between DI and APD, i.e. feedback. Fig. 4A shows representative examples of the nth RT intervals as a function of the nth TR intervals for two Healthy (open and filled circles) and two Diseased (open and filled triangles) patients. Note that for Healthy patients, the standard deviations of the TR range are higher than in the RT range, while for Diseased patients the standard deviations for both ranges are similar. To quantify this data, we calculated the sensitivity, s, according to Eq. (5), which is an indirect representation of feedback (see Fig. 4B). Note that s was significantly higher for Healthy patients (s¼5.1 70.6) than for Diseased patients (s ¼2.2 7 0.3, po 0.01), suggesting the presence of different associations between RT and TR intervals for Healthy than for Diseased patients. 3.3. The effect of HRV on alternans formation in the presence of feedback Fig. 5 shows APD as a function of BCL for the ionic model of an isolated cardiac myocyte that was paced using the periodic pacing protocol (Protocol 1) with no HRV (Fig. 5A) and HRV ¼2.5% (Fig. 5B). Note the formation of alternans at a certain range of BCLs both with and without HRV. Also note that the onset of alternans occurred at a higher BCL (BCLstart ¼ 200 ms) and ended at a lower BCL (BCLend ¼140 ms) for HRV ¼2.5% than for HRV ¼ 0% (BCLstart ¼190 ms and BCLend ¼ 150 ms, respectively). Fig. 6A illustrates the influence of HRV on the onset, 〈BCL〉start, and the end, 〈BCL〉end, of alternans. As HRV is increased, alternans
Fig. 5. APD as a function of BCL for the ionic model paced with constant-BCL pacing protocol with HRV ¼0% (A) and 2.5% (B). BCLstart and BCLend denote the start and end of APD alternans.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
6
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 6. Influence of HRV on the following parameters: (A) 〈BCL〉start and 〈BCL〉end; (B) ΔAPDmax ; (C) ΔAPDBCL ¼ 170 ms and (D) 〈Sr〉. as function of HRV. * represents statistical significance (p o 0.01).
occurred earlier in the pacing protocol and ended later in the pacing protocol. Note that HRV larger than 1.5% significantly increases 〈BCL〉start when compared to HRV ¼0%. Similarly, HRV larger than 2.5% significantly decreases 〈BCL〉end. These results suggest that HRV promotes the formation of alternans for the case of Protocol 1 with feedback. In order to investigate the effect of HRV on the amplitude of alternans, we calculated two parameters: the maximal amplitude of alternans, ΔAPDmax , over all BCLs in the run and the amplitude of alternans at BCL ¼ 170 ms, ΔAPDBCL ¼ 170 ms, where alternans was present at HRV ¼0%. The influence of HRV on ΔAPDmax and ΔAPDBCL ¼ 170 ms is shown in Fig. 6B and C, respectively. Fig. 6B and C shows that the presence of HRV did not significantly affect either ΔAPDmax or ΔAPDBCL ¼ 170 ms, when compared to values at HRV ¼0%. To further understand the effect of HRV on the onset of alternans in the presence of feedback, we calculated the maximum slope of the restitution curve, 〈Smax 〉, at the onset of alternans, r 〈BCL〉start. Fig. 6D shows that HRV 41.5% leads to a significant decrease of 〈Smax 〉 when compared to HRV ¼0%. r 3.4. Elimination of feedback suppresses the formation of alternans Fig. 7 shows APD as a function of DI for the ionic model of an isolated cardiac myocyte that was paced using Protocol 2 without feedback with HRV ¼0% (Fig. 7A) and HRV ¼ 2.5% (Fig. 7B). Note that in contrast to Fig. 5, no alternans is seen in Fig. 7 (confirmed by inspection of actual voltage traces) both for HRV ¼0% or 2.5%. To confirm these findings, we increased HRV up to 12%, and alternans was still absent (data not shown). This demonstrated that elimination of feedback prevents formation of alternans in a myocyte. Fig. 8 illustrates the presence of feedback in a cardiac myocyte that was paced either with Protocol 1 (Fig. 8A) or Protocol 2 (Fig. 8B).
Specifically, in Fig. 8A, APDn is plotted as a function of DIn for Protocol 1 with HRV ¼2.5%. To demonstrate the presence of feedback for different BCLs, two values of BCLs were chosen: BCL ¼270 ms (open circles), and BCL ¼ 170 ms (filled circles). In the case of HRV ¼2.5%, Protocol 1 led to steady state 1:1 responses at BCL ¼270 ms, and alternans at BCL ¼ 170 ms. Fig. 8A suggests that the distribution of data for BCL ¼ 270 ms was qualitatively similar to the data from Healthy patients (see Fig. 4A). However, it is important to note that the ranges of the values were very different, thus preventing us from making any distinct conclusions. On the other hand, for BCL ¼170 ms, the DIn and APDn pairs fall along a line with slope of 1, as predicted by Eq. (2). Similar simulations were performed with Protocol 2, and the results are shown in Fig. 8B. Here two values of DI were chosen, DI¼ 48 ms (open circles) and DI ¼105 ms (filled circles), as these DI values correspond to the steady state BCLs 170 ms and 270 ms, respectively. Fig. 8B demonstrates that for DI ¼105 ms the distribution of the DIn and APDn was qualitatively similar to the one from Healthy patients (see Fig. 4A). On the other hand, for DI ¼48 ms the feedback was qualitatively similar to the one from Diseased patients (see Fig. 4A). However, this observation does not allows us to make any distinct conclusions since the ranges of values were very different.
4. Conclusion and discussion In this study, we investigated the effects of HRV and feedback on the development of alternans in a physiological ionic model of a canine cardiac action potential. We demonstrated that (Eq. (1)) HRV promoted the formation of alternans in cardiac myocytes in the presence of feedback. Specifically HRV 41.5% significantly shifted the onset of alternans towards larger BCL, but does not affect the amplitude of alternans, ΔAPDmax and ΔAPDBCL ¼ 170 ms. This shift of the onset of alternans was accompanied by a significant
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Fig. 7. APD as a function of DI for the ionic model paced with constant-DI pacing protocol with HRV ¼0% (A) and 2.5% (B). Both figures show the absence of alternans.
decrease in 〈Smax 〉 value for the same HRV values. We also showed r that (Eq. (2)) the absence of feedback during pacing prevented the formation of alternans in cardiac myocytes regardless of the presence or absence of HRV. Since alternans is closely correlated with arrhythmias, prevention of alternans formation is an important area of research in cardiac electrophysiology. First, we demonstrated that in the case of periodic pacing protocol with feedback, HRV facilitated alternans formation in isolated cell. Indeed, Fig. 5A shows that the introduction of HRV causes 〈BCL〉start to occur at larger values and 〈BCL〉end to occur at lower values, suggesting a potential pro-arrhythmic effect of HRV. Interestingly, the increased range of BCL for which alternans was observed was accompanied by a decreased 〈Smax 〉 value at the r onset of alternans, suggesting that alternans can be formed in the absence of steep restitution slope, thus again confirming a potential pro-arrhythmic effect of HRV. Second, we showed that removing the feedback from the pacing protocol (Protocol 2) resulted in the elimination of alternans in isolated cardiac cells. Moreover, alternans did not appear when HRV was introduced in that case, indicating a potential anti-arrhythmic role of no-feedback pacing. However, more comprehensive numerical simulations of the cardiac tissue (instead of isolated cells) or experiments are necessary to further elucidate the role of HRV on the stability of cardiac rhythms both in healthy and diseased hearts. Previously, several attempts have been made to develop a pacing protocol that permits explicit control of DI, independent of APD (Patwardhan and Moghe, 2001; Wu and Patwardhan, 2004). It was demonstrated that the ability to pace cardiac tissue so that DIs can be selected independently provides a unique way to explore mechanisms of APD alternans by eliminating DIdependent restitution effects. Specifically, Wu and Patwardhan (Patwardhan and Moghe, 2001) demonstrated that the mechanisms of alternans include restitution-dependent and restitution-
7
Fig. 8. Feedback represented via APDn as a function of the DIn for the following pacing protocols: (A) constant-BCL pacing protocol with HRV ¼2.5% at BCL ¼270 ms (open circles) and BCL¼ 170 ms (filled circles); and (B) constant-DI pacing protocol with HRV¼ 5%, shown for the DI¼ 105 ms (open circles) and DI¼ 48 ms (filled circles). Solid line indicates a 1 slope.
independent components, and therefore, restitution may not be a necessary mechanism for alternans to exist. Later on, this group applied a constant DI pacing protocol experimentally in canine tissue (Wu et al., 2006) and in pig tissue slab preparation (Jing el al., 2012) to demonstrate that eliminating the restitutiondependent mechanism of alternans by explicitly controlling DI does not preclude from APD alternans formation in the tissue. Although their protocols are somewhat similar to our Protocol 2 in the absence of HRV, their overall results contradict our observations. This discrepancy can be explained by the fact that the authors applied 15 or 30 beats at each DI before changing it, while we applied 120 beats in order to ensure steady state. Furthermore, this discrepancy can also be attributed to the fact that the amount of short-term memory that is present in real canine or pig tissue is much more substantial in comparison to the ionic model we used. This fact was indirectly demonstrated in the numerical results of Jordan and Christini (Jordan et al., 2004), who showed the suppression of alternans in several ionic models that were paced using a constant DI pacing protocol. It is important to note, however, that none of the discussed works attempted to incorporate HRV in their pacing protocols. It is also worth mentioning that several studies have been using the concept of pro- or anti-arrhythmic pacing to facilitate or control development of alternans in cardiac tissue, respectively. For instance, Christini et al. 2001, 2006 used a protocol similar to our Protocol 2 to reduce or eliminate the presence of APD alternans by modifying the cycle length in real time based on the size of alternans. Gelzer et al. (2008) performed an experimental realization of previously developed pacing protocol (Fox et al., 2003) that consists of the delivery of three premature stimuli at various time after basic stimulus to facilitate alternans
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
formation and to induce ventricular fibrillation. In contrast, in our study, we aimed to understand how different pacing protocols affect the development of alternans, without trying to control or facilitate the development of alternans by altering pacing. Previous work by Dai and Keener (2012) investigated how it is possible to approximate the dynamics of a cardiac cell from knowing the mean of the BCLs and APDs. Similar to our study, the authors introduced some randomness through the BCL but unlike us, they used a simpler mapping model with feedback. Several other attempts to add randomness in the pacing have been made (Dai and Keener, 2012; Christini et al., 2001, 2006), but nobody has systemically investigated effects of feedback on alternans formation. We also examined the effect of adding randomness to the DI and removing the feedback relationship. Dia and Keener did use different distributions for the randomness but none of these distributions were based on data analysis of ECG data as our modeling was. Our research also examined 〈BCL〉start, 〈BCL〉end, and alternans magnitude. In this manuscript, we modeled HRV with a random number function, and we assumed that HRV does not change as the heart rate changes, which is an over-simplified approach. Indeed, it is known that HRV can be affected by, but not limited to, respiration, thermoregulation, hormonal regulation, blood pressure, etc. (Malik and Camm, 1995). In addition, the relative contribution and importance of all these physiological aspects of HRV might be altered under different disease conditions. Therefore, a more comprehensive study needs to be performed to provide a more physiologically relevant model of HRV. One of the limitations of our study is that we did not take into account the intracellular calcium cycling, which can play an important role in the development of APD alternans (Weiss et al., 2006; Chudin et al., 1998; Goldhaber et al., 2005). Indeed, alternans in intracellular calcium transient amplitude has been linked to mechanical alternans; thus the simultaneous occurrence of calcium and APD alternans, termed electromechanical alternans, is believed to be a substrate for various cardiac arrhythmias. Another limitation of our study is that numerical simulations performed were for single cardiac myocytes only. It is known that electrophysiological properties, including alternans formation, can be different at the tissue level (Fox et al., 2002; Watanabe et al., 2001; Sato et al., 2006).
Q2 Uncited references
Watanabe and Koller (2002); Jordan and Christini (2004); Otani and Gilmour (1997); Tolkacheva et al. (2003); Kalb et al. (2005); Chialvo et al. (1990); Kalb et al. (2004); Tolkacheva et al. (2004).
Acknowledgments The authors wish to thank Dr. Mironov for helpful discussions, and Dr. Xie, Mr. Ippolito, Mr. Visweswaran, and Mr. Switala for careful reading of the manuscript. This work was supported by National Science Foundation grants (E.G.T) PHY 0957468, CMMI-1233951, and NSF CAREER PHY-125541. References Banville, I., Gray, R.A., 2002. Effect of action potential duration and conduction velocity restitution and their spatial dispersion on alternans and the stability of arrhythmias. J. Cardiovasc. Electrophysiol. 13, 1141. Chialvo, D.R., Michaels, D.C., Jalife, J., 1990. Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fiber. Circ. Res. 66, 525. Christini, D.J., Stein, K.M., et al., 2001. Nonlinear-dynamical arrhythmia control in humans. Proc. Natl. Acad. Sci. USA 98, 5827–5832.
Christini, D.J., Riccio, M.L., et al., 2006. Control of electrical alternans in canine cardiac Purkinje fibers. Phys. Rev. Lett 96, 104101. Chudin, E., Garfinkel, A., et al., 1998. Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study). Prog. Biophys. Mol. Biol. 69 (2-3), 225–236. Dai, S., Keener, J.P., 2012. Using noise to determine cardiac restitution with memory. Phys. Rev. E 85 (6), 061902. Fox, J.J., McHarg, J.L., Gilmour, R.F., 2002. Ionic mechanism of electrical alternans. Am. J. Physiol. 282 (2), H516–H530. Fox, J.J., Gilmour , R.F., Bodenschatz, E., 2002. Conduction block in one-dimensional heart fiber. Phys. Rev. Lett. 89, 198101. Fox, J.J., Riccio, M.L., Hua, F., Bodenschatz, E., Gilmour, R.F., 2002. Spatiotemporal transition to conduction block in canine ventricle. Circ. Res. 90, 289–296. Fox, Jeffrey J., et al., 2003. Dynamic mechanism for conduction block in heart tissue. New J. Phys. 5 (1), 101.14. Franz, M.R., 2003. The electrical restitution curve revisited: steep or flat slope – which is better? J. Cardiovasc. Electrophysiol. 14, 140–147. Gelzer, Anna R.M., et al., 2008. Dynamic mechanism for initiation of ventricular fibrillation in vivo. Circulation 118 (11), 1123–1129. Gilmour , R.F., 2002. Electrical restitution and ventricular fibrillation: negotiating a slippery slope. J. Cardiovasc. Electropyhsiol. 13, 1150. Gilmour , R.F., Chialvo, D.R., 1999. Editorial: electrical restitution, critical mass and the riddle of fibrillation. J. Cardiovasc. Electrophysiol. 10, 1087. Goldberger, A.L., Amaral, L.A.N., Glass, L., 2000. PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101, e215–e220. Goldhaber, J.I., Xie, L.-H., Duong, T., Motter, C.h., Khuu, K., Weiss, J.N., 2005. Action potential duration restitution and alternans in rabbit ventricular myocytes. The key role of intracellular calcium cycling. Circ. Res. 96, 459–466. Guevara, M., Ward, G., Shrier, A., Glass, L., 1984. Computers in Cardiology. IEEE Computer Society, Silver Spring, MD p. 167 Hall, G.M., Gauthier, D.J., 1999. Experimental control of cardiac muscle alternans. Phys. Rev. Lett. 82, 2995. Jing, Linyuan, Abhijit Patwardhan, 2012. Hysteresis in DI independent mechanisms in threshold for transition between 1∶1 and 2∶2 rhythms in pigs. Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE, IEEE. Jordan, P.N., Christini, D.J., 2004. Determining the effects of memory and APD alternans on cardiac restitution using a constant-memory restitution protocol. Physiol. Meas. 25, 1013–1024. Jordan, Peter N., David, J. Christini, 2004. Determining the effects of memory and action potential duration alternans on cardiac restitution using a constantmemory restitution protocol. Physiol. Meas. 25 (4), 1013. Kalb, S.S., Dobrovolny, H., Tolkacheva, E.G., Idriss, S.F., Krassowska, W., Gauthier, D.J., 2004. The restitution portrait: a new method for investigating rate-dependent restitution. J. Cardiovasc. Electrophysiol. 15, 698. Kalb, S.S., Tolkacheva, E.G., Schaeffer, D.G., Gauthier, D.J., Krassowska, W., 2005. Restitution in mapping models with arbitrary amount of memory. Chaos 15, 23701. Karma, A., 1994. Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 4, 461. Kleiger, R.E., Miller, J.H.R.V., Bigger, J.T., Moss, A.J., 1987. Decreased heart rate variability and its association with increased mortality after acute myocardial infarction. Am. J. Cardiol. 59, 256. Koller, M.L., Riccio, M.L., Gilmour , R.F., 1998. Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation. Am. J. Physiol. 275, H1635. La Rovere, M.T., Pinna, G.D., et al., 2003. Short-term heart rate variability strongly predicts sudden cardiac death in chronic heart failure patients. Circulation 107, 565. Malik, M., Camm, A.J., 1995. Heart Rate Variability. Futura Pub. Co., Armonk, N.Y. Narayan, S.M., Bayer, J., et al., 2008. Action potential dynamics explain arrhythmic vulnerability in human heart failure: a clinical and modeling study implicating abnormal calcium handling. J. Am. Coll. Cardiol. 52, 1782–1792. Nolasco, J.B., Dahlen, R.W., 1968. A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25, 191. Otani, N.F., Gilmour , R.F., 1997. Memory models for the electrical properties of local cardiac systems. J. Theor. Biol. 187, 409. Patwardhan, A., Moghe, S., 2001. Novel feedback based stimulation protocol shows hysteresis in cardiac action potential duration restitution. Biomed. Sci. Instrum. 37, 505–510. Riccio, M.L., Koller, M.L., Gilmour , R.F., 1999. Electrical restitution and spatiotemporal organization during ventricular fibrillation. Circ. Res. 84, 955. Rudy, Y., Silva, J.R., 2006. Computational biology in the study of cardiac ion channels and cell electrophysiology. Q. Rev. Biophys. 39, 57–116. Sato, D., Shiferaw, Y., Garfinkel, A., Weiss, J.N., Qu, Z., Karma, A., 2006. Spatially discordant alternans in cardiac tissue role of calcium cycling. Circ. Res. 9, 520–527. Tolkacheva, E.G., Schaeffer, D.G., Gauthier, D.J., Krassowska, W., 2003. Condition for alternans and stability of the 1:1 response pattern in a “memory” model of paced cardiac dynamics. Phys. Rev. E. 67, 031904. Tolkacheva, E.G., Romeo, M.M., Guerraty, M., Gauthier, D.J., 2004. Condition for alternans and its control in a two-dimensional mapping model of paced cardiac dynamics. Phys. Rev. E. 69, 031904. Watanabe, M.A., Koller, M.L., 2002. Mathematical analysis of dynamics of cardiac memory and accommadation: theory and experiment. Am. J. Physiol. 282, H1534.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
Q5
Q6
Q7
Q8
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7
Watanabe, M.A., Otani, N.F., Gilmour , R.F., 1995. Biphasic restitution of action potential duration and complex dynamics in ventricular myocardium. Circ. Res. 76, 915. Watanabe, M.A., Fenton, F.H., Evans, S.J., Hastings, H.M., Karma, A., 2001. Mechanisms for discordant alternans. J. Cardiovasc. Electrophysiol. 12, 196–206. Weiss, J.N., Karma, A., Shiferaw, Y., Chen, P.-S., Garfinkel, A., Qu, Z., 2006. From pulsus to pulseless: the saga of cardiac alternans. Circ. Res. 98, 1244–1253.
9
Wu, R., Patwardhan, A., 2004. Restitution of action potential duration during sequential changes in diastolic intervals shows multimodal behavior. Circ. Res. 94, 634–641. Wu, Runze, Patwardhan, Abhijit, 2006. Mechanism of repolarization alternans has restitution of action potential duration dependent and independent components. J. Cardiovasc. Electrophysiol. 17.1, 87–93. Zipes, D.H.R.V., Wellens, H.J., 1998. Sudden cardiac death. Circulation 98, 2334–2351.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29Q3 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Q4 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics Stephen D. McIntyre a,c, Virendra Kakade b,c, Yoichiro Mori a, Elena G. Tolkacheva c,n a
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Electrical and Computer Engineering Department, University of Minnesota, Minneapolis, MN 55455, USA c Biomedical Engineering Department, University of Minnesota, Minneapolis, MN 55455, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 May 2013 Received in revised form 28 January 2014 Accepted 13 February 2014
A beat-to-beat alternation in the action potential duration (APD) of myocytes, i.e. alternans, is believed to be a direct precursor of ventricular fibrillation in the whole heart. A common approach for the prediction of alternans is to construct the restitution curve, which is the nonlinear functional relationship between the APD and the preceding diastolic interval (DI). It was proposed that alternans appears when the magnitude of the slope of the restitution curve exceeds one, known as the restitution hypothesis. However, this restitution hypothesis was derived under the assumption of periodic stimulation, when there is a dependence of the DI on the immediate preceding APD (i.e. feedback). However, under physiological conditions, the heart rate exhibits substantial variations in time, known as heart rate variability (HRV), which introduces deviations from periodic stimulation in the system. In this manuscript, we investigated the role of HRV on alternans formation in isolated cardiac myocytes using numerical simulations of an ionic model of the cardiac action potential. We used this model with two different pacing protocols: a periodic pacing protocol with feedback and a protocol without feedback. We show that when HRV is incorporated in the periodic pacing protocol, it facilitated alternans formation in the isolated cell, but did not significantly change the magnitude of alternans. On the other hand, in the case of the pacing protocol without feedback, alternans formation was prevented, even in the presence of HRV. & 2014 Published by Elsevier Ltd.
Keywords: Action potential Periodic pacing Cardiac dynamics Numerical simulations
1. Introduction One of the most fundamental characteristics of cardiac cells is the shortening of the action potential duration (APD) as the heart rate increases, a phenomenon known as electrical restitution. Restitution plays a vital role in heart function: for a given heart rate, a shorter APD allows for a longer diastolic interval (DI), thereby giving adequate time for the heart to refill with blood. Although important for life at moderate heart rates, at higher rates, restitution may result in life-threatening cardiac rhythms and ventricular fibrillation (VF), in particular (Zipes and Wellens, 1998; Franz, 2003). It is generally believed that T-wave alternans, defined as an alternating change in the amplitude or shape of the T-wave in the ECG, is a precursor of cardiac electrical instability (Karma, 1994; Watanabe et al., 1995; Gilmour and Chialvo, 1999; Fox et al., 2002). T-wave alternans results from APD alternans at the cellular level.
n Correspondence to: University of Minnesota, 312 Church Street SE, 6-128 Nils Hasselmo Hall, Minneapolis, MN 55455, USA. Tel.: þ1 612 626 2719. E-mail address: [email protected] (E.G. Tolkacheva).
A common technique for studying the initiation and maintenance of alternans and other complex rhythms is to analyze the restitution curve, the nonlinear functional relationship between the APD and the preceding DI. While detailed ionic models were used extensively to study the response of cardiac myocytes to stimulation, mapping models were introduced to focus on restitution (Nolasco and Dahlen, 1968; Guevara et al., 1984). Specifically, it was proposed that the APD could be determined as a function of the preceding DI, essentially forming the one-dimensional mapping model. APDn þ 1 ¼ f ðDIn Þ:
ð1Þ
here, f is the restitution curve, APDn þ 1 is the APD generated by the (nþ1)st stimulus and DIn is the nth DI, i.e., the interval during which the tissue recovers to its resting state after the end of the previous (nth) action potential. In 1968, Nolasco and Dahlen (Nolasco and Dahlen, 1968) developed a graphical method to analyze and predict APD alternans in a mapping model (Eq. (1)) under the assumption that pacing occurs at a constant rate, i.e. when the APD and DI are related through the pacing relation. APDn þ DIn ¼ BCLn
ð2Þ
http://dx.doi.org/10.1016/j.jtbi.2014.02.015 0022-5193 & 2014 Published by Elsevier Ltd.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
2
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
where the basic cycle length BCLn ¼BCL (a constant) under periodic pacing. Using Eqs. (1) and (2), it was proposed that the normal cardiac rhythm, or so-called 1:1 response, becomes unstable and alternans occurs when the magnitude of the slope of the restitution curve exceeds one, 0 df Sr ¼ ¼ f j Z1; ð3Þ dDI DI ¼ DIn known as the restitution hypothesis. This restitution hypothesis has been confirmed in some experiments (Koller et al., 1998) and has led to the assumption that flattening the restitution curve will help prevent VF (Koller et al., 1998; Hall and Gauthier, 1999; Gilmour, 2002; Banville and Gray, 2002; Riccio et al., 1999). However, recent experimental results have shown that this hypothesis is incorrect in many situations, where the normal cardiac rhythm is observed when the restitution curve is very steep or in which the transition to alternans occurs in the presence of a flat restitution curve (Koller et al., 1998; Hall and Gauthier, 1999; Gilmour, 2002; Banville and Gray, 2002; Narayan et al., 2008). One of the reasons why the restitution condition fails experimentally is that Nolasco and Dahlen's approach is valid only for periodic pacing, i.e. when BCLn ¼ BCL is a constant in Eq. (2), and cannot be applied for complex stimulation regimes, such as those which result in physiological heart rates that exhibit heart rate variability (HRV) (Malik and Camm, 1995; Goldberger et al., 2000; Kleiger et al., 1987; La Rovere et al., 2003). It is known that the heart rate is regulated by the autonomic nervous system, baroreceptors, and other factors (Malik and Camm, 1995). The sympathetic components of the autonomic nervous system increase heart rate by releasing the neural hormones catecholamine, epinephrine, and norepinephrine; while the parasympathetic components decrease heart rate through the releasing of the neurohormone acetylcholine. HRV is affected by, but not limited to, respiration, thermoregulation, hormonal regulation, blood pressure, etc. (Malik and Camm, 1995). HRV is a temporal variation between sequences of consecutive heartbeats, which reflects the balance between sympathetic and parasympathetic mediators. HRV alters pacing relation Eq. (2), in which BCLn is no longer a constant, and therefore, might affect the overall dynamics of cardiac rhythm. Despite these important consequences, the effect of HRV on alternans formation in the heart has never been investigated. The purpose of this manuscript was to determine the role of HRV on alternans formation in isolated cardiac myocytes using numerical simulations of physiological ionic model of cardiac
Fig. 1. The correspondence between APD, DI, and BCL values and the RT, TR, and RR intervals from the ECG, respectively.
action potential. First, we analyzed ECG data from Healthy and Diseased patients, separately, to determine HRV, feedback, and several other important physiological parameters used to design pacing protocols for the numerical simulations. Specifically, we designed two pacing protocols: periodic pacing with and without feedback. We then used both protocols to investigate the influence of HRV on the formation of alternans in an ionic model of cardiac action potential.
2. Methods 2.1. HRV data analysis ECG data analyses from 14 patients taken from Physionet database (Goldberger et al., 2000) were performed. Based on the information provided in Physionet, the data sets were divided in two different categories: Healthy (n ¼8), and Diseased (n ¼6). All Diseased patients were diagnosed with myocardial infarction. Each ECG trace was approximately 120 s long. We mainly applied band-pass filtering to each data set, and calculated the following parameters: RR intervals, determined as a distance between RR peaks; standard deviation (SDRR) and average (AVGRR) of RR intervals; TR and RT intervals, determined as a time between T and R peaks, and R and T peaks within RR intervals. Fig. 1 shows a direct correspondence between APD, DI, and BCL values and the RT, TR, and RR intervals from the ECG, respectively. HRV for each ECG data set was determined as HRV ¼
SDRR n100%: AVGRR
ð4Þ
The sensitivity of each ECG data set, which is an indirect representation of feedback, was calculated using the following equation s¼
SDTR AVGRT ; AVGTR SDRT
ð5Þ
where AVGTR and AVGRT are the average TR and RT values from each ECG data set, respectively. All these parameters were calculated separately for Healthy and Diseased patients and the results are presented in Tables 1 and 2, respectively. 2.2. Numerical simulations To investigate the influences of HRV on alternans formation, we used a physiological ionic model of a canine cardiac action potential (Fox et al., 2002). This model exhibits APD alternans while being periodically paced at progressively decreased BCLs, and therefore, a distinct value of BCL for the onset of alternans, BCLstart, and for the end of alternans, BCLend, can be defined. The system of ordinary differential equations was solved using a two-step Runge–Kutta method with a time step of Δt¼0.05 ms. The APD was calculated at 80% repolarization. Two different pacing protocols were used to model feedback and HRV in numerical simulations, based on ECG analysis: Protocol 1, a periodic pacing protocol with feedback, and Protocol 2, a pacing protocol without feedback. Specifically, the RR data from ECG data analysis was used for Protocol 1, and TR data was used for Protocol 2. The periodic pacing protocol is described by Eq. (2), where BCLn ¼BCL if HRV is absent. This pacing protocol entails a strong connection between APDn and DIn, and thus possesses feedback associated with periodic pacing, as described in Section 1. We started pacing at BCL ¼400 ms, and then decreased BCL by increments of 10 ms down to 100 ms. 120 stimuli were applied at each BCL in order to reach steady state. HRV was modeled by
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
3
was defined as
modifying the BCLn in Eq. (2) to the following: BCLn BCLHRV ¼ BCL þ δðHRVÞ;
ð6Þ
where δ(HRV) is a random number with a mean of zero and a standard deviation HRVnBCL, and HRV is defined by Eq. (4). In the pacing protocol without feedback, the connection between APDn and DIn was eliminated by fixing the DIn values. The DI values were set to correspond to the steady state values of DI from Protocol 1 with HRV ¼ 0% when 1:1 responses were present. When alternans was present in Protocol 1, we chose three different DI values: the high DI value, the low DI value and the DI value corresponding to the average of two consecutive stimuli. However, since the results of numerical simulations were similar for all three DI values, for demonstration purpose we chose to show results only for average DI values. Similar to Protocol 1, HRV was modeled as the following, DIn DIHRV ¼ DI þ δðHRVÞ
ð7Þ
where δ(HRV) is a random number with a mean of zero and a standard deviation HRVnBCL, and HRV is defined by Eq. (4). For both protocols, the value of HRV was varied from 0% to 6% to cover physiological range of HRV calculated from the ECG data, both for Healthy and Diseased patients. For each value of HRV, both protocols were run 10 times to mimic a stochastic process. The last 20 APDs for each BCL (or DI) for each run, representing steady state responses, were divided into even and odd beats and the mean values of 〈APD〉even and 〈APD〉odd, and corresponding standard errors were calculated separately. For each run, alternans
ΔAPD ¼ j〈APD〉even 〈APD〉odd j Z 5 ms:
ð8Þ
When HRV was present, we ensured that the magnitude of alternans is larger than the standard deviation of the mean BCL, to account for noise. Eq. (8) was also used to determine the BCL at which alternans appears, BCLstart, and the BCL at which alternans ends, BCLend, at each run. The mean 〈BCL〉start and 〈BCL〉end with their respective standard errors were then determined from the 10 runs. Once BCLstart was determined at each run, a restitution curve was constructed from the last 20 APD and DI values taken from each BCL prior to the onset of alternans, BCLstart. The restitution curve was then fitted with an exponential function, and the slope of the restitution, Smax , was calculated at the mean DI value r corresponding to the BCL value immediately preceding BCLstart. Note that Smax represents the maximum slope of the restitution r curve just prior the onset of alternans, i.e. during 1:1 behavior. The mean 〈Smax 〉 and the standard error were then determined from r the 10 runs.
2.3. Statistical analysis Statistical significance of the ECG data, presented in Tables 1 and 2, was determined by one-way ANOVA analysis. Data was considered statistically significant if p o0.05. Statistical significance of numerical data was determined using Student's t-test. Numerical results were considered statistically significant if po 0.01.
Table 1 ECG analysis of Healthy patients representing average RR (AVGRR), TR (AVGTR) and RT (AVGRT) intervals, standard deviations of the RR (SDRR), TR (SDTR) and RT (SDRT) intervals, and corresponding HRVs. Patient ID
S0306 S0303 S0292 S0304 S0308 S0452 S0457 S0460 AVG n
# $
RR
TR
RT
AVGRR (ms)
SDRR (ms)
HRVRR (%)
AVGTR (ms)
SDTR (ms)
HRVTR (%)
AVGRT (ms)
SDRT (ms)
HRVRT (%)
975.8 771.5 874.3 888.9 839.2 1047.5 1044.0 1113.8 944.3 742.1
60.9 34.5 39.3 22.6 27.4 46.0 67.7 69.4 46.0 7 6.4
6.2 4.5 4.5 2.5 3.3 4.4 6.5 6.2 4.8 7 1.5
702.3 531.6 637.1 592.6 588.8 747.1 771.1 808.4 672.4 7 93.0
59.2 34.4 38.7 23.0 27.6 45.1 67.9 68.7 45.6 7 16.7
8.4 6.5 6.1 3.9 4.7 6.0 8.8 8.5 6.6 71.7
270.6 239.9 237.3 296.4 250.3 300.5 272.9 305.3 271.7 7 25.6
4.8 3.0 3.3 3.3 3.1 4.4 3.6 3.2 3.6 7 0.6
1.8 1.2 1.4 1.1 1.3 1.5 1.3 1.0 1.3 7 0.2
n
n
n
n,#,$
n,$
n,#,$
#,$
n,#,$
n,#,$
Denotes statistical significance (p o 0.05) between Healthy and Diseased patients. Denotes statistical significance (p o 0.05) between RR and TR (RT) parameters within Healthy patients. Denotes statistical significance (po 0.05) between TR and RT parameters within Healthy patients.
Table 2 ECG Analysis of Diseased Patients representing average RR (AVGRR), TR (AVGTR) and RT (AVGRT) intervals, standard deviations of the RR (SDRR), TR (SDTR) and RT (SDRT) intervals, and corresponding HRVs. Patient ID
TWA18 TWA38 TWA31 TWA11 TWA20 TWA42 AVG n
# $
RR
TR
RT
AVGRR (ms)
SDRR (ms)
HRVRR (%)
AVGTR (ms)
SDTR (ms)
HRVTR (%)
AVGRT (ms)
SDRT (ms)
HRVRT (%)
699.7 711.7 710.0 786.0 814.8 824.0 757.2 7 23.3
19.6 18.5 13.8 20.8 25.7 34.5 22.2 7 2.9
2.8 2.6 1.9 2.6 3.2 4.2 2.9 70.8
441.2 464.3 454.2 538.4 571.4 557.6 504.57 52.6
19.8 18.8 14.2 20.5 25.5 34.5 22.2 76.4
4.5 4.1 3.1 3.8 4.5 6.2 4.4 7 0.9
258.4 247.5 256.1 247.9 243.4 266.4 253.3 7 7.8
9.3 3.2 6.2 4.2 4.0 6.1 5.5 7 2.0
3.6 1.3 2.4 1.7 1.6 2.3 2.2 70.7
n
n
n
n,#,$
n,$
n,#,$
#,$
n,#,$
n,$
Denotes statistical significance (p o 0.05) between Healthy and Diseased patients. Denotes statistical significance (p o 0.05) between RR and TR (RT) parameters within Diseased patients. Denotes statistical significance (po 0.05) between TR and RT parameters within Diseased patients.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
3. Results 3.1. The role of feedback in alternans formation: analytical insight As mentioned in Section 1, Eqs. (1) and (2) have been widely used to describe dynamics of cardiac myocytes subject to periodic pacing. Let us look at these equations a little closer, since we want to determine the behavior of the system near the 1:1 steady state value of Eq. (1). Let APDn and DIn be the steady state values of Eq. (1), such that APDn ¼ f ðBCL APDn Þ ¼ f ðDIn Þ:
ð9Þ
Taking the difference between Eqs. (1) and (9), we can write APDn þ 1 APDn ¼ f ðDIn Þ f ðDIn Þ:
ð10Þ
We can approximate the right-hand side of Eq. (10) as the following: 0
f ðDIn ÞðDIn DIn Þ ¼ Sr ðDIn DIn Þ:
ð11Þ
Therefore, combining Eqs. (10) and (11) we obtain n
n
ðAPDn þ 1 APD Þ ¼ Sr ðDIn DI Þ:
ð12Þ
Eq. (12) describes the restitution properties of the heart, suggesting that small changes in APD are proportional to small changes in preceding DI. It is important to note that Eq. (12) is linear, since Sr is a constant for any given small changes, and therefore, cannot describe complex cardiac rhythms, such as alternans. The physiological basis for restitution properties of the heart, arising from, among other factors, incomplete recovery from
inactivation of delayed rectifier ionic channels, has been discussed previously in greater details (Rudy and Silva, 2006). In the case of periodic pacing with constant BCL (pacing Protocol 1), the APD and DI are related through the pacing relation (Eq. (2)), which can be rewritten as DIn ¼ BCL APDn DIn ¼ BCL APDn ;
ð13Þ
and therefore, the following relation can be derived: APDn APDn ¼ ðDIn DIn Þ
ð14Þ
Eq. (14) describes the feedback associated with periodic pacing protocol, and suggests that small changes in DI are proportional with opposite sign to small changes in APD. Combination of the restitution properties of the myocytes (Eq. (12)) with the relationship derived based on periodic pacing (Eq. (14)) leads to a recurrence relation APDn þ 1 APDn ¼ Sr ðAPDn APDn Þ
ð15Þ
that has been used to describe different dynamical behavior of cardiac myocytes: 1:1 responses and possible presence of alternans. Indeed, for |Sr| 4 1, any small changes in APD that appear after nth stimulus is applied, will enhance the difference APDn þ 1 APDn during further pacing, i.e. at the next (n þ1)st stimulus. If |Sr| o1, relation (Eq. (15)) describes 1:1 responses, since small changes in APD will eventually die out. In the case when the connection between APDn and DIn are eliminated by fixing the DIn value (pacing Protocol 2), Eq. (13) is no longer valid, so feedback is absent. Therefore, we cannot derive Eq. (15) and as a result are unable to mathematically explain the mechanism of alternans formation in isolated cells. Indeed, mapping model (Eq. (1)) will return a single APDn þ 1 value for a fixed DIn in the absence of feedback. The difference between pacing protocols with and without feedback is illustrated in Fig. 2, where BCLs (stars), APDs (open circles), and DIs (open triangles) are shown for Protocol 1 (Fig. 2A) and Protocol 2 (Fig. 2B) as a function of beat number for the case of HRV ¼0%. For Protocol 1, the BCL was changed from 270 ms to 170 ms, and corresponding values of DI were calculated using Eq. (13). Note that the slope of the restitution curve Sr 41 at BCL ¼170 ms, leading to the formation of alternans due to the presence of feedback (Eq. (14)), and therefore, recurrence relation (Eq. (15)). In contrast, no alternans is present in Fig. 2B, when DI is changed from 105 ms to 48 ms. Note, that the calculated values of BCLs are very similar to those from Fig. 2A. 3.2. ECG data analysis
Fig. 2. BCL (stars), APD (open circles), and DI (open triangles) as a function of beat number for (A) constant-BCL pacing protocol and (B) constant-DI pacing protocol for the case of HRV ¼0%. For (A), alternans formation is shown when BCL is changed from 270 ms to 170 ms is shown. For (B), no alternans is present when DI is changed from 105 ms to 85 ms (which correspond to roughly the same BCLs).
Fig. 3A shows a representative example of RR intervals as a function of beat number taken from an ECG of a Healthy patient's data, illustrating the presence of HRV. Fig. 3B shows the histogram of RR intervals from the data in Fig. 3A. Note the approximately normal distribution of RR intervals in Fig. 3B. Both Healthy and Diseased patients' RR distributions were qualitatively similar to the one shown in Fig. 1, and therefore we chose a normal distribution for the random numbers in Eqs. (6) and (7). Fig. 1 illustrates the relationship of the RR, RT, and TR intervals to the BCL, APD, and DI respectively. Comprehensive analyses of ECG traces from the two groups of patients, Healthy and Diseased, are presented in Tables 1 and 2, respectively. Note that AVGRR, (944.38 742.08 ms vs. 757.7 7 23.25 ms, po 0.05), SDRR, (45.98 76.42 ms vs. 22.15 72.92 ms, po 0.01), and HRV (4.767 1.45% vs. 2.897 0.75%, po 0.05) are significantly larger for Healthy patients in comparison with Diseased patients.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Fig. 3. (A) An ECG trace of a 32-year-old male with normal sinus rhythm taken from Goldberger et al. (2000). (B) The histogram of the RR intervals calculated from the ECG trace shown in A.
5
Fig. 4. (A) Representative plot of RT vs. TR intervals for two Healthy (circles) and two Diseased (triangles) patients. (B) Values of s from all Healthy and Diseased patients. n represent statistical significance (p o0.01).
In order to determine whether feedback is present in the Healthy and Diseased patients, we separately calculated the nth TR (TRn) and RT (RTn) intervals from the entire ECG data set, corresponding to DIn and APDn, respectively, as indicated in Fig. 1. Therefore, the relationship between TR and RT intervals correlates with the relationship between DI and APD, i.e. feedback. Fig. 4A shows representative examples of the nth RT intervals as a function of the nth TR intervals for two Healthy (open and filled circles) and two Diseased (open and filled triangles) patients. Note that for Healthy patients, the standard deviations of the TR range are higher than in the RT range, while for Diseased patients the standard deviations for both ranges are similar. To quantify this data, we calculated the sensitivity, s, according to Eq. (5), which is an indirect representation of feedback (see Fig. 4B). Note that s was significantly higher for Healthy patients (s¼5.1 70.6) than for Diseased patients (s ¼2.2 7 0.3, po 0.01), suggesting the presence of different associations between RT and TR intervals for Healthy than for Diseased patients. 3.3. The effect of HRV on alternans formation in the presence of feedback Fig. 5 shows APD as a function of BCL for the ionic model of an isolated cardiac myocyte that was paced using the periodic pacing protocol (Protocol 1) with no HRV (Fig. 5A) and HRV ¼2.5% (Fig. 5B). Note the formation of alternans at a certain range of BCLs both with and without HRV. Also note that the onset of alternans occurred at a higher BCL (BCLstart ¼ 200 ms) and ended at a lower BCL (BCLend ¼140 ms) for HRV ¼2.5% than for HRV ¼ 0% (BCLstart ¼190 ms and BCLend ¼ 150 ms, respectively). Fig. 6A illustrates the influence of HRV on the onset, 〈BCL〉start, and the end, 〈BCL〉end, of alternans. As HRV is increased, alternans
Fig. 5. APD as a function of BCL for the ionic model paced with constant-BCL pacing protocol with HRV ¼0% (A) and 2.5% (B). BCLstart and BCLend denote the start and end of APD alternans.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
6
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 6. Influence of HRV on the following parameters: (A) 〈BCL〉start and 〈BCL〉end; (B) ΔAPDmax ; (C) ΔAPDBCL ¼ 170 ms and (D) 〈Sr〉. as function of HRV. * represents statistical significance (p o 0.01).
occurred earlier in the pacing protocol and ended later in the pacing protocol. Note that HRV larger than 1.5% significantly increases 〈BCL〉start when compared to HRV ¼0%. Similarly, HRV larger than 2.5% significantly decreases 〈BCL〉end. These results suggest that HRV promotes the formation of alternans for the case of Protocol 1 with feedback. In order to investigate the effect of HRV on the amplitude of alternans, we calculated two parameters: the maximal amplitude of alternans, ΔAPDmax , over all BCLs in the run and the amplitude of alternans at BCL ¼ 170 ms, ΔAPDBCL ¼ 170 ms, where alternans was present at HRV ¼0%. The influence of HRV on ΔAPDmax and ΔAPDBCL ¼ 170 ms is shown in Fig. 6B and C, respectively. Fig. 6B and C shows that the presence of HRV did not significantly affect either ΔAPDmax or ΔAPDBCL ¼ 170 ms, when compared to values at HRV ¼0%. To further understand the effect of HRV on the onset of alternans in the presence of feedback, we calculated the maximum slope of the restitution curve, 〈Smax 〉, at the onset of alternans, r 〈BCL〉start. Fig. 6D shows that HRV 41.5% leads to a significant decrease of 〈Smax 〉 when compared to HRV ¼0%. r 3.4. Elimination of feedback suppresses the formation of alternans Fig. 7 shows APD as a function of DI for the ionic model of an isolated cardiac myocyte that was paced using Protocol 2 without feedback with HRV ¼0% (Fig. 7A) and HRV ¼ 2.5% (Fig. 7B). Note that in contrast to Fig. 5, no alternans is seen in Fig. 7 (confirmed by inspection of actual voltage traces) both for HRV ¼0% or 2.5%. To confirm these findings, we increased HRV up to 12%, and alternans was still absent (data not shown). This demonstrated that elimination of feedback prevents formation of alternans in a myocyte. Fig. 8 illustrates the presence of feedback in a cardiac myocyte that was paced either with Protocol 1 (Fig. 8A) or Protocol 2 (Fig. 8B).
Specifically, in Fig. 8A, APDn is plotted as a function of DIn for Protocol 1 with HRV ¼2.5%. To demonstrate the presence of feedback for different BCLs, two values of BCLs were chosen: BCL ¼270 ms (open circles), and BCL ¼ 170 ms (filled circles). In the case of HRV ¼2.5%, Protocol 1 led to steady state 1:1 responses at BCL ¼270 ms, and alternans at BCL ¼ 170 ms. Fig. 8A suggests that the distribution of data for BCL ¼ 270 ms was qualitatively similar to the data from Healthy patients (see Fig. 4A). However, it is important to note that the ranges of the values were very different, thus preventing us from making any distinct conclusions. On the other hand, for BCL ¼170 ms, the DIn and APDn pairs fall along a line with slope of 1, as predicted by Eq. (2). Similar simulations were performed with Protocol 2, and the results are shown in Fig. 8B. Here two values of DI were chosen, DI¼ 48 ms (open circles) and DI ¼105 ms (filled circles), as these DI values correspond to the steady state BCLs 170 ms and 270 ms, respectively. Fig. 8B demonstrates that for DI ¼105 ms the distribution of the DIn and APDn was qualitatively similar to the one from Healthy patients (see Fig. 4A). On the other hand, for DI ¼48 ms the feedback was qualitatively similar to the one from Diseased patients (see Fig. 4A). However, this observation does not allows us to make any distinct conclusions since the ranges of values were very different.
4. Conclusion and discussion In this study, we investigated the effects of HRV and feedback on the development of alternans in a physiological ionic model of a canine cardiac action potential. We demonstrated that (Eq. (1)) HRV promoted the formation of alternans in cardiac myocytes in the presence of feedback. Specifically HRV 41.5% significantly shifted the onset of alternans towards larger BCL, but does not affect the amplitude of alternans, ΔAPDmax and ΔAPDBCL ¼ 170 ms. This shift of the onset of alternans was accompanied by a significant
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Fig. 7. APD as a function of DI for the ionic model paced with constant-DI pacing protocol with HRV ¼0% (A) and 2.5% (B). Both figures show the absence of alternans.
decrease in 〈Smax 〉 value for the same HRV values. We also showed r that (Eq. (2)) the absence of feedback during pacing prevented the formation of alternans in cardiac myocytes regardless of the presence or absence of HRV. Since alternans is closely correlated with arrhythmias, prevention of alternans formation is an important area of research in cardiac electrophysiology. First, we demonstrated that in the case of periodic pacing protocol with feedback, HRV facilitated alternans formation in isolated cell. Indeed, Fig. 5A shows that the introduction of HRV causes 〈BCL〉start to occur at larger values and 〈BCL〉end to occur at lower values, suggesting a potential pro-arrhythmic effect of HRV. Interestingly, the increased range of BCL for which alternans was observed was accompanied by a decreased 〈Smax 〉 value at the r onset of alternans, suggesting that alternans can be formed in the absence of steep restitution slope, thus again confirming a potential pro-arrhythmic effect of HRV. Second, we showed that removing the feedback from the pacing protocol (Protocol 2) resulted in the elimination of alternans in isolated cardiac cells. Moreover, alternans did not appear when HRV was introduced in that case, indicating a potential anti-arrhythmic role of no-feedback pacing. However, more comprehensive numerical simulations of the cardiac tissue (instead of isolated cells) or experiments are necessary to further elucidate the role of HRV on the stability of cardiac rhythms both in healthy and diseased hearts. Previously, several attempts have been made to develop a pacing protocol that permits explicit control of DI, independent of APD (Patwardhan and Moghe, 2001; Wu and Patwardhan, 2004). It was demonstrated that the ability to pace cardiac tissue so that DIs can be selected independently provides a unique way to explore mechanisms of APD alternans by eliminating DIdependent restitution effects. Specifically, Wu and Patwardhan (Patwardhan and Moghe, 2001) demonstrated that the mechanisms of alternans include restitution-dependent and restitution-
7
Fig. 8. Feedback represented via APDn as a function of the DIn for the following pacing protocols: (A) constant-BCL pacing protocol with HRV ¼2.5% at BCL ¼270 ms (open circles) and BCL¼ 170 ms (filled circles); and (B) constant-DI pacing protocol with HRV¼ 5%, shown for the DI¼ 105 ms (open circles) and DI¼ 48 ms (filled circles). Solid line indicates a 1 slope.
independent components, and therefore, restitution may not be a necessary mechanism for alternans to exist. Later on, this group applied a constant DI pacing protocol experimentally in canine tissue (Wu et al., 2006) and in pig tissue slab preparation (Jing el al., 2012) to demonstrate that eliminating the restitutiondependent mechanism of alternans by explicitly controlling DI does not preclude from APD alternans formation in the tissue. Although their protocols are somewhat similar to our Protocol 2 in the absence of HRV, their overall results contradict our observations. This discrepancy can be explained by the fact that the authors applied 15 or 30 beats at each DI before changing it, while we applied 120 beats in order to ensure steady state. Furthermore, this discrepancy can also be attributed to the fact that the amount of short-term memory that is present in real canine or pig tissue is much more substantial in comparison to the ionic model we used. This fact was indirectly demonstrated in the numerical results of Jordan and Christini (Jordan et al., 2004), who showed the suppression of alternans in several ionic models that were paced using a constant DI pacing protocol. It is important to note, however, that none of the discussed works attempted to incorporate HRV in their pacing protocols. It is also worth mentioning that several studies have been using the concept of pro- or anti-arrhythmic pacing to facilitate or control development of alternans in cardiac tissue, respectively. For instance, Christini et al. 2001, 2006 used a protocol similar to our Protocol 2 to reduce or eliminate the presence of APD alternans by modifying the cycle length in real time based on the size of alternans. Gelzer et al. (2008) performed an experimental realization of previously developed pacing protocol (Fox et al., 2003) that consists of the delivery of three premature stimuli at various time after basic stimulus to facilitate alternans
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
formation and to induce ventricular fibrillation. In contrast, in our study, we aimed to understand how different pacing protocols affect the development of alternans, without trying to control or facilitate the development of alternans by altering pacing. Previous work by Dai and Keener (2012) investigated how it is possible to approximate the dynamics of a cardiac cell from knowing the mean of the BCLs and APDs. Similar to our study, the authors introduced some randomness through the BCL but unlike us, they used a simpler mapping model with feedback. Several other attempts to add randomness in the pacing have been made (Dai and Keener, 2012; Christini et al., 2001, 2006), but nobody has systemically investigated effects of feedback on alternans formation. We also examined the effect of adding randomness to the DI and removing the feedback relationship. Dia and Keener did use different distributions for the randomness but none of these distributions were based on data analysis of ECG data as our modeling was. Our research also examined 〈BCL〉start, 〈BCL〉end, and alternans magnitude. In this manuscript, we modeled HRV with a random number function, and we assumed that HRV does not change as the heart rate changes, which is an over-simplified approach. Indeed, it is known that HRV can be affected by, but not limited to, respiration, thermoregulation, hormonal regulation, blood pressure, etc. (Malik and Camm, 1995). In addition, the relative contribution and importance of all these physiological aspects of HRV might be altered under different disease conditions. Therefore, a more comprehensive study needs to be performed to provide a more physiologically relevant model of HRV. One of the limitations of our study is that we did not take into account the intracellular calcium cycling, which can play an important role in the development of APD alternans (Weiss et al., 2006; Chudin et al., 1998; Goldhaber et al., 2005). Indeed, alternans in intracellular calcium transient amplitude has been linked to mechanical alternans; thus the simultaneous occurrence of calcium and APD alternans, termed electromechanical alternans, is believed to be a substrate for various cardiac arrhythmias. Another limitation of our study is that numerical simulations performed were for single cardiac myocytes only. It is known that electrophysiological properties, including alternans formation, can be different at the tissue level (Fox et al., 2002; Watanabe et al., 2001; Sato et al., 2006).
Q2 Uncited references
Watanabe and Koller (2002); Jordan and Christini (2004); Otani and Gilmour (1997); Tolkacheva et al. (2003); Kalb et al. (2005); Chialvo et al. (1990); Kalb et al. (2004); Tolkacheva et al. (2004).
Acknowledgments The authors wish to thank Dr. Mironov for helpful discussions, and Dr. Xie, Mr. Ippolito, Mr. Visweswaran, and Mr. Switala for careful reading of the manuscript. This work was supported by National Science Foundation grants (E.G.T) PHY 0957468, CMMI-1233951, and NSF CAREER PHY-125541. References Banville, I., Gray, R.A., 2002. Effect of action potential duration and conduction velocity restitution and their spatial dispersion on alternans and the stability of arrhythmias. J. Cardiovasc. Electrophysiol. 13, 1141. Chialvo, D.R., Michaels, D.C., Jalife, J., 1990. Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fiber. Circ. Res. 66, 525. Christini, D.J., Stein, K.M., et al., 2001. Nonlinear-dynamical arrhythmia control in humans. Proc. Natl. Acad. Sci. USA 98, 5827–5832.
Christini, D.J., Riccio, M.L., et al., 2006. Control of electrical alternans in canine cardiac Purkinje fibers. Phys. Rev. Lett 96, 104101. Chudin, E., Garfinkel, A., et al., 1998. Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study). Prog. Biophys. Mol. Biol. 69 (2-3), 225–236. Dai, S., Keener, J.P., 2012. Using noise to determine cardiac restitution with memory. Phys. Rev. E 85 (6), 061902. Fox, J.J., McHarg, J.L., Gilmour, R.F., 2002. Ionic mechanism of electrical alternans. Am. J. Physiol. 282 (2), H516–H530. Fox, J.J., Gilmour , R.F., Bodenschatz, E., 2002. Conduction block in one-dimensional heart fiber. Phys. Rev. Lett. 89, 198101. Fox, J.J., Riccio, M.L., Hua, F., Bodenschatz, E., Gilmour, R.F., 2002. Spatiotemporal transition to conduction block in canine ventricle. Circ. Res. 90, 289–296. Fox, Jeffrey J., et al., 2003. Dynamic mechanism for conduction block in heart tissue. New J. Phys. 5 (1), 101.14. Franz, M.R., 2003. The electrical restitution curve revisited: steep or flat slope – which is better? J. Cardiovasc. Electrophysiol. 14, 140–147. Gelzer, Anna R.M., et al., 2008. Dynamic mechanism for initiation of ventricular fibrillation in vivo. Circulation 118 (11), 1123–1129. Gilmour , R.F., 2002. Electrical restitution and ventricular fibrillation: negotiating a slippery slope. J. Cardiovasc. Electropyhsiol. 13, 1150. Gilmour , R.F., Chialvo, D.R., 1999. Editorial: electrical restitution, critical mass and the riddle of fibrillation. J. Cardiovasc. Electrophysiol. 10, 1087. Goldberger, A.L., Amaral, L.A.N., Glass, L., 2000. PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101, e215–e220. Goldhaber, J.I., Xie, L.-H., Duong, T., Motter, C.h., Khuu, K., Weiss, J.N., 2005. Action potential duration restitution and alternans in rabbit ventricular myocytes. The key role of intracellular calcium cycling. Circ. Res. 96, 459–466. Guevara, M., Ward, G., Shrier, A., Glass, L., 1984. Computers in Cardiology. IEEE Computer Society, Silver Spring, MD p. 167 Hall, G.M., Gauthier, D.J., 1999. Experimental control of cardiac muscle alternans. Phys. Rev. Lett. 82, 2995. Jing, Linyuan, Abhijit Patwardhan, 2012. Hysteresis in DI independent mechanisms in threshold for transition between 1∶1 and 2∶2 rhythms in pigs. Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE, IEEE. Jordan, P.N., Christini, D.J., 2004. Determining the effects of memory and APD alternans on cardiac restitution using a constant-memory restitution protocol. Physiol. Meas. 25, 1013–1024. Jordan, Peter N., David, J. Christini, 2004. Determining the effects of memory and action potential duration alternans on cardiac restitution using a constantmemory restitution protocol. Physiol. Meas. 25 (4), 1013. Kalb, S.S., Dobrovolny, H., Tolkacheva, E.G., Idriss, S.F., Krassowska, W., Gauthier, D.J., 2004. The restitution portrait: a new method for investigating rate-dependent restitution. J. Cardiovasc. Electrophysiol. 15, 698. Kalb, S.S., Tolkacheva, E.G., Schaeffer, D.G., Gauthier, D.J., Krassowska, W., 2005. Restitution in mapping models with arbitrary amount of memory. Chaos 15, 23701. Karma, A., 1994. Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 4, 461. Kleiger, R.E., Miller, J.H.R.V., Bigger, J.T., Moss, A.J., 1987. Decreased heart rate variability and its association with increased mortality after acute myocardial infarction. Am. J. Cardiol. 59, 256. Koller, M.L., Riccio, M.L., Gilmour , R.F., 1998. Dynamic restitution of action potential duration during electrical alternans and ventricular fibrillation. Am. J. Physiol. 275, H1635. La Rovere, M.T., Pinna, G.D., et al., 2003. Short-term heart rate variability strongly predicts sudden cardiac death in chronic heart failure patients. Circulation 107, 565. Malik, M., Camm, A.J., 1995. Heart Rate Variability. Futura Pub. Co., Armonk, N.Y. Narayan, S.M., Bayer, J., et al., 2008. Action potential dynamics explain arrhythmic vulnerability in human heart failure: a clinical and modeling study implicating abnormal calcium handling. J. Am. Coll. Cardiol. 52, 1782–1792. Nolasco, J.B., Dahlen, R.W., 1968. A graphic method for the study of alternation in cardiac action potentials. J. Appl. Physiol. 25, 191. Otani, N.F., Gilmour , R.F., 1997. Memory models for the electrical properties of local cardiac systems. J. Theor. Biol. 187, 409. Patwardhan, A., Moghe, S., 2001. Novel feedback based stimulation protocol shows hysteresis in cardiac action potential duration restitution. Biomed. Sci. Instrum. 37, 505–510. Riccio, M.L., Koller, M.L., Gilmour , R.F., 1999. Electrical restitution and spatiotemporal organization during ventricular fibrillation. Circ. Res. 84, 955. Rudy, Y., Silva, J.R., 2006. Computational biology in the study of cardiac ion channels and cell electrophysiology. Q. Rev. Biophys. 39, 57–116. Sato, D., Shiferaw, Y., Garfinkel, A., Weiss, J.N., Qu, Z., Karma, A., 2006. Spatially discordant alternans in cardiac tissue role of calcium cycling. Circ. Res. 9, 520–527. Tolkacheva, E.G., Schaeffer, D.G., Gauthier, D.J., Krassowska, W., 2003. Condition for alternans and stability of the 1:1 response pattern in a “memory” model of paced cardiac dynamics. Phys. Rev. E. 67, 031904. Tolkacheva, E.G., Romeo, M.M., Guerraty, M., Gauthier, D.J., 2004. Condition for alternans and its control in a two-dimensional mapping model of paced cardiac dynamics. Phys. Rev. E. 69, 031904. Watanabe, M.A., Koller, M.L., 2002. Mathematical analysis of dynamics of cardiac memory and accommadation: theory and experiment. Am. J. Physiol. 282, H1534.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i
Q5
Q6
Q7
Q8
S.D. McIntyre et al. / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7
Watanabe, M.A., Otani, N.F., Gilmour , R.F., 1995. Biphasic restitution of action potential duration and complex dynamics in ventricular myocardium. Circ. Res. 76, 915. Watanabe, M.A., Fenton, F.H., Evans, S.J., Hastings, H.M., Karma, A., 2001. Mechanisms for discordant alternans. J. Cardiovasc. Electrophysiol. 12, 196–206. Weiss, J.N., Karma, A., Shiferaw, Y., Chen, P.-S., Garfinkel, A., Qu, Z., 2006. From pulsus to pulseless: the saga of cardiac alternans. Circ. Res. 98, 1244–1253.
9
Wu, R., Patwardhan, A., 2004. Restitution of action potential duration during sequential changes in diastolic intervals shows multimodal behavior. Circ. Res. 94, 634–641. Wu, Runze, Patwardhan, Abhijit, 2006. Mechanism of repolarization alternans has restitution of action potential duration dependent and independent components. J. Cardiovasc. Electrophysiol. 17.1, 87–93. Zipes, D.H.R.V., Wellens, H.J., 1998. Sudden cardiac death. Circulation 98, 2334–2351.
Please cite this article as: McIntyre, S.D., et al., Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.02.015i