Generalities of MSE

MSE is a technique estimating the complexity of a time series x = {x(i), i = 1,…,N}, where i is the sample counter and N is the series length, via entropy rate at different time scales [32]. It consists of three steps performing: i) the elimination of the fast temporal scales via a low pass filtering procedure, thus obtaining the filtered series, x_f = {x_f(i), i = 1,…,N}, that focuses the frequency range of interest; ii) the downsampling of x_f with a scale factor, τ, chosen according to the cutoff of the low pass filter exploited for canceling fast oscillations, thus obtaining the filtered downsampled series, x_f^τ = {x_f^τ(j), j = 1,…,N/τ}; iii) the computation of an entropy rate over x_f^τ as a function of τ. In this work we exploited a refined version of the MSE, the refined MSE (RMSE), devised to fix two biases present in the computation of the MSE [33].

MSE and RMSE

The elimination of the fast temporal scales in MSE is carried out using a low pass finite impulse response filter performing the mean of τ samples (i.e. all coefficients of the filter are set to 1/τ) [32]. Since the frequency response of this filter is very poor being characterized by a slow roll-off of the main lobe, a large transition band and important side lobes, aliasing is not prevented when the filtered series is downsampled at a rate of one sample every τ. RMSE substitutes the low pass finite impulse response filter with a low pass Butterworth filter of order 6 having a cutoff frequency equal to 0.5/τ cycles/sample [33]. This filter has a flat response in the pass band, no side lobes in the stop band and a faster roll-off, thus being more efficient in limiting aliasing during downsampling. The low pass filtered series is downsampled at a rate of one sample every τ, thus reducing the total number of values from N to N/τ. The complexity of the low pass filtered series is estimated via the sample entropy [34]. Let us label the negative logarithm of the probability of finding two patterns of length L, x_f,L^τ(j) = [x_f^τ(j), x_f^τ(j-1),…, x_f^τ(j-L+1)] and x_f,L^τ(k) = [x_f^τ(k), x_f^τ(k-1),…, x_f^τ(k-L+1)] with L≤j,k≤N/τ at distance closer than a parameter r as Φ(L,r). The sample entropy is defined as the difference between Φ(L,r) and Φ(L-1,r), thus quantifying the probability that, if x_f,L-1^τ(j) and x_f,L-1^τ(k) are nearby in the embedding space of dimension L-1, they will remain nearby in the embedding space of dimension L [34]. r is usually referred to as tolerance for the calculation of the sample entropy and sets the level of coarse graining of the embedding space (i.e. patterns at distance closer than r cannot be distinguished as separated entities under the adopted level of discretization of the embedding space. In MSE r was set as a percentage of the standard deviation of x (usually the 15%) and it was kept constant as a function of τ [32]. Since the low pass filtering procedure reduces the standard deviation of the series, the region of the embedding space occupied by the patterns decreases more and more with τ and, consequently, the number of pairs of patterns becoming closer and closer increases with τ, thus leading to a decrease of the sample entropy. This complexity reduction is artificial because it is solely the effect of the decline of variance with τ. In RMSE r is set as a percentage of the standard deviation of x_f^τ, thus continuously updating r with τ [33].

Study population, experimental protocol and data analysis

Time domain analysis

Table 1 shows mean and variance of the HP and QT series in the NMC-MC protocol. The HP mean, μ_HP, was higher in ASYMP individuals than in NMC subjects. The HP variance, σ²_HP, was comparable in all the populations. According to the pathology, the QT mean, μ_QT, was longer in ASYMP and SYMP patients than in NMC individuals. The QT variance, σ²_QT, separated ASYMP and SYMP patients with σ²_QT larger in ASYMP individuals than in SYMP ones.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Time domain indexes derived from HP and QT series in the NMC-MC protocol.

https://doi.org/10.1371/journal.pone.0093808.t001

Table 2 shows the same parameters reported in Tab.1 computed in the DAY-NIGHT protocol. In both ASYMP and SYMP patients μ_HP and μ_QT were longer during NIGHT than during DAY. In SYMP group σ²_HP increased during NIGHT, while in the ASYMP group σ²_QT decreased during NIGHT. Remarkably, during DAY σ²_QT was able to separate ASYMP subjects from SYMP ones. Indeed, during DAY σ²_QT of the SYMP group was smaller than that of the ASYMP one.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Time domain indexes derived from HP and QT series in the DAY-NIGHT protocol.

https://doi.org/10.1371/journal.pone.0093808.t002

Table 3 shows the same parameters reported in Tabs.1,2 computed in the BBoff-BBon protocol. In both ASYMP and SYMP subjects beta-blocker therapy lengthened μ_HP but only μ_QT in SYMP patients was significantly increased. In the SYMP group σ²_HP was larger BBon than BBoff, while in ASYMP group beta-blocker therapy significantly reduced σ²_QT. Given the same experimental condition (i.e. BBoff or BBon) μ_HP, σ²_HP and σ²_QT were able to differentiate ASYMP individuals from SYMP ones. Indeed, BBon μ_HP was shorter and σ²_HP was larger in the SYMP group than in the ASYMP one and BBoff σ²_QT was smaller in the SYMP group than in the ASYMP one.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Time domain indexes derived from HP and QT series in the BBoff-BBon protocol.

https://doi.org/10.1371/journal.pone.0093808.t003

Comparison between RMSE indexes derived from HP and QT series

Figure 1 shows RMSE mean (plus standard deviation) at short time scale (i.e. τ = 1, Figs.1a,d,g), RMSE_τ = 1, at medium time scales (i.e. τ = 2–4 Figs.1b,e,h), RMSE_{τ = 2–4}, and at long time scales (i.e. τ = 5–12, Figs.1c,f,i), RMSE_{τ = 5–12}, as a function of the series (i.e. HP and QT). RMSE mean was computed by pooling together NMC, ASYMP and SYMP individuals BBoff during DAY in Figs.1a,b,c, ASYMP and SYMP subjects BBoff during NIGHT in Figs.1d,e,f, and ASYMP and SYMP individuals BBon during DAY in Figs.1g,h,i. BBoff during DAY RMSE_τ = 1 and RMSE_{τ = 2–4} were significantly larger in the QT series than in the HP one. Conversely, the reverse situation was observed in the case of RMSE_{τ = 5–12}. Similar results were found BBoff during NIGHT and BBon during DAY. However, the difference between RMSE_τ = 1 indexes assessed over HP and QT series BBoff during NIGHT (Fig.1d) and BBon during DAY (Fig.1g) was less evident than BBoff during DAY (Fig.1a). RMSE_{τ = 2–4} exhibited the same trend (Figs.1b,e,h). The difference between RMSE indexes assessed over HP and QT series remained stable in the case of RMSE_{τ = 5–12} (Figs.1c,f,i).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. Comparison between RMSE indexes derived from HP and QT series.

RMSE mean (plus standard deviation) assessed at short time scale (i.e. τ = 1), RMSE_τ = 1 (a,d,g), at medium time scales (i.e. τ = 2–4), RMSE_{τ = 2–4} (b,e,h), and at long time scales (i.e. τ = 5–12), RMSE_{τ = 5–12}, (c,f,i) is shown as a function of the time series (i.e. HP and QT). RMSE_τ = 1, RMSE_{τ = 2–4}, and RMSE_{τ = 5–12} were obtained by pooling RMSE values computed BBoff during DAY (a,b,c), BBoff during NIGHT in (d,e,f), and BBon during DAY (g,h,i), The symbols § and ° indicate a significant difference with p<0.001 and p<0.05 respectively.

https://doi.org/10.1371/journal.pone.0093808.g001

RMSE at short time scale (τ = 1) in NMC, ASYMP and SYMP groups

Figure 2 shows RMSE mean (plus standard deviation) at short time scale (i.e. τ = 1) as derived from HP and QT series (i.e. RMSE_{HP,τ = 1} and RMSE_{QT,τ = 1}) in Figs.2a,c,e and Figs.2b,d,f respectively. RMSE_τ = 1 was unable to separate groups (i.e. NMC, ASYMP and SYMP). This conclusion held for any series (i.e. HP or QT) and for any protocol (i.e. NMC-MC, DAY-NIGHT or BBoff-BBon). In both ASYMP and SYMP groups RMSE_{HP,τ = 1} significantly increased during NIGHT compared to DAY (Fig.2c) and BBon compared to BBoff (Fig.2e). While RMSE_{QT,τ = 1} significantly decreased during NIGHT compared to DAY in both ASYMP and SYMP patients (Fig.2d), beta-blocker therapy did not modify RMSE_{QT,τ = 1} (Fig.2f).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. RMSE at short time scale over HP and QT series.

RMSE mean (plus standard deviation) assessed at short time scale (i.e. τ = 1) over HP series, RMSE_{HP,τ = 1} (a,c,e), and QT series, RMSE_{QT,τ = 1} (b,d,f), is shown as a function of the experimental protocol. RMSE_{HP,τ = 1} and RMSE_{QT,τ = 1} are depicted as a function of the group of subjects in the NMC-MC protocol in (a) and (b) respectively, as a function of the period of analysis in the DAY-NIGHT protocol in (c) and (d) respectively, and as a function of the therapy in the BBoff-BBon protocol in (e) and (f) respectively. The gray, dark and white bars are relevant to NMC, ASYMP and SYMP individuals, respectively. The symbols * indicates a significant difference between experimental conditions (i.e. DAY versus NIGHT, BBoff versus BBon) within the same group with p<0.05.

https://doi.org/10.1371/journal.pone.0093808.g002

RMSE at medium time scales (τ = 2–4) in NMC, ASYMP and SYMP groups)

Figure 3 shows RMSE mean (plus standard deviation) at medium time scales (i.e. τ = 2–4) as derived from HP and QT series (i.e. RMSE_{HP,τ = 2–4} and RMSE_{QT,τ = 2–4}) in Figs.3a,c,e and Figs.3b,d,f respectively. In the NMC-MC protocol RMSE_{HP,τ = 2–4} (Fig.3a) and RMSE_{QT,τ = 2–4} (Fig.3b) were able to separate the ASYMP group from the SYMP one with both indexes smaller in the ASYMP group. RMSE_{QT,τ = 2–4} differentiated the ASYMP group from the NMC one as well, being RMSE_{QT,τ = 2–4} in the ASYMP group smaller than in NMC one (Fig.3b). In the DAY-NIGHT protocol (Figs.3c,d) and in the BBoff-BBon protocol (Figs.3e,f) RMSE_{HP,τ = 2–4} and RMSE_{QT,τ = 2–4} separated the ASYMP group from the SYMP one only during DAY with both RMSE_{HP,τ = 2–4} and RMSE_{QT,τ = 2–4} higher in SYMP patients than in ASYMP group. DAY-NIGHT variations and the effect of the therapy were observable only in RMSE_{HP,τ = 2–4} in ASYMP group with RMSE_{HP,τ = 2–4} significantly increased during NIGHT (Fig.3c) and BBon (Fig.3e) and only in RMSE_{QT,τ = 2–4} in SYMP group with RMSE_{QT,τ = 2–4} significantly decreased during NIGHT (Fig.3d) and BBon (Fig.3f).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. RMSE at medium time scales over HP and QT series.

RMSE mean (plus standard deviation) assessed at medium time scales (i.e. τ = 2–4) over HP series, RMSE_{HP,τ = 2–4} (a,c,e), and QT series, RMSE_{QT,τ = 2–4} (b,d,f), is shown as a function of the experimental protocol. RMSE_{HP,τ = 2–4} and RMSE_{QT,τ = 2–4} are depicted as a function of the group of subjects in the NMC-MC protocol in (a) and (b) respectively, as a function of the period of analysis in the DAY-NIGHT protocol in (c) and (d) respectively, and as a function of the therapy in the BBoff-BBon protocol in (e) and (f) respectively. The gray, dark and white bars are relevant to NMC, ASYMP and SYMP individuals, respectively. The symbol * indicates a significant difference between experimental conditions (i.e. DAY versus NIGHT, BBoff versus BBon) within the same group with p<0.05. The symbol ^# indicates a significant difference between groups within the same experimental conditions (i.e. DAY, NIGHT, BBoff or BBon) with p<0.05.

https://doi.org/10.1371/journal.pone.0093808.g003

RMSE at long time scales (τ = 5–12) in NMC, ASYMP and SYMP groups

Figure 4 shows RMSE mean (plus standard deviation) at long time scales (i.e. τ = 5–12) as derived from HP and QT series (i.e. RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12}) in Figs.4a,c,e and Figs.4b,d,f respectively. In the NMC-MC protocol RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} were able to separate the SYMP group from NMC and ASYMP individuals with RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} in the SYMP group larger than those in the NMC and ASYMP ones (Figs.4a,b). In the DAY-NIGHT protocol RMSE_{HP,τ = 5–12} (Fig.4c) and RMSE_{QT,τ = 5–12} (Fig.4d) separated the ASYMP group from the SYMP one only during DAY with RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} larger in SYMP individuals. Significant DAY-NIGHT variations were observed only in SYMP individuals (Figs.4c,d) with both RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} smaller during NIGHT (Figs.4c,d). In the BBoff-BBon protocol RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} distinguished ASYMP individuals from SYMP ones both BBoff and BBon with both RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} larger in SYMP individuals (Figs.4e,f). As to the effect of the beta-blocker therapy, it was visible only in SYMP patients over RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} (Fig.4e,f). The effect of the therapy was to reduce both RMSE_{HP,τ = 5–12} (Fig.4e) and RMSE_{QT,τ = 5–12} (Fig.4f).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. RMSE at long time scales over HP and QT series.

RMSE mean (plus standard deviation) assessed at long time scales (i.e. τ = 5–12) over HP series, RMSE_{HP,τ = 5–12} (a,c,e), and QT series, RMSE_{QT,τ = 5–12} (b,d,f), is shown as a function of the experimental protocol. RMSE_{HP,τ = 5–12} and RMSE_{QT,τ = 5–12} are depicted as a function of the group of subjects in the NMC-MC protocol in (a) and (b) respectively, as a function of the period of analysis in the DAY-NIGHT protocol in (c) and (d) respectively, and as a function of the therapy in the BBoff-BBon protocol in (e) and (f) respectively. The gray, dark and white bars are relevant to NMC, ASYMP and SYMP individuals, respectively. The symbol * indicates a significant difference between experimental conditions (i.e. DAY versus NIGHT, BBoff versus BBon) within the same group with p<0.05. The symbol ^# indicates a significant difference between groups within the same experimental conditions (i.e. DAY, NIGHT, BBoff or BBon) with p<0.05.

https://doi.org/10.1371/journal.pone.0093808.g004

Time domain analysis of HP and QT variability

We confirmed that ASYMP patients had longer HP [4]. This characteristic can be considered as a protective factor because, in presence of a relatively immutable duration of the T-wave, it decreases the likelihood that a new ventricular depolarization could occur in the wrong phase of the T wave. This observation substantiates the protective effect of the beta-blocker therapy in LQT1 patients because it leads to a HP lengthening [36]. Since the HP prolongation induced by the beta-blocker therapy was larger in ASYMP patients than in SYMP ones, it could be conjectured that the beta-blocker treatment is more effective in ASYMP individuals than in SYMP one. We observed that HP mean was longer during NIGHT than during DAY in both ASYMP and SYMP group. Since having a longer HP might be safer in LQT1 patients [4], this finding might explain why LQT1 individuals are less at risk during NIGHT than during DAY [3]. According to the LQT1 phenotype QT mean was longer in ASYMP and SYMP groups compared to the NMC one. Circadian rhythm of the QT mean was preserved in both ASYMP and SYMP, thus suggesting that the positive relation linking HP to QT was maintained in LQT1 patients. In SYMP patients the power of the HP variability increased during NIGHT in absence of therapy, as a likely result of a vagal enhancement [6], [8], and BBon during DAY, as a likely result of the beta-adrenergic blockade [37]. This result suggests a more reactive vagal control in SYMP patients [4], [5]. This observation is in agreement with the finding that the cardiac baroreflex sensitivity, another index of vagal modulation, was increased in SYMP patients [4] and that SYMP patients were characterized by an exaggerated bradycardia following an exercise stress test [5]. As a novel finding the ASYMP patients were characterized by a higher QT variance compared to the SYMP ones during DAY in absence of therapy, thus suggesting that a higher sympathetic modulation might be a protective factor in ASYMP patients because it is helpful in limiting the vagal control and its responsiveness to challenges. The observed reduction of the QT variance in ASYMP subjects BBoff during NIGHT and BBon during DAY confirms that the amount of the QT variability is under sympathetic control [9]–[12], [16], [17], [21], [38].

Complexity analysis of HP and QT series provides non redundant information in LQT1 patients

In LQT1 population we confirm that QT complexity indexes provide different information from the HP complexity ones [26]. Differences depend on the temporal scales under scrutiny. While at short and medium temporal scales the complexity of the QT series was larger than that of the HP series [39], [40], the reverse situation was found at long temporal scales [26]. This result was independent of the period of analysis (i.e. DAY or NIGHT) and of the therapy (i.e. BBoff or BBon). The larger complexity of QT series at short and medium temporal scales suggests that the QT variability cannot be considered merely the consequence of the HP changes inducing QT variations through the QT-HP relation [41]. Inputs modifying QT interval and its variability independently of HP changes [10], [13], [42]–[44] tend to increase the complexity of the QT series compared to that of the HP one. We suggest that the complexity of the QT series at short and medium time scales was kept high by QT dynamics unrelated to HP changes. Since these inputs increase during the sympathetic activation induced by an orthostatic challenge [10] and mental stress [13], it is not surprising to find out that the difference between QT and HP complexity tends to decrease during NIGHT and in presence of beta-blocker therapy. Also the smaller complexity of the QT series at long time scales compared to that of the HP series is incompatible with an all-pass QT-HP transfer function. It might be hypothesized that inputs at long time scales targeting the sinus node cannot reach ventricles, thus suggesting that the control of the ventricles at long time scales is much simpler than that of the sinus node.

Link between complexity indexes at short time scale and autonomic regulation in LQT1 population

Complexity of the HP series at short temporal scale is mainly under vagal control. Indeed, it decreased significantly after complete cholinergic blockade induced by a high dose administration of atropine [45] and during vagal withdrawal induced by head-up tilt [23] or active standing [46]. The present study confirms the link between complexity of the HP series at short time scale and vagal control: indeed, complexity of the HP series increased during NIGHT [47], as a likely consequence of the increased importance of vagal modulation, and in presence of beta-blocker therapy, as a likely consequence of the sympathetic blockade leading to an augmented respiratory sinus arrhythmia [37]. The increase was significant in both SYMP and ASYMP patients, thus suggesting that the circadian rhythm was preserved and effects of beta-blocker therapy were evident in both groups.

Fewer studies tried to establish an association between complexity of QT series at short time scale and autonomic modulation. Some studies suggested that an augmented complexity of QT series could be interpreted as a marker of a higher sympathetic drive [40], [48]. The present study detected a reduction of complexity of the QT series at short time scale during NIGHT. Since QT series is mainly under sympathetic control [9]–[12], [16], [17], [21], [38], this tendency suggests a simplification of the cardiac control directed to ventricles during NIGHT, as a likely result of the vagal enhancement. However, since beta-blocker therapy did not affect the complexity of the QT series at short time scale, the association between the complexity of the QT series at short time scale and sympathetic control appears to be weak, thus prompting for the search of this association at time scales more compatible with the sluggishness of the sympathetic control.

Link between complexity indexes at medium and long time scales and autonomic regulation in LQT1 population

The physiological correlates of the complexity of the HP series assessed at medium and long temporal scales are not completely identified. Since complexity indexes at medium time scales assessed from the QT series and at long time scales estimated from both HP and QT series decreased during sympathetic withdrawal occurring during NIGHT and after beta-adrenergic blockade, we suggest that these parameters are under sympathetic control. This observation is in agreement with the finding that complexity indexes at long temporal scales increased during the sympathetic activation induced by active standing [46]. It is worth noting that, although complexity indexes at medium and long time scales exhibited similar trends, the ones at long time scales appeared to be more powerful in suggesting the simplification of the sympathetic control during NIGHT and due to the beta-blocker treatment. As a consequence, we recommend the sole calculation of complexity indexes at long time scales in future applications aiming at extracting indexes linked to the sympathetic function.

Complexity analysis at short time scale was unable to distinguish SYMP patients from ASYMP ones in LQT1 population

One of the most important finding of this study is that the most commonly utilized index of complexity based on entropy (i.e. RMSE at short time scale) [34] failed to differentiate MC individuals from NMC patients and to separate MC individuals into SYMP and ASYMP subjects. This finding was robust because it did not depend on the considered variability series (i.e. HP or QT series) and on the experimental protocol (i.e. NMC-MC, DAY-NIGHT and BBoff-BBon protocols). This conclusion substantiates the need of adopting the MSE approach in this specific study. A likely explanation of this disappointing result might be the low temporal resolution of HP and QT measures derived from a historical database of Holter recordings [27], [28]. This low temporal resolution might increase the corrupting influence of noise, especially at high frequencies (i.e. close to superior limit of the HF band), thus limiting the information content of faster temporal scales and reducing the statistical power of RMSE indexes at short time scale.

Complexity analysis at medium and long time scales did distinguish SYMP patients from ASYMP ones in LQT1 population

At difference with complexity indexes at short time scale, the ones at medium and long time scales did distinguish ASYMP individuals from SYMP ones especially during DAY. Complexity of the cardiac control at medium and long time scales was larger in SYMP subjects than in ASYMP individuals. Since the increase was observed in the case of both indexes at medium and long time scales, it cannot be ascribed to oscillations in the HF band but to slower temporal scales (i.e. in the LF band or even below the inferior limit of the LF band). Differences in vagal control between SYMP and ASYMP individuals cannot fully explain this differentiation. Indeed, if vagal control was responsible for the increase of the complexity of the cardiac control at medium and long time scales, the complexity indexes at medium and long time scales derived from the QT series would have remained unchanged because QT variability is more responsive to sympathetic control and largely unaffected by the vagal control [21]. Therefore, data suggest that in SYMP patients the sympathetic control impinging both sinus node and ventricles is more complex than that of ASYMP subjects. Since ASYMP subjects are characterized by a lower probability of cardiac events, we suggest that a smaller complexity of the cardiovascular control is protective. In addition, since complexity indexes assessed from the QT series at medium time scales and assessed from HP and QT series at long time scales decreased during NIGHT and due to the beta-blocker therapy in SYMP subjects, complexity analysis confirms that NIGHT is a safer period for LQT1 patients and beta-blocker therapy is beneficial. This conclusion is supported also by the reduced differences between ASYMP and SYMP patients during NIGHT and after beta-blocker therapy.