## Abstract

Open-loop linear parametric models were exploited to describe ventricular repolarization duration (VRD) variability during graded head-up tilt. Surface ECG and thoracic movements were recorded in 15 healthy humans (age: 24–54 yr, median: 28 yr; 6 women and 9 men). Tilt table inclinations ranged from 15 to 90° and were varied in steps of 15°. All subjects underwent recordings at every step in random order. Heart period was assessed as the time difference between two consecutive R-wave peaks (RR) and the respiratory signal (R) as the sampling of the thoracic movement signal at the R-wave peaks. VRD was measured automatically as the temporal difference between the R-wave peak and T-wave apex (RT_{a}) or T-wave end (RT_{e}). The best model decomposed RT variability as due to RR changes (RR-related RT variability) to direct respiratory-related inputs (R-related RT variability) and to unknown rhythmical sources unrelated to RR changes and R (RR-R-unrelated RT variability). Using this model, RT_{e} variability was found to be less predictable than RT_{a} variability and composed of a smaller fraction of RR-related RT variability and a larger fraction of RR-R-unrelated RT variability. Predictability progressively decreased with tilt table angles, suggesting increased complexity of RT regulation. RT variance progressively increased with tilt table inclination. This increase was characterized by a gradual rise of the amount of RR-R-unrelated RT variability, whereas the amount of RR-related RT variability remained unchanged. These results suggest that the amount of RT variability, complexity of RT dynamics, and amount of RR-R-unrelated RT variability increase with the magnitude of the sympathetic drive directly related to tilt table inclination. We propose the utilization of the amount of RR-R-unrelated RT variability instead of overall RT variability as an indirect measure of autonomic regulation directed to ventricles.

- QT measurement
- QT variability
- QT-RR relationship
- modelling
- autonomic nervous system

there is increasing interest in quantifying the amount of the beat-to-beat changes of ventricular repolarization duration (VRD), i.e., VRD variability (2, 4, 6, 12, 15, 20, 27, 40). This interest is based on the original suggestion that VRD variability provides an indirect measure of the autonomic regulation directed to ventricles (7). VRD variability is usually computed from the surface ECG as the variability of the time interval between the Q-wave onset and T-wave end (QT interval) under the hypothesis that the variability of the ventricular depolarization period is negligible with respect to that of VRD. Since variability of the heart period, computed as the temporal difference between two consecutive R-wave peaks on the ECG (RR), provides indexes of the autonomic control directed to the sinus node (37), the combined use of QT and RR variability measures would permit the assessment of the autonomic regulation at the sinus node and ventricular levels (6, 7, 20).

Although QT variability depends on autonomic inputs regulating VRD independently of the heart period (8–10, 24, 28), its overall amount cannot be considered a reliable indirect index of autonomic regulation directed to ventricles. Indeed, QT variability depends on RR changes (5, 18, 21, 28, 34) and on influences synchronous with respiration (R) capable of directly modifying the QT interval [e.g., respiratory-related artifacts capable of distorting the T-wave such as cardiac axis movements (18, 30)].

The amount of QT variability can be considered as an indirect measure of the ventricular sympathetic control if it progressively increases as a function of the sympathetic drive and this augmentation is accompanied by the rise of the amount of QT variability unrelated to RR changes and respiratory-related fluctuations.

The present study was developed to verify these necessary conditions for the use of QT variability as an indirect measure of the ventricular sympathetic control. Therefore, the aim of this study was twofold. The first aim was to propose several models describing QT variability and to assess which maximized the predictability of QT interval variability (i.e., the goodness of fit of the model). This aim was accomplished by exploiting an approach based on open-loop linear parametric models belonging to the multivariate dynamic adjustment class (3, 29). The second aim was to exploit the best model to decompose overall QT variability into the fractions driven by RR changes (i.e., RR-related QT variability), driven by R (i.e., R-related QT variability), and independent of RR and R (i.e., RR-R-unrelated QT variability) and assess their relationship with sympathetic drive. This aim was achieved by performing decomposition of QT variability during graded head-up tilt, i.e., an experimental condition known to produce an increase of sympathetic tone and modulation according to tilt table inclination (11, 14, 23). Two methods for computing QT interval were compared concerning the aims of the study: QT interval was derived as the temporal difference between the R-wave peak and T-wave apex (RT_{a}) or T-wave end (RT_{e}). In the following text, we will use the acronym QT only when referring to data present in the literature.

Preliminary results were presented at the 31^{st} Annual International Conference of the IEEE Engineering in Medicine and Biology Society (33).

## METHODS

#### Experimental protocol.

The data belong to a database designed to check whether the gradual increase of sympathetic modulation produced by a graded head-up tilt protocol could be indirectly monitored via the analysis of RR variability (32).

Briefly, we studied 15 healthy nonsmoking humans (age: 24–54 yr, median: 28 yr; 9 men and 6 women). A detailed medical history and examination excluded the evidence of any disease. Subjects did not take any medication or consume any caffeine- or alcohol-containing beverages in the 24 h before the recording. They gave their informed consent to take part in the study. They were positioned on the tilt table supported by two belts at the level of the thigh and waist, respectively, and with both the feet touching the footrest of the tilt table. During the protocol, subjects breathed spontaneously but were not allowed to talk. This study adhered to the principles of the Declaration of Helsinki for medical research involving human subjects. The human research and ethical review boards of L. Sacco Hospital and of the Department of Clinical Sciences approved the protocol.

ECG (Biosignal Conditioning Device, Marazza, Monza, Italy) from lead II and R via the thoracic belt (Marazza, Monza, Italy) were recorded at rest and during head-up tilt. The signals were sampled at 1,000 Hz using an analog-to-digital board (National Instruments, Austin, TX) plugged into a PC. After 7 min at rest, subjects underwent a session (lasting 10 min) of tilt with table angles randomly chosen within the set of 15, 30, 45, 60, 75, and 90° (T_{15}, T_{30}, T_{45}, T_{60}, T_{75}, and T_{90}, respectively). Each tilt session was always preceded by a rest session and followed by 8 min of recovery. All subjects were able to complete the overall protocol without experiencing any sign of presyncope. The duration of the phases was never varied.

#### Data extraction.

ECG traces were preprocessed according to Porta et al. (30) to limit noise and cancel baseline wandering. The heart period was computed as RR interval. The R-wave peak was detected on the ECG using a derivative-threshold algorithm, and its occurrence was fixed using parabolic interpolation. The QT interval was computed as RT_{a} and RT_{e} intervals. Both were automatically derived from the ECG signal. The T-wave apex was searched in a predefined temporal window, the duration of which depended on the preceding RR interval. The T-wave apex was located using parabolic interpolation (30). The T-wave end was located according to a threshold on the first derivative set as a fraction (i.e., 30%) of the absolute maximal first derivative value computed on the T-wave downslope (30). The *i*th RT_{a} or RT_{e} intervals followed the *i*th RR interval, thus directly linking the current RT interval with the preceding RR duration. The *i*th respiratory sample [R(*i*)] was taken in correspondence of the R-wave peak starting the *i*th RR interval. All R-wave peak detections were carefully checked to avoid erroneous identifications or missed beats. RR and RT series were not corrected or filtered except in correspondence of a few premature ventricular contractions. In this case, the cubic spline interpolation technique was applied over the RR and RT values that were directly influenced by the occurrence of the premature ventricular contraction. Before the parameters of the models were identified, the series (i.e., RT_{a}, RT_{e}, RR, and R) were linearly detrended. The series length ranged from 220 to 260 beats and was kept constant while varying the experimental condition in the same subject.

#### Parametric linear open-loop model of RT variability.

The model used to describe RT variability belongs to the class of dynamic adjustment models (3), and, more specifically, it is referred to as an autoregressive (AR), double-exogenous (XX) model with AR noise (ARXXAR) (28, 31). The *i*th RT interval depends on past RT values, on the exogenous actions of current and past RR intervals and of current and past R samples, and on additive AR noise (see *ARX*_{RR}*X*_{R}*AR model of RT variability* in the appendix). The model structure accounts for RR influences, for the effects of unknown rhythmical variability sources independent of RR and R changes (modeled by AR noise, e.g., slow unknown autonomic nervous system influences directly affecting ventricles), and for the direct influences of R acting independently on RR changes (e.g., respiratory-related artifacts).

#### Goodness of fit of the model.

After the model coefficients were identified directly from the RT, RR, and R series (see *Identification and hypothesis testing procedures* in the appendix), the one step-ahead prediction of the RT interval was calculated and subtracted from the actual RT value to assess the prediction error (see *Goodness of fit of the ARX*_{RR}*X*_{R}*AR model* in the appendix). The mean squared prediction error (MSPE) was assessed to evaluate the performance of the model. MSPE was bounded between the RT variance (σ_{RT}^{2}) and 0: MSPE = σ_{RT}^{2} indicated that the model was unable to explain RT variability and MSPE = 0 indicated that the model perfectly described RT variability (i.e., RT interval changes were completely predictable). The goodness of fit (ρ_{RT}) was calculated as (σ_{RT}^{2} − MSPE)/ σ_{RT}^{2}, thus being bounded between 0 and 1 and positively correlated with the ability of the model to describe RT dynamics (the larger ρ_{RT}, the larger RT predictability, the better the performance of the model).

#### Decomposition of RT variability.

The structure of the model allowed the decomposition of RT variability into independent partial processes (3, 28), each relevant to the white noise sources (w_{RR}, w_{R}, and w_{N}; see *ARX*_{RR}*X*_{R}*AR factorization into partial processes* in the appendix). The noises w_{RR}, w_{R}, and w_{N} were filtered to provide the contributions of RR variability, R variability (e.g., respiratory-related artifacts), and unknown rhythmical sources independent of RR changes and R (e.g., unmeasured autonomic nervous system influences directly affecting ventricles) to RT variability. These partial processes represented RR-related, R-related, and RR-R-unrelated RT variability (RT_{RR}, RT_{R}, and RT_{N}, respectively_{)}. The variance of these partial processes (i.e., σ_{RT/RR}^{2}, σ_{RT/R}^{2}, and σ_{RT/N}^{2}) was calculated. They were expressed in both absolute units (i.e., ms^{2}) and dimensionless units after dividing them by σ_{RT}^{2} (i.e., χ_{RT/RR}^{2}, χ_{RT/R}^{2}, and χ_{RT/N}^{2}). Under the hypotheses of whiteness and uncorrelation of w_{RR}, w_{R}, and w_{N}, σ_{RT/RR}^{2} + σ_{RT/R}^{2} + σ_{RT/N}^{2} = σ_{RT}^{2} and χ_{RT/RR}^{2} + χ_{RT/R}^{2} + χ_{RT/N}^{2} = 1 (3, 29).

#### Comparing different model structures.

In addition to the ARX_{RR}X_{R}AR model, three simplified versions were identified (see *Customizing the ARX*_{RR}*X*_{R}*AR model* in the appendix): *1*) the AR model, *2*) the ARX_{RR} model, and *3*) the ARX_{RR}X_{R} model. The goodness of fit of these models was compared to check whether the increase of the complexity of the model structure produced a significant improvement of the goodness of fit. More specifically, the AR model was contrasted with the ARX_{RR} model, the ARX_{RR} model was contrasted with the ARX_{RR}X_{R} model, and the ARX_{RR}X_{R} model was contrasted with the ARX_{RR}X_{R}AR model to assess the relevance of accounting for RR and R exogenous influences and rhythmical noisy inputs, respectively.

#### Statistical analysis.

The Wilcoxon signed-rank test was applied to the pooled values of the goodness of fit derived from all the models to check whether parameters extracted from RT_{a} variability were different from those derived from RT_{e} variability. Friedman repeated-measures ANOVA on ranks (Dunn's test) was applied to the pooled values of the goodness of fit to compare the ability of the different models to describe RT_{a} and RT_{e} variabilities regardless the experimental condition. Friedman repeated-measures ANOVA on ranks (Dunnett's test) was applied to check whether the goodness of fit and indexes derived from the factorization of RT_{a} and RT_{e} variances changed with respect to those found at rest. Linear regression analysis was carried out to assess the degree of association with tilt table angles via the evaluation of the Pearson product-moment correlation coefficient (*r*_{P}). The null hypothesis of slope equal to 0 (i.e., no linear relationship) was tested. Global linear regression analysis was carried out by pooling together data relevant to all the experimental conditions, whereas individual linear regression analysis by considering only one subject at time. Individual linear regression analysis was carried out only if global linear regression analysis was found significant, and, in this case, we evaluated the percentage of subjects with a significant individual linear regression analysis. *P* values of <0.05 were considered significant.

## RESULTS

#### Goodness of fit of RT variability during graded head-up tilt.

The goodness of fit derived from RT_{a} variability (median: 0.59) was found to be significantly larger than that derived from RT_{e} variability (median: 0.46) independently of the experimental condition and models (Fig. 1*A*). The best model order (Fig. 1*B*) as derived from RT_{a} variability (median: 7.0) was similar to that derived from RT_{e} variability (median: 6.0).

Independently of the experimental condition, the goodness of fit of RT_{a} variability increased as a function of the complexity of the model structure. More specifically, the ARX_{RR} model was better than the AR model, the ARX_{RR}X_{R} model was better than the ARX_{RR} model, and the ARX_{RR}X_{R}AR model was better than the ARX_{RR}X_{R} model (Fig. 2*A*). The same result was found when considering the goodness of fit of RT_{e} variability (Fig. 2*B*). Since the ARX_{RR}X_{R}AR model maximized the goodness of fit in the case of both RT_{a} and RT_{e} variabilities, this model structure was selected as the best model structure to describe RT_{a} and RT_{e} variabilities and applied to decompose RT_{a} and RT_{e} variances into their components due to partial processes.

Global linear regression analysis carried out over all the data pooled together was used to assess the degree of association between the goodness of fit and tilt table angles. The goodness of fit of the ARX_{RR}X_{R}AR model was significantly correlated with the tilt table inclination (Fig. 3) regardless of the type of RT variability (i.e., RT_{a} or RT_{e} variability). The *r*_{P} value was negative [*r*_{P} = −0.26 (Fig. 3*A*) and −0.30 (Fig. 3*B*) in the case of RT_{a} and RT_{e} variabilities, respectively], thus indicating that the predictability of RT dynamics decreased as a function of the tilt table angle. Changes with respect to rest became significant during T_{60}, T_{75}, and T_{90} in the case of RT_{a} variability and during T_{75} and T_{90} in the case of RT_{e} variability. Individual linear regression analysis showed that the goodness of fit of the ARX_{RR}X_{R}AR model was significantly linked to tilt table angles in 40% and 47% of the subjects in the case of RT_{a} and RT_{e} variabilities, respectively.

#### Decomposition of RT variability during graded head-up tilt.

In Fig. 4, the amount of RR-related RT power (i.e., σ_{RT/RR}^{2}), R-related RT power (i.e., σ_{RT/R}^{2}), and RR-R-unrelated RT power (i.e., σ_{RT/N}^{2}) are pooled together independently of the experimental condition. The indexes derived from RT_{a} variability were significantly smaller than those derived from RT_{e} variability (Fig. 4, *A–C*). When indexes derived from the factorization of RT variability were normalized by RT variance, the fractional contribution of RR-related RT variance was smaller in RT_{e} variability than in RT_{a} variability (Fig. 4*D*), whereas that of RR-R-unrelated RT variance was significantly larger (Fig. 4*F*). The fraction of R-related RT variance was similar in RT_{a} and RT_{e} variabilities (Fig. 4*E*).

Results of the factorization of RT variance into the contributions of partial processes during all the phases of the experimental protocol are shown in Tables 1 and 2. RT_{a} variance (σ_{RTa}^{2}; Table 1) increased as a function of tilt table inclination, and changes became significant during T_{75} and T_{90}. The amount of RR-related RT_{a} variability (σ_{RTa/RR}^{2}) remained constant, whereas that of RR-R-unrelated RT_{a} variability (σ_{RTa/N}^{2}) increased significantly above 60° (i.e., during T_{60}, T_{75}, and T_{90}). The amount of R-related RT_{a} variability (σ_{RTa/R}^{2}) tended to increase as well: indeed, σ_{RTa/R}^{2} was significantly higher during T_{75}. Similar findings were obtained for RT_{e} variability (σ_{RTe}^{2}; Table 2). The unique remarkable difference was that R-related RT_{e} variability remained constant.

At rest, RT variability was remarkably driven by RR variations and unknown rhythmical sources: indeed, the fractional amount of RR-related RT variability (χ_{RT/RR}^{2}) and that of RR-R-unrelated RT variability (χ_{RT/N}^{2}) was 0.68 and 0.29 in the case of RT_{a} variability (Table 1) and 0.66 and 0.41 in the case of RT_{e} variability (Table 2). In contrast, at rest, the fractional contribution of R-related RT variability (χ_{RT/R}^{2}) was negligible in both RT_{a} and RT_{e} variabilities (i.e., 0.051 in Table 1 and 0.084 in Table 2, respectively). The head-up tilt protocol affected χ_{RTa/RR}^{2} and χ_{RTa/N}^{2}: indeed, χ_{RTa/RR}^{2} significantly decreased above 60° (i.e., during T_{60}, T_{75}, and T_{90}; Table 1), whereas χ_{RTa/N}^{2} increased significantly above 45° (i.e., during T_{45}, T_{60}, T_{75}, and T_{90}; Table 1). In contrast, χ_{RTa/R}^{2} was unmodified during the experimental protocol. Similar results were derived for RT_{e} variability (χ_{RTe}^{2}; Table 2).

Global linear regression analysis was carried out over all the data pooled together to assess the degree of association between RT variance and tilt table angles (Tables 3 and 4). Both RT_{a} and RT_{e} variances were significantly and positively correlated with tilt table angle. Individual linear regression analysis determined that RT_{a} and RT_{e} variances were linearly related to tilt table inclination in 40% of the subjects. Global linear regression analysis applied to the factorization of RT_{a} and RT_{e} variances into the contributions due to partial processes indicated that σ_{RTa/N}^{2} and σ_{RTe/N}^{2} were significantly and positively correlated with tilt table inclination, whereas σ_{RTa/RR}^{2} and σ_{RTe/RR}^{2} were unrelated to tilt table angle (Tables 3 and 4). The indexes σ_{RTa/N}^{2} and σ_{RTe/N}^{2} were found linearly related to tilt table inclination in 53% of the subjects. RT_{a} variability behaved differently than RT_{e} variability in terms of the contribution to R-related RT variance: indeed, σ_{RTa/R}^{2} was significantly correlated with tilt table angles (Table 3), whereas σ_{RTe/R}^{2} was unrelated to them (Table 4). When global linear regression analysis was applied to the fractional amount of RT variance due to partial processes, it was found that χ_{RTa/RR}^{2}, χ_{RTe/RR}^{2}, χ_{RTa/N}^{2}, and χ_{RTe/N}^{2} were significantly related to tilt table angles: χ_{RTa/RR}^{2} and χ_{RTe/RR}^{2} were negatively correlated (Tables 3 and 4), whereas χ_{RTa/N}^{2} and χ_{RTe/N}^{2} were positively correlated (Tables 3 and 4). In contrast, χ_{RTa/R}^{2} and χ_{RTe/R}^{2} were unrelated to tilt table angles (Tables 3 and 4). Individual linear regression analysis determined that χ_{RTa/RR}^{2}, χ_{RTe/RR}^{2}, χ_{RTa/N}^{2}, and χ_{RTe/N}^{2} were linearly related to tilt table inclination in 40, 40, 47, and 40% of the subjects.

## DISCUSSION

The main findings of this study can be summarized as follows: *1*) the model structure maximizing the goodness of fit allowed the decomposition of RT variability into fractions accounting for RR dependences (RR-related RT variability), respiratory-related influences (R-related RT variability), and unknown rhythmical sources independent of RR changes and R (RR-R-unrelated RT variability); *2*) the goodness of fit progressively decreased as a function of the tilt table inclination, thus indicating that RT dynamics became more complex and difficult to be explained using the best model; *3*) RT_{e} variability was more complex (less predictable) than RT_{a} variability and was composed of a smaller fraction of RR-related variability and by a larger fraction of RR-R-unrelated variability; *4*) RT_{a} and RT_{e} variances progressively increased with tilt table angles; and *5*) the increase of RT variances was characterized by a progressive increase of the amount of RR-R-unrelated RT variability, whereas RR-related RT variability remained constant.

#### Modeling RT variability.

The model maximizing the goodness of fit of RT variability was the ARX_{RR}X_{R}AR model. This model was capable of accounting for the influences of RR changes, the effects of R independent of RR variations [e.g., changes of the RT interval related to deformation of the T-wave due to cardiac axis movement synchronous with R (30)], and the influences of unknown rhythmical sources independent of RR changes and R (e.g., autonomic regulations direct to the ventricles capable of modifying the RT interval independently of RR changes and R).

The exploitation of the trivariate ARX_{RR}X_{R}AR linear open-loop model allowed the factorization of RT variability in terms of contributions due to RR changes (RR-related RT variability), due to R independently of RR variations (R-related RT variability), and due to unknown rhythmical sources independent of RR and R (RR-R-unrelated RT variability). At rest, the fractional amount of RR-related and RR-R-unrelated RT variabilities was significant (i.e., 0.68 and 0.29 in the case of RT_{a} variability and 0.66 and 0.41 in the case of RT_{e} variability). This result stresses the relevance of both RR-related and RR-R-unrelated portions of RT variability. It is worth noting that the fractional amount of R-related RT variability was very small (<0.1 in both the case of RT_{a} and RT_{e} variabilities), thus suggesting that regulations of RT dynamics independently of RR changes are more likely to occur at frequencies different from the respiratory frequency. Despite the smallness of this part, model structures that did not account for R as an exogenous input exhibited a significantly smaller goodness of fit, thus stressing the importance of accounting for direct respiratory-related influences.

#### Predictability of RT variability.

Although the ARX_{RR}X_{R}AR model provided the best prediction of RT variability, the goodness of fit was far from 1 at rest, especially when RT_{e} variability was considered (i.e., 0.83 and 0.69 in the case of RT_{a} and RT_{e} variabilities, respectively, at rest), thus suggesting that RT dynamics contain dynamic features that cannot be accounted by the structure of this model. Among these features, we recall nonlinear dynamics, e.g., due to effects of QT-RR hysteresis (17, 38), adjustments according to long time scales (i.e., several minutes) that might need more specific model structures to be reliably resolved (13), and feedback effects activated by cardiac neural afferents. In addition, the goodness of fit was not constant during the experimental protocol: indeed, it progressively decreased with tilt table inclination, thus suggesting that RT dynamics became more and more complex and the best model structure tended to become more and more inadequate. This result suggests that the complexity of RT dynamics depends on the magnitude of the sympathetic drive, which tends to accentuate the weight of dynamic features unaccounted by the model.

#### Comparison between RT_{e} and RT_{a} variabilities.

The comparison of the goodness of fit calculated over RT_{a} and RT_{e} variabilities suggested that RT_{a} variability is more predictable than RT_{e} variability. This result indicates that RT_{e} regulation is more complex than RT_{a} regulation, thus pointing out that the dynamic influences unaccounted by the model more predominantly affect the RT_{e} interval than the RT_{a} interval.

In addition, RT_{e} variability was characterized by a smaller fraction of RR-related RT_{e} variability and by a larger fraction of RR-R-unrelated RT_{e} variability than RT_{a} variability. This result suggests that the RT_{a} interval is more predominantly driven by RR changes, whereas the RT_{e} interval is more independent of autonomic regulations directed to the sinoatrial node. Therefore, we suggest that RT_{e} variability is more under the control of mechanisms directed to ventricles, and, thus, RT_{e} variability is more informative about autonomic control independent of sinus node regulation. Therefore, when the main aim is the characterization of RT regulation independent of heart rate control, the beat-to-beat RT_{e} measure is preferrable to the RT_{a} measure. This conclusion is in agreement with Merri et al. (22), who suggested measurement of the duration of the first part of the ventricular repolarization process (from the S-wave offset to the T-wave apex) when the main aim is to analyze the dependence of VRD on the RR interval. However, since the RT_{e} measure is less robust with regard to broad band noise than the RT_{a} measure (30), a portion of RR-R-unrelated RT_{e} variability might be simply the result of the difficulty in locating the T-wave end in the presence of broad-band noise. Specific methodological studies are needed to clarify whether measurement techniques different from the RT_{e} measure [e.g., based on the template-matching technique (7)] should be recommended to control the potential influence of broad-band noise on RT_{e} power factorization.

#### RT variability during graded head-up tilt.

This study identified a positive linear relationship between RT variance and tilt table inclination. Since sympathetic tone and modulation increase during head-up tilt (11, 14, 23, 32), the gradual increase of RT variability with tilt table inclination supports the existence of a relationship between RT variance and ventricular sympathetic tone and/or modulation (26, 39) and the opportunity of using the level of RT variability as an index proportional to the magnitude of ventricular sympathetic control. This result, combined with recent findings suggesting that QT variability is unrelated to sympathetic tone (4), suggests that the relationship between RT variance and ventricular sympathetic control should be limited to the amplitude of ventricular sympathetic modulation (i.e., the amplitude of the fluctuations of the sympathetic discharge around its mean value) and/or, as suggested by Berger (6), to ventricular sympathetic tone when its values are elevated, as occurs at the highest tilt table inclinations.

The amount of QT variability has been proposed to indirectly quantify autonomic regulation directed to ventricles (6, 20). However, since QT variability is largely influenced by RR changes, its amount depends on autonomic regulations directed to the sinus node as well. This dependence might reduce the effectiveness of the amount of QT variability as a measure of ventricular sympathetic regulation. Indeed, if and only if the RR-related fraction of QT variability remains constant between different conditions and different populations, the overall amount of QT variability might be considered a reliable index of ventricular sympathetic regulation. The proposed factorization might overcome this limitation by giving the possibility of separating the RR-related RT variability, related to autonomic regulation at the sinus node level, from the RR-R-unrelated RT variability, more likely related to autonomic regulations at the ventricular level, and from the R-related RT variability, linked to direct respiratory-related influences. Even though the assumption that RR-R-unrelated RT variability is the result of sympathetic modulations at the ventricular level is a pure speculation and should be eventually tested on animals, the quantification of its amount from surface ECG is a step toward a finer assessment of VRD regulation.

We found out that the gradual increase of variance of RT variability with tilt table inclination was due to an increase of variance of RT variability driven by unknown rhythmical sources independently of RR changes and R (i.e., RR-R-unrelated RT variability). According to this finding, we suggest using variance of RR-R-unrelated RT variability as an index of ventricular sympathetic regulation instead of the overall amount of RT variability. It is worth stressing that the method, here applied to RT variability, can be used over QT variability, and any index of autonomic regulation currently derived from QT variability (6, 20) can be derived from RR-R-unrelated QT variability after the factorization of QT variability into partial processes (see *ARX*_{RR}*X*_{R}*AR factorization into partial processes* in the appendix), thus avoiding the dependence of QT variability on RR changes. The use of the fraction of RR-R-unrelated QT power might clarify puzzling results such as the lack of variations of the QT variability index during β-adrenergic blockade (25).

In this experimental protocol, the use of the variance of RR-R-unrelated RT variability did not seem to produce any additional advantage with respect to the use of the overall level of RT variability. This result is due to the invariance of the amount of RR-related RT variability during the graded head-up tilt protocol. Conversely, the advantage of using the variance of RR-R-unrelated RT variability instead of the overall level of RT variability should become evident when the amount of RR-related RT variability varies between different conditions and/or different populations.

Additional studies are necessary to understand whether the variance of RR-R-unrelated RT variability carries complementary information with respect to the variance of the error about the regression line of the current RT interval on the previous RR duration or diastolic interval (35). However, it is worth noting that the proposed approach exploits a multiple regression analysis that reconstructs RT dynamics in a multidimensional phase space, thus accounting for the dynamic dependence of the RT interval on several RR durations (19) and the effect of respiration.

#### Conclusions.

The proposed linear parametric modelling approach to the study of RT dynamics provided quantitative indexes that can be easily derived from surface ECG and fruitfully exploited in practical applications (i.e., goodness of fit and decomposition of RT variance into portions with different meanings). The course of the goodness of fit with tilt table angles suggested that the complexity of RT regulation depends on the magnitude of sympathetic control. The course of the fraction of RT variability independent of RR and R suggested that RR-R-unrelated RT variability can be used instead of RT variability to extract more reliable indexes of sympathetic regulation directed to ventricles.

## GRANTS

This work was partially supported by Telethon Grant GGP09247 (to A. Porta) and a PRIN 2007 grant (to N. Montano).

## DISCLOSURES

No conflicts of interest are declared by the author(s).

- Copyright © 2010 the American Physiological Society

## APPENDIX

##### ARX_{RR}X_{R}AR model of RT variability.

Given the beat-to-beat series of RT = {RT(*i*), *i* = 1,…, *N*}, RR = {RR(*i*), *i* = 1,…, *N*} and R = {R(*i*), *i* = 1,…, *N*}, where *i* is the progressive cardiac beat number and *N* is the series length, they are first normalized by subtracting the mean and then by dividing the result by the SD, thus obtaining rt, rr, and r series with zero mean and unit variance. The adopted ARX_{RR}X_{R}AR model is defined as follows:
*a*_{rt − rt}(*k*), *b*_{rt − rr}(*k*), and *b*_{rt − r}(*k*) values are *p*, *p* + 1, and *p* + 1 constant coefficients and *z*^{−1} is the one-lag delay operator in the *z* domain.

The noise (n) is an AR process described by the following equation:
_{n} is a white noise with zero mean and variance of λ_{wn}^{2}.

##### Goodness of fit of the ARX_{RR}X_{R}AR model.

The one step-ahead prediction error of the ARX_{RR}X_{R}AR model is defined as the difference between rt(*i*) and the best one step-ahead prediction of rt(*i*) [rt̂(*i*)] as follows:

In the case of the ARX_{RR}X_{R}AR model (36):
*Â*_{rt − rt}(*z*), *B̂*_{rt − r}(*z*), and *D̂*_{n}(*z*) values are estimated from real data via identification procedures. The ability of the model in fitting data is measured according to MSPE as follows:

MSPE is bounded between 0 (i.e., the model perfectly fits the rt series) and the variance of the rt series (i.e., the model fails completely to describe rt variability). Due to normalization, MSPE ranges from 0 to 1.

##### ARX_{RR}X_{R}AR factorization into partial processes.

When RT variability is modeled as an ARX_{RR}X_{R}AR process, it can be factorized as sum of partial processes due to rr, r, and n as follows (29):
_{rr} and w_{r} are the white noises with zero mean and variances of λ_{wrr}^{2} and λ_{wr}^{2}, respectively. The factorization holds under the hypotheses of whiteness and uncorrelation of the residuals, w_{rr}, w_{r}, and w_{n} (29). The noises w_{rr} and w_{r}, filtered as follows,
*a*_{rr − rr}(*k*) and *a*_{r − r}(*k*) values are *p* constant coefficients. The rt_{rr}, rt_{r}, and rt_{n} processes represent RR-related, R-related, and RR-R-unrelated RT variability, respectively.

##### Customizing the ARX_{RR}X_{R}AR model.

The ARX_{RR}X_{R}AR model can be easily customized by setting all the coefficients relevant to some polynomials to zero while identifying those relevant to the remaining ones. More specifically, we can derive the AR model by imposing *B*_{rt − rr}(*z*) = 0, *B*_{rt − r}(*z*) = 0, and *D*_{n}(*z*) = 0, the ARX_{RR} model by setting *B*_{rt − r}(*z*) = 0 and *D*_{n}(*z*) = 0, and the ARX_{RR}X_{R} model by imposing *D*_{n}(*z*) = 0.

##### Identification and hypothesis testing procedures.

The coefficients of the AR, ARX_{RR}, and ARX_{RR}X_{R} models were identified using a traditional least-squares approach, whereas those relevant to ARX_{RR}X_{R}AR model were identified using a generalized least-squares approach (31, 36). The latter procedure was stopped when the current iterate did not produce a significant percent decrease of the MSPE with respect to the previous iterate (the threshold was 0.001). The solutions of both the traditional and generalized least-squares problems were found using the Cholesky decomposition method (16). The best model order was selected according to the Akaike figure of merit for multivariate processes (1) and searched in the range from 4 and 16. After the best model order had been chosen, the whiteness and uncorrelation of the residuals were tested. The hypothesis of whiteness of the residuals (i.e., w_{rr}, w_{r}, and w_{n}) was tested by checking that the autocorrelation functions divided by variance of the residuals were 0 for τ ≠ 0 and τ ≤ 40 (31). The hypothesis of uncorrelation between w_{rr} and w_{n} and between w_{rr} and w_{r} was tested by checking that the cross-correlation functions divided by the product of the SDs were 0 for τ ≤ 40 (even at τ = 0) (31).