Modeling and Analysis of Ventricular Response in Atrial Fibrillation

Abstract

This chapter begins with an overview of results reported in clinical studies on RR interval analysis during atrial fibrillation, followed by a brief overview of methods for heuristic assessment of the atrioventricular node. The main part of the chapter reviews several AV node models proposed for simulation as well as for parameter estimation. The chapter concludes with a comparison of atrioventricular node models.

7.1 Introduction

The ventricular response in atrial fibrillation (AF) is highly irregular, mainly due to the atrial impulses arriving irregularly at the atrioventricular (AV) node. As a result, the RR intervals differ dramatically in length. Despite the irregularity, the ventricular response is not completely random, but exhibits weak correlation [1] and certain short- or long-term predictability [2]. Another characteristic is that the ventricular rate is often higher in AF than in normal sinus rhythm, a characteristic explored in AF detection, see Chap. 4. The RR interval series observed in normal sinus rhythm and AF differ with respect to both variability and irregularity, two aspects which are illustrated in Fig. 7.1.

Fig. 7.1

24-h RR interval series in a normal sinus rhythm and b AF. The 24-h plots (top row) show that the dispersion is much larger in AF than in normal sinus rhythm. The zoomed-in segments (bottom row) demonstrate that the RR interval series in AF not only has larger dispersion, but it is also much more irregular

The ventricular response plays a significant role in the management of patients with AF [3]. In fact, the control of ventricular rate effectively reduces the risk of complication and improves the quality of life. Therefore, the study of factors influencing ventricular rate and its dynamics is of great importance as it may lead to strengthened decision-making in AF management.

To characterize normal sinus rhythm, a wide range of parameters have been investigated, often categorized into dispersion parameters to characterize RR variability, spectral parameters to characterize autonomic influence on the sinus node, and different types of entropy to characterize RR irregularity. To characterize AF, spectral parameters have received very limited interest since the RR interval spectrum is essentially flat, and lacks peaks which may carry physiological information [4]. On the other hand, dispersion parameters and entropy measures have conveyed clinically valuable information: for example, lower variability and/or irregularity of the RR interval series have been associated with poor outcome in patients with AF [5, 6]. Given the growing clinical interest to understand the characteristics of the RR intervals, an overview of results reported in clinical studies is provided in Sect. 7.2.

The analysis of ventricular response can be augmented with information on f waves so that the coupling between the atria and the ventricles, through the AV node, can be investigated. The AV node plays a particularly important role in AF by acting as a “filter” which blocks certain atrial impulses, with repercussions on ventricular activation. By developing methods for analyzing AV nodal properties, patient-specific information may be obtained which describes the effect of a certain antiarrhythmic drug. The properties can be studied by means of mathematical modeling, considered either for simulation of various scenarios or estimation of model parameters. In the latter case, the observed signal is acquired either invasively or from the surface ECG. Signal processing techniques are usually required to separate the atrial from the ventricular activity before parameter estimation can be performed, see Chap. 5.

Section 7.3 provides a brief overview of methods for heuristic assessment of the AV node. Section 7.4 describes a method for analyzing the relationship between atrial input and ventricular response during AF. Sections 7.5 and 7.6 review several AV node models for simulation and parameter estimation, respectively. The chapter concludes with a comparison of AV node models in Sect. 7.7.

Fig. 7.2

Illustration of the difference between variability, quantified by the standard deviation, and irregularity, quantified by the sample entropy. Each row shows a time series with identical irregularity (given by the numbers to the left of the diagrams), but increasing variability from left to right, whereas each column shows series with identical variability (given by the numbers on above the diagrams), but increasing irregularity from top to bottom. The units of the horizontal and vertical axes are arbitrary

7.2 RR Interval Analysis

Classical dispersion parameters such as the coefficient of variation and the root mean square of successive differences (RMSSD), defined in (4.2) and (4.3), respectively, have been found useful for characterizing the variability of RR intervals in AF [2, 7].¹ However, variability parameters provide an incomplete characterization of RR intervals, since they cannot characterize irregularity, i.e., the degree of unpredictability. Therefore, variability parameters have been complemented with different entropy measures to characterize irregularity, including approximate entropy $I_{\mathrm {ApEn}}$, sample entropy $I_{\mathrm {SampEn}}$, and Shannon entropy $I_{\mathrm {ShEn}}$ (see Sect. 4.2.1 for definitions), as well as conditional entropy [8]. Figure 7.2 illustrates the difference between variability and irregularity for different time series with identical variability but different irregularity, and vice versa.

Two early studies analyzed the RR interval series in patients with AF, demonstrating that reduced variability is associated with worse outcome [5, 6]. Using the 24-h ambulatory ECG, reduced RR interval irregularity was found to have independent prognostic value for cardiac mortality during long-term follow-up in patients with permanent AF [9].

The association between RR intervals and long-term clinical outcome has been evaluated in a population of ambulatory patients with mild-to-moderate heart failure and AF at baseline. Patients with symptomatic heart failure were enrolled in a multicenter study on sudden death [10]. Both $I_{\mathrm {ApEn}}$ and $I_{\mathrm {ShEn}}$ were found to be significantly lower in nonsurvivors than in survivors for all subgroups of death (total mortality, sudden death, and heart failure death). Patients with a lower $I_{\mathrm {ApEn}}$ had significantly lower survival: Kaplan–Meier analysis [11] showed that a lower $I_{\mathrm {ApEn}}$ was associated with a nearly fourfold higher total mortality (40% vs. 12%) and more than six times higher mortality due to progression of heart failure (19% vs. 3%) and sudden death (18 vs. 3%). The criterion $I_{\mathrm {ApEn}}<1.68$, where 1.68 is the lower tertile of the data set, was found to be a significant predictor of all types of mortality after adjustment for significant clinical covariates [12], leading up to the main finding that lower irregularity is associated with worse outcome in AF patients.

Results in the literature suggest that irregularity parameters may be used as risk indicators. Thus, it is of interest to investigate to what extent irregularity is affected by commonly used rate-control drugs. The effect of the selective A1-receptor agonist tecadenoson, alone as well as in combination with the beta blocker esmolol, was assessed in a small group of AF patients [13]. Tecadenoson was found to reduce heart rate and increase variability, but did not have any effect on irregularity. Beta blockade with intravenous esmolol further increased variability and decreased heart rate. In another study [14], no significant differences in RR irregularity, as quantified by $I_{\mathrm {ApEn}}$, were observed in patients with AF and congestive heart failure when treated with beta blockers, digoxin, or amiodarone. The effect of rate-control drugs on RR variability/irregularity was investigated in 60 patients with permanent AF, involving the drugs diltiazem, verapamil (both calcium channel blockers), metoprolol, and carvedilol (beta blocker) [15]. Variability was assessed by well-known parameters such as the standard deviation and the RMSSD, whereas irregularity was assessed by $I_{\mathrm {ApEn}}$, $I_{\mathrm {ShEn}}$, and a measure based on conditional entropy [16]. A significantly lower heart rate was obtained for all investigated drugs, reaching its lowest rate for the calcium channel blockers. Moreover, all drugs were found to increase RR variability significantly relative to the baseline recording, whereas only the beta blockers increased RR irregularity significantly.

Using the data set in [15], circadian variation was investigated by means of five ambulatory recordings per patient, obtained at baseline as well as during the four different drug regimens [17]. Variability and irregularity parameters were computed in nonoverlapping, 20-min segments. Circadianity was assessed using cosinor analysis of the resulting series, characterized by the 24-h mean and the excursion over the mean described by the amplitude of the cosine fitted to the data [18]. Heart rate and variability parameters, including the standard deviation and the RMSSD, exhibited significant circadian variation in most patients, whereas circadian variation in $I_{\mathrm {ApEn}}$ and $I_{\mathrm {SampEn}}$ was detected in only a few patients. When circadian variation was detected in $I_{\mathrm {ApEn}}$ at baseline, the patients had more severe symptoms. All drugs decreased the rhythm-adjusted mean of the heart rate and increased the rhythm-adjusted mean of variability parameters (the rhythm-adjusted mean is also referred to as “midline estimating statistic of rhythm”, MESOR [19]). Only carvedilol and metoprolol decreased the normalized amplitude over the 24 h of the irregularity parameters and heart rate. The results suggested that circadian variation can be observed in most patients using variability parameters, but only in a few patients using irregularity parameters.

The above-mentioned clinical studies are limited by small patient groups. Therefore, further studies are needed to better assess whether variability and irregularity parameters are predictive of patient status.

7.3 Heuristic Assessment of the Atrioventricular Node

Rate-control drugs act on atrial and/or AV nodal properties to lower the ventricular rate. During drug development, electrophysiological effects of antiarrhythmic drugs are usually assessed invasively in sinus rhythm. Since an atrial pacing protocol cannot be applied in patients with AF, the electrophysiological drug effects on the AV node are still not completely understood. When optimizing drug therapy, noninvasive assessment of AV nodal electrophysiology may help to select optimal therapy. During the early clinical phases of drug development, noninvasive characterization of the AV node may facilitate data collection from large patient cohorts and favor patient-tailored therapy. Estimation of the AV nodal refractory period using the surface ECG has been attempted in several studies, employing different heuristic approaches [20,21,22,23,24].

Heuristic assessment of AV nodal electrophysiology has relied on simple approaches to characterizing the RR intervals. Noninvasive estimation of the functional refractory period of the AV node during AF has been attempted by simply selecting the shortest RR interval or the 5-th percentile of the RR interval series [20, 23, 25]. In dogs, it was demonstrated that the shortest RR interval correlated statistically with the functional refractory period, determined using a pacing protocol. Therefore, the shortest RR interval was used as a surrogate measurement of the functional refractory period [20].

Using the Poincaré plot, where each RR interval is plotted against the preceding RR interval, an estimate of the functional refractory period can be obtained as well. The value of the lower envelope, determined as a regression line, at 1 s (“1 s intercept”) and the degree of scatter above the lower envelope have been proposed as surrogate measurements of AV nodal refractoriness and concealed AV conduction, i.e., the effect of blocked impulses on the conduction of subsequent impulses, respectively. The circadian variation of the 1-s intercept of the lower envelope was investigated in 120 patients who underwent 24-h ambulatory monitoring at baseline [24]. During an observation period of $33\pm 16$ months, there were 25 deaths, including 13 cardiac and 8 stroke deaths. All patients showed significant circadian rhythms in the lower envelope, however, patients dying subsequently from cardiac causes, but not from fatal stroke, had less pronounced circadian rhythm, with amplitudes which were less than 55% of those in surviving patients. It was suggested that blunted circadian rhythm of AV conduction represents an independent risk of cardiac death in patients with permanent AF.

The presence of clusters in the histogram-based Poincaré plot, based on the RR intervals derived from the 24-h ambulatory ECG, has been suggested as a marker of higher AF organization to predict the outcome of electrical cardioversion [26]. A cluster was considered to be present when a peak in the histogram plot could be identified visually. Later, the histogram-based Poincaré plot served as the basis for the Poincaré surface profile, i.e., a univariate histogram defined by those RR intervals which are preceded by RR intervals of approximately the same length [27], cf. Sect. 4.2.2. The Poincaré surface profile was proposed as a tool for characterizing AV nodal memory effect and detecting preferential AV nodal conduction. However, neither the Poincaré plot nor the histogram-based Poincaré plot analysis have raised much interest in the research community. This may be due to a number of reasons, including that the plot is strongly dependent on bin size, the bins must be sufficiently well-populated with points to produce meaningful results, and manual interaction is often needed to determine the lower envelope [24].

Fig. 7.3

AV synchrogram analysis of AF [28]. Atrial-to-atrial (AA) and RR interval series, normalized instantaneous ventricular phases $\varPsi _1(t_{a,k})$, $\varPsi _2(t_{a,k})$, $\varPsi _3(t_{a,k})$, defined in (7.2), and ratio of ventricular to atrial activations q / p of the synchronized epochs (top to bottom). Note that p : q is displayed inside the bottom diagram, whereas the ratio q / p is the unit of the vertical axis. (Modified from [28] with permission.)

7.4 Synchrogram Analysis

Synchrogram analysis has been introduced for exploratory analysis of the relationship between atrial input and ventricular response during AF, providing valuable insights into AV nodal function [28]. The method is purely phenomenological, and no attempt is made to account for AV nodal electrophysiological properties such as refractoriness and conduction delay. The analysis is applied to atrial activations, determined from the electrogram, and ventricular activations, determined from the ECG, to analyze AV coupling. The analysis is performed by observing the phase of the ventricular activations at time instants triggered by the atrial activations. The instantaneous ventricular phase is assumed to be a monotonically increasing, piecewise linear function, defined by

$$\begin{aligned} \phi _v(t)= 2 \pi \frac{t-t_{v,n-1}}{t_{v,n}-t_{v,n-1}} +2 \pi n, \quad t_{v,n-1} \le t < t_{v,n},\quad n=0,\ldots ,N-1, \end{aligned}$$

(7.1)

where $t_{v,n}$ is the time of n-th ventricular activation and N is the total number of ventricular activations. To be consistent with the indexing of RR intervals adopted in Chap. 4, the first RR interval, defined by $x_0 = t_{v,0}-t_{v,-1}$, requires that the time of the first ventricular activation is indexed by $-1$.

The instantaneous ventricular phase is normalized to the interval [0, q], and sampled at the time of atrial activations $t_{a,k}$ for the purpose of detecting whether p : q coupling is present,

$$\begin{aligned} \varPsi _q(t_{a,k})=\frac{1}{2\pi } \left( \phi _v(t_{a,k}) \text { mod } (2 \pi q) \right) , \quad q=1,2,3, \end{aligned}$$

(7.2)

where p is the number of atrial activations and q is the number of ventricular activations. Epochs of synchronization are automatically detected by alternately dividing the values of $\varPsi _q(t_{a,k})$ into different subgroups. The normalized phases are classified as p : q coupling whenever the absolute difference between $\varPsi _q(t_{a,{k+p}})$ and $\varPsi _q(t_{a,k})$ within a subgroup is below a predefined tolerance threshold. The synchrogram analysis is illustrated in Fig. 7.3.

The synchrogram was investigated in both atrial flutter and AF. As expected, the percentage of coupled beats and the duration of coupled epochs were significantly higher in atrial flutter than in AF [28]. Moreover, the synchrogram was used to assess the dynamics of AV coupling as a function of atrial fibrillatory rate (AFR) in a small group of patients during spontaneous acceleration of the AFR at the onset of an AF episode; in this particular assessment, the AFR was estimated from the electrogram. The results demonstrated that the occurrence and the duration of coupled epochs decreased as the AFR increased, and that the average AV conduction ratio, i.e., the ratio of ventricular to atrial activations, was significantly smaller at higher AFRs [29].

Synchrogram analysis has also been considered for investigating the effects of atrial activity and AV nodal conduction on the ventricular response in patients with paroxysmal AF [30]. The results showed that ventricular rate and RR variability are significantly correlated with the average AV conduction ratio and the variability of the atrial input. On the other hand, the AFR is not correlated with ventricular rate nor with RR variability.

7.5 Mathematical Modeling of the Atrioventricular Node

Refractoriness and concealed conduction of the AV node are important AV nodal properties which contribute to forming the ventricular response. Due to refractoriness, many atrial impulses are blocked when arriving to the AV node. Concealed conduction of a single atrial impulse, occurring when the impulse only partially penetrates into the AV node without reaching the ventricles, influences the conduction of subsequent atrial impulses. Moreover, the existence of two dominant pathways through the AV node, each with its own electrophysiological properties, is well-documented and plays an important role in AF.

The properties of AV nodal function can be studied by mathematical modeling which may be categorized into:

• Models primarily developed for simulation to provide better understanding of AV nodal properties, sometimes involving intracardiac information on atrial activity where the arrival times of the atrial impulses are known (this section).

• Models primarily developed for statistical estimation of AV node parameters, relying entirely on information derived from the surface ECG. The arrival times of atrial impulses are modeled as a random process (Sect. 7.6).

7.5.1 Modeling of Conduction Delay in Non-AF Rhythms

Conduction delay is an important AV nodal property, and has therefore received considerable attention in mathematical model building. With reference to AV nodal conduction in Wenckebach periodicity, i.e., a non-AF rhythm, the conduction delay $d_k$ related to the k-th atrial impulse depends on the AV nodal recovery time (RT) $\varDelta t_{\text {RT},k}$. The conduction delay is modeled by [21]

$$\begin{aligned} d_k = d_{\text {min}} + \alpha _{\text {max}} \exp \left[ -\frac{\varDelta t_{\text {RT},k}}{\gamma _c} \right] , \end{aligned}$$

(7.3)

where $d_{\text {min}}$ is the minimal conduction delay, $\alpha _{\text {max}}$ is the maximal prolongation of the conduction delay, and $\gamma _c$ is the time constant of the exponential conduction curve. The recovery time $\varDelta t_{\text {RT},k}$ is given by the time elapsing from the preceding ventricular activation $t_{v,n}$ to the current AV nodal activation time $t_{a,k}$,

$$\begin{aligned} \varDelta t_{\text {RT},k} = t_{a,k} - t_{v,n}, \quad t_{a,k} > t_{v,n}, \end{aligned}$$

(7.4)

where ventricular activations are indexed by n.

The basic model of conduction delay in (7.3) can be expanded to include rate-dependent shortening of the conduction delay, referred to as facilitation (fac), and rate-dependent prolongation of the recovery time, referred to as fatigue (fat) [21], see also [31]. In the expanded model, the conduction delay $d_k$ in (7.3) is denoted $d_k^{\prime }$, $\alpha _{\text {max}}$ is replaced by $\alpha _k$ to model facilitation, and the term $s_k$ is introduced to model fatigue,

$$\begin{aligned} d_k^{\prime } = d_{\text {min}} + s_k+ \alpha _k \exp \left[ -\frac{\varDelta t_{\text {RT},k}}{\gamma _c} \right] . \end{aligned}$$

(7.5)

Facilitation is incorporated into the model by assuming that the maximal prolongation $\alpha _{\text {max}}$ depends on the interval $\varDelta t_{a,k-1}$ between two successive atrial impulses immediately preceding $t_{a,k}$, commonly referred to as the AA interval,

$$\begin{aligned} \alpha _k = \alpha _{\text {max}} -\kappa \exp \left[ -\frac{\varDelta t_{a,k-1}}{\gamma _{\text {fac}}}\right] , \end{aligned}$$

(7.6)

where

$$\begin{aligned} \varDelta t_{a,k} = t_{a,k} - t_{a,k-1}, \end{aligned}$$

(7.7)

and $\alpha _{\text {max}}$, $\gamma _{\text {fac}}$, and $\kappa$ are model constants. Fatigue is incorporated by assuming that each AV nodal activation causes a slowing of the conduction delay of all subsequent impulses, modeled by

$$\begin{aligned} s_k = s_{k-1} \exp \left[ {-\frac{\varDelta t_{a,k-1}}{\gamma _{\text {fat}}}}\right] + \eta \exp \left[ -\frac{\varDelta t_{\text {RT},k}}{\gamma _{\text {fat}}}\right] , \end{aligned}$$

(7.8)

where $\eta$ and $\gamma _{\text {fat}}$ are model constants.

The conduction delay model, defined by (7.3)–(7.8), was fitted to experimental data obtained from seven autonomically blocked dogs during pacing [21]. The results showed that the model can accurately predict dynamic changes in Wenckebach periodicity. Although the model does not account for concealed conduction, it has nonetheless served as a starting point for AV node modeling in AF [32,33,34,35].

7.5.2 Modeling of Conduction Delay in AF

An AV node model accounting for conduction delay, defined by (7.3), and refractoriness in AF was proposed in [33, 34], however, fatigue and facilitation were not modeled. In that model, the AV node becomes refractory after an atrial impulse has been conducted through the AV node to the ventricles. Impulses arriving to the AV node during the refractory period are blocked (concealed), and each blocked impulse causes the refractory period to be prolonged, first with a fixed length [33], but later with a Gaussian random variable [34].

The proposed model, with fixed prolongation of the refractory period, was tested on one, single intracardiac recording from a patient with AF, exhibiting an agreement between the estimated and the observed RR series which is not particularly satisfactory, see Fig. 7.4. Using instead the model with random prolongation [34], a better fit was obtained. The significance of these two conduction delay models remain to be established on a larger set of data.

Fig. 7.4

(Reprinted from [33] with permission)

a Atrial-to-atrial (AA) and b ventricular-to-ventricular (VV) intervals obtained from an intracardiac recording. Ventricular-to-ventricular intervals were obtained using the AV node model in [33]. The vertical lines in both panels are the times of the atrial impulses, where vertical, solid lines indicate conducted impulses, and vertical, dashed lines indicate blocked impulses.

More recently, a dual-pathway AV node model has been proposed in which the conduction delay, similar to the model in [21], is assumed to be affected by the stimulation history [36]. The conduction delay is described by the model in (7.3), except that $\alpha _{\text {max}}$ and $\gamma _{c}$ are assumed to be functionally dependent on the preceding conduction delay $d_{k-1}$,

$$\begin{aligned} \alpha _{\text {max},k} = a_1 d_{k-1}^2+a_2 d_{k-1} + a_3, \end{aligned}$$

(7.9)

and

$$\begin{aligned} \gamma _{\text {c},k} = b_1+\left( \frac{d_{k-1}}{b_2}\right) ^{b_3}, \end{aligned}$$

(7.10)

respectively. The model constants $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, and $b_3$ are assumed to differ between the two pathways. Concealed conduction is modeled by a virtual conduction delay $\tilde{d}_{k}$ which depends on the AA interval $\varDelta t_{a,k}$ [36],

$$\begin{aligned} \tilde{d}_{k} = c_1 - c_2 \exp \left[ -\frac{\varDelta t_{a,k} }{c_3}\right] , \end{aligned}$$

(7.11)

implying that $\varDelta t_{\text {RT},k}$ in (7.3) is replaced by $(\varDelta t_{a,k} - \tilde{d}_{k-1})$,

$$\begin{aligned} d_{k}=d_{\text {min}}+ \alpha _{\text {max},k} \exp \left[ -\frac{(\varDelta t_{a,k} - \tilde{d}_{k-1})}{\gamma _{c,k}} \right] . \end{aligned}$$

(7.12)

The model constants $c_1$, $c_2$, and $c_3$ are assumed to differ between the two pathways. It should be noted that the effect of replacing $\varDelta t_{\text {RT},k}$ by $(\varDelta t_{a,k} - \tilde{d}_{k-1})$ is similar to that of prolongation of the refractory period due to concealed conduction, see Sect. 7.5.4.

The model parameters were estimated by fitting $\alpha _{\text {max},k}$ and $\gamma _{c,k}$ to data obtained using a pacing protocol. The fitted model could predict the conducting pathway with specificity and sensitivity exceeding 85% when AF-like random stimulation was applied to a rabbit preparation. His’ electrogram alternans was used to determine the pathway of each conducted impulse in the experimental data [37].

7.5.3 Modeling of Refractory Period in AF

In a simple model accounting for the refractory period, the atrial impulses are assumed to arrive randomly in time at the AV node according to a Gaussian distribution [38]. Each atrial impulse results in ventricular activation, unless the AV node is refractory which causes the atrial impulses to be blocked and the refractory period to be prolonged. For each blocked atrial impulse, the refractory period $\tau _k$, following the k-th atrial impulse arriving at the AV node after ventricular activation, is prolonged according to the following equation:

$$\begin{aligned} \tau _{k+1}=\tau _k+u_k (0.9-\tau _k), \quad k \ge 0. \end{aligned}$$

(7.13)

The time-dependent prolongation $u_k$ of the refractory period is described by the logistic function

$$\begin{aligned} u_k = \frac{1}{1+e^{-a (z_k-b)}}, \end{aligned}$$

(7.14)

where

$$\begin{aligned} z_k= 1 - \frac{t_{a,k}}{\tau _k}, \quad \tau _k > t_{a,k}, \end{aligned}$$

(7.15)

and a and b are positive-valued model constants. The index k is reset to 0 and $\tau _0$ is reset to 0.3 s when a ventricular activation occurs [38]. The parameters a and b define the shape of the RR interval histogram, but lack a physiological interpretation. For the model in (7.13)–(7.15), an atrial impulse arriving close in time to a conducted atrial impulse prolongs the refractory period more than an atrial impulse arriving at the end of the refractory period. The blocked atrial impulses prolong the refractory period $\tau _k$ towards its upper limit of 0.9 s.

7.5.4 Modeling of Refractory Period and Conduction Delay in AF

A much more sophisticated AV node model for the simulation of ventricular activation during AF was proposed in [35, 39], see also [40, 41], where conduction delay, prolongation of the refractory period due to concealed conduction, and ventricular pacing (VP) are also taken into account. The AV node is activated due to the combined effect of spontaneous depolarization and AF bombardment. However, the AV node can also be activated by a VP-induced, retrograde wave. The activation initiates a refractory period during which the AV node is nonresponsive to atrial impulses as well as to a VP-induced retrograde wave. When the refractory period ends, the transmembrane potential returns to its resting potential and initiates a spontaneous, linear increase in the transmembrane potential. Each time an AF impulse arrives at the AV node when not being in a refractory state, its transmembrane potential is increased by a discrete amount, see Fig. 7.5. If instead a VP-induced retrograde wave penetrates the AV node in a nonrefractory state, the transmembrane potential reaches its threshold immediately.

Fig. 7.5

a Schematic representation of the AV node model in [35], and b related modeling of the transmembrane potential $V_m$ of the AV node. The resting value $V_r$ can increase spontaneously in a linear fashion as well as by a discrete amount $\varDelta V$ when an atrial impulse arrives. When $V_m$ exceeds the threshold $V_t$, an action potential is fired and the AV node becomes refractory for a certain period of time $\tau _k^{\prime }$ (indicated by the grey area). Ventricular activation is associated with a delay $d_k$, modeled by (7.3). Prolongation of the refractory period due to a blocked atrial impulse is modeled by (7.17)

In this model, the conduction delay $d_k$ is modeled by (7.3), and the refractory period $\tau _k$ is modeled by

$$\begin{aligned} \tau _k = \tau _{\text {min}}+\beta \left( 1-\exp \left[ - \frac{\varDelta t_{\text {RT},k}}{\gamma _r} \right] \right) , \end{aligned}$$

(7.16)

where $\tau _{\text {min}}$ is the shortest refractory period, $\beta$ is the maximal prolongation of the refractory period, and $\gamma _r$ is the time constant of the exponential refractory curve. Moreover, the model accounts for prolongation of the refractory period due to concealed conduction. The prolonged refractory period is a product of two factors: one depending on the arrival time of the atrial impulse and another depending on the strength of the atrial impulse,

$$\begin{aligned} \tau '_k = \tau _k + \tau _{\text {min}}\left( \frac{t_{a,k}}{\tau _k}\right) ^{\rho _1} \left( \min \left( 1,\frac{\varDelta V}{V_{t} - V_{r}} \right) \right) ^{\rho _2}, \end{aligned}$$

(7.17)

where $\tau '_k$ is the prolonged refractory period and $\varDelta V$ is the strength of an atrial impulse. The voltages $V_t$ and $V_r$ are defined in Fig. 7.5. The two positive-valued exponents $\rho _1$ and $\rho _2$ are model constants.

Fig. 7.6

(Reprinted from [35] with permission)

Simulated RR intervals series obtained for two different settings of model parameters [35] (top row), and corresponding (b) histogram (middle row), and c autocorrelation function (bottom row). The following model parameter values were used: a $\lambda _a=8$ per second, $\varDelta V=20$ mV, $\rho _1=10, \rho _2=10$, and b $\lambda _a=4$ per second, $\varDelta V=10$ mV, $\rho _1=10, \rho _2=10$.

Figure 7.6 shows two simulated RR series generated using different model parameter settings [35, 39]. The simulation is based on the assumption that atrial impulses arrive to the AV node according to a Poisson process with mean arrival rate $\lambda _a$.

The authors stated that the simulation model may provide a quantitative framework to investigating drug effects by fitting their model to experimental data. However, no results have so far been published which investigate such effects. The problem of fitting the model to observed data is likely to be challenging since the model involves a large number of parameters.

7.5.5 Modeling of Spatial Dynamics

A radically different approach to modeling is to treat the AV node as a connected graph [42], consisting of a series of interacting nodes, rather than having a lumped structure as the above-described AV node models. An advantage of the graph approach is that it accounts for spatial propagation of atrial impulses in the AV node, implying that phenomena such as concealed conduction and retrograde conduction are intrinsic to the model structure.

The nodes in the graph model propagate impulses, corresponding to action potentials, along the graph edges, see Fig. 7.7. Each node corresponds to a localized part of the AV node with its own conduction delay and refractory period, both quantities depending on the stimulation history of the node. When an impulse arrives at a node, the conduction delay and the refractory period are updated according to (7.3) and (7.16), respectively. Each node has its own dynamics and is characterized by its own recovery time $\varDelta t_{\text {RT}}$, thus differing from the above-described models where the recovery time applies to the whole AV node.

Fig. 7.7

Schematic presentation of the spatial AV node model proposed in [42]. The graph nodes propagate impulses along the edges. Each node is characterized by its own refractory period and conduction delay, both depending on the stimulation history of the node. The full model comprises 21 nodes

The proposed model consists of 21 nodes, where 10 nodes characterize the fast pathway and 11 the slow pathway. The parameters modeling conduction delay, i.e., $d_{\text {min}}$, $\alpha _{\text {max}}$, and $\gamma _c$ in (7.3), and refractory period, i.e., $\tau _{\text {min}}$, $\beta$, and $\gamma _r$ in (7.16), are identical for all nodes of the slow pathway; the same applies to all nodes of the fast pathway. Hence, the model is defined by 12 parameters. To simulate conduction through the model, it is assumed that the first nodes on the atrial side of the slow and the fast pathways are simultaneously activated. A ventricular activation occurs when the atrial impulse reaches the rightmost end of the graph, corresponding to the bundle of His.

Simulations were performed using AA intervals determined from electrograms recorded from patients with AF, as well as from simulated AA intervals determined by a Poisson process with mean arrival rate $\lambda _a$. A genetic algorithm was used to fit the model by minimizing the difference between simulated RR series and observed RR interval series, obtained from a number of ECG recordings. In the simulations using electrogram-derived series of atrial activations, the difference was quantified by the mean square error of the times of ventricular activations. If simulated AA intervals were used and only the ECG was available, the difference was quantified based on the RR interval histogram. The model could accurately replicate the RR intervals determined from the ECG.

The model fitting was repeated 1000 times for different initial conditions, resulting in 1000 estimates of each parameter for each recording [42]. No unique solution was obtained since several different parameter sets resulted in a similar model fit, however, the ranges of the estimated model parameters were limited. For ECG data, 90% of the estimated values of $\tau _{\text {min}}$ and $\beta$ were within ${\pm }20\%$ of the median value of the estimates, whereas this did not apply to $d_{\text {min}}$ and $\alpha _{\text {max}}$.

7.6 Statistical Modeling of the Atrioventricular Node and Parameter Estimation

In the very first paper dealing with statistical modeling, the AV node was treated as a lumped structure whose behavior represents the temporal and spatial summation of the cellular electrical activity [43]. In that model, briefly described in Sect. 7.6.1, the atrial impulses are assumed to arrive randomly in time at the AV node, modeled by a Poisson process with mean rate $\lambda _a$ [44]. The conduction time is not explicitly modeled.

Many years later, an improved statistical model was proposed which also accounts for dual AV nodal conduction [45, 46], see Sect. 7.6.2. Since the model parameters can be estimated from the surface ECG, without use of any intracardiac information, noninvasive electrophysiological characterization of the AV node is made possible. In a subsequent study, the model was further improved to incorporate pathway switching, accompanied by more robust parameter estimation [47], see Sect. 7.6.3.

7.6.1 A First Statistical Model of the AV Node

In this statistical model, the AV node is always in one of two states. In the first state, the AV node is absolutely refractory to stimulation by atrial impulses. At the onset of the second state, the transmembrane potential is at its resting value, and increases spontaneously at a constant rate as well as by a discrete amount $\varDelta V$ when an atrial impulse arrives. When the transmembrane potential reaches its threshold value $V_t$ as a result of any combination of spontaneous and stepwise depolarization, the AV node fires and a new refractory period is initiated, see Fig. 7.8.

Fig. 7.8

a Schematic representation of the AV node model in [43], and b related modeling of the transmembrane potential $V_m$ of the AV node. The resting value $V_r$ can increase spontaneously in a linear fashion (defined by the slope v) as well as by a discrete amount $\varDelta V$ when an atrial impulse arrives. When $V_m$ exceeds the threshold $V_t$, an action potential is fired and the AV node becomes refractory for a certain period (indicated by the grey area). This model differs from the one in Fig. 7.5 as it does not account for the delay d associated with ventricular activation

The refractory period is assumed to be rate-dependent, implying that a longer RR interval is followed by a longer refractory period, and vice versa. The relation between refractory period and RR interval is modeled by an exponential function,

$$\begin{aligned} \tau _n=\tau _{\infty } \left( 1-\exp \left[ -\frac{x_{n}}{\tau _{\infty }}\right] \right) , \end{aligned}$$

(7.18)

where $\tau _n$ is the refractory period following the n-th ventricular activation, $\tau _{\infty }$ is the maximal refractory period, and $x_{n}$ is the RR interval preceding the n-th ventricular activation,

$$\begin{aligned} x_{n} = t_{v,n} -t_{v,n-1}. \end{aligned}$$

(7.19)

The conduction delay is not explicitly modeled.

The model is defined by the following four parameters:

• the mean arrival rate $\lambda _a$ of atrial impulses,

• the relative amplitude $\varDelta V$ of atrial impulses,

• the rate v of spontaneous AV depolarization, measured in units of $\varDelta V$, and

• the maximal refractory period $\tau _{\infty }$.

Although this model is statistical in nature, no well-established statistical estimation procedure, such as the maximum likelihood (ML) technique, was considered in [43]. Instead, the model parameters were determined using an ad hoc optimization procedure which yielded unphysiological parameter estimates.

7.6.2 Statistical Modeling of Dual AV Nodal Pathways

The improved AV node model accounts for concealed conduction, relative refractoriness, and dual AV nodal pathways [45], see Fig. 7.9. In this model, each atrial impulse is assumed to result in ventricular activation, unless the impulse is blocked by a refractory AV node. The probability of an atrial impulse passing through the AV node depends on the time elapsed since the preceding ventricular activation $t_{v,n-1}$. The refractory period is defined by the sum of a deterministic period $\tau$ and a random period, uniformly distributed in the interval $[0, \tau _p]$. The random period models prolongation due to concealed conduction and/or relative refractoriness. All atrial impulses arriving at the AV node before the end of the deterministic period $\tau$ are blocked, whereas impulses with arrival time in the interval $[\tau ,\tau +\tau _p]$ have linearly increasing likelihood of passing through the AV node. No impulses are blocked if they arrive after $\tau + \tau _p$.

Fig. 7.9

a Schematic representation of the AV node model in [45], and b related modeling of the transmembrane potential $V_m$ of the AV node. When an atrial impulse arrives at the AV node, the resting value $V_r$ increases by a discrete amount $\varDelta V$ which always makes $V_m$ exceed the threshold $V_t$, an action potential to be fired, and the AV node refractory for a certain period of time (indicated by the grey area)

The model accounts for a fast pathway with a longer refractory period, defined by $\tau _{f}$ and $\tau _{f,p}$, and a slow pathway with a shorter refractory period, defined by $\tau _{s}$ and $\tau _{s,p}$ (depending on pathway, the indices “s” and “f” are added to $\tau$ and $\tau _p$). In mathematical terms, the refractoriness of the slow pathway is defined by the function $\beta _s(t)$,

$$\begin{aligned} \beta _s(t)=\left\{ \begin{array}{ll} 0, &{} \quad 0<t<\tau _s,\\ \frac{\displaystyle t -\tau _s}{\displaystyle \tau _{s,p}}, &{} \quad \tau _s\le t <\tau _s + \tau _{s,p},\\ 1, &{} \quad t \ge \tau _s+\tau _{s,p}, \end{array} \right. \end{aligned}$$

(7.20)

where, for convenience, t is used instead of $\varDelta t_{\text {RT},k}$. The function $\beta _{f}(t)$ characterizes refractoriness of the fast pathway and is identical to $\beta _s(t)$ except that $\tau _{f}$ and $\tau _{f,p}$ are substituted for $\tau _{s}$ and $\tau _{s,p}$, respectively. The deterministic part of the refractory periods $\tau _s$ and $\tau _f$ are assumed to depend linearly on the preceding RR interval $x_{n-1}$, implying that a longer RR interval is followed by a longer refractory period, and vice versa. Moreover, it is assumed that the AV conduction time is incorporated into $\beta _s(t)$ and $\beta _f(t)$ so that a ventricular activation occurs immediately after a non-blocked atrial impulse.

Since non-blocked atrial impulses are assumed to occur according to an inhomogeneous Poisson process characterized by the intensity function $\lambda _a \beta _s(t)$, the PDF of an RR interval x, related to the slow pathway, is given by [45]

$$\begin{aligned} p_{x,s}(x)=\lambda _a \beta _s(x) \exp \left[ -\int _0^x \lambda _a \beta _s(\tau ) d\tau \right] , \end{aligned}$$

(7.21)

which, after insertion of (7.20), becomes

$$\begin{aligned} p_{x,s}(x)=\left\{ \begin{array}{ll} {\displaystyle 0}, &{} \ 0<x<\tau _s,\\ {\displaystyle \frac{\lambda _a (x-\tau _s)}{\tau _{s,p}} \exp \left[ -\frac{\lambda _a(x-\tau _s)^2}{2 \tau _{s,p}} \right] }, &{} \ \tau _s\le x<\tau _s+\tau _{s,p}, \\ {\displaystyle \lambda _a \exp \left[ -\frac{\lambda _a \tau _{s,p}}{2} -\lambda _a (x-\tau _s-\tau _{s,p})\right] }, &{} \ x\ge \tau _s+\tau _{s,p}. \end{array} \right. \end{aligned}$$

(7.22)

The PDF $p_{x,f}(x)$, related to the fast pathway, is obtained by substituting $\tau _{f}$ and $\tau _{f,p}$ for $\tau _{s}$ and $\tau _{s,p}$ in (7.22), respectively.

Conduction through the slow and fast pathways are assumed to occur with probabilities $\varepsilon$ and $1-\varepsilon$, respectively. Assuming that ventricular activations occur according to a Poisson process, the intervals between successive ventricular activations are statistically independent, and the joint probability of the RR intervals $x_0,\ldots ,x_{N-1}$ is given by

$$\begin{aligned} p_{x}(x_0,x_1,\ldots , x_{N-1})&= \prod _{n=0}^{N-1} p_{x}(x_n)\nonumber \\&= \prod _{n=0}^{N-1}( \varepsilon p_{x,s}(x_n)+(1-\varepsilon )p_{x,f}(x_n)). \end{aligned}$$

(7.23)

The mean arrival rate $\lambda _a$ is estimated from the f wave signal extracted from the ECG, but corrected to account for atrial refractoriness, using [46]

$$\begin{aligned} \hat{\lambda }_a =\frac{\lambda _{\text{ AF }}}{1-\delta \lambda _{\text{ AF }}}, \end{aligned}$$

(7.24)

where $\lambda _{\text{ AF }}$ is taken as the AFR, estimated from the ECG, and $\delta$ is the minimal time interval between successive impulses arriving to the AV node. The five model parameters $\pmb {\theta }= \begin{bmatrix} \varepsilon&\tau _{s}&\tau _{s,p}&\tau _{f}&\tau _{f,p} \end{bmatrix}$ are estimated from the observed RR interval series using the ML technique, defined by

$$\begin{aligned} {\hat{ \pmb \theta }} = \arg \max _{\pmb \theta } \log p_{x}(x_0 ,x_1, \ldots , x_{N-1} | \pmb \theta ; \hat{\lambda }_a). \end{aligned}$$

(7.25)

Since no closed-form solution can be found for the estimator ${\hat{ \pmb \theta }}$, combined with the fact that the gradient is discontinuous, the multi-swarm particle swarm optimization is used to optimize the log-likelihood function [48, 49]. It should be noted that since $\tau _s$ and $\tau _f$ depend on the preceding RR interval, the original RR interval series is subject to decorrelation before ML estimation is performed, to better comply with the assumption of statistical independence in (7.23) [45].

The parameters of the single pathway model, i.e., $\begin{bmatrix} \tau&\tau _{p} \end{bmatrix}^T$, are also estimated. The Bayes information criterion is then used to determine which of the single- and the dual-pathway model is most likely the observed [46].

The block diagram in Fig. 7.10 shows the main signal processing steps required to estimate the model parameters from the ECG. Figure 7.11 illustrates, in histogram form, the RR intervals produced by three different parameter settings of the AV node model.

Fig. 7.10

The main signal processing steps required for estimating the AV node model parameters from the ECG

Fig. 7.11

(Reprinted from [45] with permission)

RR interval histogram (area defined by grey bars) and fitted model PDF (solid line) for three different parameter settings. The histograms derive from model data with increasing probability $\varepsilon$ of an atrial impulse passing through the slow pathway, set to either 0, 0.25, or 0.5 (left to right); the other model parameters were held constant.

The AV node model was fitted to clinical data acquired during treatment with different drugs for the purpose of investigating drug-induced changes in AV nodal properties [50,51,52]. The hypothesis was that the estimates of AV nodal refractory periods would reflect the main changes in AV nodal properties previously reported in studies performed in sinus rhythm and based on invasive data. The effects of tecadenoson and esmolol were investigated in a small cohort of patients [50]. The parameters $\tau _s$ and $\tau _f$, accounting for both effective refractory period and conduction interval, were prolonged for both tecadenoson and esmolol [50]. The increase in $\tau _s$ and $\tau _f$, observed for both pathways, suggested either prolonged effective refractory period² or prolonged AV conduction, or both. In addition, tecadenoson was shown to affect heart rate but not AFR, suggesting that a decrease in heart rate may be attributed to that tecadenoson affects the AV node. These results are in agreement with previous studies demonstrating that tecadenoson prolongs the effective refractory period of the AV node and slows down its conduction [53], whereas esmolol prolongs refractoriness and conduction time in both pathways during AV nodal reentrant tachycardia [54].

Changes in AV nodal properties were investigated during administration of beta blockers (carvedilol and metoprolol) and calcium channel blockers (diltiazem and verapamil) in a controlled setting [52]. For patients with permanent AF, this study compared the effects of four once-daily drug regimens (metoprolol, diltiazem, verapamil and carvedilol) on heart rate and arrhythmia-related symptoms. While the results of this study are not directly comparable to previous studies, the changes in estimated AV nodal properties are in agreement with previous electrophysiological findings [55,56,57,58].

The results suggest that the noninvasively obtained parameter estimates reflect the expected changes in AV nodal properties for the investigated drugs. Therefore, the method should be suitable for assessing the drug effect on AV nodal electrophysiology during AF, especially for antiarrhythmic compounds aimed at rate-control during AF and tested in clinical trials during initial clinical phases of drug development.

Another application of the AV node model is to analyze data acquired during rest and head-up tilt (75$^{\circ }$) from patients with persistent AF. A shortening of the refractory periods $\tau _s$ and $\tau _f$ was observed for both pathways during adrenergic activation [59]—results which are in agreement with earlier reported results [60]. The effect of tilting on the refractory period of the AV node has not been assessed previously, but invasive studies have evaluated the effect of vagal tone on AV node refractory periods by either stimulating the vagal nerve directly [60] or by assessing the effect of vagolytic drugs.

7.6.3 Statistical Modeling of Pathway Switching

A limitation of the statistical model in [45] is the assumption that atrial impulses arriving between two ventricular activations attempt conduction through the same pathway, i.e., pathway switching is not allowed. Therefore, another model suitable for ECG-based estimation of the model parameters was proposed in [47]. Similar to the model in [45], atrial impulses are assumed to arrive at the AV node according to a Poisson process with mean arrival rate $\lambda _a$. Each impulse attempts to pass through either the slow or the fast pathway, being blocked according to the time-dependent functions $\beta _s(t)$ and $\beta _f(t)$ depending on which pathway is chosen. The choice of pathway is independent of the pathway taken by the preceding atrial impulse. Conduction through the slow pathway is attempted with probability $\varepsilon$, and consequently conduction through the fast pathway is attempted with probability $1-\varepsilon$.

Since an atrial impulse is assumed to arrive at the AV node according to a Poisson process, the PDF of the arrival time of the first atrial impulse following a ventricular activation is given by [61]

$$\begin{aligned} p_1(t) = \left\{ \begin{array}{ll} \lambda _a e^{-\lambda _at}, &{} \ t \ge 0,\\ 0, &{} \ t < 0. \end{array} \right. \end{aligned}$$

(7.26)

The first impulse attempts conduction through the slow pathway with probability $\varepsilon$, where conduction is characterized by $\beta _s(t)$, defined in (7.20). Hence, the PDF of the arrival time of the first impulse conducted through the slow pathway is given by

$$\begin{aligned} p_{1,\text {cs}}(t)=\varepsilon \beta _s(t) p_1(t). \end{aligned}$$

(7.27)

The PDF of the arrival time of the first impulse conducted through the fast pathway $p_{1,\text {cf}}(t)$ is obtained by replacing $\beta _s(t)$ with $\beta _f(t)$ and $\varepsilon$ with $1-\varepsilon$. The notations “cs” and “cf” refers to conduction through the slow and the fast pathway, respectively.

The PDF of the arrival time of the second atrial impulse depends on the arrival time of preceding blocked atrial impulses as well as the time interval between the second and the first atrial impulses, i.e., the AA interval. Since AA intervals are statistically independent in the Poisson model, the PDF of the arrival time of the second atrial impulse following a ventricular activation is given by

$$\begin{aligned} p_{2}(t)=\int _0^{\infty }p_{1}(t-\rho ) (p_{1,\text {bs}}(\rho ) + p_{1,\text {bf}}(\rho ) )d \rho , \end{aligned}$$

(7.28)

where $p_{1,\text {bs}}(t)$ and $p_{1,\text {bf}}(t)$ denotes the PDF of the arrival time of the first impulse blocked in the slow and the fast pathway, respectively. To account for pathway switching, the second atrial impulse attempts to pass through the slow pathway with probability $\varepsilon$ irrespectively of the pathway in which the first impulse was blocked. Hence, $p_{2,\text {cs}}(t)$ and $p_{2,\text {cf}}(t)$ are computed analogously to $p_{1,\text {cs}}(t)$ and $p_{1,\text {cf}}(t)$.

A general expression for recursive computation of the PDF of the arrival times is given by

$$\begin{aligned} p_{i,\text {cs}}(t)&=\varepsilon \beta _s(t) p_i(t),\end{aligned}$$

(7.29)

$$\begin{aligned} p_{i,\text {cf}}(t)&=(1-\varepsilon ) \beta _f(t) p_i(t),\end{aligned}$$

(7.30)

$$\begin{aligned} p_{i,\text {bs}}(t)&=\varepsilon (1-\beta _s(t)) p_i(t),\end{aligned}$$

(7.31)

$$\begin{aligned} p_{i,\text {bf}}(t)&=(1-\varepsilon ) (1-\beta _f(t)) p_i(t),\end{aligned}$$

(7.32)

$$\begin{aligned} p_{i+1}(t)&=\int _0^{\infty } p_1(t-\rho ) (p_{i,\text {bs}}(\rho ) + p_{i,\text {bf}}(\rho )) d \rho , \end{aligned}$$

(7.33)

where $p_{i+1}(t)$ denotes the PDF of the arrival time of the $(i+1)$:st atrial impulse following a ventricular activation, $p_{i,\text {cs}}(t)$ and $p_{i,\text {cf}}(t)$ denote the PDFs of the arrival time of the i-th atrial impulse conducted through the slow pathway and the fast pathway, respectively, and $p_{i,\text {bs}}(t)$ and $p_{i,\text {bf}}(t)$ denote the corresponding PDFs of blocked atrial impulses.

A conducted atrial impulse is assumed to immediately cause a ventricular activation. Hence, the PDF of the time intervals between ventricular activations, i.e., $x_n$, is obtained by summing the PDFs of all conducted atrial impulses,

$$\begin{aligned} p_c(x_n;J)=\sum _{i=1}^{J}(p_{i,\text {cs}}(x_n)+p_{i,\text {cf}}(x_n)), \end{aligned}$$

(7.34)

where J denotes the maximal number of blocked atrial impulses between successive ventricular activations. This number is chosen so that more than 90% of the conducted atrial impulses are accounted for [47].

When applying the model in [47] to ECG signals, the probability $\varepsilon$ of choosing the slow pathway was simply set to 0.5, whereas the remaining model parameters $\varvec{\pmb \theta }=\begin{bmatrix} \tau _s&\tau _f&\tau _{s,p}&\tau _{f,p}\end{bmatrix}$ were estimated using the ML technique,

$$\begin{aligned} {\hat{\varvec{\pmb \theta }}} = \arg \max _{\varvec{\pmb \theta }} \left( \sum _{n=0}^{N-1} \log p_{c}(x_n | \varvec{\pmb {\theta }}; \hat{\lambda }_a)\right) , \end{aligned}$$

(7.35)

where the mean arrival rate $\lambda _a$ was estimated as described in Sect. 7.6.2.

The ratio of atrial impulses conducted through the slow pathway, defined by

$$\begin{aligned} \alpha =\frac{\displaystyle \sum _{i=1}^{J} \int _0^{\infty }\displaystyle p_{i,\text {cs}}(t)dt}{\displaystyle \int _0^{\infty }{p_{c},(t;I)dt}}, \end{aligned}$$

(7.36)

can be used to quantify the reliability of the parameter estimates in $\hat{\varvec{\pmb {\theta }}}$. A small $\alpha$ indicates that few impulses are conducted through the slow pathway, and, therefore, $\hat{\tau }_s$ and $\hat{\tau }_{s,p}$ are less reliable. Conversely, a large $\alpha$ indicates that few impulses are conducted through the fast pathway, and, therefore, $\hat{\tau }_f$ and $\hat{\tau }_{f,p}$ are less reliable.

The AV node model has been fitted to each nonoverlapping, 30-min segment of 24-h ECG recordings from 60 patients in permanent AF [47]. Based on results from simulated data, a threshold was applied to $\alpha$ in order to judge whether the estimated model parameters were reliable. Figure 7.12 illustrates how the four parameters characterizing the refractory periods change over a 24 h period, using the models in [45, 47]. It is obvious from Fig. 7.12 that the estimates based on the AV node model in [47] is associated with less variation in $\hat{\tau }_{s,p}$ and $\hat{\tau }_{f,p}$ than is the model in [45]. It should be noted that the model in [47] leads to an unequally sampled series of parameter estimates, since several estimates are omitted because the reliability, determined by $\alpha$, is judged to be too low.

Fig. 7.12

Model parameter estimates obtained from a 24-h ECG recording of a patient with permanent AF, using the models in [47] (red line, “o” markers) and [45] (black line, “x” markers). Successive estimates, indicated by markers, are connected with straight lines (see text for further explanations)

7.7 Comparison of AV Models

The AV node models described in this chapter were developed for different purposes, one purpose being to simulate ventricular activation series resembling those observed during AF and to characterize AV nodal function from intracardiac recordings (Sect. 7.5), another purpose being to characterize AV nodal function from the surface ECG (Sect. 7.6). These purposes are reflected in the structure and complexity of the different models. While the early models embrace two to six parameters [33, 34, 38, 43], the more recent ones, primarily used for simulation and electrogram-based characterization are considerably more complex, embracing 12 to 16 parameters [35, 36, 42]. The statistical models for ECG-based characterization consist of four to five model parameters which make them better suited for estimation [45, 47].

The AV node models differ in their respective approach to handling the following properties:

• Atrial activation times

• Refractory period

• Conduction delay

• Ventricular activation

• Concealed conduction

• Dual pathways

The atrial activation times are usually modeled by a homogenous Poisson process, implying that the AA intervals are exponentially distributed [35, 42, 43, 45, 47]. A Gaussian distribution of the AA intervals has also been proposed [38], although the atrial activation times can no longer be treated as a Poisson process. In the very first statistical model, the mean arrival rate $\lambda _a$ of the Poisson process assumed an unphysiological value [43]—a problem which was later solved by relating $\lambda _a$ to the AFR, estimated from the f waves in the ECG [42, 45, 47]. For models where the AV node is characterized using intracardiac information, the atrial activation times are determined by the peaks of the atrial electrogram [33, 34, 36]. Positioning of the electrodes relative to the AV node entrance is particularly important during AF because of the disorganized atrial activity.

From experimental data obtained using a pacing protocol, the effective refractory period is known to be dependent on the paced atrial rate [62]. This rate dependence can be modeled in different ways. For example, the refractory period can depend on the preceding RR interval according to an exponential curve defined by the maximal refractory period $\tau _{\infty }$, cf. (7.18) [43]. Another approach is to assume that the refractory period is linearly dependent on the preceding RR interval, calling for decorrelation of the observed RR interval series before parameter estimation can be performed [45, 47]. Yet another approach is to assume that the refractory period is recovery-dependent, i.e., dependent on the time elapsed since the end of the preceding refractory period according to an exponential curve modeled by three parameters: the minimal refractory period $\tau _{\text {min}}$, the maximal prolongation $\beta$, and the time constant $\gamma _r$ of the exponential refractory curve, cf. (7.16) [35, 42]. In some models, the rate dependence of the refractory period is not explicitly modeled [33, 34, 36, 38].

The conduction delay is an important property of the AV node during normal sinus rhythm, known to be dependent on the paced atrial rate [62]. The AV nodal conduction delay may be incorporated in the refractory period so that its dynamics is not explicitly modeled [43, 45, 47]. Alternatively, the conduction delay can be made dependent on the recovery time, where recovery dependence is modeled by an exponential curve defined by three model parameters: the minimal conduction delay $d_{\text {min}}$, the maximal prolongation $\alpha _{\text {max}}$, and the time constant $\gamma _c$ of the exponential conduction curve, cf. (7.3) [33,34,35, 42]. A similar approach was considered in [36], although the maximal prolongation and the time constant of the conduction curve were assumed to depend on the preceding conduction delay.

In most models, ventricular activation is directly linked to the arrival of one atrial impulse. Each atrial impulse is assumed to result in a ventricular activation, unless it is blocked due to AV nodal refractoriness. However, more than one atrial impulse may be needed to cause a ventricular activation [35, 43]. The AV node may also fire spontaneously.

Concealed conduction is incorporated in the models in different ways. For each blocked impulse, the refractory period can be incremented by a fixed [33] or random time [34]. The refractory period prolongation can depend on the timing of the blocked impulse [38], or on both the timing and the strength of the blocked impulse [35]. Blocked impulses can alter the conduction time of the following impulse, so that a longer AA interval results in a longer conduction delay [36]. Concealed conduction can also be disregarded, assuming a refractory period which is not influenced by blocked impulses [43]. The refractory period prolongation caused by each blocked impulse is not always explicitly modeled, but concealed conduction is modeled by a random, uniformly distributed prolongation of the refractory period [45, 47]. Concealed conduction can also be an intrinsic feature of the chosen model structure [42].

Table 7.1 Comparison of atrioventricular node models with respect to structure and atrial impulse assumptions. The models are listed in chronological order

Table 7.2 Comparison of atrioventricular node models with respect to various electrophysiological properties. The functional dependence of the refractory period is indicated, where $x_{n}$ is the preceding RR interval, and $\varDelta t_{\text {RT},k}$ is the AV nodal recovery time, cf. (7.4)

The earlier models [33,34,35, 38, 43] did not account for dual pathways of the AV node, while the more recent models account for separate conduction time [36, 42] and refractory period [42, 45, 47] of the two pathways.

The model in [43] was fitted to observed RR series using an ad hoc procedure. For some models, no attempts have been made at all to fit the models to observed data [35, 38]. The models proposed for characterization of AV nodal function based on intracardiac recordings were fitted using a grid search to find the minimum error between observed and simulated RR intervals, given the observed AA intervals as input [33, 34, 42]. The model parameters in [36] were assessed by fitting data obtained using a dedicated pacing protocol. For ECG-based characterization of AV nodal function during AF, the mean arrival rate of atrial impulses is estimated from an extracted f wave signal [42, 45, 47]; the remaining model parameters are estimated from an RR interval series using ML estimation [45, 47].

Tables 7.1 and 7.2 provides a comparison of AV node models described in this chapter and their respective properties.