Statistical Modeling of Atrioventricular Nodal Function

during Atrial Fibrillation

An Update

Valentina D. A. Corino

, Frida Sandberg

, Federico Lombardi

, Luca T. Mainardi

and Leif S

ornmo

Department of Bioengineering, Politecnico di Milano, Milan, Italy

Department of Electrical and Information Technology and Center for Integrative Electrocardiology (CIEL),

Lund University, Lund, Sweden

Cardiologia, Dipartimento di Medicina, Chirurgia e Odontoiatria, Ospedale San Paolo, University of Milan, Milan, Italy

Keywords:

Atrial Fibrillation, Atrioventricular Node, Statistical Modeling, Maximum Likelihood Estimation.

Abstract:

This paper introduces a number of advancements of our recently proposed model of atrioventricular (AV)

node function during atrial ﬁbrillation (AF). The model is deﬁned by parameters characterizing the arrival

rate of atrial impulses, the probability of an impulse choosing either one of the two AV nodal pathways, the

refractory periods of these pathways, and their prolongation. In the updated model, the characterization of AV

nodal pathways is made more detailed and the number of pathways is determined by the Bayesian information

criterion. The performance is evaluated on ECG data acquired from twenty-ﬁve AF patients during rest and

head-up tilt test. The results show that the reﬁned AV node model provides signiﬁcantly better ﬁt than did the

original model. During tilt, the AF frequency increased (6.25 ± 0.58 Hz vs. 6.32 ± 0.61 Hz, p < 0.05, rest vs.

tilt) and the prolongation of the refractory periods decreased for both pathways (slow pathway: 0.23 ± 0.20 s

vs. 0.11 ± 0.10 s, p < 0.001, rest vs. tilt; fast pathway: 0.24 ± 0.31 s vs. 0.16 ±0.19 s, p < 0.05, rest vs. tilt).

These results show that AV node characteristics can be assessed noninvasively for the purpose of quantifying

changes induced by autonomic stimulation.

1 INTRODUCTION

The atrioventricular (AV) node is subjected to the

impact of atrial impulses during atrial ﬁbrillation

(AF) which leads to summation and/or cancellation

of wavefronts in the AV node and, accordingly, a high

level of disorganization of the penetrating impulses.

As a result, the ventricular rhythm becomes much

more irregular than during normal sinus rhythm. Dif-

ferent properties such as intrinsic refractoriness of

the AV node and concealed conduction determine

the characteristics of the ventricular response (Fuster

et al., 2006), but they are not routinely evaluated in

clinical practice mainly because of the lack of nonin-

vasive methodology.

Statistical model-based analysis of essential AV

nodal characteristics constitutes a powerful method

to assess AV node properties during AF. A statisti-

cal model suitable for parameter estimation was at an

early stage described in (Cohen et al., 1983): the AV

node was treated as a lumped structure whose behav-

ior represented the temporal and spatial summation

of the cellular electrical activity. The atrial impulses

were assumed to arrive randomly to the AV node ac-

cording to a Poisson process. When the AV node

was not refractory, its transmembrane potential was

assumed to increase with the contribution of each ar-

riving atrial impulse as well as to increase sponta-

neously. When the transmembrane potential reached

a certain threshold, a new action potential initiated a

ventricular beat. Despite the fact that the model was

statistical in nature, the model parameters were de-

termined from the RR series using an ad hoc proce-

dure. A serious limitation of that procedure was that

the resulting parameter estimates could assume non-

physiological values. An extension of this model was

proposed (Lian et al., 2006), however, the proposed

model was suitable for simulation purposes only.

Other AV node models have been proposed, mainly

based on the analysis of atrial electrograms recorded

during electrophysiological studies (Jørgensen et al.,

2002; Mangin et al., 2005); None of those models

D. A. Corino V., Sandberg F., Lombardi F., T. Mainardi L. and Sörnmo L..

Statistical Modeling of Atrioventricular Nodal Function during Atrial Fibrillation - An Update.

DOI: 10.5220/0004194100250029

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2013), pages 25-29

ISBN: 978-989-8565-36-5

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

were accompanied by a statistical estimation proce-

dure.

In a recent paper, we proposed a statistical model

of the AV nodal function during AF w hich lends itself

to ECG-based parameter estimation (Corino et al.,

2011). The model is deﬁned by a small set of pa-

rameters which characterizes the arrival rate of atrial

impulses, the probability of an impulse choosing ei-

ther one of the dual AV nodal pathways, the refractory

periods of the pathways, and the prolongation of re-

fractory periods. The parameters were estimated from

the RR series using maximum likelihood (ML) esti-

mation, except for the shorter refractory period which

was estimated from the Poincar

e plot of successive

RR intervals, and the mean arrival rate of atrial im-

pulses which was estimated by the AF frequency de-

rived from the f-waves of the ECG (Sandberg et al.,

2008).

Subsequent application of our AV node model

suggests that certain model properties should be ex-

tended and the estimation performance improved with

respect to robustness. Therefore, this paper introduces

a more detailed characterization of the dual pathways.

In particular, the effect of tilting on the refractory pe-

riods of the AV node was assessed by means of the

model parameters.

2 METHODS

2.1 Existing AV Node Model

In the present model, the AV node is treated as a

lumped structure which accounts for concealed con-

duction, relative refractoriness, and dual AV nodal

pathways. Atrial impulses are assumed to arrive to the

AV node according to a Poisson process with mean ar-

rival rate λ. Each arriving impulse is suprathreshold,

i.e., the impulse results in ventricular activation un-

less blocked by a refractory AV node. The probability

of an atrial impulse passing through the AV node de-

pends on the time elapsed since the previous ventric-

ular activation t. The length of the refractory period is

deﬁned by a deterministic part τ and a stochastic part

. The latter part models prolongation due to con-

cealed conduction and/or relative refractoriness, and

is assumed to be uniformly distributed in the inter-

val [0; τ

]. Hence, all atrial impulses arriving to the

AV node before the end of the refractory period τ are

blocked. Then follows an interval [τ, τ + τ

] with lin-

early increasing likelihood of penetration into the AV

node. Finally, no impulses can be blocked if they ar-

rive after the end of the maximally prolonged refrac-

tory period τ + τ

. The mathematical characterization

of refractoriness of the i:th pathway (i = 1,2) is thus

deﬁned by the positive-valued function β

(t),

(t) =











0, 0 < t < τ

t − τ

, τ

≤ t < τ

+ τ

1, t ≥ τ

+ τ

(1)

where t denotes the time elapsed since the preceding

ventricular activation.

The probability of an atrial impulse to take the

pathway with the shorter refractory period τ

is equal

to α, and accordingly the other pathway is taken with

probability (1 − α). For this model, the time intervals

between consecutive ventricular activations, i.e.,

corresponding to the RR intervals, are independent.

It can be shown that the joint PDF is given by (Corino

et al., 2011)

, x

, . . . , x

) =

∏

m=1

(αp

x,1

) + (1 − α)p

x,2

)),

(2)

where M is the total number of intervals, and

x,i

), i = 1, 2, is given by

x,i

(x) =











0, x < τ

λy

exp

{

−λy

2τ

}

, τ

≤ x < τ

+ τ

λexp

{

−λτ

− λ(y

− τ

)

}

, x ≥ τ

+ τ

(3)

where y

= x − τ

2.2 Model Parameter Estimation

2.2.1 Estimation of τ

min

Since the property of statistical independence is not

fully valid for observed RR intervals, a simple func-

tional dependence of the refractory periods related to

the previous RR interval is explored. The interdepen-

dence of consecutive RR intervals can be reduced by

preprocessing the original RR interval series, denoted

′

, with the linear transformation,

= x

′

− ˆs

′

m−1

. (4)

where ˆs

, together with an estimate of the minimal

refractory period τ

min

, is found from the line that de-

ﬁnes the lower envelope of the Poincar

e plot using the

Hough transform, see (Corino et al., 2012) for details

on how to determine this line. The interval series x

that results from (4) constitutes the data used for ML

estimation.

BIOSIGNALS2013-InternationalConferenceonBio-inspiredSystemsandSignalProcessing

An estimate of the refractory period τ

m,1

of the

slow pathway and the m:th activation may be obtained

from the linear relationship

m,1

= τ

min

+ s

′

m−1

. (5)

Assuming that s

applies to both τ

m,1

and τ

m,2

, the

problem of estimating τ

m,2

is identical to the problem

of estimating a ﬁxed duration ∆τ since

m,2

= ∆τ +τ

m,1

. (6)

2.2.2 Maximum Likelihood Estimation of Dual

Pathway Parameters

The model parameters related to the dual AV nodal

pathways and the refractory period prolongation are

estimated by maximizing the log-likelihood function

Λ(θ) with respect to θ, i.e.,

θ = argmax

Λ(θ) (7)

= arg max

log p

, x

, . . . , x

|θ;

λ,

min

where θ = [α ∆τ τ

p,i

]

. Since no closed-form so-

lution could be found for

θ, combined with the fact

that the gradient is discontinuous, simulated anneal-

ing was employed for maximization. The optimiza-

tion algorithm was initiated with 10 different, ran-

domly chosen values for each estimation. The opti-

mal

θ was then chosen as that value for which Λ(

θ)

is maximum, almost invariably being the dominant

value.

2.3 AV Node Model Update

2.3.1 Dual Pathways

To date, no evidence has been presented which sug-

gests that refractory period prolongation is identical

for the dual AV nodal pathways. Rather, it has been

found that the input regions of the atrial impulses play

an important role in determining the properties of AV

nodal conduction and refractoriness (Mazgalev et al.,

1984). Inspired by the work in (Climent et al., 2011),

our original model is here extended to account for

pathway-dependent prolongation of the refractory pe-

riod. Hence, prolongation is described by the two pa-

rameters τ

p,1

and τ

p,2

, implying that the PDF p(x

) in

(3) is modiﬁed so that τ

p,i

replaces τ

2.3.2 Estimation of λ

In the original AV node model, the atrial impulses

were assumed to arrive to the AV node according to a

Poisson process at a rate λ. An estimate of λ was ob-

tained by ﬁrst estimating the dominant AF frequency

from the ECG, independently of the AV node pa-

rameters, and then assuming that

λ = λ

. (8)

The details of the procedure for estimating λ

are de-

scribed in (Corino et al., 2011). A disadvantage with

such an approach is that it does not account for the

fact that there is a minimum time interval δ between

successive impulses arriving to the AV node. Such an

interval was included in the simulation model in (Lian

et al., 2006), where it was assumed that the atria de-

polarize again after δ = 50 ms.

Evidently, the Poisson model does not impose

a minimum time interval δ between successive im-

pulses arriving to the AV node, but they can ar-

rive immediately upon each other. Therefore, the

use of (8) produces an underestimated value of λ.

For Poisson-distributed impulses not arriving closer

to each other than a minimum interval δ, the arrival

rate λ

should be modiﬁed according to (Larsen and

Kostinski, 2009)

λ =

1 − δλ

, (9)

2.3.3 Selection of Single or Dual Pathways

The selection of single or dual pathway model is de-

termined by the Bayesian information criterion (BIC),

deﬁned by

BIC

(k) = −2Λ(

) + n

log(M), k = 1, 2, (10)

where the parameter estimate is either equal to

p,1

for a single pathway or

= [α ∆τ τ

p,1

p,2

]

for dual pathways. The number of parameters n

thus equal to 1 or 4. The number of pathways is

given by that index k which produces the lowest value

of C

BIC

(k). The original use of simulated anneal-

ing for maximization of the log-likelihood function is

here replaced with a genetic algorithm because it was

found to offer much faster maximization.

3 DATA

We analyzed 25 consecutive patients with persistent

AF (67 ± 7 years, 16 females) who underwent elec-

trical cardioversion, according to the international

guidelines, at the Cardiology department of San Paolo

Hospital, Milan, Italy. Recordings were acquired at

rest and during a passive orthostatic stimulus (75

◦

tilt-

ing). One patient was excluded from analysis due to

poor ECG quality preventing the estimation of AF fre-

quency. Hence, the results presented below are based

on 24 patients.

StatisticalModelingofAtrioventricularNodalFunctionduringAtrialFibrillation-AnUpdate

The ECG was recorded at rest for 10 min and,

when applicable, followed by tilting, using three or-

thogonal leads and a sampling rate of 1 kHz. All

recordings were performed in the morning in a quiet

environment following 15 min of adaptation. The

study was approved by the Ethics Committee, and all

patients gave their written informed consent to partic-

ipate.

4 RESULTS

To assess if the present model could provide a better

ﬁt of the data, an index taking the underlying PDF

into account is preferable. However, since the under-

lying PDF is unknown for ECG-derived RR intervals,

an empirical PDF, denoted ˜p

(x), was determined by

wavelet-based density estimation (Corino et al., 2011;

Ogden, 1997). The capability to model different RR

series was evaluated in terms of a percentage measure

of ﬁt U, deﬁned by

U = 100 ·

(

1 −

∫



(x|

;

λ,

min

) − ˜p

(x)



)

(11)

where the upper integration limit reﬂects the fact that

very few RR intervals are longer than 2 s during AF.

The reason for using wavelet-based density estima-

tion rather than comparing p

(x|

;

λ,

min

) to the RR

interval histogram is the poor statistical properties of

the histogram (Ogden, 1997).

Figure 1 indicates that the model ﬁt, as described

by U, was signiﬁcantly better with the present model,

both for rest and tilt data. Figure 2 shows an exam-

ple of histogram of the transformed RR intervals and

the estimated model PDF from the same patient using

the original and the present model. The latter model

ﬁts the histogram much better than does the original

one, U increasing from 84% to 89%. It can be noted

that the histograms of transformed RR intervals are

not exactly the same because the transformation de-

pends on

which is different in the two models. In

the following, the presented results are obtained using

the present model.

To assess whether the model parameters can cap-

ture changes occurring due to increased sympathetic

tone, such as during a tilt test, parameters during rest

were compared to those during tilt. Table 1 com-

pares the model parameter estimates obtained at rest

and during tilt, with signiﬁcant changes due to sym-

pathetic activation in

p,1

and

p,2

. The AF frequency

was found to increase signiﬁcantly during tilt. The

probability of an atrial impulse of choosing either

pathway is almost equal during rest and tilt (α = 0.5),

100

original new

(a)

U %

100

original new

(b)

U %

Figure 1: Boxplots of U for data during rest and tilt, com-

paring the original and the updated AV node model.

* p<0.05, ** p<0.01.

0 0.5 1 1.5 2

transformed RR interval (s)

# transformed RR intervals

(a)

0 0.5 1 1.5 2

transformed RR interval (s)

# transformed RR intervals

(b)

Figure 2: Histogram of transformed RR intervals and the es-

timated model PDF from the same patient during rest using

(a) the original and (b) the new AV node model.

although α spans the range from 0.05 to 1, thus mak-

ing the involvement of the pathway with slower re-

fractory period (α < 0.5) in about half of all record-

ings. The refractory periods of both pathways remain

almost unchanged during tilt, whereas their prolonga-

tion, due to concealed conduction, relative refractori-

ness etc., are signiﬁcantly shortened during tilt.

Both the mean and standard deviation of RR in-

tervals are signiﬁcantly shortened during tilt due to

sympathetic activation. The mean RR interval length

was 743 ± 146 ms vs. 680 ± 125 ms (rest vs. tilt,

p < 0.0001), and the related standard deviation was

156 ± 51 vs. 137 ± 29 ms (rest vs. tilt, p < 0.0001).

BIOSIGNALS2013-InternationalConferenceonBio-inspiredSystemsandSignalProcessing

Table 1: Comparison of rest and tilt parameters. *p<0.05,

**p<0.001.

Rest Tilt

α 0.52 ± 0.24 0.51 ± 0.30

min

(s) 0.37 ± 0.09 0.38 ± 0.10

min

(s) 0.46 ± 0.12 0.47 ± 0.09

p,1

(s) 0.23 ± 0.20 0.11 ± 0.10 **

p,2

(s) 0.24 ± 0.31 0.16 ± 0.19 *

λ (Hz) 6.25 ± 0.58 6.32 ± 0.61 *

5 CONCLUSIONS

We have proposed an updated AV node model in

which i) the characterization of the AV nodal path-

ways is made more detailed using two different pa-

rameters representing the prolongation of related re-

fractory periods, ii) the number of pathways is deter-

mined by the BIC, and iii) the arrival rate is corrected

to take into account there is a minimum time inter-

val between successive impulses arriving to the AV

node. The updated model leads to better estimation

of the PDF when two peaks with different width are

to be modeled, and also the most parsimonious model

is selected (choosing between single or dual pathway

model). Considering physiological aspects, our re-

sults indicate that tilting is associated with signiﬁcant

changes in AV conduction that are well-described by

the model and reﬂected by shortening of both τ

p,1

and

p,2

during adrenergic activation. Thus, the present

AV node model is adequate for studying and describ-

ing the functional characteristics of AV conduction in

AF patients.

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StatisticalModelingofAtrioventricularNodalFunctionduringAtrialFibrillation-AnUpdate