Theoretical Study of Modulated Parasystole with
Feedback as a Possible Mechanism for
Cyclic Bursts of Ventricular Premature Contractions

Noriaki Ikeda^a, Kan Takayanagi^b, Akihiro Takeuchi^a, Hideo Miyahara^a,
Makoto Koda^a, Atsushi Hamada^a, Hiroshi Goto^a

^aDepartment of Medical Informatics, Kitasato University, Kanagawa, Japan
^bDepartment of Cardiology, Dokkyo University Koshigaya Hospital, Saitama, Japan

Correspondence: Noriaki Ikeda, Medical Informatics, School of Allied Health Sciences, Kitasato University, Kitasato 1-15-1, Sagamihara, Kanagawa, 228-8555 Japan. E-mail: ikeda@kitasato-u.ac.jp, phone +81 42 778 9652, fax +81 42 778 7264

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1.  Introduction

Modulated parasystole model is an idea of mechanism of ventricular premature contraction (VPC). It stated that an ectopic pacemaker received an electrotonic influence from the excitation of the ventricle and generated stable patterns of VPC by the synchronization to the sinus rhythm. Mathematically, it is understood as a phenomenon of a nonlinear oscillator synchronizing to external stimuli. This idea was formulated as a mathematical equation that used the phase response curve (PRC) of an oscillating system, and detailed analysis was performed about the mathematical structure of the stable solutions [Ikeda et al., 1983].

The purpose of this study is to revise the model and derive the equation of modulated parasystole with feedback and to theoretically show that the two different solutions, each corresponding to the normal ECG and bigeminy, are existing together, which may be a mechanism of cyclic bursts of VPCs reported by Takayanagi et al. [1999].

2.  Methods

2.1. Equation of the Modulated Parasystole Model with Feedback

Assuming that the variable t_n is the n-th phase of the ventricular excitation induced by the sinus pacemaker in the oscillation of the ectopic pacemaker, the following equations are obtained between the successive phases, t_n and t_n+1 [Ikeda et al., 2002]. The functions PRCe (t) and PRCs (t) represent the PRC of the ectopic and the sinus pacemakers, respectively.

tn+1 =

tn + Ts - PRCe(tn )                                         (0 ≦ tn<Te)                                                tn - Te + Ts                                                       (Te ≦ tn≦ Te+δ)                                   tn – Te + Ts + PRCs(Te+Ts+δ- tn )      (Te+δ ≦ tn<Te+(1-γ)Ts)                            tn – Te                                                       (Te+(1-γ)Ts ≦ tn<Te+Ts)

(1N) (1F) (1C) (1S)

where, the parameters are, Ts : the intrinsic period of the sinus pacemaker, Te : the intrinsic period of the ectopic pacemaker, γTs: refractory period due to the sinus R wave, and δ: feedback delay between the sinus node and the ventricle. Equations (1N), (1F), (1C) and (1S) correspond to the normal, fusion, ectopic, and silent R waves, respectively.

2.2. Region of Coexistence of Two Different Solutions

Existing region of the stable solution was obtained theoretically from Eq. 1, and two solutions,  normal ECG and bigeminy were plotted on Ts/Te – b plane (Figure 1), where the parameter b was the amplitude of PRCe. Two solutions existed separately for the model without feedback (PRCs = 0, Figure 1A), while they had a region of coexistence for the model with feedback (Figure 1B).

Figure 4. Region of two solutions; normal ECG (hatch) and bigeminy (shade). A: the model without feedback  (PRCs = 0) and B: with feedback in which two regions have an overlapping area.

3.  Results

Two examples of the simulation results were shown in Figure 2, the flat type (A) and the dome type (B), which were recognized by Takayanagi [1999] from the clinical cases of Holter ECG recordings. Clinical ECGs that corresponded to these patterns were given in Figure 3. The parameter Ts was given a fluctuation by uniform random values (±8% in amplitude) around Ts = 1/2 Te.

Figure 2. Simulation results of the model with fluctuated Ts. Traces show ECG and RR intervals.	Figure 3. Typical clinical cases of cyclic bursts of VPC. The plot shows RR interval vs beat number.

4. Discussion

Model of modulated parasystole with feedback from the ectopic to the sinus pacemaker was investigated. Coexistence of different solutions in the model could be a mechanism for clinically reported cases of cyclic bursts of VPCs, although the feedback pathway was not precisely identified.

References

Ikeda N, Yoshizawa S, Sato T: Difference equation model of a ventricular parasystole as an interaction between cardiac pacemakers based on the phase response curve, J theor Biol, 103: 439-465, 1983.

Takayanagi T, Kamishirado H, Iwasaki Y, Fujito T, Sakai Y, Inoue T, Hayashi T, Morooka S: Cyclic bursts of ventricular premature contractions of more than one minute intervals, Jpn Heart J, 40:135-144, 1999.

Ikeda N, Takayanagi K, Takeuchi A, Miyahara H, Koda M: Model-based interpretation of arrhythmia – cyclic bursts of ventricular premature contractions. Proc BSI 2002, 91-94, 2002.