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A mathematical model for the polarization impedance of cardiac pacemaker electrodes

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Abstract

A mathematical model for the linear properties of the polarization impedance of cardiac pacemaker electrodes is presented. This model reproduces frequency and time domain behavior of the electrodes in physiological saline.

The model chosen is a special case of a more general model used to describe dielectric relaxation behavior. Equations are presented for the dependence on frequency of polarization resistance, capacitance and tanδ, and for the potential difference across the interface vs. time in response to a current step. Frequency domain measurements were made with an impedance bridge. Time domain measurements were made with constant current pulses delivered to the electrodes immersed in saline.

The experimental results exhibit the same behavior as predicted from the model, with good agreement between theory and experiment. Features proposed by this model are also exhibited by other metal electrodes.

Sommaire

L'article présente un modèle mathématique des propriétés linéaires de l'impédance de polarisation des électrodes des pacemakers cardiaques. Ce modèle reproduit la réponse en fréquence et le comportement dans le temps des électrodes plongées dans un sérum physiologique.

Le modèle choisi est un cas particulier d'un modèle plus général, utilisé pour la description des phénomènes de relaxation des diélectriques. Les équations présentent les valeurs de la résistance de polarisation, de la capacitance et de tgδ en fonction de la fréquence, et celle de la différence de potentiel à travers l'interface en fonction du temps, en réponse à un échelon de courant. Les mesures de fréquence ont été faites à l'aide d'un pont d'impédances; les mesures dans le temps ont été faites en appliquant aux électrodes immergées dans la solution saline des impulsions de courant.

Les résultats expérimentaux prouvent que le comportement est identique à celui qui était prévu par le modèle, avec une bonne concordance entre la théorie et l'expérience. Les caractéristiques proposées par ce modèle sont également présentées par d'autres électrodes métalliques.

Zusammenfassung

Ein mathematisches Modell der linearen Eigenschaften der Polarisations-impedanz von Herzschrittmacher-Elektroden wird mitgeteilt. Das Modell reproduziert Frequenzund Zeitverhalten der Elektroden in physiologischer Kochsalzlösung.

Das gewählte Modell ist ein Spezialfall eines allgemeineren Modells, welches zur Beschreibung des dielektrischen Relaxationsverhaltens benutzt wird. Gleichungen für die Frequenz des Polarisationswiderstandes, der Kapazitanz und des tan δ, und für die Potentialdifferenz über die Fläche in Abhängigkeit von der Zeit als Antwort auf eine Stromstufe werden angegeben. Die Frequenzmessungen wurden mit einer Impedanzbrücke vorgenommen. Die Zeitabhängigkeitsmessungen wurden mit Impulsen konstanter Stromstärke durchgeführt, welche an die in physiologische Kochsalzlösung getauchten Elektroden abgegeben wurden.

Die Versuchsergebnisse zeigen das von dem Modell erwartete Verhalten mit guter Übereinstimmung zwischen Theorie und Experiment. Die mit diesem Modell gezeigten Eigenschaften sind auch auf andere Metallelektroden übertragbar.

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Abbreviations

A, a, b :

constants

C, C par :

equivalent parallel capacitance

C * :

equivalent parallel complex capacitance

C′:

the real part of the complex capacitanceC *

C″:

the imaginary part of the complex capacitanceC *

C o :

equivalent parallel capacitance measured at low frequencies

C ∞ :

equivalent parallel capacitance measured at high frequencies

C p :

polarization capacitance in Fricke's model

C s :

equivalent series capacitance

C pol :

equivalent series polarization capacitance

D :

dissipation factor

f :

frequency, Hz

f o :

frequency at wichi ωτ=1

G, G par :

equivalent parallel conductance

I o :

amplitude of a current step function

f :

\(\sqrt { - 1} \)

m :

slope of the polarization capacitance vs. frequency on a log-log plot (Fricke's model)

Q :

1/D

R p :

polarization resistance in Fricke's model

R s :

equivalent series resistance

R pol :

equivalent series polarization resistance

s :

Laplace operator

t :

time

tan δ:

ωR pol C pol

V :

potential difference

Y :

admittance

Z pol :

polarization impedance

α:

a parameter describing the width of the distribution of relaxation times

β:

1−α

Г:

the gamma function

ω:

angular frequency=2πf

τ:

the average time constant

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Author notes

  1. Dov Jaron

    Present address: Surgical Research Laboratory, Maimonides Medical Center, 4802 Tenth Avenue, 11219, Brooklyn, New York

Authors and Affiliations

  1. Biomedical Engineering Department, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.

    Dov Jaron, Herman P. Schwan & David B. Geselowitz

Authors
  1. Dov Jaron
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  2. Herman P. Schwan
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  3. David B. Geselowitz
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Additional information

This work was supported in part by U.S. Public Health Service Grants HE-5239, HE-08805, HE-06352, 5R01 HE-01253 and 2T01GM-606 and was based in part on a dissertation submitted by D. Jaron in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Pennsylvania, Philadelphia, Pennsylvania.

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Jaron, D., Schwan, H.P. & Geselowitz, D.B. A mathematical model for the polarization impedance of cardiac pacemaker electrodes. Med. & biol. Engng. 6, 579–594 (1968). https://doi.org/10.1007/BF02474721

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