Propagation Versus Delayed Activation During Field Stimulation of Cardiac Muscle WANDA KRASSOWSKA,*** CANDIDO CABO,*** STEPHEN B. KNISLEY,*** and RAYMOND E. I D E K E R * * t t t From the * Department of Biomedical Engineering and the ** Duke-North Carolina NSF/ERC, Duke University; and the t Departments of Pathology and t Medicine, Duke University Medical Center, Durham, North Carolina
KRASSOWSKA, W., ET AL.: Propagation Versus Delayed Activation During Field Stimulation of Cardiac Muscle. This modeling study seeks to explain the experimentally detected delay between the application of an electric field and the recorded response of the transmembrane potential. In this experiment, conditions were deliberately set so that the field should excite all cells at once and so that no delay should be caused hy a propagating wave front. The explanation 0/ the observed delay may he in the intrinsic properties 0/ the membrane, To test this hypothesis, the strength latency curves were determined for three cases: (1) for a membrane patch modei, in which the membrane is uniformly polarized and its intrinsic properties can be studied; (2} for the cardiac strand directly excited by the electric field; and (3) for the cordiac strand excited by a propagating wave front. The models of the membrane patch and the directly excited strand yieJd excitation delays that are comparable to those observed experimentaiiy in magnitude and in the overall shape 0/ the strength latency curves. The delays resulting from propagation are, in generai, dependent on the position along the strand, although for some positions the strength latency curves for propagation are similar to those obtained from the directly activated strand and from the patch model. Therefore, the delay in excitation does not necessariiy impiy the presence 0/ propagating wave fronts and can be attributed to intrinsic membrane kinetics. (PACE, Vol. 15, February 1992) modeling, discrete cardiac strand, propagation, excitation, field stimulation
Introduction Measurements of transmembrane potential are an important source of information regarding the physiology of cardiac muscle. The traditional approach uses a microelectrode technique that, since its introduction in 1939^ and 1940,^ has significantly advanced cardiac electrophysiology. The limitation of this technique comes from the
Supported in part by the National Institute of Health research grants HL-2842g, HL-33637, HL-44066, HL-42760, and HL40092; by the American Heart Association, North Carolina Affiliate grant NC-91-G-14; and by the National Science Foundation Engineering Research Center grant CDR-86222G1. Address for reprints: Wanda Krassowska, Department of Biomedical Engineering, Duke University, Durham. NC 27706. Fax: (919) 660-5405. Received [une 27, 1991; revision September 9, 1991; accepted October 16, 1991.
PACE, Vol. 15
fact that few impalements can be made and maintained during a study. Thus, the transmembrane potential is known only at a few points of the preparation, and the state of the tissue elsewhere must be guessed. This restriction has influenced researchers, experimental and theoretical, to think about transmembrane potential primarily as a function of time and not as a function of space. In particular, any delay of activation observed in a time course of the action potential is customarily associated with propagation and is used to compute conduction velocity. However, this interpretation may not always be valid. The conditions of the in vitro experiments conducted in our laboratory (see Appendix) were deliberately set in such a way that direct excitation of the entire preparation was expected. Yet the activation, measured from the beginning of the 2msec stimulus, occurred with a delay that de-
February 1992
197
KRASSOWSKA, ET AL.
potential gradient (mV/cm) Figure 1. The delay of excitation as a function of the
stimulus strength measured in a rabbit ventricular muscJe. The delay was measured from the onsel of the stimulus to the maximum dVmldt The threshold for this preparation was 550 mV/cm.
creased with the increasing strength of the stimulus. For a representative preparation, the delay varied from 9.8 msec for the weakest stimulus of 0.5 V/cm, to 3.1 msec for the strongest one, 3.5 V/ cm (Fig. 1). One possible explanation is that not all of the preparation is directly excited by the weaker stimuli and that the site of impalement is activated by a wave front propagating from the border of the directly excited region. As the stimulus strength increases and the border of the directly excited tissue moves towards the microelectrode, the wave front travels progressively shorter distances, resulting in smaller latency. When the stimulus becomes strong enough to excite the impalement site, the latency no longer changes appreciably. If the propagation velocity is approximately 0.5 m/sec, then during the latency of 9.8 msec the impulse can travel 4.9 mm, i.e., enough fo cover most of the preparation under study. Thus, the above explanation has a sound hasis and is supported by available data. However, an alternative explanation exists for the reported experimental result, which attributes the observed latency to the membrane kinetics. If has heen ohserved in a nerve^ that the delay between the application of the stimulus and the actual activation of the membrane depends on the
198
stimulus strength. In addition, a recent modeling study of a cardiac fiber^ reports that a stimulus just above the threshold value activates fhe membrane after a latency as long as 11.7 msec. In view of these results, the latency observed in in vitro experiments may result either from prcpagafion, from delayed membrane activation, or from a combination of both. This study uses computer simulation to explore fhe pos. ibility that fhe latency results from the delayed membrane activation. First, the membrane patch models are studied to determine the range of excitation delays as a function of the stimulus strength. The same problem is then studied for a one-dimensional discrete cardiac strand, in which cells are directly excited by field stimulation. Assuming the discrete structure of cardiac muscle and the presence of extracellular rather than infracellular stimulus bring the model closer fo fhe experimental conditions; moreover, only under these two assumptions can the excitation of the entire strand with so-called field stimulation occur.^ The strength latency curves from the memhrane patch and fhe directly excited fiher are then compared fo those determined for fhe propagafing wave front. This comparison allows determination of the similarities and differences between excitation delays observed during direct excitation and during propagation and helps distinguish between them.
Methods Membrane Patch Models The membrane patch models allow invesfigating the response of a uniformly polarized membrane to the stimulating current. Although such models are remote from usual experimental conditions in which the transmembrane potential varies considerably in space, they are useful in separating the response of fhe membrane itself from the influence of factors related to the geometry of.the preparation. As this study focuses on the excitation of fully repolarized tissue, it uses primarily the membrane description developed by Ebihara and lohnson^ (EJ) which contains only sodium and leakage currents. The parameters of fhe membrane used in this study (Table I) are those modified by Spach,^
February 1992
PAGE, Vol. 15
PROPAGATION VERSUS DELAYED ACTIVATION
Table I. Membrane Patch Parameters, Ebihara-Johnson Model Membrane capacitance Maximum sodium conductivity Leakage conductivity Sodium equilibrium potential Leakage equilibrium potential
Om
G, VNS
V,
since they result in acfion pofenfials more closely resembling fhose of ventricular muscle fhan fhe paramefers of the original EJ model. The currents in a patch are governed by a differential equation dt
^s
(1)
.lion
where C^ is a membrane capacitance, V^ is transmembrane pofenfial, I^ is impressed membrane current density, and the total ionic currenf density is
+
- V,).
(2)
In equation 2, GN^ and Gi are sodium and leakage conducfivifies, m and h are gafing variables of sodium channel, and V^a and Vj are sodium and leakage equilibrium potentials. The changes of gating variables m and h are determined by the ionic model.'* Equation 1 is discretized, solved numerically using fhe Euler mefhod,^ and implemented in C on a SUN 4/110 workstation [Sun Corp., Ichinomiya, Japan). In addition to the EJ model, fhe acfivation delays resulting from other membrane models were also examined. Specifically, fhe study included Beeler-Reufer (BR). Drouhard-Roberge (DR), and DiErancesco-Noble (DN) membrane descriptions.''"" The first two models, as well as the EJ model, were adapted from the referenced Hterature, and the Euler method solufions were implemented in C on a SUN 4/110 workstation. The DN model was based on the program Hearf^^ {Oxsoft Ltd., Oxford, England) and implemented on the SUN workstation in Model of Cardiac Strands To study fhe excifafion process under conditions closer fo those of fhe in vifro experiments
PACE, Vol. 15
1 35 20
mS/cm^ mS/cm^
33.4
mV mV
-80
Weidmann^" Spach' Lieberman^^ Spach' Spach^
conducted in our laboratory, fhe second parf of fhis sfudy was based on a model of a one-dimensional cardiac strand. Studying fhe direct excifafion of fhe entire strand by an external electric field requires that fhe model accounf for fhe exisfence of high resistance junctions befween cells and allow the application of fhe sfimulafing currenf to fhe extracellular space. Only under these two conditions will the transmembrane potential contain a periodic component of sufficient magnitude to directly activate the entire strand.^''' The model represents a 6-mm strand of cardiac muscle composed of 60 cells connected by junctional regions and surrounded by a limited volume of extracellular fluid. According to recent resulfs,^^"'^ juncfions are modeled by resistance and capacitance connected in parallel. The potentials in fhe intracellular and extracellular space,
KRASSOWSKA, W., ET AL.: Propagation Versus Delayed Activation During Field Stimulation of Cardiac Muscle. This modeling study seeks to explain the experimentally detected delay between the application of an electric field and the recorded response of the transmembrane potential. In this experiment, conditions were deliberately set so that the field should excite all cells at once and so that no delay should be caused hy a propagating wave front. The explanation 0/ the observed delay may he in the intrinsic properties 0/ the membrane, To test this hypothesis, the strength latency curves were determined for three cases: (1) for a membrane patch modei, in which the membrane is uniformly polarized and its intrinsic properties can be studied; (2} for the cardiac strand directly excited by the electric field; and (3) for the cordiac strand excited by a propagating wave front. The models of the membrane patch and the directly excited strand yieJd excitation delays that are comparable to those observed experimentaiiy in magnitude and in the overall shape 0/ the strength latency curves. The delays resulting from propagation are, in generai, dependent on the position along the strand, although for some positions the strength latency curves for propagation are similar to those obtained from the directly activated strand and from the patch model. Therefore, the delay in excitation does not necessariiy impiy the presence 0/ propagating wave fronts and can be attributed to intrinsic membrane kinetics. (PACE, Vol. 15, February 1992) modeling, discrete cardiac strand, propagation, excitation, field stimulation
Introduction Measurements of transmembrane potential are an important source of information regarding the physiology of cardiac muscle. The traditional approach uses a microelectrode technique that, since its introduction in 1939^ and 1940,^ has significantly advanced cardiac electrophysiology. The limitation of this technique comes from the
Supported in part by the National Institute of Health research grants HL-2842g, HL-33637, HL-44066, HL-42760, and HL40092; by the American Heart Association, North Carolina Affiliate grant NC-91-G-14; and by the National Science Foundation Engineering Research Center grant CDR-86222G1. Address for reprints: Wanda Krassowska, Department of Biomedical Engineering, Duke University, Durham. NC 27706. Fax: (919) 660-5405. Received [une 27, 1991; revision September 9, 1991; accepted October 16, 1991.
PACE, Vol. 15
fact that few impalements can be made and maintained during a study. Thus, the transmembrane potential is known only at a few points of the preparation, and the state of the tissue elsewhere must be guessed. This restriction has influenced researchers, experimental and theoretical, to think about transmembrane potential primarily as a function of time and not as a function of space. In particular, any delay of activation observed in a time course of the action potential is customarily associated with propagation and is used to compute conduction velocity. However, this interpretation may not always be valid. The conditions of the in vitro experiments conducted in our laboratory (see Appendix) were deliberately set in such a way that direct excitation of the entire preparation was expected. Yet the activation, measured from the beginning of the 2msec stimulus, occurred with a delay that de-
February 1992
197
KRASSOWSKA, ET AL.
potential gradient (mV/cm) Figure 1. The delay of excitation as a function of the
stimulus strength measured in a rabbit ventricular muscJe. The delay was measured from the onsel of the stimulus to the maximum dVmldt The threshold for this preparation was 550 mV/cm.
creased with the increasing strength of the stimulus. For a representative preparation, the delay varied from 9.8 msec for the weakest stimulus of 0.5 V/cm, to 3.1 msec for the strongest one, 3.5 V/ cm (Fig. 1). One possible explanation is that not all of the preparation is directly excited by the weaker stimuli and that the site of impalement is activated by a wave front propagating from the border of the directly excited region. As the stimulus strength increases and the border of the directly excited tissue moves towards the microelectrode, the wave front travels progressively shorter distances, resulting in smaller latency. When the stimulus becomes strong enough to excite the impalement site, the latency no longer changes appreciably. If the propagation velocity is approximately 0.5 m/sec, then during the latency of 9.8 msec the impulse can travel 4.9 mm, i.e., enough fo cover most of the preparation under study. Thus, the above explanation has a sound hasis and is supported by available data. However, an alternative explanation exists for the reported experimental result, which attributes the observed latency to the membrane kinetics. If has heen ohserved in a nerve^ that the delay between the application of the stimulus and the actual activation of the membrane depends on the
198
stimulus strength. In addition, a recent modeling study of a cardiac fiber^ reports that a stimulus just above the threshold value activates fhe membrane after a latency as long as 11.7 msec. In view of these results, the latency observed in in vitro experiments may result either from prcpagafion, from delayed membrane activation, or from a combination of both. This study uses computer simulation to explore fhe pos. ibility that fhe latency results from the delayed membrane activation. First, the membrane patch models are studied to determine the range of excitation delays as a function of the stimulus strength. The same problem is then studied for a one-dimensional discrete cardiac strand, in which cells are directly excited by field stimulation. Assuming the discrete structure of cardiac muscle and the presence of extracellular rather than infracellular stimulus bring the model closer fo fhe experimental conditions; moreover, only under these two assumptions can the excitation of the entire strand with so-called field stimulation occur.^ The strength latency curves from the memhrane patch and fhe directly excited fiher are then compared fo those determined for fhe propagafing wave front. This comparison allows determination of the similarities and differences between excitation delays observed during direct excitation and during propagation and helps distinguish between them.
Methods Membrane Patch Models The membrane patch models allow invesfigating the response of a uniformly polarized membrane to the stimulating current. Although such models are remote from usual experimental conditions in which the transmembrane potential varies considerably in space, they are useful in separating the response of fhe membrane itself from the influence of factors related to the geometry of.the preparation. As this study focuses on the excitation of fully repolarized tissue, it uses primarily the membrane description developed by Ebihara and lohnson^ (EJ) which contains only sodium and leakage currents. The parameters of fhe membrane used in this study (Table I) are those modified by Spach,^
February 1992
PAGE, Vol. 15
PROPAGATION VERSUS DELAYED ACTIVATION
Table I. Membrane Patch Parameters, Ebihara-Johnson Model Membrane capacitance Maximum sodium conductivity Leakage conductivity Sodium equilibrium potential Leakage equilibrium potential
Om
G, VNS
V,
since they result in acfion pofenfials more closely resembling fhose of ventricular muscle fhan fhe paramefers of the original EJ model. The currents in a patch are governed by a differential equation dt
^s
(1)
.lion
where C^ is a membrane capacitance, V^ is transmembrane pofenfial, I^ is impressed membrane current density, and the total ionic currenf density is
+
- V,).
(2)
In equation 2, GN^ and Gi are sodium and leakage conducfivifies, m and h are gafing variables of sodium channel, and V^a and Vj are sodium and leakage equilibrium potentials. The changes of gating variables m and h are determined by the ionic model.'* Equation 1 is discretized, solved numerically using fhe Euler mefhod,^ and implemented in C on a SUN 4/110 workstation [Sun Corp., Ichinomiya, Japan). In addition to the EJ model, fhe acfivation delays resulting from other membrane models were also examined. Specifically, fhe study included Beeler-Reufer (BR). Drouhard-Roberge (DR), and DiErancesco-Noble (DN) membrane descriptions.''"" The first two models, as well as the EJ model, were adapted from the referenced Hterature, and the Euler method solufions were implemented in C on a SUN 4/110 workstation. The DN model was based on the program Hearf^^ {Oxsoft Ltd., Oxford, England) and implemented on the SUN workstation in Model of Cardiac Strands To study fhe excifafion process under conditions closer fo those of fhe in vifro experiments
PACE, Vol. 15
1 35 20
mS/cm^ mS/cm^
33.4
mV mV
-80
Weidmann^" Spach' Lieberman^^ Spach' Spach^
conducted in our laboratory, fhe second parf of fhis sfudy was based on a model of a one-dimensional cardiac strand. Studying fhe direct excifafion of fhe entire strand by an external electric field requires that fhe model accounf for fhe exisfence of high resistance junctions befween cells and allow the application of fhe sfimulafing currenf to fhe extracellular space. Only under these two conditions will the transmembrane potential contain a periodic component of sufficient magnitude to directly activate the entire strand.^''' The model represents a 6-mm strand of cardiac muscle composed of 60 cells connected by junctional regions and surrounded by a limited volume of extracellular fluid. According to recent resulfs,^^"'^ juncfions are modeled by resistance and capacitance connected in parallel. The potentials in fhe intracellular and extracellular space,
