Cardiac dynamics: a simplified model for action potential propagation
© Peñaranda et al.; licensee BioMed Central Ltd. 2012
Received: 31 May 2012
Accepted: 19 November 2012
Published: 29 November 2012
This paper analyzes a new semiphysiological ionic model, used recently to study reexitations and reentry in cardiac tissue [I.R. Cantalapiedra et al, PRE 82 011907 (2010)]. The aim of the model is to reproduce action potencial morphologies and restitution curves obtained, either from experimental data, or from more complex electrophysiological models. The model divides all ion currents into four groups according to their function, thus resulting into fast-slow and inward-outward currents. We show that this simplified model is flexible enough as to accurately capture the electrical properties of cardiac myocytes, having the advantage of being less computational demanding than detailed electrophysiological models. Under some conditions, it has been shown to be amenable to mathematical analysis. The model reproduces the action potential (AP) change with stimulation rate observed both experimentally and in realistic models of healthy human and guinea pig myocytes (TNNP and LRd models, respectively). When simulated in a cable it also gives the right dependence of the conduction velocity (CV) with stimulation rate. Besides reproducing correctly these restitution properties, it also gives a good fit for the morphology of the AP, including the notch typical of phase 1. Finally, we perform simulations in a realistic geometric model of the rabbit’s ventricles, finding a good qualitative agreement in AP propagation and the ECG. Thus, this simplified model represents an alternative to more complex models when studying instabilities in wave propagation.
Cardiac action potentials (APs) are electrical signals that trigger the synchronous contraction of the heart. As such, the regular propagation of the action potential is necessary for ensuring a correct heart functioning. APs are produced as a result of ion currents that cross the cell membrane, producing a net depolarization or repolarization of the membrane as different currents are invoked in response to the transmembrane voltage changes. The currents are produced by the movement of individual ions through ion channels, which are specialized pore-forming proteins that span the cell membrane, forming a pathway for ions to cross it. Each type of channel is highly selective for a specific type of ion. The most common intracellular ion concentrations considered in cardiac models are calcium, sodium, and potassium. Channelopaties due to mutations that modify the ion channel function can perturb the form of the action potential, sometimes leading to cardiac dysfunctions or altered AP propagation.
Following the pioneering mathematical description of the action potential by Hodgkin and Huxley in neuronal cells, the first cardiac models considered, in much the same way, the changes in the transmembrane potential produced by a sum of ionic gated currents. As more refined experimental data of the different currents and their dynamics became available, more complex models of the AP for specific cardiac cells have been proposed. The quest of new models has been especially noticeable for several animal species, as rabbit, guinea pig, rat, or dog; and, most recently, for humans[7–10]. Nowadays, one can find very detailed models in the literature where the description of ionic channel gating is given in terms of Markov processes, which permit to link specific mutations to the model parameters. A known drawback of the Markov formulation is the increasing complexity of the models and consequently the very high computational cost associated to the simulation of few beats in a realistic heart geometry. Clearly, a highly realistic ionic model is ideal as a test bench, allowing to safely test multiple hypotheses and making predictions without incurring any cardiac risk for the patient, or to check conditions not easily reproducible in experiments. But, despite their more realistic description, these models are often difficult to analyze, let alone amenable for deep mathematical analysis.
In order to tackle the problem of modeling AP propagation in the computer, simplified models have also been proposed, from the very simple but unphysiological[12–14], to the so-called semiphysiological models[15–19], somewhere in between the former and the extremely detailed physiological models mentioned earlier. These semi-physiological models preserve particular sets of properties of the action potential, as for instance, the dependence of the AP duration (APD, time during which the voltage is above a certain threshold, characterizing the duration of the excited state) or the conduction velocity (CV, propagation speed of the AP pulse) on the stimulation period, known as restitution curves. In this respect, these can be viewed as mesoscopic models, that bridge the gap between dynamics at the molecular level (ion channel gating) and whole heart description, being still manageable for both computation and theoretical analysis.
A widely used semiphysiological model is the one by Fenton and Karma, a three-variable model of the cardiac action potential. This model uses three transmembrane currents, i.e., fast inward, slow inward, and slow outward, which resemble the set of physiological sodium, calcium, and potassium currents, respectively. In the Fenton-Karma model, the variables are the transmembrane potential and two gating variables that are used to regulate the inactivation of the fast inward and slow inward currents, respectively. Despite its simplicity, the model can reproduce fairly the action potential duration (APD) and conduction velocity (CV) restitution curves of more complex models or experiments. However, it fails to reproduce accurately the AP morphology. This is a strong drawback of the Fenton-Karma model. In fact, the model was originally created in order to describe the propagation and properties of scroll waves. The underlying idea is that wave behavior is determined by the restitution properties of the system. Thus, it has also been useful in the study of cardiac alternans (a beat to beat modification in the duration of the AP, at fast pacing rates), where the onset and subsequent evolution of alternans has been related to the shape of the restitution curves. However, more recent results have shown that models with the same restitution properties, but different AP morphologies may give different onsets for alternans. The reason is that the dynamics of alternans is also affected by electrotonic currents, that depend crucially on the form of the action potential. Furthermore, electrotonic currents are also a determinant factor for the occurrence of re-excitation and phase 2 re-entry in cases where the duration of the action potential is non-homogeneous in the heart tissue. Typical examples are Brugada, or long QT syndromes, that may be caused by fast or delayed repolarization, respectively.
With the aim of providing models that fit better the AP morphology, a modification of the three-variable Fenton-Karma model has been proposed in. This new model includes an additional gate that modulates the slow inward current, in a fashion that resembles the effect of the fast outward potassium current (I to ) in physiological models. This allows to reproduce the notch in phase 1 of the AP typical of epicardial cells. In contrast to, in the present paper we specifically include the effect of the I to . Following the same idea as in the Fenton-Karma model, we divide the currents not by their carrier ion, but by their function: inward or outward currents, and among these, slow and fast currents, to a total of four. With this assumption, we show that it is possible to reproduce all the important characteristics of AP propagation and morphology. Here, we specifically compare the results with more realistic models of electrical activity in human myocytes and guinea pig heart ventricles, and with available experimental data, including action potential shape for different pacing rates, and different types of myocytes, i.e. epicardial; endocardial; and midmyocardial cells.
The present model has been recently used to study reexcitations in tissue presenting large dispersion of repolarization. There, it was shown that reexcitation is due to a slow pulse, induced by the L-type calcium current, that propagates from the region of long APDs to the region of short APDs until it encounters newly excitable tissue. An interesting advantage of studying this mechanism using this simplified model was the possibility to obtain the characteristics of this slow pulse analytically.
The paper is organized as follows: in the next section we introduce the equations of the model. Then, the AP morphology is studied, obtaining the proper parameters matching with experiments and two detailed models: Luo-Rudy (LRd) model for guinea-pig, and ten Tusscher et al. (TNNP) model of human ventricular cells. A discussion of the different currents and gates follows, as well as a more detailed comparison with other semiphysiological models. In the next section, we perform simulations of AP propagation in a three-dimensional model of the ventricles. In the concluding section, we provide some future lines of research opened by this paper.
Mathematical formulation of the simplified model
where ∇ is the standard nabla operator, is the anisotropic diffusion tensor (units: cm2 ms−1), σ is the electrical conductivity (units: mS cm−1), is the cell surface to volume ratio (units: cm−1) and C m is the membrane capacitance per unit area (≈1 μ F cm−2). The simulations in a one-dimensional cable are carried out by replacing the spatial derivative term by its one-dimensional counterpart in Equation (1). In the three-dimensional simulations, one must take into account the inhomogeneous and anisotropic behavior of the diffusion parameter. All computer simulations have been performed using a simple Euler forward algorithm (for time integration) and finite spatial discretization in order to ensure current conservation up to double precision arithmetic. The time step is fixed to dt = 0. 01 ms. The uniform spatial discretization mesh used in the simulations was set to dx = 0. 02 cm for 1D and 2D, and dx = 0. 025 cm for 3D simulations. The current I ion is the sum of the ionic currents flowing across the cell membrane including the ones exchanged by the pumps through active transport (units are μ A cm−2) and I stim is an applied external stimulation current. To stimulate the cell we apply it during 1 ms with an amplitude of 1.5 times the threshold stimulation value.
Realistic models include ion currents corresponding to all the existing ion channels experimentally found in the cell membrane. In the present simplified model, we decompose the total membrane current into only four components, i.e., I ion = I fi + I si + I so + I to , where the sum contains a fast inward current I fi (Na+ current plus the fast part of the Ca2 + current); a slow inward Ca2+ current, I si ; a slow outward time-independent K+ current, I so ; and a fast transient outward K+ current, I to . The model does not consider the intra- and extracellular concentrations of different ions, and therefore the effect of the pumps is implicitly included in the previous currents. At this point, let us remind that in the Fenton-Karma model, the three variables of the model are the membrane potential V(x t), the inactivation gate h(t) of the fast inward current, and the inactivation gate f(t) of the slow inward current; steady-state activation is assumed for both of these currents. In the present paper, we supplement the Fenton-Karma model variables with two additional dynamical variables associated with the activation and inactivation of the I to current. The latter current being the essential novelty with respect to the three variable model of Fenton and Karma. Let us insist that it is crucial to take into consideration this fast transient outward current to obtain accurately the characteristic notch in the phase 1 of the AP, most noticeable in epicardial myocytes. This current was experimentally studied in human hearts by Nabauer et al. and Li et al. and has been suggested to contribute significantly to regional electrophysiological heterogeneity in myocardial cells and tissue of several animal species. In particular, it is well known that the human ventricle shows substantial transmural heterogeneity in AP morphology related to transient outward properties. Furthermore, this current is sometimes related to the occurrence of phase-2 reentry[28, 29].
Values of the parameters for the present model
Comparison with experimental results and detailed electrophysiological models
One of the main goals of the present model is to reproduce the AP morphology measured in experiments, or obtained with more complex mathematical models, for different types of myocytes and different pacing rates. Hence, the present model is an extension of the Fenton-Karma model, but with the ability of reproducing the AP morphology and, particularly, the notch in phase 1 faithfully.
We have validated our model in two ways: firstly, we have fitted experimental APs for human endo-, mid-, and epicardial ventricular cells choosing the appropriate model parameters. Secondly, we have also fitted the AP morphologies of two of the most currently used theoretical models in the literature: ten Tusscher et al. for human ventricular myocytes (TNNP), and Luo-Rudy[4, 30] for guinea pig (LRd), but also widely used in other contexts. We have fitted the parameters of our model such that the AP morphologies, as well as the correct conduction velocities, at different pacing rates were obtained. The model parameters are determined by using a standard nonlinear optimization routine from Matlab based on the Levenberg- Marquardt algorithm, that minimizes the square of the difference between the model action potential and the objective template. One subtlety of the optimization process used here is that we have imposed a slightly larger weight in the zone corresponding to the notch of the AP that is mostly influenced by the J to current. The codes used for the fit, as well as more information about the model can be found in.
Comparison with experimental human cell ventricular morphologies
Let us mention that human data provided by other authors[24, 27, 32, 33] differ slightly. The cause of the discrepancies being presumably due to the variation in the temperature of experiment, the heart rate stimulation, and if the right/left heart side ventricle were used. In the present paper, we considered the experiments done by Li et al. because in the same paper the authors present the three different types of myocytes at several physiological heart rates.
Comparison with detailed ventricular electrophysiological models
Another important physiological model is the LRd model for guinea pig[4, 30] (where we consider the last version of the model, obtained from). In Table1 we provide the parameter values for the fit to the LRd model. In this case, the comparison between our model and the LRd model was again done at four following pacing periods: 100, 250, 500 and 1000 ms. The LRd dynamic model (using D = 10−3 cm2/ms) gives an approximate propagation speed of 52 cm/s at large DI. This value compares well with the actual velocity ∼50 cm/s obtained with our model with parameter values fitted to the LRd model (see Table1).
Gates dynamics and transmembrane currents
To obtain a better understanding of the newly proposed model, we will briefly discuss the shape of the different currents entering into the formulation of our model. The currents in our simplified model have all dimension of inverse of time (units ms−1). In order to perform a direct comparison with the currents in more complex models they must be multiplied by the factor to obtain the corresponding dimensional current in μ A cm−2.
J fi : fast inward current. It corresponds mainly to the Na+ current in more detailed electrophysiological models. For instance, in the TNNP model, it is expressed by(8)where m is an activation gate, h is a fast inactivation gate, and j is a slow inactivation gate. The sodium conductance in the TNNP model is equal to G Na = 14. 838 nS/pF. Straightforward dimensional analysis allows to compare this value for the conductance to the one in our model g fi = 14 ms−1(nS/pF ≡ ms−1, see the value given in the ninth row and fourth column of Table1). For the gates dynamics, some simplifications are done in the model. The dynamics of the activation gate m is fast (in realistic models τ m ∼ 0. 1 ms), so we take its steady state value, given in Figure4 (although this may lead to a decrease in the numerical accuracy of the model, see). Therefore, it opens at V > V c , (for the fit to TNNP, we set V c = 0. 35, equivalent to mV). The product of the steady state of the inactivation gates h and j is modeled by a step function Θ(V−V c ). The time constants for opening and closing are ms and ms, respectively, which again are of the order of magnitude of those found in the TNNP model. The Nernst potential E Na is typically in the range E Na ≃ [40−70] mV, while in our case we have set V fi = 1. 18, corresponding to mV. The time constant is related to the recovery of the gates from inactivation, and gives the CV restitution properties of the propagating AP.
J si : slow inward current. This corresponds approximately to the Ca2+ current in more detailed electrophysiological models, responsible for the formation of the plateau (phase 2) in the AP. The principal component is the current I CaL , that, in the TNNP model is given by(9)
This current depends on the intra- and extracellular Ca2+ concentrations denoted by C a i and C a0, respectively. The current defined by Equation (9) possesses three gates, two voltage dependent gates d and f, and a third one f Ca that depends on the calcium concentration. In the present model we do not include the calcium concentration dynamics, and therefore all the gates are only voltage dependent. We assume that voltage activated gate d is fast, therefore we neglect its dynamics and rather take its steady-state value. Formally, it means that the dynamics of calcium current is slaved to voltage dynamics, so the last part of I CaL gives an effective voltage activated gate. In the model, the dependence on voltage is given by two sigmoidal functions that determine the voltage range at which this current is active. In conclusion, all the dynamics associated with calcium lies in the voltage inactivation gate f, that operates at a time scale of ms for opening, and ms, for closing. The time constant is related with the AP duration, while determines the AP restitution properties.
J to : transient outward current. This current corresponds to the transient K+ current responsible for the notch in the action potential. In the TNNP model it is expressed by :(10)
For the reversal potential we take V to = 0, which corresponds to mV, while typical values in the literature are E K ≃ −100 mV. For the gates, once again we adopt the simplified formulation through step functions for opening (or closing) at V r = 0. 6, corresponding to a potential mV, similar to the TNNP model. In our model, the dynamics of the gate s is similar to the inactivation gate of Na+, and closes in a time scale of ∼4 ms. For the activating gate r, we have a time scale for opening of ∼13 ms. A comparison of the steady state values of the gates as a function of the transmembrane potential is shown in Figure4.
J so : slow outward current. This current corresponds to the slow (mainly K+) repolarization currents. Among the slow potassium currents, one can distinguish: I Ks , I Kr , IK 1,.. but all in all, their sum is almost constant. We therefore maintain the assumption of the FK model that this current depends on a single gate at steady state.
Comparison with other simplified models
In this section we compare our model with other existing simplified models proposed in the literature. Two of the simplest models of cardiac dynamics contain only two variables. These are the models by Mitchell and Schaeffer and by Aliev and Panfilov. In both cases the equation for the transmembrane voltage includes just two currents, one outward J out and one inward J in , such that dV/dt = J in −J out . Besides the equation for the transmembrane voltage, there is an additional equation for a gate variable that, in modulates the inward current, while in modulates the outward current. Despite their extreme simplicity, these models present restitution properties and morphologies that resemble those of cardiac cells (except for the missing notch in phase 1 of the AP). Interestingly enough to mention, in the model by Mitchell and Schaeffer it is even possible to calculate the APD restitution curve analytically, under certain constraints in the parameters. This, together with the conduction velocity at a given pacing rate (or the maximum CV) fixes univocally all the coefficients in the model. Once this is done, there is no more freedom to fit the shape of the CV-restitution curve. Unfortunately, for typical restitution curves found in human cardiac cell, the previous constraints are not fulfilled and, thus, the simple expressions for the APD-restitution do not hold.
The Mitchell-Schaeffer two variable model is a simplification of the Fenton-Karma model. In the latter the inward current is split into a fast, J fi and a slow J si component. This model is able to fit both APD and CV-restitution curves at the same time, however it does not fit properly the morphology. A further simplified model was proposed by Bueno et al, including an additional gate that modulates the slow inward current, mimicking the effect of the transient outward current I to . This modification implies the need to introduce different values of the parameters at different levels of the transmembrane voltage, through the inclusion of Heaviside functions. In the present paper, we follow a similar approach but, instead of modulating the slow inward current as it is done in the BCF model, we explicitly introduce a fast outward current similar to the physiological I to , an approach that we reckon more natural. This approach, for instance, has been useful to study Brugada syndrome[25, 29], where the loss of the dome is achieved just changing the conductance of this new current J to .
Two dimensional simulations
In this subsection, we will check that our model reproduces the complex spiral wave dynamics observed in realistic cardiac models (for a comparison of spiral wave dynamics in several cardiac models, see). In order to do so, we take a piece of uniform epicardium tissue and apply an ectopic activation to study the stability of the artificially created spiral wave and its dynamics. Spiral tips may follow different types of trajectories, from circles to flower like patterns to chaotic meanderings[38, 39]. The spiral tip is determined by the algorithm described by Fenton and Karma in the reference. The wave tip is defined as the point where the excitation wavefront meets the repolarization waveback of the AP. This point (x tip ,y tip ) is the intersection point of the lines V = V iso and ∂ t V = 0. The arbitrary choice for V iso only slightly modifies the meander trajectories. In this article, we have chosen V iso as a half of the maximum depolarization potential value, i.e. V iso ≈ (ΔV)/2.
Three dimensional simulations and ECG calculations
Cardiac arrhythmias, especially those occurring in the ventricles, are intrinsically three-dimensional phenomena whereas direct experimental observations are only mostly limited to surface recordings. Other techniques like plunge electrodes can and have been used to obtain intramural information, albeit at coarse spatial resolutions, and transillumination is another technique that can be also used in some circumstances. Therefore, it is in general difficult to identify the exact nature of the arrhythmic states and the mechanisms causing their initiations. In this respect, computer simulations of the propagating action potential throughout the heart constitute an invaluable tool to get a better insight into these mechanisms.
Conclusions and further works
In this paper we have analyzed a semi-physiological cardiac model that has been used previously to study reexcitation in a medium with dispersion of repolarization. The model reproduces well the action potential morphology of different cardiac cells from experimental data and also from more complete physiological models.
The CV-restitution and the pseudo-ECG’s in a realistic model of heart ventricles have also been satisfactorily compared. One concludes that to study AP propagation our present simplified model is a good alternative to the most costly physiological models. The gain in computer speed is around a factor of twenty and the need for computer memory (RAM) is also greatly reduced. In certain situations it also allows to obtain analytically some characteristics of propagation, which, in a realistic cardiac model, would be intractable. A limitation of the model is the absence of memory due to the lack of the slow dynamics of ion concentrations. This makes unfeasible to fit simultaneously dynamic and S1-S2 restitution curves and may not give a good comparison with realistic models under situations where the stimulation frequency is abruptly changed. In the near future, we plan to refine the present three dimensional simulation to include the important transmural heterogeneities, in order to study transmural reexcitations and to obtain more realistic ECGs from the 3D model.
This work has been financially supported by MICINN (Spanish Ministry of Science) under co-ordinated projects FIS2008-06335 and FIS2011-28820. The work by AP has also been supported under project FIS2012-37655-02-01. The authors acknowledge useful discussions with Professors S. Santidrian and I. Garcia-Bolao (UNAV) about some physiological questions. Discussions about the model and its simulation with Drs. F. Fenton, E. Cherry and N. Otani are also acknowledged.
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