0798-4065 0798-4065 S0798-40652009000100007 Venezuela France France 00 03 2009 00 03 2009 24 1 89 105

Simulation of a cardiac cell. Part II: applications

Rossany Roche1, Rosalba Lamanna1, Marisol Delgado1, François Rocaries2, Yskandar Hamam2, Françoise Pecker3

1 Universidad Simón Bolívar, Departamento de Procesos y Sistemas, Apartado 89000, Valle de Sartenejas, Edo. Miranda 9995, Venezuela.

2 Groupe ESIEE, Laboratoire Algorithmique et Architecture des Systèmes Informatiques (A2SI), Cité Descartes, BP 99 93162, Noisy-Le-Grand, Paris, France.

3 Institut National de la Santé et de la Recherche Médicale (INSERM), U581, Créteil, F-94010 France.

ABSTRACT

A simulator of the cardiac cell (E-C coupling process) is developed in this work, which solves the electro-chemical model of the cell proposed by Roche et al. (2004). It also contains other mathematical models as well, namely, those by Tang & Othmer (1994) and Fox et al. (2001), for comparison purposes. The simulator allows the modification of the model parameters in correspondence with the different cellular elements, to emulate cells of different species or cells under different conditions. The paper also contains an application of the simulator for testing the sensitivity of the E-C process to drugs and inhibitors, showing its intended use as a research tool in experiment design and cardiac pathology treatment.

Keywords: Cardiac simulation, Cardiac cell, E-C coupling, Calcium dynamics, Cellular simulation.

Simulación de la célula cardiaca. Parte II: aplicaciones

]]> RESUMEN

En este trabajo se presenta una ayuda computarizada para la simulación de la célula cardiaca que reproduce el comportamiento de contracción-relajación, resolviendo un modelo electro-químico de las células del músculo miocárdico desarrollado por Roche et al. (2004). El simulador también contiene otros modelos matemáticos como el de Tang & Othmer (1994) y Fox et al. (2001), a efectos comparativos. El simulador permite modificar con facilidad los parámetros del modelo asociados con los diferentes elementos celulares, para emular células de diferentes especies o bien células en diferentes condiciones. Se presenta también una aplicación del simulador para probar la sensibilidad del proceso de contracción-relajación ante diferentes drogas, demostrando cómo esta herramienta constituye una interesante ayuda en el diseño de experimentos o en la investigación sobre tratamientos de patologías cardiacas.

Palabras clave: Simulación cardiaca, Célula cardiaca, Excitación-contracción, Dinámica de calcio, Simulación celular.

Recibido: octubre de 2008 Recibido en forma final revisado: febrero de 2009

INTRODUCTION

In order to develop appropriate strategies for pharmacological therapy in heart diseases, the mechanisms underlying to E-C coupling process must be well understood. A large number of "in vitro" and "in vivo" experiments, mostly qualitatively focused, are routinely carried out in biological research to study the role of several sub-cellular elements. As an alternative, a computer simulator of the cell is an excellent tool to get better understanding of the E-C coupling process and allows the substitution of an important part of laboratory experiments. It can be used to design experimental protocols or test new ones, to reduce experimental variations, to design alternative experiments and to discard "a priori" non-desirable drug effects.

Cardiac models have often been focused on only one of the chemical or the electrical aspects, in spite of the evident relationship of the two phenomena. Among the electrical models, models based on the works of Luo & Rudy (1994), Fox et al. (2001) and Noble (2002) can be found. They contain a complete description of the electrophysiological aspects of the cell along with some empirical blocks, and the results have been validated using patch clamp techniques.

Among the chemical models, complete descriptions of the chemical activities of the cellular elements are presented by Tang & Othmer (1994), Hamam et al. (2000) and Rocaries et al. (2004), however these models do not provide a prediction of the membrane potential and its interdependence with the calcium homeostasis.

The cell simulator presented in this paper contains a model with the complete description of the myocyte, both chemical and electrical (Roche et al. 2004). It is based on a "hybrid" formulation (phenomenological descriptions including some empirical elements to be identified) that ensures a certain compromise between precision and complexity of the model, and also helps to adapt it to different conditions.

In the following sections a description of the simulator of the cardiac cell is presented after a short summary of the electro-chemical model. Some simulations are shown to illustrate the process and results of a sensitivity analysis carried out on the model with the aide of the simulator. Finally, a study of the effects of some drugs on the cell behaviour is presented as examples of interesting possible applications of the simulator.

 ]]> THE MODEL

In the model by Roche et al. (2004), the cardiac cell is represented as a system of two interconnected micro-chemical-reactors, namely the cytosol (or main reactor) and the sarcoplasmic reticulum. The different transfer processes take place between these two sub-systems, and between them and the external medium, by means of different valves (L-channels and R-channels) and pumps (SERCA, sarcolemma pump and Na+-Ca2+ exchanger). They are modeled using the principles of mass transfer, fluid-dynamics and the chemical kinetics of the reactions taking place in the system. In addition, the main reactor wall is electrically charged and its potential V also modulates many of the chemical reactions involved. The detailed description of the model equations can be found in the previous companion paper (Part I) by Roche et al. (2009).

The equations for the main current and mass balances allow the description of three output variables of the system: Cai2+, and V, namely:

The total formulation contains eighteen non-linear first order differential equations, since the integration of fifteen additional state-variables, namely d, x1, x2, x3, M, CaM, m, h, j, KXto, KYto, KXKr, KXKs, f, and fCa is needed to solve the model.

A fine tuning of the parameters of the model allows a good emulation of the behaviour of cardiac muscle cells from different species (rabbit, chicken, dog, human), and a better prediction when comparing the results with those from other models. These results have been reported in Roche et al. (2004, 2009) based on the dynamics of the output variables ( Cai2+ and V). Specifically, on:

- Curb shapes: cyclic, with a plateau phase for V, and bell-shaped for Cai2+.

- Time periods: oscillation period for V(tvoltage) in the range 200-400 ms and oscillation period for Cai2+, (tcalcium) in the range 800-1000 ms.

- Maximum calcium gradient ∆calcium: oscillating between 0.8 and 1.1 μM.

SIMULATION

 ]]> A software tool for the resolution of the model just presented has been developed with the aide of MatLab® and Simulink®, which also includes the models by Tang & Othmer (1994) and Fox (2001).

The simulator interface main screen features a main menu, the simulator and the graphics builder (Figure 1). It contains two commands. The View command provides the functional scheme of each model. The Help command offers information on the simulator functions and parameter settings.

Simulation (Figure 1)

▪ Model selection

The user selects the model to be simulated among the three boxes that contain the parameters of each model: Kinetic model by Tang & Othmer (1994), Electric model by Fox et al. (2001), or Electro-Chemical model by Roche et al. (2004). Selection of a model box opens the simulation secondary screen, showing the parameters of the particular model, where the simulation is to be carried out (Figure 2).

• Results

Simulation results will appear (when available) on the first list on the main screen corresponding to the model indicated by the activated selection button. It is also possible to import data contained in Excel® or in .EXE files, arranged in a unique data vector (corresponding to a unique variable), to allow real data from experimental work to be charged on the second list of this screen (Figure 3).

]]> Graphic builder (Figure 3)

Selection of the variables to be plotted on the Y axis (not more than three) and the variable on the X axis (not more than one) is made with a mouse click. The zoom capabilities of the graphics builder are activated selecting an area to detail with the right mouse button. The Draw button, (re)draws the graph for the selected X and Y values.

The simulator secondary screens (one associated with each model) allow the modification of the model parameters, the actual realization of the simulation and visualization of the results (Figure 2 for the electro-chemical model).

• Model parameters

It’s possible to modify a selected parameter by typing its new value in the box at the bottom of the list. The File menu is available to allow:

- Save: to memorize the corresponding parameter values. This command saves in an .MAT file the selected values of the model parameters. The user provides a name to correctly identify the set of modified parameters.

- Open: to use a pre-existing set of modified parameters contained in a .MAT file.

- Exit: to close the screen.

It is also possible to restore the initial values of the model coefficients using the Reset button (Figure 4).

]]> • Simulation:

- Simulation time: to be fixed by the user in the "time (sec)" box (Figure 4).

- Running: The Run button initiates the simulation process. The equations of the model are integrated with Simulink® of Matlab® (v6.5), using a variable-step numerical solver based on the algorithm by Klopfenstein (1971). This method is relatively fast and accurate, and allows the modification of the relative error tolerance and absolute error tolerance values (Shampine & Reichelt, 1997). During the simulation process the Run button appears as Processing.

- Model Perturbations: The activation of the Perturbation option allows the realization of simulations with modified parameters, to test the model. An active button designated by Set 1 appears (Figure 5) to let the user charge an initial set of parameters, and changes afterwards to Set 2 to accept the modified or "perturbed" parameters. The simulation is carried out with the initial coefficients for the first half of the total simulation time, and then switching to the changed or perturbed parameters.

- Results: Once the simulation concludes, the set of simulated variables is deployed in the abscissas and ordinate of the graphic builder, with easily identifiable names (Figure 4). Selecting each variable with the right mouse button makes available the Save option. The Save command creates an .EXE or .MAT file containing the selected variable with the name assigned by the user.

• Graphics builder

The variables to be plotted are picked up with a mouse click. The zoom capabilities of the graphics builder are activated selecting an area to detail with the right button of the mouse. It is also possible to see the graph in a separated window, by clicking the Graphic button. The Erase button eliminates a previously drawn graph. The Draw button, (re)draws the graph for the selected variables.

SIMULATOR APPLICATION: SENSITIVITY ANALYSIS OF THE MODEL

A good number of calcium and voltage time responses were obtained with the aide of the simulator, and the parameters adjusted to emulate real experimental data. The flexibility of the simulator, that deploys all the important parameters of each cell element, easily allows the parameter modifications.

]]> An important result of the extensive validation procedures carried out with the simulator, is precisely an expert knowledge of the effects of different coefficient alterations in specific model components on the shape and duration of the cell time responses. The methodical testing of these effects, namely the sensibility analysis of the model, includes two steps:

1) The simulation of the model with the standard parameters for a generalized mammal cardiac cell, up to steady-state (40 s. of simulation time).

2) The "perturbation" of the model, meaning the simulation with an altered element (one parameter at a time), being a "stimulation" of the element if the parameter is increased, or an "inhibition" if it is decreased.

The following results from the sensibility analysis can be reported:

- Some elements do not have any significant effect on the dynamics and therefore can be simplified or eliminated from the model, as in the case of perturbations of q1 parameters and ĪpCamax associated with the chemical and electrical activity of the sarcolemma pump (Figures 6 and 7).

- The model is very sensitive to some elements which description is therefore a key issue, as in the case of the Na+-K+ pump. In Figures 8 and 9 some results of parameter perturbations in this element show important alterations in the process outputs.

]]> - The emulation of abnormal behaviors (pathologies) of the cell along with the identification of the cellular elements that cause them can be achieved with the sensitivity analysis of the model. As an example, the inhibition of parameter Kd1, which is related to the myofibrils kinetics, emulates a typical situation of cardiac insufficiency, this is the decrease in chemical affinity between the myofibrils and the calcium ions and therefore the loss of contractile power of the cell (Figures 10 and 11). Another example can the seen in Figure 12, where the altered outputs obtained by changes in the chemical and electrical activities of the L-channels, through parameters Fd and PCa, correspond to cardiac arrhythmias.

Even if several kind of parametric alterations can produce similar effects, the results of this sensitivity analysis allow the identification of the cellular elements and therefore which parameters must be altered to produce changes on each phase of the calcium or the voltage dynamics. This information is summarized in Figures 13 and 14.

It is easy to foresee at this point the applications of the simulator to support pharmacological research for instance, because it can be used to study the effects of different drugs by determining which alterations in cellular elements are induced by them.

SIMULATOR APPLICATION: TESTING OF DRUGS EFFECTS

Previous works (Tang & Othmer, 1994; Hamam et al. 2000; Luo & Rudy, 1994; Fox et al. 2001) cannot explain myocyte behavior in presence of some specific substances because the models only take into account the electrical or the chemical aspects of the cell. Results of testing the effects of several drugs using the electro-chemical model are presented in this section. Reproduction of the altered behavior of the myocyte under drug injection is achieved with the aide of the simulator to methodically modify the parameters of the model, identifying specifically the cellular elements affected by each drug. This may help to clarify several questions about the abnormal E-C coupling process and to explore alternative therapies for cardiac diseases.

]]> Experimental data for chicken myocytes under the injection of Caffeine, Forskolin, Ruthenium red and Thapsigargin drugs have been provided by INSERM, following experimental protocols similar to those used by (Rocaries et al. 2004).

A study for each drug effect separately has been carried out according to the following methodology:

- The set of parameters corresponding to the drug effect are identified by taking into account the sensitivity study of the model presented in the previous section, and reported knowledge found in bibliographical reviews.

- By trial and error method, the selected set of parameters are modified in order to reproduce the altered myocyte behaviour.

- The simulation results are compared with the experimental results. The protocol for this comparison is to obtain the normal behavior in the first place and then, after of 40 minutes of simulation, to alter the parameters set (using the "Perturbation" button in the simulator secondary screen, Figure 5). This protocol also corresponds to the experimental drug injection.

Caffeine effects

Caffeine is used in muscle research to study the role of the sarcoplasmic reticulum in the E-C coupling process (Smith & Steele, 1998). Caffeine binds to a specific site on the R-channels inducing a Ca2+-independent activation of the channels, with increases in both the frequency and duration of the channels openings (Duke & Steele, 1998). It is also reported by Smith & Steele (1998), that Caffeine may lower the Ca2+ within the SR, to a sufficient level to stimulate the SERCA pump by a mechanism that is not yet clear. Bassani et al. (1998), applying rapid injections of 10mM-Caffeine observed a rate of the SERCA pump of about 3-4 times faster than that of the Na+-Ca2+-exchanger.

Experimental data provided by INSERM is obtained injecting a pulse of 10 mM-Caffeine, 40 minutes after the normal behavior of the chicken myocyte is reached. Results of this experiment show that calcium concentration values in the cytosol are increased, however the concentration gradient is slightly reduced (∆calcio= 0,9 μM whereas ∆calcio=1 μM under normal conditions) indicating that probably the SERCA pump activity is stimulated such as observed by Bassani et al. (1998) and Smith & Steele (1998). Reduction of calcium gradient in the cytosol using Caffeine injection is also observed by Zhang et al. (1999). In order to reproduce the facts previously mentioned three parameters of the model have been altered: increasing of l1 in order to increase the open probability of R-channels, stimulus of p1 (SERCA pump) and inhibition of Na+-Ca2+-exchanger activity (rate reduction of K).

The set of values of the best fit parameters are shown in the Table 1. Figure 15 and 16 present the comparison between experimental data and simulations results for calcium concentration. The model reproduces the Caffeine effects in the chicken myocyte, that is, calcium concentration value increased in the cytosol and calcium gradient decreased.

]]>

Additionally, in Figure 17 the increasing of open R-channels, from range of probability 0,1 - 0,5 to range 0,2 - 0,55 can be noticed, agreeing with SITSAPESAN (1998) and with Duke & Steele (1998). On other hand, Figure 18 shows that the results for SERCA pump and Na+-Ca2+ exchanger flow-rates are similar to results found by Bassani et al. (1998), this is the SERCA pump rate being faster than that of the Na+-Ca2+-exchanger.

Forskolin effects

Forskolin increases the level of the cell-regulating compound called cyclic adenosine monophosphate (cAMP) - one of the key regulators of ATP cycle (Zhang & Wong, 1998). The raise of cAMP concentration in the heart leads to an increased force of contraction. This may be useful in congestive heart failure and various heart diseases. Furthermore, Forskolin appears to relax the smooth muscles in the walls of the arteries. The relaxation of the arteries decreases blood pressure, pain due to angina, and strain on the heart. Tominaga et al. (1995), studied the effects of 1μM Forskolin injection on guinea-pig ventricular myocytes concluding that, during current-clamp experiments, Forskolin reduces the action potential significantly, from 250 ms to 201 ms. Also, Forskolin was found responsible for increasing the magnitude of hyperpolarized currents in mouse embryo pacemaker cells (Song et al. 2002).

Experimental data provided by the INSERM shows that 30 hours after a 1 μM Forskolin injection the cytosol calcium cycles become very unstable (Figure 19). This strong perturbation is similarly observed in the model when the Na+-Ca2+-exchanger is perturbed. Additionally, low membrane potential and unstable calcium cycles are also produced by perturbations on the INaK current. Based on the previous sensitivity study and analysis, we have selected the K parameter of the Na+-Ca2+ exchanger and the coefficient INaK max of the INaK current as the parameters linked to the Forskolin effects. After few trials, the appropriate parameters values are found as reported on Table 1. Simulations of the model after the modifications of just these two parameters are close to real Forskolin effects, as can be observed in Figure 19.

However, Forskolin effects on the model are observed immediately and not 30 hours later as with the experimental data. This may be associated to the ATP cycle dynamics which is not included in the model. In future works we suggest to add the chemical reactions of the ATP cycle in order to observe its impact on the Forskolin effects. Despite this setback, the model involving chemical and electrical aspects has proved its adaptability to test the strong Forskolin effects, where other models that do not include the electric dynamics fail (Rocaries et al. 2004). Figure 20 contains the voltage simulations results for Forskolin injection showing that the membrane potential is faster, agreeing with the results by Tominaga et al. (1995), but at the same time the voltage cycle becomes strongly unstable.

]]>

Ruthenium red effects

Ruthenium red is known as an inhibitor of sarcoplasmic reticulum Ca2+ release. Lukyanenko et al. (2000), report inhibitions of the probability of open R-channels (20-50% of reduction) in rat ventricular myocytes with concentrations of 0.1 μM, 1 μM and 5 μM on a Ruthenium red injection. Others results of this work show that Ruthenium red (5μM) increases the sarcoplasmic reticulum calcium concentration from 151 μM to 312 μM. Ruthenium red is also recognized to inhibit the mitochondrial Ca2+ uptake mechanism, which in turn produces a delay in the recovery of L-channels calcium currents reducing the muscle contractibility and generating ventricular tachycardia (Sánchez et al. 2001). Experimental Data of INSERM showing the effects of 1 μM Ruthenium red injection on calcium concentration in the cytosol can be seen in Figure 21. After 140 minutes, Ruthenium red perturbation changes enormously the calcium gradient. Resting calcium value is modified from 0.1 μM to 0.75 μM reducing calcium gradient significantly (~70%).

We have observed, during the sensitivity study of the model, that inhibition of the Na+-Ca2+ exchanger rate produces changes in the resting calcium value in the cytosol. Based on this aspect and taking into account the works by Lukyanenko et al. (2000) and Sánchez et al. (2001), we have selected the following set of parameters to be altered in order to reproduce Ruthenium red effects: l1 (reduction of open probability of R-channels), p1 (sarcoplasmic reticulum calcium concentration increasing), ICa factor (delay of L-channels) and K (resting calcium value perturbation).

Table 1 shows the new parameters values adjusted by trial and error. Simulations of Ruthenium red effects (Figure 21) are instantaneous instead of delayed as the experimental results, which occur 140 minutes later. The model does not include a mitochondrial description, and this may be the cause of the delay in the experimental Ruthenium red effects. Despite this aspect, the simulation results fit well to the experimental data. On the other hand, in Figure 22, it can be noticed that the reduction of the fraction of open R-channels agrees with Lukyanenko et al. (2000). Figure 23 shows the increasing of sarcoplasmic reticulum calcium concentration by Ruthenium red effects. This increase is larger that the values reported by Lukyanenko et al. (2000), however the cellular species in both cases are different (rat vs. chicken ventricular myocyte).

Thapsigargin effects

Thapsigargin has been used as a selective inhibitor of the SERCA pump in cardiac myocytes (Kirby et al. 1992). Gaugham et al. (1999), were able to virtually eliminate sarcoplasmic reticulum using Thapsigargin. In their experiment, they show that Thapsigargin had no effect on the action potential shape. Results of Ginsburg et al. (1998), show that, when the SERCA pump rate was slowed by 39% through a 0.4 μM Thapsigargin injection, the sarcoplasmic reticulum calcium concentration decreased by 20%. Vigne et al. (1992) report that Thapsigargin injection rapidly raises calcium concentration in the cytosol followed by a depression. Similar results are reported in the experimental data of INSERM (Figure 24) where calcium concentration in the cytosol is increased and rapidly decreased producing instable calcium cycles. Wu et al. (2001) suggest that Thapsigargin is also an inhibitor agent of the R-channels.

]]>

We have found that perturbations in ICab current produces an increase in calcium concentration in the cytosol followed by decrease. Also modifications of INaCa current and Na+-Ca2+ exchanger rate have similar effects. Taking into account these aspects we have selected the following parameters set in order to test Thapsigargin effects on chicken myocyte: l1 (R-channels inhibition), p1 (SERCA pump inhibition), K, INaCa and ICab (unstable calcium cycles). Table 1 shows the values of the parameters adjusted by trial and error. Figure 24 shows the comparison between experimental data and simulation results where it may observed that the model produces the Thapsigargin effects on myocyte close to experimental data.

Simulation results of R-channels are observed in Figure 25, where the R-channels are inhibited according to the observations by Wu et al. (2001). The simulation results of the sarcoplasmic reticulum calcium concentration are shown in Figure 26. A decreasing in calcium concentration is observed (50% for chicken myocyte) following the same trend as in Ginsburg’s report (1998) (20% for adult ferret ventricular myocytes). Voltage simulation results are presented in Figure 27, where a faster voltage cycle can be noted due to the Thapsigargin injection, although the alterations in shape reported by Gaughan et al. (1999) are not observed.

Finally, although previous works don’t state any association between Thapsigargin effects and the Na+-Ca2+ exchanger activity, the results by Gaughan et al. (1999) have established that when the SERCA pump is inhibited the Na+-Ca2+ exchanger contributes to the E-C coupling process producing altered calcium cycles similar to those present in the experimental data of INSERM. Therefore to reproduce the Thapsigargin effects, the kinetic and the electrical parameters related to the Na+-Ca2+ exchanger are to be altered. These results may be further improved by adding a more complete description of the kinetics for the Na+-Ca2+ exchanger, since Gaughan et al. (1999) have suggested that the Na+-Ca2+ exchanger works in an additional mode (reverse mode) which is not described by this model. This reverse mode may contribute to Thapsigargin delay (700 minutes), which is not reproduced at the moment by the simulator.

CONCLUSIONS

A simulator of the cardiac cell has been developed, devoted to the solution of a global electro-chemical model of the cell (Roche et al. 2004). It describes a cardiac cell integrating the electrical and chemical dynamical aspects of the different components in the cytosol, the sarcoplasmic reticulum, and the cellular membrane, producing the expected time responses of the excitation-contraction (E-C) coupling, i.e. the oscillatory-bell-shaped curve in calcium dynamics and the characteristic plateau phase in membrane potential.

The simulator has been used to carry out a sensibility analysis of the model, allowing to explore the coherent behaviour of each model component, to identify key elements (as the Na+K+ pump), or even elements that do not have influence on the process outputs (as the sarcolemma pump). Finally the simulator can be used to emulate abnormal situations (cardiac pathologies). This feature extents greatly its applications in order to test drug effects on the myocyte.

]]> In the testing of drug effets, the phenomenological structure of the model allows the identification of the cellular sub-systems that are related to specific changes in the process dynamics. The parameters modifications reproduce the experimental drug effect allowing to identify the cellular mechanism altered. This aspect is an important contribution in the development of new treatments.

Some necessary improvements of the model by Roche et al. (2004) such as an alternative kinetic description of the Na+-Ca2+, or some other additions required to well adjust the model to experimental data, have also been pointed out.

ACKNOWLEDGEMENTS

The work presented in this paper is the result of the French Venezuelan collaboration research project entitled: "Modeling of the cardiac phenomena: from the cell to the organ". The authors gratefully acknowledge the financial support of ECOS Nord (France), FONACIT, Universidad Simón Bolívar and FUNDAYACUCHO (Venezuela).

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