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УДК 616.12; 073.7

A Computational Model of Electrophysiological Properties of Cardiomyocytes

Ivanushkina N. G.1, Ivan’ko E. O.1, Prokopenko Yu. V.1, Redaelli A.2, Tymofieiev V. I.1, Visone R.2

1National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine

2Politecnico di Milano, Italy

E-mail: [email protected]

Introduction. The method of electrical analogies for the analysis of bioelectric dynamic processes in cardiomyocytes is used in the study. This method allows for replacing investigation of phenomena in non- electrical systems by research of analogous phenomena in electrical circuits. The investigation of time processes in cardiac cells is based on the solution of the system of ordinary differential equations for an electrical circuit. Electrophysiological properties of cardiomyocytes such as refractory period, maximum capture rate and electrical restitution are studied.

Mathematical modeling. Computational simulation of the action potential and currents for 𝐾+,𝑁 𝑎+, 𝐶𝑎2+ ions in cardiomyocytes is performed by using the parallel conductance model. This model is based on the assumption of the presence of independent ion channels for𝐾+,𝑁 𝑎+,𝐶𝑎2+ions, as well as leakage through the membrane of cardiac cell. Each branch of the electrical circuit reflects the contribution of one type of ions to total membrane current.

Results.The obtained electrical restitution curves for ventricular and atrial cardiomyocytes are presented in the paper. The proposed model makes it possible to identify the areas with the maximum slope on the restitution curves, which are crucial in the development of cardiac arrhythmias. Dependences of calcium current on stimulation frequency for atrial and ventricular cardiomyocytes are obtained. Analysis of the kinetics of calcium ions under various protocols of external influences can be useful for predicting the contractile force of cardiomyocytes.

Conclusion.The results of calculations can be used to interpret the experimental results obtained in investigations of cardiomyocytes using the “laboratory on a chip” technology, as well as in the design of new experiments with cardiomyocytes for drug screening, cell therapy and personalized studies of heart diseases.

Key words: method of electrical analogies; cardiomyocyte; action potential; parallel conductance model;

electrical restitution curve; lab-on-chip platform

Introduction

The method of dynamic analogies is widely used for a long time as a basis for interdisciplinary research of technical systems and physical models in medici- ne, biology, ecology [1–4]. The concept of biomimetic design [5] is the development of dynamic analogies methods.

In electrical circuits, electrical energy is transmitted through the branches containing resistors, capacitors, inductive coils and other components, and redistri- buted between branches by means of nodes. Electri- cal processes, including nonlinear ones, are investi- gated using known concepts: electric current, voltage, electromotive force. The mathematical description of electrical processes often coincides with the descri- ption of processes in objects of a different physi- cal nature, which allows us to replace the study of phenomena in non-electrical systems by studies of analogous phenomena in electrical circuits. Analyzing

the components and topological equations of various types of systems, we can detect their dynamic analogi- es. This makes it possible to use Kirchhoff’s laws of electrical engineering and also component equations, in particular, for analyzing bioelectric processes in living tissues and cells. In this case, the cell membrane is represented by a circuit model that includes a capaciti- ve element, and in which the ion channel conductivities are represented in the form of resistive linear and nonli- near components, but nonequilibrium electrochemi- cal processes are described by voltage sources. This approach was applied by Hodgkin-Huxly during the study of bioelectric processes in the nervous tissue [6].

It should be noted that Kirchhoff’s laws are basic for analysis of electrical model that reflects bioelectric processes in cells. Application of the Kirchhoff’s laws and component equations to electrical circuit leads to system of differential equations. Thus, the study of action potentials and time processes in cardiomyocytes can be based on the system of ordinary differenti-

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al equations for a nonlinear model of an electrical circuit that describes the dynamics of cardiac cells functioning.

1 Literature review and problem statement

Nowadays, lab-on-chip technology is an emerging in vitro tool to obtain and study cardiomyocytes typically with human-induced pluripotent stem cells (hiPSCs).

These cells can be differentiated into a variety of cardiomyocytes (hiPSC-CMs) and then used for the development of heart disease models, drug screening and tissue regeneration for cell-replacement therapi- es [7–10]. The key conclusion of these studies is the similarity of electrophysiological properties in hiPSC- CMs and human cardiomyocytes.

However, experimental studies using hiPSC-CMs are fairly complex and have advantages, as well as limi- tations, these latters being mainly related to the mai- ntenance of the environmental and stimulation condi- tions through time, the small numbers of myocytes produced and their immature nature. As a result information about electrophysiological properties of hiPSC-CMs is limited.

In this scenario, further improvements of methods and tools to study hiPSC-CMs’ electrical activi- ty can be provided by computational modeling of electrophysiological properties of cardiomyocytes.

In order to investigate the functional properties of hiPSC-CMs, different electrophysiological technologies are used. Studies [11,12] are based on patch-clamp technique, which is a classical approach to record intracellular electrical activity by inserting a sharp electrode into a cardiomyocyte. Alternatively, multi- electrode potential recordings [10,12] allows for studyi- ng the ion channels functions. Contactless imaging methods with use of voltage- and calcium-sensitive fluorescent dyes [13,14] provide multicellular recordi- ngs at high spatial resolution for studying morphology of transmembrane potential and intracellular calci- um transients during cardiomyocyte’s differentiation and drug discovery. The non-invasive, high-resolution method [15], based on genetically encoded calcium and voltage fluorescent reporters, allows for the long- term study of healthy or diseased hiPSC-CMs and, сonsequently, mechanisms of arrhythmia.

Noteworthy, the study of hiPSC-CM electrophysi- ological properties allows for the assessment of the functional maturity of the cells. In agreement wi- th [11,13], three action potential types (nodal-, atrial-, or ventricular-like) are generally identified.

A variety of bioengineering strategies, employed with different cell types, can prove the influence of different factors on hESC-CMs functionality. In [16,17]

the authors have recorded membrane potential of cardiomyocytes and determinated the action potenti-

al’s duration after electrical stimulation and under spontaneous beating. In [18] the design and fabri- cation of a micro-scale cell stimulator, capable of simultaneously providing mechanical, electrical and bi- ochemical stimulation, have been described.

To recapitulate physiological environment of cells in the native myocardium we have recently developed specific heart-on-chip technologies [19]. The proposed device allows for performing electrical and mechanical stimulation and for evaluating the electrophysiological properties of cardiomyocytes. Major attention is paid to the study of the functionality of micro-cardiac tissue:

the electrical functionality of control and stimulated constructs was assessed in culture by evaluating the excitation threshold and the maximum capture rate.

However, most of the quoted works describe the characteristics of cardiomyocytes at the sarcomere level, myofilament organization, ion channel expression and intercellular connections during the differentiation process.

The present study is devoted to computational modeling of cardiomyocytes’ electrical activity that supplements experimental modeling and can predict the kinetics of contractile force of heart cells at the functional level.

2 The aim and objectives of the study

The aim of the paper is to focus on the cardi- omyocyte electrophysiological properties at the functi- onal level, including the generation of action potenti- als, activativation/inactivativation processes in calci- um ions channels, the frequency-dependent changes in action potential duration and the intracellular calcium release or uptake, that enables to explain the changes in excitation-contraction coupling of cardiomyocytes.

The experimental modeling of electrical and mechanical processes in cardiac cells, realized on the lab-on-chip platform [19], was a starting point for the development of the purely computational model.

Considering the peculiarities of the research on the heart-on-chip platform, our work was focused on:

ˆ the improvement of the parameters of the mathematical model of cardiomyocytes for reflection order to recapitulate the functional maturity of the heart cells;

ˆ the study of the cardiomyocytes’ refractoriness phenomenon;

ˆ the investigation of the processes involved in the cardiomyocytes’ electrical restitution.

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3 Computational modeling

In this paper, the parallel conductance model was used to modeling of cardiomyocytes’ electrophysiologi- cal properties at the functional level. This model, based on the approach of Hodgkin-Huxly for nerve tissue, was improved by many scientists specifically to address cardiomyocytes [20,21].

The proposed model [22] accounts for main ionic currents and change of potential for cell’s membrane.

Independent conductance channels are used for 𝐾+, 𝑁 𝑎+, 𝐶𝑎2+ and leakage.

Transmembrane potential𝑉𝑚(𝑡)can be represented as the sum of alternating component of membrane potential𝑣𝑚(𝑡)and resting potential𝑉𝑚0:

𝑉𝑚(𝑡) =𝑉𝑚0+𝑣𝑚(𝑡).

As it follows from the parallel conductance model, the alternating component of the membrane potential is described as:

𝑑𝑣𝑚

𝑑𝑡 = 1

𝐶𝑚(−𝐼𝐾(𝑣𝑚, 𝑡)−

−𝐼𝑁 𝑎(𝑣𝑚, 𝑡)−𝐼𝐶𝑎(𝑣𝑚, 𝑡)−𝐼𝑙+𝐼𝑑), (1) where𝐼𝑑=

{︂𝐼𝑑0, 0< 𝑡 < 𝑇𝑑

0, 𝑡≥𝑇𝑑 is the depolarizing pulse of current (with amplitude𝐼𝑑0and duration𝑇𝑑), which is repeated with a stimulation frequency (𝐹𝑠𝑡),

𝐼𝐾(𝑣𝑚, 𝑡) =𝑔𝐾(𝑣𝑚, 𝑡)(𝑉𝑚0+𝑣𝑚+𝐸𝐾) 𝐼𝑁 𝑎(𝑣𝑚, 𝑡) =𝑔𝑁 𝑎(𝑣𝑚, 𝑡)(𝑉𝑚0+𝑣𝑚−𝐸𝑁 𝑎)

𝐼𝐶𝑎(𝑣𝑚, 𝑡) =𝑔𝐶𝑎(𝑣𝑚, 𝑡)(𝑉𝑚0+𝑣𝑚−𝐸𝐶𝑎) (2)

are currents for potassium, sodium and calcium, respectively, 𝐼𝑙=𝑔𝑙(𝑉𝑚0+𝑣𝑚 + 𝐸𝑙) is the leakage current,

𝑉𝑚0= −𝑔𝐾0𝐸𝐾+𝑔𝑁 𝑎0𝐸𝑁 𝑎+𝑔𝐶𝑎0𝐸𝐶𝑎−𝑔𝑙𝐸𝑙 𝑔𝐾0+𝑔𝑁 𝑎0+𝑔𝐶𝑎0+𝑔𝑙

is the resting potential, 𝑔𝐾0, 𝑔𝑁 𝑎0 and 𝑔𝐶𝑎0 are the conductances of potassium, sodium and calcium ions at rest; 𝐸𝐾, 𝐸𝑁 𝑎 and 𝐸𝐶𝑎 are Nernst potentials of potassium, sodium and calcium respectively; 𝑔𝑙 is the leakage conductivity through the membrane;𝐸𝑙 is the electromotive force of source, which simulates Nernst potential for chlorine ions, leakage and other factors that affect the membrane potential at rest.

Membrane conductances for 𝐾+, 𝑁 𝑎+, 𝐶𝑎2+

channels are described by the following equations:

𝑔𝐾(𝑢𝑚, 𝑡) =𝑔𝐾max𝑛4(𝑣𝑚, 𝑡),

𝑔𝑁 𝑎(𝑣𝑚, 𝑡) =𝑔𝑁 𝑎max𝑚3(𝑣𝑚, 𝑡)ℎ(𝑣𝑚, 𝑡), 𝑔𝐶𝑎(𝑣𝑚, 𝑡) =𝑔𝐶𝑎max𝑑(𝑣𝑚, 𝑡)𝑓(𝑣𝑚, 𝑡),

where𝑔𝐾max,𝑔𝑁 𝑎maxand𝑔𝐶𝑎maxare conductances for potassium, sodium and calcium ions, respectively, in the case that all the channels for this type of ions are in

the open state;𝑛is activation function of𝐾+channels;

𝑚is activation function andℎis inactivation function for 𝑁 𝑎+ channels; 𝑑 is activation function and 𝑓 is inactivation function for𝐶𝑎2+ channels.

Consequently, conductances and currents for 𝐾+, 𝑁 𝑎+,𝐶𝑎2+ions are determined by five gating variables 𝑛,𝑚,ℎ,𝑑, and𝑓, which are solutions of the differential equations:

𝑑𝑛

𝑑𝑡 =𝑛−𝑛 𝜏𝑛

, 𝑑𝑚

𝑑𝑡 =𝑚−𝑚 𝜏𝑚

, 𝑑ℎ

𝑑𝑡 =ℎ−ℎ 𝜏

, 𝑑𝑑

𝑑𝑡 =𝑑−𝑑 𝜏𝑑

, 𝑑𝑓

𝑑𝑡 =𝑓−𝑓 𝜏𝑓

,

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where 𝑛, 𝑚 and 𝑑 are the steady-state value of activation function for potassium, sodium and calcium channels respectively;ℎand𝑓are the steady-state values of inactivation function for sodium and calcium channels; 𝜏𝑛, 𝜏𝑚 and 𝜏𝑑 are the relaxation periods of activation for potassium, sodium and calcium channels;

𝜏 and𝜏𝑓 are the relaxation periods of inactivation for sodium and calcium channels.

Equations (1) and (3) define the Cauchy problem for the system of ordinary differential equations with the initial conditions:

𝑣𝑚(0) = 0; 𝑛(0) =𝑛0; 𝑚(0) =𝑚0; 𝑑(0) =𝑑0; ℎ(0) =ℎ0; 𝑓(0) =𝑓0,

where𝑛0,𝑚0,𝑑0,ℎ0and𝑓0are the steady-state values of activation and inactivation function if alternating component of membrane potential is equal to zero.

The attained system is a set of stiff differential equations. Consequently, to solve Cauchy problem, the implicit methods of integration is used [23].

Using the predictor-corrector method, step ∆𝑡 in the initial segment of integration should not be too large (∆𝑡 < 𝑇𝑑, where𝑇𝑑is duration of the depolarizing pulse).

A detailed description of the functions and numeri- cal values for parameters of the proposed model is given in [22].

4 Numerical experiments

Numerical experiments to model AP in heart cells were performed in Matlab environment. The generati- on of AP and ion currents for ventricular and atrial cardiomyocytes was simulated. The study accounts for the cardiomyocyte electrical properties such as the refractory period, the maximum capture rate and the restitution during stimulation.

4.1 Action potentials and main currents for cardiomyocytes

Simulated action potentials for ventricular and atri- al cardiomyocytes (Fig.1) were obtained in [22].

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Fig. 1. Simulated action potentials for ventricular and atrial cardiomyocytes. Modified from [22]. Arrows indi- cate the time point of 90% of AP duration (APD90).

In accordance with the theoretical information [20]

each action potential has three characteristic phases:

depolarization, plateau and repolarization.

Depolarization phase is determined by a sharp increase of AP amplitude initiated by the growth of membrane permeability for sodium ions. Plateau phase describes the slow decline of action potential, because the inward (slow calcium) and outward (potassium) currents are nearly balanced (Fig.2a). Repolarization phase is characterized by the faster decline of AP, which could be explained by inactivating of calcium channels and activating of the potassium channels [20].

According to the proposed model [22], currents and conductances for potassium, sodium, calcium ions were calculated and presented for ventricular cardi- omyocytes in Fig. 2b-d (currents of 𝐾+, 𝑁 𝑎+, 𝐶𝑎2+

ions, conductance of𝐶𝑎2+ channels and conductance of𝐾+,𝑁 𝑎+ channels, respectively).

(a) (b)

(c) (d)

Рис. 2. Simulated action potential (a), currents of𝐾+,𝑁 𝑎+,𝐶𝑎2+ions (b), conductance of𝐶𝑎2+ channels (c), conductance of𝐾+,𝑁 𝑎+ channels (d) of ventricular cardiomyocytes.

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In numerical experiments the stimulation frequency 𝐹𝑠𝑡 (stimulation cycle length (𝐶𝐿)) varied from 1 Hz (1000 ms) to 6 Hz (167 ms). Durations of action potentials (APD) were measured as the interval from beginning of action potential to the time point of 90%

of AP duration (APD90) (Fig.1).

4.2 Refractory period and maximum capture rate of cardiomyocytes

It is known [20] that refractoriness of cardi- omyocytes is determined by the refractory period (RP), which is a period of time when another AP cannot be generated as the cell needs to recover from the previous AP. This peculiarity of AP takes place due to the inactivation of𝑁 𝑎+ and𝐶𝑎2+ channels.

Three action potentials are chosen as representative APs at various stimulation frequencies (2.5 Hz, 3 Hz, 3.25 Hz, 3.27 Hz). Fig.3a-c demonstrates changing of APD with increasing𝐹𝑠𝑡(decreasing𝐶𝐿), and Fig.3d (𝐶𝐿=306 ms,𝐹𝑠𝑡=3.27 Hz) shows the case, when the myocyte fails to capture AP. Therefore, the maximum capture rate of cardiomyocyte is achieved under the stimulation frequency𝐹𝑠𝑡=3.27 Hz.

4.3 Electrical restitution of cardi- omyocytes

Different authors have studied the electrical resti- tution that is an intrinsic heart property of change of action potential duration (APD) according to heart rate (HR) [24–28]. The relationships between APD of cardiomyocytes and 𝐹𝑠𝑡 (or 𝐶𝐿) or, more correctly, preceding diastolic intervals (𝐷𝐼) are investigated.

Usually, the preceding 𝐷𝐼 is determined as the time interval between full cardiac repolarization and the growth of the next AP (during cycle length).

The APD shortens with decreasing𝐶𝐿length and thus with decreasing 𝐷𝐼. It is thought that restituti- on takes place because calcium current does not fully recover at short𝐷𝐼, which leads to short APD at short 𝐷𝐼.

Electrical restitution data were simulated using the dynamic restitution protocol (DYRT) [24], in which stimulation impulses were generated with various sti- mulation frequencies𝐹𝑠𝑡or cycle lengths𝐶𝐿. Accordi- ng to this protocol cardiomyocytes were stimulated in the physiologically determined frequency range with increasing𝐹𝑠𝑡 incrementally until the myocyte fails to capture AP, that means refractory period has been reached and maximum capture rate of cardiomyocytes has been obtained.

The responses (AP and currents) from ventricular cardiomyocytes stimulated at different𝐹𝑠𝑡were investi- gated. Electrical stimulation started at 1 Hz and increased up to 6 Hz with assessment of the maximum capture rate. The ventricular myocyte failed to capture

AP at stimulation frequency𝐹𝑠𝑡= 3.27 Hz (𝐶𝐿 =306 ms).Similar action potentials and currents in atrial cardyomyocytes were obtained varying 𝐹𝑠𝑡. However, the atrial myocyte failed to capture AP at higher stimulation frequency compared to the stimulati- on frequency of ventricular myocyte (at frequency 𝐹𝑠𝑡=5.125 Hz, when maximum capture rate is achi- eved).

APDs for atrial and ventricular cardyomyocytes shortened at increasing stimulation frequencies and decreasing cycle lengths (Fig.4). Mean APD90 at each stimulation frequency (cycle length) was obtained by averaging of 20 consecutive steady-state APs.

The restitution curves of mean APD90 for atrial and ventricular cardyomyocytes are shown as functions of stimulation frequency (Fig. 4a) and as functions of cycle length, the inverse of stimulation frequency (Fig.4b).

Сalcium current mean value for atrial and ventri- cular cardyomyocytes decreases with increasing of sti- mulation frequency (Fig. 5a) and with decreasing of cycle length (Fig. 5b). Mean value of 𝐼𝐶𝑎 at each stimulation frequency (cycle length) was obtained by averaging of𝐼𝐶𝑎 during 20 consecutive action potenti- als.

5 Discussion

Many studies have been performed in order to investigate the changes of APDs, which likely plays an important role in arrhythmogenesis. According to the overview of published data [24–28], the normal electrical restitution curve has three main phases: a steep recovery at the shortest either 𝐶𝐿𝑠 or 𝐷𝐼𝑠, a transient decline at middle 𝐶𝐿𝑠 (𝐷𝐼𝑠), and a final rise to a plateau at long 𝐶𝐿𝑠 (𝐷𝐼𝑠). Furthermore, the maximum slope of the restitution curve is cruci- al in determining the arrhythmogenic properties of cardiomyocytes. Under maximum slope of the curve more than 1 the fluctuation of cardiomyocyte’s electri- cal activity occur thus creating the heart unstability.

In addition, rate dependent alterations of APD are markers of atrial and ventricular arrhythmias.

Dependences APD against𝐹𝑠𝑡 (Fig. 4a) and APD against 𝐶𝐿 (Fig. 4b) generate the APD electrical restitution curves, which also have several phases wi- th the various steepness. The maximum slope of the curves arises due to refractoriness of cardiomyocytes and corresponds to the range of maximum capture rate. Therefore the analysis of the electrical restituti- on curves allows for explaining the difference between behavior of atrial and ventricular cardiomyocytes.

The results of the numerical experiments were analyzed and compared with the experimental results, based on the lab-on-chip technology [19]. In the computational analysis the same research protocol was used as for the experimental study. According to the

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(a) (b)

(c) (d)

Рис. 3. Change of action potential duration (APD) during stimulation process:𝐹𝑠𝑡= 2.5 Hz (𝐶𝐿=400 ms) (a), 𝐹𝑠𝑡= 3 Hz (𝐶𝐿=333 ms) (b), 𝐹𝑠𝑡= 3.25 Hz (𝐶𝐿=308 ms) (c),𝐹𝑠𝑡= 3.27 Hz (𝐶𝐿=306 ms) (d).

1 2 3 4 5 6

100 150 200 250 300 350

Stimulation frequency, Hz

APD90, ms

Restitution curve

Ventricular Atrial

100 200 300 400 500 600 700 800 900 1000

100 150 200 250 300 350

Cycle length, ms

APD90, ms

Restitution curve

Ventricular Atrial

(a) (b)

Рис. 4. Electrical restitution curves for atrial and ventricular cardiomyocytes:

APD90 against𝐹𝑠𝑡 (a), APD90 against𝐶𝐿(b).

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1 2 3 4 5 6 2

3 4 5 6 7 8 9 10 11

Stimulation frequency, Hz Ica, 10-6 A

Calcium current

Ventricular Atrial

100 200 300 400 500 600 700 800 900 1000

2 3 4 5 6 7 8 9 10 11

Cycle length, ms Ica, 10-6 A

Calcium current

Ventricular Atrial

(a) (b)

Рис. 5. Dependence of calcium current for atrial and ventricular cardiomyocytes:

ICa against𝐹𝑠𝑡 (a), ICa against𝐶𝐿(b).

dynamic restitution protocol, cardiomyocytes were sti- mulated at the physiologically determined frequency range with increasing 𝐹𝑠𝑡 incrementally until the myocyte fails to capture AP. This means that the refractory period has been reached and the maximum capture rate of cardiomyocytes has been obtained.

During experimental results, the values of maximum capture rate were obtained in the frequency range from 3 Hz to 5.5 Hz. This variability can be explained by the presence of different types of cardiomyocytes (atrial- or ventricular-like) after the differentiation process from human-induced pluripotent stem cells.

The calculated restitution curves allows for identi- fying the maximum slopes, which determine the arrhythmogenic properties of heart cells (in whi- ch cardiomyocytes can change the beating rate).

Moreover, there is supportive evidence that drugs, whi- ch reduce restitution slope, play a protective role agai- nst arrhythmias. Accordingly, computational experi- ments can effectively support the design of new experi- ments with hiPSC-CMs.

Conclusions

Computational simulation of cardiomyocytes’

electrophysiological properties may help to explain the shape of the electrical restitution curves at various repolarization levels for atrial and ventricular cardi- omyocytes and to predict the kinetics of intracellular calcium and contractile force. The proposed model allowed us to detect those regions characterized by elevated slopes, capable to confer arrhythmogenic properties of cardiomyocytes. Computational results are useful to interpret experimental results with hiPSC- CMs on the lab-on-chip platform and to propose the new design of the experiments for personalized studies of heart disease. The aforementioned research can allow monitoring the changes in AP, calcium

current properties of hiPSC-CMs and the development of arrhythmias in response to application of various drugs or different stimulation modes. The method of dynamic analogies used in the work allows us to generalize this approach for studying the electrical properties of cardiomocytes and for investigation of the electromechanical model of nonequilibrium processes in the myocardial tissues. Future directions of research related to the simulation of cardiomyocyte’s activity should be focused on electro-mechanical coupling, electrical and mechanical stimulation, and spontaneous beating

Acknowledgment

The study was supported by EU-financed Hori- zon 2020 project AMMODIT (Approximation Methods for Molecular Modeling and Diagnosis Tools) - Grant Number MSCA-RISE 645672.

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Обчислювальна модель електрофiзiо- логiчних властивостей кардiомiоцитiв

Iванушкiна Н. Г., Iванько К. О., Прокопенко Ю. В., Редаеллi А., Тимофєєв В. I., Вiсоне Р.

В даному дослiдженнi метод електричних аналогiй був використаний для аналiзу бiоелектричних динамi- чних процесiв у кардiомiоцитах. Цей метод дозволив замiнити вивчення явищ у неелектричних системах до- слiдженнями аналогiчних явищ в електричних колах.

Вивчення часових процесiв у серцевих клiтинах грун- тувалось на дослiдженнi системи звичайних диферен- цiальних рiвнянь для електричної схеми, реакцiї якої є аналогiчними процесам в клiтинах серця. У цьому до- слiдженнi основна увага придiлена комп’ютерному мо- делюванню електричної активностi серця на клiтинному рiвнi. У роботi вивчено електрофiзiологiчнi властиво- стi кардiомiоцитiв: рефрактерний перiод, максимальна швидкiсть захоплення та електрична реституцiя. Обчи- слювальне моделювання потенцiалу дiї та струмiв для iонiв 𝐾+, 𝑁 𝑎+, 𝐶𝑎2+ у кардiомiоцитах проведено за допомогою моделi паралельних провiдностей. Ця модель ґрунтується на припущеннi про наявнiсть незалежних каналiв для iонiв 𝐾+, 𝑁 𝑎+, 𝐶𝑎2+, а також їх витоку через мембрану серцевої клiтини. Кожна гiлка електри- чного кола моделi вiдображає внесок одного типу iонiв у загальний мембранний струм. У роботi дослiджено кри- вi електричної реституцiї для кардiомiоцитiв шлуночкiв та передсердь. Запропонована модель дозволила iденти- фiкувати на кривих реституцiї дiлянки з максимальним нахилом, якi мають вирiшальне значення у розвитку серцевих аритмiй. Отримано залежнiсть середнього зна- чення кальцiєвого струму вiд частоти стимуляцiї для кардiомiоцитiв передсердь та шлуночкiв. Аналiз кiне- тики iонiв кальцiю за рiзними протоколами зовнiшнiх впливiв може бути корисним для прогнозування скоро- чувальної сили кардiомiоцитiв. Результати розрахункiв можна буде застосувати для iнтерпретацiї експеримен- тальних результатiв, що отримано в дослiдженнях кар- дiомiоцитiв з використанням технологiї “лабораторiя на чiпi”, а також в розробцi нових експериментiв з кар- дiомiоцитами для скринiнгу лiкiв, клiтинної терапiї та персоналiзованих дослiджень хвороб серця.

Ключовi слова: метод електричних аналогiй; кардiо- мiоцит; потенцiал дiї; модель паралельних провiдностей;

крива електричної реституцiї; лабораторiя на чiпi

Вычислительная модель электрофизи- ологических свойств кардиомиоцитов

Иванушкина Н. Г., Иванько Е. О.,

Прокопенко Ю. В., Редаэлли А., Тимофеев В. И., Висонэ Р.

В данном исследовании метод электрических анало- гий применён для анализа биоэлектрических динамиче- ских процессов в кардиомиоцитах. Этот метод позволил заменить изучение явлений в неэлектрических системах исследованиями аналогичных явлений в электрических цепях. Изучение временных процессов в кардиомиоци- тах основано на исследовании системы обыкновенных дифференциальных уравнений для электрической цепи, рекции которой аналогичны процесам в клетках серд- ца. В данном исследовании основное внимание уделено компьютерному моделированию электрической актив- ности сердца на клеточном уровне. В работе изучены электрофизиологические свойства кардиомиоцитов: ре- фрактерный период, максимальная скорость захвата и электрическая реституция. Вычислительное моделиро- вание потенциала действия и токов для ионов𝐾+,𝑁 𝑎+, 𝐶𝑎2+ в кардиомиоцитах проведено с использованием модели параллельных проводимостей. Данная модель основана на предположении наличия независимых кана- лов для ионов𝐾+,𝑁 𝑎+,𝐶𝑎2+, а также их протекания через мембрану сердечной клетки. Каждая ветвь эле- ктрической схемы модели отражает вклад одного типа ионов в общий ток мембраны. В работе исследованы кривые электрической реституции для желудочковых и предсердных кардиомиоцитов. Предложенная модель позволила идентифицировать на кривых реституции участки с максимальным наклоном, которые имеют ре- шающее значение в развитии сердечных аритмий. Полу- чены зависимости тока кальция от частоты стимуляции для кардиомиоцитов предсердий и желудочков. Ана- лиз кинетики ионов кальция при различных протоколах внешних воздействий может быть полезен для прогно- за сократительной силы кардиомиоцитов. Результаты вычислений могут применяться для интерпретации эк- спериментальных результатов, полученных при иссле- довании кардиомиоцитов с использованием платформы

“лаборатория на чипе”, а также при проектировании но- вых экспериментов с кардиомиоцитами для скрининга лекарств, клеточной терапии и персонализированных исследований заболеваний сердца.

Ключевые слова: метод электрических аналогий;

кардиомиоцит; потенциал действия; модель параллель- ных проводимостей; кривая электрической реституции;

платформа “лаборатория на чипе”

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