potentials into the mammal Sinoatrial Node. Math models are used for investigate the ionic base that supports the pacemaker activity in central and periphery.
Implementation and Verification of Sinoatrial Node Math Model 1,2
D. A. Sierra1, C. R. Correa2 , O. L. Rueda3 Electrical, Electronic and Telecommunications Engineering School, Universidad Industrial de Santander, Colombia 3 Medicine School, Universidad Industrial de Santander, Colombia third type of cells in the SAN is the same auricular myocardium’s ordinary fibers, they are extended for the node’s border and form sinoatrial zones interlaced with myocardium fiber zones. This is especially seeing in the neighbor of the crista terminalis. Another point to remember is that electrical activity in the hearth is based in the action potentials generation: electrical potentials present in the cellular membrane due to the differences in ionic concentrations. The action potential from a myocardium’s muscular cell (not pacemaker) presents some phases:
Abstract— A math model of the Sinoatrial node
(SAN) is presented, based in last experimental results in literature. This model was proposed by H. Zhang, A. V. Holden y M.R. Boyett and was implemented and verified in order to obtain a better understand of the processes associated with generation and propagation of action potentials into the mammal Sinoatrial Node. Math models are used for investigate the ionic base that supports the pacemaker activity in central and periphery cells of SAN. This implementation is the support to develop a sensitivity analysis in the model as the next step in the investigation line. Keywords— Action potentials, ionic channels, model, propagation, sinoatrial node.
I. INTRODUCTION The Sinoatrial Node (SAN) is the hearth’s natural pacemaker. It initiates the rhythmical action potentials in the hearth. Math models of the electrical activity of SAN created before this considerate a typical SAN and ignored its heterogeneities in functions, anatomy and electrophysiology. These differences are essentials for an accurate modeling of the generation and propagation of the action potentials in the hearth. The proposed model includes regional differences en the SAN activity. The model’s authors (Zhang et al.) validated it by mean of experimental information about ionic channels kinetics, ionic current densities, action potentials in the node and its physiologic differences. Our work is related to describe the model, implemented it and for the immediate future to develop a sensitivity analysis of the model. Next sections focuses in the math model and the discussions items are in the final part.
Fig. 1. Action Potential in the Cell Membrane. No pacemaker cell
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II. SAN PHYSIOLOGY SAN is a structure composed by a fibber tissue matrix. It is placed in the topside of the right auricle, near to the top venae cavae. SAN is a heterogeneous tissue formed by three type of cells: Nodal cells (P) in the centre, they are small, with few myofilaments and poorly organized, in this region is where potential actions are generated and by this issue the central region is called the leading pacemaker site. The transition cells (T) are intermediate between the nodal and the auricular cells, and for it they present a high grade of difference, some cells are more complex than others. The
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Phase 4, Equilibrium potential of membrane: It is the electrical potential in the membrane due to the balance of ionic concentrations. In absent of external excitations the membrane will maintain this potential Phase 0, fast depolarization: when the excitation condition is obtained a significant current is present and depolarize the membrane in a fast form. Phase 1, early fast repolarization: This is due the outcome of potassium ions. Phase 2, plateau: The membrane potential is stabilized temporally near to zero milivolts, due to the effect of some contraries ionic currents and with the important inward of calcium to the interior of the cell. Phase 3, fast final repolarization: In this phase the potential is reduced until reach a minimum value called maximum diastolic potential (MDP).
For a pacemaker cell, instead of the muscular cell, it is present an automatism characteristic in the action potential generation. This means that it isn’t needed a prior excitation in order to began the depolarization. This implies changes
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in the form of an action potential from a pacemaker cell. In the former there aren’t present the 1 and 2 phases, the 0 phase of depolarization isn’t so fast due the need of a calcium ionic current, the 3 phase represents the outward of potassium and in general terms is similar in both type of cells. The 4 phase is called the pre-potential phase and is characterized by a progressive and autonomous increase until reach the excitation potential until the start of the depolarization phase. The electrical activity registers from the SAN or small tissue balls have showed that in the center of SAN the takeoff potential is less negative, the action potential upstroke velocity is fewer, the action potential is larger, the maximum diastolic potential (also resting potential in quiescent tissue) is less negative, and the intrinsic pacemaker activity is paradoxically slower than in the periphery. In normal conditions the action potential begins in a small part of the SAN, called leader pacemaker site, sited approximated in the center of the SAN. When the potential is initiated it began to be propagated in direction of the periphery and the atrial muscle. The propagation is asymmetric, the potential propagates with preference in direction of the top of the crista terminalis, but it is blocked in the septum (blocking zone). The action potential surrounds the SAN and reexcites it in the blocking zone, this zone is important in a functional point of view because protects the SAN from the arrhythmias. A key characteristic of the leader pacemaker site is to have the ability to move dynamically, p.e. in the autonomic nervous stimulation it moves from the center and in some situations it moves to the periphery.
20_pF, until periphery cells have a capacitance surrounding 65 pF. IV. THE NATURE OF IONIC CURRENTS IN THE SAN The pacemaker activity of the cells in the SAN is due to the cooperative effect of various individual currents through ionic channels. Ionic channels are macromolecular proteins that allow the flux of important amounts of several ions as Ca++, Na+, K+. The dynamics of ionic channels respond to the nature of each one and may be voltage, time and chemical dependent, one important feature of ionic channels is its selectivity to specific ions. Next we show the ionic currents present in the SAN cells and that determine the generation of action potentials: • TTX-sensitive Na+ current (iNa): This current is the responsible of the fast depolarization in the 0 phase of the action potential. It is very important in peripheral cells but not too much in the central cells. • L-type Ca++ current (iCa,L): Long lasting calcium current, this current is present in both type of cells but only the central region is sensitive to its presence. If the current is blocked the action potential never appears. • T-type Ca++ current (iCa,T): Transitory calcium current. Both regions have important amounts of this current and vary with its block. • 4-AP-sensitive currents (ito and isus), these ionic currents are associated to K+ ions and its key roll is in the repolarization phase but the influence vary regionally from center to periphery. • Rapid delayed rectifying K+ current (iK,r): Its total block eliminates the action potential because it generates the depolarization due K+ ions accumulation. This current is the responsible of generate the maximum diastolic potential. • Slow delayed rectifying K+ current (iK,s): Its total block have few effects to the action potential in both regions of the SAN • Hyperpolarization-activated current (if): This current has associated with two ions, Na+ and K+, its effects are minor but affects both regions. Added to these ionic currents through ionic channels the whole model includes another ionic currents: • Background or linear current: Include the three type of ions • Pump current: There are two type of this current, the Na+-K+ pump and the Ca++ pump current due to the ATP. • Exchanger currents due to the balance of ionic concentrations of sodium and calcium.
III. REGIONAL DIFFERENCES IN THE CURRENT DENSITIES As we said before the electrical activity changes from the center to the periphery of the SAN in a characteristic fashion. There are two radical different interpretations to explain this situation. The first is the mosaic model: This implies that the electrophysiological properties of the individual cells of the pacemaker are uniforms and that the apparent differences in electrical activity are due to a progressive increase in the hiperpolarization from the center to the periphery. Instead of that the mosaic model can’t generate the action potentials with the same characteristics of regional dependency that the experimental. The second model is the gradient model; this implies that the regional differences in the electrical activity are due to a distribution gradient of ionic current densities in the cells from the periphery to the center of the SAN. Experimental results have showed that there is a correlation between the cell’s sizes in the rabbit SAN and the ionic current form each type present during the cardiac cycle. If the cell are bigger the electrical capacitance too and the ability to carry more current are more important. Cells in the center of the SAN have a capacitance surrounding of
IV. STRUCTURE OF THE SAN MODEL The operation of the SAN isn’t only dependent of the cellular properties; the node’s multicellular nature and the
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electrotonic interaction between the node and the muscle surrounding it are important issues to take in account. The developed SAN model is a multicellular model using one partial one-dimensional differential equation for the action potential and including a set of auxiliary equations for the ionic currents that generates it and that are voltage dependent. In the model the SAN and the atrial muscle are modeled as a string of tissue with a length (L) of 12.6 mm; of this the string of SA node tissue has a length (Ls) of 3 mm [similar to the distance from the center of the SA node to the atrial muscle in the rabbit], and the string of atrial tissue has a length of 9.6 mm. The cellular capacitance was assumed to vary from 20 pF in the center of the SAN to 65 pF in the periphery, and the ionic currents are function of this capacitance. In the model the single atrial cells are represented by the EarmHilgemann-Noble equations. The electrotonic interactions between cells are modeled by the diffusive interactions of the membrane potentials. The one-dimensional model equations are: 1 s ∂V s ( x, t ) ∂V s ( x, t ) (1) =− itot ( x, t ) + D s s ∂x ∂t ( x) Cm 1.07 • ( x − 0.1) s C ( x) = 20 + (65 − 20 ) (2) m
An efficient numeric solution of the model requires different spatial steps for the SAN and for the atrial muscle, dxs and dxa respectively. In both extreme there was a nonflux frontier condition (4) and (5). The coupling in the union of the SAN with the atrial muscle is modeled by the diffusion coefficient (Ds). The value used for Da, 1.25 cm2/s, induces a conduction velocity of the unique planar wave of 6 m/s and Ds = 1.25 cm2/s gives a spatial distribution of the activation time for the SAN cells consistent with the data obtained from the rabbit hearth. Equations (1) and (3) correspond to the potential model of a SAN cell and an atrial muscle cell, respectively. Each cell is considered as an electrical capacitor that receives an ionic total current istot(x,t), with the previously described components plus a current due to the spatial variation of potential (electrotonic effects). Equation (2) Corresponds to the model of the variation of capacitance (size dependent) of the SAN cells, as early mentioned this variation is spatially dependant of the location of the cell from the center of the SAN, x. This equation has a linear component important near to the periphery and an exponential term, the denominator, important for small values of x, near the center of the SAN. IV. SOME IONIC CURRENT MODELS
Ls (1 + 0.7745e − ( x − 2.05) /( 0.295) )
1 a ∂V ( x, t ) ∂V ( x, t ) =− itot ( x, t ) + D a a ∂t ∂x C m ( x) a
a
∂V s ∂n
=0
Each one of the currents involved in the electrical activity of the SAN is generated in the model using a set of coupled equations; these are dependent form the membrane potential and the equilibrium potentials. We present only the model from one of these currents (if you want the other equations could review [9]). The next equations are the set used to determine the 4AminoPirydine sensitive currents (ito and isus)
(4)
x =0
a
∂V ∂n
(3)
=0
(5)
x=L
itot = i Na + iCa , L + iCa ,T + ito + i sus + i K , r + i K , s + it
(6)
ito = g to qr (V − E K )
+ ib , Na + ib ,Ca + ib , K + i NaCa + i p + I cap
q∞ =
Where: Va(x), Vs(x): Membrane Potential of the atrial muscle cells or SAN in the one-dimensional model of the intact SAN at a x distance from the center of the node. Cma(x) Cms(x): Capacitance of the of the atrial muscle cells or SAN in the one-dimensional model of the intact SAN at a x distance from the center of the node. x: Distance from the center of the node. itota(x) itots(x): Total ionic current in the atrial muscle cells or SAN in the one-dimensional model of the intact SAN at a x distance from the center of the node. In this equations the s superindex denotes the SAN, the a superindex denotes the atrial muscle, t is the time, istot or iatot is the total current. Ds and Da are the coupling coefficient that models the electrotonic interaction between the SAN cells or atrial muscle cells.
τ q = 10.1 *10−3 + ..
1 1+ e
(V +59.37 ) / 13.1
65.17 * 10− 3 0.57 * e− 0.08(V + 49 ) + 0.24 * 10− 4 * e0.1(V + 50.93)
r∞ =
τ r = 2.98 * 10 −3 + ..
dq q∞ − q = τq dt 1
1 + e −(V −10.93) / 19.7
15.59 *10 −3 0.09 (V + 30.61) 1.037 * e + 0.369 * e −0.12 (V + 23.84 ) dr r∞ − r = dt τr
isus = g sus r (V − E K )
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(7)
(8) (9) (10) (11) (12)
(13) (14)
potential and the equilibrium potential for the ionic type that form the current (Na+, K+, Ca++). This difference determines the impulse give to the ions and the current intensity form each type of ions. For the solution of the model it is necessary to take an initial value of the action potential, with this value each ionic current is solved, using a numerical method like 4th order Runge–Kutta–Merson for the resolution of ordinary differential equation and the model associated with each one like the 4AP sensitive currents model. After this the total current value is obtained for the cell and the membrane voltage is determined solving the partial differential equation from (1), using one explicit Euler method. The model was implemented using Matlab® and the results was contrasted with the results publicized by the model’s authors.
ito , isus: Transitory and permanent components of the 4-AP sensitive current. gto, gsus: Conductance of ito and isus q Inactivation variable of ito r Activation variable of ito V Membrane potential Equilibrium potential for K+ EK q∞ , r∞ Stationary values for q and r τq , τr Time constants for q and r Equations (7) and (14) corresponds to the compute of the transitory and sustained current, respectively. In these equations we must remark some factors: gto and gsus are the conductance provided by the ionic channel in open condition. Conductance is the measure of the facility that the channel offers to the ionic interchange. Factor r is the activation variable, it models the opening of the ionic channel for this current, the permission give by the molecule in the cellular membrane for the pass of the ions; the opening and closing of the ionic channel is dynamic and must be modeled in an accurate form. While the channel is opening the activation variable increases its value, causing the increase of current; (13) models the dynamic associated with the r variable as a first order submodel with variable time constant and final value obtained from (12) and (11) respectively. Remember that the time constant (maximum value 15 ms for this variable) is associated with the velocity of opening in the channel. The q factor corresponds to the inactivation variable and models the closing of the ionic channel for this current. While the potential increases this variable decreases causing a decrease of the current. The dynamic variation of the inactivation variable is analogue to the activation variable’s dynamics and is represented by (8), (9) and (10), its time constant increases until 130 ms aprox. And it reveals that the closing time is greater than the opening time of the 4AP sensitive current. The last factor from (7) and (14) is the difference between the membrane potential V and the equilibrium potential for potassium EK , this is due because the 4AP sensitive currents are associated with potassium ionic channels. From this equation set is obtained a current profile near to the exponential both in the activation phase and in the inactivation phase. We can observe that both currents are modeled by mean of simple ordinary differential equations of first order, instead are dependent of the membrane potential, variable for witch we the solution is searched. We only show the model of the 4AP sensitive currents due its simple fashion and because it contains the structure of all ionic currents. One point to remark is the existence in the models of a conductivity value (g) to take in account the facility for the ionic flux trough the channel. The activation and inactivation variables allow modeling the opening and closing of ionic channels, normally voltage dependant. And as a final point in each current the current value is proportional to the difference between the membrane
V. CONCLUSIONS AND FUTURE WORK A math model of the mammal SAN was presented; in agreement with its authors this model recovers the majority of the components observed in the practice. By the model implementation the university and the engineering and medicine school have obtained a tool to continue with the study of fundamental electrophysiological models of the hearth. Due to the great number of parameters involved in the modeling of the SAN the next step in our investigation is review of robustness of the model using a sensitivity analysis that allows obtains critical parameters to be obtained in a more accurate level and irrelevant parameters that may be simplified in the model REFERENCES [1] Bleeker W., Mackaay J, “Functional and morphological organization of the rabbit sinus node”, Circ. Res 46:11-22, 1980. [2] DiFrancesco D, and D. Noble, “A model of cardiac electrical activity incorporating ionic pumps and concentration changes”, Phil. Trans. R. Soc. Lond. 307:353-398, 1985. [3] Honjo H., Boyett M., Kodama I, “Correlation between electrical activity and the size of rabbit sinoatrial node cells”, J. Physiol. Heart Circ. Physi. 276:H1295-H1304, 1999. [4] Kodama I, Nikmaram, Boyett M. “Regional differences in the role of the Ca2+ and Na+ currents in pacemaker activity in the sinoatrial node”. Am. J Physiol Heart Circ Physiol 272:H2793-H2806, 1997. [5] Noble D. and Noble S, “A model of sinoatrial node electrical activity based on a modification of the DiFrancesco-noble Equations”, Proc. Roy. Soc. Lond. 222:295-304, 1984. [6] Opthof, T. “The mammalian sinoatrial node”, Cardiovasc. Drugs Ther 1: 573-597, 1988. [7] Yanagihara, A. Noma, H. Irisawa, “Reconstruction of sinoatrial node pacemaker potential based on the voltaje clamp experiments”, Jpn. J. Physiol. 30:841-857, 1980. [8] Zhang, H., Holden, A. V., and Boyett, M. R., “Computer modelling of the sinoatrial node”, Am. J. Physiology (Heart & Circulation), 279 (1): 397-421, 2000 [9] Zhang, H., Holden, A. V., Kodama, I. Honjo, H., Lei, M., Varghese,T. and Boyett, M. R., “Mathematical models of action potencials for centre and periphery sinoatrial node of rabbit heart”, Am. J. Physiology (Heart & Circulation), 279 (1): 397-421, 2000
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