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NBCR Summer Institute 2006: Multi-Scale Cardiac Modeling with Continuity 6.3 Thursday: Monodomain Modeling in Cardiac Electrophysiology Sarah Flaim Andrew McCulloch, and Fred Lionetti

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Thursday: Monodomain Modeling in Cardiac Electrophysiology Modeling cardiac action potential propagation in a monodomain - Continuity 6.3 Cardiac myocyte ionic models - MATLAB

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Cardiac myocyte ionic models: simplified vs. biophysical Biophysical: –Account for underlying physiology -> greater predictive power –Parameters (eg. ion channel conductances) relate to experimental measurements –Increasingly more complex (and more!) equations -> slow to solve Simplified: –Mathematically represent cellular properties –Fewer equations to solve -> faster! –Parameters do not map to experimental measurements Ionic modelODEsYear Fitzhugh-Nagumo21960 Beeler-Reuter81977 Puglisi-Bers Flaim-Giles-McCulloch Simplified Biophysical

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Simplified model: Modified Fitzhugh-Nagumo (MFHN) u = excitation variable v = recovery variable

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Multiple ion channels exist in the cell membrane K loss

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Review: Cardiac Action Potential VmVm Inward Na + current Inward Ca 2+ current Outward K + currents Outward K + currents Stimulus

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Rise in [Ca 2+ ] i → mechanical contraction Sarcoplasmic Reticulum Contractile Elements

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Biophysical model: Beeler-Reuter Ionic currents 4 ionic membrane currents plus a stimulus current are included Currents are functions of the independent variables of the ODE set: –6 gating variables –Calcium concentration, [Ca] i –Membrane potential, V m I ion = f (V m, [Ca] i, x1, m, h, j, d, f) Fast inward Na + current Slow inward Ca 2+ current Time & voltage dep. outward K + current Time indep. outward K + current Beeler GW, Reuter H (1977) J Physiol 268(1):

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Biophysical model: Beeler-Reuter Equations 8 time dependent ODEs 6 ODE’s describe the state of gated ion channels (y represents 6 gating conductance variables x 1, m, h, j, d, and f) –the gating parameters α y and β y are calculated from patch clamp data 1 ODE describes intracellular Ca 2+ concentration 1 ODE describes membrane voltage –Statement of charge conservation

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Biophysical model: Beeler-Reuter Solution Method % Initial conditions Vm(1) = ; % Membrane Voltage Ca_i(1) = 1.782e-7; % Intracellular Calcium … % Gating parameters… p = [Vm(1) Ca_i(1) x1(1) …]; % Initial condition vector % Solve tspan = [ ]; % Time span: 0 to 300 ms [t, new_p] = tspan, p); % Use MATLAB stiff solver Vm = new_p(:,1); % Save solution Ca_i = new_p(:,2); … % Function file function pdot = BR_deriv_vts(t,p) …% Intermediate variables i_Ca = g_s * d * f * (Vm - E_Ca); % Ionic currents i_Na = (g_Na * m^3 * h * j + g_NaC) * (Vm - E_Na); … pdot(1,1) = -(1/C_m) * (i_K1 + i_x1 + i_Na + i_Ca - STIMULUS); %dVm/dt pdot(2,1) = -1e-7 * i_Ca * (1e-7 - Ca_i); %d[Ca]/dt...

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Question: How can we make these models more biophysically detailed?

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Luo CH, Rudy Y (1994) Circ Res 74(6): Functional integration: Ionic currents + [Ca 2+ ] i handling

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Bluhm WF et al (1998) Am J Physiol 274(3 Pt 2): H Functional integration: Excitation-contraction coupling

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Clancy CE, Rudy Y (1999) Nature 400(6744): Functional integration: Ion channel kinetics and gene mutations

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Jafri MS, Rice JJ, et al. (1998) Biophys J 74(3): Winslow RL, Rice JJ, et al. (1999) Circ Res 84(5): Functional integration: E-C coupling + Ca 2+ subspaces

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Michailova AP, McCulloch AD (2001) Biophys J 81(2): Functional integration: Metabolic regulation of E-C coupling

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Saucerman, JJ et al., J Biol Chem 278: (2003) -Adrenergic regulation of excitation- contraction coupling

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Thursday: Monodomain Modeling in Cardiac Electrophysiology Modeling cardiac action potential propagation in a monodomain - Continuity 6.3 Cardiac myocyte ionic models - MATLAB

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Precise sequence of electrical activation → well- coordinated & efficient contraction An electrocardiogram (ECG) is used to measure the electrical activity of the heart and can detect “arrhythmias” (conduction abnormalities) 3. Right and left ventricles recover 1. Right and left atria activate 2. Right and left ventricles activate Einthoven (1912) Sinoatrial node Atrioventricular node Right atrium Left atrium Right ventricle Left ventricle

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Multiscale Tissue/Organ Models A multistep process: –Step 1 – Anatomy model –Step 2 – Governing Equation Derivation FE formulation –Step 3 – Cell Model

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Derivation of governing equation: cable theory – 1D Consider a cell as a cable with a conductive interior (cytoplasm) surrounded by an insulator (cell membrane) with: –axial current, I a (mA) –membrane current, I m (mA/cm) –resistance, R (m /cm) Using Ohm’s Law and conservation of charge we get: Assume I m is both capacitive and ionic: I m = I c + I ion where Therefore: V m (x,t) IaIa ImIm

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Derivation of governing equation: cable theory – 3D Consider a cell as a cable with a conductive interior (cytoplasm), a separated by an insulator (cell membrane) with: – intracellular potential, i (mV) –extracellular potential, e (mV) () –a 3D conductivity tensor, G (mS/cm) Electric field vector, E (mV/cm), is defined as a potential drop maintained spatially in a material By Ohm’s Law (in 3D), flux vector, J (µA/cm 2 ), is proportional to the electric field vector Physically, current flux in a cable occurs in the direction of greatest potential drop V m (x,t) = i - e i (x,t) e (x,t) Assume e = 0 mV for monodomain

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Derivation of governing equation: cable theory – 3D Current entering a section (volume) of cable (dvolume) must equal current that leaves the section of cable. Inward currents are positive Outward currents are negative Flux in through darea Flux out through darea Sum of currents through membrane dsurf –= JJ + dJ IcIc I ion End Area = darea Membrane surface Area = dsurf dvolume

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Derivation of governing equation: cable theory – 3D Total current traveling into and out of the cable through neighboring conductive volume, i.e. the ends (µA/cm2): Change in flux in the x 1 direction [ (1/cm)(µA/cm 2 )(cm 3 ) = µA ]: Total change in flux for general 3D cable (µA): darea

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Derivation of governing equation: cable theory – 3D Total current leaving through the membrane surface dS [ (µA/cm 2 )(cm 2 ) = µA ]: Note that I ion is calculated by the system of ODE’s described earlier and depends on V m Conservation of charge results in: J J + dJ IcIc I ion Membrane surface Area = dsurf With dsurf/dvolume set equal to the surface to volume ratio of a cell (S v ) and flux expressed in terms of potential, we have:

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V m = Transmembrane voltage, coupled across finite element degrees of freedom D = Diffusion tensor, represents anisotropic resistivity with respect to local fiber and transverse axes I ion = Transmembrane ionic current, determined by choice of cellular ionic model Ionic models are increasingly detailed: –Modified FitzHugh-Nagumo 1 – 2 ODEs –Luo-Rudy 2 – 9 ODEs –Flaim-Giles-McCulloch 3 – 87 ODEs Summary: Monodomain model of impulse propagation 1 Rogers, JM et al. (1994). 2 Luo, CH et al. (1994). 3 Flaim et al.(2006). D has units of diffusion (cm 2 /msec), by combining G with S v (1/cm) and C m (µF/cm 2 )

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Solution of monodomain model: finite element method Divide 2D domain into 4 sided elements (or 3D domain into 6- faced elements) with “nodes” at the vertices Geometry, local fiber orientation and material properties defined using linear Lagrange or cubic Hermite interpolation Spatial variation of V m is approximated with cubic Hermite interpolation Allows complex domain Must convert between coordinate systems to solve governing equation x1x1 x2x2 11 22 11 22 Global coordinates: x i Local element coordinates: i Fiber coordinates: i

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Review of finite element method: Weighted residual methods

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Collocation-Galerkin FE Method Collocation uses a weighted residual formulation, but the weights are Dirac delta functions. It solves strong form of PDEs Therefore needs high-order elements to interpolate second derivatives in D V m Cubic Hermite interpolation of V m 4 DOF/node in 2D 8 DOF/node in 3D Collocation points are Gauss-Legendre quadrature points Need one collocation point for each nodal degree of freedom Galerkin approximation of no-flux boundary condition Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:

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Collocation-Galerkin FE Method Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:

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Collocation Galerkin FE Method: Governing equation is a non-linear reaction diffusion equation: We seek an approximate solution to the reaction- diffusion equation in the form: We begin by rewriting the governing equation in component form: are the element basis functions (functions of the element coordinates i ) Here D ij are functions of the global coordinates, x i. It is more convenient for us to express them with respect to the local fiber coordinate system, v p, as the diffusion tensor then becomes diagonal:

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Collocation Galerkin FE Method: We then transform the spatial derivatives of Vm to the local finite element coordinate system, : In order to evolve a solution in time, a system of ODEs must be derived from this equation. Here we use the collocation method to satisfy the PDE at a discrete set of points.

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Collocation Galerkin FE Method: Thus we end up with a system of equations of the form: We must also discretize in time as well as space. Here we use a finite difference scheme: is a weighting symbol. If = 1, the method is termed “fully implicit”. When = 0, the method is termed “explicit”. Rearrange to yield:

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Collocation Galerkin FE Method: Boundary no-flux condition: In component form (Galerkin): Rearranging and coordinate transformations:

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Subcellular Clancy & Rudy (1999) Open Inactivated Closed Markov model 1. Prescribe transition rates 2. Calculate the probabilities Cellular 1. Solve for transmembrane potential Tissue 1. Solve resulting reaction- diffusion equation Finite elements Implicit time- stepping Operator splitting Qu Z, Garfinkel A (1999)

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Subcellular Saucerman et al (2004) Cellular Saucerman et al (2004) Tissue G589D mutation prevents PKA- mediated phosphorylation of KCNQ1 (I Ks ) Action potential duration shortens in WT but prolongs in G589D with ISO APD prolongation is greatest on the endocardium → increased TDR

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Subcellular Clancy et al (2003) UIC3 UC3 LC3 UO UIC2UIM1UIF UC2 UC1 LOLC2LC1 UIM2 Closed states “Inactivated states” states UIC3 UC3 LC3 UO UIC2UIM1UIF UC2 UC1 LOLC2LC1 UIM2 UIC3 UC3 LC3 UO UIC2UIM1UIF UC2 UC1 LOLC2LC1 UIM2 UO UIC2UIM1UIF UC2 UC1 LOLC2LC1 UIM2 Open states Closed states “Inactivated states” states Cellular Flaim et al, in press Flaim et al, in preparation Tissue SCN5A-I1768V mutation augments the late Na+ current (I NaL ) EADs occur in midmyocardial and endocardial (but not epicardial) myocytes Endocardial EADs trigger epicardial APs resulting in “R on T” extrasystoles and polymorphic VT

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Examples: Continuity Example: –https://nbcr.net/pub/wiki/index.php?title=Electrophysiologyhttps://nbcr.net/pub/wiki/index.php?title=Electrophysiology Suggested Experimentation: –https://nbcr.net/pub/wiki/index.php?title=Suggested_experimentationhttps://nbcr.net/pub/wiki/index.php?title=Suggested_experimentation MATLAB example (Beeler-Reuter): –https://nbcr.net/pub/wiki/index.php?title=Ionic_model_example_1:_Beeler_Reuterhttps://nbcr.net/pub/wiki/index.php?title=Ionic_model_example_1:_Beeler_Reuter

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A Note on Units Unit of conductivity (mS/cm), Siemens are inverse of resistivity: Therefore the units of flux are: From conductivity to diffusion: All terms in the cable equation have units of (mV/msec) including:

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