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Thursday: Monodomain Modeling in Cardiac Electrophysiology

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. Thursday: Monodomain Modeling in Cardiac Electrophysiology.

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Thursday: Monodomain Modeling in Cardiac Electrophysiology

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  1. NBCR Summer Institute 2006:Multi-Scale Cardiac Modeling with Continuity 6.3Thursday:Monodomain Modeling in Cardiac ElectrophysiologySarah FlaimAndrew McCulloch, and Fred Lionetti

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

  3. Cardiac myocyte ionic models: simplifiedvs. biophysical • Simplified: • Mathematically represent cellular properties • Fewer equations to solve -> faster! • Parameters do not map to experimental measurements • 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 Biophysical

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

  5. Multiple ion channels exist in the cell membrane K loss

  6. Review: Cardiac Action Potential Inward Ca2+ current Outward K+ currents Outward K+ currents Inward Na+ current Vm Stimulus

  7. Rise in [Ca2+]i→ mechanical contraction Sarcoplasmic Reticulum Contractile Elements

  8. Slow inward Ca2+ current Fast inward Na+ current Time & voltage dep. outward K+ current Time indep. outward K+ current Biophysical model: Beeler-ReuterIonic 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, Vm Iion = f (Vm, [Ca]i, x1, m, h, j, d, f) Beeler GW, Reuter H (1977) J Physiol 268(1): 177-210

  9. Biophysical model: Beeler-ReuterEquations • 8 time dependent ODEs • 6 ODE’s describe the state of gated ion channels (y represents 6 gating conductance variables x1, m, h, j, d, and f) • the gating parameters αy and βy are calculated from patch clamp data • 1 ODE describes intracellular Ca2+ concentration • 1 ODE describes membrane voltage • Statement of charge conservation

  10. Biophysical model: Beeler-ReuterSolution Method % Initial conditions Vm(1) = -84.5732; % Membrane Voltage Ca_i(1) = 1.782e-7; % Intracellular Calcium … % Gating parameters… p = [Vm(1) Ca_i(1) x1(1) …]; % Initial condition vector % 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 + 0.07 * (1e-7 - Ca_i); %d[Ca]/dt ... % Solve tspan = [0.0 300.0]; % Time span: 0 to 300 ms [t, new_p] = ode15s(@BR_deriv_vts, tspan, p); % Use MATLAB stiff solver Vm = new_p(:,1); % Save solution Ca_i = new_p(:,2); …

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

  12. Functional integration: Ionic currents + [Ca2+]i handling Luo CH, Rudy Y (1994) Circ Res74(6): 1071-96

  13. Functional integration:Excitation-contraction coupling Bluhm WF et al (1998) Am J Physiol274(3 Pt 2): H1032-40

  14. Functional integration: Ion channel kinetics and gene mutations Clancy CE, Rudy Y (1999) Nature400(6744): 566-9

  15. Functional integration:E-C coupling + Ca2+ subspaces Jafri MS, Rice JJ, et al. (1998) Biophys J74(3): 1149-68 Winslow RL, Rice JJ, et al. (1999) Circ Res84(5): 571-86

  16. Functional integration: Metabolic regulation of E-C coupling Michailova AP, McCulloch AD (2001) Biophys J81(2): 614-29

  17. b-Adrenergic regulation of excitation-contraction coupling Saucerman, JJ et al., J Biol Chem 278: 47997 (2003)

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

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

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

  21. 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, Ia (mA) • membrane current, Im (mA/cm) • resistance, R (m/cm) • Using Ohm’s Law and conservation of charge we get: • Assume Im is both capacitive and ionic: Im = Ic + Iion where • Therefore: Im Vm(x,t) Ia

  22. 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/cm2), is proportional to the electric field vector i(x,t) e(x,t) Vm(x,t) = i - e Assume e = 0 mV for monodomain Physically, current flux in a cable occurs in the direction of greatest potential drop

  23. Iion Ic End Area = darea J J + dJ dvolume Membrane surface Area = dsurf 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 – =

  24. darea 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 x1 direction [ (1/cm)(µA/cm2)(cm3) = µA ]: • Total change in flux for general 3D cable (µA):

  25. Iion Ic J + dJ J Membrane surface Area = dsurf Derivation of governing equation:cable theory – 3D • Total current leaving through the membrane surface dS [ (µA/cm2)(cm2) = µA ]: Note that Iion is calculated by the system of ODE’s described earlier and depends on Vm • Conservation of charge results in: • With dsurf/dvolume set equal to the surface to volume ratio of a cell (Sv) and flux expressed in terms of potential, we have:

  26. Summary: Monodomain model of impulse propagation D has units of diffusion (cm2/msec), by combining G with Sv (1/cm) and Cm (µF/cm2) • Vm = Transmembrane voltage, coupled across finite element degrees of freedom • D = Diffusion tensor, represents anisotropic resistivity with respect to local fiber and transverse axes • Iion = 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 1 Rogers, JM et al. (1994). 2 Luo, CH et al. (1994). 3 Flaim et al.(2006).

  27. 2 1 2 x2 1 x1 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 Vm is approximated with cubic Hermite interpolation • Allows complex domain • Must convert between coordinate systems to solve governing equation Global coordinates: xi Local element coordinates:i Fiber coordinates: i

  28. Review of finite element method:Weighted residual methods

  29. 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Vm • Cubic Hermite interpolation of Vm4 DOF/node in 2D8 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; 41:743-757.

  30. Collocation-Galerkin FE Method Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:743-757.

  31. 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:  are the element basis functions (functions of the element coordinates i) We begin by rewriting the governing equation in component form: Here Dij are functions of the global coordinates, xi. It is more convenient for us to express them with respect to the local fiber coordinate system, vp, as the diffusion tensor then becomes diagonal:

  32. 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.

  33. 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: Rearrange to yield:  is a weighting symbol. If  = 1, the method is termed “fully implicit”. When  = 0, the method is termed “explicit”.

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

  35. 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) Subcellular Markov model 1. Prescribe transition rates 2. Calculate the probabilities Inactivated Open Clancy & Rudy (1999) Closed

  36. Cellular Saucerman et al (2004) Tissue Saucerman et al (2004) Subcellular G589D mutation prevents PKA-mediated phosphorylation of KCNQ1 (IKs) Action potential duration shortens in WT but prolongs in G589D with ISO APD prolongationis greatest onthe endocardium → increased TDR Saucerman et al (2004)

  37. UIC3 UIC2 UIC2 UIF UIF UIM1 UIM1 UIM2 UIM2 UC3 UC2 UC2 UO UO UC1 UC1 LC3 LC2 LC2 LC1 LC1 LO LO Cellular Flaim et al, in press Tissue Flaim et al, in preparation Subcellular “ “ Inactivated states” states Inactivated states” states SCN5A-I1768V mutation augments the late Na+ current (INaL) EADs occur in midmyocardial and endocardial (but not epicardial) myocytes Endocardial EADs trigger epicardial APs resulting in “R on T” extrasystoles and polymorphic VT UIC3 UIC3 UIC2 UIC2 UIF UIF UIM1 UIM1 UIM2 UIM2 UC3 UC3 UC2 UC2 UO UO UC1 UC1 Open states LC3 LC3 LC2 LC2 LC1 LC1 LO LO Clancy et al (2003) Closed states Closed states

  38. Examples: • Continuity Example: • https://nbcr.net/pub/wiki/index.php?title=Electrophysiology • Suggested Experimentation: • https://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_Reuter

  39. 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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