NBCR Summer Institute 2006:Multi-Scale Cardiac Modeling with

Continuity 6.3

Thursday:Monodomain Modeling in Cardiac

Electrophysiology

Sarah FlaimAndrew McCulloch, and Fred Lionetti

Thursday: Monodomain Modeling in Cardiac Electrophysiology

Modeling cardiac action potential propagation in a monodomain - Continuity 6.3

Cardiac myocyte ionic models - MATLAB

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 model ODEs Year

Fitzhugh-Nagumo 2 1960

Beeler-Reuter 8 1977

Puglisi-Bers 21 2000

Flaim-Giles-McCulloch 87 2006

Simplified

Biophysical

0 50 100 150 200 250 300-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

u

v

Simplified model: Modified Fitzhugh-Nagumo (MFHN)

1 2( )(1 )du

c u u a u c vdt

( )dv

b u dvdt

u = excitation variablev = recovery variable

Multiple ion channels exist in the cell membrane

K loss

Review: Cardiac Action Potential

Vm

Inward

Na+ current

Inward Ca2+ current

Outward K+ currentsOutward

K+ currents

Stimulus

Rise in [Ca2+]i → mechanical contraction

Sarcoplasmic Reticulum

Contractile Elements

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)

1KICas II NaI

1xI

Fast inward Na+ current

Slow inward Ca2+ current

Time & voltage dep. outward K+ current

Time indep. outward K+ current

stimsNaxKm

m IIIIICdt

dV

11

1

Beeler GW, Reuter H (1977) J Physiol 268(1): 177-210

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

yyy

yy

y

y

andywith

yy

dt

dy

1

7

541

36

32

ce

cVcecor cVc

mcVc

yy m

m

isi I

dt

dCa1007.010

Ca 77

stimsNaxKm

m IIIIICdt

dV

11

1

Biophysical model: Beeler-ReuterSolution Method

% Initial conditionsVm(1) = -84.5732; % Membrane VoltageCa_i(1) = 1.782e-7; % Intracellular Calcium… % Gating parameters…p = [Vm(1) Ca_i(1) x1(1) …]; % Initial condition vector

% Solve tspan = [0.0 300.0]; % Time span: 0 to 300 ms[t, new_p] = ode15s(@BR_deriv_vts, tspan, p); % Use MATLAB stiff solverVm = new_p(:,1); % Save solution Ca_i = new_p(:,2); …

% Function filefunction pdot = BR_deriv_vts(t,p)… % Intermediate variablesi_Ca = g_s * d * f * (Vm - E_Ca); % Ionic currentsi_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/dtpdot(2,1) = -1e-7 * i_Ca + 0.07 * (1e-7 - Ca_i); %d[Ca]/dt...

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

Luo CH, Rudy Y (1994) Circ Res 74(6): 1071-96

Functional integration: Ionic currents + [Ca2+]i handling

Bluhm WF et al (1998) Am J Physiol 274(3 Pt 2): H1032-40

Functional integration:Excitation-contraction coupling

Clancy CE, Rudy Y (1999) Nature 400(6744): 566-9

Functional integration: Ion channel kinetics and gene mutations

Jafri MS, Rice JJ, et al. (1998) Biophys J 74(3): 1149-68Winslow RL, Rice JJ, et al. (1999) Circ Res 84(5): 571-86

Functional integration:E-C coupling + Ca2+ subspaces

Michailova AP, McCulloch AD (2001) Biophys J 81(2): 614-29

Functional integration: Metabolic regulation of E-C coupling

Saucerman, JJ et al., J Biol Chem 278: 47997 (2003)

-Adrenergic regulation of excitation-contraction coupling

Thursday: Monodomain Modeling in Cardiac Electrophysiology

Modeling cardiac action potential propagation in a monodomain - Continuity 6.3

Cardiac myocyte ionic models - MATLAB

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 atriumLeft atrium

Right ventricle

Left ventricle

Multiscale Tissue/Organ Models

• A multistep process:– Step 1 – Anatomy model

– Step 2 – Governing Equation• Derivation

• FE formulation

– Step 3 – Cell Model

m ionm

m

V ID V

t C

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:

Vm(x,t)Ia

Im

ma

VRI

x

a

m

II

x

2

2m a

m

V IR RI

x x

m

c m

VI C

t

2

2

1 1m mion

m m

V VI

t C C R x

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

kx

jx

ix

where ˆˆˆ321

E

GEJ Physically, current flux in a cable occurs in the direction of greatest potential drop

Vm(x,t) = i - e

i(x,t)

e(x,t)

mV J G

Assume e = 0 mV for monodomain

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 positiveOutward currents are negative

surfareaarea dIIdd ionc JJJ

Flux in through darea

Flux out through darea

Sum of currents through membrane dsurf– =

J J + dJ

IcIionEnd Area = darea

Membrane surface Area = dsurf

dvolume

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):

areaarea dddd JJ)(JJ

• Change in flux in the x1 direction [ (1/cm)(µA/cm2)(cm3) = µA ]:

• Total change in flux for general 3D cable (µA): volumevolume

JJJ

321

321 ddxxx

d xxx JJ

volumeJJ

1321

1

11 dx

dxdxdxx

xx

darea

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:

J J + dJ

IcIion

Membrane surface Area = dsurf

• With dsurf/dvolume set equal to the surface to volume ratio of a cell (Sv) and flux expressed in terms of potential, we have:

( ) mm v m ion

VG V S C I

t

volume surfmm ion

Vd C I d

t

J

surf surf surfmm c ion m ion

VI d I I d C I d

t

• 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

Summary: Monodomain model of impulse propagation

1 Rogers, JM et al. (1994).2 Luo, CH et al. (1994).3 Flaim et al.(2006).

m ionm

m

V ID V

t C

D has units of diffusion (cm2/msec), by combining G with Sv (1/cm) and Cm (µF/cm2)

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

x1

x2 1

21

2

Global coordinates: xi

Local element coordinates:i

Fiber coordinates: i

Review of finite element method:Weighted residual methods

2 2

[ ]

( ) d 0

( )

( )

Let ( ) and minimize:

Galerkin's Method

Least Squares Method

Collocatio

( ) d 2 d 2 d 0

Let (

n Me

) - a Dirac delta funct

thod

i

i ii

i

i i i

i

Lu f w

u U

w

w Lu f R

RLu f R R

U U U

w

x

x

x

x x x

[ ] [ ]

[ ]

[ ]

on, where

- d

( ) - d 0

0

i i

i

i

f f

Lu f

R

x x x x

x x

x

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 DVm

• Cubic Hermite interpolation of Vm

4 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 conditionRogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:743-757.

Collocation-Galerkin FE Method

Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:743-757.

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 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:

p jqrm m ion

j p r q m

V V Iv xD

t x v v x C

ijm m ion

i j m

V V ID

t x x C

^

m m i miV V V

m ionm

m

V ID V

t C

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.

2l lmm m m ion

l l m m

V V V IA B

t C

2 2 2j n m p l ll lm qr qr

n m j p r m q q r

x vA B D D

x v v v v v

l mlm qr

q r

B Dv v

2( )( ) ( )( )

( )l lmm ion

b b ml l m m

dV IA B V

dt C

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”.

m ionm

m

dV IM KV

dt C

11 (1 )

n nn nm m ion

m mm

V V IM K V V

t C

1 1n n ionm m

m

IM tK V M tK V t

C Rearrange to yield:

Collocation Galerkin FE Method:

0mV

n

Boundary no-flux condition:

0l m

ijm

i j m

V Vm

t n t x x

In component form (Galerkin):

( ) ( ) ( )( ) 0

l mij m

b ci j m

VdS

x x t

Rearranging and coordinate transformations:

Subcellular

Clancy & Rudy (1999)

( )o m revNa NaI P G V E

Open

Inactivated

Closed

Markov model1. Prescribe transition rates

2. Calculate the probabilities

Cellular

ionm

m

IdVdt C

1. Solve for transmembrane potential

Tissue

ionm m

m m

IV D Vt C S C

1. Solve resulting reaction-diffusion equation •Finite elements

•Implicit time-stepping•Operator splitting

Qu Z, Garfinkel A (1999)

Subcellular

Saucerman et al (2004)

Cellular

Saucerman et al (2004)

Saucerman et al (2004)

Endo

10

0 m

V

M Cell

Epi

ECG0.5

mV

200 ms

Endo

10

0 m

V

M Cell

Epi

ECG0.5

mV

200 ms0 0.25 0.5 0.75 1

150

200

250

300

Endo EpiTransmural Position

G589D+Iso

WT+Iso

WT, G589D

AP

D90

(ms)

A BTissue

G589D mutation prevents PKA-

mediated phosphorylation

of KCNQ1 (IKs)

Action potential duration shortens

in WT but prolongs in G589D

with ISO

APD prolongation is greatest on the endocardium → increased TDR

Subcellular

Clancy et al (2003)

UIC3

UC3

LC3

UO

UIC2 UIM1UIF

UC2 UC1

LOLC2 LC1

UIM2

Closed states

“Inactivated states” states

UIC3

UC3

LC3

UO

UIC2 UIM1UIF

UC2 UC1

LOLC2 LC1

UIM2UIC3

UC3

LC3

UO

UIC2 UIM1UIF

UC2 UC1

LOLC2 LC1

UIM2

UO

UIC2 UIM1UIF

UC2 UC1

LOLC2 LC1

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 (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

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

A Note on Units

• Unit of conductivity (mS/cm), Siemens are inverse of resistivity:

secμ1000μμ1

2222

2m

cm

mF

cm

F

cm

F

cmmS

cmF

cm

cmmS

CS mV

G

• Therefore the units of flux are:

• From conductivity to diffusion:

V

A1S

22 cm

μA

cm1,000,000

A

mV1000

V

cm

mV

cm1000

1

cm

mV

cm

mS

GEJ

• All terms in the cable equation have units of (mV/msec) including:

secsecμ

μ

11

m

mVV

F

V

F

AA

FI

C ionm