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, , multi-scale i
2


, MULTI-SCALE , LARGE-SCALE
" , , ."() :
· · · (); ; ( ).

- , .
3


M odel

The rate of change of membrane potential is given by

dV dt = ( - I
where

ion

+I

stim

)/C

I I

ion

- the total ionic current density, - a stimulus current, - the membrane capacitance
4

stim

C


Luo-Rudy model
Transmembrane voltage V (mV)

C. Luo and Y. Rudy. Circ. Res. Vol. 68 N66,11501-1526 (1991) °° , 501

I

io n

dV ( - I ion + I stim ) = dt Cm = I Na + I Ca + I K + I K1 + I

Kp

+I

b

Ionic cur rents I Na = 23m 3h j (V - E Na ) I
Ca

Fast sodium current, I Na

Slow Background Potassium inward current, currents, current, Ib K, Kp, K1 I Ca

= GCa d f (V - ECa ),

I k1 = G k1 K1 (V - Ek1 ), I Kp= 0.0183 K p (V - E Kp ),

I k = Gk X X i (V - Ek ),

I b = 0.03921 (V + 59.87
time, ms

)
5


Ionic gates
m h

dy dt = ( y - y ) /
y =
1 y +

y

y =
y

a

y y

y +

y = f (V ), y = f (V )
I
Na

= 23m 3h j (V - E Na )
= GCa d f (V - ECa ),

j

I

d

Ca

f

I k = Gk X X i (V - Ek ),
d ([Ca
time, ms

X

Ca

]i )

/ dt = - 10 - 4 I

Ca

+ 0.07 10 -4 - [Ca
6

(

]i )


dy dt = ( y - y ) / y I Na = 23m 3h j (V - E Na )
For all range of V
0.32(V + 47.13) 1 - exp[-0.1(V + 47.13) V m = 0.08 exp( - ) 11 For V > - 40 mV

y =

a

y y

y +
I
Ca

1 y = y +

y

= GCa d f (V - ECa ),

m =

ECa = 7.7 - 13.0287 Ln([Ca ]i

h = a j = 0
0.3 exp(-2.535 10-7 V ) j = 1 + exp[-0.1(V + 32)] 1 h = 0.13(1 + exp[V + 10.66)/ - 11.1])

) 0.095 exp[- 0.01(V - 5)] d = 1 + exp[- 0.072(V - 5)] 0.07 exp[- 0.017(V + 44)] d =
1 + exp[0.05(v + 44)]

f =

0.012 exp[- 0.008(V + 28 1 + exp[0.15(V + 28)]

)]

f =
d ([Ca

For V < - 40 mV
h = 0 .1 3 5 e x p [ ( 8 0 + V ) / - 6 .8 ]
h = 3.56 exp(0.079 V ) + 3.1105 exp(0.35 V )

0.0065 exp[- 0.02(V + 30 1 + exp[- 0.2(V + 30 )]

)]

]i )

/ d t = - 1 0 -4 I

Ca

+ 0.07 10 -4 - [Ca

(

]i )

[-1.2714 105 exp(0.2444 V ) - 3.474 105 exp(-0.04391 V )] (V + 37.78) j = 1 + exp[0.311 (V + 79.23) 0 . 1 2 1 2 ex p ( - 0 . 0 1 0 5 2 V ) j = 1 + exp( -0.1378 (V + 40.14))

Luo-Rudy model ...
7


dy dt = ( y - y ) /

y

y =

a

y y

I k1 = G k1 K1 (V - Ek1 ),
G k1 = 0.6047

y +

1 y = y +

y

[K ]0

/ 5.4

I Kp= 0.0183 K p (V - E
E
Kp

=E

Kp

),
]}

K1



K1

=

1.02 1 + exp[0.2385 (V - EK 1 - 59.215)]

K p = 1 /{ + exp[(7.488 - V ) / 5.98 1

k1 =

0.49124 exp[0.08032 (V - Ek1 + 5.476 )] + exp[0.06175 (V - Ek1 - 594.31)] 1 + exp[- 0.5143 (V - Ek1 + 4.753)]

I k = Gk X X i (V - Ek ),
G k = 0.282

[K ]0

/ 5.4. V > -100mV

For

Xi =

X i = 1 forV -100mV 0.0005 exp[0.083(V + 50 = 1 + exp[0.057(V + 50)]

2.837 {exp[0.04(V + 77 )] - 1 (V + 77 ) exp( 0.04(V + 35))

)]

=

0.0013 exp[- 0.06(V + 20 1 + exp[ -0.04(V + 20)]

)]
8

Luo-Rudy model


ACTION POTENTIAL
Transmembrane voltage
mV +50 0 t, sec

dV dt = ( - I

ion

+I

stim

)/C

APD
-7 2 -8 4

DI
Threshold Rest Time

PCL APD - action potential duration DI - diastolic interval PCL - pacing cycle length

I

stim - stimulus current

PCL

I

s tim



9

s tim


Steady states of action potential duration
PCL = 180 ms

APD = 120

Transmembrane voltage, mV

PCL = 152 ms

APD = 108

1:1

PCL = 140 ms

2 :1
APD = 178
-72

Time, ms
10

*

Action potential duration, ms

1:1


100 50 0 -50 -100 0 100 50 0 -50 -100 0

APD

=113ms

1:1

P CL = 1 6 0
50 100

150

200

250

300

350

400

450

500

APD1 =151ms APD =33ms
2

2:2

P CL = 1 5 2

50

100

150

200

250

300

350

400

450

500

100

Bifurcation diagram
Steady values of action potential duration
2:1 4:3 2:2

50 0 -50 -100 0

APD=179ms

2:1

50

100

150

200

250

300

350

400

450

500

100 50 0

PCL = 140
APD1 =183ms APD =165ms
2

4:2

(APD=QT), ms

4:2 1:1

-50 -100 0

50

100

150

200

250

300

350

400

450

500

100 50

PCL = 132
APD1 =181ms APD =118ms
2

4:3
= 32m s

0
-50 -100

APD
1

0

50

PCL = 120

100

150

200

250

300

350

400

450

500

Pacing cycle length (PCL=RR), ms

I st = 30 µA / cm

2

11


Michael Rubart and Douglas P. Zipes, Mechanisms of sudden cardiac death, J. Clin. Invest. 115,2305 (2005). "While the implantable cardioverter defibrillator (ICD) improves survival in high-risk patients , standard antiarrhythmic drug therapy has failed to reduce, and in some instances has increased, the incidence of SCD. In fact, the greatest reduction in cardiovascular mortality (including SCD) in patients with clinically manifest heart disease has resulted from the use of beta blockers and nonantiarrhythmic drugs"

12


Noisy Unmaskers of Multistability
Steady values of action potential duration, (APD) ms

f =0

I st = 30 µA / cm

f = 1 ms
2

f = 0.5 ms
Pacing cycle length, ms

f = 5 ms
1 Pacing cycle length, (PCL) ms3


Experimental observations of bi-stability in cardio dynamics

G. R. Mines, ``On dynamic equilibrium in the heart,'' J. Physiol. London 46, 349­383 ,1913.

Yehia, A.R., Jeandupeux, D., Alonso, F., Guevara, M.R., Hysteresis and bi-stability in the direct transition from 1:1 to 2:1 rhythm in periodically driven single ventricular cells. Chaos 9 (4) (1999), 916­931. Guevara M.R., Ward G., Shrier A., Glass L., Electrical alternans and period-doubling bifurcations. In: Computers in Cardiology. IEEE Computer Society, Silver Spring, MD, (1984) 167­170. Goldhaber, J. I., Xie, L. H., Duong, T, Motter, C., Khuu, K, and Weiss, J. N. (2005). Action potential duration restitution and alternans in rabbit ventricular myocytes: the key role of intracellular calcium cycling, Circ. Res. 96, pp. 459­466.

14


15


Double-stage protocol of stimulation
PCL=180
A

CI

PCL=152
APD

Transmembrane voltage, mV

PCL=180
CI

PCL=152
APD
1

B

APD

2

PCL=180
CI

PCL=152

APD APD

1

C

2

Multistability -

the coexistence of different dynamical regimes of cardiac cell-model at a fixed set of stimulation parameters 16

*

Action potential duration, ms


Multistability as lability invariant
Steady values of action potential duration, (APD) ms
I st = 30 µA / cm
2: 1 4: 2 1: 1 4:3 1:1 2: 2 4:3
2

I s t = 4 0 µ A / cm
2:1 4:2

2

M U LTI STAB I LITY

2: 2

M U LTI STAB I LITY

1:1

4:3

1:1

4:3

Pacing cycle length, (PCL) ms
17 Preliminary stimulation: PCL=180

CI=150 ms

CI=120 ms


Multistability as variability invariant
Steady values of action potential duration, (APD) ms G
Ca 0 = 0.06; d = 0.75 d ; f = 0.75
2: 1 4: 2 1: 1 4:3 1:1

0 f

G

Ca

0 = 0.08; d = 0.5 d ; f = 0.5
2:1

0 f

M U LTI STAB I LITY

CI=120 ms

M U LTI STAB I LITY

V,mV

2: 2

2:1

1:1

CI=150 ms

4:3

1:1

Time, ms

I st = 30 µA / cm

Pacing cycle length, (PCL) ms
2

Preliminary stimulation: PCL=180

18


Basins of attraction - Vulnerable windows
Steady values of action potential duration, ms
2: 1

PCL=180 ms

PCL=152 ms

4:2 4: 2 1: 1 4:3

2: 2

CI=120 ms

Transmembrane voltage, mV

-84

1:1

4:3

I st = 3 0 µ A / c m

2
100 100

CI=150 ms

50 0

APD1 =151ms APD =33ms
2

2:2

50 0 -50 -100 500 0

=113ms

1:1

Pacing cycle length, ms
100 50 0 -50 -100 0 100

-50 -100 0

APD
50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350 100 50

400

450

APD=179ms

2:1

50 0 -50

APD1 =183ms

APD =165ms
2

4:2

APD1 =181ms

APD =118ms
2 1

4:3
APD =32ms

0
-50 -100 500

50

100

150

200

250

300

350

400

450

500

-100 0

50

100

150

200

250

300

350

400

450

0

5 0

100

15 0

200

25 0

30 0

350

400

450

19 500


20



. . , - . , , .
21


Stay in good health! Listen to your heart....

Thank you!
22