High order time-stepping methods for cardiac electrophysiology models

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High order time-stepping methods for cardiac electrophysiology models

Charlie Douanla Lontsi, Yves Coudière, Charles Pierre

To cite this version:

Charlie Douanla Lontsi, Yves Coudière, Charles Pierre. High order time-stepping methods for cardiac

electrophysiology models. IHU Liryc-Workshop 2016, Sep 2016, Bordeaux, France. �hal-01406536�

High order time-stepping methods for cardiac electrophysiology models

C. Douanla Lontsi ^{1, 2, 3, 4} , Yves Coudière ^{1, 2, 3, 4} , C. Pierre ⁵

1 Carmen Research Team, Inria, ² IHU Liryc, ³ Univ. Bordeaux, IMB, ⁴ CNRS, IMB,

5 CNRS, LMA Pau

Problematic of the time numerical integration in cardiac electrophysiology

Ionic models :

• Stiff differential equations

• Large system

• Highly nonlinear

Numerical contraints :

• Stability + accuracy require fine grid

Numerical consequences :

• High CPU cost

• Lack of precision aaa

• Conditioning problems

Questions:

• How can we cope this situation?

• Can high order solver (s) be a solution?

The problem

EXAMPLE IONIC MODEL (TNNP)

T RANSMEMBRANE POTENTIAL V.

-100 -80 -60 -40 -20 20 40 0

0 200 400 600 800 1000

V oltage (mV)

Time (ms)

F AST INWARD SODIUM CURRENT .

-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

0 200 400 600 800 1000 Curr ent ( A/F /cm 2 )

Time (ms)

I _{N a}

Behavior of the physiological parameters :

• Different time and space scales.

• Fast and slow variables.

• Stiff wave fronts.

Numerical consequences :

• Numerical instabilities.

• High computational cost.

• Significant loss of accuracy

State of the art

S TABILIZATION PRINCIPLE AND SOME STABILIZED SOLVERS

Equation on gating variables : dW

dt = W _∞ (V ) − W τ (V ) .

Forward Backward scheme : W ⁿ⁺¹ − W ⁿ

∆t = W _∞ (V ⁿ ) − W ⁿ⁺¹ τ (V ⁿ ) .

This scheme allows large ∆t , linear part = _τ ⁻¹ _{(V )} . Linear part known at time t _n ⇒ Stabilization.

O RDER ONE SOLVERS :

• Forward backward Euler

• RL1: Rush Larsen ( 1978 ).

O RDER TWO SOLVER :

• RL2: Perego and Veneziani ( 2009 ).

H IGH ORDER METHODS :

• Exponential integrator of Adams type: Norsett ( 1969 ).

• Exponential Runge kutta and exponential multi-step (the last ten years).

An innovative method

H ^IGH - ^ORDER R ^USH L ARSEN METHODS

Consider the problem,

y ⁰ = a(t, y )y + b(t, y ), y (0) = y ₀ ,

transformed in each time discretization interval (t _n , t _n+1 ] into,

y ⁰ = α _n y + c _n (t, y ),

with c _n (t, y ) = (a(t, y ) − α _n )y + b(t, y ) and α _n a sta- bilizer. For t ∈ (t _n , t _n+1 ] , the exact solution of this problem satisfies the variation of constant formula

y (t) = e ^A(t) (y _n +

^Z

_t ^t

n e ^{−A(τ )} c _n (τ, y)dτ ),

with A(t) = α _n (t − t _n ) . If we set t = t _n+1 and approx- imate c _n by a constant β _n , we obtain the following definition of RL _k

y _n+1 = y _n + hϕ ₁ (α _n h)(α _n y _n + β _n ).

ϕ ₁ (z ) = ^{e z} _z ⁻¹ , α _n and β _n are set so that the conver- gence order k is ensured.

R ^USH L ARSEN ORDER 3 α _n = 1

12 (23a _n − 16a _n−1 + 5a _n−2 ), β _n = 1

12 (23b _n − 16b _n−1 + 5b _n−2 ) + h

12 (a _n b _n−1 − a _n−1 b _n ).

Scheme properties

A DVANTAGES AND CONVERGENCE

• Explicit k -multi-step method:

y _n , y _n−1 , . . . , y _n−k+1 → y _n+1 .

• Stability: The same critical time-step as the implicit schemes.

• Easy to implement.

• Cost: one evaluation of the model at every time step (minimal).

T ^HEOREM . The RL _k scheme is stable under per- turbation and converges with an error of order k , under the assumptions that a(t, y ) and b(t, y ) in the previous problem are C ^k functions and its solution y (t) is defined in [0, T ) .

N UMERICAL ILLUSTRATION (Relative error for the BR model in Log/Log scale)

1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

0.001 0.01 0.1 1

e ( h )

Time step h (ms) RL1 RL2

RL3 RL4 slope 1 slope 2 slope 3 slope 4

CPU cost

C ^OMPARISONS

CPU time plotted in Log/Log scale against the er- ror for RL _k , RK ₄ and CN

0.01

0.0001 0.01 1

Cost in CPU time ( s )

e(h) RL ₁

RL ₂ RL ₃ RL ₄ RK CN ₄

Discussion and conclusion

D ISCUSSION AND CONCLUSION

D ^ISCUSSION :

• Ionic models are very stiff ⇒ classical explicit numerical solver must use small time-steps for stability ⇒ Large CPU time.

• High order RL _k remain stable for large

time-steps. Their critical time-steps don’t

depend on the stiffness but on the stabilizer’s choice.

C ^ONCLUSION :

• We found that high order RL _k is a good alternative to improve the accuracy with

neglibible additional cost w/r to RL1 scheme.

• High order RL _k allow the use of large time steps unlike classical explicit schemes. They are

suitable for solving stiff ODE.

• High order schemes in cardiac electrophysiology may allow more reliable simulations of long

lasting events.

References

[1] Coudiere, Y. and Douanla Lontsi, C. and Pierre, C.

High order Rush Larsen solver for stiff ODEs HAL Preprint (2015).

[2] Rush and Larsen.

A practical algorithm for solving dynamic membrane equations.

IEEE Trans Biomed Eng 1978.

[3] Hochbruck. and Ostermann.

Exponential multistep methods of Adams-type BIT Numer Math (2011).

[4] Beeler, G.W. and Reuter, H.

Reconstruction of the Action Potential of Ventricular Myocardial Fibers J. Physiol. (1977).

[5] Douanla Lontsi, C. and Coudière, Y. and Pierre, C.

Efficient High Order Schemes for Stiff ODEs in Cardiac Electrophysiology

CARI (2016).