REDUCED ORDER MODEL IN CARDIAC ELECTROPHYSIOLOGY€¦ · Elisa SCHENONE Inria Junior Seminar 17th June 2014 10" Infarcted tissue: damaged area which cannot be activated! to build

Elisa SCHENONE 17th June 2014Inria Junior Seminar

REDUCED ORDER MODEL INCARDIAC ELECTROPHYSIOLOGY

Elisa SCHENONE

1

REO team, Inria Paris-RocquencourtLaboratoire Jacques-Louis Lions

‡,✻

✻

‡

Elisa SCHENONE 17th June 2014Inria Junior Seminar 2

teamNumerical simulation of biological flows

0 200 400 600�2

�1

0

1

2I

0 200 400 600�2

�1

0

1

2aVR

0 200 400 600�3

�2

�1

0

1

2

3V1

0 200 400 600�3

�2

�1

0

1

2

3V4

0 200 400 600�2

�1

0

1

2II

0 200 400 600�2

�1

0

1

2aVL

0 200 400 600�3

�2

�1

0

1

2

3V2

0 200 400 600�3

�2

�1

0

1

2

3V5

0 200 400 600�2

�1

0

1

2III

0 200 400 600�2

�1

0

1

2aVF

0 200 400 600�3

�2

�1

0

1

2

3V3

0 200 400 600�3

�2

�1

0

1

2

3V6

Blood flow modellingPast talks: Cristobal Bertoglio,Saverio Smaldone, Gregory Arbia

Respiration modellingPast talks: Jessica Oakes

Cardiac electrophysiology


teamNumerical simulation of biological flows

0 200 400 600�2

�1

0

1

2I

0 200 400 600�2

�1

0

1

2aVR

0 200 400 600�3

�2

�1

0

1

2

3V1

0 200 400 600�3

�2

�1

0

1

2

3V4

0 200 400 600�2

�1

0

1

2II

0 200 400 600�2

�1

0

1

2aVL

0 200 400 600�3

�2

�1

0

1

2

3V2

0 200 400 600�3

�2

�1

0

1

2

3V5

0 200 400 600�2

�1

0

1

2III

0 200 400 600�2

�1

0

1

2aVF

0 200 400 600�3

�2

�1

0

1

2

3V3

0 200 400 600�3

�2

�1

0

1

2

3V6

Blood flow modellingPast talks: Cristobal Bertoglio,Saverio Smaldone, Gregory Arbia

Respiration modellingPast talks: Jessica Oakes

Cardiac electrophysiology


✓Supervisors:

Muriel BOULAKIA and Jean-Frédéric GERBEAU

✓Subjects:➡Simulation of realistic Electrocardiograms (with Annabelle Collin)

➡Parameters estimation

➡Reduced order models and inverse problems with POD

➡Reduced order models with Approximated Lax Pairs (with Damiano Lombardi)

➡Inverse problems

3

PhD project:Inverse problems and reduced models in cardiac electrophysiology


✓Supervisors:

Muriel BOULAKIA and Jean-Frédéric GERBEAU

✓Subjects:➡Simulation of realistic Electrocardiograms (with Annabelle Collin)

➡Parameters estimation

➡Reduced order models and inverse problems with POD

➡Reduced order models with Approximated Lax Pairs (with Damiano Lombardi)

➡Inverse problems

3

PhD project:Inverse problems and reduced models in cardiac electrophysiology


Summary

4

✓ Bidomain Model equations

✓ PDEs numerical approximation

➡Finite Element Methods (overview)

➡Reduced Order Methods

✓ Numerical Results

✓ Conclusions and Perspectives


R

V3

F

V1 V2V4V5V6

L

Cardiac electrophysiology modelBidomain equations

5

[Collin, Gerbeau, Schenone - 2014]

I = uT (L)� uT (R) aVR = 1.5(uT (R)� uw) V1 = uT (V1)� uw

II = uT (F )� uT (R) aVL = 1.5(uT (L)� uw)...

III = uT (F )� uT (L) aVF = 1.5(uT (F )� uw) V6 = uT (V6)� uw

From 9 measures to 12 leads

Atria

Ventricles



5








5








‣ Bidomain equations

Am

�C

m

@vm

@t+ I

ion

(vm

, w)�� div(¯�

I

rvm

)� div(¯�I

ruE

) = Am

Iapp

�div((¯�I

+ ¯�E

)ruE

)� div(¯�I

rvm

) = 0@w

@t� g(v

m

, w) = 0



‣ Bidomain equations

Am

�C

m

@vm

@t+ I

ion

(vm

, w)�� div(¯�

I

rvm

)� div(¯�I

ruE

) = Am

Iapp

�div((¯�I

+ ¯�E

)ruE

)� div(¯�I

rvm

) = 0@w

@t� g(v

m

, w) = 0

‣ Ionic model

Iion

(u,w) = su(u� a)(u� 1) + wg(u,w) = ✏(�u� w)

๏ Fitzhugh-Nagumo (FhN) ๏ Mitchell-Schaeffer (MS)

Iion

(u,w) = wu2(1� u)

⌧in

� u

⌧out

g(u,w) =

8><

>:

1� w

⌧open

u ugate

� w

⌧close

u > ugate


A numerical approximationOverview on Finite Element Method (FEM)

7

FEM space

u = u(x)



7

xixi�1 xi+1

FEM space

u = u(x)

ui�1

ui+1ui



7

xixi�1 xi+1

vi+1vi�1 vi

xixi�1 xi+1

FEM space

u = u(x)

ui�1

ui+1ui

u(x) 'NX

i

uivi(x)



7

xixi�1 xi+1

FEM space

u = u(x)

ui�1

ui+1ui

u(x) 'NX

i

uivi(x)

'1(x)

'2(x)

u(x) ' u1'1(x) + u2'2(x)


FEM versus ROM

FEM space ROM space

�N = ['1 . . .'N ] 2 RN⇥N

eu = �TNu

VN ⌘ span{v1, . . . , vN } ⇢ Vdim(VN ) = N

eu =NX

i

ui'i(x), ui = hu,'iiu =NX

i

uivi(x), ui = hu, vii

VN ⌘ span{'1, . . . ,'N} ⇢ VN

x1

x2

x3

dim(VN ) = N ⌧ N


Proper Orthogonal Decomposition (POD)A “classical” approach to Reduced Order Models

‣Use information given by FEM solution(s) to build a suitable basis



‣Use information given by FEM solution(s) to build a suitable basis1.Solve with FEM the problem(s)

@t

u = F (u, @(n)x

,⇡), in ⌦⇥ [0, T ] ⇢ Rd ⇥ R+



‣Use information given by FEM solution(s) to build a suitable basis1.Solve with FEM the problem(s)

@t

u = F (u, @(n)x

,⇡), in ⌦⇥ [0, T ] ⇢ Rd ⇥ R+

2.Collect snapshots of FEM solution(s) in a matrix

S = (u1, . . . ,up) 2 RN⇥p ui =

2

64u(x1, ti)

...u(xN , ti)

3

75 2 RN, 8i = 1, . . . ,N



‣Use information given by FEM solution(s) to build a suitable basis

‣Can reproduce events described by the FEM solution(s)

1.Solve with FEM the problem(s)@t

u = F (u, @(n)x

,⇡), in ⌦⇥ [0, T ] ⇢ Rd ⇥ R+

2.Collect snapshots of FEM solution(s) in a matrix

S = (u1, . . . ,up) 2 RN⇥p ui =

2

64u(x1, ti)

...u(xN , ti)

3

75 2 RN, 8i = 1, . . . ,N

3.Compute Singular Value Decomposition (SVD) of the matrixS = �⌃ T , ⌃ = diag(�i)

4.Keep first eigenvectors as ROM basis functions


‣ Infarcted tissue: damaged area which cannot be activated➡to build the POD basis, we use a FEM solution with NO infarction

POD application in Cardiac ElectrophysiologySimulation of a myocardial infarction

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FEM (79,537 basis) POD (100 basis)

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0 100 200 300 400

�4

�2

0

2

4

I

0 100 200 300 400

�4

�2

0

2

4

aVR

0 100 200 300 400

�4

�2

0

2

4

V1

0 100 200 300 400

�4

�2

0

2

4

V4

0 100 200 300 400

�4

�2

0

2

4

II

0 100 200 300 400

�4

�2

0

2

4

aVL

0 100 200 300 400

�4

�2

0

2

4

V2

0 100 200 300 400

�4

�2

0

2

4

V5

0 100 200 300 400

�4

�2

0

2

4

III

0 100 200 300 400

�4

�2

0

2

4

aVF

0 100 200 300 400

�4

�2

0

2

4

V3

0 100 200 300 400

�4

�2

0

2

4

V6

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2]FEM



‣ An efficient POD to simulate an infarction in any point of the heart➡to build the POD basis, we use many FEM solutions with infarction

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SI1 =⇥u1I1 |u

2I1 | . . . |u

NTI1

⇤2 RN⇥NT

SI2 =⇥u1I2 |u

2I2 | . . . |u

NTI2

⇤2 RN⇥NT

Sh =⇥u1h|u2

h| . . . |uNTh

⇤2 RN⇥NT

S =hSh|SI1 |SI2 | . . . |SIm

i2 RN⇥(m+1)NT

๏ healthy FEM solution

๏ infarct 1 FEM solution

๏ infarct 2 FEM solution

๏ infarct m FEM solution

. . . . . . . . .

SIm =⇥u1Im |u2

Im | . . . |uNTIm

⇤2 RN⇥NT

➡compute the SVD on an enlarged matrix

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0 100 200 300 400

�4

�2

0

2

4

I

0 100 200 300 400

�4

�2

0

2

4

aVR

0 100 200 300 400

�4

�2

0

2

4

V1

0 100 200 300 400

�4

�2

0

2

4

V4

0 100 200 300 400

�4

�2

0

2

4

II

0 100 200 300 400

�4

�2

0

2

4

aVL

0 100 200 300 400

�4

�2

0

2

4

V2

0 100 200 300 400

�4

�2

0

2

4

V5

0 100 200 300 400

�4

�2

0

2

4

III

0 100 200 300 400

�4

�2

0

2

4

aVF

0 100 200 300 400

�4

�2

0

2

4

V3

0 100 200 300 400

�4

�2

0

2

4

V6

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‣ Solve a generic PDE system for any set of parameters

‣ Define a time evolving modal expansion

12

ALP (Approximated Lax Pairs) methodA new approach to Reduced Order Models

u =NX

i=1

�i(t)'i(x, t)

@t

u = F (u, @(n)x

,⇡), in ⌦⇥ [0, T ] ⇢ Rd ⇥ R+

x1

x2

x3

x1

x2

x3


ALP methodSummary

‣ Solve a generic PDE system for any set of parameters

‣ Method

1.Choose an initial basis

2.Define the basis evolution

3.Representation in the reduced space

⇢@tu = F (u), ⌦⇥ [0, T ]u(t = 0) = u0, ⌦


ALP method1.Choose an initial basis, 2. Define the basis evolution

1. Definition of the basis using PDE solution

@tu = F (u)Solution of

1.1) Choose of an operator SchrödingerL

1.2) Solve the eigenvalue problem

1.3) The initial basis is ('m)m�1

L�(u)'m = �m'm

L�(u)' = ��'� �u'

L(u)'m(t) = �m(t)'m(t)

M @t'm(t) = M(t)'m(t)2. Find an operator s.t.


ALP method in electrophysiologyDiscretization of bidomain equations

‣ Bidomain equations + FhN model

where

‣Write the solution in the RO space

f(vm, uE , w) = Amsvm(vm � a)(1� vm)�Amw + div(�Irvm) + div(�IruE) +AmIappg(vm, w) = ✏(�vm � w)q(vm, uE) = �div((�I + �E)ruE)� div(�Irvm)

AmCm@tvm = f(vm, uE , w)@tw = g(vm, w)

q(vm, uE) = 0

vm =NX

i=1

�i(t)'i(x, t) w =NX

i=1

µi(t)'i(x, t) uE =NX

i=1

⇠i(t)'i(x, t)

f(vm, uE , w) =NX

i=1

�i(t)'i(x, t) g(vm, w) =NX

i=1

⌘i(t)'i(x, t) q(vm, uE) = Evm +QuE


ALP method in electrophysiologyDiscretization of bidomain equations

Update of ALP solution

Update of eigenvalue

Update of matrices/tensors

Update of evolution operator

Compute new equation RHS

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

� +M� � � = 0µ+Mµ� ⌘ = 0E� +Q⇠ = 0

�i + �NX

m=1

Tiim�m = 0, i = 1 . . . N

B = [M,B] T = {M,T}(3)E = [M,E] T (s) = {M,T (s)}(3)Q = [M,Q] Y = {M,Y }(4)

Mij =�

�j � �i

NX

m=1

Tijm�m, i, j = 1 . . . N

� = �(�, ⇠, µ)⌘ = ⌘(�, µ)


‣ FEM solution➡ 2D mesh with 5978 vertices

‣ ALP solution➡ Initial basis computed with FEM solution (at t = 5 msec)➡ Number of ROM modes N=25

‣ POD solution➡ POD built from a homogeneous parameters simulation➡ Number of ROM modes N=25➡ Number of FEM snapshots = 100, with sampling time 0.5 msec

17

Numerical results (1)Bidomain equations in a 2D mesh

[Gerbeau, Lombardi, Schenone - 2014]


Numerical results (1)Bidomain equations in a 2D mesh

FEM (5,978 basis) ALP(25 basis) POD (25 basis)



‣ FEM solution➡ 2D mesh with 5978 vertices➡ heterogeneous parameter s=s(x)

‣ ALP solution➡ Initial basis computed with FEM solution (at t = 5 msec)➡ Number of ROM modes N=25➡ heterogeneous parameter s=s(x)


18

Numerical results (2)Bidomain equations in 2D - heterogeneous parameters

I(Vm, w) = s(x)u(u� a)(u� 1) + w



Numerical results (2)Bidomain equations in 2D - heterogeneous parameters




Numerical results (3)Bidomain equations in 2D - source term

‣ FEM solution➡ 2D mesh with 5978 vertices➡ ectopic pacemaker (source at t = 0 msec + t = 60 msec)

‣ ALP solution➡ Initial basis computed with FEM solution (at t = 5 msec)➡ Number of ROM modes N=25➡ ectopic pacemaker (source at t = 60 msec)




Numerical results (3)Bidomain equations in 2D - source term




Conclusions

‣Numerical simulations make presentations nice and colorful‣ Cardiac electrophysiology equations are complicated‣ FEM are boring‣ ROM are much more fun,

... but use them carefully!

๏POD does not always work and often need many FEM solutions to give good results

๏ALP does not need an a priori knowledge of the solution, no data-base is needed

‣ A lot of things are still to do ... maybe during my post-doc!


A special thank to the organizers for these two years working together!

Documents

REDUCED ORDER MODEL IN CARDIAC ELECTROPHYSIOLOGY€¦ · Elisa SCHENONE Inria Junior Seminar 17th June 2014 10" Infarcted tissue: damaged area which cannot be activated! to build