Some inverse problems in electrocardiography Jocelyne Erhel and - - PowerPoint PPT Presentation

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Some inverse problems in electrocardiography Jocelyne Erhel and - - PowerPoint PPT Presentation

Jan 02, 2023 187 likes •339 views

Some inverse problems in electrocardiography Jocelyne Erhel and Edouard Canot Projects from M. Hayek and E. Deriaz ALADIN team - INRIA-Rennes Models in electrocardiography and discretisation Discrete ill-posed general least-squares

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Some inverse problems in electrocardiography Jocelyne Erhel and Edouard Canot Projects from M. Hayek and E. Deriaz ALADIN team - INRIA-Rennes

  • Models in electrocardiography and discretisation
  • Discrete ill-posed general least-squares problem
  • Time regularisation

ERCIM workshop - February 2002

  • J. Erhel - 02/02

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Electrocardiography

50 100 150 200 −160 −140 −120 −100 −80 −60 −40 −20

source of electrical current at the surface of the heart : ΓE propagation through the chest, bioelectric volume conductor : Ω measure of potential at the surface of the torso : ΓT

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Direct and inverse models in ECG

Φ is the electrostatic potentiel σ is the electrical conductivity tensor Direct model ∇.(σ∇Φ) = 0 in Ω

∂Φ ∂n = 0 on ΓT

Φ = ΦE on ΓE (1) Well-posed problem Inverse model ∇.(σ∇Φ) = 0 in Ω

∂Φ ∂n = 0 on ΓT

Φ = ΦT on Σ ⊂ ΓT (2) Unique solution but not continuous : ill-posed problem

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Discretisation by Boundary Element Methods

Well-suited for homogeneous and isotropic case (σ scalar and constant) n nodes on ΓE and m ≥ n nodes on ΓT x1 : discretisation of ΦE x2 : discretisation of ∂ΦE

∂n

y : discretisation of ΦT

  • A11

A12 A13 A21 A22 A23

  

x1 x2 y

   = 0

(3) Dense small linear system

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Discretisation by Finite Element Methods

Well-suited for heterogeneous or anisotropic case (σ tensor) n nodes on ΓE, N nodes in Ω and m ≥ n nodes on ΓT x1 : discretisation of ΦE x2 : discretisation of Φ in Ω y : discretisation of ΦT

  • A11

A12 A13 A21 A22 A23

  

x1 x2 y

   = 0

(4) Sparse large linear system (N large)

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Direct problem

In BEM, we set N = n Square linear system of order N + m

  • A12

A13 A22 A23 x2 y

  • =
  • −A11

−A21

  • x1

(5) which can be solved by (A23 − A22A−1

12 A13)y = (A22A−1 12 A11 − A21)x1

x2 = −A−1

12 (A11x1 + A13y)

(6) Transfer matrix : y = Tx1, T ∈ Rm×n

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Inverse problem

N + n unknowns and N + m equations : least-squares problem y : measure of potential on the torso : errors y + e with e blank noise (0,µ2I) Approach used in most papers and software min

x1 y − Tx1

Our proposal min

x,e e subject to Ax + Be = By

x = (x1,x2) ∈ Rn+N, y ∈ Rm A ∈ R(N+m)×(n+N), B ∈ R(N+m)×m General Gauss-Markov linear model

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Regularisation

Algorithms for general linear models Generalised Singular Value Decomposition GSVD Generalised QR factorisation GQR (Paige ’s algorithm) Iterative methods of type LSQR? Regularisation for discrete ill-posed problem min

x,e (e2 + λ2Cx2) subject to Ax + Be = By

Ref : H. Zua and P.C. Hansen, 1990 Restricted SVD Algorithm similar to GQR Iterative methods of type LSQR?

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Some very preliminary results

Discretisation by BEM - homogeneous and isotropic case - 2D Linear model minx z − Bx with z = By Tychonov regularisation and parameter selection using an L-curve

20 40 60 80 100 120 140 −1.5 −1 −0.5 0.5 1 1.5 nombre total de noeuds = 128 (φc, qc) Régularisation du problème inverse perturbé par un bruit d’amplitude 1e−3 solution régularisée solution analytique Le potentiel électrique à la surface du coeur (φc) La dérivée normale du potentiel électrique à la surface du coeur (qc)

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Time dependent problem

Time dependent model Measures during a time interval Time discretisation (t1, . . . ,tp) Measure Y = (y(t1), . . . ,y(tp)) Unknown X = (x(t1), . . . ,x(tp)) Matrices A and B independent of time AX = B(Y + E) where A ∈ R(N+m)×n+N, B ∈ R(N+m)×m, X ∈ R(n+N)×p, Y ∈ Rm×p (Ip ⊗ A)vec(X) = (Ip ⊗ B)(vec(Y ) + vec(E)) Case of total blank noise p independent problems Ax(tk) = B(y(tk) + e(tk)), k = 1, . . . ,p

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Some very preliminary results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −80 −60 −40 −20 20 40 60 80 100

Same algorithm as before for each time step

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Time dependent regularisation

Approach based on the SVD of Y with m ≤ t Ref : F. Greensite and G. Huiskamp, 1998 Y = P(S 0)QT and Q = (Q1 Q2) AXQ1 = PS + EQ1 and AXQ2 = EQ2 Preliminary method Solve minZ AZ − PS and take X = ZQT

1

Solve p independent problems min

z(tk) Az(tk) − (PS)(tk), k = 1, . . . ,p

Regularisation for each (PS)(tk) Solution discarded if the Discrete Picard condition is not satisfied

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Some very preliminary results

10 20 30 40 50 60 70 80 −70 −60 −50 −40 −30 −20 −10 10 10 20 30 40 50 60 70 80 −70 −60 −50 −40 −30 −20 −10 10

Discrete Picard condition satisfied Discrete Picard condition not satisfied

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Some very preliminary results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −60 −40 −20 20 40 60 80 100

Solution using time regularisation

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Some perspectives

  • Regularisation of general linear model
  • Analysis of time regularisation
  • General time regularisation
  • Comparison with the classical approach
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