# Un modèle moyenné en épaisseur pour le problème inverse de l'imagerie électrocardiographique

Yves Bourgault, Emma Lagracie, Yves Coudière, Lisl Weynans

► To cite this version:

Yves Bourgault, Emma Lagracie, Yves Coudière, Lisl Weynans. Un modèle moyenné en épaisseur pour le problème inverse de l'imagerie électrocardiographique. CANUM 2024 - 46ème Congrès National d'Analyse Numérique, May 2024, Le-Bois-Plage-en-Ré, France. hal-04906803

HAL Id: hal-04906803

https://hal.science/hal-04906803v1

Submitted on 22 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# A depth-averaged heart model for the inverse problem of cardiac electrophysiology

## CANUM 2024

### Ile de Ré, 29/05/2024

Yves Bourgault, Yves Coudière, Emma Lagracie, Lisl Weynans
► Context

► Forward averaged model

► Inverse problem

► Numerical results
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

* Cardiac rhythm disorders: pathologies of heart's electrical activity

* Electrical activation: gives the signal that precedes the contraction

<figure><img src="image_4.png" /></figure>

* Action potential

* Activation maps

* Goal: detect pathologies from torso potentials, ECG imaging. Very ill posed inverse problem 😞

<figure><img src="image_5.png" /></figure>
Potential $u \in H^1(\Omega_T \cup \Omega_H)$, voltage
$v \in H^1(\Omega_H)$.

$$ \mathrm{div}(\sigma_i \nabla (u + v)) = \partial_t v + I_{\mathrm{ion}}(v, t, h) \qquad \Omega_H $$

$$ \mathrm{div}((\sigma_e + \sigma_i)\nabla u) = -\mathrm{div}(\sigma_i \nabla v) \qquad \Omega_H $$

$$ \mathrm{div}(\sigma_T \nabla u) = 0 \qquad \Omega_T $$

$$ \sigma_i \nabla (u + v) \cdot n = 0 \qquad \partial \Omega_H $$

$$ \sigma_T \nabla u \cdot n = 0 \qquad \partial \Omega_T $$

<figure><img src="image_6.png" /></figure>
Find $u$ and/or $v$ that satisfy (part of) the bidomain equations, matching torso data $z_T$.

*Find the activation map.*

$$ \operatorname{div}((\sigma_e + \sigma_i)\nabla u) = -\operatorname{div}(\sigma_i \nabla v) \qquad \Omega_H $$

$$ \operatorname{div}(\sigma_T \nabla u) = 0 \qquad \Omega_T $$

$$ \sigma_i \nabla (u + v) \cdot n = 0 \qquad \partial \Omega_H $$

$$ \sigma_T \nabla u \cdot n = 0 \qquad \partial \Omega_T $$

* In the heart surface: $u$: uniqueness (up to a constant) but non existence, $v$ can be found up to a constant

* In the heart volume: infinite-dimensional kernel for $v$

<figure><img src="image_7.png" /></figure>
Cauchy problem:

$$ -\Delta u = 0 \quad \Omega_T $$

$$ u = z_T \quad \Gamma_T $$

$$ \partial_n u = 0 \quad \Gamma_T $$

* No interactions with the heart

* Does not allow to reconstruct $v$

* No information on the heart volume

<figure><img src="image_8.png" /></figure>
* Context

* Forward averaged model

* Inverse problem

* Numerical results
• 2D + smooth

• $\Omega_H$ separates blood cavity from torso

• Depth $h \ll L$, $h'(x) \ll 1$

• $\Gamma_H$ smoothe curve embedded in $\Omega_H$

• Parallel fibers:

$$ \sigma_{i/e} = \begin{pmatrix} \sigma_{i/e, \ell} & 0 \\ 0 & \sigma_{i/e, p} \end{pmatrix} $$

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>
$$ \sigma_{\ell} h \frac{\partial^2}{\partial s^2} \bar{u} + \sigma_{T} \frac{\partial u_{T}}{\partial n} |_{\Gamma_{H}} + \sigma_{B} \frac{\partial u_{B}}{\partial n} |_{\Gamma_{H}} + \sigma_{i\ell} h \frac{\partial^2}{\partial s^2} \bar{v} = 0 \quad \Gamma_{H} $$

$$ \operatorname{div}(\sigma_T \nabla u_T) = 0 \qquad \Omega_T $$

$$ \operatorname{div}(\sigma_B \nabla u_B) = 0 \qquad \Omega_B $$

+BC:

$$ \sigma_T \partial_n u = 0 \qquad \Gamma_T $$

$$ \sigma_{ep} \frac{u_T - \bar{u}}{h} = (1 - \alpha) \sigma_T \partial_n u_T \quad \Gamma_H $$

$$ \sigma_{ep} \frac{u_B - \bar{u}}{h} = \alpha \sigma_B \partial_n u_B \quad \Gamma_H $$

$\Omega_T$

<figure><img src="image_11.png" /></figure>

$\Omega_T$

<figure><img src="image_12.png" /></figure>
Find $(\bar{u}, u_T, u_B)$ in $V = H^1(\Gamma_H) \times H^1(\Omega_T)/\mathbb{R} \times H^1(\Omega_B)$ such that

$$ \begin{aligned} & \sigma_{\ell} h \int_{\Gamma_H} \frac{\partial}{\partial s} \bar{u} \frac{\partial}{\partial s} \bar{\varphi} + \sigma_T \int_{\Omega_T} \nabla u_T \cdot \nabla \varphi_T + \sigma_B \int_{\Omega_T} \nabla u_B \cdot \nabla \varphi_B \\ & + \frac{\sigma_{e,p}}{(1-\alpha)h} \int_{\Gamma_H} (\bar{u} - u_T)(\bar{\varphi} - \varphi_T) + \frac{\sigma_{e,p}}{\alpha h} \int_{\Gamma_H} (\bar{u} - u_B)(\bar{\varphi} - \varphi_B) = \int_{\Gamma_H} F \bar{\varphi}, \quad \forall (\bar{\varphi}, \varphi_T, \varphi_B) \in V \end{aligned} $$

$$ F = h \sigma_{i,\ell} \partial_{ss} \bar{v} $$

Well posed, all Lax-Milgram hypotheses are satisfied !
## Comparison with the bidomain

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>
* Context

* Forward averaged model

* Inverse problem

* Numerical results
$$
\min_{\mathcal{E}} J(U[F], F) = \frac{1}{2} \int_{\Gamma_T} |u_T - z_T|^2 + \frac{\varepsilon_{\text{regul}}}{2} |F|_{H^1(\Gamma_H)}^2
$$

Over space $\mathcal{E} = \{(U, F) | U = (u_T, u_B, \bar{u}) = AF + \mathbb{R}, F \in H^1(\Gamma_H)/\mathbb{R}\}$ with operator $A$:

$$
\sigma_{\ell} h \int_{\Gamma_H} \frac{\partial}{\partial s} \bar{u} \frac{\partial}{\partial s} \bar{\varphi} + \sigma_T \int_{\Omega_T} \nabla u_T \cdot \nabla \varphi_T + \sigma_B \int_{\Omega_T} \nabla u_B \cdot \nabla \varphi_B \\
+ \frac{\sigma_{e,p}}{(1-\alpha)h} \int_{\Gamma_H} (\bar{u} - u_T)(\bar{\varphi} - \varphi_T) + \frac{\sigma_{e,p}}{\alpha h} \int_{\Gamma_H} (\bar{u} - u_B)(\bar{\varphi} - \varphi_B) = \int_{\Gamma_H} F \bar{\varphi}
$$
Steps of the proof:

■ Existence:

1. Minimizing sequence

2. Weakly convergent subsequences

3. Use of lower semi continuity

■ Uniqueness: Strict convexity

$$J(U, F) = \frac{1}{2} \int_{\Gamma_T} |u_T - z_T|^2 + \frac{\varepsilon_{\text{regul}}}{2} |F|_{H^1}^2$$
$$
\begin{aligned}
\mathcal{L}(\bar{u}, u_T, u_B, \bar{\lambda}, \lambda_T, \lambda_B, F) = & \frac{1}{2} \int_{\Gamma_T} |u_T - z_T|^2 + \frac{\varepsilon_{\text{regul}}}{2} \int_{\Gamma_H} |\partial_s F|^2 \\
& + \sigma_{\ell} h \int_{\Gamma_H} \partial_s \bar{u} \partial_s \bar{\lambda} + \frac{\sigma_{e,p}}{(1-\alpha)h} \int_{\Gamma_H} (\bar{u} - u_T)(\bar{\lambda} - \lambda_T) \\
& + \frac{\sigma_{e,p}}{\alpha h} \int_{\Gamma_H} (\bar{u} - u_B)(\bar{\lambda} - \lambda_B) \\
& + \sigma_T \int_{\Omega_T} \nabla u_T \cdot \nabla \lambda_T + \sigma_B \sigma_B \int_{\Omega_B} \nabla u_B \cdot \nabla \lambda_B - \int_{\Gamma_H} F \bar{\lambda}
\end{aligned}
$$

1st order optimality conditions → linear system to solve 😊
► Context

► Forward averaged model

► Inverse problem

► Numerical results
<figure><img src="image_17.png" /><figcaption>(a) Pacing protocol 1</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b) Pacing protocol 2</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(c) Pacing protocol 3</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(d) Pacing protocol 4</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>(a) Protocol 2. t=6 ms</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(b) Protocol 3. t=6 ms</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(c) Protocol 4. t=6 ms</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(d) Protocol 2. t=13 ms</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(e) Protocol 3. t=13 ms</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(f) Protocol 4. t=13 ms</figcaption></figure>
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /><figcaption>(a) Protocol 1</figcaption></figure>

<caption>(b) Protocol 2</caption>

<caption>(c) Protocol 3</caption>

<caption>(d) Protocol 4</caption>
* Accurate activation maps, bounded by endocardial & epicardial ATs

* We recover an average $v$, some information on the volume

* Comparison with the usual epicardial reconstruction

* Perspective of 3D + more realistic geometry

Thank you for your attention !

<figure><img src="image_35.png" /></figure>

Emma Lagracie, Yves Bourgault, Yves Coudière and Lisl Weynans, *A depth-averaged heart model for the inverse problem of cardiac electrophysiology*, Submitted
Operator $T: AF|_{\Gamma_T} = ([\text{Trace}] \circ A)F = TF$ continuous Banach to Banach

* **Minimizing sequence:** $m = \inf_{\mathcal{E}} J(U, F) \ge 0$, minimizing sequence $(U_n, F_n) \in \mathcal{E}_2$ s.t. $J(U_n, F_n) \xrightarrow{n \to \infty} m$.

* **Weakly convergent subsequences**

$$ \frac{1}{2} \int_{\Gamma_T} |u_{n_T} - z_T|^2 + \frac{\varepsilon_{\text{regul}}}{2} |F_n|_{H^1}^2 \le J(0, 0) = Cte. \quad (1) $$

Subsequences

— $(F_n) \xrightarrow{n \to \infty} \bar{F}$

— $(C_n) \xrightarrow{n \to \infty} \bar{C}$

— (T weakly continuous) $TF_n \xrightarrow{n \to \infty} T\bar{F}$

— $(F_n)$ bounded

— $(u_{nT}|_{\Gamma_T}) = (TF_n + C_n)$ bounded + $(TF_n)$ bounded (continuity) $\Rightarrow$ $(C_n)$ bounded
• use of lower semi continuity $F \longmapsto \|F\|^2$ weakly l.s.c.

$$ \frac{\varepsilon_{\text{regul}}}{2} \|\bar{F}\|^2 \le \liminf_{n \to \infty} \frac{\varepsilon_{\text{regul}}}{2} \|F_n\|^2, \\ \frac{1}{2} \int_{\Gamma_T} |T\bar{F} + \bar{C} - z_T|^2 \le \liminf_{n \to \infty} \frac{1}{2} \int_{\Gamma_T} |u_n T - z_T|^2 $$

Finally,

$$ J(\bar{U}, \bar{F}) \le \liminf_{n \to \infty} J(U_n, F_n) = m. $$

**Uniqueness:** $J$ strictly convex.