<figure><img src="image_1.png" /></figure>

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# Bidomain Model Coupled To A Realistic Multi-Electrode Device In Cardiac Electrophysiology

Valentin Pannetier, Yves Coudière, Michael Leguèbe

► To cite this version:

Valentin Pannetier, Yves Coudière, Michael Leguèbe. Bidomain Model Coupled To A Realistic Multi-Electrode Device In Cardiac Electrophysiology. CANUM 2024 - Congrès National d'Analyse Numérique, May 2024, Le Bois-Plage-en-Ré, Île de Ré, France. hal-04610592

HAL Id: hal-04610592

https://inria.hal.science/hal-04610592

Submitted on 13 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Bidomain Model Coupled To A Realistic Multi-Electrode Device In Cardiac Electrophysiology

Valentin Pannetier, Yves Coudière, Michael Leguèbe

► To cite this version:

Valentin Pannetier, Yves Coudière, Michael Leguèbe. Bidomain Model Coupled To A Realistic Multi-Electrode Device In Cardiac Electrophysiology. CANUM 2024 - Congrès National d'Analyse Numérique, May 2024, Le Bois-Plage-en-Ré, Île de Ré, France. hal-04610592

HAL Id: hal-04610592

https://inria.hal.science/hal-04610592v1

Submitted on 13 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

SIM
CARDIO
TEST

# Bidomain Model Coupled To
A Realistic Multi-Electrode Device
In Cardiac Electrophysiology

## SimCardioTest European Project

### Valentin PANNETIER, Yves COUDIÈRE and Michael LEGUÈBE

May 29, 2024

This project received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101016496 (SimCardioTest).

université
de BORDEAUX

liryc
Institut des maies du rythme cardiaque

iria

m
Institut de
Mathématiques de
Bordeaux
<figure><img src="image_2.png" /></figure>

SIM
CARDIO
TEST

# Bidomain Model Coupled To
A Realistic Multi-Electrode Device
In Cardiac Electrophysiology

## SimCardioTest European Project

### Valentin PANNETIER, Yves COUDIÈRE and Michael LEGUÈBE

May 29, 2024

This project received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101016496 (SimCardioTest).

université
de BORDEAUX

liryc
Institut des maies du rythme cardiaque

iria

m
Institut de
Mathématiques de
Bordeaux
<figure><img src="image_4.png" /></figure>

SIM
CARDIO
TEST

Context & Motivation

WP2 of SimCardioTest focuses on Pacing Leads and Catheters, especially on energy required to
trigger a cardiac beat with a pacemaker.

<figure><img src="image_5.png" /><figcaption>(a) BOREA™ pacemaker</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Lead implanted in the septum</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(c) Simulation of electrical field</figcaption></figure>
<figure><img src="image_3.png" /></figure>

SIM
CARDIO
TEST

Context & Motivation

WP2 of SimCardioTest focuses on Pacing Leads and Catheters, especially on energy required to
trigger a cardiac beat with a pacemaker.

<figure><img src="image_4.png" /><figcaption>(a) BOREA™ pacemaker</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Lead implanted in the septum</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Simulation of electrical field</figcaption></figure>
<figure><img src="image_8.png" /></figure>

SIM
CARDIO
TEST

## Problematic

Electrophysiological propagation models already exist, but we need to describe the stimulation

* Three discontinuous parts by stimulation ⇒ **Device circuit must be modeled**
  Structural analysis of electric circuits and consequences for MNA,
  D. Estévez Schwarz, C. Tischendorf, 2000

* Presence of a reactive layer at electrode boundaries ⇒ **Layer must be modeled too**
  Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography,
  E. Somersalo, M. Cheney and D. Isaacson, 1992

<figure><img src="image_9.png" /><figcaption>(a) Geometric setup</figcaption></figure>

+ Contacts +

<figure><img src="image_10.png" /><figcaption>(b) BOREA™</figcaption></figure>
<figure><img src="image_7.png" /></figure>

SIM
CARDIO
TEST

## Problematic

Electrophysiological propagation models already exist, but we need to describe the stimulation

* Three discontinuous parts by stimulation ⇒ **Device circuit must be modeled**
  Structural analysis of electric circuits and consequences for MNA,
  D. Estévez Schwarz, C. Tischendorf, 2000

* Presence of a reactive layer at electrode boundaries ⇒ **Layer must be modeled too**
  Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography,
  E. Somersalo, M. Cheney and D. Isaacson, 1992

<figure><img src="image_8.png" /><figcaption>(a) Geometric setup</figcaption></figure>

+ Contacts +

<figure><img src="image_9.png" /><figcaption>(b) BOREA™</figcaption></figure>
<figure><img src="image_11.png" /></figure>

Principle & Notations - I

* An oriented graph $\mathcal{G} = (S, B)$ with

- $N_n$ nodes,

- $N_b$ branches (dipoles)

* Potentials $U$ in Volt and currents $J$ in Ampere

* Incidence matrix $A = (a_{s,b})_{s \in S, b \in B}$ of size $N_n \times N_b$

$$a_{s,b} = \begin{cases} -1 & \text{if } s \text{ is an end point of } b \\ +1 & \text{if } s \text{ is a start point of } b \\ 0 & \text{elsewhere} \end{cases}$$

* Kirchhoff's current law $AJ = 0$ and voltages $V := A^T U$.

<figure><img src="image_12.png" /></figure>

$$U = (U_{S1}, U_{S2}, U_{S3}, U_{S4})$$

$$J = (J_{B1}, J_{B2}, J_{B3}, J_{B4}, J_{B5})$$

$$ \begin{array}{c} \begin{array}{ccccc} & & B_1 & B_2 & B_3 & B_4 & B_5 \end{array} \\ \begin{array}{c|ccccc} s_1 & 1 & 0 & 0 & -1 & 1 \\ s_2 & -1 & 1 & 0 & 0 & 0 \\ s_3 & 0 & -1 & 1 & 0 & -1 \\ s_4 & 0 & 0 & -1 & 1 & 0 \end{array} \end{array} $$
<figure><img src="image_10.png" /></figure>

Principle & Notations - I

* An oriented graph $\mathcal{G} = (S, B)$ with

- $N_n$ nodes,

- $N_b$ branches (dipoles)

* Potentials $U$ in Volt and currents $J$ in Ampere

* Incidence matrix $A = (a_{s,b})_{s \in S, b \in B}$ of size $N_n \times N_b$

$$ a_{s,b} = \begin{cases} -1 & \text{if } s \text{ is an end point of } b \\ +1 & \text{if } s \text{ is a start point of } b \\ 0 & \text{elsewhere} \end{cases} $$

* Kirchhoff's current law $AJ = 0$ and voltages $V := A^T U$.

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

$$ U = (U_{S1}, U_{S2}, U_{S3}, U_{S4}) $$

$$ J = (J_{B1}, J_{B2}, J_{B3}, J_{B4}, J_{B5}) $$

$$ \begin{array}{c} \begin{array}{ccccc} B_1 & B_2 & B_3 & B_4 & B_5 \end{array} \\ \begin{array}{c|cccc} s_1 & 1 & 0 & 0 & -1 \end{array} \\ \begin{array}{c|cccc} s_2 & -1 & 1 & 0 & 0 \end{array} \\ \begin{array}{c|cccc} s_3 & 0 & -1 & 1 & 0 \end{array} \\ \begin{array}{c|cccc} s_4 & 0 & 0 & -1 & 1 \end{array} \end{array} $$
<figure><img src="image_13.png" /></figure>

Principle & Notations - II

* Only contains capacitors, resistors and voltage sources

* Split graph and matrix accordingly:

$$
B = B_c \cup B_g \cup B_s \text{ and } A = [A_c, A_g, A_s]
$$

$$
\bullet A J = 0 \Rightarrow \boxed{A_c J_c + A_g J_g + A_s J_s = 0}
$$

$$
\bullet V = A^T U \Rightarrow V_c = A_c^T U, V_g = A_g^T U \text{ and } V_s(t) = A_s^T U
$$

$$
\bullet \text{ Dipole laws: } J_c = C \frac{dV_c}{dt} = C \frac{dA_c^T U}{dt} \text{ and } J_g = G V_g = G A_g^T U
$$

* Differential algebraic equations (DAE)

$$
(U, J_s) \quad \text{solution of}
\begin{cases}
A_c C \frac{dA_c^T U}{dt} + A_g G A_g^T U + A_s J_s = 0 \\
A_s^T U = V_s
\end{cases}
$$

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

$$
U = (U_{S1}, U_{S2}, U_{S3}, U_{S4})
$$

$$
J = (J_{B1}, J_{B2}, J_{B3}, J_{B4}, J_{B5})
$$

$$
\begin{array}{ccccc}
 & B_1 & B_2 & B_3 & B_4 & B_5 \\
s_1 & \begin{bmatrix} 1 & 0 & 0 & -1 & 1 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & -1 \\ 0 & 0 & -1 & 1 & 0 \end{bmatrix} & & & & \\
s_2 & & & & & \\
s_3 & & & & & \\
s_4 & & & & & \\
\end{array}
$$
<figure><img src="image_12.png" /></figure>

Principle & Notations - II

• Only contains capacitors, resistors and voltage sources

• Split graph and matrix accordingly:

$$
B = B_c \cup B_g \cup B_s \text{ and } A = [A_c, A_g, A_s]
$$

$$
\bullet A J = 0 \Rightarrow \boxed{A_c J_c + A_g J_g + A_s J_s = 0}
$$

$$
\bullet V = A^T U \Rightarrow V_c = A_c^T U, V_g = A_g^T U \text{ and } V_s(t) = A_s^T U
$$

$$
\bullet \text{ Dipole laws: } J_c = C \frac{dV_c}{dt} = C \frac{dA_c^T U}{dt} \text{ and } J_g = G V_g = G A_g^T U
$$

• Differential algebraic equations (DAE)

$$
(U, J_s) \quad \text{solution of}
\begin{cases}
A_c C \frac{dA_c^T U}{dt} + A_g G A_g^T U + A_s J_s = 0 \\
A_s^T U = V_s
\end{cases}
$$

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

$$
U = (U_{S1}, U_{S2}, U_{S3}, U_{S4})
$$

$$
J = (J_{B1}, J_{B2}, J_{B3}, J_{B4}, J_{B5})
$$

$$
\begin{array}{ccccc}
 & B_1 & B_2 & B_3 & B_4 & B_5 \\
s_1 & \begin{bmatrix} 1 & 0 & 0 & -1 & 1 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & -1 \\ 0 & 0 & -1 & 1 & 0 \end{bmatrix} & & & & \\
s_2 & & & & & \\
s_3 & & & & & \\
s_4 & & & & & \\
\end{array}
$$
<figure><img src="image_15.png" /></figure>

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

* Graph ($S, B$) oriented with $L + 1$ specialized nodes

* Potentials split as $U = (U^\circ, \gamma U)$ of size $N_n$, internal potentials $U^\circ$

* Dirichlet boundary condition, $\gamma_0 U = 0$ (ground)

* Coupling boundary potentials $\gamma_1 U, \dots, \gamma_L U$

* Coupling output currents $\gamma J = (\gamma_1 J, \dots, \gamma_L J)$

$$ (U, J_s) \quad \text{solution of DAE} \qquad \left\{ \begin{array}{l} A_c C \frac{dA_s^T U}{dt} + A_g G A_g^T U + A_s J_s + \gamma A \gamma J = 0 \\ A_s^T U = V_s \end{array} \right. $$

with $\gamma A := [0, \mathrm{Id}_L]^T$.
<figure><img src="image_14.png" /></figure>

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

* Graph (S, B) oriented with L + 1 specialized nodes

* Potentials split as $U = (U^\circ, \gamma U)$ of size $N_n$, internal potentials $U^\circ$

* Dirichlet boundary condition, $\gamma_0 U = 0$ (ground)

* Coupling boundary potentials $\gamma_1 U, \dots, \gamma_L U$

* Coupling output currents $\gamma J = (\gamma_1 J, \dots, \gamma_L J)$

$$ (U, J_s) \quad \text{solution of DAE} \qquad \left\{ \begin{array}{l} A_c C \frac{dA_s^T U}{dt} + A_g G A_g^T U + A_s J_s + \gamma A \gamma J = 0 \\ A_s^T U = V_s \end{array} \right. $$

with $\gamma A := [0, \mathrm{Id}_L]^T$.
<figure><img src="image_17.png" /></figure>

On each electrode $ℓ = 1, \dots, L$ (without voltage sources)

* **Defined spatially:** $\forall x \in \Gamma_ℓ$

* **Potentials split as** $u_ℓ = (u_ℓ^o, \gamma u_ℓ)$ of size $N_ℓ$

* **Coupling boundary potentials** $\gamma_1 u_ℓ, \gamma_2 u_ℓ$

* **Coupling output currents** $\gamma j_ℓ = (\gamma_1 j_ℓ, \gamma_2 j_ℓ)$

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

$$ u_ℓ \quad \text{solution of x-ODE} \qquad A_{ℓ,c} C_ℓ \frac{\partial A_{ℓ,c}^T}{\partial t} u_ℓ + A_{ℓ,g} G_ℓ A_{ℓ,g}^T u_ℓ + \gamma A_ℓ \gamma j_ℓ = 0 \quad \text{on } \Gamma_ℓ $$

with $\gamma A_ℓ := [0, \mathrm{Id}_2]^T$.
<figure><img src="image_16.png" /></figure>

On each electrode $ℓ = 1, \dots, L$ (without voltage sources)

* **Defined spatially:** $\forall x \in \Gamma_ℓ$

* **Potentials split as** $u_ℓ = (u_ℓ^o, \gamma u_ℓ)$ of size $N_ℓ$

* **Coupling boundary potentials** $\gamma_1 u_ℓ, \gamma_2 u_ℓ$

* **Coupling output currents** $\gamma j_ℓ = (\gamma_1 j_ℓ, \gamma_2 j_ℓ)$

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

$$ u_ℓ \quad \text{solution of x-ODE} \qquad A_{ℓ,c} C_ℓ \frac{\partial A_{ℓ,c}^T}{\partial t} u_ℓ + A_{ℓ,g} G_ℓ A_{ℓ,g}^T u_ℓ + \gamma A_ℓ \gamma j_ℓ = 0 \quad \text{on } \Gamma_ℓ $$

with $\gamma A_ℓ := [0, \mathrm{Id}_2]^T$.
<figure><img src="image_21.png" /></figure>

# Generic Models

## Bidomain Model (PDE)

* Tissue $\Omega_M$ - intracellular and extracellular

* Bath/Blood $\Omega_B$ - passive media

* Transmembrane voltage $[u] = u_i - u$

* Conductivity tensors $\sigma$ and $\sigma_i$

* L electrodes at boundary $\bigcup_{\ell=1}^L \Gamma_\ell \subset \partial (\overline{\Omega_M} \cup \overline{\Omega_B})$

* $\gamma_\ell u = u_{|\Gamma_\ell}$, and $-\sigma \nabla u \cdot n = j_{\Gamma_\ell}$ on $\Gamma_\ell$

<figure><img src="image_22.png" /></figure>

$$ (u, u_i) \text{ solution of PDE} \quad - \operatorname{div}(\sigma_i \nabla u_i) = -j_m \quad \text{in } \Omega_M \text{ and} \quad \begin{cases} - \operatorname{div}(\sigma \nabla u) = j_m & \text{in } \Omega_M \\ - \operatorname{div}(\sigma \nabla u) = 0 & \text{in } \Omega_B \end{cases} $$

with membrane current density $j_m = \chi (c_m \partial_t [u] + I_{\text{ion}}([u], h, t))$, and Ionic Model on $h$.
<figure><img src="image_20.png" /></figure>

# Generic Models

## Bidomain Model (PDE)

* Tissue $\Omega_M$ - intracellular and extracellular

* Bath/Blood $\Omega_B$ - passive media

* Transmembrane voltage $[u] = u_i - u$

* Conductivity tensors $\sigma$ and $\sigma_i$

* L electrodes at boundary $\bigcup_{\ell=1}^L \Gamma_\ell \subset \partial (\overline{\Omega_M} \cup \overline{\Omega_B})$

* $\gamma_\ell u = u_{|\Gamma_\ell}$, and $-\sigma \nabla u \cdot n = j_{\Gamma_\ell}$ on $\Gamma_\ell$

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

$$ (u, u_i) \text{ solution of PDE} \quad - \operatorname{div}(\sigma_i \nabla u_i) = -j_m \quad \text{in } \Omega_M \text{ and} \quad \begin{cases} - \operatorname{div}(\sigma \nabla u) = j_m & \text{in } \Omega_M \\ - \operatorname{div}(\sigma \nabla u) = 0 & \text{in } \Omega_B \end{cases} $$

with membrane current density $j_m = \chi (c_m \partial_t [u] + I_{\text{ion}}([u], h, t))$, and Ionic Model on $h$.
<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

— Potentials

$\gamma_{\ell}U = \gamma_{1}u_{\ell}(x)$, for $x \in \Gamma_{\ell}$
$\gamma_{2}u_{\ell}(x) = \gamma_{\ell}u(x)$, for $x \in \Gamma_{\ell}$

— Currents

$\gamma_{\ell}J + \int_{\Gamma_{\ell}} \gamma_{1}j_{\ell}(x) = 0$
$\gamma_{2}j_{\ell}(x) + j_{\Gamma_{\ell}}(x) = 0, \text{ for } x \in \Gamma_{\ell}$
<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

### — Potentials

$$ 
\begin{aligned}
\gamma_{\ell} U &= \gamma_{1} u_{\ell}(x), \quad \text{for } x \in \Gamma_{\ell} \\
\gamma_{2} u_{\ell}(x) &= \gamma_{\ell} u(x), \quad \text{for } x \in \Gamma_{\ell}
\end{aligned}
 $$

### — Currents

$$ 
\begin{aligned}
\gamma_{\ell} J + \int_{\Gamma_{\ell}} \gamma_{1} j_{\ell}(x) &= 0 \\
\gamma_{2} j_{\ell}(x) + j_{\Gamma_{\ell}}(x) &= 0, \quad \text{for } x \in \Gamma_{\ell}
\end{aligned}
 $$
<figure><img src="image_25.png" /></figure>

For all $\phi = (\phi, \phi_i, \phi_1, \dots, \phi_L, \Phi) \in C_c^\infty$ of $[0, T) \times \{\overline{\Omega}, \overline{\Omega_M}, (\overline{\Gamma}_\ell)^{N_\ell}, \mathbb{R}^{N_\ell}\}$,

find $\mathbf{u} := (u, u_i, u_1, \dots, u_L, U)$ such that $\gamma_0 U = 0 + \text{transmission conditions on potentials, and } J_s$, verifying

$$ 
\left\{
\begin{array}{l}
\displaystyle \int_0^T \int_{\Omega_M} \sigma_i \nabla u_i \cdot \nabla \phi_i + \int_0^T \int_{\Omega} \sigma \nabla u \cdot \nabla \phi + \int_0^T \int_{\Omega_M} \left( c_m \frac{\partial [u]}{\partial t} [\phi] + f([u], \mathbf{h}, t) [\phi] \right) \\
\displaystyle \qquad + \int_0^T \left( c \frac{dA_c^\mathrm{T} U}{dt} \cdot A_c^\mathrm{T} \Phi + G A_g^\mathrm{T} U \cdot A_g^\mathrm{T} \Phi + A_s J_s \cdot \Phi \right) \\
\displaystyle \qquad + \sum_{\ell=1}^L \int_0^T \int_{\Gamma_\ell} \left( c_\ell \frac{\partial A_{\ell,c}^\mathrm{T} u_\ell}{\partial t} \cdot A_{\ell,c}^\mathrm{T} \phi_\ell + G_\ell A_{\ell,g}^\mathrm{T} u_\ell \cdot A_{\ell,g}^\mathrm{T} \phi_\ell \right) = 0, \\
\\
A_s^\mathrm{T} U = V_s
\end{array}
\right.
$$

+ IPP $\Rightarrow$ weak solution
<figure><img src="image_26.png" /></figure>

For all $\phi = (\phi, \phi_i, \phi_1, \dots, \phi_L, \Phi) \in C_c^\infty$ of $[0, T) \times \{\overline{\Omega}, \overline{\Omega_M}, (\overline{\Gamma}_\ell)^{N_\ell}, \mathbb{R}^{N_\ell}\}$,

find $\mathbf{u} := (u, u_i, u_1, \dots, u_L, U)$ such that $\gamma_0 U = 0 + \text{transmission conditions on potentials, and } J_s$, verifying

$$ 
\left\{
\begin{array}{l}
\displaystyle \int_0^T \int_{\Omega_M} \sigma_i \nabla u_i \cdot \nabla \phi_i + \int_0^T \int_{\Omega} \sigma \nabla u \cdot \nabla \phi + \int_0^T \int_{\Omega_M} \left( c_m \frac{\partial [u]}{\partial t} [\phi] + f([u], \mathbf{h}, t) [\phi] \right) \\
\displaystyle \qquad + \int_0^T \left( c \frac{dA_c^\mathrm{T} U}{dt} \cdot A_c^\mathrm{T} \Phi + G A_g^\mathrm{T} U \cdot A_g^\mathrm{T} \Phi + A_s J_s \cdot \Phi \right) \\
\displaystyle \qquad + \sum_{\ell=1}^L \int_0^T \int_{\Gamma_\ell} \left( c_\ell \frac{\partial A_{\ell,c}^\mathrm{T} u_\ell}{\partial t} \cdot A_{\ell,c}^\mathrm{T} \phi_\ell + G_\ell A_{\ell,g}^\mathrm{T} u_\ell \cdot A_{\ell,g}^\mathrm{T} \phi_\ell \right) = 0, \\
\\
A_s^\mathrm{T} U = V_s
\end{array}
\right.
$$

+ IPP $\Rightarrow$ weak solution
<figure><img src="image_26.png" /></figure>

Find $\mathbf{u}^n := (u^n, u_1^n, u_2^n, \dots, u_L^n, U^n) \in \mathcal{H}_0$, and $J_s^n \in \mathbb{R}^{N_s}$ with

$$ \mathcal{H} := H^1(\Omega_M) \times H^1(\Omega) \times \Pi_{\ell=1}^L L^2(\Gamma_\ell)^{N_\ell} \times \mathbb{R}^{N_s}, $$

$$ \mathcal{H}_0 := \{ \mathbf{u} \in \mathcal{H} \text{ s.t. } \gamma_0 U = 0, \gamma_\ell U = \gamma_1 u_\ell, \gamma \mathbf{u} = \gamma_2 u_\ell, \ell = 1, \dots, L \}, $$

verifying, for all $\phi := (\phi_1, \phi_2, \dots, \phi_L, \Phi) \in \mathcal{H}_0$,

$$ a \left( \frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t}, \phi \right) + b(\mathbf{u}^n, \phi) + \left( f([\mathbf{u}^n]), [\phi] \right)_{\Omega_M} + A_s J_s^n \cdot \Phi = 0, \\ A_s^T U^n = V_s^n $$

$$ a(\delta \mathbf{u}, \phi) = (c_m [\delta \mathbf{u}], [\phi])_{\Omega_M} + \sum_{\ell=1}^{L} (C_\ell A_{\ell,c}^T \delta u_\ell, A_{\ell,c}^T \phi_\ell)_{\Gamma_\ell} + C A_c^T \delta U \cdot A_c^T \Phi, $$

$$ b(\mathbf{u}, \phi) = (\sigma_i \nabla u_i, \nabla \phi_i)_{\Omega_M} + (\sigma \nabla u, \nabla \phi)_\Omega + \sum_{\ell=1}^{L} (G_\ell A_{\ell,g}^T u_\ell, A_{\ell,g}^T \phi_\ell)_{\Gamma_\ell} + G A_g^T U \cdot A_g^T \Phi. $$
<figure><img src="image_27.png" /></figure>

Find $\mathbf{u}^n := (u^n, u_1^n, u_2^n, \dots, u_L^n, U^n) \in \mathcal{H}_0$, and $J_s^n \in \mathbb{R}^{N_s}$ with

$$ \mathcal{H} := H^1(\Omega_M) \times H^1(\Omega) \times \Pi_{\ell=1}^L L^2(\Gamma_\ell)^{N_\ell} \times \mathbb{R}^{N_s}, \\ \mathcal{H}_0 := \{\mathbf{u} \in \mathcal{H} \text{ s.t. } \gamma_0 U = 0, \gamma_\ell U = \gamma_1 u_\ell, \gamma \mathbf{u} = \gamma_2 u_\ell, \ell = 1, \dots, L\}, $$

verifying, for all $\phi := (\phi, \phi_1, \phi_2, \dots, \phi_L, \Phi) \in \mathcal{H}_0$,

$$ a\left(\frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t}, \phi\right) + b(\mathbf{u}^n, \phi) + \left(f([\mathbf{u}^n]), [\phi]\right)_{\Omega_M} + A_s J_s^n \cdot \Phi = 0, \\ A_s^\top U^n = V_s^n $$

$$ a(\delta \mathbf{u}, \phi) = (c_m [\delta \mathbf{u}], [\phi])_{\Omega_M} + \sum_{\ell=1}^{L} \left( C_{\ell} A_{\ell,c}^{\top} \delta u_{\ell}, A_{\ell,c}^{\top} \phi_{\ell} \right)_{\Gamma_{\ell}} + C A_{c}^{\top} \delta U \cdot A_{c}^{\top} \Phi, $$

$$ b(\mathbf{u}, \phi) = (\sigma_i \nabla u_i, \nabla \phi_i)_{\Omega_M} + (\sigma \nabla u, \nabla \phi)_\Omega + \sum_{\ell=1}^L \left( G_\ell A_{\ell,g}^\top u_\ell, A_{\ell,g}^\top \phi_\ell \right)_{\Gamma_\ell} + G A_g^\top U \cdot A_g^\top \Phi. $$
<figure><img src="image_27.png" /></figure>

Sketch of the proof – III

Existence

— Hypothesis

A_s has full column rank

— Consequence

ker (A_s) = {0} and Im (A_s^T) = ker (A_s)^⊥ = ℝ^{N_s}, ⇒ ∀V_s^n ∈ ℝ^{N_s} there exists U_0^n ∈ ℝ^{N_n} such as A_s^T U_0^n = V_s^n and ||u_0^n||_H ≤ cst ||V_s^n||_{ℝ^{N_n}}.

— Variational formulation in the constraint space

Find **u**\*_n ∈ H_0^{ker}, with **u** = **u**\*_n + **u**_0^n and H_0^{ker} := {**u** ∈ H_0 | **u** ∈ ker (A_s^T)}, verifying for all
**φ**_\* ∈ H_0^{ker},

$$
a \left( \frac{\mathbf{u}_{\star}^{n} + \mathbf{u}_{0}^{n} - \mathbf{u}^{n-1}}{\Delta t}, \phi_{\star} \right) - b \left( \mathbf{u}_{\star}^{n} + \mathbf{u}_{0}^{n}, \phi_{\star} \right) + \left( f([\mathbf{u}^{n}]), [\phi] \right)_{\Omega_{M}} = 0
$$

Lax-Milgram and Banach fixed-point theorems: existence of **u**\*_n in H_0^{ker}.

— Resolution of $J_s^n$ equation in $\mathbb{R}^{N_s}$

* $\forall \{e_k\}$ basis of $\mathbb{R}^{N_s}$, $\exists \{\phi_k\} \subset \mathcal{H}_0$ s.t. $e_k = A_s^T \Phi_k$, and $J_s^n$ solves the linear systems $J_s^n \cdot e_k = w_k$,

* $w_k$ are independent of the choice of $\{\phi_k\}$.
<figure><img src="image_28.png" /></figure>

Sketch of the proof – III

Existence

— Hypothesis

A_s has full column rank

— Consequence

ker (A_s) = {0} and Im (A_s^T) = ker (A_s)^⊥ = ℝ^{N_s}, ⇒ ∀V_s^n ∈ ℝ^{N_s} there exists U_0^n ∈ ℝ^{N_n} such as A_s^T U_0^n = V_s^n and ||u_0^n||_H ≤ cst ||V_s^n||_{ℝ^{N_n}}.

— Variational formulation in the constraint space

Find **u**\*_n ∈ H_0^{ker}, with **u** = **u**\*_n + **u**\*_0 and H_0^{ker} := {**u** ∈ H_0 | **u** ∈ ker (A\*_s)}, verifying for all

φ\_ ∈ H_0^{ker},

$$
a \left( \frac{\mathbf{u}_{*}^{n} + \mathbf{u}_{0}^{n} - \mathbf{u}^{n-1}}{\Delta t}, \phi_{*} \right) - b \left( \mathbf{u}_{*}^{n} + \mathbf{u}_{0}^{n}, \phi_{*} \right) + \left( f([\mathbf{u}^{n}]), [\phi] \right)_{\Omega_{M}} = 0
$$

Lax-Milgram and Banach fixed-point theorems: existence of **u**\*_n in H_0^{ker}.

— Resolution of $J_s^n$ equation in $\mathbb{R}^{N_s}$

* $\forall \{e_k\}$ basis of $\mathbb{R}^{N_s}$, $\exists \{\phi_k\} \subset \mathcal{H}_0$ s.t. $e_k = A_s^T \Phi_k$, and $J_s^n$ solves the linear systems $J_s^n \cdot e_k = w_k$,

* $w_k$ are independent of the choice of $\{\phi_k\}$.
<figure><img src="image_28.png" /></figure>

Sketch of the proof – IV

Compactness & passing to the limit

We have a sequence $(\mathbf{u}_0^n)$ bounded by data, and $(\mathbf{u}_\star^n)$ in $\mathcal{H}_0^{\text{ker}}$ with a priori estimates.

— A priori estimates on

$$a(\mathbf{u}^n, \mathbf{u}^n) \qquad \sum_n \Delta t b(\mathbf{u}^n, \mathbf{u}^n) \qquad \sum_n \Delta t a \left( \frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t}, \frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t} \right) \\ \text{for compactness (as } f \text{ is nonlinear)}$$

— **Compactness:** discrete Aubin-Simon (Gallouët-Latché theorem)

⇒ up to subsequence $\mathbf{u}^n \to \mathbf{u}$ (strong convergence)

— Passing to the limit

⇒ **u** solution of the continuous problem.
<figure><img src="image_29.png" /></figure>

We have a sequence $(\mathbf{u}_0^n)$ bounded by data, and $(\mathbf{u}_\star^n)$ in $H_0^{\text{ker}}$ with a priori estimates.

— A priori estimates on

$$a(\mathbf{u}^n, \mathbf{u}^n) \quad \sum_n \Delta t b(\mathbf{u}^n, \mathbf{u}^n) \quad \sum_n \Delta t a \left( \frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t}, \frac{\mathbf{u}^n - \mathbf{u}^{n-1}}{\Delta t} \right) \quad \text{for compactness (as f is nonlinear)}$$

— Compactness: discrete Aubin-Simon (Gallouët-Latché theorem)

⇒ up to subsequence $\mathbf{u}^n \to \mathbf{u}$ (strong convergence)

— Passing to the limit

⇒ **u** solution of the continuous problem.
<figure><img src="image_29.png" /></figure>

SIM
CARDIO
TEST

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

Figure: Pacemaker + 2 R//C contacts + geometry inspired of a standard VEGA lead
<figure><img src="image_30.png" /></figure>

SIM
CARDIO
TEST

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

Figure: Pacemaker + 2 R//C contacts + geometry inspired of a standard VEGA lead