Computer model of cardiac pacing devices

Yves Coudière

► To cite this version:

Yves Coudière. Computer model of cardiac pacing devices. Journées Math Bio Santé 2024, CNRS RT Math Bio Santé et SMAI GT Mabiome, Jun 2024, Nantes, France. hal-04905933

HAL Id: hal-04905933

https://hal.science/hal-04905933v1

Submitted on 22 Jan 2025

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
Computer model of cardiac pacing devices

Y. Coudière

IMB - UMR 5251, Liryc, Inria Carmen – Université de Bordeaux – CNRS – Bordeaux-INP

June 28, 2024

M. Leguèbe, V. Pannetier, G. Ravon (IMB),
R. Walton (manip), J. Magat (IRM), G. Faure (Microport-CRM)

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

SIM
CARDIO
TEST

Digital transformation in Health and Care SC1-DTH-06-2020 Grant Agreement No. 101016496.

université
"BORDEAUX

Inria

CNRS

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

Inserm

Institut national
de la santé et de la recherche médicale

CHU
BDX

CENTRE
HOSPITALIER
UNIVERSITAIRE
BORDEAUX

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

RÉGION
Nouvelle-
Aquitaine
# Pacemakers and electrical questions

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

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

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

<figure><img src="image_9.png" /><figcaption>(b) Focus: bradychardia leads</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
Question of interest

What are the stimulation pulse characteristics (voltage amplitude in V and pulse duration in ms) required for a bradychardia lead in bipolar (tip/ring) mode to capture (stimulate) healthy cardiac tissue ?

Context: threshold test in animal experiments.

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

Computer model
and simulation

Verification and
validation<fnref n="1" />

Input: design parameters

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

Output: Lapicque curve

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

<footnote n="1">¹ASME V&V 40-2018, Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices.</footnote>
# Web-based computational platform

Ideal objective: run trials on a web based platform, and obtain *credible results*

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

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

InSilicoTrials

Credible according to VV40² : Qol, Cou, Verification, Validation...

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

<footnote n="2">²ASME V&V 40-2018, *Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices*.</footnote>
# Time course of an electrical pulse

1. Pacing: for duration T (ms), with charge Q (eq. V)

2. Interruption (switches): 122µs

3. Output Capacitor Discharge (OCD): ≈ 13ms

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

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

electrolytic bath → bench exp.

excitable tissue → animal exp.

<figure><img src="image_20.png" /></figure>
# Time course of an electrical pulse

1. Pacing: for duration T (ms), with charge Q (eq. V)

2. Interruption (switches): 122µs

3. Output Capacitor Discharge (OCD): ≈ 13ms

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

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

contact

charges

electrolytic bath → bench exp.

excitable tissue → animal exp.

contact

charges

<figure><img src="image_23.png" /></figure>
# Time course of an electrical pulse

1. **Pacing**: for duration T (ms), with charge Q (eq. V)

2. **Interruption (switches)**: 122µs

3. **Output Capacitor Discharge (OCD)**: ≈ 13ms

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

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

contact
discharges

contact
discharges

electrolytic bath → bench exp.

excitable tissue → animal exp.

<figure><img src="image_26.png" /></figure>
# Time course of an electrical pulse

1. **Pacing**: for duration T (ms), with charge Q (eq. V)

2. **Interruption (switches)**: 122µs

3. **Output Capacitor Discharge (OCD)**: ≈ 13ms

<figure><img src="image_27.png" /></figure>

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

contact
discharges

contact
discharges

electrolytic bath → bench exp.

excitable tissue → animal exp.

<figure><img src="image_29.png" /></figure>
# Time course of an electrical pulse

1. Pacing: for duration T (ms), with charge Q (eq. V)

2. Interruption (switches): 122µs

3. Output Capacitor Discharge (OCD): ≈ 13ms

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

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

contact ?

electrolytic bath → bench exp.

excitable tissue → animal exp.

contact ?

<figure><img src="image_32.png" /></figure>
Contact impedance model

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

* At pacing, creation of a “double layer” impedance

* Boundary layer related to the Debye length

* Literature is huge³

Parallel R-C equivalent circuit

³Grimnes and Martinsen, *Bioimpedance and Bioelectricity Basics* – Third Edition.

<figure><img src="image_34.png" /></figure>
Bath experiment / model

<figure><img src="image_35.png" /><figcaption>Figure: Passive bath → def of contact model and parameter identification</figcaption></figure>

<figure><img src="image_36.png" /></figure>
Bath experiment / model

<figure><img src="image_37.png" /></figure>

Equation for a passive electrolyte ?

● Poisson equation = linear PDE

Contact model ⇔ Debye BL

● Linear equivalent resistance Rbio

Figure: Passive bath → def of contact model and parameter identification

<figure><img src="image_38.png" /></figure>
The passive bath model

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

$$R_{\text{tot}} = R_{\text{pulse}} + R_{\text{gnd}} + r_{\text{ring}} + r_{\text{tip}} + R_{\text{bio}}$$

$$\frac{1}{C_{\text{tot}}} = \frac{1}{C_{\text{ets}}} + \frac{1}{C_{\text{ts}}} + \frac{1}{C_{\text{tip}}} + \frac{1}{C_{\text{ring}}}$$

$$\frac{dI}{dt} + \frac{1}{\tau}I = \frac{1}{R_{\text{tot}}} \left( \frac{V_{\text{tip}}}{\tau_{\text{tip}}} + \frac{V_{\text{ring}}}{\tau_{\text{ring}}} \right) \\ \frac{dV_i}{dt} + \frac{V_i}{\tau_i} = \frac{I}{C_i}, \quad i \in \{\text{tip, ring}\}$$

<figure><img src="image_41.png" /></figure>
# The passive bath experiments

* MICROPORT CRM BOREA pacemaker + VEGA leads

* Saline solution measured at $\sigma = 2.85\,\text{mS/cm}$

<figure><img src="image_42.png" /><figcaption>(a) Bath experiment</figcaption></figure>

<figure><img src="image_43.png" /><figcaption>(b) Measured / model voltage $V_{\text{meas}}$</figcaption></figure>

* 9 datasets: $(0.25, 0.5, 1) \, \text{ms} \times (1, 2, 4) \, \text{V}$

<figure><img src="image_44.png" /></figure>
Method to calibrate the contact parameters

* Bath equivalent resistance computed from the initial data ($V_{tip} = V_{ring} = 0$):

$$R_{bio} = \frac{V_{meas}^{*}(t=0)}{I^{*}(t=0)} - r_{tip} - r_{ring}$$

* 4 parameters (device components are known):

$$\theta = (R_{tip}, R_{ring}, \tau_{tip} := R_{tip} C_{tip}, \tau_{ring} := R_{tip} C_{tip})$$

* Cost function weighted by step $s \in \{\text{pulse, interruption, OCD}\}$

$$J(\theta) = \sum_{s} \omega_{s} \mathbb{1}_{s}(t) \left( \frac{|V_{\text{meas}}(\theta) - V_{\text{meas}}^{*}|l^{2}}{|V_{\text{meas}}^{*}|l^{2}} + \frac{|I(\theta) - I^{*}|l^{2}}{|I^{*}|l^{2}} \right), \quad \sum_{s} \omega_{s} = 1$$

* L-BFGS-B from Python Scipy, restricted to a subset of $\mathbb{R}^4$

* ODE solved semi-analytically (i.e. no approx. error)

* First step in Log scale, $R \in (1, 1 \times 10^5)$, $\tau \in (1 \times 10^{-6}, 1)$

* 1 optimisation for each dataset $\rightarrow 9$ parameter sets

<figure><img src="image_45.png" /></figure>
Calibration results – contact parameters

$$R_{\text{ring}} = 26.0 \pm 1.77\Omega, \quad \tau_{\text{ring}} = 54.8 \pm 6.93\mu\text{s}$$

$$R_{\text{tip}} = 4.0 \pm 2.0\text{k}\Omega, \quad \tau_{\text{tip}} = 50 \pm 24\text{ms}$$

Impedance measured by the pacemaker: 940 Ω

<figure><img src="image_46.png" /><figcaption>Figure: Current and voltage in each step, measures and model (left), and parameters (right)</figcaption></figure>

<figure><img src="image_47.png" /></figure>
Tissue experiment / model

<figure><img src="image_48.png" /></figure>

Figure: Animal experiment → validation data

<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>
Tissue experiment / model

<figure><img src="image_55.png" /></figure>

Equation for an excitable tissue ?

* Bidomain equation
extracellular current

* Boundary conditions ?

Figure: Animal experiment → validation data

<figure><img src="image_56.png" /></figure>
Thresehold detection tests in animal experiments

<figure><img src="image_57.png" /></figure>

* Ventricular wedge preparation, 3 + 4 sheeps

* Healthy (5) and with scars (2)

* 3 sites: apex (X), septum (→), base (+)

* screw fully deployed, perpendicular

* Recording of $V_{meas}^*$, like in the bath

* Recording of distant EGMs with needle electrodes

* Optical signals + 9.4T MRI

Automatic threshold search by the pacemaker:

* For a fixed duration $T$ (ms) – 5 pulses at 1.5 Hz

* Decrease voltage to locate the interval : last capturing → first non-capturing

* Capture is controlled through EGMs, or may be detected by the device

<figure><img src="image_58.png" /></figure>
Bidomain coupled to the pacemaker – CEM<fnref n="4" />

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

$$j_i = -\sigma \nabla u \cdot n_i = c_i \partial_t (U_i(t) - u_{|\Gamma_i}(t, x)) + \frac{1}{r_i} (U_i(t) - u_{|\Gamma_i}(t, x)), \quad i \in \{\text{tip, ring}\}$$

<figure><img src="image_61.png" /></figure>

<footnote n="4">⁴Somersalo, Cheney, and Isaacson, “Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography”.</footnote>
Complete system of equations

$$
\tau_{\text{tip}} \frac{dU_{\text{ring}}}{dt} + U_{\text{ring}} = \tau \frac{dU_{\text{tip}}}{dt},
$$

$$
\tau_{\text{tip}} = r_{\text{tip}} C \ll \tau = RC,
$$

$$
R = \textcolor{red}{R_{\text{pulse}}} + R_{\text{gnd}} + r_{\text{ring}},
$$

$$
\frac{1}{C} = \frac{1}{C_{\text{ets}}} + \frac{1}{cts}
$$

$$
- \operatorname{div} (\sigma_i \nabla u_i) = -c_m \partial_t v_m - l_{\text{ion}}(v_m, h) \quad \text{in } \Omega_M
$$

$$
- \operatorname{div} (\sigma \nabla u) = \begin{cases} c_m \partial_t v_m + l_{\text{ion}}(v_m, h) & \text{in } \Omega_M \\ 0 & \text{in } \Omega_B \end{cases}
$$

$$
\partial_t h + g_{\text{ion}}(v_m, h) = 0 \quad \text{in } \Omega_M
$$

With the CEM transmission conditions

$$
j_i = -\sigma \nabla u \cdot n_i = c_i \partial_t (U_i - u_{|\Gamma_i}) + \frac{1}{r_i} (U_i - u_{|\Gamma_i}), \quad i \in \{\text{tip, ring}\}
$$

<figure><img src="image_62.png" /></figure>
Energy estimate

* There is an inequality of energy, which states the energy associated to each (equivalent) dipole in the overall electrical circuit

For test functions $w_i(x)$, $w(x) \in H^1$, and $W_{\text{tip}}$, $W_{\text{ring}} \in \mathbb{R}$,

$$ \int_{\Omega_M} \sigma_i \nabla u_i \cdot \nabla w_i + \int_{\Omega} \sigma \nabla u \cdot \nabla w + \int_{\Omega_M} (c_m \partial_t v_m + l_{\text{ion}}) w_m \\ + \sum_{k \in \{\text{tip}, \text{ring}\}} \int_{\Gamma_k} \left( c_k \partial_t (U_k - u|_{\Gamma_k}) + \frac{1}{r_k} (U_k - u|_{\Gamma_k}) \right) (W_i - w|_{\Gamma_k}) \\ + C \frac{dU_{\text{tip}}}{dt} W_{\text{tip}} + \frac{1}{R} U_{\text{ring}} W_{\text{ring}} = 0 $$

* proof of existence of a solution

<figure><img src="image_63.png" /></figure>
A surrogate 0D model

<figure><img src="image_64.png" /></figure>

We need to rescale the current per unit of surface (ionic) to a current:

$$I = S \left( C_m \frac{dV_m}{dt} + I_{\text{ion}} + g_e V_m \right)$$

<figure><img src="image_65.png" /></figure>
Surrogate 0D model ODEs

$$ \frac{dI}{dt} + \frac{1}{\tau}I = \frac{1}{R_{\text{tot}}} \left( \frac{V_{\text{tip}}}{\tau_{\text{tip}}} + \frac{V_{\text{ring}}}{\tau_{\text{ring}}} + \frac{l_{\text{ion}}(V_m, h)}{C_m} + \frac{V_m}{\tau_m} \right) \quad (1) $$

$$ \frac{dV_i}{dt} + \frac{V_i}{\tau_i} = \frac{I}{C_i}, \quad i \in \{\text{tip}, \text{ring}\} \qquad (2) $$

$$ \frac{dV_m}{dt} + \frac{l_{\text{ion}}(V_m, h)}{C_m} + \frac{V_m}{\tau_m} = \frac{I}{SC_m}, \qquad (3) $$

$$ \frac{dh}{dt} + g(V_m, h) = 0, \qquad (4) $$

* How to choose S and $g_e$ ?

* Our starting choice: $S = 100\text{cm}^2$ and $g_e = 10^{-2}\text{mS}$, obtained from a volume of $1\text{mm}^3$

<figure><img src="image_66.png" /></figure>
Threshold detection in the model

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>

* Capture := more than 50% successful captures, seen on $V_m$

* a 2D dichotomy algorithm → intervals ($V_{min}(T)$, $V_{max}(T)$)

* Rheobase $V_*$ (0.372 V) and chronaxie $T_*$ (1.18 ms) from the Lapicque law (LS fitting):

$$V_{\text{Lapicque}}(T) := V_* \left(1 + \frac{T_*}{T}\right), \quad \text{an minimize} \quad \left| V_{\text{Lapicque}}(t) - \frac{1}{2}(V_{\text{min}}(t) + V_{\text{max}}(t)) \right|^2$$

<figure><img src="image_69.png" /></figure>
First comparisons – experiments and 0D model, see<fnref n="6" />

* Contact parameters fixed from the bath experiments

* Lapicque<fnref n="5" /> law is: $V = a + b/T$

<figure><img src="image_70.png" /></figure>

Figure: Comparison of Lapicque curves (solid lines) computed with different values of $S$ (left) and $R_{\text{tip}}$ (right). The shaded regions indicate where the experimental capture threshold is located.

<figure><img src="image_71.png" /></figure>

<footnote n="5">⁵Louis Lapicque, “Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation”.</footnote>

<footnote n="6">⁶Pannetier, Leguèbe, Coudière, Walton, Dhiver, Feuerstein, and Amaro, “Modeling Cardiac Stimulation by a Pacemaker, with Accurate Tissue-Electrode Interface”.</footnote>
* Pacemakers have very sparse covering of the Laplacian plane,

$T \in \{0.12, 0.25, 0.35, 0.5, 0.75, 0.85, 1.00\}\text{ms}, \quad V \in \{2.0, 1.75, 1.5, 1.0, 0.75, 0.5, 0.25\}\text{V}$

* Laplacian experimental law and the model fit very well

* Laplacian law do not fit within the experimental regions

* (Too ?) many modeling simplification ?

* 0D – 1 cell (now 2 cells)

* $g_e$ and $S$: influence of $g_e \ll$ influence of $S \to 3D$ ?

* Contact properties in bath *versus* tissue ? Influence of $R_{\text{tip}}$

* BR model with CellML std parameters

* What about the V&V40 guidelines

* Fast computations, OK for prototyping ideas / computational procedures

<figure><img src="image_72.png" /></figure>
Some recent 3D results I

<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /><figcaption>Mesh resolution 6µm – 1.5mm, time-step 10µs, P1-Lagrange FEM, BDF2+RL2, BiCGStab
Supra-threshold pacing, *T* = 1ms and *V* = 2.5V</figcaption></figure>

<figure><img src="image_78.png" /></figure>
Some recent 3D results II

Standard CellML BR model + electrical conductivity coefficients

<figure><img src="image_79.png" /><figcaption>(a) Simulated voltage $V_{meas}$, capture case</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>(b) voltage during OCD, capture case</figcaption></figure>

<figure><img src="image_81.png" /></figure>
Some recent 3D results III

<figure><img src="image_82.png" /><figcaption>(c) Laplacque curve, 2 lead designs, the commercial and a custom ones</figcaption></figure>

<figure><img src="image_83.png" /></figure>
# The V&V40 framework

Next: how to make it useful and credible ?

**Qol:** What is the energy level needed to capture a heart beat ?

**CoU:** As it would be reported by threshold detection procedures during animal experiments

<table><caption>Figure 2: Model Risk Matrix (cf. ASME VV40) evaluating the COU included in UCI.</caption><thead><tr><th>Model</th><th>high</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><th>influence</th><th>medium</th><td>2 COU</td><td>3</td><td>4</td></tr><tr><th></th><th>low</th><td>1</td><td>2</td><td>3</td></tr><tr><th></th><th></th><th>low</th><th>medium</th><th>high</th></tr><tr><th></th><th></th><th colspan="2">Decision</th><th>consequence</th></tr></tbody></table>

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.png" /></figure>

Technical Report
D/A-1: VERIFICATION & UNIFORMITY QUANTIFICATION FOR THE COU
CODE OF VV40

Technical Report
D/A-2: VALUATION OF THE MODEL FOR THE COU FOR THE CODES OF VV40

Work Product: 7/90
VERIFICATION, VALUATION, UNIFORMITY, QUANTIFICATION &
CRITERION

Work Product: 7/90
VERIFICATION, VALUATION, UNIFORMITY, QUANTIFICATION &
CRITERION

Task Level: U, P, T, F, P, T, F

Task Level: U, P, T, F, P, T, F

T

F

(a) Risk matrix foll. ASME VV40

(b) V: 231 pages

(c) V: 151 pages

<figure><img src="image_88.png" /></figure>
<table><caption>Table 6: Verification Credibility Factors Coverage Level for Use Case 1 (cf. ASME VV40); * indicates verification activities not yet completed.</caption><thead><tr><th>Model Risk<br>Credibility Factor Coverage Level</th><th></th><th>X</th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Code Verification: Software Quality Assurance</td><td>V</td><td>X</td><td></td><td></td><td></td></tr><tr><td>Code Verification: Numerical Code Verification - NCV</td><td>V</td><td>X</td><td></td><td></td><td></td></tr><tr><td>Calculation Verification - Discretization Error *</td><td>V</td><td>X</td><td></td><td></td><td></td></tr><tr><td>Calculation Verification - Numerical Solver Error *</td><td>V</td><td>X</td><td></td><td></td><td></td></tr><tr><td>Calculation Verification - Use Error</td><td>IV</td><td>X</td><td></td><td></td><td></td></tr></tbody></table>

SQA: software code CEPS, thorough CI, SonarQube analysis, ...

NCV, discretization error, solver error: that is numerical analysis.

We provide many automatic tests (interpolation, quadrature, ODE solvers) and convergence tests on PDEs (Laplace, heat, reaction-diffusion) w/r to manufactured and reference solutions

Use error: on the web-based platform + within SQA

We cover credibility up to 4 or 5, while 2 is needed (risk analysis).

<figure><img src="image_89.png" /></figure>
# On-going work on validation

<table><caption>Table 6: Validation and Applicability Credibility Factors Coverage Level for Use Case 1 (cf. ASME VV40); * indicates validation activities not yet completed; N/A indicates activities which requires completion to be evaluated.</caption><thead><tr><th rowspan="2">Credibility Factor Coverage Level</th><th colspan="5">Model Risk</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>Validation - Model [Form]</td><td>III</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Validation - Model [Inputs] *</td><td>III</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Validation - Comparator [Test Samples] *</td><td>IV</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Validation - Comparator [Test Conditions] *</td><td>II</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Validation - Assessment [Input Parameters] *</td><td>III</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Validation - Assessment [Output Comparison]</td><td>N.A.</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Applicability: Relevance of the Quantities of Interest *</td><td>V</td><td></td><td>x</td><td></td><td></td></tr><tr><td>Applicability: Relevance of the Validation Activities to the COU *</td><td>V</td><td></td><td>x</td><td></td><td></td></tr></tbody></table>

* To be done in view of the Qol and CoU

* For UC1 (Simcardiotest), we could design and run our own experiments

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

<figure><img src="image_92.png" /></figure>

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>
Path to calibration, before validation

<figure><img src="image_96.png" /></figure>

Calibrate model's parameters

* Contact parameters → done (bath experiments)

* Ionic parameters → optical signal, pixel-wise, done

* Conductivity coefficients → 9.4T MRI (fat, fibrosis) + activation maps, partially done

Compare to experiments for validation

* Lapicque curves are within the experiment Lapicque regions ?

<figure><img src="image_97.png" /></figure>
Ionic parameters for the BR model

<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

* BR has 2 parameters for the Na⁺ current: $g_{Na}$ and $g_{Na,c}$

* Optimisation technique developed inª: compass search algorithm

* Automatic search in each pixel of each camera (2 camera)

* Using 6 consecutive pnings, beginning of the experiment

* Data has to be normalized (either baseline only or baseline + peak)

<figure><img src="image_101.png" /></figure>

<footnote>ªPongui Ngoma, Bourgault, Pop, and Nkounkou, "Adjustment of Parameters in Ionic Models Using Optimal Control Problems".</footnote>
Identification of parameters for 1 pixel

<figure><img src="image_102.png" /></figure>

Figure: Exemple identification match. Raw signal (blue), filtered one (orange), and model (black).

<figure><img src="image_103.png" /></figure>
Spatial distribution of g<sub>Na</sub> and g<sub>Na,c</sub>

<figure><img src="image_104.png" /><figcaption>Figure: g<sub>Na</sub> (left) and g<sub>Na,c</sub> (right) on endo (top) and epi (bottom) – 2 acquisitions.</figcaption></figure>

<figure><img src="image_105.png" /></figure>
Endo / epi distribution of parameters

<figure><img src="image_106.png" /><figcaption>Figure: Statistical distribution of identified $g_{Na}$ and $g_{Na,c}$</figcaption></figure>

<figure><img src="image_107.png" /></figure>
Microstructure around the tip : conductivity coefficients

<figure><img src="image_108.png" /><figcaption>(a) volume
fraction</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>(b) optical map
window</figcaption></figure>

<figure><img src="image_110.png" /><figcaption>(c) homogenization principle<sup>7</sup></figcaption></figure>

<figure><img src="image_111.png" /><figcaption>(d) conductivity
coefficients</figcaption></figure>

* Segmentation and fiber direction → done (thanks to V. Ozenne, J. Magat)

* Volume fraction → done

* How to register the optical window to the MR image (folded sample) → in progress

<footnote n="7">⁷Davidović, Coudière, and Bourgault, "Modelling the action potential propagation in a heart with structural heterogeneities: From high-resolution MRI to numerical simulations".</footnote>
Optical image registered to the MRI

<figure><img src="image_112.png" /><figcaption>Figure: Registration with landmarks</figcaption></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>

<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>
Conclusion

* Bridge between bioimpedance (signal processing) and bidomain (PDE modeling)

* Validation following V&V40 is difficult, a necessity, and *useful*

* Part of it is cumbersome (SQA)

* Validation (on-going), experiment were specifically designed

* Sensitivity analysis, uncertainty quantification ?
  → **postdoc**

* What really is capture, what criteria in the model ?

* Series of pulses in 3D, faster Laplacque in 3D (continuation methods) ?

<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /></figure>
Thanks for your attention!

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /></figure>
<figure><img src="image_127.png" /></figure>

ASME V&V 40-2018. Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices.

American Society of Mechanical Engineers New York, NY, 2018.

<figure><img src="image_128.png" /></figure>

Davidović, Andela, Yves Coudière, and Yves Bourgault. "Modelling the action potential propagation in a heart with structural heterogeneities:

From high-resolution MRI to numerical simulations". In: *International Journal for Numerical Methods in Biomedical Engineering* 37.11 (2021), e3322. DOI: 10.1002/cnm.3322.

<figure><img src="image_129.png" /></figure>

Grimnes, Sverre and Ørjan G. Martinsen. *Bioimpedance and Bioelectricity Basics – Third Edition*. Elsevier, 2015. DOI: 10.1016/c2012-0-06951-7.

<figure><img src="image_130.png" /></figure>

Louis Lapicque, Louis. "Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation". In: *Journal de physiologie et de pathologie générale* 9 (1907), pp. 620-635.

<figure><img src="image_131.png" /></figure>

Pannetier, Valentin, Michael Leguèbe, Yves Coudière, Richard Walton, Philippe Dhiver, Delphine Feuerstein, and Diego Amaro. "Modeling

Cardiac Stimulation by a Pacemaker, with Accurate Tissue-Electrode Interface". In: *Functional Imaging and Modeling of the Heart*. Springer Nature Switzerland, June 2023, pp. 194-203. DOI: 10.1007/978-3-031-35302-4_20.

<figure><img src="image_132.png" /></figure>

Pongui Ngoma, Diogène Vianney, Yves Bourgault, Mihaela Pop, and Hilaire Nkounkou. "Adjustment of Parameters in Ionic Models Using

Optimal Control Problems". In: *Functional Imaging and modeling of the Heart*. Ed. by Mihaela Pop and Graham A. Wright. Vol. 10263. Lecture Notes in Computer Science. Springer, 2017, 322 ff.

<figure><img src="image_133.png" /></figure>

Somersalo, Erkki, Margaret Cheney, and David Isaacson. "Existence and Uniqueness for Electrode Models for Electric Current Computed

Tomography". In: *SIAM Journal on Applied Mathematics* 52.4 (1992), pp. 1023-1040. DOI: 10.1137/0152060. eprint: https://doi.org/10.1137/0152060.

<figure><img src="image_134.png" /></figure>