# Deep modelling of electric field distribution for clinical electroporation ablation therapies

Kylian Desier, Luc Lafitte, Laurent Facq, Olivier Sutter, Olivier Seror, Clair

Poignard, Baudouin Denis de Senneville

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Kylian Desier, Luc Lafitte, Laurent Facq, Olivier Sutter, Olivier Seror, et al.. Deep modelling of electric field distribution for clinical electroporation ablation therapies. ISBI 2025 - International Symposium on Biomedical Imaging, IEEE, Apr 2025, Houston (TX), United States. pp.1-5. hal-05060090

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Distributed under a Creative Commons Attribution 4.0 International License
DEEP MODELLING OF ELECTRIC FIELD DISTRIBUTION FOR CLINICAL
ELECTROPORATION ABLATION THERAPIES

Kylian Desier¹, Luc Lafitte¹, Laurent Facq¹, Olivier Sutter¹,², Olivier Seror²,
Clair Poignard³, Baudouin Denis de Senneville¹

¹ University of Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251, F-33405 Talence, France

² Interventional Radiology Unit, Avicenne Hospital, APHP, Paris Nord University,
F-93000 Bobigny, France

³ INRIA Research Center at Rennes University, F-35042 Rennes, France

ABSTRACT

Electroporation ablation is a promising non-surgical and minimally invasive tumor ablation technique. The objective of this paper is to propose a deep learning strategy for fast and accurate 3D computation of the electric field distribution during the ablation procedure. The computation of the electrostatic potential is accelerated using a well-designed convolutional neural network, which serves to initialize the iterative solver for matrix inversion

The proposed approach, which combine deep learning and more
standard numerical strategies, is compared with state-of-the-art nu-
merical schemes, evaluating its performance in terms of error on the
solution, residuals of the numerical scheme, and computation time.
As a proof of concept, we demonstrate its potential on a clinical case
of electroporation ablation of a liver tumor, showing that the pro-
posed approach can be advantageously combined with clinical data
to provide real-time insights into the effective electroporation abla-
tion.

***Index Terms***— Deep Neural Network, Electric field distribu-
tion, Electroporation, Interventional imaging.

# 1. INTRODUCTION

Irreversible electroporation (IRE) is a minimally invasive ablation
technique commonly employed for the treatment of deep-seated tu-
mors, in particular liver tumors [1]. It consists in delivering local-
ized high-voltage short duration electrical pulses through conduc-
tive needles, in order to permeabilize irreversibly the cell membranes
and induce mainly apoptotic cell death [2]. This minimally invasive
technique precisely targets the tumor site while minimizing damage
to surrounding healthy tissue [3, 4, 5]. As a predominantly non-
thermal method, IRE is well-suited for tumors located near major
blood vessels and critical organs, minimizing the risk of damage to
surrounding vital structures. However, precise targeting is critical, as
incorrect application can lead to incomplete ablation or unintended
damage. In the treatment of liver tumors, 3 to 6 electrodes are in-
serted in the organ, and high-voltage electrical pulses are delivered
between them [6]. This method has been shown to induce cell death
in a certain zone of the liver, but careful assessment of the affected
tissues is essential and challenging to control the ablation area effec-
tively [7, 8]. Due to the complexity of the IRE procedure, detailed
treatment planning and per procedural evaluation based on medical
imaging are essential. In practice, the target tumor is first localized
using high-resolution imaging, followed by the percutaneous inser-
tion of needles. Computed Tomography (CT), Cone Beam Com-

puted Tomography (CBCT), ultrasound, or angiography can be used
to verify needle positioning [9]. Finally, the IRE pulses are deliv-
ered.

Several mathematical models exist for simulating the electric
distribution of the deposited dose, with varying levels of complex-
ity [10]. However, computation time increases as the models become
more refined. Due to the nonlinearity of these models and the numer-
ical methods typically employed — mainly finite element or finite
difference methods — the computations involve the manipulation
and inversion of large matrices, particularly in three-dimensional
cases. This results in high computational costs, rendering these mod-
els impractical for meeting the time constraints of clinical work-
flows. For per-procedural purposes, the simple linear electrostatic
model has proven to provide valuable information for physicians
[11]. However, the computation of this relatively simple model re-
mains too time-consuming for routine clinical use. Recently, deep
learning strategies have been proposed to leverage neural networks
for approximating partial differential equations (PDEs), often re-
ducing the computational burden of traditional numerical methods
[12, 13, 14]. However, without incorporating physical laws, these
methods can yield solutions that are physically inaccurate or incon-
sistent in certain regions. Achieving a balance between speed and
reliability remains a key challenge.

The computation of the 3D electric field distribution during the
IRE interventional procedure can be achieved within the linear elec-
trostatic model as a linear combination of carefully designed float-
ing potentials, as proposed in [15]. However, the 3D computation
of these basis functions in a clinical context requires a large number
of iterations to achieve satisfactory accuracy, significantly increasing
computation time. In this paper, we propose an accelerated method
that leverages deep learning to accelerate the 3D computation of the
electric field during the ablation procedure. Our contribution is four-
fold:

1. We propose a novel deep learning approach to predict the 3D electric field distribution during the IRE interventional procedure by computing the floating potential basis functions using a dedicated convolutional neural network.

2. A numerical scheme incorporating the proposed network is de-
veloped to ensure the mathematical accuracy of the final solu-
tion, while utilizing the network's output to reduce the number
of solver iterations, thereby accelerating the overall process.

3. The proposed deep learning strategy is compared to a state-of-the-art multi-resolution approach in terms of solution error, numerical scheme residuals (which quantify the accuracy of solv-
ing the electrostatic equation), and computation time.

4. We demonstrate that the proposed method complies with clinical constraints in the context of a patient undergoing IRE intervention for liver tumor treatment.

## 2. MATERIAL AND METHODS

### 2.1. Electrostatic potential in an IRE procedure

Let first recall the problem to be solved during an IRE procedure within a spatial domain $\Omega$ with $N$ inserted needles denoted as $(\Gamma_k)_{k=1,N}$ (see [8, 15, 11]). The pulse delivery is achieved by imposing a voltage (potential difference) $V_{k_+,k_-}$ between a pair of needles $\Gamma_{k_+}, \Gamma_{k_-}$, where $(k_+, k_-) \in \{1, \dots, N\} \times \{k_+ + 1, \dots, N\}$. Consequently, the potential $\phi^{k_+,k_-}$ satisfies the following problem:

$$ \left\{ \begin{array}{l} -\nabla \cdot (\sigma \nabla \phi^{k_+,k_-}) = 0, \text{ in } \Omega, \\ \partial_n \phi^{k_+,k_-}|_{\partial \Omega} = 0, \\ \phi^{k_+,k_-}|_{\Gamma_{k_\pm}} = \pm \frac{1}{2} V_{k_+,k_-}, \\ \phi^{k_+,k_-}|_{\Gamma_\ell} - \alpha_\ell^{k_+,k_-} = 0, \forall \ell \in \{1, \dots, N\}, \ell \neq k_+, k_-, \\ \text{where } \alpha_\ell^{k_+,k_-} \text{ are such that } \int_{\Gamma_\ell} \sigma \nabla \phi^{k_+,k_-} \cdot \mathbf{n} \, ds = 0. \end{array} \right. \quad (1) $$

From this electrostatic potential, the magnitude of the electric field is given as:

$$ E^{k_+,k_-}(x) = \|\nabla\phi^{k_+,k_-}(x)\|, \quad \forall x \in \Omega, $$

and the dose distribution of the procedure $D$ is the map defined as:

$$ D(x) = \max_{(k_+, k_-) \in \{1, \dots, N\} \times \{k_+ + 1, \dots, N\}} (\|\nabla \phi^{k_+, k_-}(x)\|). \quad (2) $$

Initially, $N(N-1)/2$ computations of Equation 1 are required. However, it is known that $\phi^{k_+,k_-}$ can be expressed as a linear combination of $N$ appropriately chosen basis functions [15]. More precisely, let $v_l \forall l \in \{1, N\}$ satisfy:

$$ \left\{ \begin{array}{l} -\nabla \cdot (\sigma \nabla v_l) = 0, \text{ in } \Omega, \\ v_l|_{\Gamma_l} = 1, \\ \partial_n v_l|_{\partial \Omega} = 0, v_l|_{\Gamma_k} = 0, \quad \forall k \neq l. \end{array} \right. \quad (3) $$

Then we have:

$$ \phi^{k_+,k_-} = \frac{1}{2} V_{k_+,k_-} v_{k_+} - \frac{1}{2} V_{k_+,k_-} v_{k_-} + \sum_{\ell \neq k_+,k_-} \beta_\ell v_\ell, $$

where $\beta_\ell$ are constants values chosen such that the last equality in Equation 1 is satisfied. This formulation reduces the problem to solving only $N$ Poisson-type equations, along with the inversion of an $(N-2) \times (N-2)$ matrix, which is computationally efficient since $N \le 10$. The main challenge then lies in solving Equation 3 as quickly as possible. Our purpose is to use convolutional neural networks (CNNs) to accelerate this computation, as outlined in the next section.

### 2.2. Deep calculation of coarse basis functions $(v_l)_{l=1}^N$

Figure 1 illustrates the 3D CNN architecture designed to generate, through supervised learning, the set of $N$ basis functions of Equation 3 based on the position, orientation, and activity of the inserted needles.

The network input relies on the position, orientation, and activation of the inserted needles, structured in a matrix format to meet the typical input requirements of CNNs. The input matrix was populated as follows: active needle voxels were assigned a value of 1, inactive needle voxels were assigned a value of -1, and background voxels were set to 0.

The network output comprises basis functions corresponding to the input needle map, generated by solving Equation 3 with a direct implicit scheme that incorporates the relevant needle positions, orientations, and activations. Second-order Neumann boundary conditions were applied during this computation.

The network training data were generated by randomly sampling the 3D coordinates of the two extremities of needle segments using a uniform distribution within the domain $\Omega$. For each pair of 3D coordinates representing the needle extremities, a 3D Bresenham algorithm was employed to map the active (resp. inactive) needle segments onto the input matrix.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of the used U-Net [16] architecture for the prediction of the $N$ basis functions $(v_l)_{l=1}^N$ (Equation 3).</figcaption></figure>

### 2.3. Fine calculation of basis functions $(v_l)_{l=1}^N$

The coarse basis functions predicted by the network were used as an initial guess for an iterative stabilized bi-conjugate gradient scheme. This approach guarantees the mathematical correctness of the final solution while utilizing the network's output to decrease the number of solver iterations, thus accelerating the overall computation process.

### 2.4. Experimental Setup

#### 2.4.1. Training parameters

The training dataset consisted of 100 basis function maps (dimension $32 \times 32 \times 32$), generated from the simulation of three needles positioned randomly without restrictions. Each of these maps was paired with its corresponding basis function, providing a comprehensive dataset for training the model.

#### 2.4.2. Assessment of coarse-to-fine calculation of basis functions

An independent set of 100 basis function maps, generated using a strategy similar to that of the training dataset, was employed to evaluate the performance of the coarse-to-fine calculation of basis functions. This separate testing set allowed for an objective assessment
of the model's accuracy and effectiveness in generating basis functions.

### 2.4.3. Preliminary assessment of an IRE procedure

During IRE of a liver tumor, a cone-beam computed tomography (CBCT) scan of the abdominal cavity was obtained after needle insertion (4 needles) and prior to pulse delivery. The dimensions of the scan are 512 pixels in the right-left and anterior-posterior directions, and between 195 and 512 pixels in the superior-inferior direction. The voxel sizes are 0.45 mm in all directions. The conductivity was set to $\sigma = 0.7$ S/m, in the conductivity range of liver tumor as specified by [17].

### 2.4.4. Benchmarking of the proposed approach

The proposed deep learning approach was benchmarked against a state-of-the-art stabilized bi-conjugate gradient method, integrated within a multi-resolution framework. A coarse solution is first computed on a grid that is downsampled by a factor of two. This coarse solution is then upsampled and used as an initial guess for solving the problem at the original resolution. The efficiency of the proposed and the competing multi-resolution approaches to solve the electrostatic problem in Equation 3 was assessed through the following dedicated metrics:

**Error on the solution:** The L2 norm (Euclidean distance) between the computed solutions $v_l$ and the solution of the direct implicit scheme $v_l^*$ as follows:

$$ \|v_l - v_l^*\|_2 = \frac{1}{|\Omega|} \sum_{(x,y,z) \in \Omega} (v_l(x,y,z) - v_l^*(x,y,z))^2 \quad (4) $$

$(x, y, z)$ being the voxel coordinates and $|\Omega|$ the cardinal of $\Omega$. Voxel-wise absolute error maps were also calculated.

**Residual of the numerical scheme:** The continuous partial differential problem in Equation 3 could be expressed as a linear system (form $Ax = b$) through the finite difference approach. The residual $\epsilon$ of the electrostatic equation was computed as $\epsilon = ||Ax - b||_2$ to quantify the mathematical accuracy of the solution.

### 2.4.5. Hardware and implementation

All trainings have been completed on a computer running Ubuntu 20.04.6 LTS with the Intel(R) Xeon(R) Gold 6248R CPU and a NVIDIA RTX A6000 GPU with 48 GB of VRAM. Time benchmarks were conducted on standard laptop hardware equipped with an Intel(R) Core(TM) i9-10980HK CPU and a NVIDIA GeForce RTX 3080 GPU. All implementation were performed using Python 3.10. We employed SciPy's sparse library for all matrix operations. Specifically, we used the Bi-Conjugate Gradient Stabilized "bicgstab" function, while the reference solutions (implicit scheme) were computed using the "spsolve" function.

## 3. RESULTS

### 3.1. Assessment of coarse-to-fine calculation of basis functions

Figure 2 displays a typical basis function $v_l$ ($l \le N$) obtained using the direct iterative scheme bi-conjugate gradient scheme, the proposed

deep learning network, and the deep learning prediction utilized as an initial guess for a stabilized bi-conjugate gradient scheme. It can be observed that the output of the deep learning model (Figure 2b) appears smoother than that of the reference method (Figure 2a), particularly in the regions around the needles (see Figure 2). When employed as an initial guess for the stabilized bi-conjugate gradient scheme, this approach yields a rapid solution with residual errors (Figure 2c). As shown in Figure 2, our proposed network, when combined with the bi-conjugate gradient scheme, is more than eight times faster than utilizing an implicit scheme.

Quantitative metrics presented in Figure 3 corroborate these visual observations: the proposed coarse-to-fine calculation of the basis function, indicated by the curve representing our method, consistently lies below that of the competing multi-resolution approach across all tested metrics (including solution error and residuals of the numerical scheme). This results in a faster convergence rate and, consequently, reduced computation times.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Typical example of a basis function $v_l$, $l \le N$ (3 needles simulated) computed using: (a) the direct implicit scheme used as reference, (b) the proposed deep learning network, (c) the proposed deep learning prediction as an initial guess for a stabilized bi-conjugate gradient scheme (100 iterations). Yellowish values correspond to voxel in the vicinity of the active needle ($v_l|_{\Gamma_l} = 1$) and blackish values correspond to voxel in the vicinity of the two other inactive needles ($v_l|_{\Gamma_k} = 0, \forall k \ne l$). The bottom right inserts in (b) and (c) show the corresponding voxel-wise absolute error (colormap ranged from 0 to 0.1 for enhanced visualization), using the direct implicit scheme in (a) as reference. The time (averaged on 100 testing cases) required by the different methods to compute their solutions is displayed in the top right corner.</figcaption></figure>

### 3.2. Numerical assessment of the IRE procedure

Figure 4 illustrates the potential of the proposed approach to compute the dose map of a specific IRE procedure as defined by Equation 2 using the applied voltage given by the IRE data. While the coarse deep learning basis functions exhibit lower accuracy in modeling the delivered electric field in the immediate vicinity of the needles (Figure 4b) compared to the reference (Figure 4a), the coarse-to-fine approach facilitates fast and accurate modeling solutions (Figure 4c). A nearly five-fold increase in speed is observed when comparing the implicit scheme to the complete pipeline.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Quantitative assessment of the proposed deep learning strategy. The error on the solution (a) (i.e L2 norm using the implicit solution as reference) and the residual of the numerical scheme (b) are displayed along the iteration of the bi-conjugate gradient stabilized algorithm. The proposed deep learning strategy and the competing multi-resolution strategy are displayed in blue and orange, respectively (3 needles/100 random needles positioning trials/solid curve=mean, thickness of the light area=standard deviation).</figcaption></figure>

## 4. DISCUSSION

The proposed method demonstrates the significant potential of deep learning approach for accelerating the 3D computation of the electric field during an IRE ablation procedure. It is worth noting that although our model was trained using only three needles, we find that it produces coherent results for configurations with four needles in the IRE ablation procedure.

To prioritize inference speed in our study, we did not investigate diffusion-based models. Instead, we employed a U-Net architecture [16] with deep supervision [18], as it provides an optimal balance between ease of training, performance, and inference efficiency. We experimented with several loss functions and found that mean squared error (MSE) yielded the best results.

While our study is based on a model assuming constant conductivity throughout the tissue, future work will extend this approach by implementing tissue-specific conductivities for both the tumor, organs at risk, and electrodes. This refinement will allow for more accurate computation of the electric field distribution and better capture the heterogeneous nature of biological tissues during the IRE procedure.

Our study is conducted within the framework of a linear model, which assumes that tissue conductivity remains constant and does

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Computation of the delivered electric field super-imposed on the CBCT scan during IRE of a liver tumor (4 needles inserted). Electric field using basis functions $(v_l)_{l=1}^N$ calculated using: (a) the direct implicit scheme, (b) the proposed deep learning network, (c) the proposed deep learning as an initial guess for a stabilized bi-conjugate gradient scheme (100 iterations). The bottom right inserts in (b) and (c) show the corresponding voxel-wise absolute error, using the direct implicit scheme in (a) as reference.</figcaption></figure>

not vary with the applied electric current. In future work, we will also explore non-linear models that account for the dynamic changes in conductivity as a function of the electric field. These advanced models will provide a more realistic representation of tissue behavior under high-intensity electric pulses, enhancing the accuracy of electric field simulations during the IRE procedure.

## 5. CONCLUSION

The proposed approach paves the way for the computation of large-scale datasets, ultimately resulting in more precise simulations of the electric field distribution. Enhanced accuracy in these simulations is crucial for optimizing ablation techniques and improving patient outcomes.

Future work will integrate physical constraints into the network, thereby adopting a physics-informed approach to enhance the model's generalizability and predictive capability in accommodating dynamic tissue-specific conductivities.

## 6. COMPLIANCE WITH ETHICAL STANDARDS

This retrospective study is in accordance with ethical principals of the Declaration of Helsinki and has been approved by the local committee on human research of the University Hospital J. Verdier.

## 7. ACKNOWLEDGMENTS

Computational analysis was carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d'Aquitaine (see https://www.plafrim.fr/). This research has been conducted with the partial financial support of the Plan Cancer MECI PC_MECI_21CM119_00 and the Institut National du Cancer (INCa) (PLBIO n°2023-156). C. Poignard was also partly granted by the ANR projects IMITATE and MIRE4VTACH. The Bernoulli lab between AP-HP and Inria yielded a crucial help to O. Séror, O. Sutter, B. Denis de Senneville and C. Poignard, through the projects NEPA and EVALHEP.
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