# A deep learning-assisted hybrid model for electric dosimetry in electroporation therapies

Kylian Desier, Luc Lafitte, Laurent Facq, Olivier Sutter, Clair Poignard, Baudouin Denis de Senneville

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Kylian Desier, Luc Lafitte, Laurent Facq, Olivier Sutter, Clair Poignard, et al. A deep learning-assisted hybrid model for electric dosimetry in electroporation therapies. CBMS 2025 - International Symposium on Computer-Based Medical Systems, IEEE, Jun 2025, Madrid, Spain. pp.1-4. hal-05060101

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# A deep learning-assisted hybrid model for electric dosimetry in electroporation therapies

Kylian Desier*, Luc Lafitte*, Laurent Facq*, Olivier Sutter*†, Olivier Seror†,
Clair Poignard‡§ and Baudouin Denis de Senneville*§

*Univ. of Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251, 351 cours de la Libération, Talence, F-33400, France
Email: {kylian.desier, bdenisde}@math.u-bordeaux.fr

†Interventional Radiology Unit, Avicenne Hospital, APHP, 125 Rue de Stalingrad, F-93000 Bobigny, France

‡INRIA Research Center at Rennes Univ., Campus de Beaulieu, 263 Av. Général Leclerc, F-35042 Rennes, France
§ These two authors jointly supervised this work.

***Abstract***—Accurate electric dosimetry is essential for predicting treatment outcomes in Irreversible Electroporation (IRE), a promising non-thermal tumor ablation technique. However, real-time 3D electric field computation remains computationally expensive, limiting its practical use in clinical settings.

In this study, we propose a hybrid approach combining deep learning (DL)-based initialization with an iterative numerical solver to accelerate dose map calculation. Specifically, we employ a convolutional neural network (CNN) to predict basis functions, which are subsequently refined using a Biconjugate Gradient Stabilized (Bi-CGSTAB) solver to maintain physical accuracy.

We evaluate our method on data from 10 patients undergoing IRE liver ablation under real-time clinical conditions. Our results demonstrate a 10× speedup compared to conventional solvers while maintaining comparable accuracy. This hybrid method offers a promising pathway toward fast and reliable electric dosimetry for on-line IRE procedures.

***Index Terms***—Deep Learning, Electroporation, Dose Map Estimation, Hybrid Methods, Biomedical Imaging.

## I. INTRODUCTION

Irreversible Electroporation (IRE) is a promising non-thermal tissue ablation technique that uses high-voltage electrical pulses to create permanent nanopores in cell membranes [1], leading to cell death while preserving the extracellular matrix and critical structures such as blood vessels and nerves [2]. This makes IRE particularly advantageous for treating tumors located near vital structures, where traditional thermal ablation methods pose significant risks [2].

The success of IRE depends heavily on the precise placement of electrodes, as their positioning directly influences the electric field (EF) distribution and treatment efficacy [3]. To optimize outcomes, computational models are employed to generate dose maps, predicting the extent of cell death based on factors such as tissue conductivity, pulse parameters, and tumor geometry [4]. Mathematical modeling holds great potential in simulating these EF distributions, by ensuring adequate tumor coverage while minimizing harm to surrounding healthy

tissue. By refining treatment protocols and personalizing therapy, these models may enhance the overall effectiveness of IRE in liver cancer ablation.

An on-line evaluation of the delivered dose during IRE is highly valuable, as it allows clinicians to adjust treatment parameters on the fly, ensuring complete tumor ablation while minimizing unintended damage. However, full 3D computation of electric fields and dose maps is computationally demanding, making real-time adjustments impractical in clinical settings [5]. To address this challenge, deep learning (DL) has emerged as a promising alternative, offering near-instantaneous predictions of dose distributions based on pretrained models. Despite its speed, DL can be unreliable, particularly when applied to patient-specific cases that differ from its training data [6]. To overcome these limitations, hybrid approaches that integrate DL with physics-based simulations are gaining traction. These methods leverage AI-generated estimates while refining them with computational modeling, striking a balance between speed and accuracy [7]. As hybrid solutions continue to evolve, they hold great potential for making IRE treatment planning both more efficient and more precise. Our proposed hybrid methods show promising results in achieving an optimal balance between computational speed and solution accuracy, paving the way for real-time, patient-specific treatment planning. While our previous study [8] demonstrated promising results at a lower resolution (spatial domain of 32 × 32 × 32 voxels), its scalability to clinically relevant resolutions remained uncertain. However, achieving higher resolution is essential in a clinical setting to precisely delineate the ablated region and ensure accurate treatment outcomes.

In this study, our contribution is threefold:

1. We implemented a DL-assisted hybrid numerical model for electric dosimetry, designed to comply with the clinical constraints of IRE procedures. The method was evaluated using data from 10 patients undergoing IRE liver ablation under real-time clinical conditions.

2. We demonstrate the scalability of this approach to a spatial domain of 100 × 100 × 100 voxels, enabling EF computations at a voxel resolution of 1 mm³.

3. We establish dedicated quantitative metrics based on

<footnote>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.</footnote>
EF thresholds and achieve a 10× speedup compared to conventional multi-resolution solvers while maintaining similar EF accuracy.

## II. METHODS

### A. IRE problem formulation

In this paper, we focus on estimating the dose map delivered during IRE. Given a configuration with N inserted needles, the dose map is defined as:

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

where $\phi^{k+,k-}$ represents the electrostatic potential associated with a given pair of needles. Building on the electrostatic potential formulation [9], [10], the potential can be decomposed as a linear combination of N basis functions $v_l$ [11] where:

$$\forall l \in \{1, \dots, N\}, \begin{cases} -\nabla \cdot (\sigma \nabla v_l) = 0, & \text{in } \Omega, \\ v_l|_{\Gamma_l} = 1, \\ \partial_n v_l|_{\partial \Omega} = 0, v_l|_{\Gamma_k} = 0, & \forall k \neq l. \end{cases} \quad (2)$$

where $(\Gamma_k)_{k=1,N}$ denotes the N inserted needles.

Thus, computing the dose map reduces to solving for these basis functions.

### B. Numerical calculation of basis functions

The proposed approach combines DL, which is used to generate a coarse initial guess for each basis function, with numerical solvers for solution refinement. Specifically, a U-Net model [12] is employed to predict an initial estimate of the aforementioned basis functions from a rasterized map of needle positions. The solution is then refined using the Bi-Conjugate Gradient Stabilized algorithm [13] (referred to as Bi-CGSTAB). This hybrid approach leverages the speed of DL while maintaining physical accuracy through the iterative numerical solution of the original equations, as described in [8].

### C. Experimental validation

1) **Patient database:** The proposed approach was retrospectively evaluated on data from 10 patients undergoing IRE liver ablation. The dataset includes needle positions for 10 patients, with one patient having two distinct needle configurations, resulting in a total of 11 IRE procedures. Each configuration consists of 3 to 7 inserted needles. This dataset originates from a series of identical interventional procedures, as described in [9]. During each intervention, a fixed number of needles are inserted into the patient. Immediately afterward, a CBCT scan is performed, followed by manual segmentation to accurately determine needle positions before the pulses are delivered. The need for real-time dose map computation naturally arises from this clinical workflow. This dataset offers a diverse and clinically relevant benchmark for assessing the reliability and effectiveness of our method.

2) **Training strategy:** To train the U-Net model designed to predict basis functions from needle position maps, a simulated dataset consisting of 100 training basis functions and 100 unseen validation samples (grid size = 100 × 100 × 100) was generated. These samples were derived from the simulation of three needles positioned randomly without any restrictions. Full convergence of the Bi-CGSTAB solver (rtol = $10^{-20}$), embedded in a multi-resolution scheme, was employed to ensure the validity of the generated basis functions.

3) **Comparison with competing approaches:** The implemented hybrid approach was challenged in terms of both accuracy and computation time, against the Bi-CGSTAB solver iterating from scratch (no initial guess), and the Bi-CGSTAB solver embedded in a multi-resolution scheme. In the latter, the solver iterates from a 25 × 25 × 25 solution, progressively refining to the original dimension with an upsample factor of 2 (cubic splines were used to upsample solutions).

4) **Assessment of the proposed hybrid approach:** Full convergence of the Bi-CGSTAB solver (rtol = $10^{-20}$), embedded in a multi-resolution scheme, was employed to generate the groundtruth data.

Let X and Y denote the regions enclosed by an isovalue in an EF estimate and a EF ground truth. The spatial similarity and worst-case discrepancy between these regions were assessed using the Dice coefficient (referred to as DICE, see Eq. (II-C4)) and the Hausdorff distance (referred to as HD, see Eq. (II-C4)), respectively.

$$\text{DICE} = \frac{2|X \cap Y|}{|X| + |Y|}$$

where $| \cdot |$ denotes the cardinality of a set.

$$HD = \max(\max_{x \in X} \min_{y \in Y} d(x, y), \max_{y \in Y} \min_{x \in X} d(x, y))$$

where $d(\cdot, \cdot)$ is the Euclidean distance between two points.

These two metrics were computed for the following three representative EF thresholds: 200 V/cm, 400 V/cm, and 600 V/cm.

5) **Hardware and implementation details:** Our implementation is written in Python 3.10, utilizing the SciPy library for all sparse matrix operations. Specifically, we use the bicgstab function from SciPy's sparse linear algebra module to perform iterative refinement.

Time benchmarks were conducted on a standard consumer-grade laptop equipped with an Intel(R) Core(TM) i9-10980HK CPU and a NVIDIA GeForce RTX 3080 GPU.

## III. RESULTS

### A. Qualitative assessment

Fig. 1 displays a representative example of EF dose maps obtained from data acquired during the IRE liver ablation of a patient. The EF dose map computed from direct inference of the DL network (no Bi-CGSTAB refinement) preserves the overall shape of the ground truth solution. However, small
irregularities are observed along the boundaries due to accumulated errors and artifacts from the learned basis functions. The proposed hybrid approach, in which the DL output serves as an initial guess for the Bi-CGSTAB solver (100 iterations), effectively corrects these irregularities, producing isovalue-enclosed regions that closely match the ground truth solution.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Typical EF dosimetry results obtained during the IRE procedure of a patient undergoing liver ablation. Cross-sections of various estimates of the 3D EF dose maps are superimposed on the corresponding CBCT scan slice. (a) Ground truth solution obtained using the Bi-CGSTAB method with full convergence. (b) EF dose computed using DL-generated basis functions without Bi-CGSTAB refinement. (c) Hybrid DL approach, where the initial guess is provided by the DL output, followed by 100 iterations of the Bi-CGSTAB solver. Colored regions represent EF values in the following ranges: 200–400 V/m (blue), 400–600 V/m (green), and >600 V/m (red).</figcaption></figure>

## B. Quantitative assessment

Table I provides a quantitative comparison of the EF dose obtained using the implemented hybrid approach, the Bi-CGSTAB solver initialized from scratch (without an initial guess), and the Bi-CGSTAB solver integrated into a multi-resolution scheme. Without initial guess, the Bi-CGSTAB solver (full convergence) consistently exhibited the poorest performance. The EF dose map computed from direct inference of the DL network (no Bi-CGSTAB refinement) resulted in poor Hausdorff distance (HD) values, primarily due to errors located at the field of view (FOV) boundaries and near needles. The initial guess generated by the proposed network significantly accelerated early convergence (10–100 iterations), whereas the multi-resolution approach gradually caught up after 1000 iterations. Notably, the solutions obtained using DL initialization/100 Bi-CGSTAB iterations surpassed the accuracy of the multi-resolution approach/1000 Bi-CGSTAB iterations.

## C. Benchmark

Using our experimental setup, the complete calculation of an initial guess, represented by a 50 × 50 × 50 solution for the multi-resolution approach, required around 7.25 seconds. In comparison, the calculation of an initial guess, in the form of a 100 × 100 × 100 output from the proposed DL network, took 0.71 seconds for inference. Using a 100 × 100 × 100 grid, a single iteration of the Bi-CGSTAB solver took an average of 0.04 seconds.

As a result, including overhead, our hybrid method requires approximately 4.7 seconds per needle, compared to 47 seconds for the multi-resolution approach, achieving a tenfold speedup.

## IV. DISCUSSION

A hybrid approach, which relies on DL-based initial guess for each basis function which are refined by an iterative numerical solver, was proposed to accelerate dose map calculation during IRE interventions.

The obtained results underscore the critical importance of a well-chosen initialization and demonstrate the advantages of integrating DL for coarse basis function estimation, enabling faster and more reliable dose map computations. While the DL model provides a strong initial estimate, a small number of numerical iterations significantly enhances accuracy, effectively bridging the gap to full convergence.

Using the proposed approach, it is important to emphasize that dose maps are derived from the gradient of the electrostatic potential rather than the potential itself. Consequently, small deviations in the potential field can be amplified, leading to increased errors in EF estimates. One limitation of the proposed model is that it does not explicitly enforce gradient consistency during training, which likely contributes to the observed artifacts in the FOV boundaries of DL predictions.

## V. CONCLUSION

By leveraging DL to generate an initial guess for each basis function which are subsequently refined using a dedicated numerical solver, we achieved a tenfold speedup compared to conventional numerical approaches, without any loss of accuracy, on clinically relevant data.

Future work may focus on developing physics-informed loss functions [14], [15] to enforce smooth gradient transitions, incorporating larger grid sizes in training data, and implementing an adaptive refinement scheme to correct dose gradient errors before numerical refinement. We hope these advancements will improve gradient consistency and reduce initialization artifacts, further pushing the boundaries of real-time electrostatic potential computation.

## 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.

## 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/).
<caption>TABLE I</caption>

QUANTITATIVE ASSESSMENT THE EF DOSE ACCURACY OBTAINED USING THE IMPLEMENTED HYBRID APPROACH, THE BI-CGSTAB SOLVER ITERATING FROM SCRATCH (NO INITIAL GUESS), AND THE BI-CGSTAB SOLVER EMBEDDED IN A MULTI-RESOLUTION SCHEME. THE SPATIAL SIMILARITY (THROUGH THE DICE COEFFICIENT) AND WORST-CASE DISCREPANCY (THROUGH THE HAUSDORFF DISTANCE COEFFICIENT) BETWEEN THE GROUNDTRUTH AND EF REGIONS ENCLOSED BY ISOVALUES 200 V/CM, 400 V/CM AND 600 V/CM.

<table><thead><tr><th rowspan="2">Initial guess</th><th colspan="2">0 iteration</th><th colspan="2">10 iterations</th><th colspan="2">50 iterations</th><th colspan="2">100 iterations</th><th colspan="2">1000 iterations</th></tr><tr><th>HD ↓</th><th>DICE ↑</th><th>HD ↓</th><th>DICE ↑</th><th>HD ↓</th><th>DICE ↑</th><th>HD ↓</th><th>DICE ↑</th><th>HD ↓</th><th>DICE ↑</th></tr></thead><tbody><tr><th colspan="11" style="text-align: center;">200 V/cm level</th></tr><tr><td>DL</td><td>56.5 ± 19.2</td><td><b>0.90 ± 0.04</b></td><td>4.23 ± 0.95</td><td><b>0.93 ± 0.03</b></td><td>2.57 ± 0.54</td><td><b>0.96 ± 0.03</b></td><td>2.07 ± 0.58</td><td><b>0.97 ± 0.02</b></td><td>1.50 ± 0.62</td><td><b>0.98 ± 0.01</b></td></tr><tr><td>Scratch</td><td>X</td><td>X</td><td>14.0 ± 1.9</td><td>0.11 ± 0.03</td><td>10.4 ± 2.0</td><td>0.52 ± 0.11</td><td>8.6 ± 2.0</td><td>0.75 ± 0.08</td><td>5.5 ± 2.5</td><td>0.88 ± 0.08</td></tr><tr><td>Multi-resolution</td><td><b>4.05 ± 0.61</b></td><td>0.88 ± 0.02</td><td><b>4.05 ± 0.60</b></td><td>0.88 ± 0.02</td><td>3.94 ± 0.63</td><td>0.88 ± 0.02</td><td>3.32 ± 0.68</td><td>0.92 ± 0.02</td><td>2.61 ± 1.13</td><td>0.95 ± 0.03</td></tr><tr><th colspan="11" style="text-align: center;">400 V/cm level</th></tr><tr><td>DL</td><td>19.2 ± 21.1</td><td><b>0.91 ± 0.02</b></td><td>3.19 ± 0.40</td><td><b>0.94 ± 0.02</b></td><td>2.18 ± 0.39</td><td><b>0.97 ± 0.01</b></td><td>1.64 ± 0.47</td><td><b>0.98 ± 0.01</b></td><td>1.27 ± 0.44</td><td><b>0.99 ± 0.01</b></td></tr><tr><td>Scratch</td><td>X</td><td>X</td><td>9.3 ± 1.1</td><td>0.19 ± 0.05</td><td>6.5 ± 1.2</td><td>0.65 ± 0.10</td><td>5.4 ± 1.4</td><td>0.82 ± 0.09</td><td>4.0 ± 1.7</td><td>0.88 ± 0.11</td></tr><tr><td>Multi-resolution</td><td><b>4.12 ± 0.60</b></td><td>0.87 ± 0.02</td><td>4.05 ± 0.63</td><td>0.87 ± 0.02</td><td>3.86 ± 0.78</td><td>0.88 ± 0.02</td><td>3.1 ± 1.2</td><td>0.95 ± 0.01</td><td>1.66 ± 0.58</td><td>0.96 ± 0.02</td></tr><tr><th colspan="11" style="text-align: center;">600 V/cm level</th></tr><tr><td>DL</td><td>5.41 ± 0.93</td><td><b>0.88 ± 0.02</b></td><td>2.77 ± 0.47</td><td><b>0.93 ± 0.01</b></td><td>1.73 ± 0.62</td><td><b>0.97 ± 0.01</b></td><td>1.41 ± 0.62</td><td><b>0.98 ± 0.01</b></td><td>1.10 ± 0.23</td><td><b>0.99 ± 0.01</b></td></tr><tr><td>Scratch</td><td>X</td><td>X</td><td>7.03 ± 0.85</td><td>0.28 ± 0.06</td><td>5.00 ± 0.97</td><td>0.74 ± 0.07</td><td>4.0 ± 1.0</td><td>0.84 ± 0.07</td><td>3.0 ± 1.4</td><td>0.89 ± 0.09</td></tr><tr><td>Multi-resolution</td><td><b>4.05 ± 0.61</b></td><td>0.83 ± 0.03</td><td>3.93 ± 0.60</td><td>0.83 ± 0.03</td><td>3.71 ± 0.78</td><td>0.88 ± 0.02</td><td>2.26 ± 0.82</td><td>0.95 ± 0.01</td><td>1.42 ± 0.41</td><td>0.95 ± 0.03</td></tr></tbody></table>

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