arXiv subjects

Michel Duprez

Publications and source records attributed to Michel Duprez.

At least 19 recordsLinked to original sources

arXiv2026

A Differentiable Simulation of the Eye for Patient-Specific Strabismus Surgery Planning

Purpose: Up to 4% of adults will develop strabismus in their lifetime. The most common surgical intervention involves adjusting the length of one or more extraocular muscles to correct the angular deviation. This correction depends on surgical expertise and statistical reference tables, which often fail to yield optimal results for patients with atypical eye morphology. Our work proposes a physics-based modeling approach to personalized surgical planning, accounting for patient-specific eye anatomy. Methods: We built a physics-based simulator of the eye and its muscles, incorporating patient-specific geometry and Hill-type muscle biomechanics. We solve an optimization problem to find the surgical dosage that minimizes angular deviation. The model is implemented as a fully differentiable simulation, enabling efficient optimization. We validated the framework by comparing its predictions with standard surgical tables for emmetropic eyes before applying it to anatomically atypical virtual patients. Results: Our model's predictions for emmetropic eyes were first validated, demonstrating a strong fit with standard surgical tables. More importantly, for high-myopia models, the framework computed a clinically significant increase in the required surgical dosage compared to standard eyes. This computed recession difference is highly relevant as surgical plans are adjusted in 0.5 mm increments. Conclusion: Our results show that our model provides a calibrated surgical plan that, unlike standard tables, also accounts for pathologies involving atypical eye shapes. This patient-specific model represents a step toward personalized surgical planning, with the potential to improve dosage accuracy and surgical outcomes for atypical cases.

math.NA↗

arXiv2026

A multigrid and neural network approach to reduce the computational cost of phi-FEM

In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.

math.NA↗

arXiv2026

A digital twin for microwave liver treatment replanning

Purpose: MicroWave Ablation (MWA) modeling and simulation bear great potential for loco-regional treatment of liver tumors. However, accurately positioning the antenna according to a planned orientation/location is technically challenging. In cases of misplacement, maintaining the original plan may cause incomplete ablation, while repositioning the antenna may induce tumor seeding. In this work, we propose (i) a digital twin of MWA that simulates ablation outcomes, and (ii) an optimizer that suggests corrections to MWA parameters without antenna reinsertion, while ensuring complete tumor ablations. Methods: A finite element scheme was used to solve the coupled microwave propagation and heat transfer equations governing MWA, with personalized dielectric and thermal properties determined from preoperative CT and MRI images. We then proposed an optimization algorithm able to adjust power input, ablation duration, and antenna position to correct for antenna misplacement. Results: The simulator and optimizer were evaluated against in vivo swine experimental data. Three ablations were performed in liver regions with varying vascularization. The simulations accurately predicted the ablation zones despite the presence of large vessels near the antenna, achieving Dice scores of 0.82, 0.81, and 0.79. In the case of replanning scenarios, our optimizer predicted new parameter sets that led to Dice scores of 0.83, 0.83, 0.80, a corresponding improvement of 20.3%, 40.7% and 48.1% in average over the initial ablation result. Conclusion: This paper is the first to address intra-operative replanning of thermal ablation therapy. It demonstrates that optimal ablation results can be achieved without requiring antenna reinsertion by optimizing specific ablation parameters.

physics.med-ph↗

arXiv2026

A penalized {\phi}-FEM scheme for the Poisson Dirichlet problem

In this work, we analyze a penalized variant of the {\phi}-FEM scheme for the Poisson equation with Dirichlet boundary conditions. The {\phi}-FEM is a recently introduced unfitted finite element method based on a level-set description of the geometry, which avoids the need for boundary-fitted meshes. Unlike the original {\phi}-FEM formulation, the method proposed here enforces boundary conditions through a penalization term. This approach has the advantage that the level-set function is required only on the cells adjacent to the boundary in the variational formulation. The scheme is stabilized using a ghost penalty technique. We derive a priori error estimates, showing optimal convergence in the H1 semi-norm and quasi-optimal convergence in the L2 norm under suitable regularity assumptions. Numerical experiments are presented to validate the theoretical results and to compare the proposed method with both the original {\phi}-FEM and the standard fitted finite element method.

math.NA↗

arXiv2025

Dual Weighted Residual-driven adaptive mesh refinement to enhance biomechanical simulations

This chapter describes how a posteriori error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique, can be easily applied for biomechanical simulations in current engineering practice. The proposed method considers a very general setting that encompasses complex geometries, model non-linearities (hyperelasticity, fluid-structure interaction) and multi-goal oriented techniques. The developments are substantiated with some numerical tests.

math.NA↗

arXiv2025

Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping operator. The purpose of this paper is to provide numerical evidence to show the effectiveness of this technique. We will focus here on the resolution of two equations: the Poisson-Dirichlet equation and the non-linear elasticity equations. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. The considered domains are defined by level-set functions due to the use of the Phi-FEM approach. We will first recall the idea of $\varphi$-FEM and of the Fourier Neural Operator. Then, we will explain how to combine these two methods. We will finally illustrate the efficiency of this combination with some numerical results on three test cases. In addition, in the last test case, we propose a new numerical scheme for hyperelastic materials following the Phi-FEM paradigm.

math.NA↗

arXiv2025

Enriching continuous Lagrange finite element approximation spaces using neural networks

In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.

math.NA↗

arXiv2024

{\varphi}-FD : A well-conditioned finite difference method inspired by {\varphi}-FEM for general geometries on elliptic PDEs

This paper presents a new finite difference method, called {\varphi}-FD, inspired by the {\phi}-FEM approach for solving elliptic partial differential equations (PDEs) on general geometries. The proposed method uses Cartesian grids, ensuring simplicity in implementation. Moreover, contrary to the previous finite difference scheme on non-rectangular domain, the associated matrix is well-conditioned. The use of a level-set function for the geometry description makes this approach relatively flexible. We prove the quasi-optimal convergence rates in several norms and the fact that the matrix is well-conditioned. Additionally, the paper explores the use of multigrid techniques to further accelerate the computation. Finally, numerical experiments in both 2D and 3D validate the performance of the {\varphi}-FD method compared to standard finite element methods and the Shortley-Weller approach.

math.NA↗

arXiv2024

A Zero-Shot Reinforcement Learning Strategy for Autonomous Guidewire Navigation

Purpose: The treatment of cardiovascular diseases requires complex and challenging navigation of a guidewire and catheter. This often leads to lengthy interventions during which the patient and clinician are exposed to X-ray radiation. Deep Reinforcement Learning approaches have shown promise in learning this task and may be the key to automating catheter navigation during robotized interventions. Yet, existing training methods show limited capabilities at generalizing to unseen vascular anatomies, requiring to be retrained each time the geometry changes. Methods: In this paper, we propose a zero-shot learning strategy for three-dimensional autonomous endovascular navigation. Using a very small training set of branching patterns, our reinforcement learning algorithm is able to learn a control that can then be applied to unseen vascular anatomies without retraining. Results: We demonstrate our method on 4 different vascular systems, with an average success rate of 95% at reaching random targets on these anatomies. Our strategy is also computationally efficient, allowing the training of our controller to be performed in only 2 hours. Conclusion: Our training method proved its ability to navigate unseen geometries with different characteristics, thanks to a nearly shape-invariant observation space.

cs.LG↗

arXiv2024

Enhancing Biomechanical Simulations Based on A Posteriori Error Estimates: The Potential of Dual Weighted Residual-Driven Adaptive Mesh Refinement

The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with a discretisation error that can be an be quantified with \emph{a posteriori} error estimates. The practical relevance of error estimates for biomechanics problems, especially for soft tissue where the response is governed by large strains, is rarely addressed. In this contribution, we propose an implementation of \emph{a posteriori} error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique tailored to biomechanics. The proposed method considers a general setting that encompasses three-dimensional geometries and model non-linearities, which appear in hyperelastic soft tissues. We take advantage of the automatic differentiation capabilities embedded in modern finite element software, which allows the error estimates to be computed generically for a large class of models and constitutive laws. First we validate our methodology using experimental measurements from silicone samples, and then illustrate its applicability for patient-specific computations of pressure ulcers on a human heel.

math.NA↗

arXiv2023

Modeling the impact of rainfall and temperature on sterile insect control strategies in a Tropical environment

The sterile insect technique (SIT) is a biological control technique that can be used either to eliminate or decay a wild mosquito population under a given threshold to reduce the nuisance or the epidemiological risk. In this work, we propose a model using a differential system that takes into account the variations of rainfall and temperature over time and study their impacts on sterile males releases strategies. Our model is as simple as possible to avoid complexity while being able to capture the temporal variations of an Aedes albopictus mosquito population in a domain treated by SIT, located in R{é}union island. The main objective is to determine what period of the year is the most suitable to start a SIT control to minimize the duration of massive releases and the amount of sterile males to release, either to reduce the mosquito nuisance, or to reduce the epidemiological risk. Since sterilization is not 100% efficient, we also study the impact of different levels of residual fertility within the released sterile male population. Our study shows that rainfall plays a major role in the dynamics of the mosquito and the SIT control, that the best period to start a massive SIT treatment lasts from July to December, that residual fertility has to be as small as possible, and that increasing the size of the releases is not always necessarily interesting. We also highlight the importance of combining SIT with mechanical control, i.e. the removal of breeding sites, in particular when the initial mosquito population is large.

math.OC↗

arXiv2023

phi-FEM for the heat equation: optimal convergence on unfitted meshes in space

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the phi-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l2(H1) and linf(L2) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of phi-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

math.NA↗

arXiv2023

Delayed closed-loop neurostimulation for the treatment of pathological brain rhythms in mental disorders

Mental disorders (MD) are among the top most demanding challenges in world-wide health. According to the World Health Organization, the burden of MDs continues to grow with significant impact on health and major social and human rights. A large number of MDs exhibit pathological rhythms, which serve as the disorders characteristic biomarkers. These rhythms are the targets for neurostimulation techniques. Open-loop neurostimulation employs stimulation protocols, which are rather independent of the patients health and brain state in the moment of treatment. Most alternative closed-loop stimulation protocols consider real-time brain activity observations but appear as adaptive open-loop protocols, where e.g. pre-defined stimulation sets in if observations fulfil pre-defined criteria. The present theoretical work proposes a fully-adaptive closed-loop neurostimulation setup, that tunes the brain activities power spectral density (PSD) according to a user-defined PSD. The utilized brain model is non-parametric and estimated from the observations via magnitude fitting in a pre-stimulus setup phase. Moreover, the algorithm takes into account possible conduction delays in the feedback connection between observation and stimulation electrode. All involved features are illustrated on pathological alpha- and gamma-rhythms known from psychosis. To this end, we simulate numerically a linear neural population brain model and a non-linear cortico-thalamic feedback loop model recently derived to explain brain activity in psychosis.

q-bio.NC↗

arXiv2023

Closed-loop neurostimulation in real-time for the treatment of pathological brain rhythms

Mental disorders may exhibit pathological brain rhythms and neurostimulation promises to alleviate of patients' symptoms by modifying these rhythms. Today, most neurostimulation schemes are open-loop, i.e. administer experimental stimulation protocols independent of the patients brain activity which may yield a sub-optimal treatment. We propose a closed-loop feedback control scheme estimating an optimal stimulation based on observed brain activity. The optimal stimulation is chosen according to a user-defined target frequency distribution, which permits frequency tuning of the brain activity in real-time. The mathematical description details the major control elements and applications to biologically realistic simulated brain activity illustrate the scheme's possible power in medical practice. Clinical relevance - The proposed neurostimulation control theme promises to permit the medical personnel to tune a patient's brain activity in real-time.

q-bio.NC↗

arXiv2023

Phi-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

We present an immersed boundary method to simulate the creeping motion of a rigid particle in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called Phi-FEM, that uses the description of the solid with a level-set function. One of the advantages of our method is the use of standard finite element spaces and classical integration tools, while maintaining the optimal convergence (theoretically in the H1 norm for the velocity and L2 for pressure; numerically also in the L2 norm for the velocity).

math.NA↗

arXiv2022

Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix

The distributed null controllability for coupled parabolic systems with non-diagonalizable diffusion matrices with a reduced number of controls has been studied in the case of constant matrices. On the other hand, boundary controllability issues and distributed controllability with non-constant coefficients for this kind of systems is not completely understood. In this paper, we analyze the boundary controllability properties of a class of coupled parabolic systems with non-diagonalizable diffusion matrices in the constant case and the distributed controllability of a 2 x 2 non-diagonalizable parabolic system with space-dependent coefficients. For the boundary controllability problem, our strategy relies on the moment method. For the distributed controllability problem, our findings provide positive and negative control results by using the Fattorini-Hautus test and a fictitious control strategy.

math.OC↗

arXiv2021

Optimal control strategies for the sterile mosquitoes technique

Mosquitoes are responsible for the transmission of many diseases such as dengue fever, zika or chigungunya. One way to control the spread of these diseases is to use the sterile insect technique (SIT), which consists in a massive release of sterilized male mosquitoes. This strategy aims at reducing the total population over time, and has the advantage being specific to the targeted species, unlike the use of pesticides.In this article, we study the optimal release strategies in order to maximize the efficiency of this technique.We consider simplified models that describe the dynamics of eggs, males, females and sterile males in order to optimize the release protocol. We determine in a precise way optimal strategies, which allows us to tackle numerically the underlying optimization problem in a very simple way. We also present some numerical simulations to illustrate our results.

math.AP↗

arXiv2021

$ϕ$-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer

One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called $ϕ$-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the $ϕ$-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditions co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of $ϕ$-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of $ϕ$-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by $ϕ$-FEM in both the accuracy and the computational time.

math.NA↗