Mathematics of electrical imaging: modeling, theory and implementation

12 juin 2023, 12:30 → 14 juin 2023, 18:00 Europe/Paris

Salle K. Johnson (1R3-1er étage) (Institut de Mathématiques)

Jeremi Darde, Jérémy Heleine (Institut de Mathématiques de Toulouse)

The workshop aims at gathering experts in the mathematical study of different medical imaging techniques that share the common trait of using electrodes to acquire data on an electromagnetic phenomenon. To name a few, these include:

- Electrical Impedance Tomography (EIT), planned to be used, e.g., in tumor detection. This is a very active field of research in the mathematical inverse problems community, particularly in Finland. 
- Electroencephalography (EEG), used in brain monitoring. France counts several mathematical specialists of this imaging modality. 
- Noninvasive electrocardiographic imaging (ECGi), used in the prevention of cardiac

Each of these imaging methods has been studied deeply from a mathematical point of view, from the mathematical modeling of the occurring phenomena to the development of efficient numerical methods to solve the corresponding inverse problems. However, even though these modalities share similar features, the scientific communities that work on one of these techniques are often not aware of the developments regarding the related imaging modalities. The primary objective of the workshop is therefore to gather mathematicians working on the aforementioned medical imaging techniques in order to discuss the similarities and differences between the methods as well as related mathematical problems and recent theoretical and numerical developments.
 

Affiche.pdf

lundi 12 juin

12:30 → 13:45 Welcome buffet lunch

14:00 → 15:00 Subspace surrogates for the parametric diffusion equation

In this talk, I consider efficient solution of the parametric diffusion equation for several parameter values. First, I study the dependency of the solution u from the parameter and establish that u can be approximated by low-dimensional subspace for any admissible parameter value. Then I propose an efficient method for approximately computing this subspace. I conclude with numerical examples investigating the performance of the proposed method.

Orateur: Antti Hannukainen

15:00 → 15:30 Addressing Challenges in Modeling Human Head Anatomy for Stroke Monitoring with EIT

In this talk, we explore the challenges encountered while implementing a computational framework for stroke monitoring using Electrical Impedance Tomography (EIT). Our focus is on accurately modeling human head anatomy based on a library of MRI scans and integrating real-world data obtained from physical experiments.

Accurately representing anatomical surfaces, such as the brain, skull, and scalp, presents significant challenges in stroke monitoring with EIT. We discuss the complexities involved in capturing the intricate geometry of these surfaces and present meshing techniques that ensure faithful representation of the underlying anatomy.

An important milestone in our project is the successful reconstruction of real-world data acquired from physical experiments, assuming prior knowledge of the geometry. This proof of concept demonstrates the effectiveness of our computational framework in stroke monitoring.

By addressing these challenges, our project contributes to advancing the field of stroke monitoring with EIT, with the ultimate goal of improving early detection and treatment outcomes. In this talk, we discuss insights into the implementation aspects of modeling human head anatomy and utilizing real-world data, underscoring the potential of EIT as a tool for stroke monitoring.

Orateur: Anton Vavilov

15:30 → 16:00 Coffee Break

16:00 → 17:00 Electrocardiology modeling after pulsed field ablation relying on asymptotic analysis

In healthy hearts, the propagation of electrical waves follows a predictable pattern, whereas in people suffering from arrhythmia, the electrical waves can become chaotic and directly affect the pumping function of the heart. One of the main treatments for these arrhythmias is catheter ablation, which destroys small areas of heart tissue to isolate or eliminate the cause of the rapid and irregular heartbeats. Most catheter ablation therapies are performed thermally through the application of a radiofrequency electromagnetic field (RFA). Some limitations and disadvantages are observed in clinical application, such as damage to adjacent structures.
In this work, we focus on the study of a novel and mainly non-thermal ablation technique: pulsed electric field ablation (PFA), which takes advantage of irreversible electroporation, a complex phenomenon of cell membrane rupture that occurs when biological tissue is exposed to very intense electric pulses. This technique has been used in oncology for more than a decade, but it is still in its infancy in cardiology. Preclinical evaluations of PFA in atrial fibrillation and ventricular ablation in large animal studies show successful results with possible transmural lesions. Despite these promising results, the application of PFA in routine clinical practice still presents some difficulties because of the technical complexity of this novel approach. Mathematical models and numerical strategies could be developed to improve the understanding of PFA on the cardiac signal and to develop numerical criteria for treatment assessment based on clinical data. The aim of this work is to derive a cardiac electrophysiological model of a cardiac domain containing a region ablated by PFA. In doing so, we start from the following three points: (1) the electroporated region is small compared with the entire domain, (2) the intra-cellular conductivity within the electroporated zone is also very small, and (3) a linearization of the ionic current is assumed in the electroporated region. The last two assumptions can be explained by the fact that the ablated tissue contains few cells and the gap junctions connecting the cells are severely compromised. This leads to a model that depends on a small parameter which could lead to numerical problems. To address this, an asymptotic analysis is performed in a static context and convergence analysis is explored. The strategy allows to determine the transmission conditions at the interface between the two regions, yielding a suitable model of a cardiac domain containing an ablation region by PFA. Well-designed Schwarz algorithms are developed to numerically solve the obtained PDE system.

Orateur: Annabelle Collin

workshopToulouse2023.pdf

17:00 → 17:30 EEG source localisation and cortical mapping

We present a theoretical and numerical framework for performing source lo-
calisation and cortical mapping using EEG data with various source models.
This framework is built on the idea that the problems of source localisation and
cortical mapping are different aspects of the same inverse problem rather than
independent inverse problems. This approach promises to improve numerical ac-
curacy and to provide context to the numerical results obtained both for source
localisation and cortical mapping. We provide theoretical results which allow
the sources to be modelled as vector-fields from a wide range of functional and
distributional spaces. The study of the various source models in motivated by a
need to take advantage of advances in medical imaging such as diffusion MRI.
This opens the possibility to restrict the support of the sources to anatomical
structures of the brain that we now have access to thanks to these advances.
We present numerical results to demonstrate the proposed alternating minimi-
sation algorithm for solving the source location and cortical mapping problems
simultaneously. Such an alternating minimisation procedure provides practical
advantages especially in computer memory usage.

Orateur: Masimba Nemaire

Source localisation and cortical mapping.pdf

17:30 → 19:00 Buffet

mardi 13 juin

09:15 → 10:15 Liver tumor ablation by electroporation: mathematical modeling for clinical applications

Electropermeabilization (also called electroporation--EP) is a significant increase in the electrical conductivity and permeability of the cell membrane that occurs when pulses of large amplitude (a few hundred volts per centimeter) are applied to the cells: due to the electric field, the cell membrane is permeabilized. If the pulse duration is sufficiently short (a few milliseconds or a few microseconds, depending on the pulse amplitude), the cell membrane reseals within several tens of minutes: reversible EP, preserves the cell viability and is used in electrochemotherapy to vectorize the drugs until the cell inside. If the pulses are too long, too numerous or if their amplitude is too high, the cell membrane is irreversibly destroyed and the cells are killed. Irreversible EP provides thus a novel non thermal and minimally invasive ablation therapy.
In this talk I will present some recent results of the Inria Team MONC on the mathematical EP modeling combining PDE models and image registration techniques, in order to help interventional radiologists in their practice of percutaneous liver tumor ablation by irreversible electroporation.

Orateur: Clair Poignard (INRIA & Université de Bordeaux)

10:15 → 10:45 EIT reconstruction using virtual X-rays and machine learning

We introduce a new reconstruction algorithm for EIT, which provides a connection between EIT and traditional X-ray tomography, based on the idea of "virtual X-rays". We divide the exponentially ill-posed and nonlinear inverse problem of EIT into separate steps. We start by mathematically calculating so-called virtual X-ray projection data from the DN map. Then, we perform explicit algebraic operations and one-dimensional integration, ending up with a blurry and nonlinearly transformed Radon sinogram. We use a neural network to learn the nonlinear deconvolution-like operation. Finally, we can compute a reconstruction of the conductivity using the inverse Radon transform. We demonstrate the method with simulated data examples.

This is a joint work with Samuli Siltanen, Matti Lassas, Rashmi Murthy, Fernando Silva de Moura, Juan Pablo Agnelli, and Melody Alsaker.

Orateur: Siiri Rautio

10:45 → 11:15 Coffee Break -- Shared with the Institute Coffee Break

11:15 → 12:15 Experimental Validation and Clinical Applications of ECGI

Cardiac electrical diseases are a major cause of mortality in the world today. Many individuals are not identified as at risk, or incorrectly identified for treatment. These suboptimal results are mostly due to the limitations of current clinical methods to detect cardiac electrical activity. Real-time and panoramic images of cardiac electrical activity can be noninvasively reconstructed through electrocardiographic imaging (ECGI) This modality fills the gap between the noninvasive (low-resolution) 12-lead ECG and invasive (high-resolution) electrophysiology studies. Much progress has been made to establish ECGI meaning today it is available for use in a clinical setting. This presentation will demonstrate the capabilities and limitations of current clinical ECGI methods through experimental validation studies, and present the latest clinical findings using ECGI to understand the mechanisms underlying unexplained sudden cardiac death.

Orateur: Laura Bear

12:30 → 14:00 Lunch

14:00 → 15:00 Numerical resolution of the inverse source problem for EEG

In the talk, I will present a numerical method for solving the EEG inverse source problem which is able to take into account the heterogeneity of the head tissues. One of the applications is to consider fontanels in the skull layer for neonates. The approach consists in firstly transmitting the recorded data from the scalp to the cortex using the
quasi-reversibility method. The second stage solves the inverse source problem within the brain by applying the method developed in the software tool FindSources3D.

Orateur: Marion Darbas

15:00 → 15:30 Immersed boundary method for electrical impedance tomography in the frame of Electrocardiography.

Orateur: Niami Nasr

15:30 → 16:00 Coffee Break

16:00 → 17:00 Monitoring of hemorrhagic stroke using electrical impedance tomography

Orateur: Ville Kolehmainen

17:00 → 18:00 Parameter estimation in the eikonal equation

Electrical waves propagate at the surface of the heart, the simplest modelling is to describe them as a front propagating according to the eikonal equation. The propagation velocity is anisotropic since it occurs along fibers. For medical diagnosis purposes, we aim at finding the source location and the parameters of the velocity, either from measurements directly on the surface of the heart, or through an observation operator that models an electrocardiogram. The parameters to be recovered are thus a point x on a manifold and a small number of scalar quantities. We implemented a least squares minimization using Gauss-Newton algorithm on a manifold. It is necessary to obtain Exp_x the exponential map based at x on the manifold (or more precisely its reciprocal Log_x). This quantity can be efficiently computed using techniques borrowed from the field of image synthesis. We illustrate our approach using sunthetic data.

Orateur: Jérôme Fehrenbach (IMT)

20:00 → 23:00 Social event

mercredi 14 juin

09:15 → 10:15 Reconstruction of inclusions and cracks in Calderón's inverse conductivity problem

I will talk about the exact reconstruction of general inclusions in Calderón's inverse conductivity problem from boundary electrical measurements in the form of a local Neumann-to-Dirichlet map. Here "inclusion" means the support of perturbations to a known conductivity coefficient.

I will briefly cover the cases on open sets, where the perturbed coefficient can have finite positive and negative perturbations, can have perfectly conducting parts and have perfectly insulating parts, and may also have parts given as restrictions of Muckenhoupt coefficients with singular and degenerate behavior (enabling continuous growth to infinity or decay to zero).

I will give a more detailed account of new results on reconstructing general cracks given as unions of Lipschitz hypersurfaces, including both perfectly conducting and perfectly insulating cracks.

Orateur: Henrik Garde

Garde.pdf

10:15 → 10:45 Multimodal analysis and inverse problems for brain imaging

The deeper understanding of brain activity is a major challenge for clinical and
mathematical research. In case of neurological disease, such as epilepsy, the main
objective is to better understand and diagnose the origin of the crisis in order to pro-
pose a proper treatment for each patient. Among the non-invasive imaging methods
used to study these pathologies, we are interested in the coupling of two of them :
the electroencephalography (EEG) and the diffuse optical tomography (DOT). On
one hand, the EEG measures the electrical potential generated by the neuronal ac-
tivity. On the other hand, the DOT measures the absorption and diffusion of the
light in the near-infrared spectrum by biological tissues, attesting of a change in the
concentrations of oxyhemoglobin and deoxyhemoglobin.

EEG measures the electric potential difference between a reference electrode and
electrodes placed on the scalp. There can be up to 128 electrods, arranged on a head
cap placed on the head of the patient. Regarding the DOT, it is an imaging technique
based on the absorption and diffusion of the light by the tissues. A head cap equiped
with transmitters and receivers is placed on the head of the patient. A transmitter
sends out light which is measured by the receptors after passing through the various
layers of the head. Thanks to these measurements, it is then possible to reconstruct
the optical parameters of the tissues in the head. Co-registration already exists for
these two modalities, which represents a source of motivation for the mathematical
modeling of this coupling.

In this communication, we present a time-dependent model for the acquisition of
coupled measurements of these two modalities. Indeed, time is an important com-
ponent of the neurovascular coupling. On the one hand, EEG measures the neural
activity and on the other hand, the DOT highlights the change of optical parame-
ters of brain tissues induced by the local increase in blood volume during cerebral
activity. We will also present the inverse problems associated with these two ima-
ging modalities, the inverse problem of the EEG being a source localization problem
whearas the DOT problem is a parameter identification problem. Finally, we will
explain how these modalities can be used together to improve the resolution of the
inverse problems from a numerical point of view.

Orateur: Benjamin Sulis

expose_Toulouse.pdf

10:45 → 11:15 Coffee Break

11:15 → 12:15 Some remarks on the quadratic nature of the matching to data operator

We study the inverse problem of the reconstruction of an obstacle in an media from boundary measurements. We assume that the data is noised and that a statistical model for the data is at hand. We propose and study a reconstruction algorithm based a weighted combination of the first two moments of the Kohn-Vogelius criterion. By numerical results in dimension two, the applicability and feasibility of our approach is demonstrated.

Orateur: Marc Dambrine

12:15 → 13:45 Lunch