Abstracts

Stéphane Chrétien (University of Lyon 2)
Blood Sample Based Early Cancer Detection

Our work develops a method that can detect malignant cancer cells at an early stage by analyzing blood samples. To achieve this, we  devise a new simple statistical method based on a stochastic model of the biomarker's spread in the blood. In a second part of the talk, we will show how conformal prediction can be leveraged to control the various uncertainties associated with our methodology.  

Jacques Demongeot (Univ. Grenoble Alpes)
A new tool for signal processing: the Dynalet transform

The linear functional analysis has been historically founded by Fourier and Legendre (the supervisor of Fourier) has provided a new vision of the mathematical transformations between vector spaces of functions. A new way to define efficient functional transforms from the approximation by a linear combination of elementary functions from an orthogonal or not basis lies on the definition of simple models linked to the physical or biological problem to model. Fourier, Wavelet transforms are respectively defined from the simple and damped and but there exist other toy problems in real world, where the modelling approach is not linked to the pendulum. This extension is called Dynalet transform and gives good results of approximation in the case of relaxation signals given for example by periodic biphasic organs in human physiology.

Christian Engwer (Uni Münster)
Linking growth models and image data to estimating the extent of glioblastoma tumors

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis.  There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection-diffusion-reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate.

In a first step we discuss how multi-scale modelling allows to incorporate patient specific MRI and DTI data into numerical simulations of tumor growth. This yields an effective instationary macroscopic GBM growth model, falling into the wide class of advection-diffusion-reaction equations.

To actually compute information about the tumor structure at the time of image acquisition, we introduce a stationalized approach. We observe that instationary model exhibits a travelling wave type behaviour. This observation is employed to derive a suitable penalty term that stationalizes the solution. We derive this term from the analytic solution of the Fisher-KPP equation, the simplest model in the considered model class and apply the approach to more realistic patient data, using inhomogeneous growth models, derived from a real 3D DTI-dataset.

Axel Hutt (INRIA, Strasbourg)
Impact of tDCS on brain networks in the context of  a ketamine-animal model  of psychotic disorder

Patients suffering from psychosis exhibit a characteristic brain activity power spectrum. Transcranial Direct Current Stimulation (tDCS) permit to modify the brain activity power spectrum and ameliorate the patients health state. To understand this effect, we employ an animal model for psychosis based on ketamine. This model demonstrates experimentally that  tDCS can affect the animals brain activity in certain frequency bands. To explain this  neurostimulation impact, we consider a brain network model subjected to stochastic input. This model allows to reproduce the experimental effects and thus provides an explanation how neurostimulation affects brain networks.

Yvon Maday (Sorbonne Université)
A new epidemiological indicator for SARS-CoV-2 using wastewater analysis

A sentinel network, “réseau Obépine", has been designed to monitor SARS-CoV-2 viral load in wastewaters arriving at wastewater treatment plants (WWTPs) in France as an indirect macro-epidemiological parameter. The sources of uncertainty in such monitoring system are numerous. The concentration measurements it provides are moreover left-censored (due to quantification limits) and contain outliers, which biases the results of usual smoothing methods. Hence the  need for an adapted pre-processing in order to evaluate the real daily amount of virus arriving to  each WWTP.

In the interdisciplinary group research Obepine, we have elaborated a smoother based on an auto-regressive model adapted to censored data with outliers. Inference and smoothing are produced via a discretised algorithm which makes it a very flexible tool. This method is both validated on simulations and on real data from Obépine.  The resulting smoothed signal shows a good correlation with other epidemiological indicators and has been used by Obépine to provide an estimate of virus circulation over the watersheds corresponding to about 200 WWTPs for surveillance purpose from summer 2020 to spring 2022.

Qiyao Peng (Leiden University)
Enhancing cell-based modeling of wound contraction after skin injuries using neural networks

Deep tissue injuries, such as severe burns, often go with the contraction of skin. This contraction can become problematic if it causes a loss of mobility of the  oint of the patient, which is known as contracture. This morbid contraction results from the cellular traction forces exerted by the (myo)fibroblasts during the  proliferation phase. In Peng and Vermolen (2020), an agent-based model, in which every single cell is treated individually as a fixed circle in two dimensions, is developed to mimic the wound healing process. To model the cellular forces exerted by the (myo)fibroblasts, the cell membrane is discretized by a polygonal chain and the point forces are applied at the midpoint of each line segment in the form of the Dirac delta distribution. Finally, the nonlinear morphoelasticity equation is solved with the finite-element methods. Since our final objective is to predict the likelihood that such a contracture occurs and to analyze the impact of various treatments on the likelihood of the development of a serious contraction, a computationally efficient model is required. Hence, a neural network (NN)-based machine learning model is applied to replace the force balance part of the wound healing model. in order to improve the computational efficiency. Moreover, the machine learning model allows for the application to complex cell shapes; in particular myofibroblasts are known as a dendritic shape with lamellipodia developed from the cell membrane. While a polygonal discretization, as in Peng and Vermolen (2020), may increase the algorithm complexity and computational cost significantly, the machine learning approach employs a local sampling process, in which only the sampling grid may need to be adjusted. In this presentation, the new hybrid FEM-NN approach is presented and numerical results are for various cell shapes presented.

Cordula Reisch (TU Braunschweig)

Model selection focusing on longtime behavior as qualitative data

Biological processes often involve many different agents through complex mechanisms. Mathematical models may abstract from the microscopic nature and propose macro-scale dynamics for comparison with observations. One example is chronic liver inflammation for which the underlying mechanisms on a cellular level leading to chronic inflammation are not fully known. Typical observations are measured inflammation markers in the blood that are only indirectly linked to the micro-scale mechanisms. One crucial step in mathematical modeling is to formulate a model that links the information on both length scales. 

We present a step-by-step approach for selecting abstract mathematical models from a model family. The aim is twofold: 

First, to identify abstract mechanisms that confine large-scale observations, and second, to determine parameter values for these abstract mechanisms. These parameter values are not directly connected to experimental data but serve as a crucial link between known mechanisms on the micro-scale and observed phenomena on the macro-scale.

Our framework integrates machine learning techniques with the characteristic solution behavior of differential equations. The approach provides valuable insights into the challenges posed by limited data, a situation that is oftentimes appearing in mathematical biology.

Dumitru Trucu (University of Dundee)
Multiscale Moving Boundary Modelling and Analysis for Glioblastoma Invasion

Despite recent mathematical modelling advances, the understanding of the biologically multiscale process of cancer invasion remains an open question. In this project we explore novel mathematical multiscale moving boundary and structural analytical and computational approaches for cancer invasion. These will explores the dynamic interactions that the migratory cancer cells population and the accompanying matrix degrading enzymes (MDEs) have with the extracellular matrix (ECM) components, and in particular with the ECM fibres. These will investigate complex two-scale interactions enabled by a series of multiscale systems that share (and contribute to) the same tumour macro-dynamics but have independent in nature micro-dynamics. For instance, on the bulk of the tumour, of major interest is the dynamics of fibres degradation and structural realignment occurring at cell-scale (micro-scale) as well as the immediate impact that this continuously changing field of oriented ECM fibres has over the tumour macro-dynamics. On the other hand, the cell-scale proteolytic micro-dynamics occurring at the tumour invasive edge interacts with the peritumoural ECM fibres through the molecular fluxes of MDEs. This interfacial cell-scale interaction not only results in changes of the micro-scale structural distribution of peritumoural ECM fibres, but also influences directly the changes of the overall tumour morphology. The new mathematical multiscale modelling framework will involve state-of-the-art numerical methods and will be explored through the development of a major musicale computational simulations platform. Furthermore, this research paves the way for new multiscale analysis research avenues that build on the novel concept of three-scale convergence that I established and introduced a while ago. 

Fred Vermolen (University of Hasselt)
Simulation of post-burned skin using principles from morphoelasticity

Each year the lives of hundreds of thousands people are heavily impacted by severe burn injuries. Although nowadays clinical technologies allow most patients to survive heavy burn traumas, nevertheless these burn injuries often come with hypertrophic scars and contractures, which impair the mobility of patients. In order to minimize the impact to the patients, therapies based on principles such as dressings, ointments, splinting and skin grafting (skin transplantation) are applied. In order to optimize treatment, a quantitative description of the underlying biological mechanisms is needed and for this reason, mathematical models have been constructed. In this talk, we present a continuum-based model that is constructed with principles from morphoelasticity. Morphoelasticity is a mathematical formalism that simultaneously deals with elasticity and microstructural changes in the tissue. We will show some mathematical results regarding stability of the model, as well as neural network simulations that reproduce the simulations at very high computational speed.

Olusegun Ekundayo Adebayo (LMB, Besançon)
Mathematical modellling for wound healing: local and non-local models

The movement of cells during (normal and abnormal) wound healing is the result of biomechanical interactions that combine cell responses with growth factors as well as cell-cell and cell-matrix interactions (adhesion and remodelling). It is known that cells can communicate and interact locally and non-locally with other cells inside the tissues through mechanical forces that act locally and at a distance, as well as through long, non-conventional cell protrusions. In this study, we consider a non-local partial differential equation model for the interactions between fibroblasts, macrophages, and the extracellular matrix via a growth factor in the context of wound healing. For the non-local interactions, we consider two types of kernels: a Gaussian kernel and a cone-shaped kernel; two types of cell-ECM adhesion functions; and two types of cell proliferation terms, with and without decay due to overcrowding. We investigate numerically the dynamics of this non-local model as well as the dynamics of the localised versions of this model.

Maxime Dalery (LMB, Besançon)
Nonlinear reduced basis using mixture Wasserstein barycenters

We are interested in the computation of the lowest energy solution of a (possibly nonlinear) eigenvalue problem, describing the behavior of electrons in a given molecular system with M nuclei characterized by their positions in space and their electric charges.

Finding such solution is in general very expensive, especially when done for many different geometries, as is the case in molecular dynamics and geometry optimization. In this talk, I will focus on the one-dimensionnal version of this problem, which is a linear eigenvalue problem with Dirac delta potentials placed at atomic positions and for which analytic solutions are fully characterized and can be seen as mixtures of Slater distributions (convex sum of Slater distributions). 

An interpolation method between solutions for different parameters based on optimal transport was proposed in a different context, however, the cost of Wasserstein barycenters tends to explode with the dimension, even with most recent algorithms. In this work, we propose a new approach based on a decomposition of the solution as a mixture of Slater functions, for which modified Wasserstein barycenters, preserving the mixture of Slater distributions shape of the solutions, can be computed efficiently. The method is based on a selection of a few representative solutions using a greedy algorithm in the so-called offline phase, followed by a so-called online phase to compute optimal parameters representing the current solution as a barycenter of selected solutions. Finally, some numerical results will be presented.

Michel Duprez (Inria Nancy Grand Est)
Phi-FEM : an immersed boundary finite element method for domains defined by level-set functions

In this talk, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and a new 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 the Poisson equation with non-homogeneous Dirichlet boundary conditions. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. Since we use the phi-FEM approach, we will consider domains defined by level-set functions. We will first recall the idea of phi-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 two test cases.

Mathilde Massard (LMB, Besançon)
Modelling and investigating memory immune responses in acute infectious disease. Application to Influenza A virus and SARS-CoV-2 reinfections

Understanding effector and memory immune responses against IAV and SARS-CoV-2 infections is extremely important in the control of these infections.
In this study, we investigate the role of memory cells and antibodies in the immune response against acute IAV and SARS-CoV-2 reinfection with the help of a within-host mathematical model. To this end, we adapt a previously published within-host model for the immune response against the SARS-CoV-2 by including two types of memory cells: memory CD$8^+$ T-cells and memory B-cells. We simulate different viral reinfections (20 days, 60 days and 400 days after the first encounter with the pathogen) which allow us to highlight which memory components have the greatest impact on the immune response and the effectiveness of the immune response depending on the time of reinfection. We observe that memory immune responses have a greatest effect in the case of IAV compared to SARS-CoV-2. Moreover, we observe that the immune response after a secondary infection is more efficient when the reinfection occurs at a shorter time.

Simone Nati Poltri (IMB, Inria Bordeaux)
Asymptotic Analysis of Electrocardiology Modeling after Pulsed Field Ablation

In this work, we focus on the mathematical study of pulsed electric field ablation (PFA), an innovative cardiac ablation technique for the treatment of cardiac arrhythmias. In particular, we would like to compare it with radiofrequency ablation (RFA), a thermal ablation that is currently the most commonly used technique. Whereas RFA is known to result in coagulation necrosis with complete loss of cellular and vascular architecture, leaving a scar of fibrotic tissue, PFA takes advantage of irreversible electroporation - a complex cell death phenomenon that occurs when a biological tissue is subjected to very intense electric pulses - and it is known to destroy mainly cardiomyocytes while preserving the tissue scaffold.

This work aims to modify the classical bidomain model - which describes the propagation of intracellular and extracellular potentials in the heart - to introduce a region ablated by RFA or PFA. Both types of ablation involve isolation of a pathological area, but we describe them differently by using appropriate transmission conditions at the interface between the ablated area and the not ablated area.

In the case of RFA, we assume that both intracellular and extracellular potentials are affected, resulting in Kedem-Katchalsky-type conditions at the interface. In contrast, in the case of PFA, we study the static bidomain model and we assume that the thickness of the electroporated (EP) region is small compared with the whole domain. Moreover, we assume that the intracellular conductivity within the EP region scales with a factor proportional to the square of the thickness parameter. We provide a formal asymptotic analysis at any order by considering an asymptotic expansion of the intracellular and extracellular potentials both outside and inside the EP area. This allows us to derive transmission conditions at the interface for PFA at any order, that read as non-homogeneous boundary conditions for the jump of the extra-cellular potential and its normal derivative, and as Neumann conditions for the intracellular potentials. Moreover, we give a proof of the asymptotic expansion by deriving estimates of H1- and L2- norms of the errors of an expansion with a given number of terms. The asymptotic expansion was validated by numerical convergence tests. In particular, zero- and first-order expansions were compared and errors were computed for different values of the thickness parameter tending to zero.

Finally, we propose physical simulations in the context of atrial fibrillation (AF). We consider the isolation of one pulmonary vein of a synthetic geometry of the left atrium. In the case of PFA, we consider transmission conditions deduced from the zero-order asymptotic analysis of the static case. Numerical simulations of AF and the long-term effects of RFA and PFA show that both models lead to isolation of the pulmonary vein. Our modeling also enables to propose a numerical explanation for the higher rate of fibrillation recurrence after RFA compared with PFA.

Thomas Saigre (IRMA, Strasbourg)
Model order reduction and sensitivity analysis for complex heat transfer simulations inside the human eyeball

The ocular contribution is devoted to modeling the complex interplay between tissue perfusion, biomechanics, fluid dynamics, and heat transfer within the eye. These different aspects of the same physical problem have to be properly connected and every step has to be verified and validated in the interest of a medical application. The models require the knowledge of various parameters and some may be important factors in the development of pathologies. However, despite recent significant advances in medical data acquisition, only some parameters and their variability are known, but others cannot be directly measured. To identify the main factors that influence the biomechanical behavior of the eye, we, therefore, need to study the influence of these parameters through an uncertainty quantification (UQ) process which requires many evaluations of the models. Since 3D models are not amenable directly to UQ, a reduction step is needed to mitigate the computational cost.

In the present talk, we propose a strategy to carry out further analysis using reduced order methods and in particular the certified reduced basis method, allowing reliable outputs several order of magnitude faster than the high fidelity model. We discuss then the implementation with the library Feel++. Finally, we apply the methodology to some advanced ocular models and report our findings in the context of heat transfer within the eye.