Our studies on collagen tissues have shown that the collagen fibers are able to elongate inelastically under stretch. In tendons, this effect has been attributed to the non-permanent cross-bridges that connect the different collagen fibrils (to assemble a fiber). This sliding effect appears experimentally to be reversible (at least partially) if the tissue is left long enough at its initial resting length. However, this sliding is classically included as an irreversible plastic response, or as a damage of the tissue. We are building a model based on a stochastic description of the binding and unbinding of the cross-bridges. This approach will enable us to have a microscopically based picture of the sliding, which will be able to explain some alterations in case of aging or pathologies of the tissue. At the moment, we have shown the importance of the density of cross-bridges in the cooperative response of the system. A publication is in preparation on the topic.

This work aims at proposing a set of models of the muscular contraction targeting different scales in time and space and that can be used in the context of heart simulation. To this end, we developed so far two models using different approaches for the modeling of the force generating process at the molecular level called power stroke. First, we revised the standard chemo-mechanical models, which see the power stroke as a series of chemical states. Following the idea introduced by Truskinovsky and collaborators describing the power stroke as a continuum of mechanical states with the dynamics of the myosin head in the prescribed energy landscape governed by Langevin equations, we incorporated the attachment and detachment dynamics in the form of jump processes. In a second step, noting that the power stroke time scale is much shorter than that of heart contraction, we eliminated the power strokes dynamics and derived a two state – attached and detached –  simplified model, each state being in fact associated with a statistical distribution of myosin head configurations. Both models have been integrated into our simulation framework CardiacLab, in order to benefit from the other modeling compartments available in the code, such as the geometrically reduced model of the heart left ventricle also developed in the team. These modeling elements will be confronted with experiments performed on cardiac muscle cells by collaborators in the team of Professor Lombardi at the University of Florence.

Shear acoustic waves remotely induced by the acoustic radiation force (ARF) of a focused ultrasound beam generated by piezoelectric sensors have been recently used in biomedical applications, e.g. in transient elastography techniques. By measuring the propagation velocity of generated shear waves in biological tissues, it is possible to locally assess biomechanical properties highly sensitive to structural changes corresponding to physiological and pathological processes. Recent experimental studies show the feasibility of applying transient elastography to the cardiac setting. In this context, the wave propagation induced by the ARF is superposed with the nonlinear mechanics associated with the heart deformation during the cardiac cycle. The aim of this work is to mathematically justify an original expression of the excitation induced by the ARF in nonlinear solids, based on energy considerations and asymptotic analysis. In soft media the propagation velocity of shear waves ($1--10{\text{m.s}}^{-1}$) is much smaller than the velocity of pressure waves ($1500{\text{m.s}}^{-1}$). The approach we propose consists in considering a family of problems parametrised by a small parameter $\epsilon$ related to the velocity ratio between the two wave propagation phenomena, the high frequency of the piezoelectric source term and the viscosity. In order to derive a simplified model for the expression of ARF, we investigate the limit behavior of the solution for $\epsilon \to 0$. We show that the leading term of the expansion is related to the underlying nonlinear mechanics of the heart deformation, and the first two correction terms correspond to a fast-oscillating pressure wave excited by the probe, and an elastic field having as source term a nonlinear function of the first corrector. This field corresponds to the shear acoustic wave induced by the ARF.

The aim of this work is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in cardiac electrophysiology. We consider space-dependent and tensorial electric conductivities as well as space-dependent physiological and phenomenological non-linear ionic models. We provide the nondimensionalization of the bidomain equations and derive uniform estimates of the solutions. The homogenization procedure is done using 2-scale convergence theory which enables us to study the behavior of the non-linear ionic models in the homogenization process.

Seismocardiography (SCG) is the study of low-frequency ($<60$ Hz) vibrations of the thorax caused by the beating heart. Although it is assumed that SCG signals are caused by forces applied on the interior of the thorax by the heart, no comprehensive model exists to describe the parameters and relationships that govern the system. The main goal of this study is to describe in some detail the filter applied by the thorax on cardiac forces, taking into account its zone of contact with the heart as well as the zone of measurement, i.e. the location of the accelerometer on a participant's chest. A secondary goal is to identify the smallest set of parameters capable of reproducing the filter, reducing the model while retaining its capacity to lend itself to physiological interpretation. Finally, we described a method to use the reduced model to estimate cardiac forces from measured thoracic accelerations. The overall aim of the study is to develop numerical methods that can augment the existing SCG interpretations to include mechanical indices of the heart muscle, and do so in real time.

This work is focused on two techniques for fast computing of the steady cyclic states of evolution problems in non-linear mechanics with space-time periodicity conditions. This kind of problems can be faced in various applications, for instance in the rolling of a tyre with periodic sculptures as well as in a beating heart. Direct solvers for such problems are not very convenient, since they require the inversion of very large matrices. In order to avoid this, a cyclic solution is usually computed as an asymptotic limit of the associated initial value problem with prescribed initial data. However, when the relaxation time is high, convergence to the limit cycle can be very slow. The first technique considered is the Newton-Krylov method, looking for the unknown initial state that provides the space-time periodic solution. This initial state is defined by the space-time periodicity condition, solved with the Newton-Raphson technique. Since the associated Jacobian cannot be expressed explicitly, the method uses one of the matrix-free Krylov iterative solvers. Using information stored while computing the residual to solve the linear system makes its calculation time negligible with respect to the residual calculation time. The second method is the delayed feedback control: an observer-controller type modification of the standard evolution to the limit cycle by introducing a feedback control term, based on the periodicity error. The main result is the optimal form of the control term for a very general class of linear evolution problems, providing the fastest convergence to the cyclic solution. This control has also been adapted and tested for nonlinear problems. The methods discussed have been assessed using academic applications and they have also been implemented into the Michelin industrial code – applied to the rolling tyre model – as well as into the M3DISIM code for the cardiac contraction problem.

This work has the potential to provide an original efficient method for the computations of elastic waves propagation in soft media (such as biological tissues), based on the property that pressure and shear waves decouple in isotropic media. Towards this direction, we considered the numerical solution of 2D elastodynamics isotropic equations using the decomposition of the displacement fields into potentials. This appears as a challenge for finite element methods, and we have addressed here the particular question of free boundary conditions. A stable (mixed) variational formulation of the evolution problem is proposed.

In this work we extend the Arlequin method to overlapping domain decomposition technique for transient wave equation scattering by small obstacles. The main contribution of this work is to construct and analyze some variants of the Arlequin method from the continuous level to the fully discrete level. The constructed discretizations allow to solve wave propagation problems while using non-conforming and overlapping meshes for the background propagating medium and the surrounding of the obstacle, respectively. Hence we obtain a flexible and stable method in terms of the space discretization – an inf-sup condition is proven – while the stability of the time discretization is ensured by energy identities.

This works deals with the construction of a fourth-order, energy preserving, explicit time discretization for dissipative linear wave equations. This discretization is obtained by replacing the inversion of a matrix – that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations – by an explicit approximation of its inverse. The stability of the scheme is studied first using an energy analysis, then an eigenvalue analysis. Numerical results in 1D illustrate the good behavior regarding space/time convergence and the efficiency of the newly derived scheme compared to more classical time discretizations. A loss of accuracy is observed for non-smooth profiles of dissipation, and we propose an extension of the method that fixes this issue. Finally, we assess the good performance of the scheme for a realistic dissipation phenomenon in Lorentz materials.

In this work, we develop a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated with a distinct variational formulation coming from a mutually well-posed family of so-called broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in some part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different illustrative examples. This work has resulted in the publication .

Cardiac electrophysiology simulations are numerically challenging due to the propagation of a steep electrochemical wave front, and thus require discretizations with small mesh sizes to obtain accurate results. In this work – in collaboration with the Institute for Computational Mechanics, Technical University Munich and published in – we present an approach based on the Hybridizable Discontinuous Galerkin method (HDG), which allows an efficient implementation of high-order discretizations into a computational framework. In particular using the advantage of the discontinuous function space, we present an efficient p-adaptive strategy for accurately tracking the wave front. HDG allows to reduce the overall degrees of freedom in the final linear system to those only on the element interfaces. Additionally, we propose a rule for a suitable integration accuracy for the ionic current term depending on the polynomial order and the cell model to handle high-order polynomials. Our results show that for the same number of degrees of freedom coarse high-order elements provide more accurate results than fine low-order elements. Introducing p-adaptivity further reduces computational costs while maintaining accuracy by restricting the use of high-order elements to resolve the wave front. For a patient-specific simulation of a cardiac cycle, p-adaptivity reduces the average number of degrees of freedom by 95% compared to the non-adaptive model. In addition to reducing computational costs, using coarse meshes with our p-adaptive high-order HDG method also simplifies practical aspects of mesh generation and postprocessing.

In this work, we seek exact formulations of the optimal estimator and filter for a non-linear framework, as the Kalman filter is for a linear framework. The solution is well established with the Mortensen filter in a continuous-time setting, but we seek here its counterpart in a discrete-time context. We demonstrate that it is possible to pursue at the discrete-time level an exact dynamic programming strategy and we find an optimal estimator combining a prediction step using the model and a correction step using the data. This optimal estimator reduces to the discrete-time Kalman estimator when the operators are in fact linear. Furthermore, the strategy that consists of discretizing the least square criterion and then finding the exact estimator at the discrete level allows to determine a new time-scheme for the Mortensen filter which is proven to be consistent and unconditionally stable, with also a consistent and stable discretization of the underlying Hamilton-Jacobi-Bellman equation. This work has resulted in the publication .

Continued advances in computational power and methods have enabled image-based biomechanical modeling to become an important tool in basic science, diagnostic and therapeutic medicine, and medical device design. One of the many challenges of this approach, however, is identification of a stress-free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs. Fortunately, iterative methods have been proposed to solve this inverse problem, among them Sellier’s method. This method is particularly appealing because it is easy to implement, converges reasonably fast, and can be coupled to nearly any finite element package. By means of several practical examples, however, we demonstrate that in its original formulation Sellier’s method is not optimally fast and may not converge for problems with large deformations. Nevertheless, we can also show that a simple, inexpensive augmentation of Sellier’s method based on Aitken’s delta-squared process can not only ensure convergence but also significantly accelerate the method. This work has resulted in the publication .

We propose a novel continuum finite strain formulation of the equilibrium gap principle – originally introduced in [Claire, Hild and Roux, 2004, Int. J. Num. Meth. Eng.] at the discrete level for linearized elasticity – used as a regularizer for finite element-based image correlation problems. Consistent linearization and finite element discretization is provided. The method is implemented using FEniCS & VTK, in a freely available Python library. The equilibrium gap constraint regularizes the image correlation problem, even in the presence of noise, and without affecting strain measurements.

We present a shape-oriented data assimilation strategy suitable for front-tracking problems through the example of wildfire. The concept of “front” is used to model, at regional scales, the burning area delimitation that moves and undergoes shape and topological changes under heterogeneous orography, biomass fuel and micrometeorology. The simulation-observation discrepancies are represented using a front shape similarity measure inspired from image processing and based on the Chan-Vese contour fitting functional. We show that consistent corrections of the front location and uncertain physical parameters can be obtained using this measure applied on a level-set fire growth model solving for an eikonal equation. This study involves a Luenberger observer for state estimation, including a topological gradient term to track multiple fronts, and a reduced-order Kalman filter for joint parameter estimation. We also highlight the need – prior to parameter estimation – for sensitivity analysis based on the same discrepancy measure, and for instance using polynomial chaos metamodels, to ensure that a meaningful inverse solution is achieved. The performance of the shape-oriented data assimilation strategy is assessed on a synthetic configuration subject to uncertainties in front initial position, near-surface wind magnitude and direction. The use of a robust front shape similarity measure paves the way toward the direct assimilation of infrared images and is a valuable asset in the perspective of data-driven wildfire modeling. This work has resulted in the publication .

In mammals, Prion pathology refers to a class of infectious neuropathologies whose mechanism is based on the self-perpetuation of structural information stored in the pathological conformer. The characterisation of the PrP folding landscape has revealed the existence of a plethora of pathways conducing to the formation of structurally different assemblies with different biological properties. However, the biochemical interconnection between these diverse assemblies remains unclear. The PrP oligomerisation process leads to the formation of neurotoxic and soluble assemblies called O1 oligomers with a high size heterodispersity. By combining the measurements in time of size distribution and average size with kinetic models and data assimilation, we revealed the existence of at least two structurally distinct sets of assemblies, termed Oa and Ob, forming O1 assemblies. We propose a kinetic model representing the main processes in prion aggregation pathway: polymerisation, depolymerisation, and disintegration. The two groups interact by exchanging monomers through a disintegration process that increases the size of Oa. Our observations suggest that PrP oligomers constitute a highly dynamic population. This work has resulted in the publication .

Various methods have been used in the statistical field to estimate parameters in mechanistic models. In particular, an approach based on penalised likelihood for the estimation of parameters in ordinary differential equations with non linear models on parameters (ODE-NLME) has proven successful. For instance, we consider the NIMROD program as a benchmark for estimation in these models. However, such an approach is time consuming. To circumvent this problem, we consider data assimilation approaches that historically arose in the context of geophysics. Here, we propose a Luenberger (also called nudging) state observer coupled with a parameter Kalman-based observer (RoUKF filter, also called SEIK filter) to perform a joint state and parameter estimation on a dataset composed of longitudinal observations of biomarkers for multiple patients. We compare these methods in terms of performances and computation time. We discuss how the concept of random effect can be modeled using Kalman-based filter and its limitations. We illustrate both methods in simulation and on two datasets (the ALBI ANRS 070 trial and the Aquitaine cohort observational data) using an HIV mechanistic model.

Skin is a complex, multi-layered organ, with important functions in the protection of the body. The dermis provides structural support to the epidermal barrier, and thus has attracted a large number of mechanical studies. As the dermis is made of a mixture of stiff fibres embedded in a soft non-fibrillar matrix, it is classically considered that its mechanical response is based on an initial alignment of the fibres, followed by the stretching of the aligned fibres. Using a recently developed set-up combining multiphoton microscopy with mechanical assay, we imaged the fibres network evolution during dermis stretching. These observations, combined with a wide set of mechanical tests, allowed us to challenge the classical microstructural interpretation of the mechanical properties of the dermis: we observed a continuous alignment of the collagen fibres along the stretching. All our results can be explained if each fibre contributes by a given stress to the global response. This plastic response is likely due to inner sliding inside each fibre. The non-linear mechanical response is due to structural effects of the fibres network in interaction with the surrounding non-linear matrix. This multiscale interpretation explains our results on genetically-modified mice with a simple alteration of the dermis microstructure. This work has resulted in the publication .

The affine transformation hypothesis is usually adopted in order to link the tissue scale with the fibers scale in structural constitutive models of fibrous tissues. Thanks to the recent advances in imaging techniques, such as multiphoton microscopy, the microstructural behavior and kinematics of fibrous tissues can now be monitored at different stretching within the same sample. Therefore, the validity of the affine hypothesis can be investigated. In this study, the fiber reorientation predicted by the affine assumption is compared with experimental data obtained during mechanical tests on skin and liver capsule coupled with microstructural imaging using multiphoton microscopy. The values of local strains and the collagen fibers orientation measured at increasing loading levels are used to compute a theoretical estimation of the affine reorientation of collagen fibers. The experimentally measured reorientation of collagen fibers during loading could not be successfully reproduced with this simple affine model. It suggests that other phenomena occur in the stretching process of planar fibrous connective tissues, which should be included in structural constitutive modeling approaches. This work has resulted in the publication .

Mechanical properties of the skin, the external organ of the human body, are important for many applications such as surgery or cosmetics. Due to the highly hierarchical structure of the tissue, it is interesting to develop microstructural models that have better predictability and should reduce the consequences of sample variability. However, these models generally include a quite large number of mechanical parameters. Therefore, complex assays are required to achieve a proper identification of the microstructural models. We investigated in this study the best experimental protocol to identify a nonlinear, anisotropic, model of skin behavior, namely, the Holzapfel law, using displacement field and force measurements. This was done through a sensitivity analysis of the different parameters. We determined first the optimal assay, which appears to be a biaxial test with an alternated loading: first a stretch in one direction, then in the perpendicular one, and so on. To further improve the quality of the assay, we also determined the optimal geometry. Interestingly, slightly asymmetric geometries are more adequate than symmetric ones, while being easier to realise. This work has resulted in the publication .

Skin aging is a complex process that strongly affects the mechanical behavior of skin. This study aims at deciphering the relationship between age-related changes in dermis mechanical behavior and the underlying changes in dermis microstructure. To that end, we use multiphoton microscopy to monitor the reorganization of dermal collagen during mechanical traction assays in ex vivo skin from young and old mice. The simultaneous variations of a full set of mechanical and microstructural parameters are analyzed in the framework of a multiscale mechanical interpretation. They show consistent results for wild-type mice as well as for genetically-modified mice with modified collagen V synthesis. We mainly observe an increase of the tangent modulus and a lengthening of the heel region in old murine skin from all strains, which is attributed to two different origins that may act together: (i) increased cross-linking of collagen fibers and (ii) loss of water due to proteoglycans deterioration, which impedes inner sliding within these fibers. In contrast, the microstructure reorganization upon stretching shows no age-related difference, which can be attributed to opposite effects of the decrease of collagen content and of the increase of collagen cross-linking in old mice. This work has resulted in the publication .

Bacterial biofilms correspond to surface-associated bacterial communities embedded in hydrogel-like matrix, in which high cell density, reduced diffusion and physico-chemical heterogeneity play a protective role and induce novel behaviors. We made a summary of the recent advances on the understanding of how bacterial mechanical properties, from single cell to high-cell density community, determine biofilm three-dimensional growth and eventual dispersion, and we attempt to draw a parallel between these properties and the mechanical properties of other well-studied hydrogels and living systems. This work has resulted in the publication .

In this work, we introduce the modeling of atrioventricular valve regurgitation in a spatially reduced-order biomechanical heart model. The model can be fast calibrated using non-invasive data of cardiac magnetic resonance imaging and provides an objective measure of contractile properties of the myocardium in the volume overloaded ventricle, for which the real systolic function may be masked by the significant level of the atrioventricular valve regurgitation. After demonstrating such diagnostic capabilities, we show the potential of modeling to address some clinical questions concerning possible therapeutic interventions for specific patients. The fast running of the model allows targeting specific questions of referring clinicians in a clinically acceptable time. The work was presented at the “Functional Imaging and Modeling of the Heart” conference (FIMH 2017, Toronto, Canada) and is included in the conference proceedings .

Understanding physiological phenomena and mechanisms of failure in congenital heart diseases is often challenging due to the complex hemodynamics and high inter-patient variations in anatomy and function. Computational modeling techniques have the potential to greatly improve the understanding of these complex diseases and provide patient-specific clues on mechanisms of deterioration and impact of treatments. This work employs a reduced 0D biomechanical heart model coupled with venous return to capture various key pathophysiological phenomena observed in patients with completed Fontan circulation – a complex surgically established circulation used to palliate patients in whom only one of the two ventricles is functionally able to support the vascular system – with exercise-induced heart failure during dobutamine stress. The framework we propose is fast, efficient and well-suited to the type of pathology and available clinical data obtained by a combined cardiac catheterization and magnetic resonance imaging exam. We demonstrate that the outcomes of modeling are a valuable addition to the current clinical diagnostic investigations and explain patient-specific exercise hemodynamics, identify potential mechanisms of Fontan failure, and enable evaluation of a potential new therapy – selective heart rate modulation – in the treatment of patients with Fontan circulation. The paper is currently in preparation.

By using mathematical models of heart and vessels developed in the team, we aim at improving intra-operative cardio-vascular safety of anesthetized patients. The patient-specific models, calibrated by echocardiography images and fed by continuous monitoring of aortic arterial pressure and aortic cardiac outflow would allow us to: 1) diagnose pathophysiological modifications associated with changes in the cardio-vascular state; 2) predict the drug response of the patient before the administration of the vaso-active treatment.

General anesthesia leads to alterations of the cardiovascular system. Intra-operative arterial hypotension is linked to post-operative complications, but using vasopressors to treat arterial hypotension has shown conflicting results. Vasopressors act mainly by elevating cardiac afterload, which could be deleterious in fragile patients, in case of excessive response. Moreover, differences among the most used vasopressors have been observed in vivo . The choice of vasopressor could be important to improve our patients' care. Consequently, we proposed a tool (Velocity-Pressure Loops) to continuously quantify changes in cardiac afterload . Although the first work involves invasive measurement of aortic blood pressure and cardiac outflow, consistent results have been observed when Velocity-Pressure Loops were obtained by a radial arterial catheter with a mathematical transform function . Those findings allow the usage of the Velocity-Pressure Loop without addition of any invasive device.

This clinical review aims at describing the issues of the management of extra-corporeal membrane oxygenation (ECMO) in the Intensive Care Unit (ICU). From pathophysiology to the description of the impact on mortality, this document shows a global picture of current clinical practices.

With the contribution of Denis Chemla, professor of Cardiology at Bicêtre Hospital, we presented this work at the CinC conference in Rennes . In this work, we emphasize the need for a consistent method to measure systolic period duration, which is related to cardiac afterload and could be used to quantify arterial pressure amplification phenomenon.