Section: New Results

Mathematical and Mechanical Modeling

Stochastic modeling of chemical-mechanical coupling in striated muscles

Participants : Matthieu Caruel, Philippe Moireau, Dominique Chapelle [correspondant] .

In [18] we propose a chemical–mechanical model of myosin heads in sarcomeres, within the classical description of rigid sliding filaments. In our case, myosin heads have two mechanical degrees-of-freedom (dofs)—one of which associated with the so-called power stroke—and two possible chemical states, i.e., bound to an actin site or not. Our major motivations are twofold: (1) to derive a multiscale coupled chemical–mechanical model and (2) to thus account—at the macroscopic scale—for mechanical phenomena that are out of reach for classical muscle models. This model is first written in the form of Langevin stochastic equations, and we are then able to obtain the corresponding Fokker–Planck partial differential equations governing the probability density functions associated with the mechanical dofs and chemical states. This second form is important, as it allows to monitor muscle energetics and also to compare our model with classical ones, such as the Huxley’57 model to which our equations are shown to reduce under two different types of simplifying assumptions. This provides insight and gives a Langevin form for Huxley’57. We then show how we can calibrate our model based on experimental data—taken here for skeletal muscles—and numerical simulations demonstrate the adequacy of the model to represent complex physiological phenomena, in particular the fast isometric transients in which the power stroke is known to have a crucial role, thus circumventing a limitation of many classical models.

Upscaling of elastic network models

Participant : Patrick Le Tallec.

This work is done in collaboration with Julie Diani from École Polytechnique. The purpose of the approach is to develop general upscaling strategy for deriving macroscopic constitutive laws for rubberlike materials from the knowledge of the network distribution and a mechanical description of the individual chains and of their free energy. It is based on a variational approach in which the microscopic configuration is described by the position of the crosslinks and is obtained not by an affine assumption but by minimizing the corresponding free energy on stochastic large Representative Volume Elements with adequate boundary conditions. This general framework is then approximated by using a microsphere (directional) description of the network and by performing a local minimisation of the network free energy on this simplified configuration space under a maximal advance path kinematic constraint. This approximation framework takes into account anisotropic damage and is extended to handle situations with tube like constraints and stress induced cristallisation. For more detail see [23].

Stochastic construction of surrogate multiphase materials

Participant : Patrick Le Tallec.

Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigue analysis usually employ simple microstructure, consisting of ideal geometries such as ellipsoidal inclusions, which generally does not capture complex geometries. In our work , we introduce a simple but flexible surrogate microstructure model for two-phase materials through a level-cut of a Gaussian random field with covariance of Matern class. In addition to the traditional morphology descriptors such as porosity, size and aspect ratio of inclusions, our approach provides control of the regularity of the inclusions interface and sphericity. These parameters are estimated from a small number of real material images using Bayesian inversion. An efficient process of evaluating the samples, based on the Fast Fourier Transform, makes possible the use of Monte-Carlo methods to estimate statistical properties for the quantities of interest in a given material class. This work in progress is done in collaboration with Andrei Constantinescu (École Polytechnique), Ustim Khristenko and Barbara Wohlmuth (Technical University Munich) and Tinsley Oden (University of Texas at Austin). This work has been submitted for publication in an international journal.

Apprehending the effects of mechanical deformations in cardiac electrophysiology – A homogenization approach

Participants : Annabelle Collin [MONC] , Sébastien Imperiale, Philippe Moireau, Jean-Frédéric Gerbeau [Inria Siège] , Dominique Chapelle [correspondant] .

In this work [22], we follow a formal homogenization approach to investigate the effects of mechanical de- formations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the frame- work of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure – and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors – in which mechanical deformations play a non-trivial role, i.e. do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical-mechanical simulations – all the way to ECG simulations – albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

Patient-specific pulmonary mechanics - Modelling and estimation. Application to pulmonary fibrosis.

Participants : Cecile Patte [correspondant] , Martin Genet, Dominique Chapelle.

Interstitial pulmonary diseases, like Idiopathic Pulmonary Fibrosis (IPF), affect the alveolar structure of lung tissue, which impacts lung mechanical properties and pulmonary functions. In this work [43], we aim to better understand the pulmonary mechanics in order to improve IPF diagnosis. We developped a poromechanical model for the lung at the organ scale and at the breathing scale. This model is then used to estimate regional mechanical parameters based on clinical data. In the future, this process can be used as an augmented diagnosis tool for clinicians. This work has been presented at the CSMA conference.

Energy preserving cardiac circulation models: formulation, reduction, coupling, inversion, and discretization

Participants : Jessica Manganotti, Philippe Moireau, Sébastien Imperiale [correspondant] , Miguel Fernandez [Inria Paris, COMMEDIA] .

The modeling of the heart cannot be satisfying if not coupled to the body circulation, and at least to the arterial circulation, which is its direct output “boundary condition”. But more importantly in the clinical context, it is still difficult – ant very invasive – to access the ventricular pressure, which is absolutely necessary for specifying the heart activity. By comparison, more and more devices allow to register non-invasively a distal pressure, for instance at the wrist or the finger, which could be used to estimate the ventricular pressure by inversion of a well adapted arterial circulation model. Such relation is of major interest for clinicians, for example anesthetists, since it could allow real-time monitoring and prediction of the effects of injected drugs during a clinical intervention. Models of the arterial circulation is a well-known subject where dimension reduction has been widely studied for more than half a century. However, the question remains of formulating energy-consistent formulations that can be consistently maintained during the reduction, when coupled to a heart model, and also when discretized. Yet the question is crucial for a better understanding of the physical phenomena of blood flow ejection from the heart as well as the propagation in the arterial network. Moreover, as these models are non-linear, energy-preserving approaches are one of the few tools at our disposal to mathematically justify modeling, discretization or inversion approaches. Finally, inverting this unsteady model for estimation purposes of medical data also benefits from energy-preserving formulation as the inverse approach should also satisfy some stability properties. The subject here is twofold and part of the thesis of J. Manganotti. First we plan to develop accurate models, coupling strategies and robust numerical methods of the arterial network propagation coupled to the heart. Second, we want to develop observer-based strategies that will allow to easily feed these models with measurements in order to perform state estimation of hidden variables or identify key biophysical parameters.

Hierarchical modeling of force generation in cardiac muscle

Participants : Matthieu Caruel, François Kimmig [correspondant] .

Performing physiologically relevant simulations of the beating heart in clinical context requires to develop detailed models of the microscale force generation process. These models however may be difficult to implement in practice due to their high computational costs and complex calibration. We propose a hierarchy of three interconnected cardiac muscle contraction models – from the more refined to the more simplified – that are rigorously and systematically related with each other, offering a way to select, for a specific application, the model that yields the best trade-off between physiological fidelity, computational cost and calibration complexity. Our starting model takes into account the stochastic dynamics of the molecular motors force producing conformational changes– and in particular the power stroke – and captures all the timescales of appearing in classical experimental isotonic responses of a heart papillary muscle submitted to rapid load changes. Adiabatic elimination of fast relaxing variables of the stochastic model yields a formulation based on partial differential equations (PDEs) that falls into the family of the Huxley'57 model, while embedding some properties of the process occurring at the fastest timescales. The third family of models is deduced from the PDE model by making minimal assumptions on the parameters, which leads to a computationally light formulation based on ordinary differential equations only. The three models families are compared to the same set of experimental data to systematically assess what physiological indicators can be reproduced or not and how these indicators constrain the model parameters. Finally, we discuss the applicability of these models for heart simulation. This work has been submitted for publication in an international journal.

A relaxed growth modeling framework for controlling growth-induced residual stresses

Participant : Martin Genet.

Background Constitutive models of the mechanical response of soft tissues have been established and are widely accepted, but models of soft tissues remodeling are more controversial. Specifically for growth, one important question arises pertaining to residual stresses: existing growth models inevitably introduce residual stresses, but it is not entirely clear if this is physiological or merely an artifact of the modeling framework. As a consequence, in simulating growth, some authors have chosen to keep growth-induced residual stresses, and others have chosen to remove them. Methods In this work, we introduce a novel “relaxed growth” framework allowing for a fine control of the amount of residual stresses generated during tissue growth. It is a direct extension of the classical framework of the multiplicative decomposition of the transformation gradient, to which an additional sub-transformation is introduced in order to let the original unloaded configuration evolve, hence relieving some residual stresses. We provide multiple illustrations of the framework mechanical response, on time-driven constrained growth as well as the strain-driven growth problem of the artery under internal pressure, including the opening angle experiment. Findings The novel relaxed growth modeling framework introduced in this paper allows for a better control of growth-induced residual stresses compared to standard growth models based on the multiplicative decomposition of the transformation gradient. Interpretation Growth-induced residual stresses should be better handled in soft tissues biomechanical models, especially in patient-specific models of diseased organs that are aimed at augmented diagnosis and treatment optimization. See [27] for more detail.

Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models

Participants : Frédérique Clément [correspondant] , Romain Yvinec.

During the supervision of a CEMRACS2018 project performed by Céline Bonnet (CMAP) and Keltoum Chahour (LERMA and JLAD), we have described (with Marie Postel, Sorbonne Université and Romain Yvinec, INRA) different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles [50]. We have put a special focus on representing the population-level feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We have taken advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We have first studied the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we have provided detailed numerical evidence of convergence to the limit behavior. We have reproduced the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging.

Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation

Participants : Frédérique Clément [correspondant] , Frédérique Robin, Romain Yvinec.

In mammals, female germ cells are sheltered within somatic structures called ovarian follicles, which remain in a quiescent state until they get activated, all along reproductive life. We have investigated the sequence of somatic cell events occurring just after follicle activation [54]. We have introduced a nonlinear stochastic model accounting for the joint dynamics of two cell types, either precursor or proliferative cells. The initial precursor cell population transitions progressively to a proliferative cell population, by both spontaneous and self-amplified processes. In the meantime, the proliferative cell population may start either a linear or exponential growing phase. A key issue is to determine whether cell proliferation is concomitant or posterior to cell transition, and to assess both the time needed for all precursor cells to complete transition and the corresponding increase in the cell number with respect to the initial cell number. Using the probabilistic theory of first passage times, we have designed a numerical scheme based on a rigorous Finite State Projection and coupling techniques to assess the mean extinction time and the cell number at extinction time. We have also obtained analytical formulas for an approximating branching process. We have calibrated the model parameters using an exact likelihood approach using both experimental and in-silico datasets. We have carried out a comprehensive comparison between the initial model and a series of submodels, which help to select the critical cell events taking place during activation. We have finally interpreted these results from a biological viewpoint.

A multiscale mathematical model of cell dynamics during neurogenesis in the mouse cerebral cortex

Participant : Frédérique Clément.

This work is a collaboration with Marie Postel and Sylvie Schneider-Maunoury (Sorbonne Université), Alice Karam (Sorbonne Universités), Guillaume Pézeron (MNHN).

Neurogenesis in the murine cerebral cortex involves the coordinated divisions of two main types of progenitor cells, whose numbers, division modes and cell cycle durations set up the final neuronal output. In this work [33] we aim at understanding the respective roles of these factors in the neurogenesis process, we have combined experimental in vivo studies with mathematical modeling and numerical simulations of the dynamics of neural progenitor cells. A special focus is put on the population of intermediate progenitors (IPs), a transit amplifying progenitor type critically involved in the size of the final neuron pool. A multiscale formalism describing IP dynamics allows one to track the progression of cells along the subsequent phases of the cell cycle, as well as the temporal evolution of the different cell numbers. Our model takes into account the dividing apical progenitors (AP) engaged into neurogenesis, both neurogenic and proliferative IPs, and the newborn neurons. The transfer rates from one population to another are subject to the mode of division (symmetric, asymmetric, neurogenic) and may be time-varying. The model outputs have been successfully fitted to experimental cell numbers from mouse embryos at different stages of cortical development, taking into account IPs and neurons, in order to adjust the numerical parameters. Applying the model to a mouse mutant for Ftm/Rpgrip1l, a gene involved in human ciliopathies with severe brain abnormalities, reveals a shortening of the neurogenic period associated with an increased influx of newborn IPs from apical progenitors at mid-neurogenesis. Additional information is provided on cell kinetics, such as the mitotic and S phase indexes, and neurogenic fraction. Our model can be used to study other mouse mutants with cortical neurogenesis defects and can be adapted to study the importance of progenitor dynamics in cortical evolution and human diseases.