A Finite Element Approach For Modeling Biomembranes In Incompressible Power-Law Flow

We present a numerical method to model the dynamics of inextensible biomembranes in a quasi-Newtonian incompressible flow, which better describes hemorheology in the small vasculature. We consider a level set model for the fluid-membrane coupling, while the local inextensibility condition is relaxed by introducing a penalty term. The penalty method is straightforward to implement from any Navier-Stokes/level set solver and allows substantial computational savings over a mixed formulation. A standard Galerkin finite element framework is used with an arbitrarily high order polynomial approximation for better accuracy in computing the bending force. The PDE system is solved using a partitioned strongly coupled scheme based on Crank-Nicolson time integration. Numerical experiments are provided to validate and assess the main features of the method

Numerical Approach Based on the Composition of One-Step Time-Integration Schemes For Highly Deformable Interfaces

In this work, we propose a numerical approach for simulations of large deformations of interfaces in a level set framework. To obtain a fast and viable numerical solution in both time and space, temporal discretization is based on the composition of one-step methods exhibiting higher orders and stability, especially in the case of stiff problems with strongly oscillatory solutions. Numerical results are provided in the case of ordinary and partial differential equations to show the main features and demonstrate the performance of the method. Convergence properties and efficiency in terms of computational cost are also investigated

Mathematical modelling of active contraction in isolated cardiomyocytes

We investigate the interaction of intracellular calcium spatio-temporal variations with the self-sustained contractions in cardiac myocytes. A consistent mathematical model is presented considering a hyperelastic description of the passive mechanical properties of the cell, combined with an active-strain framework to explain the active shortening of myocytes and its coupling with cytosolic and sarcoplasmic calcium dynamics. A finite element method based on a Taylor-Hood discretization is employed to approximate the nonlinear elasticity equations, whereas the calcium concentration and mechanical activation variables are discretized by piecewise linear finite elements. Several numerical tests illustrate the ability of the model in predicting key experimentally established characteristics including: (i) calcium propagation patterns and contractility, (ii) the influence of boundary conditions and cell shape on the onset of structural and active anisotropy and (iii) the high localized stress distributions at the focal adhesions. Besides, they also highlight the potential of the method in elucidating some important subcellular mechanisms affecting, e.g. cardiac repolarizatio

Mathematical modelling of active contraction in isolated cardiomyocytes

We investigate the interaction of intracellular calcium spatio-temporal variations with the self-sustained contractions in cardiac myocytes. A consistent mathematical model is presented considering a hyperelastic description of the passive mechanical properties of the cell, combined with an active-strain framework to explain the active shortening of myocytes and its coupling with cytosolic and sarcoplasmic calcium dynamics. A finite element method based on a Taylor-Hood discretization is employed to approximate the nonlinear elasticity equations, whereas the calcium concentration and mechanical activation variables are discretized by piecewise linear finite elements. Several numerical tests illustrate the ability of the model in predicting key experimentally established characteristics including: (i) calcium propagation patterns and contractility, (ii) the influence of boundary conditions and cell shape on the onset of structural and active anisotropy and (iii) the high localized stress distributions at the focal adhesions. Besides, they also highlight the potential of the method in elucidating some important subcellular mechanisms affecting, e.g. cardiac repolarization

Numerical modelling of the dynamics of red blood cells using the level set method

Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char.This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion

An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes

ISSN:1307-689

Finite-Element Method for the Simulation of Lipid Vesicle/Fluid Interactions in a Quasi–Newtonian Fluid Flow

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty which allows computational savings and facilitates implementation. A high-order Galerkin finite element approximation allows accurate calculations of the membrane force with high-order derivatives. The time discretization is based on the double composition of the one-step backward Euler scheme, while the time step size is flexibly controlled using a time integration error estimation. Numerical examples are presented with particular attention paid to the validation and assessment of the model’s relevance in terms of physiological significance. Optimal convergence rates of the time discretization are obtained

Modélisation numérique de la dynamique des globules rouges par la méthode des fonctions de niveau

This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char

Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive Time-Stepping Strategies

In this article, we present a finite element method for studying the dynamic behavior of deformable vesicles, which mimic red blood cells, in a non-Newtonian Casson fluid. The fluid membrane, represented by an implicit level-set function, adheres to the Canham–Helfrich model and maintains surface inextensibility constraint through penalty. We propose a two-step time integration scheme that incorporates higher-order accuracy by using an asymmetric composition of discrete flow based on the second-order backward difference formula, followed by a projection onto the real axis. Our framework incorporates variable time steps generated by an appropriate adaptation criterion. We validate our model through numerical simulations against existing experimental and numerical results in the case of purely Newtonian flow. Furthermore, we provide preliminary results demonstrating the influence of the non-Newtonian fluid model on membrane regimes