Following red blood cells in a pulmonary capillary

The red blood cells or erythrocytes are biconcave shaped cells and consist mostly in a membrane delimiting a cytosol with a high concentration in hemoglobin. This membrane is highly deformable and allows the cells to go through narrow passages like the capillaries which diameters can be much smaller than red blood cells one. They carry oxygen thanks to hemoglobin, a complex molecule that have very high affinity for oxygen. The capacity of erythrocytes to load and unload oxygen is thus a determinant factor in their efficacy. In this paper, we will focus on the pulmonary capillary where red blood cells capture oxygen. We propose a camera method in order to numerically study the behavior of the red blood cell along a whole capillary. Our goal is to understand how erythrocytes geometrical changes along the capillary can affect its capacity to capture oxygen. The first part of this document presents the model chosen for the red blood cells along with the numerical method used to determine and follow their shapes along the capillary. The membrane of the red blood cell is complex and has been modelled by an hyper-elastic approach coming from Mills et al (2004). This camera method is then validated and confronted with a standard ALE method. Some geometrical properties of the red blood cells observed in our simulations are then studied and discussed. The second part of this paper deals with the modeling of oxygen and hemoglobin chemistry in the geometries obtained in the first part. We have implemented a full complex hemoglobin behavior with allosteric states inspired from Czerlinski et al (1999).Comment: 17 page

A conservative MURD scheme on moving domains: Application to three-dimensional free surface flows

We introduce a monotonic and conservative numerical scheme for the resolution of the linear advection problem set on a moving domain. The scheme is based on an extension to moving domains of the Multidimensional Upwind Residual Distributive (MURD) approach. The properties ensured by the residual distribution schemes set on fixed domains are not altered by the domain movement, except for the conservation of the advected quantity. We introduce in this paper an additional condition which ensures the conservation properties of the MURD scheme on amoving domain. Several numerical tests have been performed, providing satisfying results in terms of conservation of the advected quantity

Partitioned FSI strategy for simulations of a thin elastic valve

We present a Fictitious Domain (FD)/Lagrange multiplier method to approximate a thin valve movement immersed in an incompressible fluid. We developed a partitioned FSI algorithm that is able to keep the fluid and structure codes independent and which involve only minor modifications to each one . We propose a quantitative comparison between FD and ALE, when the displacement is moderated. We show that the two approaches give results in good agreement. We also illustrate FD in a simulation involving very large displacements

A Coupled System of PDEs and ODEs Arising in Electrocardiograms Modeling

International audienceWe study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure

Mathematical modeling of electrocardiograms: a numerical study

International audienceThis report deals with the numerical simulation of electrocardiograms (ECG). Our aim is to devise a mathematical model, based on partial differential equations, which is able to provide realistic 12-lead ECGs. The main ingredients of this model are classical: the bidomain equations coupled to a phenomenological ionic model in the heart, and a generalized Laplace equation in the torso. The obtention of realistic ECGs relies on other important features --- including heart-torso transmission conditions, anisotropy, cell heterogeneity and His bundle modeling --- that are discussed in detail. The numerical implementation is based on state-of-the-art numerical methods: domain decomposition techniques and second order semi-implicit time marching schemes, offering a good compromise between accuracy, stability and efficiency. The numerical ECGs obtained with this approach show correct amplitudes, shapes and polarities, in all the 12 standard leads. The relevance of every modeling choice is carefully discussed and the numerical ECG sensitivity to the model parameters investigated

From intracardiac electrograms to electrocardiograms: Models and metamodels

International audienceWe consider the problem of building a standard electrocardiogram (ECG) from the electrical potential provided by a pacemaker in a few points of the heart (electrogram). We use a 3D mathematical model of the heart and the torso electrical activity, able to generate "computational ECG", and a "metamodel" based on a kernel ridge regression. The input of the metamodel is the electrogram, its output is the ECG. The 3D model is used to train and test the metamodel. We illustrate the performance of the proposed strategy on simulated bundle branch blocks of various severities and a few clinical dat