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. In order to numerically study the behavior of red blood cells along a whole capillary, we propose a camera method that consists in working in a reference frame that follows the red blood cells. More precisely, the domain of study is reduced to a neighborhood of the red blood cells and moves along at erythrocytes mean velocity. This method avoids too large mesh deformation. 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 J.P. et al., MCB, vol. 1, no. 3, pp. 169-180, 2004]. This camera method is then validated and confronted with a standard Arbitrary Lagrangian Eulerian (ALE) method in which the displacements of the red blood cells are correlated with the deformation of an initial mesh of the whole capillary with red blood cells at start positions. 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 G. et al., J. Theor. Biol., vol. 199, pp. 25-44, 1999].
