[en] The mechanics and elasticity of red blood cells (RBCs) determine the capability to deform of these cells when passing through the thinnest capillaries, where the delivery of oxygen takes place. The understanding of the elastic properties of RBCs is fundamental for improving our knowledge about microcirculation and it also has important biomedical applications, such as control of blood storage, or cell manipulation for pathology diagnosis. In this Thesis, we study the elasticity of RBCs under different conditions, understanding their mechanical response to different type of perturbations. In a first Part, we study the shape morphologies observed in the disco-echinocyte transition, when the cell is subjected to an imbalance in the membrane asymmetry, for instance after ATP depletion when lipids flip from the inner to the outer leaflet. Affected cells deform, adopting crenated morphologies known as echinocytes. We develop a theoretical study which allows us to identify and quantify the relevant aspects that trigger the shape transition. The lipid bilayer tries to expand its outer leaflet in order to accommodate the excess area, whereas the cytoskeleton opposes resistance to this type of deformations, preserving more compact shapes. The subtle interplay between both membrane structures determines the equilibrium morphology of the cell. The cytoskeleton is fundamental to ensure the stability of the healthy shape, the discocyte, against changes in the membrane composition. However, it is not severely stressed under weak deformations in which low curvatures are involved. Our results show that the energetic scale of these shape transitions is of hundreds of kbT, demonstrating the large stability of these shapes. Based on the knowledge gained from the theoretical study we also analyze a series of experiments in which echinocytes are mechanically perturbed by a AFM tip, inducing shape transitions towards the healthy discocyte in a controlled manner. In the second Part, we derive a phase-field method for membrane modeling. Phase-field methods have been extensively used for the study of interface phenomena, though with few applications to membranes. We present a new model which accounts for the membrane elasticity, and couples the membrane dynamics with an external fluid, whose hydrodynamics is dictated by the Navier-Stokes equation. We derive the expression of the stress tensor which allows us to recover the stress profile of the membrane. We also obtain the membrane equilibrium equations, proving that in the macroscopic limit our phase-field model recovers the correct expressions given by the elastic theory of membranes. In the third Part we make use of this phase-field model to study the behaviour of RBCs in flow in narrow channels, of width similar to that of the cell. We consider pressure-driven flows as they relevant for both in vivo and in vitro circulation. We carry out simulations by means of a lattice-Bolztmann method. Our study highlights the crucial role of the RBC shape, softness and deformability to explain its complex behaviour and rheological properties. RBCs flowing at low concentratrions, when they do not interact with other cells and the dynamics is governed by the interaction with the cell, are shown to migrate lateral towards the wall, avoiding the axial position. The RBC assumes an asymmetric shape and orients with the flow, reducing the viscosity of the fluid which presents a shear-thinning behaviour. The lateral position can be controlled by tuning the channel geometry and flow velocity, and it is also dependent on the shape of the cell, as sherical cells as shown to occupy and axial position. The control of these factors is important for the manipulation of different cell species, such as RBCs and leukocytes, in microfluidic devices. Finally, we study the behaviour of RBC suspensions at intermediate concentrations, when hydrodynamic interactions between RBCs govern the dynamics. The focusing to lateral positions induced by the walls is inhibited and cells are shown to order along the channel section, occupying the core of the channel. RBCs adopt and horizontal inclination, forming a relatively ordered structure of parallel rows. The rheology of the suspension is also affected, as the interactions between cells attenuate the orientation with the flux and higher flow velocities are required to induce the shear-thinning decay of the viscosity. The results presented in this Thesis highlight the delicate dependence of the cell mechanics in the balance of the cell membrane composition and elastic properties. They also demonstrate that the elastic behaviour of the cell, determined by its membrane, is also crucial for the rheological behaviour of blood, and any process of membrane damage or stiffening can substantially alter the correct blood functioning.

[es] El estudio del comportamiento mecánico de los glóbulos rojos es fundamental para entender aspectos relevantes acerca de la elasticidad de membranas y reología de la sangre, incluyendo importantes aplicaciones biomédicas. En esta tesis se aborda la respuesta elástica de estas células bajo diferentes tipos de deformaciones morfológicas. Por un lado, se estudia el efecto de la microestructura de la membrana en las formas de equilibrio de los glóbulos, identificando la función del citoesqueleto celular cuando la asimetría en la bicapa lipídica es alterada (por ejemplo, reduciendo los niveles de ATP). Nuestros resultados muestran que la bicapa tiende a expandirse formando estructuras puntiagudas, mientras que el citoesqueleto se opone a estas deformaciones y mantiene formas más compactas cercanas al discocito. El citoesqueleto aparece como un elemento fundamental para estabilizar la célula en su conformación de equilibrio. En la segunda parte de la tesis, se deriva un modelo de interfase difusa para membranas. Para ello obtenemos el perfil de esfuerzos que muestra cómo el modelo captura correctamente las propiedades elásticas de las membranas. También se obtienen las ecuaciones macroscópicas que definen el comportamiento de equilibrio y dinámico del modelo, y que convergen correctamente a los resultados clásicos de la teoría general de membranas. Finalmente, en la tercera parte realizamos simulaciones haciendo uso de este modelo de interfase difusa para estudiar el comportamiento de glóbulos rojos fluyendo en canales confinados. El estudio refleja la compleja respuesta de las células, en las que la elasticidad y deformabilidad forman un papel clave. Los glóbulos a bajas concentraciones evitan la posición central del canal y se desplazan hacia un lateral, adquiriendo morfologías asimétricas y orientándose con el flujo. Esto permite que la viscosidad del fluído disminuya. En cambio, a mayores concentraciones, cuando varias células fluyen juntas, la interacciones hidrodinámicas inhiben este comportamiento, y las células fluyen alineadas con una orientación horizontal, organizadas en filas tanto en los laterales como en el centro del canal. La interacción y apantallamiento entre las células hace que el decaimiento en la viscosidad requiera de velocidades considerablemente mayores.