## 6 Modeling the Microrheological Deformation Behavior of Blood Cells
[55] provides a comprehensive overview of the mechanisms and conditions that govern how blood cells adhere to the vascular endothelium under flow conditions [54; 56]. The authors investigated the adhesion of circulating leukocytes and platelets to the vessel wall. Their study focuses on the dynamics of cell adhesion and includes both experimental approaches and conceptual frameworks. The main methods and instruments in the study include the use of intravital microscopy and in vitro flow-based rheological systems to study adhesion in dynamic conditions; specific cellular receptors and molecular mechanisms involved in leukocyte and platelet adhesion; the influence of physical factors (e.g., shear rate, flow conditions) on adhesion behavior; and a comparison between physiological (normal) and abnormal (pathological) adhesion processes. [PERSON] et al. [49] review physical models of leukocytes, focusing on the ability of leukocytes to explain the deformation and recovery behavior of these cells. This determination is modeled in several ways: the liquid drop model treats leukocytes as Newtonian fluid droplets with surface tension and captures many experimental features; non-Newtonian models incorporate the elastic and viscous properties of the cytoplasm, offering a more detailed but still limited representation; the compound drop model helps to reconcile inconsistencies in measured cell viscosity due to limited data on internal cell structure; and the tensegrity model sees the cell as a structure with microfilaments under tension and compression. This technique explains how mechanical signals are transmitted and converted into biochemical responses--though it is more effective for adherent cells than suspended ones. Dynamic testing methods and the modeling of cells as viscoelastic solids (with spring constants and dashpots) allow the determination of complex shear modulus (G)**, though relating the results to intrinsic cell mechanics remains challenging. The mathematical modeling and numerical simulation of the motion and deformation of red blood cells (RBC) and vesicles subject to an external incompressible flow in a microchannel are performed in [57]. RBC and vesicles are viscoelastic bodies consisting of a deformable elastic membrane enclosing an incompressible fluid. The authors provide an extension of the Finite Element Immersed Boundary Method (FE/BE FE-IBM) [11; 13] based on a model for the membrane and that additionally accounts for bending energy and also consider in flow/out flow conditions for the external fluid flow. The stability analysis requires both the approximation of the membrane by cubic splines (instead of linear splines without bending energy) and an upper bound on the inflow velocity. The FE/BE FE-IBM has been applied to the numerical simulation of RBC and vesicles in shear flow with emphases on tank treading motion, the impact of the membrane stiffness, and the motion of RBC through a thin capillary. Comparisons of the simulation results with experimentally available data show very good agreement and indicate the reliability of the approach. Overall, while these models provide insights, limitations persist due to incomplete knowledge of leukocyte substructure and internal rheology.