# In Silico Investigation of the RBC Velocity Fluctuations in Ex Vivo Capillaries
## 1 Introduction  
The importance of capillary blood flow was stated back in 1922 by [PERSON] and [PERSON] as follows: \"It is evident that the rest of the cardiovascular system exist only to regulate the blood flow through the capillaries\" [1]. Indeed, the high and dynamic energy requirements of the central nervous system (CNS) require a continuous supply of oxygen and nutrients, which must be provided by a highly efficient microcirculation. As most acute and chronic neurodegenerative conditions have a component of microcirculatory dysfunction in their pathogenesis, it is necessary to understand the physical mechanisms that govern cell transport in capillaries. Capillary dysfunction arises when there is extremely high capillary transit time heterogeneity (CTTH) in the microcirculatory network [2]. High CTTH results in rapid oxygenated RBC shunting, leading to suboptimal tissue oxygenation, even when there is no flow-limiting condition in the large arteries that feed the capillary network [3, 4]. The underlying causes of high CTTH are subject to active investigation. At any given time, cell fluxes,flow velocities, and hematocrit distributions can vary significantly between individual interconnected parenchymal capillaries. Many in vivo studies have suggested prominent fluctuations in cell flux and hematocrit level in a given capillary [5; 6; 7]. Observed heterogeneity, studied by utilizing ex vivo, in vitro, and in silico techniques, suggests possible contributors such as RBC lingering at the bifurcation apex, vessel plugging by leukocytes or platelets, and changes in the diameters of vessels upstream and downstream [6; 8; 9; 10; 11; 12]. However, despite its importance, the detailed biomechanics of RBC flow in relation to natural luminal irregularities and its contribution to flow heterogeneity are still relatively underexplored. In silico investigations therefore present a significant opportunity to deepen this understanding. Several numerical approaches have been developed over the years to investigate blood flow in silico. For large vessels, where the diameter is sufficient to create a cell-free layer (CFL), also known as the Fahraeus-Lindqvist effect, it is possible to determine the hematocrit and/or the bulk velocity of the blood using correlation-based models and non-Newtonian viscosity assumptions [13; 14; 15; 16]. In correlation-based models, the diameter of the vessels and the inlet hematocrit (Ht) ratio determine the local viscosity, and the capillary networks can be modeled and solved with Hardy-Cross-like methods [17]. Similarly, a single-phase non-Newtonian viscosity assumption can be a valid and useful approach if a CFL is present [16]. In this approach, it can be assumed that the viscosity of the single-phase fluid can be assumed to be shear-thinning (such as a Casson fluid), and the viscosity of the bulk fluid will be lower near the walls, where the shear rate is high and the Fahraeus-Lindqvist effect prevails. Both of these approaches yield good results that match the experimental data, as long as the capillary diameter is large enough to create a CFL. However, these approaches ignore the local dynamics of RBC transport, such as RBC-to-RBC and RBC-to-wall interactions, as well as the non-linear behavior of RBCs due to their hyperelastic membrane when a CFL is not present. In flow configurations where RBCs move in a single-file manner [18], such as narrow capillaries, the assumptions of these models are not valid. Hence, these are low-fidelity approaches for hemodynamics in small channels and cannot be used to understand the local dynamics of RBCs. For small and realistic capillaries, it is essential to model the RBCs explicitly, since the capillary diameters are between 5 and 10 \(\upmu\)m, which means that a CFL is not present. To the best of our best knowledge, the literature review does not show a direct comparison of the differences in RBC transit time between realistic capillaries that harbor luminal irregularities and their equivalent tube geometries.