A constitutive hemorheological model addressing both the deformability and aggregation of red blood cells

Abstract

Red blood cells (RBCs) in physiological conditions are capable of deforming and aggregating. However, both deformation and aggregation are seldom considered together when modeling the rheological behavior of blood. This is particularly important since each mechanism is dominant under specific conditions. To address this void, we herein propose a new model that accounts for the deformability of red blood cells, by modeling them as deformed droplets with a constant volume, and of their aggregation, by properly characterizing the network formed by red blood cells under small shear rates. To derive the model, we employ non-equilibrium thermodynamics that allows us to consistently couple the two mechanisms and guarantees model admissibility with the thermodynamic laws. Relative to our previous model, which addresses the rheological behavior of non-aggregating deformable red blood cells, one additional structural variable, λ, to properly characterize the network formed by RBCs, and another additional parameter, ε, that quantifies the relative importance between the regeneration/buildup and flow-induced breakup of the network, are considered here. The new model predicts a yield shear stress, in accord with experimental data, but also predicts non-vanishing yield normal stresses. Although no rheological measurements of yield normal stresses of blood have been reported in the literature, the recent measurement of yield normal stresses of other yield stress fluids indicates their potential existence in blood as well. We show that the new model is in complete accord with the experimental rheological behavior of normal blood in both steady-state and transient (step-change in shear-rate) simple shear.

Elucidating the rheological implications of adding particles in blood

“…Throughout this work, we consider an isothermal and incompressible flow, meaning that both the total mass density, ρ , and the entropy density (or temperature) are excluded from the vector of state variables. At first, to describe the rheology and microstructure of nanoblood, we follow previous works (Stephanou 2020 , 2021 ; Stephanou and Tsimouri 2020 ) and consider RBCs as emulsions with a droplet morphology by using a constrained contravariant second-rank tensor, , such that det is equal to the squared volume of a RBC. We also define the dimensionless tensor so that .…”

“…At equilibrium, the tensor is equal to where and are the principal semi-axes of the RBC (see Fig. 1 of Stephanou and Tsimouri ( 2020 )), whose determinant is ; thus, (Stephanou 2020 ; Stephanou and Tsimouri 2020 ). If we then consider the typical values = 7.5 μm and 90 μm 3 , we must consider 3.06 μm (Stephanou and Tsimouri 2020 ).…”

“…The mere use of the conformation tensor allows us to consider the deformability of RBCs but not their aggregation in normal blood, which leads to the exhibition of a yield stress at sufficiently low shear rates as a result of the network formed. We have recently proposed (Stephanou 2020 ) that to accomplish this we need to employ one additional scalar structural variable, , to properly characterize the network formed by RBCs. When the network is fully formed, then =1, while in a completely broken state, i.e., when only single RBCs remain, it vanishes (Mewis and Wagner 2009 ; Stephanou and Georgiou 2018 ).…”

“…In this work, we will employ non-equilibrium thermodynamics (NET) to develop a sophisticated mathematical model addressing the rheological behavior of nanoblood. To accomplish this, we will develop the model using the generalized bracket formalism (Beris and Edwards 1994 ) of NET (Beris and Edwards 1994 ; Grmela and Öttinger 1997 ; Öttinger and Grmela 1997 ; Edwards et al 2003 ; Öttinger 2005 ), by means of which several microstructured systems have been addressed up to date, such as, but not limited to, immiscible complex fluids (Edwards and Dressler 2003 ; Dressler and Edwards 2004 ; Dressler et al 2008 ; Grmela et al 2014 ; Mwasame et al 2017 ), polymer melts and solutions (Beris and Edwards 1994 ; Grmela and Öttinger 1997 ; Öttinger and Grmela 1997 ; Öttinger 2005 ; Stephanou et al 2009 , 2016 , 2020b ), polymer nanocomposites (Rajabian et al 2005 ; Eslami et al 2010 ; Stephanou et al 2014 ; Stephanou 2015 ), micellar systems (Germann et al 2013 ; Stephanou et al 2020a ), blood (Tsimouri et al 2018 ; Stephanou 2020 ; Stephanou and Tsimouri 2020 ), drilling fluids (Stephanou 2018 ), and thixotropic fluids (Stephanou and Georgiou 2018 ). The use of a NET formalism has the compelling advantage, over other approaches, that the constitutive model as a whole is guaranteed consistency with the laws of thermodynamics (as extended for beyond equilibrium systems).…”

Modeling the Deformation of Shear Thinning Droplets Suspended in a Newtonian Fluid

“…The last term in the above equation has been proven and validated by Stephanou et al [42] and Stephanou [43]. σ * αβ is the stress value of the droplet phase, Go is the elastic modulus, τ B is the relaxation time of the Bautista model, and k is a structural relaxation parameter of the droplet phase.…”