Analysis of the variation in the determination of the shear modulus of the erythrocyte membrane: Effects of the constitutive law and membrane modeling

Abstract

Despite research spanning several decades, the exact value of the shear modulus Gs of the erythrocyte membrane is still ambiguous, and a wealth of studies, using measurements based on micropipette aspirations, ektacytometry systems and other flow chambers, and optical tweezers as well as application of several models have found different average values in the range 2–10 µN/m. Our study shows that different methodologies have predicted the correct shear modulus for the specific membrane modeling employed, i.e. the variation in the shear modulus determination results from the specific membrane modeling. Available experimental findings from ektacytometry systems and optical tweezers suggest that the dynamics of the erythrocyte membrane is strain-hardening at both moderate and large deformations. Thus the erythrocyte shear modulus cannot be determined accurately using strain-softening models (such as the neo-Hookean and Evans laws) or strain-softening/strain-hardening models (such as the Yeoh law) which overestimate the erythrocyte shear modulus. According to our analysis, the only available strain-hardening constitutive law, the Skalak et al. law, is able to match well both deformation-shear rate data from ektacytometry and force-extension data from optical tweezers at moderate and large strains, using an average value of the shear modulus of Gs = 2.4–2.75 µN/m, i.e. very close to that found in the linear regime of deformations via force-extension data from optical tweezers, Gs = 2.5±0.4 µN/m. In addition, our analysis suggests that a standard deviation in Gs of 0.4–0.5 µN/m (owing to the inherent differences between erythrocytes within a large population) describes well the findings from optical tweezers at small and large strains as well as from micro-pipette aspirations.

Threshold shear stress for the transition between tumbling and tank-treading of red blood cells in shear flow: dependence on the viscosity of the suspending medium

“…The ratios of the rates of energy dissipation (D m ) in the membrane and cytoplasm are in keeping with the range of values in the literature (Fischer 1980;Secomb & Skalak 1982;Tran-Son-Tay, Sutera & Rao 1984). The values of G m s , on the other hand, are appreciably lower than the well-established experimental value (see references in Dimitrakopoulos 2012). Another way to decrease E 0 while keeping the accepted value of G s = 2.5 µN m −1 is discussed in § 5.7.…”

“…First, we used η cyt = 10 mPa s, the value at room temperature, and G s = 2.5 µN m −1 , the best choice when the constitutive equation of Skalak et al (1973) is used (Dimitrakopoulos 2012). Among the values for K b used by both authors, we selected the one resulting in κ b = 4 × 10 −19 J.…”

“…Note that E s = 2G s , where G s is the shear modulus as defined by Dimitrakopoulos (2012). With V = 94 µm 3 , η cyt , and G s , we calculated the values of η 0 ,γ and κ b .…”

Effects of membrane hardness and scaling analysis for capsules in planar extensional flows

“…For example, for millisize capsules made from aminomethacrylate membranes with shear modulus G s = O(10 −2 ) N m −1 , flow rates Ca = O(1) require shear stress µG = O(10) Pa (Pieper et al 1998); these flow rates require shear stress µG = O(1) Pa for microcapsules with similar size and shear modulus as those of the erythrocyte, i.e. a = 2.8 µm and G s = O(10 −6 ) N m −1 (Popel & Johnson 2005;Dimitrakopoulos 2012). It is of interest to note that the required flow rates for large deformations of millisize and microsize capsules have long been achieved in four-roll mill experiments with droplets and bubbles (with size and fluids/interfacial properties similar to those of the capsules) under Stokes flow conditions (Bentley & Leal 1986;Ha & Leal 2001).…”

A novel strain energy relationship for red blood cell membrane skeleton based on spectrin stiffness and its application to micropipette deformation

“…In one common approach, the skeleton elastic properties are described on the basis of assumed constitutive relations defined at the level of an elastic continuum (for example, Waugh and Evans 1979; Skalak et al 1973; Mohandas and Evans 1994; Mukhopadhyay et al 2002; Kuzman et al 2004). The different versions of constitutive relations presented in the literature typically describe small deformations equally well, but differ in describing the behavior of the system at relatively large deformations (Dimitrakopoulos 2012). Alternatively, skeletal energies have been estimated based on molecular-level models of the spectrin network either using simple elastic springs (Hansen et al 1996) or a worm-like chain potential (Discher et al 1998; Li et al 2005).…”