22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Viscoelastic transient of confined red blood cells
G. PRADOa,b , A. FARUTINa,b,c , C. MISBAHa,b , L. BUREAUa,b
a. LIPhy, CNRS, F-38000, Grenoble, France
b. LIPhy, Univ. Grenoble Alpes, F-38000, Grenoble, France
c. Experimental Physics, Saarland University, 66123 Saarbrücken, Germany
Résumé : (16 gras)
Nous présentons ici des résultats concernant le temps de réponse de globules rouges dans un écoulement
confiné. Les expériences montrent que le temps de réponse dépend de la vitesse de l’écoulement imposé ;
nous proposons une explication théorique pour cette dépendance nous permettant de déduire une viscosité effective et le module élastique rentrant en jeu dans ce phénomène. Les résultats expérimentaux et
le modèle théorique sont validés par des simulations numériques, ce qui nous permet d’obtenir la valeur
2D = 10−7 N.s.m−1 ).
de la viscosité du complexe membrane-cytosquelette des globules rouges (ηmem
Abstract : (16 gras)
We present the results of a study of the transient startup time of red blood cells confined in microchannels.
We show that this response time depends on the imposed flow velocity and offer a theoretical model to
explain this dependence. Our model allows us to determine the effective viscosity as well as the elastic
modulus involved in the phenomenon. The experimental results and the theoretical model are validated
by numerical simulations which we use to obtain the value of the viscosity of the membrane-cytoskeleton
2D = 10−7 N.s.m−1 ).
complex (ηmem
Mots clefs : Microfluidique, Globules rouges, Viscosité, Sang
1
Introduction
Understanding the flow properties of blood and its components is highly relevant both from a medical and
a physical point of view, for example by relating blood disorders to the modification of the biophysical
parameters of the red blood cells.
Blood is commonly described as a suspension of various biological components (red and white blood
cells, platelets) in a carrier fluid (plasma). The non-Newtonian behaviour of blood is mainly due to the
presence of red blood cells (RBCs) which compose the majority of the suspended cells. Thus, the flow
properties of whole blood are mostly governed by the concentration of RBCs, their mutual interactions
as well as their individual mechanical properties.
The viscoelastic properties of RBCs are of utmost relevance in the microcirculation of blood, when
the cells experience a high degree of deformation due to the confinement in narrow capillaries. In this
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
context, in vitro studies appear as a appropriate tool to characterize the various biophysical parameters
of RBCs, develop theoretical models as well as diagnose healthy and pathological RBCs.
Even though there is a consensus in the description of RBCs from a physical point of view, there remains
discrepancies in the value of the membrane viscosity, an issue we will address in the following.
2
Theoretical Model
The heuristic model we propose describes the behavior of RBCs submitted to deformation in a shear
flow and is obtained by balancing the external viscous stresses imposed on the RBC with the stresses
occurring in the cell due to its deformation (with an elastic, viscous and tension contribution). Without
a thorough derivation (which can be found in [1]), our model yields a characteristic deformation time
expressed as :
Kef f + ηout V
τc−1 =
(1)
ηef f R
where Kef f – the effective elastic modulus – is a function of the shear and bending elastic moduli
(µS and κ respectively), ηef f – the effective viscosity – is a linear combination of the viscosities at play
(cytoplasm viscosity ηin , external fluid viscosity ηout and membrane viscosity ηmem ) and R is the radius
of the sphere having the same surface area as a RBC.
To test the validity of the model, RBCs are submitted to a Poiseuille flow at different velocities and their
characteristic time of deformation measured, in vitro and in silico.
3
Experimental Setup
Experiments were conducted in a home-made microfluidic device allowing us to observe RBCs flowing
in short pieces of silica capillaries (inner diameter 10µm). To control the startup and stop of the flow, a
solenoid valve with a short response time was used in the inlet circuit. The range of velocities accessible
in the experiment was close to the physiological velocities in the microcirculation (0.5 − 5mm.s−1 ).
RBCs were isolated from whole blood obtained through the Établissement Français du Sang and diluted in Phosphate Buffer Saline (PBS) at a low enough hematocrit to observe individual cells in the
capillaries.
Image sequences of the RBCs were acquired through a 100x magnification oil-immersion objective and
a fast camera.
Details about the numerical simulations can be found in [6] and [7].
4
Results and discussion
Image sequences of the RBCs from their rest position (at zero flow velocity) to their stationary velocity
are recorded at different imposed flow strengths. From these image sequences, two transient response
times can be defined : one based on the stationary shape of the flowing RBC, the other based on the
stationary velocity reached by the flowing RBC. The latter is chosen as the characteristic time here, as
the former relies on a subjective evaluation of the shape of the RBC and depends on the orientation of
the cell in the capillary ; nevertheless, we observe that the two transient times presents a good correlation
over the range of velocities studied. Moreover, the order of magnitude of τc (τc . 100ms) is in agreement
with previous works, for example [3]. As illustrated in Fig.1a, over the range of velocities considered, a
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
linear dependence of 1/τc on Vrbc appears, in agreement with our heuristic model. The linear behavior
is conserved when increasing ηout , though with a different slope, again as expected from our model.
Figure 1 – a) Experimental results : Inverse transient time as a function of RBC steady-state velocity. Black circles are from RBCs in PBS solution (ηout ≈ 1.5mP a.s), white squares in PBS+Dextran
(ηout ≈ 5mP a.s). b) Numerical results : Inverse transient time as a function of RBC steady-state velocity for different viscosity contrasts (λ = ηin /ηout with ηin = 6mP a.s in the simulations).
From the linear fit of the experimental data (in the form of eq.1), we can extract the value of ηef f and
Kef f ; as well as the value of ηmem from the value of ηef f (results in Table 1).
Kef f
ηef f
ηmem
3 ± 0.8 µN.m−1
122 ± 21 mP a.s
28 ± 8 mP a.s
Table 1 : Physical parameters of a healthy RBC measured from the experimental data.
When considering the results of the numerical simulations, the criterion used to determine τc is similar
to the one used in the experimental results. The inverse relaxation time is shown to vary linearly with
the steady-state velocity and to depend as well on the value of the external viscosity ηout (as shown in
Fig.1b). In qualitative agreement with the experiments, the linear relation is observed on the whole range
of velocities considered and viscosity contrasts.
The main difference between experiments and simulations here is that membrane viscosity is not accounted for in the simulations. Thus, using Eq.1 will not give the correct values of the biophysical parameters
Kef f and ηef f with the numerical results. However, comparing the slopes of the linear fits of τc−1 (V )
from the experimental and numerical data yields another way of determining ηmem ; while comparing
the intercepts at zero velocity gives the value of Kef f (see Table 2).
Kef f
ηef f
ηmem
2.5 µN.m−1
–
28 mP a.s
Table 2 : Physical parameters of a healthy RBC calculated from the comparison between numerical and
experimental data.
22ème Congrès Français de Mécanique
Lyon, 24 au 28 Août 2015
Both methods of investigation give the same value for the membrane viscosity of healthy RBCs, which
2D ≈ 10−7 N.s.m−1 . While this value is in agreement with [5] and
leads to a 2D membrane viscosity ηmem
[4], it is one order of magnitude lower than the value reported in [2, 3, 8, 9]. This apparent discrepancy
can be explained by the fact that previous authors made the assumption that ηef f ≡ ηmem which leads
to an overestimation of ηmem by up to one order of magnitude.
The value of the effective elastic modulus we obtain is of the same order of magnitude as the elastic
shear modulus measured in other studies [10, 11].
5
Conclusion
We have obtained information on the response time of deformation of RBCs when submitted to an
external flow. By measuring this characteristic time and with the help of a simple model, we extract the
values of the effective elastic modulus as well as the effective viscosity of the RBC. Our model then
allows us to determine the values of the elastic shear modulus and the membrane viscosity of the RBC.
We furthermore offer an explanation as to the different values of the membrane viscosity determined
in the past : the dissipation processes taking place in a deformed RBC involve the external and internal
viscosities as well as the membrane viscosity.
Références
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