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Blood Cell Biomechanics Evaluated by the Single-Cell Micromanipulation
J. Lelièvre, C. Bucherer, S. Geiger, C. Lacombe, V. Vereycken
To cite this version:
J. Lelièvre, C. Bucherer, S. Geiger, C. Lacombe, V. Vereycken. Blood Cell Biomechanics Evaluated by the Single-Cell Micromanipulation. Journal de Physique III, EDP Sciences, 1995, 5 (10), pp.1689-1706.
�10.1051/jp3:1995218�. �jpa-00249409�
J. Phys. III France 5 (1995) 1689-1706 OCTOBER1995, PAGE 1689
Classification Physics Abstracts
46.10 87.45
Blood Cell Biomechanics Evaluated by the Single-Cell
Micromanipulation
J-C- LeliAvre, C. Bucherer, S. Geiger, C. Lacombe and V. Vereycken
Unitd de Biorhdologie, Universit+ Paris VI et CNRS LBHP URA 343 Centre Hospitalier Universitaire, 91 Boulevard de l'H6pital, 75013 Paris, France
(Received 27 December 1994, revised 24 May 1995, accepted 7 July 1995)
Abstract. Assessed by the micromanipulation technique, the main mechanical properties of blood cells (red cell and white cell) are reviewed. Hyperelasticity of red blood cell membrane is
characterized and corresponding material coefficients
are given. Surface tension and cell viscosity of leukocyte are discussed and related values
are stated. Some aspects of cell contact mechanics
are developed. The fundamentals of mechanics of two elastic cell in contact when adhesion is taken into account is specified.
1. Introduction
Human blood is a suspension of cells (red cells, white cells and platelets) in plasma which is
composed of 90% water w/w, carries ions, organic substances and proteins. In a physiological state, cell number per mm~ ranges from 4.0 x 10~ to S-g x 10~ for red cells, from 4.0 x 103 to 10.0 x 103 for white cells and from 2.0 x 105 to 4.0
x 10~ for platelets respectively [I].
It is well-known that blood flow conditions in the different circulatory compartments are
extremely variable. In arteries and veins whose diameters are larger than 500 pm, blood cells
are small enough for the blood flow to be considered as that of a homogeneous non Newtonian fluid. In arterioles and venules of diameters between 50 pm and 100 pm, flow is viewed as biphasic due to the formation of a red cell plasmatic layer. In capillaries whose diameters are smaller than 10 pm, red cells individually move at very low speed and deform or link together
to form aggregates whose length can vary during the motion [2, 3]. White cells exert a non
negligible influence on the microcirculation hemodynamics and undergo deformations when they squeeze through the vessel and tissue openings.
Regulated by hormonal, metabolic and neurologic factors, blood flow characteristics depend
not only on hemodynamics factors such as cardiac frequency, peripheral resistance, vessel geometry, but also on mechanical properties of cells themselves. The cell shape and structure confer to the different kinds of cells specific mechanical behaviors.
It is the aim of this paper to review the red and white blood cell mechanical properties when evaluated with the micromanipulation technique [4]. It is a very powerful tool, particularly
@ Les Editions de Physique 1995
adapted to the cell level, which provides an experimental and direct evaluation of deformation when interpreted with adequate models. In this approach, the cell (two cells in the case of
adhesion) is aspirated into a (or two) calibrated cylindrical glass micropipette(s) under a (or two) well controlled aspiration pressure(s).
2. The Red Cell
The red cell is an anucleated cell. It has the essential role of carrying oxygen from lungs to tissues and carbon dioxyde from tissues to lungs. It is a biconcave disc of 8 pm in diameter,
I.4 pm thick at the center and with a maximum thickness of 2.8 pm.
The red cell membrane consists of a lipid bilayer containing proteins and whose thickness of about 10 nm cannot be easily changed. Under microscope observation, the cell surface appears to be smooth. The red cell cytoplasm contains an hemoglobin solution whose Newtonian viscosity at 37 °C is around 6 mPas [5].
Hooked to the membrane inner bilayer surface, the red cell protein network is mainly made up of spectrin (the basic unit is a hexagonal lattice composed of six spectrin molecules) and
actin linked to proteins 4.9 and 4.I. The spectrin (with its o and fl chains) bridges connecting complexes of proteins between themselves. Each spectrin dimer head interacts with an another dimer. The spectrin network is hung on protein 3 and on sialoglycoproteins [6].
At a constant volume (about 90 pm3) and surface (about 140 pm~) the red cell travels several hundred kilometers through the organism. Such a journey requires the red cell to have both an exceptional mechanical resistance and a good deformation ability. The lipid bilayer of the membrane has been proved to be responsible for cell area restriction and cell membrane
bending stillness and the protein scaffolding underneath the bilayer for the cell membrane extensional rigidity [7-10]. In the following section, only the elastic behavior of the red cell will be taken into consideration and the so-called "membrane" will refer to the set of lipid bilayer and cytoskeleton proteins.
As it is very thin compared to overall cell dimensions, the cell membrane can be mechanically regarded as a thin shell of quasi-constant thickness h. However, membrane material cannot be
considered as usual 3-D material because of its strong transverse anisotropy. Furthermore, like
numerous other biological soft tissues the cell membrane is considered as an incompressible
material.
Let us consider at M a small membrane plane element submitted to a biaxial load test. ii,
lz and 13 measure the principal deformations (Fig. I). The Fingers' tensor:
j21
j2 jJf) j~)
2
j2
~
allows the characterization of elastic finite deformations of the cell membrane. For each mate- rial point M, a strain energy function Ws(ii,12,13 defined per unit volume of the non-deformed
body is conjectured. ii, 12, 13 are invariant tensors. The material is assumed to be incompress- ible (13 " 1). As the membrane thickness da3 " h cannot be easily changed, 13 and therefore the product Ii12 remain close to one. It is convenient to introduce here a new invariant set 11 121
Ii "
I( +1( 2 12 =1(1( (2)
N°10 BLOOD CELL BIOMECHANICS 1691
~ ~J
1 ~ lid,"h
~/@ '
J
~
lid+
~
i
Fig. 1. Biaxial load test of a red cell membrane element.
where12 is directly linked to membrane area changes. Skalak et al. [7] adopted the following
strain energy function Ws Iii,12), defined per unit surface area of non-deformed membrane:
~~~~'~~~ ~~~ ~ ~ ~~~ ~ ~~~ ~~~
where p and K are constants which characterize the material. The corresponding resultant tensions Ti and T2 are given by:
T2 "
~i 2
where To is an
isotropic
tension which is not
sidered here in order to maintain aterial incompressibility [8].
Material onstants p and K can be
determined with ppropriate mechanical ests
such as icropipette A is a very narrow lass tube (a few pm
inner iameter 2Rp) in which a
constant pressure P induces a local deformation [11].
pressure P alues, a
normal red cell ndergoes
constant surface area deformation while for
high pressure P values an motically
swollen red
cell undergoes deformations ith an area
increase. An lastic solid behavior of the cell is hown
the pressure in the pipette is reset IF = 0).
In the
case
of low pressure P values (Fig. 2a) the laccid cell membrane is supposed
eformed at a onstant area. Thus in quation (3), 12 " 0. For moderate deformations
from the reference
state, ii and12 are close to
~ ~
2
~
~~~~ ~ ~~~ ~~~ ~~~~ ~~~
where p is the membrane shear elasticity modulus. The membrane shear elasticity modulus is mainly due to the cytoskeleton protein scaffolding [10].
p p
L L
a) b)
Fig. 2. Principle of a micropipette experiment performed on a red cell. Membrane deformation
occurs, a) at constant area for a flaccid cell, b) with a surface area increase for a sphered cell. P is
the pressure Ill] which creates the deformation of the I X I dotted squared figure. The corresponding
resultant tensions are also shown.
On the other hand, Evans [9] and Chien et al. [12] demonstrated that the static pressure P
required to aspirate a cell part of length L can be written as:
P =
~ l~~ 1 + Log
2 ~
L > Rp (6)
Rp Rp Rp
Therefore, by measuring Rp and measuring L as a function of P, ~ may be determined. In the second case (Fig. 2b), experiments show that deforming the swollen cell from an initial
spherical shape requires much higher pressures P in the pipette than in the previous case. Thus,
K is much higher than ~. Taking into account only the term involving K in the equation (4), choosing the suitable reference state, and considering moderate deformations for which both ii and 12 are close to one and equal to the common value I, the isotropic tension T can be
expressed as:
T To "
K(l~ 1) (7)
K is therefore the elastic area compressibility modulus of the cell membrane. Using Rand equation (13), the pressure P required inside the pipette to deform the cell is found to be:
P = 2(T To) () )) (8)
c P
where Rc is the swollen cell radius.
As a result, knowing Rp, controlling P and determining I by measuring cell area variations, K may be determined [14].
Figure 3 represents a typical sequence of a dimple flaccid cell aspiration. To each aspiration
pressure inside the pipette, an equilibrium deformation state is reached. The cell projected length in the pipette is then measured for the corresponding aspiration pressure.
Figure 4 shows typical results obtained in our laboratory at the room temperature of 20 °C for a red cell suspended in a butler (pH = 7.4, osmolality = 300 mosni/kg). Values for ~ and
N°10 BLOOD CELL BIOMECHANICS 1693
fl ~. L~ .1 15 .Z o
[. j~'k~'"
b "~2 §,Z,1
~
'-'i'
", ~'"
-T,'
Fig. 3. Typical sequence of
a dimple cell aspiration for a flaccid red blood cell. To each aspiration
pressure inside the pipette, an equilibrium deformation state is reached. The cell projected length in
the pipette is then measured for the corresponding aspiration pressure P. The pipette diameter is 2Rp = 1 ~m.
s to
PRp(~N/m) ~j T-To) (mN/m) ~j
4 8
Rp=0,5~m
3 6
.
2 4
2
[(2L/Rp)-I+log(2L/Rp)] (l~-I)
x 100
0 0
2 4 6 8 lo o-O O-S I-O I-S 2.0 2.5 3.0
Fig. 4. Typical results of micropipette experiments performed on red cells. a) Determination of the shear elasticity modulus ~J. Using the formula (6) (see text) we obtained ~
= 4.3 ~N/m.
b) Determination of the area compressibility elasticity modulus K. Using the formulas (7) et (8) (see text) we obtained Ii
= 301 mN/m (from Bucherer cl al. [15-16]). In both cases the membrane 1,hickness is taken as h
= 10 nm.
K are determined within a relative accuracy of 10% including possible error on pipette radius determination. Bucherer et al. [15,16] obtained the following values for ~ and K:
Jt = 4.5 + 0.8 JtN/m 129 Cells) K j~~
= 300 + 60 mN/m (12 cells)
Our results for ~ are included in the range of 4 ~N/m to 10 ~N/m given in the review paper of Hochmuth et al. [17]. Our results for K are between those of Evans et al. [14] and Waugh
et al. [18] who respectively found a value of 288 + 50 mN/m and 450 mN/m. In the case
of constant area deformations, a membrane element undergoes increasing compression as the
pressure P in the pipette increases. This leads to the cell membrane buckling.
Before this occurs, the membrane stability is ensured by the cell bending stillness dominated by the lipid bilayer [10]. In the case of experiments in which cell area variations occur, a cell
membrane element undergoes isotropic tensions. If these tensions tend to become P too high in intensity, the cell breaks and cell lysis occurs when the relative local area deformation exceeds
a few percents.
For a plate made up of an isotropic elastic material, the elastic bending energy per unit area is given by:
WB(Ml
=
~ (K) + K( + 2uKiK2 (10)
where Ki = (I/Ri) and K2 = (1/R2) are the principal curvatures of the membrane element at M and D the bending stillness:
D = jEh3j/j12ji u)j iii)
where E and n are respectively the material Young modulus and the Poisson ratio. Taking
the following values u
= 0.5 (incompressibility), E
= 2(1+ u)~ (~ is given from Eq. (9)) and
taking h
= 10 nm, the value of D is computed to be equal to D
= I-S x 10~~~ N-m-
No experimental method allowing the direct determination of red cell bending stillness is yet available. Nevertheless, based on observation of the buckling phenomenon in a micropipette experiment, D can be evaluated [19]:
D =1.8 + 0.2 x 10~~~ N-m (12)
The reason why the likely value indicated in equation (12) is quite different from the cal-
culated one using equation ill must be looked for in the initial hypothesis of a 3-D material which has led to adopt the function
as the bending energy (10). Finally, Evans [20] proposed:
WBIM)
= lKi + K2 Ko)~ l13)
where D takes the value of equation (12). Ko is called a spontaneous curvature which is a ficti- tious curvature introduced by Evans to take into account chemical and electrical effects internal to biological membranes. Corresponding bending moments are given by the relationships:
MI " )(Ki + K2 Ko) M2
= )(Ki + K2 Ko) (14)
2 2
The essential question which arises now is how to relate the material constants ~, K and D of the cell membrane to its molecular composition and architecture. Conformational rearrange-
ment in which spectrin molecules fold and unfold would be necessary for the cell to deform
Na10 BLOOD CELL BIOMECHANICS 1695
normally [21]. Red cell pathologies have highlighted various hypotheses about the structural basis for the red cell membrane mechanical properties such as lateral skeletal protein linkages
and bilayer-skeletal protein network linkages [10].
3. The White Cell
White cells play a determinant role in protecting body identity and integrity and in the fight against infections and food poisoning. The aggressor can be either consumed or killed by direct
contact with the white cell or be executed by antibodies released by the white cell itself.
White cells are dragged along the vessels by blood flow. They are permanently watchful,
more or less clinging to the vessel walls on which they roll, ready to be activated for leaving
the blood circulation and migrating through the tissues. In the capillaries white cells can temporarily block all red cell movement [22].
White cells contain a nucleus whose the shape schematically allows two main groups to be considered [23, 24]. The first white cell group, the larger one, contains from 40 to 83% of the white cells whose nuclei appear to be in several subparts. Those are called polynucle-
ars. According to the nature of the granules, polynuclear neutrophils (the most numerous), polynuclear eosinophils and polynuclear basophils can be distinguished. In its passive state a
polynuclear neutrophil is a sphere of 8 ~m to 10 ~m in diameter. The second white cell group contains from 17 to 60% of the white cells whose nuclei appear as spheres. Lymphocytes are
the most numerous. A passive lymphocyte is a sphere of 6 ~m to 9 ~m in diameter.
Like the red cell membrane, the white cell membrane is a lipid bilayer with proteins but under microscope appears ruffled and wrinkled. An actin, myosin and other protein gel is in contact with the membrane inner surface. This gel is involved in cell activation processes when immune reactions occur. In its passive state the gel which occupies from 20 to 50% of the cytoplasm explains the overall spherical shape of the cell. The white cell cytoplasm contains
enzymes necessary for the cell functions.
From a rheological point of view, Evans and Yeung [25] have considered a white cell as a
spherical liquid drop of Newtonian viscosity q. The spherical shape is maintained due to an
qquivalent surface tension To called cortical tension resulting from the gel structure previously
mentioned.
The viscosity q and the cortical tension To can be determined by observing a white cell flow into a micropipette in which a pressure P is maintained (Fig. 5). The fluid behavior of the cell becomes obvious since it does not recover its initial shape when the pressure P is reset.
There is a critical pressure P~r above which the cell flow is initiated. Below P~r the cell does not flow and adopts a static equilibrium shape due to the surface tension. Experimentally the
aspirated length inside the pipette is observed to be close to Rp. Then, using Laplace's law the cortical tension can be calculated as follows:
To = P~r ~ jls)
2 RP Rc
~
Figures 6a and 6b show results obtained in our laboratory by Geiger et al. [26] for 32 polynuclear neutrophils and for 35 lymphocytes respectively. For polynuclear neutrophils, Laplace's law from which the cortical tension is evaluated fits the experimental data:
To " 36 + 9 ~N/m (32 Neutrophils) (16)
Our value is iii close agreement with that of Evans et al. [25] who found 35 + 9 pN/m and
higher than that of Needham et al. [27] who found 24 + 3 ~N/m.
2R, ~~
p~ P>P~
'
"
L<R L(t)>R,
a) b)
Fig. 5. Principle of a micropipette experiment performed on a white cell, a) for P < Pcr, the cell adopts a static equilibrium state (no flow), b) for P > Pcr, the cell flows into the pipette.
6 2.0
P~r(Pa)x lo-' al
p~~(Pa) x io.2 b)
5
1.s 4
,
from equation(15)
/ /
3 1. 0
2
0.5 l
~l
2 3 4 5 6 7 ~"~i
2 3 4 5 6
2Rp(~m) 2Rp(~m)
Fig. 6. Critical pressure Pcr as a function of the pipette diameter 2Rp, a) for the polynuclear neutrophils and b) for the lymphocytes. Solid line is a fit of equation (15) (see text) to the experimental data with a
mean value for Rc equal to 4.5 ~Jm. Bar represents the standard deviation obtained from
a series of measurements (from Geiger d al. [26]).
However, as it can be seen in Figure 6b, the equation (15) it is not appropriate for evaluating
a single value for the lymphocyte cortical tension. Nevertheless, the cortical tension of lym- phocytes and of monocytes has been evaluated by Geiger [28] using an indirect method based
on the work of Tran-Son-Tay et al. [29]. The obtained values are respectively lls ~N/m for the
lymphocytes and 90 ~N/m for the monocytes. These higher values have not been explained yet.
At P > P~r, the white cell flows into the pipette. In the flow modeled by Yeung and Evans [30], two compartments are considered: one is the cell part outside the pipette, the
N°10 BLOOD CELL BIOMECHANICS 1697
Plpetwall P>P~,
L(t)>R,
Fig. 7. Geometry of the flow of a white cell outside and inside the pipette.
other the cell part inside the pipette (Fig. 7). Flow is extremely slow, so the stationary, incompressible and Newtonian Stokes flow hypothesis can be adopted. Thus, the equations of
motion are:
Vp = qAV (17)
V.V=0 (18)
V, p and q are respectively velocity, pressure and fluid viscosity. Outside the pipette, the flow geometry in the cell is spherical and axisymmetrical. For the cell flow outside the pipette, the general solution of the equation set (Eqs. ill) and (18)) is thus formulated in terms of
stream function [31]:
~b(R, 9) = ~(AnR"2 + CnR"+~)ln(cos9) (19)
An and Cn are the coefficients of the order n of the development and In(cos 9) the Gegenbauer
function of order n.
Inside the pipette, the fluid flow is regarded as a plug flow of both uniform velocity dL(t)/dt
and pressure P~ with no wall friction. The internal flow pressure P~ is given by Laplace law:
fl = P +
~~ (20)
Rp
Using equation jig), the velocity V(R,9) and stress tensor Y(R,9) fields are calculated on the cell surface R
= Rc outside the pipette and the best set of the coefficients An and Cn are
obtained with a minimization procedure and the boundary conditions:
~~~~~~' ~~~~' ~~~~ ~~' ~~°~~
9p < 9 < 7r, R
= Rc, aRR
= f, aRo
= 0 (21)
Then the fluid flow rate entering the pipette is calculated from the integral:
/ V NdS;
=
~~~~Sp
(22)
s, t
