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micromachines
Article
Dielectrophoresis Testing of Nonlinear Viscoelastic
Behaviors of Human Red Blood Cells
Yuhao Qiang, Jia Liu and E Du *
Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA;
yqiang2015@fau.edu (Y.Q.); jliu2015@fau.edu (J.L.)
*Correspondence: edu@fau.edu; Tel.: +1-561-297-3441
Received: 17 November 2017; Accepted: 8 January 2018; Published: 9 January 2018
Abstract:
Dielectrophoresis in microfluidics provides a useful tool to test biomechanics of living
cells, regardless of surface charges on cell membranes. We have designed an experimental method
to characterize the nonlinear viscoelastic behaviors of single cells using dielectrophoresis in a
microfluidic channel. This method uses radio frequency, low voltage excitations through interdigitated
microelectrodes, allowing probing multiple cells simultaneously with controllable load levels.
Dielectrophoretic force was calibrated using a triaxial ellipsoid model. Using a Kelvin–Voigt model,
the nonlinear shear moduli of cell membranes were determined from the steady-state deformations
of red blood cells in response to a series of electric field strengths. The nonlinear elastic moduli of cell
membranes ranged from 6.05
µ
N/m to up to 20.85
µ
N/m, which were identified as a function of
extension ratio, rather than the lumped-parameter models as reported in the literature. Value of the
characteristic time of the extensional recovery of cell membranes initially deformed to varied extent
was found to be about 0.14 s. Shear viscosity of cell membrane was estimated to be 0.8–2.9 (
µ
N/m)
·
s.
This method is particularly valuable for rapid, non-invasive probing of mechanical properties of
living cells.
Keywords: biomechanics; viscoelasticity; red blood cells; dielectrophoresis; microfluidics
1. Introduction
Mature red blood cells (RBCs) in humans consist of cell membrane and cytoplasm, mainly
hemoglobin. Its unique bi-concave disc shape, membrane viscoelastic properties, and cytoplasmic
viscosity are the primary factors that affect the ability of a RBC to flow through the microvasculature.
Characterization of the mechanical properties of RBC membranes has allowed us to better understand
and accurately model the physiological processes, such as blood rheology [
1
], splenic filtration [
2
], cell
adhesion [
3
], and cell-cell interactions [
4
]. Viscoelasticity of RBC membranes is of particular interest
for the studies of blood diseases [5].
Microfluidics has been recognized as a versatile tool to develop experimental models [
6
] for
characterization of cell biomechanics and rheology, complementary to those classical physics and
engineering techniques [
7
]. Passive shear flow [
3
] and dielectrophoresis (DEP) [
8
] are the two main
mechanisms implemented in microfluidic environment for the biomechanical studies of living cells.
The DEP polarization force [
9
] exerted on a cell is along with the field lines. This effect has been
utilized to study electrodeformation of single cells, such as RBCs [
10
,
11
], mammalian cells [
12
], plant
protoplasts [
13
], and cervical cancer cells [
14
]. Comparing to other experimental methods for cell
biomechanics testing, such as classical optical tweezers [
15
] or micropipette aspirations [
16
], which
are typically complex systems and are limited to measure one cell at a time, DEP can provide much
higher throughput analysis through the interdigitated microelectrodes in microfluidic platform [
11
].
As DEP force is generated by the electrical field in microfluidics, it offers control in both the stress
Micromachines 2018,9, 21; doi:10.3390/mi9010021 www.mdpi.com/journal/micromachines
Micromachines 2018,9, 21 2 of 8
level and loading waveforms [
17
]. Furthermore, using a theory of membrane viscoelasticity developed
by Evans and Hochmuth [
18
], shear elastic modulus and viscosity of RBC membranes can be
determined [
19
–
21
]. Other models, such as three-stage elastic moduli [
15
,
22
] and with an additional
dual-viscosity dash pots [
19
] can be used to describe the nonlinear elastic and viscoelastic behaviors of
RBC membranes. The reported values of membrane shear modulus for healthy RBCs were in a range
of 2.4–13.3 µN/m [23,24].
In a recent article [
17
], we reported an experimental methodology to study the fatigue behavior
of human RBCs in response to a low-cycle DEP fatigue load with a fixed field strength. Cumulative
mechanical degradation in cell membranes was observed in both healthy RBCs and Adenosine
triphosphate (ATP)-depleted counterparts in response to such cyclic DEP load. To further study the
fatigue failure of cell membranes in response to cyclic stress scenarios mimicking the
in vivo
complexity,
it is important to probe the nonlinear elasticity and viscoelastic behavior of RBC membranes using
DEP method. Therefore, it is important to investigate the electrodeformation of RBCs that exhibit
both small and large deformations. In this paper, we present a method to characterize stress-strain
relationship of cell membranes in response to different levels of field strength as well as to extract the
nonlinear shear elastic moduli and viscosity.
2. Materials and Methods
The schematic of the microfluidic device for the biomechanical testing is shown in Figure 1a.
It consists of a Polydimethylsiloxane (PDMS) microfluidic channel and a glass substrate printed with
thin-film interdigitated gold electrodes (20
µ
m gap and 20
µ
m band width). Healthy RBCs were
diluted in an isotonic working buffer [
25
] (8.5% w/vsucrose and 0.3% w/vdextrose with the electrical
conductivity of 0.018 S/m) for testing. Before the testing, the microfluidic channel was primed with
the working buffer containing 5% bovine serum albumin to prevent cell adhesion, then washed with
the working buffer to remove excess serum. All experiments were performed at room temperature
and under stationary condition. As the direction and magnitude of DEP force are both dependent on
the frequency of the electrical signal (Equations (1) and (2)), in current setup, when electrical frequency
is below 100 kHz, cells were observed to be pushed away from the electrodes, which failed to provide
necessary condition to deform cells; as electrical frequency goes beyond 100 kHz to 1.58 MHz, cells
were observed to move towards electrodes and magnitude of DEP force increased; beyond 1.58 MHz,
magnitude of DEP force decreases. Therefore, to generate a favorable positive DEP force to deform
cell membranes in current setup using lowest voltage levels, sinusoid waveform of 1.58 MHz was
selected. Varied voltage levels (0.5–1.0 V
rms
with an interval of 0.1 V
rms
, 1.5 V
rms
, 2.0 V
rms
, 2.5 V
rms
,
3 V
rms
, and 3.5 V
rms
) were supplied to the electrode array to deform cell membranes. The DEP force
field aligned individual RBCs to the field lines and stretched uniaxially (Figure 1a inset and Figure 1b).
Each excitation was maintained for 10 s, allowing all cells reach to a steady-state equilibrium state. Cell
extension and relaxation process was recorded by a charge-coupled device (CCD) camera attached to
an inverted Olympus IX81 microscope (Olympus, Tokyo, Japan) with bright-field imaging. A bandpass
filter 414 ±46 nm was inserted in the light path to improve the image contrast of RBCs.
RBCs moved toward the higher electric field strength due to the net DEP force from the two
electrical force components, F
1
and F
2
exerted on the induced dipole on cell membranes (Figure 1a
inset). This net DEP force, F
DEP
, has been the primary mechanism widely used for cell trapping and
sorting. As cells approach to the edges of electrodes with highest field strength, reaction force from the
electrode balances with the DEP force where cells are in transient equilibrium. In this case, RBCs are
deformed uniaxially due to the force equilibrium from three components, the reaction force and the
electrical force, F
2
in negative x-direction, as well as the electrical force, F
1
in the positive x-direction.
Assuming a stretched RBC as an ellipsoid, the time-averaged DEP force exerted on cell membranes
Micromachines 2018,9, 21 3 of 8
depends on the relative polarizability of the cell and surrounding medium, field strength, cell shape
and size and can be determined from [9],
hFDEPi=2πabc ·εm·Re(fC M )· ∇E2
rms (1)
where a,b, and care the semi-principal axes of the ellipsoid,
εm
is the permittivity of the surrounding
medium, and
∇E2
rms
is the root-mean-square value of the gradient of electric field strength square.
The value of the real part of the Clausius–Mossotti factor,
fCM
is determined from the complex
permittivity and size of a RBC and its surrounding medium, with a simplified single-shell ellipsoid
model [26] according to the concentric multi-shell theory [27],
fCM =1
3
(ε∗
mem−ε∗
m)[ε∗
mem+A1(ε∗
cyto−ε∗
mem)]+ρ(ε∗
cyto−ε∗
mem)[ε∗
mem−A1(ε∗
mem−ε∗
m)]
(ε∗
m+A1(ε∗
mem−ε∗
m))[ε∗
mem+A1(ε∗
cyto−ε∗
mem)]+ρA2(1−A2)(ε∗
cyto−ε∗
mem)(ε∗
mem−ε∗
m)(2)
where the subscripts cyto, mem and m stand for cytoplasm, membrane and medium, respectively.
ε∗=ε−jσ/ω
with
ω
as the angular frequency,
ε
and
σ
as the dielectric permittivity and conductivity,
respectively.
ρ=a0b0c0
abc
, with
a0=a−t
,
b0=b−t
,
c0=c−t
,tas the thickness of cell membrane,
4.5 nm, c= 1.3
µ
m.
εmem =
4.44,
εcyto =
59,
σmem =
10
−6S/m
,
σcyto =
0.31
S/m
. The depolarization
factor along the extensional direction, Ai=1,2 are defined by
A1=1
2abc Z∞
0
ds
(s+a2)B(3)
A2=1
2a0b0c0Z∞
0
ds
s+a02B0(4)
where
B=p(s+a2)(s+b2)(s+c2)
,
B0=rs+a02s+b02s+c02
and sis an arbitrary
distance for integration.
Micromachines 2018, 9, 21 3 of 8
where a, b, and c are the semi-principal axes of the ellipsoid, is the permittivity of the
surrounding medium, and ∇
is the root-mean-square value of the gradient of electric field
strength square. The value of the real part of the Clausius–Mossotti factor, is determined from
the complex permittivity and size of a RBC and its surrounding medium, with a simplified single-
shell ellipsoid model [26] according to the concentric multi-shell theory [27],
=
(
∗
∗)
∗(
∗
∗)(
∗
∗)
∗(
∗
∗)
(
∗(
∗
∗))
∗(
∗
∗) ( )(
∗
∗)(
∗
∗) (2)
where the subscripts cyto, mem and m stand for cytoplasm, membrane and medium, respectively.
∗=−/ with as the angular frequency, ε and σ as the dielectric permittivity and
conductivity, respectively. =
, with =−,=−,=−, t as the thickness of cell
membrane, 4.5 nm, c = 1.3 µm. =4.44, =59, =10
S/m, =0.31S/m. The
depolarization factor along the extensional direction, , are defined by
=
()
(3)
=
()
(4)
where =(+)(+)(+),=( +)(+)( +) and s is an arbitrary
distance for integration.
Figure 1. Biomechanics testing of live cells using dielectrophoresis (DEP) in microfluidics. (a)
Microfluidic device with inset of microscopic view of cellular deformation. Dark strips represent
interdigitated electrodes. (b) A triaxial ellipsoid model for description of positive DEP-induced
uniaxial cell deformation. (c) Surface plot of the gradient of field strength square, ∇
in the
microfluidic device, from simulation by COMSOL Multiphysics (COMSOL, Inc., Burlington, MA,
USA).
RBC exhibits viscoelastic behavior in response to an external force. The shear stress–strain
relationship of cell membranes was considered to be linear for small deformations and nonlinear for
large deformations [28]. The membrane shear modulus is relevant to both shear strain and strain rate,
especially for large deformations [15,29]. When the external load is removed, the deformed cell
recovers to its original shape in a time-dependent manner, due to the elastic energy storage and
viscous dissipation in the membrane. A classical model to explain the viscoelastic behavior of the cell
membranes was developed as an analogue of the classical Kelvin–Voigt solid model [30],
=
()+
(5)
Figure 1.
Biomechanics testing of live cells using dielectrophoresis (DEP) in microfluidics. (
a
) Microfluidic
device with inset of microscopic view of cellular deformation. Dark strips represent interdigitated
electrodes. (
b
) A triaxial ellipsoid model for description of positive DEP-induced uniaxial cell
deformation. (
c
) Surface plot of the gradient of field strength square,
∇E2
rms
in the microfluidic device,
from simulation by COMSOL Multiphysics (COMSOL, Inc., Burlington, MA, USA).
RBC exhibits viscoelastic behavior in response to an external force. The shear stress–strain
relationship of cell membranes was considered to be linear for small deformations and nonlinear for
Micromachines 2018,9, 21 4 of 8
large deformations [
28
]. The membrane shear modulus is relevant to both shear strain and strain
rate, especially for large deformations [
15
,
29
]. When the external load is removed, the deformed cell
recovers to its original shape in a time-dependent manner, due to the elastic energy storage and
viscous dissipation in the membrane. A classical model to explain the viscoelastic behavior of the cell
membranes was developed as an analogue of the classical Kelvin–Voigt solid model [30],
Ts=µ
2(λ2−λ−2)+2η
λ
∂λ
∂t(5)
where
Ts
is the shear stress,
λ
is the stretch ratio,
µ
and
η
are membrane shear elastic modulus and
viscosity. In this study, the value of Tsis calculated by F/2b, where Fis assumed as FDEP/2 [12,31].
3. Results
Cell membrane deformation was extracted by a custom, MATLAB-based (The MathWorks, Natick,
MA, USA) imaging processing program. Two parameters were utilized to quantify the membrane
deformation, including extensional index, EI%= (a
−
b)/(a+b) and extension ratio,
λ
=b/b
t= 0
.
The values of EI measured at the stationary state for RBCs (n= 84) increased with the strength of the
applied electric field (Figure 2). The average values of EI were 5.6%
±
3.9% to 17.5%
±
7.1%, 29.6%
±
8.9%, 39.5% ±8.4%, 46.4% ±7.09%, 52.4% ±5.6%, respectively.
Micromachines 2018, 9, 21 4 of 8
where is the shear stress, is the stretch ratio, and are membrane shear elastic modulus and
viscosity. In this study, the value of is calculated by F/2b, where F is assumed as FDEP/2 [12,31].
3. Results
Cell membrane deformation was extracted by a custom, MATLAB-based (The MathWorks,
Natick, MA, USA) imaging processing program. Two parameters were utilized to quantify the
membrane deformation, including extensional index, EI%= (a − b)/(a + b) and extension ratio, λ = b/b t = 0.
The values of EI measured at the stationary state for RBCs (n = 84) increased with the strength of the
applied electric field (Figure 2). The average values of EI were 5.6% ± 3.9% to 17.5% ± 7.1%, 29.6% ±
8.9%, 39.5% ± 8.4%, 46.4% ± 7.09%, 52.4% ± 5.6%, respectively.
Figure 2. Electrodeformation EI values of red blood cells (RBCs) at a series of voltage levels with insets
of representative deformed cells.
The magnitude of DEP force was determined using Equation (1), where a, b and c are the semi-
principal axes of the ellipsoid (Figure 1b), ∇
is the gradient of the root-mean-square (rms) value
of the field strength square (Figure 1c). We tracked 84 cells of their equilibrium deformations
individually at each field strength in order to determine the DEP forces exerted on the individual
cells. As a cell deforms in response to electric excitation, the value of varies with time and the
electrical frequency according to Equation (2). A representative value of () for a deformed RBC
is shown in Figure 3. The maximum value of () was found to be around the selected electrical
frequency, 1.58 MHz. Values of extension ratio, λ as a function of time for individual RBCs were
characterized experimentally. Mean values of the DEP force magnitude and the corresponding
extension ratio, λ were plotted as functions of applied voltages (Figure 4). Then membrane shear
modulus, µ was determined from the relationship between the maximum stretch ratio and the
corresponding shear stress using Equations (1) and (5), when
approaches to zero.
Figure 2.
Electrodeformation EI values of red blood cells (RBCs) at a series of voltage levels with insets
of representative deformed cells.
The magnitude of DEP force was determined using Equation (1), where a,band care the
semi-principal axes of the ellipsoid (Figure 1b),
∇E2
rms
is the gradient of the root-mean-square (rms)
value of the field strength square (Figure 1c). We tracked 84 cells of their equilibrium deformations
individually at each field strength in order to determine the DEP forces exerted on the individual cells.
As a cell deforms in response to electric excitation, the value of
fCM
varies with time and the electrical
frequency according to Equation (2). A representative value of
Re(fC M )
for a deformed RBC is shown
in Figure 3. The maximum value of
Re(fC M )
was found to be around the selected electrical frequency,
Micromachines 2018,9, 21 5 of 8
1.58 MHz. Values of extension ratio,
λ
as a function of time for individual RBCs were characterized
experimentally. Mean values of the DEP force magnitude and the corresponding extension ratio,
λ
were plotted as functions of applied voltages (Figure 4). Then membrane shear modulus,
µ
was
determined from the relationship between the maximum stretch ratio and the corresponding shear
stress using Equations (1) and (5), when ∂λ
∂tapproaches to zero.
Figure 3.
A representative Clausius-Mossotti (CM) factor of a deformed RBC based on the triaxial
ellipsoid multi-shell model. The semi-principal axes of the cell are a= 5
µ
m, b= 1.5
µ
m, and c= 1.3
µ
m.
Nonlinearity in membrane shear moduli was observed, which can be expressed as a combination
of two separate functions (Figure 5). For small deformations
λ
< 1.4, membrane shear modulus,
µ
is
approximately a constant, 6.05
±
0.33
µ
N/m. For large deformation
λ
> 1.4. The value of
µ
can be
well fitted with an exponential function with respect to the extension ratio,
µ=
0.28
×e1.98λ+
1.96.
When
λ
approaches to 2.1, value of
µ
was found to be 20.85
µ
N/m. These results are in the range
of reported literature values, obtained by other independent studies. Membrane shear modulus
obtained by other DEP study [
32
] was in the lower range of 1.4–2.5
µ
N/m. The nonlinear shear
moduli of RBC membranes determined from a sophisticated 3D finite element cell model and optical
tweezers experiments [
15
] were 2.4–5.0
µ
N/m for small deformations, and 5.3–11.3
µ
N/m for large
deformations, and 13.9–29.6
µ
N/m prior to failure. The discrepancy between our measurements and
other studies may be attributed to several factors, such as the constitutive models used, variations in
samples, experimental conditions, optical resolution of the measuring systems, and force calibration
among various studies.
Micromachines 2018, 9, 21 5 of 8
Figure 3. A representative Clausius-Mossotti (CM) factor of a deformed RBC based on the triaxial
ellipsoid multi-shell model. The semi-principal axes of the cell are a = 5 µm, b = 1.5 µm, and c = 1.3
µm.
Nonlinearity in membrane shear moduli was observed, which can be expressed as a combination
of two separate functions (Figure 5). For small deformations λ < 1.4, membrane shear modulus, µ is
approximately a constant, 6.05 ± 0.33 µN/m. For large deformation λ > 1.4. The value of µ can be well
fitted with an exponential function with respect to the extension ratio, = 0.28×. +1.96.
When λ approaches to 2.1, value of µ was found to be 20.85 µN/m. These results are in the range of
reported literature values, obtained by other independent studies. Membrane shear modulus
obtained by other DEP study [32] was in the lower range of 1.4–2.5 µN/m. The nonlinear shear moduli
of RBC membranes determined from a sophisticated 3D finite element cell model and optical
tweezers experiments [15] were 2.4–5.0 µN/m for small deformations, and 5.3–11.3 µN/m for large
deformations, and 13.9–29.6 µN/m prior to failure. The discrepancy between our measurements and
other studies may be attributed to several factors, such as the constitutive models used, variations in
samples, experimental conditions, optical resolution of the measuring systems, and force calibration
among various studies.
Figure 4. Electrodeformation of RBCs (n = 84) with estimated DEP force magnitude and the
corresponding extension ratio, λ as functions of applied voltage levels. Error bar indicates standard
deviations.
Upon a sudden release of DEP load, stretched RBCs recovered to their stress-free state at a time-
dependent rate. Membrane viscosity, η was determined from the extensional recovery when
was deactivated (TS = 0), based on Equation (5) following the normalization method developed by
Hochmuth [30]. Figure 6 compares the tc values of cell membranes determined from RBCs initially
stretched at different levels of field strength. Value of tc was 0.14 ± 0.05 s, which compares well with
the value 0.19 s measured using optical tweezers method [33]. From tc ≡ η/µ, shear viscosity of cell
membrane can be determined, which is 0.8–2.9 (µN/m)·s, which are consistent with the reported
standard values obtained by micropipette aspiration, 0.6–2.7 (µN/m)·s [24] and by optical tweezers,
0.3–2.8 (µN/m)·s [15].
Figure 4.
Electrodeformation of RBCs (n= 84) with estimated DEP force magnitude and the
corresponding extension ratio,
λ
as functions of applied voltage levels. Error bar indicates
standard deviations.
Micromachines 2018,9, 21 6 of 8
Upon a sudden release of DEP load, stretched RBCs recovered to their stress-free state at a
time-dependent rate. Membrane viscosity,
η
was determined from the extensional recovery when
FDEP
was deactivated (T
S
= 0), based on Equation (5) following the normalization method developed
by Hochmuth [
30
]. Figure 6compares the t
c
values of cell membranes determined from RBCs initially
stretched at different levels of field strength. Value of t
c
was 0.14
±
0.05 s, which compares well with
the value 0.19 s measured using optical tweezers method [
33
]. From t
c≡η
/
µ
, shear viscosity of cell
membrane can be determined, which is 0.8–2.9 (
µ
N/m)
·
s, which are consistent with the reported
standard values obtained by micropipette aspiration, 0.6–2.7 (
µ
N/m)
·
s [
24
] and by optical tweezers,
0.3–2.8 (µN/m)·s [15].
Micromachines 2018, 9, 21 6 of 8
Figure 5. Mean values of membrane shear modulus and best fit functions for nonlinear elastic moduli:
for small deformation (λ < 1.4), membrane shear modulus, µ = 6.05 µN/m; for large deformation
(λ > 1.4), = 0.28×.+1.96.
Figure 6. Best fit to the extensional recovery of RBC membranes (n = 15). The averaged characteristic
time was found to be tc = 0.14 ± 0.05 s for the three voltage levels.
4. Discussion
Comparing to other DEP studies, discrepancy may mainly arise from assumptions of the shape
of deformed cells. Spherical model (a = b = c) has been widely used to estimate the force exerted on
bioparticles. Considering many bioparticles are highly nonspherical, such as DNA strands and E. coli,
ellipsoid models, such as prolate and oblate, have been developed which can provide an improved
accuracy for force calibration [34]. In the case of RBC deformation, spherical model is no long valid;
the prolate assumption may be valid for large deformations but may overestimate the DEP force for
small deformations. Considering the magnitude of DEP force is proportional to the volume of the
deformed cell and usually b value is higher than c in uniaxially stretched RBCs, an ellipsoid (≠≠
) model can provide an improved accuracy for fore calibration, comparing to the spherical and
prolate models. On the other hand, tip formation was noticed in RBCs, which exhibited large
deformations at high electric field strengths (Figure 2). For the same reason, even though the triaxial
ellipsoid model may be very close to an uniaxially deformed RBC, it can still overestimate the DEP
force to some extent. Consequently, the characterized shear modulus for large deformations may be
slightly higher than the true values. A numerical solution using finite element model shall be used to
account for such variation, as pointed out by an earlier study [20].
5. Conclusions
In summary, we performed a systematic electrodeformation study of human RBCs and
characterized the nonlinear elastic moduli using the relationship between shear stress and membrane
deformation of cell membranes stretched by different electric strengths. The extensional recovery
characteristic time of cell membranes was found to be a constant from the extensional recovery
processes of RBCs that were initially deformed to different levels. We envision that the general
Figure 5.
Mean values of membrane shear modulus and best fit functions for nonlinear elastic moduli:
for small deformation (
λ
< 1.4), membrane shear modulus,
µ
= 6.05
µ
N/m; for large deformation
(λ> 1.4), µ=0.28 ×e1.98λ+1.96
Micromachines 2018, 9, 21 6 of 8
Figure 5. Mean values of membrane shear modulus and best fit functions for nonlinear elastic moduli:
for small deformation (λ < 1.4), membrane shear modulus, µ = 6.05 µN/m; for large deformation
(λ > 1.4), = 0.28×.+1.96.
Figure 6. Best fit to the extensional recovery of RBC membranes (n = 15). The averaged characteristic
time was found to be tc = 0.14 ± 0.05 s for the three voltage levels.
4. Discussion
Comparing to other DEP studies, discrepancy may mainly arise from assumptions of the shape
of deformed cells. Spherical model (a = b = c) has been widely used to estimate the force exerted on
bioparticles. Considering many bioparticles are highly nonspherical, such as DNA strands and E. coli,
ellipsoid models, such as prolate and oblate, have been developed which can provide an improved
accuracy for force calibration [34]. In the case of RBC deformation, spherical model is no long valid;
the prolate assumption may be valid for large deformations but may overestimate the DEP force for
small deformations. Considering the magnitude of DEP force is proportional to the volume of the
deformed cell and usually b value is higher than c in uniaxially stretched RBCs, an ellipsoid (≠≠
) model can provide an improved accuracy for fore calibration, comparing to the spherical and
prolate models. On the other hand, tip formation was noticed in RBCs, which exhibited large
deformations at high electric field strengths (Figure 2). For the same reason, even though the triaxial
ellipsoid model may be very close to an uniaxially deformed RBC, it can still overestimate the DEP
force to some extent. Consequently, the characterized shear modulus for large deformations may be
slightly higher than the true values. A numerical solution using finite element model shall be used to
account for such variation, as pointed out by an earlier study [20].
5. Conclusions
In summary, we performed a systematic electrodeformation study of human RBCs and
characterized the nonlinear elastic moduli using the relationship between shear stress and membrane
deformation of cell membranes stretched by different electric strengths. The extensional recovery
characteristic time of cell membranes was found to be a constant from the extensional recovery
processes of RBCs that were initially deformed to different levels. We envision that the general
Figure 6.
Best fit to the extensional recovery of RBC membranes (n= 15). The averaged characteristic
time was found to be tc= 0.14 ±0.05 s for the three voltage levels.
4. Discussion
Comparing to other DEP studies, discrepancy may mainly arise from assumptions of the shape
of deformed cells. Spherical model (a=b=c) has been widely used to estimate the force exerted on
bioparticles. Considering many bioparticles are highly nonspherical, such as DNA strands and E. coli,
ellipsoid models, such as prolate and oblate, have been developed which can provide an improved
accuracy for force calibration [
34
]. In the case of RBC deformation, spherical model is no long valid; the
prolate assumption may be valid for large deformations but may overestimate the DEP force for small
deformations. Considering the magnitude of DEP force is proportional to the volume of the deformed
cell and usually b value is higher than c in uniaxially stretched RBCs, an ellipsoid (
a6=b6=c)
model
can provide an improved accuracy for fore calibration, comparing to the spherical and prolate models.
On the other hand, tip formation was noticed in RBCs, which exhibited large deformations at high
Micromachines 2018,9, 21 7 of 8
electric field strengths (Figure 2). For the same reason, even though the triaxial ellipsoid model may
be very close to an uniaxially deformed RBC, it can still overestimate the DEP force to some extent.
Consequently, the characterized shear modulus for large deformations may be slightly higher than
the true values. A numerical solution using finite element model shall be used to account for such
variation, as pointed out by an earlier study [20].
5. Conclusions
In summary, we performed a systematic electrodeformation study of human RBCs and
characterized the nonlinear elastic moduli using the relationship between shear stress and membrane
deformation of cell membranes stretched by different electric strengths. The extensional recovery
characteristic time of cell membranes was found to be a constant from the extensional recovery
processes of RBCs that were initially deformed to different levels. We envision that the general
ellipsoid model and the microfluidic DEP platform can benefit biomechanical studies of other cell
types and diseased cells.
Acknowledgments:
This material is based upon work supported by the National Science Foundation under
Grant No. 1635312 and No. 1464102. E Du acknowledges for Florida Atlantic University faculty startup grant.
Author Contributions:
Yuhao Qiang, Jia Liu and E Du conceived and designed the experiments; Yuhao Qiang
and Jia Liu performed the experiments; Yuhao Qiang and E Du analyzed the data; Yuhao Qiang and E Du wrote
the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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