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Analysis of the equivalent dipole moment of red blood cell by using the boundary element method
Engineering Analysis with Boundary Elements 112 (2020) 68–76
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Analysis of the equivalent dipole moment of red blood cell by using the boundary element method Nitipong Panklang a, Boonchai Techaumnat a,b,∗, Anurat Wisitsoraat c a
Faculty of Engineering, Department of Electrical Engineering, Chulalongkorn University, 254 Phyathai road, Pathumwan, Bangkok 10330, Thailand Biomedical Engineering Research Center, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand c Nanoelectronics and MEMS Laboratory, National Electronics and Computer Technology Center, Pathumthani 12120, Thailand b
a r t i c l e
i n f o
Keyword: Red blood cell Boundary element method Equivalent dipole moment Membrane capacitance Intracellular conductivity
a b s t r a c t Equivalent dipole moment of biological cells under electric field is an important parameter for various applications such as analysis and manipulation of cells in biomedical samples. The dipole moment depends on cell geometries as well as electrical parameters of media involved. Unfortunately, the analytical expression of equivalent dipole is available only for simple geometries. This work numerically studies the variation of the dipole moment of a red blood cell with cell geometries and electrical parameters. The cell is modelled as a sphere, an oblate spheroid or a biconcave disc. The authors apply the boundary element method to electric field calculation and use reexpansion formulae to compute the equivalent dipole moment of the cell. The numerical results agree well with the analytical one for the spherical model. The effects of cell geometries are clarified for two directions of the electric field, which are parallel or normal to the axis of symmetry of the cell. Using the biconcave disc model, we perform iterative calculation to estimate the intracellular conductivity and specific membrane capacitance of red blood cells from experimental results.
1. Introduction Dynamics of particles under electric field are used in various applications. Electrostatic precipitator, electrostatic painting and coating utilize the Coulomb force exerted on charged particles to control their movement or trajectories. Uncharged particles also exhibit electromechanical responses. A suspension of dielectric particles can be employed as an electrorheological fluid for applications such as active valves or suspensions [1]. There are also a variety of biomedical applications based on the electrical force on biological cells. The dielectrophoretic (DEP) force [2], which is the force acts on an uncharged object due to polarization difference between the object and its surrounding medium, is widely used in microfluidic platforms for immobilization, isolation, and analysis of biological cells [3–5]. For uncharged particles or biological cells under electric field E, electromechanical characteristics of cells depend primarily on the induced charges on the particles or cells. The simplest approach for estimating the contributions of the induced charges is to represent the charges with an equivalent dipole moment p. The DEP force FDEP resulting from the interaction between the dipole moment and the electric field can be expressed as 𝐅𝐷𝐸𝑃 = (𝐩 ⋅ ∇)𝐄.
(1)
It is clear from the equation that the DEP force exists where the electric field is nonuniform, and the force direction depends on the dipole moment p and the spatial variation of E. Positive and negative dielectrophoresis (p-DEP and n-DEP) are defined for the cases in which the force acts toward the region of stronger and weaker electric field, respectively. Under a harmonic electric field Ecos(𝜔t) of frequency f = 𝜔/(2𝜋), the equivalent dipole moment is also in a form of temporal harmonic pcos(𝜔t + 𝜑) where 𝜑 is the phase angle. The phase difference between the field and the dipole moment may exist according to the conductivity and the permittivity of media involved. It is convenient to write the field and the dipole moment in the phasor domain as 𝐄̇ and 𝐩̇ where the dot symbol denotes phasor variables. Taking the phase of 𝐄̇ as the reference (i.e., 𝐄̇ = 𝐄), the time averageof the DEP force is determined from the real part of the 𝐩̇ [6]. 1( [ ] ) ⟨𝐅𝐷𝐸𝑃 ⟩ = (2) Re 𝐩̇ ⋅ ∇ 𝐄. 2 For a rotating electric field, the phase difference between the electric field and the dipole moment gives rise to the DEP torque TDEP . The time average DEP torque ⟨TDEP ⟩ is related to the imaginary part of 𝐩̇ as [6] ⟨𝐓𝐷𝐸𝑃 ⟩ =
1 [ ] Im 𝐩̇ × 𝐄. 2
(3)
∗ Corresponding author at: Faculty of Engineering, Department of Electrical Engineering, Chulalongkorn University, 254 Phyathai road, Pathumwan, Bangkok 10330, Thailand. E-mail address: [email protected] (B. Techaumnat).
https://doi.org/10.1016/j.enganabound.2019.12.002 Received 6 August 2019; Received in revised form 8 December 2019; Accepted 8 December 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
N. Panklang, B. Techaumnat and A. Wisitsoraat
Engineering Analysis with Boundary Elements 112 (2020) 68–76
Table 1 Dimension of spherical and spheroidal cells for calculation.
It is obvious from Eqs. (2) and (3) that the equivalent dipole moment is a critical parameter governing the electromechanical characteristics of cells or particles. The dipole moment depends on the electrical properties of the intracellular and extracellular media. The analytical expression of 𝐩̇ is available for cases of simple geometries such as spherical and spheroidal profiles. For complicated profiles, numerical methods are inevitably needed for the determination of 𝐩̇ . Numerical calculation of effective moments was presented for particles where the moments result from single layer charge on particle surface [7]. However, the influence of the double layer charge must be also taken in account for biological cells. In this work, the authors apply the boundary element method (BEM) to calculate electric field for configurations of a red blood cell in the 3D condition. There exist a number of works on the analysis of the dielectrophoresis by using numerical methods. For example, the BEM was used to analyses dielectrophoresis of particles on which typical boundary conditions of continuous potential and normal flux density hold [8,9]. For biological cells, the method was used to simulate the transmembrane voltage of a cell under dc electric field [10,11]. In this article, we determine the solution of electric field in the ac steady state by using the BEM. To our knowledge, the application of BEM to the 3D electric field analysis for non-spherical cells has not been presented yet in existing works. An advantage of the BEM is that the relationship between the transmembrane potential and electric field can be applied to the calculation without the reduction of accuracy because the normal electric field on boundaries are a primary variable for the BEM. From the field calculation results, we determine the equivalent dipole moment of the cell by using multipole re-expansion formulae. The dipole moment of non-spherical cells have been obtained analytically only for limited cases such as prolate or oblate spheroidal cells [6]. The aim of this work is to clarify the effects of cell geometries and electrical parameters on the equivalent dipole moment by using the quantitative investigation. In addition, we also demonstrate the determination of electrical parameters of red blood cell based on results obtained from DEP experiments. As red blood cells are a biomedical sample that is extensively used for various disease detection, the determination of cell parameters can be an important tool for diagnosis.
√ ( )( ) 0.862 1 − 𝜉 2 𝑐0 + 𝑐1 𝜉 2 + 𝑐2 𝜉 4 ,
b (μm)
Sphere Oblate spheroid A Oblate spheroid B
2.796 4.2 7.172
2.796 1.239 0.425
3. Calculation method 3.1. Electric field calculation The boundary element method (BEM) [15] is used to calculate electric field in configurations of an isolated cell under a uniform ac electric field 𝐄̇ 0 . Although domain-dividing methods such as the finite difference or the finite element method are also applicable to the calculation, very fine meshes are required to yield accurate result as the thickness of cell membrane is significantly smaller than the cell size. One of the authors has also applied the BEM to the calculation of transmembrane potential of cell pairs under dc energization [11]. The use of the BEM eliminates the need for domain subdivision and is convenient for the treatment of cell membrane. For the BEM, the cell membrane is modeled by using zero-thickness elements of which electric potential can be represented independently on both sides. Let 𝜙̇ 𝐼 and 𝜙̇ 𝐸 denote the potential in the interior and that in the exterior of the cell, respectively. The normal component of electric field on the cell membrane is denoted as 𝐸̇ 𝑛𝐼 in the interior side and 𝐸̇ 𝑛𝐸 in the exterior side. The potential at point r is related to the integrals of the potential and the normal component of the electric field on cell surface Scell as
The configurations of analysis are a single red blood cell under a uniform ac electric field 𝐄̇ 0 . We consider the externally applied electric field 𝐄̇ 0 in two directions. The electric field is either parallel or perpendicular to the symmetrical axis of the cell. For the latter case, the potential distribution is three-dimensional although the cell geometries are axisymmetric ones. Based on the results from those field directions, a solution for an arbitrary field direction can be deduced. The cell is modeled as a sphere, an oblate spheroid, or a biconcave disc (the normal shape of mature red blood cells). The spherical model is used to verify the numerical calculation of equivalent dipole moment with the analytical ones and also used to investigate the effects of the cell profile on the dipole moment. For the spheroidal model, a and b denote the semi-axial length normal and parallel to the axis of rotating symmetry (z axis), respectively. The contour of the biconcave disc is given (in μm) by the following equations [12].
𝑧(𝜉) = ±4.2
a (μm)
spheroid A has the length a = 4.2 μm, which is equal to that of the biconcave disc on the z = 0 plane. Oblate spheroid B has the semi-axial length b equal to 0.425 μm, making the same thickness on the symmetrical axis as that of the biconcave disc model. Dimensions of the models are summarized in Table 1. Fig. 1 presents all models approximately on the same scale. Unless specified otherwise, the following electrical parameters are used for the red blood cell in the calculation [13]. Intracellular conductivity 𝜎 I = 0.3 S/m and permittivity 𝜀I = 59𝜀0 where 𝜀0 is the permittivity of free space. The specific capacitance of cell membrane Cm = 8.7 mF/m2 based on 4.5 nm membrane thickness, and the membrane conductance is neglected in typical cases. Specific membrane conductance Gm values between 5 and 50 kS/m2 are applied to the calculation in some cases for studying the variation of dipole moment due to membrane modification which may be, for example, a result from an invasion of parasites [14].
2. Configuration
𝜌(𝜉) = 4.2𝜉,
Cell geometry
𝛼 𝐼 (𝐫 )𝜙̇ 𝐼 (𝐫 ) =
∫𝑆𝑐𝑒𝑙𝑙
𝐸̇ 𝑛𝐼 (𝐱)𝑤(𝐫, 𝐱)d𝑠+
∫𝑆𝑐𝑒𝑙𝑙
𝜕 𝜙̇ 𝐼 (𝐱) 𝑤(𝐫, 𝐱)d𝑠 𝜕𝑛
(6)
for the interior of the cell, and 𝛼 𝐸 (𝐫 )𝜙̇ 𝐸 (𝐫 ) = −
∫𝑆𝑐𝑒𝑙𝑙
𝐸̇ 𝑛𝐸 (𝐱)𝑤(𝐫, 𝐱)d𝑠−
∫𝑆𝑐𝑒𝑙𝑙
𝜕 𝜙̇ 𝐸 (𝐱) 𝑤(𝐫, 𝐱)d𝑠 − 𝐄̇ 0 ⋅ 𝐫 𝜕𝑛 (7)
for the exterior of the cell. In the above equations, the potential reference is taken at the origin. x is the point on Scell , w is the fundamental solution of the potential problem, and n denotes the unit normal at x from the interior to the exterior of the cell. The 𝛼 I and 𝛼 E values depend on the surface geometry at x. For smooth surface, 𝛼 I = 𝛼 E = 1/2. The fundamental solution for 3D configurations is expressed as
(4)
(5)
𝑤(𝐫, 𝐱) =
where c0 = 0.01384083, c1 = 0.2842917, c2 = 0.01306932, and the parameter 𝜉 varies from 0 to 1. The dimensions of the spherical and spheroidal models are determined so as to have the same volume as the biconcave disc. From the contour given by Eqs. (4) and (5), the cell volume Vcell is equal to 91.6 fL. For oblate spheroidal models, two sets of a and b values are used. Oblate
1 . 4𝜋‖𝐫 − 𝐱‖
(8)
It should be noted here that transmembrane potential arises if charges accumulate on both sides of the cell membrane, i.e., double layer charge. Hence, the potential are denoted as 𝜙̇ 𝐼 (𝐱) and 𝜙̇ 𝐸 (𝐱) in the surface integral equations. The relationship between the transmembrane potential 69
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. 1. Models of a red blood cell: (a) sphere, (b) Oblate spheroid A, (c) Oblate spheroid B, and (d) biconcave disc.
𝜙̇ 𝐼 (𝐱) − 𝜙̇ 𝐸 (𝐱) and the normal component of the electric field can be written as ( )( ) ( ) ( ) (9) 𝐺𝑚 + 𝑗𝜔𝐶𝑚 𝜙̇ 𝐼 − 𝜙̇ 𝐸 = 𝜎𝐼 + 𝑗𝜔𝜀𝐼 𝐸̇ 𝑛𝐼 = 𝜎𝐸 + 𝑗𝜔𝜀𝐸 𝐸̇ 𝑛𝐸 ,
duced dipole moment. 𝐩̇ 𝑖𝑛𝑑 =
∫𝑆𝑐𝑒𝑙𝑙
𝜌̇ 𝑑 𝐚𝑛 d𝑠,
(12)
4. Results and discussion 4.1. Equivalent dipole moment of spherical cell For the spherical model, we can determine 𝐩̇ 𝑖𝑛𝑑 analytically [18]. Define complex permittivity 𝜀̇ for electrical frequency f as 𝜎 𝜀̇ = 𝜀 − 𝑗 , (13) 2𝜋𝑓
3.2. Equivalent dipole calculation
and complex specific membrane capacitance 𝐶̇ 𝑚 as
For the configurations under consideration, the equivalent induced dipole is parallel with the direction of the applied field 𝐄̇ 0 . The dipole moment 𝐩̇ 𝑖𝑛𝑑 can be determined from the discontinuities of the electric field and potential across the cell membrane. On the cell membrane, single layer charge contributes to the discontinuity of the normal component 𝐸̇ 𝑛 of electric field. We determine the single layer charge density 𝜌̇ 𝑠 by using the following equation [16].
𝐶̇ 𝑚 = 𝐶𝑚 − 𝑗
𝐺𝑚 , 2𝜋𝑓
(14)
where 𝜀 is the permittivity and 𝜎 is the conductivity. The analytical expression of the equivalent complex permittivity 𝜀̇ 𝑠𝑝ℎ for a spherical cell of radius R and complex intracellular permittivity 𝜀̇ 𝐼 is [19,20] 𝜀̇ 𝑠𝑝ℎ =
(10)
𝑅𝐶̇ 𝑚 𝜀̇ 𝐼
𝑅𝐶̇ 𝑚 + 𝜀̇ 𝐼
.
(15)
From the complex permittivity of the cell and that of the extracellular medium, we can determine the equivalent dipole moment
On the other hand, the double layer charge (dipole) density 𝜌̇ 𝑑 is determined from the potential discontinuity, i.e., the transmembrane potential. The direction of dipole is normal to the cell membrane, and the magnitude follows [16] 𝜌̇ 𝑑 = 𝜙̇ 𝐸 − 𝜙̇ 𝐼 . 𝜀𝐸
𝜌̇ 𝑠 (𝐱 − 𝐜)d𝑠 +
where an is the outward unit normal vector at x. Note that it is also possible to determine the higher-order multipole magnitudes using appropriate re-expansion formulae. However, for characterizing a single cell, it is usually adequate to consider only the equivalent dipole representation.
where Gm is the specific conductance, Cm is the specific capacitance of the membrane, 𝜎 E is the extracellular conductivity, 𝜀E is the extracellular permittivity, and j is the imaginary unit. A linear equation system is constructed by applying Eqs. (6) and (7) to the nodes on the interior side and to those on the exterior side of the cell. The linear system is then solved to satisfy the boundary conditions in Eq. (9). We simulate all the boundaries with second-order curved triangular or rectangular elements. The shape, electric potential and electric field on the elements are interpolated by the same secondorder functions. The computation is done by an in-house Matlab program.
𝜌̇ 𝑠 = 𝐸̇ 𝑛𝐸 − 𝐸̇ 𝑛𝐼 . 𝜀𝐸
∫𝑆𝑐𝑒𝑙𝑙
𝐩̇ 𝑖𝑛𝑑 = 4𝜋𝜀𝐸 𝑅3 𝐾̇ 𝐄̇ ,
(16)
where 𝐾̇ is the Clausius-Mossotti factor, a complex-number constant depending on 𝜀̇ 𝑠𝑝ℎ and complex permittivity 𝜀̇ 𝐸 of the extracellular medium. 𝜀̇ 𝑠𝑝ℎ − 𝜀̇ 𝐸 𝐾̇ = . (17) 𝜀̇ 𝑠𝑝ℎ + 2𝜀̇ 𝐸
(11)
Utilizing the multipole re-expansion [17] of the single and double layer charges on the membrane to the center c of the cell, we obtain the in70
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
4.2. Effects of cell profiles on the equivalent dipole moment This section compares the equivalent dipole moments obtained by using different cell profiles. The two oblate spheroidal models represent the cases where one of the semi-axial length is equal to that of the biconcave disc model. Although it is possible to obtain analytical expression of the induced dipole for a spheroidal model under an external electric field, the analytical solution is based on a confocal model with varying membrane thickness. An analytical approach for constant membrane thickness requires a mathematical modification of the spheroidal harmonics [21]. Here, we present the induced dipole obtained for cells with constant membrane thickness, i.e., uniform specific membrane capacitance and conductance. Fig. 3 shows the calculated equivalent dipole moments where the electric field E0 is parallel with the axis of symmetry. Note that we utilize spline functions to interpolate the calculated dipole moments in Fig. 3 and other figures hereafter. The frequencies of the dipole-moment calculation for all models are the same as those shown by the symbols in Fig. 2. It is clear from the figure that the magnitudes of the equivalent dipole moment of spheroidal B model are significantly different from those of the other models. The negative real part of the dipole moment at low frequencies and the positive imaginary part are smallest for the spherical model and largest for the spheroidal B model. On the other hand, the positive real part and the negative imaginary part of the dipole moment exhibit the opposite tendency with the model geometries. The peak values of the real and the imaginary parts are related to in the maximum force and torque that can be observed in the experiments. For an electric field parallel with the axis of symmetry, fDEP 0 and fROT 0 of the oblate spheroidal and biconcave disc models are higher than those of the spherical model. Fig. 4 shows the calculated equivalent dipole moment where the electric field E0 is perpendicular to the axis of symmetry. Note that the results from the oblate spheroidal A and the biconcave disc models are so close to each other that the difference is hardly noticeable from the figure. The variation of the real part of the dipole moment with the model geometries in Fig. 4(a) follows the opposite tendency in comparison with that in Fig. 3(a). That is, the negative real part is largest for the spherical model, but the positive real part is largest for the oblate spheroidal B model. For the imaginary part of the dipole moment in Fig. 4(b), the highest values of the positive and negative peaks are obtained from the oblate spheroidal B model. From Figs. 3 and 4, we can conclude that the oblate spheroidal A model yields the equivalent dipole moment that resembles to that of the biconcave-disc model for both electric field directions. The critical frequencies (fDEP 0 and fROT 0 ) of the biconcave disc and the spheroidal
Fig. 2. Real (Re) and imaginary (Im) parts of the equivalent dipole moment of a spherical cell. Calculated and analytical values are presented as symbols and lines, respectively.
For the numerical calculation, the surface of the spherical cell is discretized into 384 second-order curved elements. After solving the potential and electric field on the cell surface, the equivalent dipole moment 𝐩̇ 𝑖𝑛𝑑 is determined by using Eq. (12). Fig. 2 shows the calculated real and imaginary parts of the dipole moment as a function of the electric field frequency f between 10 kHz and 1 GHz. For the purpose of comparison with other geometrical models, the ordinates are normalized by p0 = 𝜀E E0 Vc to be dimensionless. The numerical results are presented as symbols on the graphs, and the analytical values are plotted as the lines. The real part of the dipole moment is negative at low frequencies, and increases with frequency to the positive peak at an intermediate frequency. With further increasing f, the real part of the dipole moment decreases to small negative values at high frequencies. The imaginary part behaves in accordance with the real part, having a positive and a negative peak in the considered frequency range. It is clear from the figure that the numerical results agree very well with the analytical values. The errors of the numerical 𝐩̇ 𝑖𝑛𝑑 are smaller than 1%, except where the values are very close to zero. In this work, we are interested in critical frequencies at which the DEP force or torque vanishes. There are two frequencies where the DEP force becomes zero in Fig. 2. However, we focus on the lower frequency fDEP 0 (≈ 430 kHz), which is possible to determine by our experiments. The frequency of zero torque is denoted as fROT 0 (≈ 5 MHz in Fig. 2).
Fig. 3. Equivalent dipole moment of cell modeled with different profiles where E0 is parallel to the symmetrical axis of the cell: (a) real part and (b) imaginary part. 71
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. 4. Equivalent dipole moment of cell modeled with different profiles where E0 is normal to the symmetrical axis of the cell: (a) real part and (b) imaginary part.
Fig. 5. Variation of the equivalent dipole moment due to intracellular conductivity 𝜎 I where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part.
Fig. 6. Variation of the equivalent dipole moment due to intracellular conductivity 𝜎 I where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
72
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Fig. 7. Variation of the equivalent dipole moments due to Cm where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part.
Fig. 8. Variation of the equivalent dipole moments due to Cm where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
models are higher than those of the spherical model when the electric field is parallel with the axis of symmetry, but lower when the field is normal with the axis.
ment. Figs. 7 and 8 present the variation of the dipole moment with the specific membrane capacitance Cm when E0 is parallel and normal to the symmetrical axis of the cell, respectively. We consider the change of Cm between 0.6 and 1.0 μF/cm2 . As can be seen from Figs. 7(a) and 8(a), the effect of the capacitance on the real part of the dipole moment is observed clearly at low frequencies up to the frequency of the positive plateaus. With increasing Cm , the magnitude of the peak is slightly enhanced, and the frequency fDEP 0 decreases. For the imaginary part of the dipole moment, a change in the positive peak magnitude is small in Figs. 7(b) and 8(b). The frequency of positive peak shifts to a slightly lower value with increasing Cm . The change of zero-torque frequency fROT 0 is, however, less pronounced in comparison to that of fDEP 0 . Figs. 9 and 10 present the dipole moment for different membrane conductance Gm when E0 is parallel and normal to the symmetrical axis of the cell, respectively. The conductance reflects the increased permeability of the cell membrane. The specific membrane conductance Gm ranges from 0 to 20 kS/m2 whereas the other parameters are assumed to be unchanged. It can be seen that both figures exhibit similar effects of Gm on the variation of the dipole moment with the field frequency. In Figs. 9(a) and 10(a), the introduction of Gm affects the equivalent dipole moment at low frequencies approximately up to the frequency of the peak of the positive real part. With increasing Gm , the real part of the dipole moment at low frequencies becomes higher, implying the weakened negative DEP force. On the other hand, the positive peak is more or less unchanged. As can be seen from Fig. 10(a), a negative real
4.3. Effects of cell parameters on the equivalent dipole moment Electrical parameters of cells may be altered due to a number of causes such as culture environment and ageing. Abnormality and infection also possibly affect the parameters. The change in electrical parameters should be reflected by the equivalent dipole moment. Using the biconcave disc model, we investigate the variation of the dipole moment with an aim to clarify the role of each parameter quantitatively. Figs. 5 and 6 present the variation of the equivalent dipole moment with the intracellular conductivity 𝜎 I . Form the figures, the effect of 𝜎 I on the equivalent dipole moment is negligible at low frequencies where the real part of the dipole moment is negative. The lower conductivity decreases the peak of the positive real part and imaginary part. That is, weaker DEP force and torque are expected with the lower conductivity. Anyhow, the change is somewhat mild when the electric field is parallel with the axis of symmetry in Fig. 5. The frequency fDEP 0 is virtually unchanged with the intracellular conductivity whereas fROT 0 decreases with lower conductivity. It is well known that the invasion of parasites alters the structure and properties of cell membrane [22]. Here we examine the effects of the electrical properties of cell membrane on the equivalent dipole mo73
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Fig. 9. Variation of the equivalent dipole moments due to conductivity Gm of the cell membrane where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part of pind .
Fig. 10. Variation of the equivalent dipole moments due to conductivity Gm of the membrane where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
part of the dipole moment may not even exist at low frequencies if Gm is high for the electric field normal to the axis of symmetry. Due to such behavior, the frequency fDEP 0 (if exists) becomes lower in the figure. For the imaginary part of the dipole moment, the positive peak values in Figs. 9(b) and 10(b) decrease monotonically with increasing Gm . Hence, the DEP torque is mitigated on this frequency range due to the membrane conductance. The frequency of positive peak becomes higher with increasing membrane conductance. However, the change in frequency fROT 0 with the conductance is practically negligible.
The frequency ranges on which 68% of the experimental results fell were 70–120 kHz for fDEP 0 and 1.68–2.35 MHz for fROT 0 . Numerical calculation was then performed to determine the intracellular conductivity 𝜎 I and the specific membrane capacitance Cm . For normal red blood cells, the conductance of cell membrane is negligible. The intracellular permittivity 𝜀I was set to 59𝜀0 . Electrical characterization of cells usually employs the frequency of zero DEP force fDEP 0 and frequency fROTM of maximum rotation, where the positive imaginary part of equivalent dipole moment takes the peak value. However, our calculation results indicate that the membrane capacitance has clearer effects on fDEP 0 and fROTM values than on fROT 0 . On the other hand, the fROT 0 and fROTM values are dependent on the intracellular conductivity. Therefore, in this work we chose fDEP 0 and fROT 0 so as to obtain more specific influences of 𝜎 I and Cm on our target frequencies, and apply the following linear approximation:
4.4. Determination of cell parameters from experimental results One of the objectives of this work is to apply the numerical analysis to the determination of cell parameters. Electromechanical methods have been used for cell characterization [23–25]. We set up experiments to measure the critical frequency fDEP 0 and fROT 0 where the DEP force and the DEP torque on a red blood cell vanish, respectively. Details of the experimental setup are described in the Appendix. Although the cells have a simple structure of cell membrane and intracellular media without nucleus, they typically possess a biconcave shape, making the analytical approach not applicable. From the measurement of the critical frequencies of red blood cells, the median value of fDEP 0 was 95 kHz and that of fROT 0 was 1.95 MHz.
[
] ⎡ 𝜕 𝑓𝐷𝐸𝑃 0 Δ𝑓𝐷𝐸𝑃 0 𝜎𝐼 = ⎢ 𝜕 𝑓𝜕𝑅𝑂𝑇 0 ⎢ Δ𝑓𝑅𝑂𝑇 0 ⎣ 𝜕 𝜎𝐼
𝜕 𝑓𝐷𝐸𝑃 0 ⎤[ ] 𝜕 𝐶𝑚 ⎥ Δ𝜎𝐼 , 𝜕 𝑓𝑅𝑂𝑇 0 ⎥ Δ𝐶𝑚 𝜕 𝐶𝑚 ⎦
(18)
where the diagonal elements of the square matrix are dominant. The Newton’s method is then used to determine the 𝜎 I and Cm values. The configuration of electric field normal to the axis of symmetry is applied to determine 𝜎 I and Cm , as it resembles the orientation of cells with electric field in the experiments. The initial values 𝜎 I = 0.5 74
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. A1. Electrodes for experiments on (a) DEP force and (b) DEP torque.
S/m and Cm = 1 μF/cm2 were used. For each iteration, the values were updated by solving Eq. (18). From the median values of fDEP 0 and fROT 0 , we obtained 𝜎 I = 0.328 S/m and Cm = 0.912 μF/cm2 . The calculation converged in a few iterations with |ΔfDEP 0 | < 0.05 kHz and |ΔfROT 0 | < 0.5 kHz. Taking the distribution of the measured critical frequencies and assuming that both frequencies vary in the same way on the cells, our calculation determined 68% of the 𝜎 I and Cm values in the ranges of 0.327–0.389 S/m and 0.729–1.24 μF/cm2 , respectively.
Appendix Red blood cells were used for the experiments to measure the critical frequency fDEP 0 and fROT 0 . From the critical frequencies, we carried out a repetitive calculation to determine the intracellular conductivity and the specific membrane capacitance (See Section 4.4). Cultured red blood cells were collected by centrifugal force (1500 rpm, 5 min). The cells were then suspended in a solution of sucrose (8.5% w/v) and glucose (0.3% w/v). An appropriate amount of phosphate buffer saline (PBS) was added to the cell suspension for adjusting the extracellular conductivity to be 20 mS/m. Two microelectrode configurations were fabricated on separated glass slides. The electrodes were made of Cr-Au or Cr-Al layers and patterned by using photolithography processes. The first configuration, used for DEP force experiments, is the interdigitated electrodes, as shown in Fig. A1(a). The electrode widths are 50 μm and the gaps are 25 μm. The second configuration, for the DEP torque experiments, is a 4-pole electrode system where the gaps between the opposite electrodes are 50 μm, as shown in Fig. A1(b). For the experiments to determine the frequency fDEP 0 of zero DEP force, the cell suspension was pipetted on the interdigitated electrode system. A sinusoidal ac voltage was then applied to the electrodes by using a signal generator (AFG-3021B, Tektronix). The frequency of the applied voltage was reduced from a sufficiently high value until the cell under observation exhibited a change from positive to negative DEP. Electro-rotation experiment was done for determining the frequency fROT 0 of zero rotation. 4-phase ac voltages were applied to the electrodes in Fig. A1(b) to generate a rotating electric field in the center gap. The frequency of the voltages was increased from a low value until the observed cell stopped its rotation and reversed the rotating direction. We performed 100 and 70 experimental runs for the measurement of fDEP 0 and fROT 0 , respectively.
5. Conclusion In this paper, the boundary element method is applied to calculate the electric field for the configuration of a single cell subjected to an ac electric field parallel or normal to the symmetrical axis of the cell. The cell is modelled by using different geometries. The calculation results clearly show the dependency of the equivalent dipole on the model geometries. The use of spherical model underestimates negative dielectrophoresis, but overestimates positive dielectrophoresis when electric field is parallel with the axis symmetry. The tendency is reversed for electric field normal to the axis symmetry. For both field directions, good approximation can be obtained by using the oblate spheroidal model based on the radius of the biconcave disc. We have examined numerically the variation of the equivalent dipole moment with the electrical cell parameters and demonstrated the application of the numerical method to determine the intracellular conductivity and specific membrane capacitance from the results of experiments on the DEP force and torque. The information of the electrical parameter values can be used for the biomedical analysis of blood cell samples.
Acknowledgments N. Panklang is supported by The 90th Anniversary of Chulalongkorn University Fund (Ratchadaphisek somphot Endowment Fund). The authors acknowledge the financial support from the National Research Council of Thailand (NRCT). BT acknowledges the support from the Thailand Science Research and Innovation (TSRI). We want to thank for the helps and discussion from Prof. M. Washizu (Department of Mechanical Engineering/Bioengineering, University of Tokyo, Japan), Prof. K. Chotivanich (Department of Clinical Tropical Medicine, Mahidol University), and Assoc. Prof. C. Putaporntip (Department of Parasitology, Chulalongkorn University). The Hitachi Asia (Thailand) and VDEC Electron Beam Lithography System at the University of Tokyo supported the fabrication works.
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[6] Jones TB. Electromechanics of particles. Cambridge University Press; 2005. [7] Green NG, Jones TB. Numerical determination of the effective moments of non– spherical particles. J Phys D Appl Phys Jan 7 2007;40(1):78–85. [8] Sugioka H. Attitude and positioning control of elliptical particle using induced-charge electrophoresis for reading information. Colloids Surf A: Physicochem Eng Asp 2014;440:27–33 Article. [9] Javidi R, Zand MM, Dastani K. Dielectrophoretic interaction of two particles in a uniform electric field. Microsyst Technol 2019;25(7):2699–711 journal article. [10] Techaumnat B, Washizu M. Analysis of the effects of an orifice plate on the membrane potential in electroporation and electrofusion of cells. J Phys D Appl Phys 2007;40(6):1831. [11] Techaumnat B. Numerical analysis of DC-field-induced transmembrane potential of spheroidal cells in axisymmetric orientations. IEEE Trans Dielectectr Electr Insulation 2013;20(5):1567–76. [12] Skalak R, Tozeren A, Zarda RP, Chien S. Strain energy function of red blood cell membranes. Biophys J 1973;13(3):245–64. [13] Gascoyne P, Mahidol C, Ruchirawat M, Satayavivad J, Watcharasit P, Becker FF. Microsample preparation by dielectrophoresis: isolation of malaria. Lab Chip 2002;2(2):70–5. doi:10.1039/B110990C. [14] Cowman AF, Tonkin CJ, Tham W-H, Duraisingh MT. The molecular basis of erythrocyte invasion by malaria parasites. Cell Host Microbe 2017;22(2):232–45. [15] Brebbia CA, Wrobel LC, Telles JCF. Boundary element techniques: theory and applications in engineering. Berlin: Springer-Verlag; 1984. [16] Jackson JD. Classical electrodynamics. New York: John Wiley & Sons; 1999.
[17] Washizu M, Jones TB. Dielectrophoretic interaction of two spherical particles calculated by equivalent multipole-moment method. IEEE Trans Ind Appl 1996;32(2):233–42. [18] Jones TB. Basic theory of dielectrophoresis and electrorotation. Eng Med Biol Mag IEEE 2003;22(6):33–42. [19] Gascoyne PRC, Becker FF, Wang XB. Numerical analysis of the influence of experimental conditions on the accuracy of dielectric parameters derived from electrorotation measurements. Bioelectrochem Bioenerg 1995;36(2):115–25. [20] Irimajiri A, Hanai T, Inouye A. A dielectric theory of “multi-stratified shell” model with its application to a lymphoma cell. J Theor Biol 1979;78(2):251–69. [21] Washizu M, Techaumnat B. Polarisation and membrane voltage of ellipsoidal particle with a constant membrane thickness: a series expansion approach. IET Nanobiotechnol 2008;2(3):62–71. [22] Miller LH, Baruch DI, Marsh K, Doumbo OK. The pathogenic basis of malaria. Nature 2002;415(6872):673–9. [23] Arnold WM, Zimmermann U. Electro-rotation: development of a technique for dielectric measurements on individual cells and particles. J Electrostat 1988;21(2–3):151–91. [24] Kriegmaier M, Zimmermann M, Wolf K, Zimmermann U, Sukhorukov VL. Dielectric spectroscopy of schizosaccharomyces pombe using electrorotation and electroorientation. Biochim Biophys Acta (BBA) - Gen Subj 2001;1568(2):135–46. [25] Gascoyne P, Pethig R, Satayavivad J, Becker FF, Ruchirawat M. Dielectrophoretic detection of changes in erythrocyte membranes following malarial infection. Biochim Biophys Acta (BBA) - Biomembr 1997;1323(2):240–52.
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Analysis of the equivalent dipole moment of red blood cell by using the boundary element method Nitipong Panklang a, Boonchai Techaumnat a,b,∗, Anurat Wisitsoraat c a
Faculty of Engineering, Department of Electrical Engineering, Chulalongkorn University, 254 Phyathai road, Pathumwan, Bangkok 10330, Thailand Biomedical Engineering Research Center, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand c Nanoelectronics and MEMS Laboratory, National Electronics and Computer Technology Center, Pathumthani 12120, Thailand b
a r t i c l e
i n f o
Keyword: Red blood cell Boundary element method Equivalent dipole moment Membrane capacitance Intracellular conductivity
a b s t r a c t Equivalent dipole moment of biological cells under electric field is an important parameter for various applications such as analysis and manipulation of cells in biomedical samples. The dipole moment depends on cell geometries as well as electrical parameters of media involved. Unfortunately, the analytical expression of equivalent dipole is available only for simple geometries. This work numerically studies the variation of the dipole moment of a red blood cell with cell geometries and electrical parameters. The cell is modelled as a sphere, an oblate spheroid or a biconcave disc. The authors apply the boundary element method to electric field calculation and use reexpansion formulae to compute the equivalent dipole moment of the cell. The numerical results agree well with the analytical one for the spherical model. The effects of cell geometries are clarified for two directions of the electric field, which are parallel or normal to the axis of symmetry of the cell. Using the biconcave disc model, we perform iterative calculation to estimate the intracellular conductivity and specific membrane capacitance of red blood cells from experimental results.
1. Introduction Dynamics of particles under electric field are used in various applications. Electrostatic precipitator, electrostatic painting and coating utilize the Coulomb force exerted on charged particles to control their movement or trajectories. Uncharged particles also exhibit electromechanical responses. A suspension of dielectric particles can be employed as an electrorheological fluid for applications such as active valves or suspensions [1]. There are also a variety of biomedical applications based on the electrical force on biological cells. The dielectrophoretic (DEP) force [2], which is the force acts on an uncharged object due to polarization difference between the object and its surrounding medium, is widely used in microfluidic platforms for immobilization, isolation, and analysis of biological cells [3–5]. For uncharged particles or biological cells under electric field E, electromechanical characteristics of cells depend primarily on the induced charges on the particles or cells. The simplest approach for estimating the contributions of the induced charges is to represent the charges with an equivalent dipole moment p. The DEP force FDEP resulting from the interaction between the dipole moment and the electric field can be expressed as 𝐅𝐷𝐸𝑃 = (𝐩 ⋅ ∇)𝐄.
(1)
It is clear from the equation that the DEP force exists where the electric field is nonuniform, and the force direction depends on the dipole moment p and the spatial variation of E. Positive and negative dielectrophoresis (p-DEP and n-DEP) are defined for the cases in which the force acts toward the region of stronger and weaker electric field, respectively. Under a harmonic electric field Ecos(𝜔t) of frequency f = 𝜔/(2𝜋), the equivalent dipole moment is also in a form of temporal harmonic pcos(𝜔t + 𝜑) where 𝜑 is the phase angle. The phase difference between the field and the dipole moment may exist according to the conductivity and the permittivity of media involved. It is convenient to write the field and the dipole moment in the phasor domain as 𝐄̇ and 𝐩̇ where the dot symbol denotes phasor variables. Taking the phase of 𝐄̇ as the reference (i.e., 𝐄̇ = 𝐄), the time average
1 [ ] Im 𝐩̇ × 𝐄. 2
(3)
∗ Corresponding author at: Faculty of Engineering, Department of Electrical Engineering, Chulalongkorn University, 254 Phyathai road, Pathumwan, Bangkok 10330, Thailand. E-mail address: [email protected] (B. Techaumnat).
https://doi.org/10.1016/j.enganabound.2019.12.002 Received 6 August 2019; Received in revised form 8 December 2019; Accepted 8 December 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Table 1 Dimension of spherical and spheroidal cells for calculation.
It is obvious from Eqs. (2) and (3) that the equivalent dipole moment is a critical parameter governing the electromechanical characteristics of cells or particles. The dipole moment depends on the electrical properties of the intracellular and extracellular media. The analytical expression of 𝐩̇ is available for cases of simple geometries such as spherical and spheroidal profiles. For complicated profiles, numerical methods are inevitably needed for the determination of 𝐩̇ . Numerical calculation of effective moments was presented for particles where the moments result from single layer charge on particle surface [7]. However, the influence of the double layer charge must be also taken in account for biological cells. In this work, the authors apply the boundary element method (BEM) to calculate electric field for configurations of a red blood cell in the 3D condition. There exist a number of works on the analysis of the dielectrophoresis by using numerical methods. For example, the BEM was used to analyses dielectrophoresis of particles on which typical boundary conditions of continuous potential and normal flux density hold [8,9]. For biological cells, the method was used to simulate the transmembrane voltage of a cell under dc electric field [10,11]. In this article, we determine the solution of electric field in the ac steady state by using the BEM. To our knowledge, the application of BEM to the 3D electric field analysis for non-spherical cells has not been presented yet in existing works. An advantage of the BEM is that the relationship between the transmembrane potential and electric field can be applied to the calculation without the reduction of accuracy because the normal electric field on boundaries are a primary variable for the BEM. From the field calculation results, we determine the equivalent dipole moment of the cell by using multipole re-expansion formulae. The dipole moment of non-spherical cells have been obtained analytically only for limited cases such as prolate or oblate spheroidal cells [6]. The aim of this work is to clarify the effects of cell geometries and electrical parameters on the equivalent dipole moment by using the quantitative investigation. In addition, we also demonstrate the determination of electrical parameters of red blood cell based on results obtained from DEP experiments. As red blood cells are a biomedical sample that is extensively used for various disease detection, the determination of cell parameters can be an important tool for diagnosis.
√ ( )( ) 0.862 1 − 𝜉 2 𝑐0 + 𝑐1 𝜉 2 + 𝑐2 𝜉 4 ,
b (μm)
Sphere Oblate spheroid A Oblate spheroid B
2.796 4.2 7.172
2.796 1.239 0.425
3. Calculation method 3.1. Electric field calculation The boundary element method (BEM) [15] is used to calculate electric field in configurations of an isolated cell under a uniform ac electric field 𝐄̇ 0 . Although domain-dividing methods such as the finite difference or the finite element method are also applicable to the calculation, very fine meshes are required to yield accurate result as the thickness of cell membrane is significantly smaller than the cell size. One of the authors has also applied the BEM to the calculation of transmembrane potential of cell pairs under dc energization [11]. The use of the BEM eliminates the need for domain subdivision and is convenient for the treatment of cell membrane. For the BEM, the cell membrane is modeled by using zero-thickness elements of which electric potential can be represented independently on both sides. Let 𝜙̇ 𝐼 and 𝜙̇ 𝐸 denote the potential in the interior and that in the exterior of the cell, respectively. The normal component of electric field on the cell membrane is denoted as 𝐸̇ 𝑛𝐼 in the interior side and 𝐸̇ 𝑛𝐸 in the exterior side. The potential at point r is related to the integrals of the potential and the normal component of the electric field on cell surface Scell as
The configurations of analysis are a single red blood cell under a uniform ac electric field 𝐄̇ 0 . We consider the externally applied electric field 𝐄̇ 0 in two directions. The electric field is either parallel or perpendicular to the symmetrical axis of the cell. For the latter case, the potential distribution is three-dimensional although the cell geometries are axisymmetric ones. Based on the results from those field directions, a solution for an arbitrary field direction can be deduced. The cell is modeled as a sphere, an oblate spheroid, or a biconcave disc (the normal shape of mature red blood cells). The spherical model is used to verify the numerical calculation of equivalent dipole moment with the analytical ones and also used to investigate the effects of the cell profile on the dipole moment. For the spheroidal model, a and b denote the semi-axial length normal and parallel to the axis of rotating symmetry (z axis), respectively. The contour of the biconcave disc is given (in μm) by the following equations [12].
𝑧(𝜉) = ±4.2
a (μm)
spheroid A has the length a = 4.2 μm, which is equal to that of the biconcave disc on the z = 0 plane. Oblate spheroid B has the semi-axial length b equal to 0.425 μm, making the same thickness on the symmetrical axis as that of the biconcave disc model. Dimensions of the models are summarized in Table 1. Fig. 1 presents all models approximately on the same scale. Unless specified otherwise, the following electrical parameters are used for the red blood cell in the calculation [13]. Intracellular conductivity 𝜎 I = 0.3 S/m and permittivity 𝜀I = 59𝜀0 where 𝜀0 is the permittivity of free space. The specific capacitance of cell membrane Cm = 8.7 mF/m2 based on 4.5 nm membrane thickness, and the membrane conductance is neglected in typical cases. Specific membrane conductance Gm values between 5 and 50 kS/m2 are applied to the calculation in some cases for studying the variation of dipole moment due to membrane modification which may be, for example, a result from an invasion of parasites [14].
2. Configuration
𝜌(𝜉) = 4.2𝜉,
Cell geometry
𝛼 𝐼 (𝐫 )𝜙̇ 𝐼 (𝐫 ) =
∫𝑆𝑐𝑒𝑙𝑙
𝐸̇ 𝑛𝐼 (𝐱)𝑤(𝐫, 𝐱)d𝑠+
∫𝑆𝑐𝑒𝑙𝑙
𝜕 𝜙̇ 𝐼 (𝐱) 𝑤(𝐫, 𝐱)d𝑠 𝜕𝑛
(6)
for the interior of the cell, and 𝛼 𝐸 (𝐫 )𝜙̇ 𝐸 (𝐫 ) = −
∫𝑆𝑐𝑒𝑙𝑙
𝐸̇ 𝑛𝐸 (𝐱)𝑤(𝐫, 𝐱)d𝑠−
∫𝑆𝑐𝑒𝑙𝑙
𝜕 𝜙̇ 𝐸 (𝐱) 𝑤(𝐫, 𝐱)d𝑠 − 𝐄̇ 0 ⋅ 𝐫 𝜕𝑛 (7)
for the exterior of the cell. In the above equations, the potential reference is taken at the origin. x is the point on Scell , w is the fundamental solution of the potential problem, and n denotes the unit normal at x from the interior to the exterior of the cell. The 𝛼 I and 𝛼 E values depend on the surface geometry at x. For smooth surface, 𝛼 I = 𝛼 E = 1/2. The fundamental solution for 3D configurations is expressed as
(4)
(5)
𝑤(𝐫, 𝐱) =
where c0 = 0.01384083, c1 = 0.2842917, c2 = 0.01306932, and the parameter 𝜉 varies from 0 to 1. The dimensions of the spherical and spheroidal models are determined so as to have the same volume as the biconcave disc. From the contour given by Eqs. (4) and (5), the cell volume Vcell is equal to 91.6 fL. For oblate spheroidal models, two sets of a and b values are used. Oblate
1 . 4𝜋‖𝐫 − 𝐱‖
(8)
It should be noted here that transmembrane potential arises if charges accumulate on both sides of the cell membrane, i.e., double layer charge. Hence, the potential are denoted as 𝜙̇ 𝐼 (𝐱) and 𝜙̇ 𝐸 (𝐱) in the surface integral equations. The relationship between the transmembrane potential 69
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. 1. Models of a red blood cell: (a) sphere, (b) Oblate spheroid A, (c) Oblate spheroid B, and (d) biconcave disc.
𝜙̇ 𝐼 (𝐱) − 𝜙̇ 𝐸 (𝐱) and the normal component of the electric field can be written as ( )( ) ( ) ( ) (9) 𝐺𝑚 + 𝑗𝜔𝐶𝑚 𝜙̇ 𝐼 − 𝜙̇ 𝐸 = 𝜎𝐼 + 𝑗𝜔𝜀𝐼 𝐸̇ 𝑛𝐼 = 𝜎𝐸 + 𝑗𝜔𝜀𝐸 𝐸̇ 𝑛𝐸 ,
duced dipole moment. 𝐩̇ 𝑖𝑛𝑑 =
∫𝑆𝑐𝑒𝑙𝑙
𝜌̇ 𝑑 𝐚𝑛 d𝑠,
(12)
4. Results and discussion 4.1. Equivalent dipole moment of spherical cell For the spherical model, we can determine 𝐩̇ 𝑖𝑛𝑑 analytically [18]. Define complex permittivity 𝜀̇ for electrical frequency f as 𝜎 𝜀̇ = 𝜀 − 𝑗 , (13) 2𝜋𝑓
3.2. Equivalent dipole calculation
and complex specific membrane capacitance 𝐶̇ 𝑚 as
For the configurations under consideration, the equivalent induced dipole is parallel with the direction of the applied field 𝐄̇ 0 . The dipole moment 𝐩̇ 𝑖𝑛𝑑 can be determined from the discontinuities of the electric field and potential across the cell membrane. On the cell membrane, single layer charge contributes to the discontinuity of the normal component 𝐸̇ 𝑛 of electric field. We determine the single layer charge density 𝜌̇ 𝑠 by using the following equation [16].
𝐶̇ 𝑚 = 𝐶𝑚 − 𝑗
𝐺𝑚 , 2𝜋𝑓
(14)
where 𝜀 is the permittivity and 𝜎 is the conductivity. The analytical expression of the equivalent complex permittivity 𝜀̇ 𝑠𝑝ℎ for a spherical cell of radius R and complex intracellular permittivity 𝜀̇ 𝐼 is [19,20] 𝜀̇ 𝑠𝑝ℎ =
(10)
𝑅𝐶̇ 𝑚 𝜀̇ 𝐼
𝑅𝐶̇ 𝑚 + 𝜀̇ 𝐼
.
(15)
From the complex permittivity of the cell and that of the extracellular medium, we can determine the equivalent dipole moment
On the other hand, the double layer charge (dipole) density 𝜌̇ 𝑑 is determined from the potential discontinuity, i.e., the transmembrane potential. The direction of dipole is normal to the cell membrane, and the magnitude follows [16] 𝜌̇ 𝑑 = 𝜙̇ 𝐸 − 𝜙̇ 𝐼 . 𝜀𝐸
𝜌̇ 𝑠 (𝐱 − 𝐜)d𝑠 +
where an is the outward unit normal vector at x. Note that it is also possible to determine the higher-order multipole magnitudes using appropriate re-expansion formulae. However, for characterizing a single cell, it is usually adequate to consider only the equivalent dipole representation.
where Gm is the specific conductance, Cm is the specific capacitance of the membrane, 𝜎 E is the extracellular conductivity, 𝜀E is the extracellular permittivity, and j is the imaginary unit. A linear equation system is constructed by applying Eqs. (6) and (7) to the nodes on the interior side and to those on the exterior side of the cell. The linear system is then solved to satisfy the boundary conditions in Eq. (9). We simulate all the boundaries with second-order curved triangular or rectangular elements. The shape, electric potential and electric field on the elements are interpolated by the same secondorder functions. The computation is done by an in-house Matlab program.
𝜌̇ 𝑠 = 𝐸̇ 𝑛𝐸 − 𝐸̇ 𝑛𝐼 . 𝜀𝐸
∫𝑆𝑐𝑒𝑙𝑙
𝐩̇ 𝑖𝑛𝑑 = 4𝜋𝜀𝐸 𝑅3 𝐾̇ 𝐄̇ ,
(16)
where 𝐾̇ is the Clausius-Mossotti factor, a complex-number constant depending on 𝜀̇ 𝑠𝑝ℎ and complex permittivity 𝜀̇ 𝐸 of the extracellular medium. 𝜀̇ 𝑠𝑝ℎ − 𝜀̇ 𝐸 𝐾̇ = . (17) 𝜀̇ 𝑠𝑝ℎ + 2𝜀̇ 𝐸
(11)
Utilizing the multipole re-expansion [17] of the single and double layer charges on the membrane to the center c of the cell, we obtain the in70
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
4.2. Effects of cell profiles on the equivalent dipole moment This section compares the equivalent dipole moments obtained by using different cell profiles. The two oblate spheroidal models represent the cases where one of the semi-axial length is equal to that of the biconcave disc model. Although it is possible to obtain analytical expression of the induced dipole for a spheroidal model under an external electric field, the analytical solution is based on a confocal model with varying membrane thickness. An analytical approach for constant membrane thickness requires a mathematical modification of the spheroidal harmonics [21]. Here, we present the induced dipole obtained for cells with constant membrane thickness, i.e., uniform specific membrane capacitance and conductance. Fig. 3 shows the calculated equivalent dipole moments where the electric field E0 is parallel with the axis of symmetry. Note that we utilize spline functions to interpolate the calculated dipole moments in Fig. 3 and other figures hereafter. The frequencies of the dipole-moment calculation for all models are the same as those shown by the symbols in Fig. 2. It is clear from the figure that the magnitudes of the equivalent dipole moment of spheroidal B model are significantly different from those of the other models. The negative real part of the dipole moment at low frequencies and the positive imaginary part are smallest for the spherical model and largest for the spheroidal B model. On the other hand, the positive real part and the negative imaginary part of the dipole moment exhibit the opposite tendency with the model geometries. The peak values of the real and the imaginary parts are related to in the maximum force and torque that can be observed in the experiments. For an electric field parallel with the axis of symmetry, fDEP 0 and fROT 0 of the oblate spheroidal and biconcave disc models are higher than those of the spherical model. Fig. 4 shows the calculated equivalent dipole moment where the electric field E0 is perpendicular to the axis of symmetry. Note that the results from the oblate spheroidal A and the biconcave disc models are so close to each other that the difference is hardly noticeable from the figure. The variation of the real part of the dipole moment with the model geometries in Fig. 4(a) follows the opposite tendency in comparison with that in Fig. 3(a). That is, the negative real part is largest for the spherical model, but the positive real part is largest for the oblate spheroidal B model. For the imaginary part of the dipole moment in Fig. 4(b), the highest values of the positive and negative peaks are obtained from the oblate spheroidal B model. From Figs. 3 and 4, we can conclude that the oblate spheroidal A model yields the equivalent dipole moment that resembles to that of the biconcave-disc model for both electric field directions. The critical frequencies (fDEP 0 and fROT 0 ) of the biconcave disc and the spheroidal
Fig. 2. Real (Re) and imaginary (Im) parts of the equivalent dipole moment of a spherical cell. Calculated and analytical values are presented as symbols and lines, respectively.
For the numerical calculation, the surface of the spherical cell is discretized into 384 second-order curved elements. After solving the potential and electric field on the cell surface, the equivalent dipole moment 𝐩̇ 𝑖𝑛𝑑 is determined by using Eq. (12). Fig. 2 shows the calculated real and imaginary parts of the dipole moment as a function of the electric field frequency f between 10 kHz and 1 GHz. For the purpose of comparison with other geometrical models, the ordinates are normalized by p0 = 𝜀E E0 Vc to be dimensionless. The numerical results are presented as symbols on the graphs, and the analytical values are plotted as the lines. The real part of the dipole moment is negative at low frequencies, and increases with frequency to the positive peak at an intermediate frequency. With further increasing f, the real part of the dipole moment decreases to small negative values at high frequencies. The imaginary part behaves in accordance with the real part, having a positive and a negative peak in the considered frequency range. It is clear from the figure that the numerical results agree very well with the analytical values. The errors of the numerical 𝐩̇ 𝑖𝑛𝑑 are smaller than 1%, except where the values are very close to zero. In this work, we are interested in critical frequencies at which the DEP force or torque vanishes. There are two frequencies where the DEP force becomes zero in Fig. 2. However, we focus on the lower frequency fDEP 0 (≈ 430 kHz), which is possible to determine by our experiments. The frequency of zero torque is denoted as fROT 0 (≈ 5 MHz in Fig. 2).
Fig. 3. Equivalent dipole moment of cell modeled with different profiles where E0 is parallel to the symmetrical axis of the cell: (a) real part and (b) imaginary part. 71
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. 4. Equivalent dipole moment of cell modeled with different profiles where E0 is normal to the symmetrical axis of the cell: (a) real part and (b) imaginary part.
Fig. 5. Variation of the equivalent dipole moment due to intracellular conductivity 𝜎 I where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part.
Fig. 6. Variation of the equivalent dipole moment due to intracellular conductivity 𝜎 I where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
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Fig. 7. Variation of the equivalent dipole moments due to Cm where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part.
Fig. 8. Variation of the equivalent dipole moments due to Cm where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
models are higher than those of the spherical model when the electric field is parallel with the axis of symmetry, but lower when the field is normal with the axis.
ment. Figs. 7 and 8 present the variation of the dipole moment with the specific membrane capacitance Cm when E0 is parallel and normal to the symmetrical axis of the cell, respectively. We consider the change of Cm between 0.6 and 1.0 μF/cm2 . As can be seen from Figs. 7(a) and 8(a), the effect of the capacitance on the real part of the dipole moment is observed clearly at low frequencies up to the frequency of the positive plateaus. With increasing Cm , the magnitude of the peak is slightly enhanced, and the frequency fDEP 0 decreases. For the imaginary part of the dipole moment, a change in the positive peak magnitude is small in Figs. 7(b) and 8(b). The frequency of positive peak shifts to a slightly lower value with increasing Cm . The change of zero-torque frequency fROT 0 is, however, less pronounced in comparison to that of fDEP 0 . Figs. 9 and 10 present the dipole moment for different membrane conductance Gm when E0 is parallel and normal to the symmetrical axis of the cell, respectively. The conductance reflects the increased permeability of the cell membrane. The specific membrane conductance Gm ranges from 0 to 20 kS/m2 whereas the other parameters are assumed to be unchanged. It can be seen that both figures exhibit similar effects of Gm on the variation of the dipole moment with the field frequency. In Figs. 9(a) and 10(a), the introduction of Gm affects the equivalent dipole moment at low frequencies approximately up to the frequency of the peak of the positive real part. With increasing Gm , the real part of the dipole moment at low frequencies becomes higher, implying the weakened negative DEP force. On the other hand, the positive peak is more or less unchanged. As can be seen from Fig. 10(a), a negative real
4.3. Effects of cell parameters on the equivalent dipole moment Electrical parameters of cells may be altered due to a number of causes such as culture environment and ageing. Abnormality and infection also possibly affect the parameters. The change in electrical parameters should be reflected by the equivalent dipole moment. Using the biconcave disc model, we investigate the variation of the dipole moment with an aim to clarify the role of each parameter quantitatively. Figs. 5 and 6 present the variation of the equivalent dipole moment with the intracellular conductivity 𝜎 I . Form the figures, the effect of 𝜎 I on the equivalent dipole moment is negligible at low frequencies where the real part of the dipole moment is negative. The lower conductivity decreases the peak of the positive real part and imaginary part. That is, weaker DEP force and torque are expected with the lower conductivity. Anyhow, the change is somewhat mild when the electric field is parallel with the axis of symmetry in Fig. 5. The frequency fDEP 0 is virtually unchanged with the intracellular conductivity whereas fROT 0 decreases with lower conductivity. It is well known that the invasion of parasites alters the structure and properties of cell membrane [22]. Here we examine the effects of the electrical properties of cell membrane on the equivalent dipole mo73
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Fig. 9. Variation of the equivalent dipole moments due to conductivity Gm of the cell membrane where the electric field is parallel to the symmetrical axis: (a) real part and (b) imaginary part of pind .
Fig. 10. Variation of the equivalent dipole moments due to conductivity Gm of the membrane where the electric field is normal to the symmetrical axis: (a) real part and (b) imaginary part.
part of the dipole moment may not even exist at low frequencies if Gm is high for the electric field normal to the axis of symmetry. Due to such behavior, the frequency fDEP 0 (if exists) becomes lower in the figure. For the imaginary part of the dipole moment, the positive peak values in Figs. 9(b) and 10(b) decrease monotonically with increasing Gm . Hence, the DEP torque is mitigated on this frequency range due to the membrane conductance. The frequency of positive peak becomes higher with increasing membrane conductance. However, the change in frequency fROT 0 with the conductance is practically negligible.
The frequency ranges on which 68% of the experimental results fell were 70–120 kHz for fDEP 0 and 1.68–2.35 MHz for fROT 0 . Numerical calculation was then performed to determine the intracellular conductivity 𝜎 I and the specific membrane capacitance Cm . For normal red blood cells, the conductance of cell membrane is negligible. The intracellular permittivity 𝜀I was set to 59𝜀0 . Electrical characterization of cells usually employs the frequency of zero DEP force fDEP 0 and frequency fROTM of maximum rotation, where the positive imaginary part of equivalent dipole moment takes the peak value. However, our calculation results indicate that the membrane capacitance has clearer effects on fDEP 0 and fROTM values than on fROT 0 . On the other hand, the fROT 0 and fROTM values are dependent on the intracellular conductivity. Therefore, in this work we chose fDEP 0 and fROT 0 so as to obtain more specific influences of 𝜎 I and Cm on our target frequencies, and apply the following linear approximation:
4.4. Determination of cell parameters from experimental results One of the objectives of this work is to apply the numerical analysis to the determination of cell parameters. Electromechanical methods have been used for cell characterization [23–25]. We set up experiments to measure the critical frequency fDEP 0 and fROT 0 where the DEP force and the DEP torque on a red blood cell vanish, respectively. Details of the experimental setup are described in the Appendix. Although the cells have a simple structure of cell membrane and intracellular media without nucleus, they typically possess a biconcave shape, making the analytical approach not applicable. From the measurement of the critical frequencies of red blood cells, the median value of fDEP 0 was 95 kHz and that of fROT 0 was 1.95 MHz.
[
] ⎡ 𝜕 𝑓𝐷𝐸𝑃 0 Δ𝑓𝐷𝐸𝑃 0 𝜎𝐼 = ⎢ 𝜕 𝑓𝜕𝑅𝑂𝑇 0 ⎢ Δ𝑓𝑅𝑂𝑇 0 ⎣ 𝜕 𝜎𝐼
𝜕 𝑓𝐷𝐸𝑃 0 ⎤[ ] 𝜕 𝐶𝑚 ⎥ Δ𝜎𝐼 , 𝜕 𝑓𝑅𝑂𝑇 0 ⎥ Δ𝐶𝑚 𝜕 𝐶𝑚 ⎦
(18)
where the diagonal elements of the square matrix are dominant. The Newton’s method is then used to determine the 𝜎 I and Cm values. The configuration of electric field normal to the axis of symmetry is applied to determine 𝜎 I and Cm , as it resembles the orientation of cells with electric field in the experiments. The initial values 𝜎 I = 0.5 74
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Engineering Analysis with Boundary Elements 112 (2020) 68–76
Fig. A1. Electrodes for experiments on (a) DEP force and (b) DEP torque.
S/m and Cm = 1 μF/cm2 were used. For each iteration, the values were updated by solving Eq. (18). From the median values of fDEP 0 and fROT 0 , we obtained 𝜎 I = 0.328 S/m and Cm = 0.912 μF/cm2 . The calculation converged in a few iterations with |ΔfDEP 0 | < 0.05 kHz and |ΔfROT 0 | < 0.5 kHz. Taking the distribution of the measured critical frequencies and assuming that both frequencies vary in the same way on the cells, our calculation determined 68% of the 𝜎 I and Cm values in the ranges of 0.327–0.389 S/m and 0.729–1.24 μF/cm2 , respectively.
Appendix Red blood cells were used for the experiments to measure the critical frequency fDEP 0 and fROT 0 . From the critical frequencies, we carried out a repetitive calculation to determine the intracellular conductivity and the specific membrane capacitance (See Section 4.4). Cultured red blood cells were collected by centrifugal force (1500 rpm, 5 min). The cells were then suspended in a solution of sucrose (8.5% w/v) and glucose (0.3% w/v). An appropriate amount of phosphate buffer saline (PBS) was added to the cell suspension for adjusting the extracellular conductivity to be 20 mS/m. Two microelectrode configurations were fabricated on separated glass slides. The electrodes were made of Cr-Au or Cr-Al layers and patterned by using photolithography processes. The first configuration, used for DEP force experiments, is the interdigitated electrodes, as shown in Fig. A1(a). The electrode widths are 50 μm and the gaps are 25 μm. The second configuration, for the DEP torque experiments, is a 4-pole electrode system where the gaps between the opposite electrodes are 50 μm, as shown in Fig. A1(b). For the experiments to determine the frequency fDEP 0 of zero DEP force, the cell suspension was pipetted on the interdigitated electrode system. A sinusoidal ac voltage was then applied to the electrodes by using a signal generator (AFG-3021B, Tektronix). The frequency of the applied voltage was reduced from a sufficiently high value until the cell under observation exhibited a change from positive to negative DEP. Electro-rotation experiment was done for determining the frequency fROT 0 of zero rotation. 4-phase ac voltages were applied to the electrodes in Fig. A1(b) to generate a rotating electric field in the center gap. The frequency of the voltages was increased from a low value until the observed cell stopped its rotation and reversed the rotating direction. We performed 100 and 70 experimental runs for the measurement of fDEP 0 and fROT 0 , respectively.
5. Conclusion In this paper, the boundary element method is applied to calculate the electric field for the configuration of a single cell subjected to an ac electric field parallel or normal to the symmetrical axis of the cell. The cell is modelled by using different geometries. The calculation results clearly show the dependency of the equivalent dipole on the model geometries. The use of spherical model underestimates negative dielectrophoresis, but overestimates positive dielectrophoresis when electric field is parallel with the axis symmetry. The tendency is reversed for electric field normal to the axis symmetry. For both field directions, good approximation can be obtained by using the oblate spheroidal model based on the radius of the biconcave disc. We have examined numerically the variation of the equivalent dipole moment with the electrical cell parameters and demonstrated the application of the numerical method to determine the intracellular conductivity and specific membrane capacitance from the results of experiments on the DEP force and torque. The information of the electrical parameter values can be used for the biomedical analysis of blood cell samples.
Acknowledgments N. Panklang is supported by The 90th Anniversary of Chulalongkorn University Fund (Ratchadaphisek somphot Endowment Fund). The authors acknowledge the financial support from the National Research Council of Thailand (NRCT). BT acknowledges the support from the Thailand Science Research and Innovation (TSRI). We want to thank for the helps and discussion from Prof. M. Washizu (Department of Mechanical Engineering/Bioengineering, University of Tokyo, Japan), Prof. K. Chotivanich (Department of Clinical Tropical Medicine, Mahidol University), and Assoc. Prof. C. Putaporntip (Department of Parasitology, Chulalongkorn University). The Hitachi Asia (Thailand) and VDEC Electron Beam Lithography System at the University of Tokyo supported the fabrication works.
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