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Analysis for relationship of transmembrane potential–pulsed electric field frequency
food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
Contents lists available at ScienceDirect
Food and Bioproducts Processing journal homepage: www.elsevier.com/locate/fbp
Analysis for relationship of transmembrane potential–pulsed electric field frequency Li Jiahui, Wei Xinlao ∗ , Wang Yonghong, Liu Gongqiang College of Electrical & Electronic Engineering, Harbin University of Science and Technology, Harbin, China
a b s t r a c t The paper presents a model of a spherical cell. The transmembrane potential on cell membrane is obtained by solving the Laplace’s equation. The frequency dependence of the transmembrane potential in pulsed electric fields is described. The value of transmembrane potential decreases as the frequency of external electric field increases. And there is a range of frequency for the value of transmembrane potential to decrease fast. It is shown that there is a strong relationship between the value of transmembrane potential and frequency components contained in the pulse. With more low-frequency components, the value of transmembrane potential is increasing and thus a better sterilization effect can be obtained. By comparing the frequency components contained in square wave pulse, exponentially decaying pulse, oscillatory pulse and their sterilization effect respectively, the analysis results about the relationship of transmembrane potential-frequency presented in this paper is validated. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Bacteria; Electric field analysis; Transmembrane potential; Frequency components
1.
Introduction
Pulsed electric field food processing is one of non-thermal food preservation methods. In this method, food is exposed to a pulsed electric field. Transmembrane potential is induced on the cell membrane of bacteria by the pulsed electric field applied. The value of the transmembrane potential increases as the external electric field intensity increases. When electric field intensity exceeds a critical value, cell membrane may breakdown. Consequently, the object of food sterilization is achieved (Xu and Wang, 2005). Comparing to the thermal food processing that is widely used, protein is not damaged, vitamins and volatile flavors are not loss and the sensory and nutritional properties remain. Assuming that the geometric parameters of bacteria are not affected by frequency and intensity of pulsed electric field, the transmembrane potential depends on the frequency of the electric field and its value varied significantly. For the three types of pulsed electric field mostly used in food processing, the transmembrane potential on cell membrane of bacteria caused by frequency components contained in the pulses are different.
∗
2.
Theory
The spherical model of a bacterium is shown in Fig. 1 (Pavlin and Miklavcic, 2003). The inside radius and outside radius of the membrane are denoted by ri and re respectively. The complex conductivity of cytoplasm, cell membrane and ∗ and ∗ respecextracellular solution are denoted by i∗ , im e tively. Assuming the extracellular solution, cytoplasm and cell membrane are linear, isotropic, and homogeneous dielectric medium. The electric potential at any point is satisfies the Laplace’s equation. The Laplace’s equation can be expressed as Eq. (1) in spherical coordinate (Guru and Hiziroglu, 2004) ∇2 =
1 ∂ r2 ∂r
∂ 2 r
∂r
+
r2
1 ∂ sin ∂
sin
∂ ∂
+
1 ∂2 =0 r2 sin ∂ϕ2 (1)
Taking the spherical center as the original point, there is symmetry about ϕ co-ordinate, that is, solutions depend on r and but not on ϕ. Then Eq. (1) reduces to
Corresponding author. E-mail address: [email protected] (W. Xinlao). Received 17 April 2008; Received in revised form 11 October 2008; Accepted 28 October 2008 0960-3085/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fbp.2008.10.003
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food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
Nomenclature re ri
outside radius inside radius
Greek symbols εi permittivity of cytoplasm εim permittivity of cell membrane permittivity of extracellular solution εe conductivity e∗ complex conductivity of extracellular solution complex conductivity of cytoplasm i∗ ∗ im complex conductivity of cell membrane electric potential i electric potential in cytoplasm electric potential in cell membrane im e electric potential in extracellular solution transmembrane potential of cell membrane m ε permittivity i conductivity of cytoplasm conductivity of cell membrane im e conductivity of extracellular solution
Fig. 1 – The spherical model for bacterium.
Therefore, the general potential solutions associated with the three media are
i = 1 ∂ r2 ∂r
∂ 2 r
∂r
+
r2
1 ∂ sin ∂
sin
∂ ∂
r2
∂2 ∂ + 2r ∂r ∂r2
=−
1
=0
1
r2
∂ ∂2 + 2r ∂r ∂r2
∂2
∂ 2
+ ctg
∂2
∂
+ ctg ∂ ∂ 2
(3)
d2 d2 r
+ 2r
=˛
(4)
= −˛
d − n(n + 1) = 0 dr
[Cn rn + Dn r−(n+1) ]Pn (cos ),
= a1 rn + a2 r−(n+1)
e =
(11)
∞
[En rn + Fn r−(n+1) ]Pn (cos ),
r > re
(12)
where An , Bn , Cn , Dn , En , Fn are constants to be determined; Pn (cos ) is Legendre Polynomials; n is any positive integer including 0. The intracellular potential remains infinite as r → 0
i = (5)
∞
An rn Pn (cos )
(13)
n=0
The extracellular potential e tends to applied electric field potential as r → ∞ (6)
∞
Fn r−(n+1) Pn (cos )
(14)
n=0
(7)
In terms of ˛ = n(n + 1), Eq. (5) takes the form of
(8)
Substituting x = cos and = P(x) into Eq. (8), it can be written as d2 P dP − 2x + n(n + 1)P = 0 dx dx2
ri < r ≤ re
n=0
Eq. (6) has the solution
(1 − x2 )
∞
e = −E0 (t)r cos +
d
d2
+ ctg + n(n + 1) = 0 d d 2
(10)
n=0
Let ˛ = n(n + 1). Eq. (4) then satisfies the Euler equation r2
im =
∂
∂
0 < r ≤ ri
(2)
The left-hand side of Eq. (3) depends only on r, while the righthand side depends only on . This can only occur when both sides are equal to a constant, denoted by ˛ 1
[An rn + Bn r−(n+1) ]Pn (cos ),
n=0
By solving for in a separable form (r,) = (r) () (Sun and Liu, 2000), Eq. (2) becomes 1
∞
(9)
The solution of Eq. (9) is given by the Legendre polynomials Pn (x).
The electric potential and the normal components of current densities at the interface between dielectrics are continuous under time-varying electric field (Vladimir et al., 2001). Therefore, the boundary conditions at each interface can be expressed as follows:
i (ri , ) = im (ri , ) ∂ ∗ ∂im = im i∗ i ∂r |r=ri ∂r |r=ri
(15)
im (re , ) = e (re , ) ∂e ∗ ∂im = e∗ im ∂r |r=re ∂r |r=re
(16)
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food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
⎧ l l ⎪ ⎪ ⎪ An ri n Pn (cos ) = [Cn ri n + Dn ri −(n+1) ]Pn (cos ) ⎪ ⎨ n=0
n=0
l−1 l−1 ⎪ ⎪ ⎪ n ⎪ A r P (cos ) = [Cn ri n + Dn ri −(n+1) ]Pn (cos ) n n i ⎩ n=0
The transmembrane potential of cell membrane is defined (17)
m = im (r, cos )|r=re − i (r, cos )|r=ri Substituting the solution of (27) into (28), we obtain
n=0
⎧ l ⎪ ⎪ ⎪ ni∗ An ri n−1 Pn (cos ) ⎪ ⎪ ⎪ ⎪ n=0 ⎪ ⎪ ⎪ ⎪ l ⎪ ⎪ ⎪ ∗ ∗ ⎪ = [nim Cn ri n−1 − (n + 1)im Dn ri −(n+2) ]Pn (cos ) ⎪ ⎨ n=0
l−1 ⎪ ⎪ ⎪ ⎪ ni∗ An ri n−1 Pn (cos ) ⎪ ⎪ ⎪ ⎪ n=0 ⎪ ⎪ ⎪ l−1 ⎪ ⎪ ⎪ ∗ ∗ ⎪ = [nim Cn ri n−1 − (n + 1)im Dn ri −(n+2) ]Pn (cos ) ⎪ ⎩
m =
(18)
(19)
Eq. (18) can be written as (20)
Cn re + Dn re
−(n+1)
n
= En re + Fn re
−(n+1)
∗ C r n−1 − (n + 1) ∗ D r −(n+2) nim n e im n e
=
ne∗ En re n−1
− (n + 1)e∗ Fn re −(n+2)
⎪ ⎪ Cn re n + Dn re −(n+1) = Fn re −(n+1) ⎪ ⎪ ⎪ ⎪ ⎩
(21)
(23)
By solving (23), we have that An = Cn = Dn = Fn = 0, and the following representations are obtained: 0 ≤ r ≤ ri
im = (C1 r + D1 r−2 ) cos ,
cos
∗ Z2 = −3e∗ ri 3 re 3 E0 (t)(im − i∗ )
(31)
∗ ∗ ∗ ∗ M = 2ri 3 (im − i∗ )(e∗ − im ) + re3 (i∗ + 2im )(im + 2e∗ )
(32)
Analysis
The complex conductivity can be expressed as * = + j2 fε under time-varying electric field (Kotnik and Miklavcic, 2006). Where is conductivity, ε is permittivity. The geometric and electrical properties used in this paper are listed as follows (Mi et al., 2007; Kotnik and Miklavcic, 2000; Alcamo, 1998): Radius (m): re = 5.04
i = 0.5;
im = 1 × 10−5 ;
e = 0.6
εi = 5 × 10−10 ;
εim = 8.8 × 10−11 ;
εe = 0.7 × 10−9
By substituting these parameters into (30), transmembrane potential of cell membrane can be obtained. Fig. 2 shows the value of the transmembrane potential as a function of frequency. It can be seen from Fig. 2 that when frequency increases, m (f) decreases gradually from 0.95 V to 0.92 V as f < 106 Hz. In the range of 106 Hz < f < 108 Hz, m (f) decreases from 0.92 V to 0.06 V. And m (f) decreases from 0.06 V to 0.03 V as f > 108 Hz. At present, the widely used pulsed voltage waves for pulsed electric field sterilization are three types: square wave pulse, exponentially decaying pulse and oscillatory pulse. Fig. 3(a)
(24) ri < r ≤ re
(25)
r > re
(26)
e = (−E0 (t)r + F1 r−2 ) cos ,
From Eqs. (15) and (16), we obtain
⎧ D1 ⎪ A1 ri − C1 ri − 2 = 0 ⎪ ⎪ ri ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ D1 = 0 ⎪ i A1 − im C1 + 2im ⎨ 3 rn
D1 F1 ⎪ ⎪ C1 re + 2 − 2 = −E0 (t)re ⎪ ⎪ r re e ⎪ ⎪ ⎪ D ⎪ ⎩ ∗ C1 − 2 ∗ 1 + 2e∗ F1 = −E0 (t)e∗ im im 3 3 re
re
(29)
Permittivity (F/m): (22)
∗ C r n−1 − (n + 1) ∗ D r −(n+2) = −(n + 1) ∗ F r −(n+2) nim n e e n e im n e
i = A1 r cos ,
1 ri 2
Conductivity (S/m):
From Eq. (14) it is apparent that E1 = −E0 (t) for n = 1 and En = 0 for n = / 1. Therefore, for n = / 1, we get
⎧ An ri n = Cn ri n + Dn ri −(n+1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ C r n−1 − (n + 1) ∗ D r −(n+2) ⎨ ni∗ An ri n−1 = nim n i im n i
re 2
−
(30)
ri = 5; Similarly, from (16), it can be deduced that n
1
∗ + i∗ ) Z1 = −3e∗ re 3 E0 (t)(2im
3.
where l is a positive integer. Eq. (17) can be written as
∗ ∗ ni∗ An ri n−1 = nim Cn ri n−1 − (n + 1)im Dn ri −(n+2)
Z1 Z2 (re − ri ) cos + M M
where
n=0
An ri n = Cn ri n + Dn ri −(n+1)
(28)
(27)
Fig. 2 – Curves of transmembrane potential–frequency.
264
food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
Fig. 3 – The three types of pulse waves used in PEF.
shows a square wave with duration of 15 s, Fig. 3(b) shows an exponentially decaying pulse with duration of 15 s and Fig. 3(c) shows an oscillatory pulse with duration of 15 s. Previous experiment results show that the square wave pulse is the most effective in sterilization, the exponentially decaying pulse is less effective and the oscillatory pulse is the least. The experiment results can be well explained by the research in this paper. With the same pulse width and amplitude, the three types of pulsed waves (square wave pulse, exponentially decaying pulse and oscillatory pulse) are analyzed with FFT analysis. The frequency spectrum distributions obtained by FFT are shown in Fig. 4. The one shown in Fig. 4(a) illustrates frequency spectrum of square wave pulse. The one shown in Fig. 4(b) illustrates frequency spectrum of exponentially decaying pulse. And the one shown in Fig. 4(c) illustrates frequency spectrum of oscillatory pulse. It can be seen from the figures that the dominant frequency components of square wave pulse with peak value of 753 mV are in the range of 0–160 kHz. The dominant frequency components of exponentially decaying pulse with peak value of 237 mV are in the
Fig. 4 – FFT analysis result of the three types of pulse waves used in PEF.
range of 0–425 kHz. The dominant frequency components of exponentially decaying pulse with peak value of 205 mV are in the range of 720 kHz to 1.2 MHz
4.
Discussion
From Fig. 2, it can be seen that if the electric field intensity is constant, the value of transmembrane potential is increasing inverse proportion to the frequency components of pulses applied. Therefore, in theory, the most efficient method for sterilization is to apply DC voltage to liquid foods. However, in general, the liquid foods are dielectrics with high conductivities (Wouters et al., 1999). It is difficult to produce DC electric field of sufficient intensity to inactivate bacteria. The electric source has to generate a very large conduction current to fulfil the requirement. And the heat produced by the conduction current will be significant. Therefore, the pulsed electric field is usually adopted for liquid food sterilization (Barbosa-Cánovas and Rodríguez, 2002). Pulse rising edge and falling edge contain significant high frequency components. Therefore, mainly depend on pulse width but not
food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
pulse rising edge and falling edge, the bacteria are inactivated. Previous studies have demonstrated that the square wave pulse is the most effective in sterilization, the exponentially decaying pulse is less effective and the oscillatory pulse is the least (Jeyamkondan et al., 1999). By comparing the three types of pulse waves, it can be seen from Fig. 4 that the low-frequency components contained in the square wave pulse are the most significant. And the oscillatory pulse contains the most significant high frequency components. It can be known from the above analysis that the value of transmembrane potential is the highest if a square pulsed electric field is applied. And the value is lower than the other two if an oscillatory pulsed electric field is applied.
5.
Conclusion
The value of transmembrane potential decreases as the frequency of external electric field increases. And there is a range of frequency for the value of transmembrane potential to decrease fast. It is shown that there is a strong relationship between the value of transmembrane potential and frequency components contained in the pulse. With more lowfrequency components, the value of transmembrane potential is increasing and thus a better sterilization effect can be obtained. The bacteria have various shapes. Therefore, an additional error is introduced with the simplified model. Electroporation may result in changing of equivalent conductivity and equivalent permittivity of cell membrane, which may influence the value of transmembrane potential. In subsequent studies, the model will be modified so it more accurately reflects shapes of bacteria. And the functional relationship of electric field intensity and electrical parameters of membrane will be derived.
265
References Alcamo I.E., 1998, Schaum’s Outline of Theory and Problems of Microbiology (McGraw-Hill Companies, Inc., New York, USA), pp. 72. Barbosa-Cánovas, G.V., Rodríguez, J.J., 2002, Update on nonthermal food processing technologies: pulsed electric field, high hydrostatic pressure, irradiation and ultrasound. Food Aust, 54(11): 513–520. Guru, B.S., Hiziroglu, H.R., 2004, Electromagnetic Field Theory Fundamentals (Cambridge University Press, Cambridge, England), pp. 113. Jeyamkondan, S., Jayas, D.S., Holley, R.A., 1999, Pulsed electric field processing of foods: a review. J Food Protect, 62(9): 1088–1096. Kotnik, T., Miklavcic, D., 2000, Second-order model of membrane electric field induced by alternating external electric fields. IEEE Trans Bio-med Eng, 47(8): 1074–1081. Kotnik, T., Miklavcic, D., 2006, Theoretical evaluation of voltage inducement on internal membranes of biological cells exposed to electric fields. Biophys J, 90(2), 480–491. Mi, Y., Sun, C., Yao, C., 2007, Frequency response of transmembrane potential on cell inner and outer membrane based on equivalent circuit model. Trans China Electrotech Soc, 22(6): 6–11. Pavlin, M., Miklavcic, D., 2003, Effective conductivity of a suspension of permeabilized cells: a theoretical analysis. Biophys J, 85(2): 719–729. Vladimir, L.S., Gustav, M., Markus, K., Ulrich, Z., 2001, A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane. J Electrostat, 50(3): 191–204. Sun, M., Liu, F., 2000, Dielectric Physics Fundamentals (South China University of Technology Press, Guangzhou, China), pp. 27. Wouters, P.C., Dutreux, N., Smelt, J.P.P.M., 1999, Effects of pulsed electric fields on inactivation kinetics of Listeria innocua, AEM, 65(12): 5364–5371. Xu, H., Wang, Y., 2005, New Technology for Food Sterilization (Scientific and Technical Documents Publishing House, Beijing, China), pp. 293.
Contents lists available at ScienceDirect
Food and Bioproducts Processing journal homepage: www.elsevier.com/locate/fbp
Analysis for relationship of transmembrane potential–pulsed electric field frequency Li Jiahui, Wei Xinlao ∗ , Wang Yonghong, Liu Gongqiang College of Electrical & Electronic Engineering, Harbin University of Science and Technology, Harbin, China
a b s t r a c t The paper presents a model of a spherical cell. The transmembrane potential on cell membrane is obtained by solving the Laplace’s equation. The frequency dependence of the transmembrane potential in pulsed electric fields is described. The value of transmembrane potential decreases as the frequency of external electric field increases. And there is a range of frequency for the value of transmembrane potential to decrease fast. It is shown that there is a strong relationship between the value of transmembrane potential and frequency components contained in the pulse. With more low-frequency components, the value of transmembrane potential is increasing and thus a better sterilization effect can be obtained. By comparing the frequency components contained in square wave pulse, exponentially decaying pulse, oscillatory pulse and their sterilization effect respectively, the analysis results about the relationship of transmembrane potential-frequency presented in this paper is validated. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Bacteria; Electric field analysis; Transmembrane potential; Frequency components
1.
Introduction
Pulsed electric field food processing is one of non-thermal food preservation methods. In this method, food is exposed to a pulsed electric field. Transmembrane potential is induced on the cell membrane of bacteria by the pulsed electric field applied. The value of the transmembrane potential increases as the external electric field intensity increases. When electric field intensity exceeds a critical value, cell membrane may breakdown. Consequently, the object of food sterilization is achieved (Xu and Wang, 2005). Comparing to the thermal food processing that is widely used, protein is not damaged, vitamins and volatile flavors are not loss and the sensory and nutritional properties remain. Assuming that the geometric parameters of bacteria are not affected by frequency and intensity of pulsed electric field, the transmembrane potential depends on the frequency of the electric field and its value varied significantly. For the three types of pulsed electric field mostly used in food processing, the transmembrane potential on cell membrane of bacteria caused by frequency components contained in the pulses are different.
∗
2.
Theory
The spherical model of a bacterium is shown in Fig. 1 (Pavlin and Miklavcic, 2003). The inside radius and outside radius of the membrane are denoted by ri and re respectively. The complex conductivity of cytoplasm, cell membrane and ∗ and ∗ respecextracellular solution are denoted by i∗ , im e tively. Assuming the extracellular solution, cytoplasm and cell membrane are linear, isotropic, and homogeneous dielectric medium. The electric potential at any point is satisfies the Laplace’s equation. The Laplace’s equation can be expressed as Eq. (1) in spherical coordinate (Guru and Hiziroglu, 2004) ∇2 =
1 ∂ r2 ∂r
∂ 2 r
∂r
+
r2
1 ∂ sin ∂
sin
∂ ∂
+
1 ∂2 =0 r2 sin ∂ϕ2 (1)
Taking the spherical center as the original point, there is symmetry about ϕ co-ordinate, that is, solutions depend on r and but not on ϕ. Then Eq. (1) reduces to
Corresponding author. E-mail address: [email protected] (W. Xinlao). Received 17 April 2008; Received in revised form 11 October 2008; Accepted 28 October 2008 0960-3085/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fbp.2008.10.003
262
food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
Nomenclature re ri
outside radius inside radius
Greek symbols εi permittivity of cytoplasm εim permittivity of cell membrane permittivity of extracellular solution εe conductivity e∗ complex conductivity of extracellular solution complex conductivity of cytoplasm i∗ ∗ im complex conductivity of cell membrane electric potential i electric potential in cytoplasm electric potential in cell membrane im e electric potential in extracellular solution transmembrane potential of cell membrane m ε permittivity i conductivity of cytoplasm conductivity of cell membrane im e conductivity of extracellular solution
Fig. 1 – The spherical model for bacterium.
Therefore, the general potential solutions associated with the three media are
i = 1 ∂ r2 ∂r
∂ 2 r
∂r
+
r2
1 ∂ sin ∂
sin
∂ ∂
r2
∂2 ∂ + 2r ∂r ∂r2
=−
1
=0
1
r2
∂ ∂2 + 2r ∂r ∂r2
∂2
∂ 2
+ ctg
∂2
∂
+ ctg ∂ ∂ 2
(3)
d2 d2 r
+ 2r
=˛
(4)
= −˛
d − n(n + 1) = 0 dr
[Cn rn + Dn r−(n+1) ]Pn (cos ),
= a1 rn + a2 r−(n+1)
e =
(11)
∞
[En rn + Fn r−(n+1) ]Pn (cos ),
r > re
(12)
where An , Bn , Cn , Dn , En , Fn are constants to be determined; Pn (cos ) is Legendre Polynomials; n is any positive integer including 0. The intracellular potential remains infinite as r → 0
i = (5)
∞
An rn Pn (cos )
(13)
n=0
The extracellular potential e tends to applied electric field potential as r → ∞ (6)
∞
Fn r−(n+1) Pn (cos )
(14)
n=0
(7)
In terms of ˛ = n(n + 1), Eq. (5) takes the form of
(8)
Substituting x = cos and = P(x) into Eq. (8), it can be written as d2 P dP − 2x + n(n + 1)P = 0 dx dx2
ri < r ≤ re
n=0
Eq. (6) has the solution
(1 − x2 )
∞
e = −E0 (t)r cos +
d
d2
+ ctg + n(n + 1) = 0 d d 2
(10)
n=0
Let ˛ = n(n + 1). Eq. (4) then satisfies the Euler equation r2
im =
∂
∂
0 < r ≤ ri
(2)
The left-hand side of Eq. (3) depends only on r, while the righthand side depends only on . This can only occur when both sides are equal to a constant, denoted by ˛ 1
[An rn + Bn r−(n+1) ]Pn (cos ),
n=0
By solving for in a separable form (r,) = (r) () (Sun and Liu, 2000), Eq. (2) becomes 1
∞
(9)
The solution of Eq. (9) is given by the Legendre polynomials Pn (x).
The electric potential and the normal components of current densities at the interface between dielectrics are continuous under time-varying electric field (Vladimir et al., 2001). Therefore, the boundary conditions at each interface can be expressed as follows:
i (ri , ) = im (ri , ) ∂ ∗ ∂im = im i∗ i ∂r |r=ri ∂r |r=ri
(15)
im (re , ) = e (re , ) ∂e ∗ ∂im = e∗ im ∂r |r=re ∂r |r=re
(16)
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food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
⎧ l l ⎪ ⎪ ⎪ An ri n Pn (cos ) = [Cn ri n + Dn ri −(n+1) ]Pn (cos ) ⎪ ⎨ n=0
n=0
l−1 l−1 ⎪ ⎪ ⎪ n ⎪ A r P (cos ) = [Cn ri n + Dn ri −(n+1) ]Pn (cos ) n n i ⎩ n=0
The transmembrane potential of cell membrane is defined (17)
m = im (r, cos )|r=re − i (r, cos )|r=ri Substituting the solution of (27) into (28), we obtain
n=0
⎧ l ⎪ ⎪ ⎪ ni∗ An ri n−1 Pn (cos ) ⎪ ⎪ ⎪ ⎪ n=0 ⎪ ⎪ ⎪ ⎪ l ⎪ ⎪ ⎪ ∗ ∗ ⎪ = [nim Cn ri n−1 − (n + 1)im Dn ri −(n+2) ]Pn (cos ) ⎪ ⎨ n=0
l−1 ⎪ ⎪ ⎪ ⎪ ni∗ An ri n−1 Pn (cos ) ⎪ ⎪ ⎪ ⎪ n=0 ⎪ ⎪ ⎪ l−1 ⎪ ⎪ ⎪ ∗ ∗ ⎪ = [nim Cn ri n−1 − (n + 1)im Dn ri −(n+2) ]Pn (cos ) ⎪ ⎩
m =
(18)
(19)
Eq. (18) can be written as (20)
Cn re + Dn re
−(n+1)
n
= En re + Fn re
−(n+1)
∗ C r n−1 − (n + 1) ∗ D r −(n+2) nim n e im n e
=
ne∗ En re n−1
− (n + 1)e∗ Fn re −(n+2)
⎪ ⎪ Cn re n + Dn re −(n+1) = Fn re −(n+1) ⎪ ⎪ ⎪ ⎪ ⎩
(21)
(23)
By solving (23), we have that An = Cn = Dn = Fn = 0, and the following representations are obtained: 0 ≤ r ≤ ri
im = (C1 r + D1 r−2 ) cos ,
cos
∗ Z2 = −3e∗ ri 3 re 3 E0 (t)(im − i∗ )
(31)
∗ ∗ ∗ ∗ M = 2ri 3 (im − i∗ )(e∗ − im ) + re3 (i∗ + 2im )(im + 2e∗ )
(32)
Analysis
The complex conductivity can be expressed as * = + j2 fε under time-varying electric field (Kotnik and Miklavcic, 2006). Where is conductivity, ε is permittivity. The geometric and electrical properties used in this paper are listed as follows (Mi et al., 2007; Kotnik and Miklavcic, 2000; Alcamo, 1998): Radius (m): re = 5.04
i = 0.5;
im = 1 × 10−5 ;
e = 0.6
εi = 5 × 10−10 ;
εim = 8.8 × 10−11 ;
εe = 0.7 × 10−9
By substituting these parameters into (30), transmembrane potential of cell membrane can be obtained. Fig. 2 shows the value of the transmembrane potential as a function of frequency. It can be seen from Fig. 2 that when frequency increases, m (f) decreases gradually from 0.95 V to 0.92 V as f < 106 Hz. In the range of 106 Hz < f < 108 Hz, m (f) decreases from 0.92 V to 0.06 V. And m (f) decreases from 0.06 V to 0.03 V as f > 108 Hz. At present, the widely used pulsed voltage waves for pulsed electric field sterilization are three types: square wave pulse, exponentially decaying pulse and oscillatory pulse. Fig. 3(a)
(24) ri < r ≤ re
(25)
r > re
(26)
e = (−E0 (t)r + F1 r−2 ) cos ,
From Eqs. (15) and (16), we obtain
⎧ D1 ⎪ A1 ri − C1 ri − 2 = 0 ⎪ ⎪ ri ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ D1 = 0 ⎪ i A1 − im C1 + 2im ⎨ 3 rn
D1 F1 ⎪ ⎪ C1 re + 2 − 2 = −E0 (t)re ⎪ ⎪ r re e ⎪ ⎪ ⎪ D ⎪ ⎩ ∗ C1 − 2 ∗ 1 + 2e∗ F1 = −E0 (t)e∗ im im 3 3 re
re
(29)
Permittivity (F/m): (22)
∗ C r n−1 − (n + 1) ∗ D r −(n+2) = −(n + 1) ∗ F r −(n+2) nim n e e n e im n e
i = A1 r cos ,
1 ri 2
Conductivity (S/m):
From Eq. (14) it is apparent that E1 = −E0 (t) for n = 1 and En = 0 for n = / 1. Therefore, for n = / 1, we get
⎧ An ri n = Cn ri n + Dn ri −(n+1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ C r n−1 − (n + 1) ∗ D r −(n+2) ⎨ ni∗ An ri n−1 = nim n i im n i
re 2
−
(30)
ri = 5; Similarly, from (16), it can be deduced that n
1
∗ + i∗ ) Z1 = −3e∗ re 3 E0 (t)(2im
3.
where l is a positive integer. Eq. (17) can be written as
∗ ∗ ni∗ An ri n−1 = nim Cn ri n−1 − (n + 1)im Dn ri −(n+2)
Z1 Z2 (re − ri ) cos + M M
where
n=0
An ri n = Cn ri n + Dn ri −(n+1)
(28)
(27)
Fig. 2 – Curves of transmembrane potential–frequency.
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food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
Fig. 3 – The three types of pulse waves used in PEF.
shows a square wave with duration of 15 s, Fig. 3(b) shows an exponentially decaying pulse with duration of 15 s and Fig. 3(c) shows an oscillatory pulse with duration of 15 s. Previous experiment results show that the square wave pulse is the most effective in sterilization, the exponentially decaying pulse is less effective and the oscillatory pulse is the least. The experiment results can be well explained by the research in this paper. With the same pulse width and amplitude, the three types of pulsed waves (square wave pulse, exponentially decaying pulse and oscillatory pulse) are analyzed with FFT analysis. The frequency spectrum distributions obtained by FFT are shown in Fig. 4. The one shown in Fig. 4(a) illustrates frequency spectrum of square wave pulse. The one shown in Fig. 4(b) illustrates frequency spectrum of exponentially decaying pulse. And the one shown in Fig. 4(c) illustrates frequency spectrum of oscillatory pulse. It can be seen from the figures that the dominant frequency components of square wave pulse with peak value of 753 mV are in the range of 0–160 kHz. The dominant frequency components of exponentially decaying pulse with peak value of 237 mV are in the
Fig. 4 – FFT analysis result of the three types of pulse waves used in PEF.
range of 0–425 kHz. The dominant frequency components of exponentially decaying pulse with peak value of 205 mV are in the range of 720 kHz to 1.2 MHz
4.
Discussion
From Fig. 2, it can be seen that if the electric field intensity is constant, the value of transmembrane potential is increasing inverse proportion to the frequency components of pulses applied. Therefore, in theory, the most efficient method for sterilization is to apply DC voltage to liquid foods. However, in general, the liquid foods are dielectrics with high conductivities (Wouters et al., 1999). It is difficult to produce DC electric field of sufficient intensity to inactivate bacteria. The electric source has to generate a very large conduction current to fulfil the requirement. And the heat produced by the conduction current will be significant. Therefore, the pulsed electric field is usually adopted for liquid food sterilization (Barbosa-Cánovas and Rodríguez, 2002). Pulse rising edge and falling edge contain significant high frequency components. Therefore, mainly depend on pulse width but not
food and bioproducts processing 8 7 ( 2 0 0 9 ) 261–265
pulse rising edge and falling edge, the bacteria are inactivated. Previous studies have demonstrated that the square wave pulse is the most effective in sterilization, the exponentially decaying pulse is less effective and the oscillatory pulse is the least (Jeyamkondan et al., 1999). By comparing the three types of pulse waves, it can be seen from Fig. 4 that the low-frequency components contained in the square wave pulse are the most significant. And the oscillatory pulse contains the most significant high frequency components. It can be known from the above analysis that the value of transmembrane potential is the highest if a square pulsed electric field is applied. And the value is lower than the other two if an oscillatory pulsed electric field is applied.
5.
Conclusion
The value of transmembrane potential decreases as the frequency of external electric field increases. And there is a range of frequency for the value of transmembrane potential to decrease fast. It is shown that there is a strong relationship between the value of transmembrane potential and frequency components contained in the pulse. With more lowfrequency components, the value of transmembrane potential is increasing and thus a better sterilization effect can be obtained. The bacteria have various shapes. Therefore, an additional error is introduced with the simplified model. Electroporation may result in changing of equivalent conductivity and equivalent permittivity of cell membrane, which may influence the value of transmembrane potential. In subsequent studies, the model will be modified so it more accurately reflects shapes of bacteria. And the functional relationship of electric field intensity and electrical parameters of membrane will be derived.
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