Pulsed Electric Fields in Biological Cells and Membranes
89
φ
x
⎡
⎛
u ⎞ ⎤
om (u ,v,ϕ,t ) = ⎢ I (
−1
om t ) + J om (t ) ⎜ ⎜ cot (u )−
2
+1 1−
2
cosϕ,
⎟ ×
⎥ u
v
u
⎣
⎝
u +1 ⎠ ⎦
4 < u ≤ 3
2
u
(2.13b)
x
⎡
⎛
u ⎞ ⎤
φ c (u,v,ϕ,t ) = ⎢ I c (t ) + J c (t ) ⎜ cot
−1
(u )− 2 ⎟ ⎟ ×
⎥ u
2
+1 1− v
2
cosϕ, u 3 < u ≤ u 2 (2.13c)
⎣
⎝
u +1 ⎠ ⎦
⎡
u ⎞ ⎤
φ
x
⎛
m (u, v,ϕ,t ) = ⎢ I m (t)+ J m (t ) ⎜ cot
−1
(u )−
⎟ ×
⎥ u
2
+1 1−
2
cosϕ
2
⎟
v
, u 2 < u ≤ u
⎣
⎝
u +1 ⎠ ⎦
1
(2.13d)
⎡ ⎡
⎛
u ⎤
φ
x
⎞
o (u, v,ϕ,t ) = −cE x +
⎢
J o (t) ⎜ cot
−1
(u )−
⎟ ×
⎥ u
2
+1 1− v
2
cosϕ, u > u 1
2
(2.13e)
⎣
⎝
u +1 ⎠ ⎦
Here, I oc (t), I om (t), I c (t), I m (t), J om (t), J c (t), J m (t), and J o (t) are the coefficients that can be
solved by the Equation (2.12).
Using the analyses described earlier, it was shown that the response of prolate spheroids is faster than that of the sphere, with the outer membrane reaching its steady-state
value in about 2 μs. The simulation result also shows that the TMP across an inner
organelle could exceed the value across the plasma membrane at least over the first 0.4 μs
or so, indicating a possibility of intracellular electromanipulation. The TMP induced by
pulsed external voltages is predicted to be higher in oblate spheroids in comparison to
both spherical and prolate spheroidal cells and to occur sooner. This results from the
flattening of the surface area in this oblate spheroidal geometry. The result suggests that
field-assisted drug delivery and molecular/ionic uptake for chemotherapy could work
best for flatter-shaped cells. Hence, external fields could conceivably be applied, in addition to an e lectroporation pulse, to first orient the cells for optimal uptake.
A representative result, in this context, is shown in Figure 2.8 for a double-shelled
prolate spheroidal cell (Hu and Joshi 2009). The outer shell was taken to have an equatorial radius a 1 , an inner equatorial radius a 2 , an outer polar radius b 1 , an inner polar
radius b 2 , and focal distance c = (b 2
2
1 − a 2 ½
2 ½
1 ) = (b 2 − a 2 ) . Similarly, the inner shell had
an equatorial radius a 3 , an inner equatorial radius a 4 , an outer polar radius b 3 , and an
inner polar radius b 4 . Using a 1 :b 1 = 3:5 with b 1 = 5 μm and b 3 = 4 μm, predictions of the
TMP versus time are shown in Figure 2.8 at polar angle θ = 15° for both fixed membrane conductivity (i.e., without considering electroporation) and variable m embrane
conductivity. The electrical excitation pulse was trapezoidal with rise, fall, and ON
times of 1.5, 1.5, and 10 ns, respectively, with a maximum intensity of 100 kV/cm. With
variable membrane conductivity, the TMP goes to zero quickly because the membrane
acts as a “leaky capacitor.” On the other hand, the TMP for membrane with constant
conductivity takes a much longer time to fall back to zero because of the larger time
constant.
Numerical microdosimetry techniques based on the finite-element method (FEM)
have been utilized to evaluate the electric field and potential distributions on the
r ealistic-shaped cells (Wang, Alfadhl, and Chen 2008). It was found that the shape and
orientation of cells can strongly affect the field strength and distribution.
89
φ
x
⎡
⎛
u ⎞ ⎤
om (u ,v,ϕ,t ) = ⎢ I (
−1
om t ) + J om (t ) ⎜ ⎜ cot (u )−
2
+1 1−
2
cosϕ,
⎟ ×
⎥ u
v
u
⎣
⎝
u +1 ⎠ ⎦
4 < u ≤ 3
2
u
(2.13b)
x
⎡
⎛
u ⎞ ⎤
φ c (u,v,ϕ,t ) = ⎢ I c (t ) + J c (t ) ⎜ cot
−1
(u )− 2 ⎟ ⎟ ×
⎥ u
2
+1 1− v
2
cosϕ, u 3 < u ≤ u 2 (2.13c)
⎣
⎝
u +1 ⎠ ⎦
⎡
u ⎞ ⎤
φ
x
⎛
m (u, v,ϕ,t ) = ⎢ I m (t)+ J m (t ) ⎜ cot
−1
(u )−
⎟ ×
⎥ u
2
+1 1−
2
cosϕ
2
⎟
v
, u 2 < u ≤ u
⎣
⎝
u +1 ⎠ ⎦
1
(2.13d)
⎡ ⎡
⎛
u ⎤
φ
x
⎞
o (u, v,ϕ,t ) = −cE x +
⎢
J o (t) ⎜ cot
−1
(u )−
⎟ ×
⎥ u
2
+1 1− v
2
cosϕ, u > u 1
2
(2.13e)
⎣
⎝
u +1 ⎠ ⎦
Here, I oc (t), I om (t), I c (t), I m (t), J om (t), J c (t), J m (t), and J o (t) are the coefficients that can be
solved by the Equation (2.12).
Using the analyses described earlier, it was shown that the response of prolate spheroids is faster than that of the sphere, with the outer membrane reaching its steady-state
value in about 2 μs. The simulation result also shows that the TMP across an inner
organelle could exceed the value across the plasma membrane at least over the first 0.4 μs
or so, indicating a possibility of intracellular electromanipulation. The TMP induced by
pulsed external voltages is predicted to be higher in oblate spheroids in comparison to
both spherical and prolate spheroidal cells and to occur sooner. This results from the
flattening of the surface area in this oblate spheroidal geometry. The result suggests that
field-assisted drug delivery and molecular/ionic uptake for chemotherapy could work
best for flatter-shaped cells. Hence, external fields could conceivably be applied, in addition to an e lectroporation pulse, to first orient the cells for optimal uptake.
A representative result, in this context, is shown in Figure 2.8 for a double-shelled
prolate spheroidal cell (Hu and Joshi 2009). The outer shell was taken to have an equatorial radius a 1 , an inner equatorial radius a 2 , an outer polar radius b 1 , an inner polar
radius b 2 , and focal distance c = (b 2
2
1 − a 2 ½
2 ½
1 ) = (b 2 − a 2 ) . Similarly, the inner shell had
an equatorial radius a 3 , an inner equatorial radius a 4 , an outer polar radius b 3 , and an
inner polar radius b 4 . Using a 1 :b 1 = 3:5 with b 1 = 5 μm and b 3 = 4 μm, predictions of the
TMP versus time are shown in Figure 2.8 at polar angle θ = 15° for both fixed membrane conductivity (i.e., without considering electroporation) and variable m embrane
conductivity. The electrical excitation pulse was trapezoidal with rise, fall, and ON
times of 1.5, 1.5, and 10 ns, respectively, with a maximum intensity of 100 kV/cm. With
variable membrane conductivity, the TMP goes to zero quickly because the membrane
acts as a “leaky capacitor.” On the other hand, the TMP for membrane with constant
conductivity takes a much longer time to fall back to zero because of the larger time
constant.
Numerical microdosimetry techniques based on the finite-element method (FEM)
have been utilized to evaluate the electric field and potential distributions on the
r ealistic-shaped cells (Wang, Alfadhl, and Chen 2008). It was found that the shape and
orientation of cells can strongly affect the field strength and distribution.