87
Pulsed Electric Fields in Biological Cells and Membranes
angle φ, the last term in Equation 2.7 can be omitted when only E z is present in the system. Hence, one gets the following:
2
∂ ⎡ 2
∂Φ ⎤ ∂ ⎡
∂ Φ ⎤
∇ Φ =
(u −1)
+
(1 − v
2
)
⎢
⎥
⎢
= = 0
(2.8)
∂ ⎣
∂ ⎦ ∂
⎥
u
u
v ⎣
∂ v ⎦
By separation of variables, Equation 2.8 can be easily solved to yield the following:
φ
z
oc (u, v,t ) = A oc (t )uv, u ≤ u 4
(2.9a)
φ
z
⎡
⎛ u u +1 ⎞ ⎤
om (u ,v,t) = v A m (t)u +
⎢
B
⎣
o
om (t ) ⎜ ln
−1 ⎟ ⎥ , u
⎝ 2 u −1 ⎠ ⎠ ⎦
4 < u < u 3
(2.9b)
z
⎡
⎛ u u +1 ⎞ ⎤
φ c (u, v,t ) = v ⎢ A c (t)u + B c (t ) ⎜ ln
−1 ⎟
<
⎥ , u u < u
(2.9c)
⎣ ⎣
⎝ 2 u −1 ⎠ ⎦
3
2
⎡
⎛ u u +1 ⎞ ⎤
φ
z
m (u ,v,t) = v ⎢ A m (t )u + B m (t ) ⎜ ln
−1 ⎟
2 < <
⎥ , u u u 1
(2.9d)
⎣ ⎣
⎝ 2 u −1 ⎠ ⎦
⎡
⎛ u u +1
z
⎞ ⎤
φ o (u,v,t) = v −
⎢ c o E
⎣
z (t)u + B o (t ) ⎜ ln
−1 ⎟ ⎥ , u > u
(2.9e)
⎝ 2 u −1 ⎠ ⎠ ⎦
1
Here φ
z
oc (u, v,t), φ
z
z
z
om (u, v,t), φ c (u, v,t), φ m (u, v,t), and φ
z
o (u , v, t) are the potentials
induced by E z at the organelle cytoplasm, the organelle membrane, the cellular cytoplasm,
the plasma membrane, and the environment, respectively. Also, E z (t ) = E 0 (t )cos(α) and
u i = b i /c , i = 1, 2 ,3, 4.
Finally, A oc (t), A om (t), A c (t), A m (t), B om (t), B c (t), B m (t), and B o (t) are the coefficients that
can be determined by applying matching boundary conditions at the interfaces of the
five regions. The TMP Δφ
z
induced by an electric field E z can be obtained as follows:
Δφ
z
= φ
z
m
m (u 2 ,v,ϕ,t ) − φ
z
m (u 1 ,v,ϕ,t )
(2.10a)
Δφ
z
z
om = φ om (u 4 ,v,ϕ,t ) − φ
z
om (u 3 ,v,ϕ,t )
(2.10b)
In order to obtain the TMP induced by an electric field E x alone, one can express
Φ(u,v,φ) = U(u)V(v)F(φ). Algebraic manipulation then leads to the following final
solution:
φ
x
oc (u,v,ϕ,t ) = S (t) u
2
−1 1− v
2
oc
cosϕ, u ≤ u 4
(2.11a)
x
⎡
⎛ ⎛ 1 u +1
u ⎞ ⎤
φ om (u ,v,ϕ,t ) = S om (t )+T om (t ) ⎜ ln
−
⎟ × u
2
−1 1−
ϕ
⎢
2
v
2
cos , u 4 < u ≤
⎥
u 3
⎣
⎝ 2 u −1 u −1 ⎠ ⎦
(2.11b)
⎡
⎛ 1
x
u +1
u ⎞ ⎞ ⎤
φ c (u ,v,ϕ,t ) = S c (t )+T c (t ) ⎜ ln
−
⎟ ×
ϕ
⎢
u
2
−1 1− v
2
cos , u < u ≤ u
⎣
⎝
2
⎠ ⎥
2 u −1 u −1 ⎦
3
2
(2.11c)
Pulsed Electric Fields in Biological Cells and Membranes
angle φ, the last term in Equation 2.7 can be omitted when only E z is present in the system. Hence, one gets the following:
2
∂ ⎡ 2
∂Φ ⎤ ∂ ⎡
∂ Φ ⎤
∇ Φ =
(u −1)
+
(1 − v
2
)
⎢
⎥
⎢
= = 0
(2.8)
∂ ⎣
∂ ⎦ ∂
⎥
u
u
v ⎣
∂ v ⎦
By separation of variables, Equation 2.8 can be easily solved to yield the following:
φ
z
oc (u, v,t ) = A oc (t )uv, u ≤ u 4
(2.9a)
φ
z
⎡
⎛ u u +1 ⎞ ⎤
om (u ,v,t) = v A m (t)u +
⎢
B
⎣
o
om (t ) ⎜ ln
−1 ⎟ ⎥ , u
⎝ 2 u −1 ⎠ ⎠ ⎦
4 < u < u 3
(2.9b)
z
⎡
⎛ u u +1 ⎞ ⎤
φ c (u, v,t ) = v ⎢ A c (t)u + B c (t ) ⎜ ln
−1 ⎟
<
⎥ , u u < u
(2.9c)
⎣ ⎣
⎝ 2 u −1 ⎠ ⎦
3
2
⎡
⎛ u u +1 ⎞ ⎤
φ
z
m (u ,v,t) = v ⎢ A m (t )u + B m (t ) ⎜ ln
−1 ⎟
2 < <
⎥ , u u u 1
(2.9d)
⎣ ⎣
⎝ 2 u −1 ⎠ ⎦
⎡
⎛ u u +1
z
⎞ ⎤
φ o (u,v,t) = v −
⎢ c o E
⎣
z (t)u + B o (t ) ⎜ ln
−1 ⎟ ⎥ , u > u
(2.9e)
⎝ 2 u −1 ⎠ ⎠ ⎦
1
Here φ
z
oc (u, v,t), φ
z
z
z
om (u, v,t), φ c (u, v,t), φ m (u, v,t), and φ
z
o (u , v, t) are the potentials
induced by E z at the organelle cytoplasm, the organelle membrane, the cellular cytoplasm,
the plasma membrane, and the environment, respectively. Also, E z (t ) = E 0 (t )cos(α) and
u i = b i /c , i = 1, 2 ,3, 4.
Finally, A oc (t), A om (t), A c (t), A m (t), B om (t), B c (t), B m (t), and B o (t) are the coefficients that
can be determined by applying matching boundary conditions at the interfaces of the
five regions. The TMP Δφ
z
induced by an electric field E z can be obtained as follows:
Δφ
z
= φ
z
m
m (u 2 ,v,ϕ,t ) − φ
z
m (u 1 ,v,ϕ,t )
(2.10a)
Δφ
z
z
om = φ om (u 4 ,v,ϕ,t ) − φ
z
om (u 3 ,v,ϕ,t )
(2.10b)
In order to obtain the TMP induced by an electric field E x alone, one can express
Φ(u,v,φ) = U(u)V(v)F(φ). Algebraic manipulation then leads to the following final
solution:
φ
x
oc (u,v,ϕ,t ) = S (t) u
2
−1 1− v
2
oc
cosϕ, u ≤ u 4
(2.11a)
x
⎡
⎛ ⎛ 1 u +1
u ⎞ ⎤
φ om (u ,v,ϕ,t ) = S om (t )+T om (t ) ⎜ ln
−
⎟ × u
2
−1 1−
ϕ
⎢
2
v
2
cos , u 4 < u ≤
⎥
u 3
⎣
⎝ 2 u −1 u −1 ⎠ ⎦
(2.11b)
⎡
⎛ 1
x
u +1
u ⎞ ⎞ ⎤
φ c (u ,v,ϕ,t ) = S c (t )+T c (t ) ⎜ ln
−
⎟ ×
ϕ
⎢
u
2
−1 1− v
2
cos , u < u ≤ u
⎣
⎝
2
⎠ ⎥
2 u −1 u −1 ⎦
3
2
(2.11c)