⎡
⎛ 1 u +1
u ⎞ ⎞ ⎤
x
2
2
φ u v,ϕ, ) = S ( )
t +T t
ln
−
u −1 1− v cosϕ, u 2 < ≤ u 1
( ,
t
( )
u
m
m
m
⎣
⎢
⎝ ⎜
2
⎠ ⎟ ⎦
⎥ ×
2 u −1 u −1
(2.11d)
⎡
⎛ ⎛ 1 u +1
u ⎞ ⎤
x
2
2
o
o x
o
( ,
t
( )
φ u v,ϕ, ) = −c E ( )
t +T t
ln
−
u −1 1− v cosϕ, u > u 1 (2.11e)
⎣
⎢
⎝ ⎜
2
⎠ ⎟ ⎦
⎥ ×
2 u −1 u −1
x
x
x
x
x
where φ oc u v t φ om u v t φ c u v t φ m u v t and φ o ( , , ) are
( , , ),
( , , ),
( , , ),
( , , ),
u v t
the potentials
induced by E x (t) at organelle cytoplasm, organelle membrane, cellular cytoplasm, the
plasma membrane, and outer region, respectively. Here E x ( )
t = E 0 ( )sin( )
t
α , while S oc (t),
S om (t), S c (t), S m (t), T om (t), T c (t), T m (t), and T o (t) are the time-varying coefficients that can
be determined by applying matching boundary conditions at the interfaces of the five
regions. Invoking continuity in potential and current density at the membrane interfaces leads to the following boundary conditions:
φ ( , ,ϕ t
u v , ) = φ ( , ,ϕ t u
(2.12a)
u v , )
o
m
u
1
1
φ ( , ,ϕ t
u v , ) u = φ ( , ,ϕ t
u v , ) u
(2.12b)
m
c
2
2
φ ( , ,ϕ, )
t
c u v
φ ( , ,ϕ, )
t
= om u v
(2.12c)
u
u
3
3
φ ( , ,ϕ, )
t
om u v
u
φ ( , ,ϕ, )
t
= oc u v
u
(2.12d)
4
4
o
o
∂E ( )
t
m
∂E
m m
( )
t
σ E ( )
t + ε
u
= σ ( )
t E ( )
t + ε ( )
t
u
(2.12e)
o u
o
m
u
m
∂t u
∂t u
1
1
m
m
∂E ( )
t
c
∂E
c c
( )
t
σ ( )
t E ( )
t + ε ( )
t
u
= σ E ( )
t + ε
u
(2.12f)
m
u
m
c u
c
∂t u
∂t u
2
2
c
c
∂E ( )
t
om
∂ ∂E
om
( )
t
σ E ( )
t + ε
u
= σ ( )
t E ( )
t + ε ( )
t
u
(2.12g)
c u
c
om
u
om
∂t u
∂t u
3
3
om
E
oc
om
∂E ( )
t
oc
∂ ( )
t
σ ( )
t E ( )
t + ε ( )
t
u
= σ oc E u ( )
t + + ε oc
u
(2.12h)
om
u
om
∂t u
∂t u
4
4
Here E
φ/ u
= −∂ ∂ is the electric field along the u direction.
The TMP, Φ, for an oblate spheroidal cell can similarly be obtained by solving the
Laplace equation in oblate spheroid coordinates with respect to E x (t) and E z (t). This leads
to the following:
u
x
2
2
φ u v,ϕ, ) = I ( )
t u +1 1− v cosϕ, u ≤ u 4
(2.13a)
( ,
t
oc
oc
88
Electromagnetic Fields in Biological Systems
⎛ 1 u +1
u ⎞ ⎞ ⎤
x
2
2
φ u v,ϕ, ) = S ( )
t +T t
ln
−
u −1 1− v cosϕ, u 2 < ≤ u 1
( ,
t
( )
u
m
m
m
⎣
⎢
⎝ ⎜
2
⎠ ⎟ ⎦
⎥ ×
2 u −1 u −1
(2.11d)
⎡
⎛ ⎛ 1 u +1
u ⎞ ⎤
x
2
2
o
o x
o
( ,
t
( )
φ u v,ϕ, ) = −c E ( )
t +T t
ln
−
u −1 1− v cosϕ, u > u 1 (2.11e)
⎣
⎢
⎝ ⎜
2
⎠ ⎟ ⎦
⎥ ×
2 u −1 u −1
x
x
x
x
x
where φ oc u v t φ om u v t φ c u v t φ m u v t and φ o ( , , ) are
( , , ),
( , , ),
( , , ),
( , , ),
u v t
the potentials
induced by E x (t) at organelle cytoplasm, organelle membrane, cellular cytoplasm, the
plasma membrane, and outer region, respectively. Here E x ( )
t = E 0 ( )sin( )
t
α , while S oc (t),
S om (t), S c (t), S m (t), T om (t), T c (t), T m (t), and T o (t) are the time-varying coefficients that can
be determined by applying matching boundary conditions at the interfaces of the five
regions. Invoking continuity in potential and current density at the membrane interfaces leads to the following boundary conditions:
φ ( , ,ϕ t
u v , ) = φ ( , ,ϕ t u
(2.12a)
u v , )
o
m
u
1
1
φ ( , ,ϕ t
u v , ) u = φ ( , ,ϕ t
u v , ) u
(2.12b)
m
c
2
2
φ ( , ,ϕ, )
t
c u v
φ ( , ,ϕ, )
t
= om u v
(2.12c)
u
u
3
3
φ ( , ,ϕ, )
t
om u v
u
φ ( , ,ϕ, )
t
= oc u v
u
(2.12d)
4
4
o
o
∂E ( )
t
m
∂E
m m
( )
t
σ E ( )
t + ε
u
= σ ( )
t E ( )
t + ε ( )
t
u
(2.12e)
o u
o
m
u
m
∂t u
∂t u
1
1
m
m
∂E ( )
t
c
∂E
c c
( )
t
σ ( )
t E ( )
t + ε ( )
t
u
= σ E ( )
t + ε
u
(2.12f)
m
u
m
c u
c
∂t u
∂t u
2
2
c
c
∂E ( )
t
om
∂ ∂E
om
( )
t
σ E ( )
t + ε
u
= σ ( )
t E ( )
t + ε ( )
t
u
(2.12g)
c u
c
om
u
om
∂t u
∂t u
3
3
om
E
oc
om
∂E ( )
t
oc
∂ ( )
t
σ ( )
t E ( )
t + ε ( )
t
u
= σ oc E u ( )
t + + ε oc
u
(2.12h)
om
u
om
∂t u
∂t u
4
4
Here E
φ/ u
= −∂ ∂ is the electric field along the u direction.
The TMP, Φ, for an oblate spheroidal cell can similarly be obtained by solving the
Laplace equation in oblate spheroid coordinates with respect to E x (t) and E z (t). This leads
to the following:
u
x
2
2
φ u v,ϕ, ) = I ( )
t u +1 1− v cosϕ, u ≤ u 4
(2.13a)
( ,
t
oc
oc
88
Electromagnetic Fields in Biological Systems