Biomedical Engineering Reference
In-Depth Information
pore creation and growth to
V
m
and lipid-lipid interaction energies within the
bilayer.
It is believed that most pores begin as tiny (
r<
1 nm) hydrophobic pores. If
these pores are to last, they must grow to become hydrophilic by allowing the
hydrophilic heads to fill in the pore sides. However, many newly created pores
are so small that they are quickly destroyed by lipid molecular fluctuations.
If the pore radii are smaller than some minimum pore radius
r<r
∗
they may
safely be neglected as they are quickly destroyed by these lipid fluctuations.
If lipids are created at a radius
r
r
∗
then they are able to transition to sus-
tainable hydrophilic and grow slightly to a minimum sustainable hydrophilic
radius,
r
m
.
In studies by Smith et al. (2004) the transient evolution of electropores is
described in such a way as to neglect the creation of the quickly destroyed
pores of
r<r
∗
:
≥
=
α
exp[
V
m
/V
ep
]
2
dN
dt
N
N
O
exp[
r
m
/
r
∗
]
−
(9.21)
where
N
is pore density,
N
O
is equilibrium pore density,
V
ep
is characteristic
electroporation voltage, and
α
is a rate coecient.
The important aspect of equation (9.21) to keep in mind is that the rate
at which pores are created is exponentially related to the square of the trans-
membrane voltage.
This method then keeps track of the time rate of growth of each of the
n
pore radii by the equation
V
m
F
max
1+
r
h
/(
r
+
r
t
)
+4
β
r
∗
r
4
1
r
−
dr
j
dt
D
kT
=
2
πγ
+2
πσ
eff
r,
j
=1
,
2
,...,n
(9.22)
where
D
is the diffusion coecient;
k
is the Boltzmann constant;
T
is the abso-
lute temperature;
F
max
,
r
h
,
r
t
,
β
; and
γ
are constants; and
σ
eff
is a geometric
function relating the membrane tension to the pore size.
In the rate equation, equation (9.22), the first term accounts for the trans-
membrane contribution to bilayer energy, and the second and third and fourth
terms denote geometric and spatial membrane effects (repulsion of lipid heads,
membrane tension, and the pore perimeter energy, respectively). The domi-
nating influence of pore growth during the application of the electroporation
pulse is the first term, which is a function of the square of the transmembrane
potential.
Typically, in membrane electroporation studies, lateral variations of per-
meability are neglected, and only transient effects are modeled. In the study
by Smith et al. (2004) the membrane is assumed to be a homogenous perfusion
and the transient behavior of the transmembrane potential is approximated
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