81
Pulsed Electric Fields in Biological Cells and Membranes
Brownian motion across the energy landscape. Continuum Smoluchowski theory yields
the pore density distribution function n(r, t), with r being the pore radius (Pastushenko
and Chizmadzhev 1983; Joshi and Schienbach 2000; Freeman, Wang, and Weaver 1994):
∂n(r,θ,t)/∂t – {D/(kT)} ∂[n(r,θ,t) {∂E(r,t)/∂r}]/∂r – D ∂ 2 n(r,θ,t)/∂r 2 = S(r,t)
(2.6)
where S(r,t) is the source (i.e., pore formation) term, D is a pore diffusion constant, k the
Boltzmann constant, T the absolute temperature, and E(r,t) the pore formation energy. In
Equation (2.6), n(r,t) denotes the pore density, with n(r,t)dr representing the pore number
of membrane pores lying within a radial spacing of r and r + dr at time t. The pore energy
E(r,t) is a sum of mechanical and electrical contributions. The mechanical contribution,
E m (r), includes the edge energy per unit length γ and the surface tension Τ of the membrane–water interface, while the electrical contribution results from the radial displacement of the lipid cylindrical walls of the pore by the aqueous electrolyte. Physically, the
radial displacement of the pore walls is facilitated by the electric field–driven Maxwell
stress tensor created at the membrane–electrolyte interface. Thus, traditionally, E(r,t) =
2πγ − πr 2 Τ − π{∆Φ(θ,t)} 2 r 2 . More sophisticated expressions for this pore energy E(r) have
also been discussed in the literature (Vasilkoski et al. 2006; Joshi et al. 2002).
The development of membrane pores is a time- and position-dependent sequence
of events. Pore formation and dynamics influence both the membrane conductivity
and permittivity and both can be taken into account at each time step. Solution for
n(r,θ,t) from Equation 2.6 then yields the dynamic conductance and capacitance of the
membrane as a function of time and angular position.
Results for the temporal development of membrane voltage across the outer and inner
cell membranes obtained using the distributed approach for a double-shelled model
(Joshi et al. 2004) are shown in Figure 2.3a and b. Two trapezoidal pulses, one having a
300-ns duration and the other having an 11-ns pulse, were chosen. For the 300-ns pulse,
membrane voltage across the inner organelle membrane exceeds that of the outer one
during the first 75 ns. For the 11-ns pulse, on the other hand, the membrane voltage
across the inner organelle exceeds that across the outer membrane for the entire pulse
duration. This clearly shows that by reducing the pulse width, it becomes possible to
charge inner membranes to higher potentials and thus trigger intracellular effects. Most
conventional electroporation pulses have fairly long durations (typically in the microsecond to millisecond range) and do not afford strong internal charging. Furthermore,
the use of extremely high electric field magnitudes is crucial in the present context
because only such strong excitation can drive bioeffects despite the short duration of the
external forcing function.
Traditionally, the neglect of the intracellular regions for potential bioelectric phenomena can be attributed to the notion that intracellular organelles are generally
shielded from external electric fields by the outer membranes (Adair 1991; Foster
and Schwan 1986). This is certainly true for low-frequency voltage excitations or for
timescales exceeding the cell membrane charging times that are roughly in the 1-ms
range (Krassowska and Neu 1994; Beeler and Reuter 1977). However, results of electric field penetration with high-frequency pulsing (similar conceptually to the results of
Figure 2.3b) have also been reported by other groups using a continuum model. As an
Pulsed Electric Fields in Biological Cells and Membranes
Brownian motion across the energy landscape. Continuum Smoluchowski theory yields
the pore density distribution function n(r, t), with r being the pore radius (Pastushenko
and Chizmadzhev 1983; Joshi and Schienbach 2000; Freeman, Wang, and Weaver 1994):
∂n(r,θ,t)/∂t – {D/(kT)} ∂[n(r,θ,t) {∂E(r,t)/∂r}]/∂r – D ∂ 2 n(r,θ,t)/∂r 2 = S(r,t)
(2.6)
where S(r,t) is the source (i.e., pore formation) term, D is a pore diffusion constant, k the
Boltzmann constant, T the absolute temperature, and E(r,t) the pore formation energy. In
Equation (2.6), n(r,t) denotes the pore density, with n(r,t)dr representing the pore number
of membrane pores lying within a radial spacing of r and r + dr at time t. The pore energy
E(r,t) is a sum of mechanical and electrical contributions. The mechanical contribution,
E m (r), includes the edge energy per unit length γ and the surface tension Τ of the membrane–water interface, while the electrical contribution results from the radial displacement of the lipid cylindrical walls of the pore by the aqueous electrolyte. Physically, the
radial displacement of the pore walls is facilitated by the electric field–driven Maxwell
stress tensor created at the membrane–electrolyte interface. Thus, traditionally, E(r,t) =
2πγ − πr 2 Τ − π{∆Φ(θ,t)} 2 r 2 . More sophisticated expressions for this pore energy E(r) have
also been discussed in the literature (Vasilkoski et al. 2006; Joshi et al. 2002).
The development of membrane pores is a time- and position-dependent sequence
of events. Pore formation and dynamics influence both the membrane conductivity
and permittivity and both can be taken into account at each time step. Solution for
n(r,θ,t) from Equation 2.6 then yields the dynamic conductance and capacitance of the
membrane as a function of time and angular position.
Results for the temporal development of membrane voltage across the outer and inner
cell membranes obtained using the distributed approach for a double-shelled model
(Joshi et al. 2004) are shown in Figure 2.3a and b. Two trapezoidal pulses, one having a
300-ns duration and the other having an 11-ns pulse, were chosen. For the 300-ns pulse,
membrane voltage across the inner organelle membrane exceeds that of the outer one
during the first 75 ns. For the 11-ns pulse, on the other hand, the membrane voltage
across the inner organelle exceeds that across the outer membrane for the entire pulse
duration. This clearly shows that by reducing the pulse width, it becomes possible to
charge inner membranes to higher potentials and thus trigger intracellular effects. Most
conventional electroporation pulses have fairly long durations (typically in the microsecond to millisecond range) and do not afford strong internal charging. Furthermore,
the use of extremely high electric field magnitudes is crucial in the present context
because only such strong excitation can drive bioeffects despite the short duration of the
external forcing function.
Traditionally, the neglect of the intracellular regions for potential bioelectric phenomena can be attributed to the notion that intracellular organelles are generally
shielded from external electric fields by the outer membranes (Adair 1991; Foster
and Schwan 1986). This is certainly true for low-frequency voltage excitations or for
timescales exceeding the cell membrane charging times that are roughly in the 1-ms
range (Krassowska and Neu 1994; Beeler and Reuter 1977). However, results of electric field penetration with high-frequency pulsing (similar conceptually to the results of
Figure 2.3b) have also been reported by other groups using a continuum model. As an