Membrane-Based Nanotechnology and Drug Delivery - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 128

exerts natural effects on all particle flow across the membrane. The second agent is
the gradient of solute concentration between the two compartments separated by a
membrane. The physiological pressure gradient is organ-specific, and the transport
of particles naturally needs to deal with this second agent, playing the role of an
input condition in the case of nanoparticle transport across the membrane. With
an appropriate technique, we can control the number density of nanoparticles just
outside the membrane (extracellular regions) and consider that, in the beginning,
the number density of nanoparticles beyond the membrane (intracellular regions)
is zero. Besides these two agents, there is a very important mechanism that should
be considered. This mechanism determines the free energy of nanoparticles while
crossing the membrane. This section is mainly dedicated to a better understanding
of the physical phenomena involved in the membrane transport of nanoparticles.
This involves the energetics of nanoparticles inside the membrane, and based on this
understandingwe wish to develop novel nanotechnology able to deliver nanoparticles
beyond membranes.
Understanding the membrane transport of nanoparticles requires specific infor-
mation about the geometry and constituents of the membrane, which determine the
partition either against or in favor of transport of any particles across the membrane.
The membrane's partition coefficient ( K m ) and the particles' experimentally mea-
surable free energy of interaction with the membrane ( G m ) are related through the

following equation:

G m =− RT ln ( K m ) (6.1)

Here R is the universal gas constant (8 31 J/K) and T is the absolute temperature.

It is very important to note that the values of K m play a key role in determining
the membrane permeability ( P m ) for any drug transport across membranes. A linear
relation is generally assumed between P m and K m [ 23 ], which follows from:

.

D m K m
L

P m = (6.2)

where D m is the membrane diffusion coefficient of the particles involved, and L is
the bilayer membrane thickness. The membrane permeability coefficient of particles
( P ) or drugs is equal to the linear velocity (nm/s) of the drugs through the membrane.
This is, in fact, the rate of particle transport through the membrane. Derivation of
P requires a very accurate consideration of all components of D m and K m . We can
instead consider the fractional release of the particles by the membrane into the
cellular interior, considering that the number density of particles ( ρ np , ext ) at the entry
level into the membrane (just outside the membrane) is known. Let us assume that
the number density of particles at the release level beyond the membrane (just inside
the cell) is ρ np , int . We can then propose an analytical relation between these two
number densities following the equation

ρ np , int = f ( L , H , m , ρ np , ext ) (6.3)