**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**3.2.2 Passive Diffusive
Intake**

Medical nanodevices will frequently be called upon to absorb some particular material from the external aqueous operating environment. Molecular diffusion presents a fundamental limit to the speed at which this absorption can occur. (Once a block of solution has passed into the interior of a nanodevice, it may be divergently subdivided and transported at ~0.01-1 m/sec along internal pathways of characteristic dimension ~1 micron far faster than the <1 mm/sec bulk diffusion velocity across 1 micron distances; Section 9.2.7.5.)

For a spherical nanodevice of radius R, the maximum diffusive intake current is:

where J is the number of molecules/sec presented to the entire
surface of the device, assumed to be 100% absorbed (but see Section
4.2.5), D (m^{2}/sec) is the translational Brownian diffusion coefficient
for the molecule to be absorbed, and C (molecules/m^{3}) is the steady-state
concentration of the molecule far from the device.^{337} (Blood concentrations
in gm/cm^{3} from Appendix B are converted
to molecules/m^{3} by multiplying Appendix B
figures by (10^{6} x N_{A}/MW), where N_{A} = 6.023
x 10^{23} molecules/mole (Avogadro's number) and MW = molecular weight
in gm/mole or daltons.) For rigid spherical particles of radius R, where R >>
R_{H2O}, the EinsteinStokes equation^{363}
gives:

though this is only an approximation because D varies slightly with concentration, with departure from molecular sphericalness, and other factors.

Measured diffusion coefficients in water for various molecules
of physiological interest, converted to 310 K, are in Table
3.3. Diffusion coefficient data for ionic salts such as NaCl and KCl, which
dissociate in water and diffuse as independent ions, are for solvated electrolytes.
A 1-micron (diameter) spherical nanodevice suspended in arterial blood plasma
at 310 K, with C = 7.3 x 10^{22} molecules/m^{3} of oxygen and
D = 2.0 x 10^{-9} m^{2}/sec, encounters a flow rate of J = 9.2
x 10^{8} molecules/sec of O_{2} impinging upon its surface.
(The same calculation applied to serum glucose yields J = 1.3 x 10^{10}
molecules/sec.) The characteristic time for change mediated by diffusion in
a region of size L scales as ~L^{2}/D (Eqn.
3.9, below). Across the diameter of an L = 1 micron nanodevice, small molecules
such as glucose diffuse in ~0.001 sec, small proteins like hemoglobin in ~0.01
sec, and virus particles diffuse in ~0.1 sec. Diffusion coefficients of the
same molecules in air at room temperature are a factor of ~60 higher, because
h_{air} ~ 183 micropoise at 20°C.

In blood, the diffusivity of larger particles is significantly
elevated because local fluid motions generated by individual red cell rotation
lead to greater random excursions of the particles.^{388}
The effective diffusivity D_{e} = D + D_{r}, where the rotation-induced
increase in diffusivity D_{r} ~ 0.25 R_{rbc}^{2} 'g,
with red cell radius R_{rbc} ~ 2.8 microns (taken for convenience as
a spherical volume equivalent) and a typical blood shear rate (Section
9.4.1.1) 'g ~ 500 sec^{-1}, giving D_{r}
~10^{-9} m^{2}/sec in normal whole blood. The elevation of diffusivity
caused by red cell stirring is just 50% for O_{2} molecules. However,
for large proteins and viruses the effective diffusivity increases 10-100 times,
and the effective diffusivity of particles the size of platelets is a factor
of 10,000 higher than for Brownian molecular diffusion.

The diffusion current to the surface of a nanodevice can also
be estimated for various nonspherical configurations.^{337}
For instance, the diffusion current to both sides of an isolated thin disk of
radius R is given by J = 8 R D C. The two-sided current to a square thin plate
of area L^{2} is J = (8/p^{1/2})
L D C. The steady-state diffusion current to an isolated cylinder of length
L_{c} and radius R is approximated by J = 2 p
L_{c} DC / (ln(2L_{c}/R) - 1 ), for L_{c} >> R.
The diffusion current through a circular hole of radius R in an impermeable
wall separating regions of concentration c_{1} and c_{2} is
J = 4 R D (c_{1} - c_{2}).

Last updated on 7 February 2003