**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

**9.4.2.6 Additional Considerations**

The need for safe shear stresses (Section
9.4.2.2), and nanorobot power requirements for swimming (Section
9.4.2.4), both suggest that the maximum sanguinatation velocity to be employed
by nanorobots in the human body in normal circumstances should be <~1 cm/sec.
Also, from Eqn. 9.65, a 1-micron nanorobot
in water has a Reynolds number N_{R} ~ v; hence, to remain in the purely
viscous regime, v << 1 m/sec. The following analysis provides additional
support for this speed limit.

Consider a spherical nanorobot of radius R_{nano}
swimming at velocity v_{nano} in the bloodstream, that impacts a passing
blood cell. Virtually all cells encountered in this way will be erythrocytes.
The elastic modulus for the viscoelastic red cell plasma membrane is E_{cell}
~ 10^{3} N/m^{2} for isoareal deformation (pure shear), ~10^{5}
N/m^{2} at low areal strain (elastic area compressibility modulus),
and the rupture strength is ~10^{6} N/m^{2}.^{1325}
The force that is driving the sphere F_{nano} (Eqn.
9.73) may be distributed over the smallest possible impact area ~p
R_{nano}^{2}, and the resulting stress must be less than the
rupture strength or elastic modulus to prevent significant damage or deformation
of the cell surface, hence:

Taking h ~ 7 x 10^{-3}
kg/m-sec (Table
9.4) for the red cell contents (~0.33 gm/cm^{3} hemoglobin solution)
and R_{nano} = 1 micron, then the impacted erythrocyte deforms slightly
with no change in surface area when v_{nano} <~ 2 cm/sec, deforms
significantly with some change in surface area when v_{nano }~ 2 m/sec,
and finally ruptures when v_{nano} >~ 20 m/sec. This suggests a conservative
maximum velocity of <~2 cm/sec for medical nanorobots traversing the human
bloodstream, consistent with our proposed ~1 cm/sec speed limit.

Nanorobot biocompatibility^{3234}
is an issue of crucial importance in nanomedicine (Chapter
15). Consider a fleet of N_{nano} = 3 V_{blood} Nct / 4
p R_{nano}^{3} nanorobots uniformly
deployed in a blood volume V_{blood} populated by red blood cells of
typical length L_{RBC}. Each nanorobot swims past ~v_{nano}/L_{RBC}
red cells per second, of which some small fraction k_{x}
are injured as a result of the encounter. If the iatrogenic injury rate is conservatively
set equal to the natural rate of erythrocyte loss in the human body, or K_{0}
~ 3 x 10^{6} sec^{-1}, then:

Taking v_{nano} = 1 cm/sec, L_{RBC} ~ 7 microns,
V_{blood} = 5400 cm^{3}, and a 1 cm^{3} therapeutic
dose of R_{nano} = 1 micron medical nanorobots (which implies N_{nano}
~ 2 x 10^{11} nanorobots and Nct ~ 0.02%), then k_{x}
<~ 10^{-8}. This amounts to a net erythrocyte injury rate of only
L_{RBC} / (v_{nano} k_{x})
~ 1 cell/day per nanorobot -- a challenging but probably attainable goal. (K_{0}
~ 2 x 10^{6} sec^{-1} for platelets, 0.3-2 x 10^{6}
sec^{-1} for blood leukocytes.) Higher rates of red cell damage in theory
could be accommodated by administering compensatory erythropoietin to stimulate
RBC production (Chapter 22), but this approach would
violate the general nanomedical design principle of avoiding iatrogenic harm
whenever possible (Chapter 11).

Another velocity-related consideration involves the largely
cell-free plasmatic zone (the "nanorobot freeway") that extends 2-4 microns
from noncapillary blood vessel walls (Section 9.4.1.4),
with the larger width occurring at higher shear rates.^{362}
Even in very narrow capillary blood vessels, moving red cells never come into
solid-to-solid contact with the endothelium of the blood vessel. There is always
a thin fluid layer in between, that serves as a lubrication layer.^{362}
In experiments with glass capillaries 7.6-8.5 microns in diameter, the apparent
plasma layer thickness d_{plasma} ~ 0.6 microns
for a red cell velocity v_{RBC} ~ 0 mm/sec, d_{plasma}
~ 1.0 microns at v_{RBC} ~ 0.5 mm/sec, and d_{plasma}
~ 1.4 microns at v_{RBC} ~ 1.5 mm/sec.^{1399}
This layer will usually be wide enough to accommodate most bloodborne nanorobots,
which are expected to be 2 microns or smaller in diameter.

Note that a 1 cm^{3} therapeutic dose of 1 micron^{3}
nanorobots includes ~10^{12} devices, each of cross-sectional area ~1
micron^{2}. Such a fleet would occupy ~1 m^{2} if spread out
uniformly on a surface, adjacent and one layer thick. The total surface of the
entire human vascular system is ~313 m^{2}, but the summed area of all
large veins and main arterial branches alone totals ~1 m^{2} (Table
8.1). Thus, small therapeutic doses of medical nanorobots may safely travel
the cell-free plasmatic "freeway" at somewhat higher speeds than previously
estimated, though possibly at the expense of significantly greater power consumption
(Section 9.4.2.4) and possibly, at the highest number
densities, interfering with the lubrication effect normally provided by the
plasmatic layer, especially in the capillaries.

The need to maintain moderate viscosity of nanorobot-rich blood (Section 9.4.1.4) and to avoid complete plug flow in the narrowest human capillaries (Section 9.4.1.5) implies that the maximum nanocrit to be employed by nanorobots in the human bloodstream in normal circumstances should be Nct < 10%. The following simple analysis provides another constraint that is consistent with this estimate, and is valid for rigid and metamorphic nanorobots alike.

The frequency of close encounters and collisions among nanorobots
may be taken as a conservative metric of the likelihood of physical jamming,
mission interference, and other pathological effects of crowding. Consider a
spherical nanorobot of radius R_{nano} and volume V_{nano} =
(4/3) p R_{nano}^{3}, and a second
spherical nanorobot of identical size that approaches the first until the two
are in contact, defining a "collision" event. If the number density n_{nano}
= 3 Nct / [4 p R_{nano}^{3} (1 -
Hct)] of a uniformly distributed population of such nanorobots exceeds 2 devices
within a spherical volume of radius 2R_{nano}, then, on average, all
devices are in collision. This defines a maximum upper bound for nanocrit of
Nct_{maxHi} >~ (1 - Hct) / 4, representing the onset of a highly
collisional state; Nct_{maxHi} = 13.5% for Hct = 46% in the arteries,
22.5% for Hct ~ 10% in the capillaries. Similarly, if the nanorobot number density
is less than 1 device within a spherical volume of radius 2R_{nano},
then each device, on average, is surrounded by a nanorobot-free zone the width
of a single nanorobot radius, and collisions in a uniformly distributed population
are relatively infrequent,* defining Nct_{maxLo} ~ (1 - Hct) / 8. For
Nct >~ Nct_{maxLo}, the nanorobot population begins transitioning
to an increasingly collisional state; Nct_{maxLo} = 6.8% for Hct = 46%
in the arteries, 11.3% for Hct ~ 10% in the capillaries. Thus in the arteries,
an Nct < 6.8% is relatively noncollisional,* an Nct > 13.5% is highly
collisional, and a 6.8% < Nct < 13.5% (midpoint Nct ~ 10%) is transitional.

* But only "relatively" -- for
example, the multi-body collision frequency is appreciable among red cells even
at hematocrits as low as 5%.^{1358}

Last updated on 21 February 2003