**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

**9.4.1.4 Viscosity of
Nanorobot-Rich Blood**

The presence of large numbers of relatively rigid nanorobots
dramatically alters bloodstream viscosity. Figure
9.13 shows the relative viscosity h_{a}
/ h_{plasma} for human blood at 298 K with
shear rate >100 sec^{-1} as a function of particle volume fraction,
compared to the relative viscosity of suspensions of latex rigid spheres, rigid
disks, emulsion droplets, and sickled erythrocytes (which are virtually nondeformable),
as determined experimentally.^{1312}
Figure
9.13 reveals that a 50% suspension of micron-sized rigid nanorobots will
increase blood viscosity by a factor of ~350, seriously impeding flow especially
in the smaller vessels. However, a plasma suspension of microspheres at a ~10%
particle volume fraction has a relative viscosity indistinguishable from Hct
= 10% whole blood. This suggests a conservative 10% volume-fraction limit for
the maximum bloodstream concentration of medical nanorobots (e.g., a maximum
"nanocrit" or Nct = 10%), a limit that may also ensure free flow of the fluid
(see also Sections 9.4.1.5 and 9.4.2.6).

Relative viscosity also depends on particle size, though the
effect due to the presence of medical nanorobots is usually minor. As a conservative
upper limit in the smallest vessels:^{1315}

where D_{nano} (= 2 R_{nano}) is the maximum
nanorobot diameter (radius) and d_{tube} (= 2 r_{tube}) is the
blood vessel diameter (radius). Taking d_{tube} = 8 microns (capillaries),
then h_{a} / h_{plasma}
= 1.0002 for D_{nano} = 1 micron, or 1.07 for D_{nano} = 4 microns
(the largest bloodborne nanorobot; see Sections 5.2.1
and 8.2.1.2). In larger blood vessels, this effect
is even smaller for micron-sized medical nanorobots.

Relative blood viscosity also depends on nanorobot shape.
Chien's measurements^{1314} of
effective viscosity as a function of particle shape in dilute suspensions found
that minimum viscosity is achieved by hard spheres or by 1:1 hard cylinders.
Thin disks or long cylinders have higher viscosity. For example, 10:1 rods (10
times longer than wide) produce a suspension with ~10 times higher viscosity
than a suspension containing an equal volume of spheres; for 100:1 rods, viscosity
increases ~2500-fold. The implication for nanodevice design is that a large
population of bloodborne medical nanorobots will have the minimum impact on
blood viscosity if each nanorobot is closest to spherical in shape. Long rod
or flat disk shapes will greatly increase blood viscosity, in comparison to
spheres, although at the lowest nanodevice number densities the total impact
on blood viscosity may be negligible.

Chien's results^{1314}
also suggest that metamorphic nanorobots capable of continuous surface deformations
in response to flow conditions (like RBCs) may further reduce their contribution
to blood viscosity by at least a factor of 2-6, depending on shear rate (Fig.
9.12). Goldsmith and Turitto^{386}
show that at shear rates over 200 sec^{-1}, typical in physiological
blood, the optimum shape for red cells is ellipsoidal, positioned at an angle
to the flow, with the surface rotating in the direction of flow in a tank-tread-like
motion. In experiments with flowing emulsions, the deformation of a liquid droplet
results in its migration across the streamlines away from the tube wall. Thus
in physiological blood over the whole range of normal hematocrits and typical
flow rates, there is a plasma-rich (blood-cell-rare) or "plasmatic" zone d_{plasma}
~ 2-4 microns deep at the walls of vessels whose diameters exceed 100 microns.^{362,1319}

Such lateral migration is not observed with small rigid particles
of any shape at high concentrations and at low Reynolds numbers (Section
9.4.2.1) N_{R} <~ 10^{3} (e.g., arterioles and smaller
vessels; Table
8.2).^{1319} However, for vessels
with N_{R} >~ 1 (e.g., arteries and veins; Table
8.2), inertial effects do come into play and rigid free-floating nanorobots
will be pushed away from the wall to produce a particle-free zone. The thickness
of this "plasmatic" zone d_{nano} decreases
sharply with increasing nanorobot concentration (Nct). For example, at Nct =
2%, d_{nano} ~ 0.3 r_{tube}; at Nct
= 10%, d_{nano} ~ 0.1 r_{tube}; at
Nct = 30%, d_{nano} ~ 0.01 r_{tube}.^{1320}

In terms of individual nanorobot motion, a rigid sphere initially
placed near the tube wall migrates inward, while a rigid sphere placed near
the tube axis migrates outward. Known as the "tubular pinch effect,"^{1321}
rigid spheres started in either position converge to an intermediate equilibrium
radius position (as measured from the tube axis) of r_{eq} ~ (0.6-0.7)
r_{tube} for R_{nano}/r_{tube} << 1, or r_{eq}
~ 0.5 r_{tube} (farther from the wall) for R_{nano}/r_{tube}
~0.25.^{1320,1321}
By analogy with Brownian translational diffusion and Eqn.
3.1, a radial dispersion coefficient D_{r} may be defined as Dr
= (2 t D_{r})^{1/2} (meters), where
Dr is the RMS radial displacement of a bloodborne
object in an observation time t. The analogy is imperfect
because these radial motions are not random, but are due to multibody collisions
determined by the local velocity gradient, particle concentration, and surface
deformations of the objects. At any given concentration, displacements are greatest
at radial distances between 0.5-0.8 r_{tube}. For local shear rates
of 5-20 sec^{-1} and volume concentrations from 20%-70%, D_{r}
= 1-20 x 10^{-12} m^{2}/sec both for red cells and for rigid
2-micron diameter microspheres,^{386}
and 3-86 x 10^{-11} m^{2}/sec for platelets in whole blood,^{1398}
as determined experimentally.^{1358}
Thus the mean time for a nanorobot (R_{nano} = 1 micron) to migrate
a radial distance Dr ~1 micron is t
~25-500 millisec -- about an order of magnitude faster than simple Brownian
diffusion (Section 3.2.1).

Last updated on 21 February 2003