© 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

{Eqn. 9.61}

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 measurements¹³¹⁴ 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 results¹³¹⁴ 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 Turitto³⁸⁶ 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 <~ 10³ (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 m²/sec both for red cells and for rigid 2-micron diameter microspheres,³⁸⁶ and 3-86 x 10^-11 m²/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