© 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 ~ 10³ N/m² for isoareal deformation (pure shear), ~10⁵ N/m² at low areal strain (elastic area compressibility modulus), and the rupture strength is ~10⁶ N/m².¹³²⁵ The force that is driving the sphere F_nano (Eqn. 9.73) may be distributed over the smallest possible impact area ~p R_nano², 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:

{Eqn. 9.77}

Taking h ~ 7 x 10^-3 kg/m-sec (Table 9.4) for the red cell contents (~0.33 gm/cm³ 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 biocompatibility³²³⁴ is an issue of crucial importance in nanomedicine (Chapter 15). Consider a fleet of N_nano = 3 V_blood Nct / 4 p R_nano³ 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 K₀ ~ 3 x 10⁶ sec^-1, then:

{Eqn. 9.78}

Taking v_nano = 1 cm/sec, L_RBC ~ 7 microns, V_blood = 5400 cm³, and a 1 cm³ therapeutic dose of R_nano = 1 micron medical nanorobots (which implies N_nano ~ 2 x 10¹¹ 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. (K₀ ~ 2 x 10⁶ sec^-1 for platelets, 0.3-2 x 10⁶ 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 cm³ therapeutic dose of 1 micron³ nanorobots includes ~10¹² devices, each of cross-sectional area ~1 micron². Such a fleet would occupy ~1 m² if spread out uniformly on a surface, adjacent and one layer thick. The total surface of the entire human vascular system is ~313 m², but the summed area of all large veins and main arterial branches alone totals ~1 m² (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_nano³, 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_nano³ (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