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 Additional Considerations

The need for safe shear stresses (Section, and nanorobot power requirements for swimming (Section, 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 NR ~ 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 Rnano swimming at velocity vnano 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 Ecell ~ 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 Fnano (Eqn. 9.73) may be distributed over the smallest possible impact area ~p Rnano2, 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 Rnano = 1 micron, then the impacted erythrocyte deforms slightly with no change in surface area when vnano <~ 2 cm/sec, deforms significantly with some change in surface area when vnano ~ 2 m/sec, and finally ruptures when vnano >~ 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 Nnano = 3 Vblood Nct / 4 p Rnano3 nanorobots uniformly deployed in a blood volume Vblood populated by red blood cells of typical length LRBC. Each nanorobot swims past ~vnano/LRBC red cells per second, of which some small fraction kx 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 vnano = 1 cm/sec, LRBC ~ 7 microns, Vblood = 5400 cm3, and a 1 cm3 therapeutic dose of Rnano = 1 micron medical nanorobots (which implies Nnano ~ 2 x 1011 nanorobots and Nct ~ 0.02%), then kx <~ 10-8. This amounts to a net erythrocyte injury rate of only LRBC / (vnano kx) ~ 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, 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 dplasma ~ 0.6 microns for a red cell velocity vRBC ~ 0 mm/sec, dplasma ~ 1.0 microns at vRBC ~ 0.5 mm/sec, and dplasma ~ 1.4 microns at vRBC ~ 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 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 and to avoid complete plug flow in the narrowest human capillaries (Section 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 Rnano and volume Vnano = (4/3) p Rnano3, 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 nnano = 3 Nct / [4 p Rnano3 (1 - Hct)] of a uniformly distributed population of such nanorobots exceeds 2 devices within a spherical volume of radius 2Rnano, then, on average, all devices are in collision. This defines a maximum upper bound for nanocrit of NctmaxHi >~ (1 - Hct) / 4, representing the onset of a highly collisional state; NctmaxHi = 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 2Rnano, 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 NctmaxLo ~ (1 - Hct) / 8. For Nct >~ NctmaxLo, the nanorobot population begins transitioning to an increasingly collisional state; NctmaxLo = 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