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
22.214.171.124 Bloodstream Velocity Profiles
Consider a Newtonian fluid of viscosity h flowing through a cylindrical tube of length ltube and radius rtube with pressure differential Dp between the ends. The average fluid velocity (vflow) for laminar or Poiseuille flow* is given by Eqn. 9.26, but imposing a no-slip condition at the vessel wall produces a radius-dependent parabolic velocity profile:362
where r is radial distance from the tube axis and kp = Dp / 4 h ltube = 2 vflow / rtube2. Maximum flow velocity vmax = 2 vflow, and occurs at the centerline (r = 0).
* For well-developed turbulent flows, the velocity profile may be expressed as vturb = vmax (1 - r/rtube)(1/m) away from the laminar sublayer near the wall, where m ~ 7 for a wide range of Reynolds numbers.1390
Of course, as fluid enters a tube from a large reservoir, there is an entrance region called the inlet length (linlet) which lies between the entrance and a point downstream, where the parabolic profile is in the process of being established asymptotically (Fig. 9.14). For NR <~ 1, linlet ~ 1.3 rtube, while for NR >~ 30, linlet ~ 0.16 NR rtube, producing, post-inlet, a <1% deviation from an ideal Poiseuille (parabolic) profile.361,1331 Inlet conditions prevail throughout the entire length of the aorta and most of the major arteries, but entrance effects are minimal in the smaller vessels.
Now suppose that red blood cells are added to the plasma. Due to the inward migration of red cells and other factors, the velocity profile of whole blood is affected by flow rate and hematocrit, particularly in vessels <500 microns in diameter, becoming blunted (Fig. 9.15). The degree of blunting decreases with increasing flow rate and increases with rising hematocrit. But the ability of RBCs to deform under the influences of cell crowding and fluid stresses in shear flow allows whole blood to continue to flow up to Hct ~ 98%.1319 It is theoretically possible that metamorphic nanorobots could approach this level of performance, though this issue has not yet been studied extensively.
What if rigid spherical nanorobots are added to the plasma instead of red cells? At small Rnano and low Nct, the flow profile remains parabolic. Onset of velocity profile blunting generally occurs either when Nct >~ 20% or when Rnano >~ 0.05 rtube.1322 Such blunted flow is sometimes called partial plug flow. Partial plug flow was not observed experimentally when (Nct Rnano/rtube) <~ 0.6%. Thus, non-plug, purely Poiseuille flow can probably be maintained for 2-micron diameter bloodborne nanorobots even while passing through the smallest dtube = 4 micron human capillary by holding Nct <~ 1.2%.
Figures 9.16A, 9.16B, 9.16C, and 9.16D show the experimentally-determined effects of Nct, Rnano/rtube, and fluid flow rate on the velocity profile of suspensions of rigid spheres and disks.1317,1322 As Nct and Rnano/rtube continue to rise, complete plug flow eventually ensues (Fig. 9.16B), wherein the entire mass of fluid moves stiffly at constant velocity, like toothpaste squeezed from a tube. Plug flow requires much higher pumping power (and pumping pressure) than laminar flow. Complete plug flow was observed at Nct = 38% and Rnano = 0.112 rtube, that is, when (Nct Rnano / rtube) >~ 3.8%. If complete plug flow can be avoided by holding Rnano/rtube < 0.38 at our assumed maximum Nct ~ 10% (Section 126.96.36.199), then for the smallest dtube = 4 micron human capillary, Rnano < 0.76 microns, allowing up to Dnano = 1.5 micron diameter for bloodborne nanodevices at the maximum nanocrit.
Further analysis that might consider the effects (on bloodborne nanorobot velocity profiles) of vessel geometry (including bifurcations, nozzling, and curved paths), vessel wall elasticity (including collapsible tubes), pulsatile flow, and adding nanorobots of various mixtures of sizes and shapes to whole blood rather than plasma, would be useful but is beyond the scope of this book.
Last updated on 21 February 2003