© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

7.2.1.7 Continuous Stationary Source in Nonstationary Medium

A nonstationary transporting medium does more than just move message molecules in the direction of flow -- it also may create turbulence in the medium, adding a component of turbulent diffusivity that overwhelms simple Brownian diffusion, giving a total diffusivity that is more a property of the flow structure of the medium and the boundary surfaces than of the substance of the medium, a complex analytical problem. Consider a continuous point source emitting a constant 'Q_message (molecules/sec) from a fixed position near the luminal surface of a blood vessel of radius R_vasc. From the relations given in⁷⁰³ and as a crude approximation assuming plume width << boundary wall separation, 'Q_message ~ (0.0396) v_vasc c_minX_fadeout^7/4 to create an ellipsoidal message molecule plume of maximum detectable length X_fadeout in a medium flowing along the x-axis with fluid velocity v_vasc. The widest part of the plume occurs at X_max ~ (3.57) F^4/7, Y_max ~ (0.686) F^1/2, and Z_max ~ (0.341) F^1/2, where F = 'Q_message / v_vasc c_min. Message receipt time t_rec ~ X_fadeout / v_vasc.

Thus a simple message (I_message = 100 bits, c_min ~ 10^-9 molecules/nm³ for t_sensor = 1 sec) broadcast from the luminal surface of the common carotid artery (v_vasc ~ 0.2 m/sec, R_vasc ~ 3 mm; Table 8.2) with a desired plume length X_fadeout = 100 microns requires at least 'Q_message ~ 10⁹ molecules/sec, producing a very narrow plume with Y_max ~ 40 microns, Z_max ~ 20 microns (the radial direction), and t_rec ~ 0.5 millisec ('I ~ 100 bits/sec, limited by t_sensor = 1 sec). For a complex message (I_message = 10⁹ bits, c_min ~ 9 x 10^-11 molecules/nm³ for t_sensor = 1 sec), 'Q_message ~ 7 x 10⁷ molecules/sec at t_rec ~ 0.5 millisec ('I ~ 10⁹ bits/sec, limited by t_sensor = 1 sec).

Last updated on 18 February 2003