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
126.96.36.199 Instantaneous Stationary Source in Stationary Medium
Consider a chemomessaging nanorobot that releases Qmessage messenger molecules (each carrying Imessage bits/molecule) in a single puff as a point source at time t = 0. This is the ideal design for an alarm-type message or for messages requiring rapid fadeout. For simplicity, the human body is taken to be a continuous, isotropic, unbounded, stationary aqueous medium; a more detailed treatment is beyond the scope of this book. The spatial density of message molecules as a function of time t and distance r from the point source is
where D = the translational diffusion coefficient for message molecules, which are assumed to be roughly spherically packed, estimated from Eqns. 3.5 and 7.2 as
For Imessage = 100 bits, D = 2.2 x 10-11 m2/sec; for Imessage = 109 bits, D ~ 1.0 x 10-13 m2/sec. Because the detection of molecules by receptor-based chemical sensors requires a concentration-dependent minimum sensor cycle time tEQ approximated by Eqn. 4.3, then there exists some minimum threshold concentration (cmin) of message molecules that can be detected by a chemical sensor in some minimum waiting time tsensor = tEQ, given by
For Nencounters ~ 100,10 NA = 6.023 x 1023 molecules/mole (Avogadro's number), rmessage from Eqn. 7.2 and taking tsensor = 1 sec, then cmin ~ 1 x 10-9 molecules/nm3 for Imessage = 100 bits and cmin ~ 9 x 10-11 molecules/nm3 for Imessage = 109 bits. The concentration of message molecules exceeds cmin within an expanding diffusive sphere. In a time trec = (0.0293 / D) (Qmessage / cmin)2/3 this expanding sphere reaches a maximum size Rmax = (0.419) (Qmessage / cmin)1/3, after which it begins to contract as the puff dissipates;703 the concentration eventually falls below cmin everywhere at tfadeout = e trec, where e = 2.718...
1012 nanorobots uniformly distributed throughout a 0.1 m3 human body have a mean interdevice separation of ~50 microns. For simple messages (Imessage = 100 bits) and Rmax = 50 microns, then Qmessage ~ 2 x 106 message molecules emitted, trec ~ 20 sec ('I ~ 5 bits/sec), and tfadeout ~ 50 sec. If Rmax = 1 mm, then Qmessage = 2 x 1010 molecules and trec ~ 8000 sec to receive the signal ('I ~ 0.01 bits/sec). For complex messages (Imessage = 109 bits), Qmessage ~ 1 x 105 molecules but trec ~ 4000 sec (~1 hour) for the message to be transported Rmax = 50 microns ('I ~ 3 x 105 bits/sec).
A simple alarm signal (Imessage = 100 bits) released within an individual tissue cell is received everywhere throughout the cytosol -- an assumed ~(20 micron)3 cell volume -- in trec ~ 0.8 sec ('I ~ 100 bits/sec) with tfadeout ~ 2 sec, using a single alarm puff of Qmessage ~ 17,000 message molecules (computed with Rmax ~ 10 microns) which molecules may be stored in a (40 nm)3 volume per puff. As a practical matter, the cytosol is quite crowded with macromolecules and cytoskeletal components (Section 8.5.3); exclusion effects will increase diffusion times by as yet unknown amounts, to be determined experimentally.
Last updated on 18 February 2003