**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

**7.2.1.3 Instantaneous
Stationary Source in Stationary Medium**

Consider a chemomessaging nanorobot that releases Q_{message}
messenger molecules (each carrying I_{message} 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

_{}
{Eqn. 7.3}

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 I_{message} = 100 bits, D = 2.2 x 10^{-11}
m^{2}/sec; for I_{message} = 10^{9} bits, D ~ 1.0 x
10^{-13} m^{2}/sec. Because the detection of molecules by receptor-based
chemical sensors requires a concentration-dependent minimum sensor cycle time
t_{EQ} approximated by Eqn. 4.3, then
there exists some minimum threshold concentration (c_{min}) of message
molecules that can be detected by a chemical sensor in some minimum waiting
time t_{sensor} = t_{EQ}, given by

_{}
{Eqn. 7.5}

For N_{encounters} ~ 100,^{10}
N_{A} = 6.023 x 10^{23} molecules/mole (Avogadro's number),
r_{message} from Eqn. 7.2 and taking
t_{sensor} = 1 sec, then c_{min} ~ 1 x 10^{-9} molecules/nm^{3}
for I_{message} = 100 bits and c_{min} ~ 9 x 10^{-11}
molecules/nm^{3} for I_{message} = 10^{9} bits. The
concentration of message molecules exceeds c_{min} within an expanding
diffusive sphere. In a time t_{rec} = (0.0293 / D) (Q_{message}
/ c_{min})^{2/3} this expanding sphere reaches a maximum size
R_{max} = (0.419) (Q_{message} / c_{min})^{1/3,}
after which it begins to contract as the puff dissipates;^{703} the
concentration eventually falls below c_{min} everywhere at t_{fadeout}
= e t_{rec}, where e = 2.718...

10^{12} nanorobots uniformly distributed throughout
a 0.1 m^{3} human body have a mean interdevice separation of ~50 microns.
For simple messages (I_{message} = 100 bits) and R_{max} = 50
microns, then Q_{message} ~ 2 x 10^{6} message molecules emitted,
t_{rec} ~ 20 sec ('I ~ 5 bits/sec), and t_{fadeout} ~ 50 sec.
If R_{max} = 1 mm, then Q_{message} = 2 x 10^{10} molecules
and t_{rec} ~ 8000 sec to receive the signal ('I ~ 0.01 bits/sec). For
complex messages (I_{message} = 10^{9} bits), Q_{message}
~ 1 x 10^{5} molecules but t_{rec} ~ 4000 sec (~1 hour) for
the message to be transported R_{max} = 50 microns ('I ~ 3 x 10^{5}
bits/sec).

A simple alarm signal (I_{message} = 100 bits) released
within an individual tissue cell is received everywhere throughout the cytosol
-- an assumed ~(20 micron)^{3 }cell volume -- in t_{rec} ~ 0.8
sec ('I ~ 100 bits/sec) with t_{fadeout} ~ 2 sec, using a single alarm
puff of Q_{message} ~ 17,000 message molecules (computed with R_{max}
~ 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