**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

**8.3.3 Microtransponder
Networks **

The archetypal high-resolution (~3-micron) internal navigational
network may be described as a set of ~10^{11} mobile acoustic transponder
nanodevices, or navicytes, uniformly deployed throughout a ~0.1 m^{3}
human body volume with an average 100-micron spacing. (Power consumption rises
sharply for much larger spacings.) Total navicyte fleet volume is a relatively
unobtrusive ~1 cm^{3}. Navicytes emitting omnidirectional ~100 MHz acoustic
signal packets using r = 1 micron radiators are ~50% energy efficient (Section
7.2.2). Navigational signal packets are ~1 microsec in duration (conveying
~100 bits/packet) and repeat at ~1 millisec intervals (~1 KHz), giving a duty
cycle of f_{duty} ~ 0.1% which leaves plenty of clear air time for positional
information inquiries from nonnavigational nanorobots, communications and sensor
traffic, local acoustic microscopy and the like, and also to allow staggered
time slots to avoid noise and crosstalk.

How is a 100 MHz packet signal detected in a 1 nanosec signal
processing time? At SNR = 2 (Section 4.5.1), a navicyte
acoustic sensor must receive at least kT e^{SNR} ~ 30 zJ within a 1
nanosec integration, an energy influx rate of ~30 pW at the receiver. To produce
30 pW at a 1-micron receiver located 100 microns away, a transmitter of equal
area must radiate ~300,000 pW of acoustic output (see Section
4.9.1.5) or ~10^{5} watts/m^{2} for ~1 nanosec. Thus, each
navigational signal packet is prefaced by a triangular pulse ~1 nanosec in duration.
This triangle pulse is used for ranging. Because the transmitter is ~50% efficient,
each triangle pulse requires ~600,000 zJ of input energy to the transmitter
in order to produce an acoustic output energy of 300,000 zJ/pulse at the transmitter
surface. Broadcasting ~1000 pulses/sec brings the required transmitter input
power to ~600 million zJ/sec or ~0.6 pW continuous. Transmitting the non-pulse
portions of all packets costs ~60 pW (Section 7.2.2.2).
Note that the 1 nanosec triangular ranging pulse has the highest frequency that
can be used without significant absorption. Given a pulse which deposits a small
fixed multiple of kT at the receiver, then using the highest frequency produces
the least uncertainty in distance. Very roughly, a pulse which is just barely
detectable can have its timing localized to ~1 pulse width.

The triangle pulses make ~9 atm acoustic spikes in water,
probably low enough to avoid transient cavitation at this frequency and too
brief for shock wave formation or stable cavitation (Section
6.4.1) -- though it might be a good precaution to vary the time intervals
between packets to preclude any possibility of unexpected resonances. Acoustic
torque and related effects (Section 6.4.1) cannot yet
be ruled out in this application and should be investigated further. Each triangular
pulse represents an energy discharge event requiring a momentary power density
of ~10^{12} watts/m^{3} inside a ~(0.8 micron)^{3} powerplant,
comparable to power densities available in mechanical (Section
6.3.2), chemical (Section 6.3.4.5), and electrical
(Section 6.3.5) systems. At ~100 pW/navicyte, total
navicyte network system power consumption is ~10 watts, well within proposed
in vivo thermogenic limits (Section 6.5.2).

How is the navigational network established? For convenience, a navigational Prime Centrum is established on the ventral surface of the 10th thoracic vertebra (T-10) at the midsagittal plane of the vertebral body (see Figures 8.22 and 8.24), defining a permanent origin for a body-centered coordinate system. This origin is centrally located, lying directly posterior to the xiphoid process (Fig. 8.22) and the liver, directly inferior to the heart and between the two lungs. The site is easily accessible by bloodborne nanorobots, well-supplied with oxygen and glucose for power, securely anchorable due to the dense bone, and reasonably well-protected from injury; also, movements in the thoracic vertebrae are the most restricted because the ribs and costal cartilages resist distortion.

The Prime Centrum is comprised of four "monument" type bloodborne
navicytes which are injected, or migrate stochastically using cytoidentification
(Section 8.5.2), or are directed to the site by demarcation
or other means. These nanorobots congregate around the chosen navigational origin
approximately at the vertices of a square. Each monument navicyte attaches one
end of a retractable fullerene cable rule to each of its three brethren and
crawls backward on an approximately radial course, paying out the slippery cable
and adjusting mutual positions until the two diagonals measure exactly 141.42
microns center-to-center and the four normals measure exactly 100.00 microns
center-to-center, guaranteeing a perfect 100-micron square with exact 90.00°
corners. The four navicytes then secure themselves to the bone beneath the periosteum
using biocompatible permanent anchors, becoming fully
sessile, and retract the cable rules. If vertebral curvature radius R ~ 2 cm
and diagonal navicyte separation S = 141.42 microns, then maximum anteroposterior
geometric deviation due to off-sagittal positioning is R(1-sin((p/2)-(S/R)))
= 0.5 micron. The square may not be precisely aligned with the conventional
anatomical coordinate axes. At least 10 independent navicyte monument quartets
should be established as alternate or backup sites on T-10 to satisfy customary
redundancy requirements (Chapter 13). Additionally,
~10^{4} regional monuments are established at ~cm intervals at fixed
positions on all major skeletal surfaces of the body. Relative positions of
regional monuments vary only within well-defined envelopes depending upon macroscopic
joint rotations and limb flexures, or almost not at all on inflexible surfaces
such as the diaphysis of the long bones.

Mobile navicytes deployed throughout the remaining body volume receive message packets emitted by neighboring devices, which in turn have received packets from their neighbors, ultimately stretching back in an unbroken chain to a regional monument or to the Prime Centrum. Assuming a simple cubical array, each stationary navicyte has two neighbors per directional axis -- a total of six neighbors within acoustic communication range (100 microns). Navicyte positional stability is enhanced by stationkeeping activities to avoid drift (~1 micron/sec for a 1-micron nanorobot due to Brownian motion; Section 3.2.1), and anchorage to nonsignalling elements of the omnipresent extracellular matrix.

Each navicyte possesses an onboard clock capable of continuous
Dt ~ 1 nanosec temporal accuracy between recalibrations
or over mission times of >~10^{3} sec (Section
10.1). Message packets received periodically from each neighbor include
data describing the exact universal time of packet transmission. Since each
recipient has a synchronized clock (Section 10.1.3),
the triangle pulse travel time between navicytes (t
~ 65 nanosec) is known to an accuracy of ~1 nanosec. The speed of sound v_{sound}
~ 1540 in soft tissue (Table
6.7), so the ~1 nanosec temporal uncertainty adds (v_{sound} Dt)
~ 1.5 microns of uncertainty to the range estimate.

The speed of sound varies between 1400-1600 m/sec for most
nonosseous tissues. This speed is reasonably uniform within specific tissues
(Table
6.7) over time, and is essentially frequency-independent over the nanomedically-relevant
range. Knowledge of its own approximate histological location (based on chemical
sampling, etc.) allows a navicyte to consult an onboard data table or a previously-compiled
low-resolution map (see Chapter 19) to estimate local
sound velocity to within ~25 m/sec. Also, two navicytes (at least one of which
is mobile) can directly measure the local speed of sound by briefly conjugating
(Section 9.4.4.4), extending a cable rule of known
length between them, then transmitting and timing a test pulse traveling through
the medium. A 100-micron rule length allows local sound velocity to be measured
to an average Dv_{sound} ~ 25 m/sec accuracy
using a 1-nanosec clock. A measurement uncertainty of ~25 m/sec in the local
speed of sound adds another (Dv_{sound} t)
~ 1.6 micron of positional uncertainty, giving a total of DX_{min}
~ 3 microns uncertainty in each range estimate. Additional small uncertainties
in sound velocity and in acoustic reflection and refraction power losses may
occur when capillaries or other microvessels cross the line of sight between
two navicytes.

Only four of the six neighbors are absolutely required for positional triangulation. The data packet received by a navicyte from its first neighbor (containing that neighbor's correct three-dimensional coordinates) narrows the navicyte's possible position to a spherical surface of radius equal to the computed range, centered on the first neighbor. The data packet received from the second neighbor defines a second geometric sphere, further narrowing the navicyte's possible position to a circle formed by the intersection of the first and second spheres. The data packet from the third neighbor adds a third sphere, reducing the possibilities to two points on the intersection circle, and the signal packet from the fourth neighbor selects one of these two points as the navicyte's true position. Additional minor corrections may be made for the bending of sound waves crossing known thermal, pressure, or salinity gradients (Section 6.4.1).

What positional accuracy can this system achieve? Consider
the simplest case with many parallel coplanar rows of N navicytes lying ~X_{rowi}
apart, each row of total length L_{row} ~ S
X_{rowi} emanating from a common tangential surface and extending deep
into the tissues. The terminal navicyte of each row estimates its cumulative
length as L_{row} which also contains an unknown range error e_{row}.
The series of randomly distributed errors in the positions of each navicyte
in the row, individually of magnitude ± DX_{min},
do not cancel to zero but instead constitute a random walk with maximum error
excursion from the mean of e_{row} ~ 2 N^{1/2}
DX_{min}. The longest rows will occur in
the viscera, farthest from any bony surface. Taking L_{row} = 15 cm,
DX_{min} = 3 microns and X_{row}
= 100 microns, then N = L_{row}/X_{row} = 1500 navicytes per
row and e_{row} ~230 microns. Hence in this
simple case, the minimum accuracy at the terminus of each row is 15 cm ± 230
microns, or ~0.2%. Shorter rows accumulate less error at the terminus -- the
cumulative error at ~2 mm from the common surface is at worst ~27 microns (~1
cell width). The average error per navicyte in a 15-cm row is only e_{row}
/ N ~ 0.2 microns. Even for L_{row} = 2 meters (~longest possible transverse
path length in the human body), N = 20,000 and e_{row}
= 800 microns. Note that an additional small systematic error in position may
accrue if there is a systematic change in local sound velocity between recalibrations
and if the subject viscera have a free surface unbounded by bone. For example,
bruised tissue in which foreign fluids are accumulating will produce a slightly
warped coordinate system; sound velocity differentials between, say, blood and
interstitial fluid may be as large as 70 m/sec (Table
6.7).

Accuracy within each row may be significantly improved by
introducing active error checking and continuous recalibration. Consider the
terminal navicytes of two parallel rows A and B. As before, the rows lie ~X_{row}
apart in their common plane. Each terminal navicyte computes that its total
length is precisely L_{row}, but in fact navicyte B is in error by a
distance e_{B} along the row. Given the minimum
range error DX_{min}, terminal navicyte B
can only detect that an error exists when its range measurement to neighboring
terminal navicyte A increases from X_{row} to X_{row} + DX_{min},
whereupon from simple geometry:

For X_{row} = 100 microns and DX_{min}
= 3 microns, then e_{B} ~25 microns or ~1
cell width. This range error is relatively insensitive to X_{row}; for
instance, if X_{row} increases to 110 microns then e_{B}
only rises to ~26 microns.

However, row terminus errors (measured in integral units
of e_{B}) are likely to be normally distributed
between ± e_{row} ~ 230 microns in the earlier
simplest-case example. The error correction procedure involves nearest neighbors
collectively polling the nearest n_{row} ~ 6000 rows (a bundle with
termini covering ~0.6 cm^{2} when X_{row} = 100 microns) for
their measured e_{B}'s and then computing
the mean of the distribution, which has an uncertainty of e_{row}
/ n_{row}^{1/2} ~ 3 microns = DX_{min},
matching the uncertainty of a single range measurement, the smallest possible
error. The estimated mean is used to produce a correction factor which is applied
to the erroneous estimated value of L_{row}, yielding a corrected L_{row}
accurate to ~ DX_{min} or ~3 microns. This
corrective process is repeated:

b. for planar cross-sections at ~2 mm increments along the entire row length (to eliminate error compression or rarefaction waves), not just along the terminal plane, and

c. at regular time intervals (~3 sec) to ensure continuous recalibration to ~3 microns at all points in each row.

Note that parts of the human body frequently may deform up
to ~30% during normal activities, so X_{row} is not constant but may
vary between 85-115 microns in as little as 0.1-1 sec in working situations.

Detailed specification of a complete recalibration protocol^{1624}
is beyond the scope of this book. One procedure to further improve row-length
measurement accuracy is to use the navicyte grid as a phonon gain medium by
configuring each navicyte as a repeater station during the calibration cycle,
effectively allowing the propagation of row-long pulses thus permitting independent
row-length measurements. Multiple measurements can reduce independent row-length
measurement error to arbitrarily low levels. Alternatively, J. Soreff suggests
treating the entire calibration bundle as a coherent amplifier using lower acoustic
frequencies to increase range. This enables phase detection to within ~1 nanosec
if phase-locked detection is used and if a few kT of energy are present in the
time resolution window, allowing navicytes to hear a propagating plane wave
averaged across an entire bundle. An analysis of soliton-like systemic effects
analogous to intrinsic local modes in lattices^{3036
}is beyond the scope of this text.

The ability to make the corrections described above assumes
that navicytes can distinguish -e_{B} from
+e_{B}, an angular spread of ± 14° for e_{B}
~ 25 microns and X = 100 microns. Since sound crosses the width of a micron-sized
nanorobot in ~1 nanosec, the relative angular location of a distant acoustic
source can be directly measured by placing two sensors on either side of the
nanodevice at precisely calibrated separations. For angles of acoustic wave
incidence <~q, measured from the axis joining
the two sensors which are separated by a width x_{sensor} on either
side of the nanorobot, the wave arrival times will differ by more than Dt
~ 1 nanosec and thus will be received as two distinct pulses, rather than one.
For incidence angles >q, the wave arrival times
at either side cannot be distinguished. q thus defines
a permissive angular detection cone of size

For Dt = 1 nanosec and v_{sound}
= 1540 m/sec, a sensor separation of x_{sensor} = 1.59 microns allows
up to q = 0.25 radian = 14° to be distinguished.
The uncertainty in v_{sound} of ~25 m/sec imposes a minimum measurement
uncertainty of Dq ~ 3°. The permissive detection
cone shrinks to q = 0° at x_{sensor} = 1.54
microns.

Relative navicyte angle can also be computed from the coordinates
of neighbors by simple geometry. Using the coordinates of a nearest neighbor
with range X_{range} = X_{row} ~ 100 microns and lateral positional
uncertainty X_{error} = DX_{min}
~ 3 microns, then relative navicyte angle can only be computed to an accuracy
of Dq ~ sin^{-1} (X_{error} / X_{range})
~ 1.7° (~30 milliradians). However, relative angle uncertainty is reduced if
the coordinates of more distant neighbors are available to the navicyte. For
example, using a neighbor located 15 rows away (X_{range} = 15 X_{row})
and again taking the minimum X_{error}, then Dq
~ 0.1° (~2 milliradians). Coordinates from a distant-neighbor navicyte with
X_{range} = L_{row} ~ 15 cm with an uncorrected lateral positional
uncertainty X_{error} = e_{row} ~
230 microns would give Dq ~ 0.09° (~2 milliradians);
taking X_{error} = DX_{min} = 3 microns
in the ideal case, Dq ~ 0.001° of arc (~0.02 milliradian).
By comparison, from Eqn. 3.2 Brownian tumbling
of a micronscale nanorobot amounts to ~10^{6}° (~0.02 microradian) in
~1 nanosec, or ~1° (~20 milliradians) between ~1 millisec signal packet repeat
intervals.

Each navicyte obtains its own orientation relative to gravity
using onboard gravity sensors, which are accurate to within ~2 milliradians
of verticality, with new measurements available every ~0.1 millisec (Section
4.9.2.2) or up to the limits of the communication network capacity (e.g.,
recalibration protocols). Regional monuments monitor and disseminate the local
grid's angular orientation relative to the gravity field at ~1 millisec intervals,
thus fixing each navicyte's orientation to the local grid on a virtually continuous
basis. Absolute spatial orientation relative to a fixed onboard standard such
as a nanogyroscope (Section 4.3.4.1) may be determined
to 1-100 microradian accuracy during non-recalibrated deployment lifetimes of
10^{3}-10^{7} sec. Direct gyrostabilization may be possible
in some applications (Section 9.4.2.2).

Last updated on 19 February 2003