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 ~1011 mobile acoustic transponder nanodevices, or navicytes, uniformly deployed throughout a ~0.1 m3 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 cm3. 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 fduty ~ 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 eSNR ~ 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 188.8.131.52) or ~105 watts/m2 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 184.108.40.206). 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 ~1012 watts/m3 inside a ~(0.8 micron)3 powerplant, comparable to power densities available in mechanical (Section 6.3.2), chemical (Section 220.127.116.11), 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, ~104 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 >~103 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 vsound ~ 1540 in soft tissue (Table 6.7), so the ~1 nanosec temporal uncertainty adds (vsound 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 18.104.22.168), 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 Dvsound ~ 25 m/sec accuracy using a 1-nanosec clock. A measurement uncertainty of ~25 m/sec in the local speed of sound adds another (Dvsound t) ~ 1.6 micron of positional uncertainty, giving a total of DXmin ~ 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 ~Xrowi apart, each row of total length Lrow ~ S Xrowi emanating from a common tangential surface and extending deep into the tissues. The terminal navicyte of each row estimates its cumulative length as Lrow which also contains an unknown range error erow. The series of randomly distributed errors in the positions of each navicyte in the row, individually of magnitude ± DXmin, do not cancel to zero but instead constitute a random walk with maximum error excursion from the mean of erow ~ 2 N1/2 DXmin. The longest rows will occur in the viscera, farthest from any bony surface. Taking Lrow = 15 cm, DXmin = 3 microns and Xrow = 100 microns, then N = Lrow/Xrow = 1500 navicytes per row and erow ~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 erow / N ~ 0.2 microns. Even for Lrow = 2 meters (~longest possible transverse path length in the human body), N = 20,000 and erow = 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 ~Xrow apart in their common plane. Each terminal navicyte computes that its total length is precisely Lrow, but in fact navicyte B is in error by a distance eB along the row. Given the minimum range error DXmin, terminal navicyte B can only detect that an error exists when its range measurement to neighboring terminal navicyte A increases from Xrow to Xrow + DXmin, whereupon from simple geometry:
For Xrow = 100 microns and DXmin = 3 microns, then eB ~25 microns or ~1 cell width. This range error is relatively insensitive to Xrow; for instance, if Xrow increases to 110 microns then eB only rises to ~26 microns.
However, row terminus errors (measured in integral units of eB) are likely to be normally distributed between ± erow ~ 230 microns in the earlier simplest-case example. The error correction procedure involves nearest neighbors collectively polling the nearest nrow ~ 6000 rows (a bundle with termini covering ~0.6 cm2 when Xrow = 100 microns) for their measured eB's and then computing the mean of the distribution, which has an uncertainty of erow / nrow1/2 ~ 3 microns = DXmin, 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 Lrow, yielding a corrected Lrow accurate to ~ DXmin or ~3 microns. This corrective process is repeated:
a. for each row,
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 Xrow 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 protocol1624 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 lattices3036 is beyond the scope of this text.
The ability to make the corrections described above assumes that navicytes can distinguish -eB from +eB, an angular spread of ± 14° for eB ~ 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 xsensor 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 vsound = 1540 m/sec, a sensor separation of xsensor = 1.59 microns allows up to q = 0.25 radian = 14° to be distinguished. The uncertainty in vsound of ~25 m/sec imposes a minimum measurement uncertainty of Dq ~ 3°. The permissive detection cone shrinks to q = 0° at xsensor = 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 Xrange = Xrow ~ 100 microns and lateral positional uncertainty Xerror = DXmin ~ 3 microns, then relative navicyte angle can only be computed to an accuracy of Dq ~ sin-1 (Xerror / Xrange) ~ 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 (Xrange = 15 Xrow) and again taking the minimum Xerror, then Dq ~ 0.1° (~2 milliradians). Coordinates from a distant-neighbor navicyte with Xrange = Lrow ~ 15 cm with an uncorrected lateral positional uncertainty Xerror = erow ~ 230 microns would give Dq ~ 0.09° (~2 milliradians); taking Xerror = DXmin = 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 ~106° (~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 22.214.171.124) 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 126.96.36.199) may be determined to 1-100 microradian accuracy during non-recalibrated deployment lifetimes of 103-107 sec. Direct gyrostabilization may be possible in some applications (Section 188.8.131.52).
Last updated on 19 February 2003