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

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

6.4.2 Inductive and Radiofrequency Power Transmission

Electrically-powered nanorobots may tap natural electric fields already present in the body (Section 4.7.1) or may seek to acquire electromagnetic power from external sources. Transdermal magnetic induction and radiofrequency (rf) energy transfer has been employed in radio tracking and biomedical telemetry since at least the 1920s,^{634,635,637,638,3510} and commercially available miniaturized implantable transmitters are now commonplace in laboratory work,⁶³⁹ but using this technique to power submillimeter devices is a relatively recent concept.⁶³⁰

The biological effects of radio waves, microwaves, and infrared rays are usually equivalent to the effects of heating, although nonthermal rf-induced vasodilation, possibly due to Ca⁺⁺ flow manipulation, apparently has been observed in frogs.³³²⁸ Radio waves mainly induce thermal agitation of molecules and excitation of molecular rotations, while infrared rays excite vibrational modes of large molecules and may release fluorescent emission as well as heat. Both types of radiation are preferentially absorbed by unsaturated fats. In the United States the maximum permissible continuous occupational exposure level to microwave radiation has been 50 watts/m² for the testes and 100 watts/m² for the whole body -- essentially double the thermal radiance of a 100-watt (~2000 Kcal/day) ~2 m² human body, e.g., ~50 watts/m² across the skin, or roughly the intensity of direct sunlight on the skin (Section 4.9.4). Figure 6.12 compares several generally accepted international occupational and population exposure limits that have been adopted.⁸²⁴ Michaelson⁸²³ carried out extensive investigations and found no evidence for hazard at 100 watts/m², though there is conclusive evidence of potentially hazardous effects at levels above 1000 watts/m².

Single exposures to levels up to ~100 times these values may be briefly tolerated without injury, as for example photosetting dental fillings which require ~20 sec exposures to visible intensities of ~10⁵ watts/m². Optical tweezers also demonstrate that monochromatic exposures of ~10¹¹ watts/m² are tolerated by biological macromolecules immersed in an optically transparent aqueous medium due to fast thermal equilibrium at the micron scale.^1630,1631 On the other hand, the threshold of pain for ink-blackened human skin exposed to a radiant heat stimulus is ~10,000 watts/m² for 3-second exposures.⁵⁸⁵

An electromagnetic wave passing through the human body declines in intensity as it heats tissues and is attenuated approximately according to:

{Eqn. 6.32}

where I = power per unit area transmitted through the tissue, I₀ < 100 watts/m² (maximum safe incident intensity), d_x is depth of tissue penetration (meters), and a_E (meter^-1) is the total attenuation factor including scattering and absorption, the inverse of the mean free path. For radio frequency and microwave radiation, a_E = a_e n_E^1/2, where n_E is incident frequency (Hz) and a_e ranges from 2 x 10^-3 sec^1/2 m^-1 for muscle to 10 x 10^-3 sec^1/2 m^-1 for vitreous humor but averages ~5 x 10^-3 sec^1/2 m^-1 for soft tissue.⁶³⁵ At 100 MHz, 99% of incident power is removed in a 5-cm path length of soft tissue; at 1000 MHz (microwaves), 99% attenuation occurs within ~15 mm; at 10⁶ MHz (far infrared), ~0.5 mm of tissue removes 99% of the energy. On the other hand, 53% of the incident energy at 0.1 MHz passes through a 20-cm thick human body unscattered and unabsorbed, thus remains available for utilization by medical nanodevices seeking electrical power. The dependence of a_E on n_E varies significantly across the electromagnetic spectrum;^509,567,727 additionally, a_E for different tissues may deviate by up to a factor of 10 from the average values shown in Figure 6.13, especially in the IR-UV region.

Heetderks⁶³⁰ has analyzed the feasibility of transmitting electrical power into the body using submillimeter receiver coils -- a magnetic transcutaneous link, which is preferred when large power-transfer rates are required. In one of several designs, Heetderks starts with a 14-cm diameter transmitter coil (~diameter of the neck) with 11 turns (~40 microhenries inductance) and a 10 volt peak driving voltage at a frequency of 2 MHz delivering 4 watts RMS into the coil, a high but probably acceptable incident intensity of ~260 watts/m² at the skin. Inside the body, Heetderks uses a 400 micron diameter ferrite-core receiver coil with 60 turns spaced 50 microns apart, making a total receiver solenoid length of 3000 microns with ~0.89 microhenries inductance and Q = 40 for both transmitter and receiver, giving a fractional energy loss per cycle of 2p / Q ~ 16%. The receiver produces 1.1 volts and 320 microwatts in the ideal case where the receiver lies coplanar within the circumference of the transmitter coil. Large noncoplanar displacements may produce significant performance declines.

Each of Heetderks' receivers develops a power density of ~10⁶ watts/m³. Up to 600 receivers may be present within the volume of operation of each transmitter, collectively drawing up to 10% of unloaded transmitter coil power consistent with maintaining Q ~40. In a nanomedical application, each receiver could act as a microscopic power station, absorbing the externally supplied electromagnetic energy and then transducing it into other forms of energy which may more directly be tapped by in vivo medical nanorobots. In theory, a 100 watt/m² flux imposed on the entire ~2 m² human body surface should allow the transdermal importation of ~20 watts, enough to support a population of ~10¹² 10-pW nanorobots assuming 50% energy transduction efficiency among the widely dispersed "power stations".

How much smaller could Heetderk's receivers be made? A detailed design analysis is beyond the scope of this book, but a simple scaling estimate may be made as follows. For a receiver coil of magnetic inductance L_m (henries), resistance R_c (ohms), and rf waves of angular frequency w_E (rad/sec), a coupling coefficient (k_coupling) between transmitter and receiver coils may be crudely approximated as the ratio of the power stored in the magnetic field of the LR oscillator to the total power delivered to the receiver coil by the transmitter, or (dividing out the terms for current) k_coupling ~ w_E L_m / (R_c + w_E L_m). For n_E = w_E / 2p, m₀ = 1.26 x 10^-6 henry/m, m_f ~5 (a self-inductance factor when the receiver coil of N_coil loops contains a ferrite core⁶³⁰), r_wire = wire resistivity (~3 x 10^-8 ohm-m), and L is the characteristic size of the receiver (meters), then L_m ~ m₀ m_f N_coil² L³, R ~ r_wire N_coil² / L, and

{Eqn. 6.33}

where k_m = r_wire / 2p m₀ m_f ~ 10^-3. For discrete-component (macroscopic) radios that receive broadcast signals, L ~ 1 cm and n_E = 1-100 MHz, giving k_coupling = 0.9-0.999, which is very good. For Heetderks' submillimeter receiver designs, k_coupling = 10^-3 - 10^-5; for example, the receiver described above has L ~ 700 microns, n_E = 2 MHz, giving k_coupling ~ 5 x 10^-4, which is very poor. With such poor coupling, very large transmitter intensities are required to induce very small receiver power levels. Troyk and Schwan⁶³⁶ have suggested that printed thin-film coils integrated directly with sensor circuitry driven by a special transmitter coil topology could provide adequate amounts of power to implanted coils having coupling coefficients up to two orders of magnitude lower than Heetderks' designs. However, even k_coupling ~ 10^-7 at 2 MHz only reduces receiver size to L ~ 500 microns, which is not much improvement. The L⁴ scaling of k_coupling suggests that magnetic transcutaneous power receivers smaller than ~500 microns may be impractical, given the negligible amounts of magnetic field energy that can be stored in nanoscale volumes (Section 6.2.4) and the likelihood that small AC circuits will be heavily damped.¹⁰ (See also Section 7.2.3.)

However, efficient micron-scale MHz-frequency rf antennas may still be feasible for individual nanorobots, as suggested by the following simplistic analysis. Consider a simple spring pendulum with a bob of mass m_bob, density r (~diamondoid), thickness h_bob, and face area L², with n_q charges and total charge Q_bob = n_q q_e = L² C_q (coul) permanently embedded on the bob surface, where C_q is bob surface charge density. An oscillating rf field of strength E_e (volts/m) and frequency n_E drives the charges with a force F = E_e Q_bob = m_bob a_bob at an acceleration a_bob (m/sec²). The bob mass accelerates for a half-cycle of duration t = (2 n_E)^-1, periodically displacing a distance x_bob = (1/2) a t² = Q_bob E_e/ 8 m_bob n_E² and traveling at a mean velocity v_bob ~ 2 x_bob n_E. The local energy density produced by the transmitter field is D_E = (1/2) k_e e₀ E_e² (joules/m³) (variables defined in Section 4.7.1, for water), with transmitter power intensity I_t = c D_E / n (watts/m²) where c = 3 x 10⁸ m/sec (speed of light) and n = k_e^1/2 is the refractive index for a nonmagnetic material at low frequency; hence E_e = (2 I_t / k_e^1/2 e₀ c)^1/2 (volts/m). At the end of each half-cycle the bob receives an energy E_1/2 = (1/2) m_bob v_bob². Assuming mechanical rectification, a lever arm connected to the charged bob in the receiver transmits to the nanorobot a received power of P_n ~ 2 E_1/2 n_E (watts); making the appropriate substitutions we have:

{Eqn. 6.34}

An additional constraint is that bob travel x_bob must not exceed the space available within a receiver housing of depth L_trav. The bob travels slower but farther at lower rf frequencies, and the lowest possible operating frequency minimizes attenuation in tissues. Taking x_bob <~ n_# h_bob = L_trav as a limit establishes a minimum operating frequency that will not peg the bob against its stops, given by:

{Eqn. 6.35}

We require a high charge density and thus assume C_q = 0.1 coul/m² (~0.6 charge/nm²), somewhat less than the ~0.16 coul/m² for charged amino acid molecules and the ~0.3 coul/m² for a "fully ionized surface",¹¹⁴⁹ but well above the 0.005-0.02 coul/m² recorded experimentally for SiO₂ on mica,¹¹⁵⁴ the 2 x 10^-3 coul/m² for the electrodes in the electrostatic motor (Section 6.3.5) designed by Drexler,¹⁰ and the 2 x 10^-4 coul/m² figure given by Lowell and Rose-Innes¹¹⁵¹ cited as typical for "highly charged surfaces." For a water-immersed bob, the DC surface field from this 0.1 coul/m² charge density is a sustainable E_DC = C_q / 2 k_e e₀ = 76 megavolts/meter.

Signal attenuation is minimal below ~1 MHz (Eqn. 6.32). Taking I_t = 100 watts/m², L = 1 micron, r = 2000 kg/m³, h_bob = 20 nm, n_# = L_trav / h_bob = 50 for L_trav = L, k_e = 74.31 for 310 K water, and e₀ = 8.85 x 10^-12 farad/m, then minimum n_E = 0.17 MHz giving P_n ~ 0.8 pW delivered to the nanorobot producing a power density within the housing of D_n = P_n / L_trav L² ~ 10⁶ watts/m³. In a practical system, these receivers must be aligned precisely with the incident field to maximize energy coupling, possibly requiring dynamic orientation control of each onboard receiver. Resonating beam structures 8-30 microns in size that can serve as rf filtering and oscillator elements have been demonstrated experimentally at ~15 MHz.⁸⁷⁹ High-frequency submicron electrometers with charge per unit bandwidth sensitivity of ~0.1 electron/Hz^1/2 have been demonstrated,²⁹²⁶ and submicron rf mechanical resonators have been discussed by Cleland and Roukes.²⁹²⁷

In 1999, Pharma Seq³⁴³² was developing laser-photon-powered cubic ~250-micron rf microtransponders for use with oligonucleotide probes in DNA assays; suspended in slurry, each device had an integrated circuit storing the sequence of the oligo attached to it, and emitted a coded signal (the probe's serial number) to a nearby receiver when a fluorophore-labeled target DNA molecule binded to a complementary probe. However, direct visible-spectrum photonic powering of medical nanorobots in vivo is not promising except within a few millimeters of the unclothed epidermis and only in the presence of direct sunlight or strong artificial sources of illumination (>10-100 watts/m²) near maximum safe exposure limits, due to the poor detection efficiency (low photon count rate sec^-1 m^-2) and rapid attenuation of photons in human tissue (Section 4.9.4).

Last updated on 18 February 2003