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

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

4.9.2.4 Gravitational Geographic Macrosensing

Medical nanodevices can measure variations in the gravity field to ~10^-6 g's for L = 20 micron gravimeters in a measurement time t_meas = 2-9 millisec (Section 4.4.2). This implies that in vivo nanodevices can take precise measurements of their latitude and altitude relative to sea level ~100 times every second. Gravity increases toward the poles and at lower altitudes. Specifically, using the formula of Cassinis (accounting for rotational and polar flattening effects on the Earth) with the Bouguer correction to the free air variation by altitude (assuming flat topography), measured gravity g_meas is given approximately by

{Eqn. 4.56}

where q_L = terrestrial latitude (equator = 0°), h = height above sea level in meters, g₀ = 9.78039 m/sec² (equatorial sea-level value ofg), k₁ = 5.2884 x 10^-3, k₂ = 5.9 x 10^-6, k₃ = 3.086 x 10^-6 sec^-2, k₄ = 4.185 x 10^-7, and r_earth = 5522 kg/m³.

Since sea-level g varies from 9.78039 m/sec² at the equator to 9.83217 m/sec² at the north pole, a 20-micron gravimeter (Dg = 10^-6 g) detects a change in position of 1 arcmin of latitude or ~1900 meters north/south along the Earth's surface. Similarly, since at 45° latitude g varies from 9.806 m/sec² at sea level to 9.803 m/sec² at 1000 meters altitude, a 20-micron gravimeter detects a change in altitude of ~3.3 meters (e.g., upstairs vs. downstairs in a house). For comparison, in 1998 high-quality commercial gravity gradiometers measured gradients of ~10^-9 g/meter and allowed the compilation of micro-g (~1 milligal) resolution aerial gravity maps;¹⁵²⁷ atom interferometers measured the gravitational acceleration of atoms to a precision of 10^-10.

To achieve such phenomenal positional accuracies, the nanodevice must be able to computationally resolve several complicating factors. First, localized mass concentrations representing nonuniformities in crustal density produce residuals of up to ±0.0006 m/sec², which may be removed from the data using a standard map of known terrestrial isostatic variations and anomalies. Indeed, matching observations to such a map could provide useful longitudinal information as well. Another complication is the variation in gravity due to tidal forces amounting to ~3 x 10^-7 g's, twice daily, which lies at the limits of detectability for a 20-micron gravimeter. Other minor geodesic and terrain-related corrections, too complicated for discussion here, may also need to be applied in certain circumstances. Note that the presence of nearby heavy objects does not influence measurement accuracy: a 100-ton building 10 meters away adds a lateral acceleration of only 7 x 10^-9 g's to a human body.

One final complication is that patient movements create kinematic accelerations that must be distinguished from the gravitational accelerations. Gravity readings can be corrected by taking derivatives of the signals from kinesthetic monitoring to give gravity in the reference frame of the patient's room, and many individual measurements may be averaged to improve accuracy since the gravity vector normally changes only very slowly.

Last updated on 17 February 2003