**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

**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

where q_{L} = terrestrial
latitude (equator = 0°), h = height above sea level in meters, g_{0}
= 9.78039 m/sec^{2} (equatorial sea-level value ofg), k_{1}
= 5.2884 x 10^{-3}, k_{2 }= 5.9 x 10^{-6}, k_{3}
= 3.086 x 10^{-6} sec^{-2}, k_{4} = 4.185 x 10^{-7},
and r_{earth} = 5522 kg/m^{3}.

Since sea-level g varies from 9.78039 m/sec^{2} at
the equator to 9.83217 m/sec^{2} 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/sec^{2} at sea
level to 9.803 m/sec^{2} 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/sec^{2}, 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