High-precision mapping of the magnetic field utilizing the harmonic function mean value property

The spatial distributions of the static magnetic field components and MR phase maps in space with homogeneous magnetic susceptibility are shown to be harmonic functions satisfying Laplace's equation. A mean value property is derived and experimentally confirmed on phase maps: the mean value on a spherical surface in space is equal to the value at the center of the sphere. Based on this property, a method is implemented for significantly improving the precision of MR phase or field mapping. Three-dimensional mappings of the static magnetic field with a precision of 10(-11) similar to 10(-12) T are obtained in phantoms by a 1.5-T clinical MR scanner, with about three-orders-of-magnitude precision improvement over the conventional phase mapping technique. In vivo application of the method is also demonstrated on human leg phase maps. (C) 2001 Academic Press.