Harmonic functions are defined as satisfying the Laplace equation ∇2u = 0. Whereas these functions have been applied to scalar quantities such as temperature and conserved potentials in physical sciences, it is valid to extend the harmonic function concept and basic properties to vectors such as electrostatic field, magnetostatic field, current density, and gravitational force. We identify these physical fields as Laplace fields and develop the Laplace field theory. One of the basic properties of Laplace fields is that the mean field value on a spherical surface in space is equal to the field value at the center. Based on this, a spherical mean value method is proposed to reduce the noise in the measurement of these fields. Methods to quantify related variables such as magnetic susceptibility under inhomogeneous field conditions for objects of arbitrary shapes are also developed. Nuclear magnetic resonance phase imaging (MRPI) can non-invasively map the three-dimensional magnetic field z component and the temperature distribution. We show theoretically and experimentally that the MR phase map is a Laplace field and employ it as a convenient basis for the validation of the Laplace field theory and methodology. We implement the spherical mean value method to map out the magnetic field (z component) and relative temperature change with high precision, i.e., 10-11 ∼ 10-12 Tesla for magnetic field and 10 -3 ∼10-4 °C for temperature. The random noise is reduced by over three orders of magnitude without cost of any extra experimental time. Furthermore, with MRPI, we measure the magnetic susceptibilities for CuSO4-solution phantoms of irregular shapes without the presumption of the homogeneity of the applied field. Numerical simulations of the boundary value problem of Laplace equation extend the methodology of precision improvement and magnetic susceptibility quantitation to nonspherical boundaries. The methodology developed in this work can be generalized to applications for any physical fields whose distributions in space are Laplace fields.
Adviser: John S. Leigh, Jr. Thesis (Ph.D. in Biochemistry and Molecular Biophysics) -- University of Pennsylvania, 2000. Includes bibliographical references and index.