A mathematical model of Bloch NMR equations for quantitative analysis of blood flow in blood vessels of changing cross-section—PART II

Abstract

Unlike most medical imaging modalities, magnetic resonance imaging (MRI) is based on effects that cross multiple biological levels: contrast depends on interactions between the local chemistry, water mobility, microscopic magnetic environment at the subcellular, cellular or vascular level, cellular integrity, etc. These interactions potentially allow for imaging functional changes in the same reference frame as the anatomic information. However, to tap this potential, we need methodologies that robustly incorporate the best models of the underlying physics interactions in order to extract the best possible interaction obtainable on flow velocity and rates. Due to the fundamental role the Bloch NMR equations play in the analysis of the properties of magnetic resonance imaging, this presentation will focus on mathematical modeling of the Bloch NMR flow equations into the harmonic differential equation. This simplification allows us to explain qualitatively, the effects of coriolis force on the motion of flowing fluid. The Transverse magnetization M y, is introduced as a stream function. Our choice of conditions has led to a linear equation for M y. We derived the stream function as a form of solution which contains the linearity property demanded by conditions at x=0. The resulting flow reveals some interesting wave-like properties which were examined directly. The existence of the waves is associated with the non-uniformity of the Coriolis parameter, and it is not difficult to see the general mechanism. The quantum mechanical models of Bloch NMR equations describe dynamical states of particles in flowing fluid. We introduce the basic background for understanding some of the applications of quantum mechanics to NMR and explain their significance and potentials. It also describes the behavior of the rF B 1 field when the fluid particles flow under physiological and some modeled pathological conditions. The wave function is explored to determine the minimum energy, a function of the rF B 1 field for the fluid particle to be located in the non-classical region. These models can be invaluable to understand the basic Physics of extracting the relevant flow parameters by which velocity quantification can be made in Blood vessels with changing cross-section.