ABSTRACT:The recent analytical solutions to the Bloch NMR equations for a general RF excitation have opened many possibilities for further investigations to NMR theory and experiments even at the molecular level. Fortunately, many of the most important but hidden applications of blood flow and general physiological fluid flow parameters can be revealed without too much difficulty if appropriate mathematical techniques are used to explore the new NMR equations derived from the Bloch equations. Generally, we should be very much concerned with analytical results that the Bloch NMR flow equations can provide for different physical, biomedical, geophysical, medical, and environmental situations especially at the molecular level for the purpose of interdisciplinary approach to solve difficult problems. It can be motivating, exciting, and rewarding if attention are focused on the possible application of these analytical techniques and methods suitable for describing each of the various normal and pathological biological conditions. Most solutions presented in this study are described both in isotropic and anisotropic geometries with minimum mathematical assumptions. We discussed a general expression for the diffusion coefficients in the common geometries. These analytical results can prove to be very invaluable in the analysis of restricted flows. It is so much special because it could tell us when restricted flows occur and also reveal the causes of such restriction. Such knowledge can help in finding the causes of many diseases (whose causes are yet unknown) and suggest the best treatment for them.

Most cardiovascular emergencies are directly caused by coronary artery disease. Coronary arteries can become clogged or occluded, leading to damage to the heart muscle supplied by the artery. Modem cardiovascular medicine can certainly be improved by meticulous analysis of geometrical factors closely associated with the degenerative disease that results in narrowing of the coronary arteries. There are, however, inherent difficulties in developing this type of mathematical models to completely describe the real or ideal geometries that are very critical in plaque formation and thickening of the vessel wall. Neither the mathematical models of the blood vessels with arthrosclerosis generated by the heart and blood flow or the NMR/MRI data to construct them are available. In this study, a mathematical formulation for the geometrical factors that are very critical for the understanding of coronary artery disease is presented. Based on the Bloch NMR flow equations, we derive analytical expressions to describe in detail the NMR transverse magnetizations and signals as a function of some NMR flow and geometrical parameters which are invaluable for the analysis of blood flow in restricted blood vessels. The procedure would apply to the situations in which the geometry of the fatty deposits, (plague) on the interior walls of the coronary arteries is spherical. The boundary conditions are introduced based on Bessel, Boubaker and Legendre polynomials.

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