Mathematical analysis of adiabatic model of Bloch NMR flow equations for functional magnetic resonance imaging

Abstract

In this contribution, solutions to the Bloch NMR flow equations in the form of polynomials are presented. The polynomials are obtained in terms of trigonometric, algebraic, ordinary and special functions. By means of these polynomials, appropriate mathematical algorithms are developed to consider the motion of a blood molecule of mass m as a function of coordinate x alone. We derived the first integral (constant of the motion) and energy integral for the blood molecule. The energy integral gives us specific information about the motion of the blood molecule (oxyhemoglobin molecule or molecule of some other chemicals in the brain that take part in NMR signal production) depending on the energy required by neurons when subjected to various degree of excitation. This mathematical model and analysis may be very significant to explain the motion of the molecules of each fluid substance that is responsible for specific task performed by the brain. The dynamics of NMR signal as developed in terms of sine and cosine functions can be additional important tools that can help to see the direct relationship between changes in blood flow to the brain and neural communication.