Analytical Solution of the Bloch NMR Flow Equations for Fluid Flow

Various biological and physiological properties of living tissues can be studied by means of NMR techniques. However, the basic physics of extracting the relevant information from the solution of Bloch NMR equations to accurately monitor the clinical state of biological systems is still not fully understood. Presently, there are no simple closed solutions known to the Bloch equations for a general RF excitation. Therefore, an exponential type of solution of the equations presented in this study, which can be taken as definitions of new functions to be studied in detail, may reveal very crucial information from which various NMR flow parameters can be derived. In this study, we are concerned with finding a solution of the form to the Equations. We shall restrict our attention to cases where the radio frequency field can be treated by simple analytical methods. First, we shall derive a time-dependent second-order non-homogenous linear differential equation from the equations in term of the equilibrium magnetization Mo, RF B1(t) field, T1 and T2 relaxation times. Then, we would solve the differential equation for the cases when RF B1(t) = 0, and when RF B1(t) 0.