Generalized Fractional Order Bloch Equation With Extended Delay

Abstract

The fundamental description of relaxation (T-1 and T-2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time-and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T-1 and T-2 relaxation. The T-1 decay is stable for the range of delays tested (1 mu sec to 200 mu sec), while the T-2 relaxation in this extended model exhibits a critical delay (typically 100 mu sec to 200 mu sec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.