Significance of the point of expansion in interpretation of gradient moments and motion sensitivity

The relationship between magnetic field gradient waveform moments and the motion sensitivity of magnetic resonance imaging was explored analytically and by computer simulation. The analysis and simulations revealed several key points. In general, waveform time moments define sensitivity to the time derivatives of position of moving material only at a single time point: the time about which the moments are computed. A Taylor series description of instantaneous position is expanded about this same time point to compute the phase acquired due to specific derivatives of position. A moment is proportional to phase sensitivity to a particular derivative of position throughout the waveform only when sensitivity to all lower-order derivatives is zero. Under restricted conditions of waveform symmetry and motion characteristics, the phase due to motion may be expressed in terms of the average value of a derivative of position over the duration of the waveform. The choice of the moment center, or point of expansion, adds a degree of freedom that may be used advantageously in the design of motion-compensating and motion phase-encoding gradient waveforms. These results facilitate a more complete understanding of the effects of motion through a magnetic field gradient.