Motion Compensation in MR Imaging

Mark S. Cohen, Ph.D.

Introduction

For the past several years methods have been known which compensate for motion-induced distortions in Magnetic Resonance (MR) imaging. This paper summarizes some of that literature and attempts to put it into easily digestible form. In addition, a computationally simple method is demonstrated for the analysis of a variety of time-varying gradient waveforms and the complete solution for a second order motion compensation scheme is given. The general schemes for motion compensation used by all MR manufacturers are similar, though the acronyms used to describe them differ. Siemens uses the term, "GMR" (gradient moment rephasing), General Electric describes the technology simply as "Flow Compensation", where Picker describes the "MAST" (motion artifact suppression technique). The implementations differ somewhat between manufacturers, some companies null only the lower order terms (see below) whereas others use more sophisticated higher order moment nulling. In this paper we will first analyze the effects of motion on MR images, We will then discuss and derive gradient moment nulling schemes and finally develop a mathematically simple method of calculating and minimizing gradient-induced phase shifts.

Motion Effects in the Presence of Field Gradients

Magnetic Resonance images frequently suffer from artifacts and signal contrast errors due to the dispersion signal which occurs as a consequence of patient or sample movement during scanning. Beyond the expected blurring which results from movement the acquisition period, the unique process of positional encoding with magnetic field gradients used in two dimensional Fourier transform (2DFT) MRI produces a special sensitivity to motion. The position of the MR signals is mapped in conventional 2DFT images as a function of phase and frequency through the use of magnetic field gradients that alter the resonance frequency, ƒ, of the nuclear spins as a function of space according to the simple formula:

where g is the Larmor constant (42.58 MHz/Tesla for protons) and B is the magnetic field. Assuming that the field is spatially constant, all nuclear spins will have the same precessional frequency. If through the use of gradients, the field is made to vary with position, the spins at different positions will develop a phase difference, Dj, proportional to the magnitude and duration of the field non-uniformity:

where DB is the variation in magnetic field and T is the duration of the presence of the non-uniformity. In the typical case of imaging with time-varying gradients a phase dispersion is calculated according to the formula below:

in which the variation in the magnetic field is represented by the time-varying gradient, G_x(t), T is the duration of the applied gradient, and the position of the spins along that gradient is represented as x(t) to indicate that the spins themselves may be in motion.

FIGURE 1

Let us now examine the phase effects of a few simple gradient pulses. Figure 1 shows a "bipolar" gradient pulse having positive and negative lobes of equal magnitude and duration. To compute its effects we will first expand the position term for x(t) in its Taylor series as the sum:

We can solve the phase integral easily as the sum of the definite integrals for the two lobes of the gradient waveform:

The first lobe adds a total phase of: while the second contributes a phase of . The net phase is thus equal to - (in addition to acceleration and higher order terms). Note that there is no net effect on the non-moving terms. The bipolar waveform thus produces a velocity (and acceleration)-dependent phase shift. The consequence of this is that in normal MR imaging sequences, which use gradient waveforms largely of this sort, spins which move during gradient application will show a net phase shift which is later transformed to a positional shift along the phase-encoding axis of the images. Note also that the phase shift is linear in amplitude but quadratic in duration and thus can be minimized, though not eliminated, by the use of short duration, high amplitude, gradient pulses.

FIGURE 2

Compensation of Simple Velocity

Figure 2 shows a gradient waveform having three lobes. In effect it is the gradient waveform of Figure 1 concatenated with its time reverse. Using the same procedure as above we note that the total phase effects of the three lobe waveform can be computed as the sum:

The first term adds a phase shift, as before, . The second, negative, gradient pulse results in a phase shift of , while the third gradient pulse adds a net phase of . The summed effect of all three gradient pulses is to leave no position- or velocity-dependent phase shift. The reader may be interested to verify that an acceleration-dependent term of

remains, together with any higher order terms..

Motion compensation of the "phase-encoding" axis requires that after the phase encoding interval the spins acquire a net phase change independent of velocity. While this is in principle possible, it is typically impractical with today's scanners to produce motion-compensated phase-encoding pulses.

It is notable that the second (negative-going) of these three gradient lobes must have and integral equal to the sum of the two positive-going lobes but can be of any combination of amplitude and duration which retains this area, while maintaining flow insensitivity; very short pulse trains may therefore be used. There exist, indeed, an infinite variety of three pulse sets which will have a null effect on moving spins. The pulse sequence designer may choose from among these according to other boundary constraints such as gradient rise times and duty cycles or, more likely, to minimize eddy current induction as discussed at the end of this section. Generally, with motion compensated sequences it is advisable to use the shortest duration gradients possible as the phase has a quadratic or higher dependence on the duration of the gradient pulses for moving spins. The phase encoding gradient should also be kept short for this reason.

A Velocity Compensated FLASH-type Sequence

The methods outlined above are sufficient to form a complete MR imaging sequence with velocity (or "first order") motion compensation. In general, compensating for higher orders of motion (e.g. acceleration) will require additional gradient pulses. Thus, where TE must be kept to a minimum, it is reasonable to apply only first order motion correction.

Figure 3 is the timing diagram for a FLASH imaging sequence which uses the simple gradient waveform of figure 2 for rephasing of spins along the slice selection and readout axes. The phase dispersion along each axis is linearly independent, thus the slice selection (GS) and readout (GR) gradients each have the shape of figure 2. The phase encoding gradient (GP) is left without moment compensation.

FIGURE 3

In this example all gradient pulses are of unit amplitude, but the middle pulse on each gradient is of double the duration of the initial pulse. As required, at the center of the readout period (marked with an arrow), the stationary and moving spins will be in phase, but some phase dispersion necessarily will occur during readout. As the readout gradient remains on past the middle of the readout period, the spins will necessarily be dispersed at the end of readout.

Even Echo Rephasing

It has been recognized for some time that the standard double echo imaging technique results in first order motion compensation for the second echo, when TE is chosen appropriately. Figure 4 illustrates the double echo pulse sequence. One effect of the 180° pulse is to invert the phases of any spins. As a consequence, a positive polarity gradient pulse following a 180° pulse acts like a negative polarity gradient applied without the RF pulse. In the figure, the gradient pulses occurring during the period highlighted in gray act as though they are of opposite polarity. Notice that, for the readout gradient, this is as though the three lobed gradient waveform described above were applied. As a result of this phase reversal effect at the middle of the second readout period, again indicated by an arrow, there is no velocity-dependent phase shift. Thus, along the readout axis at least, the double echo sequence is motion compensated as long as the TE for the second echo is twice that of the first echo. This phenomenon is commonly known as "even echo rephasing" and results in relatively high signal from flowing spins in double echo imaging sequences.

Figure 4. Conventional double echo pulse sequence. The effect of the 180° pulses is to reverse the phase of the nuclear spins. As a result, in the period indicated in gray the phase effects of the gradients are also reversed. In particular, the middle pulse of GR (the readout gradient) has a negative phase effect as compared to the gradient prior to the RF pulse. At the point indicated by the arrow moving and stationary spins are in phase along GR.

Generalized Method for Phase Calculation

Extending these analyses to the compensation of higher moments, such as acceleration and its derivatives, and to more complex gradient waveforms may be done as outlined above, but the calculations become computationally tedious. To minimize the calculation complexity it is useful to solve for the phase shift of a variety of simple gradient pulse "elements" in the general case. The summed effects of these pulses can then be calculated algebraically.

FIGURE 5

The figures 5a, 5b and 5c are the constituents of most typically used gradient pulse shapes. When concatenated in that order they form the trapezoidal pulse shape favored in commercial scanning equipment. Here, M is the gradient amplitude, t0 is the onset time of the gradient pulse and d is the pulse duration.

The phase contributions of each of the three elementary pulses may now be solved in the general form. Using the methods outlined above the reader will quickly verify that that the phase shift resulting from the waveform of 5b, including the acceleration term, is:

It is not generally necessary to compensate for terms beyond acceleration as these are typically of small magnitude in human imaging.

The gradient pulse of figure 5a has the analytical form:

Solving for the phase dispersion of that pulse, is straightforward and results in:

Finally, the pulse of 5c, whose time domain function in the range from t₀ < t < t₀ + d can be shown to result in a phase shift of:

It is interesting to note that the sum of the phase effects of the pulses 5a and 5c is equal, as expected, to the net effect of the square pulse 5b.

Compensation for Velocity and Acceleration

Solving for the general class of pulses, to second order terms, results in a considerable simplification in calculation. For example, let us solve for the four pulse series shown in Figure 6 with the goal of producing a pulse series having no phase effects for stationary, moving and/or accelerating spins. Here, since the system will otherwise be under-constrained, we will assume

FIGURE 6

that all of the pulses will have a fixed duration, T, and that the final pulse will have unit amplitude. This pulse train might be used in the design of an acceleration compensated readout gradient.

Here we can solve for the phase effects of each of the four components separately where the first pulse has amplitude A from t = 0 to T, the second pulse has amplitude B from t = T to 2T and so on. We will solve for the phase at the time t = 4T as indicated by the arrow.

Noting that each of these pulses is of the form shown in figure 5b, we can write down the phase immediately. In the first pulse, for example, the duration, d, is equal to T, the amplitude, M, is equal to A and the pulse begins at t0 = 0. It therefore contributes a net phase effect of:

Similarly, the phase effect of the second pulse is:

and the combined effect of all four pulses is:

For complete compensation each of these terms, in x, v and a must equal zero. The structure of the equations above is such that solving for the static spins simplifies the equation for the moving spins; solving for the moving spins simplifies the equation for the accelerating spins. The entire system of equations reduces, in the case of second order moment compensation, to:

which is easily solved. For this example, A = -1, B = 3 and C = -3. In this example, the method of direct integration is sufficiently straightforward. Verification of the results is left as an exercise to the reader. In the spirit of keeping the reader well exercised it is also interesting to verify that the three pulse series shown in Figure 2, when made with trapezoidal pulse shapes, also results in complete rephasing; square gradient pulses are not necessary.†

Application to Complete Imaging Sequences

A complete second order, flow and acceleration, compensated sequence may be built on this model as shown in Figure 7. In this example, an FID (or FLASH-type) sequence using an excitation pulse of a° is shown. For clarity only the activity on GR, the readout gradient, is shown although the events on the selection gradient would be much the same. The numbers adjacent to GR indicate the amplitudes, as derived above, for the gradient events.

Figure 7. FID, or FLASH-type MR pulse sequence with second order moment compensation.

Figure 8. Hahn spin echo MR pulse sequence with second order moment compensation.

Recognizing that the 180° RF pulse acts to induce a phase reversal it is easy to see that the Hahn echo sequence in Figure 8 will also result in velocity and acceleration compensation.

In practice there are other considerations in the design of motion compensated sequences. In particular, gradient eddy currents may pose a substantial problem when large gradients are used.

This is especially apparent when asymmetrical pulsing patterns are utilized. The FLASH-type sequence show above is inherently resistant to eddy current effects due to its internal symmetry: for every positive-going gradient pulse there exists a corresponding negative-going pulse. Such a pattern reduces the buildup of long time constant eddy currents in the magnet. The Hahn spin echo sequence, however, is highly asymmetrical. The combined effects of the two large negative gradient pulses may, if not properly compensated, result in the formation of large amplitude eddy currents.

Summary

The phase dispersing effects of gradients used in MRI are, at least to a first approximation, easily analyzed by integrating the effects of the applied time-dependent magnetic fields. While the integrals themselves are generally trivial, there solution in all but the simplest cases may be extremely tedious. A generalized solution can be derived for common gradient pulses and results in a reduction in calculation complexity. Using this technique the magnitudes or durations of gradient events may be calculated by matrix algebra which, being machine-calculable, may be easily automated. Application of the motion compensation methods to standard MR pulsing sequences is shown for both gradient echo and RF echo methods. While the analytic solutions presented are straightforward enough their application may be limited by non-idealities of the imaging hardware.

References

Haacke, E.M. (1987). "Improving MR image quality in the presence of motion by using rephasing of gradients." American Journal of Roentgenology. 148: 1251-1258.

Nishimura, D., A. Macovski and J. Pauly. (1986). "Magnetic resonance angiography." IEEE Transactions on Medical Imaging MI-5(3): 140-151.

Wood, M., V. Runge and R. Henkelman. (1988). "Overcoming motion in abdominal MR imaging." American Journal of Roentgenology. 150: 513-522.