A bipolar gradient is defined as a pair of gradient lobes with opposite polarity. For the discussion here, the amplitude of each lobe is equal but opposite in sign so that the areas under each lobe are assumed to be equal. As reviewed in Figure II3-1, a bipolar gradient causes no net phase shift for stationary protons (neglecting T2^* decay).

Now consider the effects of the bipolar gradient on protons moving in the direction of the gradient (Figure II3-2).

If a proton moves while the bipolar gradient is applied, then the amount of dephasing experienced will not equal the amount of rephasing. For example, consider a proton that moves from the neck to the pelvis during the course of the bipolar gradient application (Figure II3-2, top). During the positive lobe of the bipolar gradient, the proton in the neck will experience a stronger magnetic field, which causes a relatively large positive phase shift relative to the center of the magnet. It moves during this gradient, and by the time the gradient is reversed, it is in the lower abdomen and pelvis, where again it experiences a magnetic field that is greater than 1.5 T. Hence, during the reversed gradient, the proton continues to accumulate positive phase. At the end of the bipolar pulse, the moving proton will have a very large positive phase shift relative to the stationary protons in the field. The phase gain is directly proportional to the velocity along the direction of the gradient.

What happens if the proton moves half as fast and travels from the neck to the abdomen during the course of the bipolar gradient? Again, during the positive lobe of the gradient, the proton will accumulate a positive phase shift. But during the reversed gradient, the proton will experience little rephasing in the center of the magnet. This means that at the end of the bipolar pulse, this slower proton will still have a positive phase shift relative to the stationary protons, but it will not be as great as the faster-moving proton.

What if the proton moves in the opposite direction, for example a proton in the veins (Figure II3-2, bottom)? The relative phase of the proton will now be negative compared to the center of the magnet. During the reversed gradient, the rephasing will be minimal, and hence the venous proton will then have accumulated a negative phase shift.

These observations can be summarized as follows:

The exact relationship between phase shift and velocity of the protons can be predicted for a given bipolar gradient. In this way, phase contrast MR imaging can be used to convert maps of phase shifts into velocity maps.

Phase shifts accumulate for moving protons only if the protons are moving in the direction of the gradients. The direction of the bipolar gradients is referred to as the flow sensitivity direction or the velocity-encoding direction. Protons moving perpendicular to the gradients will not experience any phase accumulation.

For a single bipolar gradient to provide accurate measurements of velocity based on phase differences, all phase shifts must be solely attributable to movement of protons during the gradients. The magnetic field must be perfectly homogeneous and the gradients perfectly linear.

Significant phase shifts are caused by magnetic field inhomogeneities, and these shifts are indistinguishable from those caused by motion. Consider the following example. If the main magnetic field is 1.5 T and the magnetic field gradient is intended to make the field at a particular location 1.51 T during the positive lobe and 1.49 T during the negative lobe, then a stationary proton would have no net phase shift after these two lobes of the bipolar gradient are applied. But what if the field is inhomogeneous, so that the magnetic field at a given location is 1.6 T instead of 1.5 T? Now the same bipolar gradient will cause the field at that location to be 1.61 T and 1.59 T during the two lobes. Relative to the protons that are at 1.5 T in the center of the magnet, the stationary protons in this region will experience a positive phase shift—a very positive phase shift—after the bipolar gradient. Thus, as a consequence of magnetic field inhomogeneity, these stationary protons would be misinterpreted as very fast-moving protons!

Therefore, without perfect field homogeneity it is impossible to do accurate phase contrast flow quantification using a single bipolar gradient. To correct for other causes of phase shifts, the collection of phase data with one bipolar gradient is repeated with following application of the mirror image of that bipolar gradient, and the two datasets subtracted (Figure II3-3).

The effect of field inhomogeneity on the net phase shift from each bipolar gradient will not be zero, but it will be the same for each bipolar gradient. In the example just described, the bipolar gradient that causes 1.61 T and 1.59 T would lead to a positive phase shift of, say, +160°. Then the second bipolar gradient, a mirror image of the first, would cause the field to be 1.59 T and 1.61 T, again resulting in a phase shift of +160°. Consequently, if the phase shifts from the two bipolar gradients are subtracted, then the net phase shift will be zero and the effects of an inhomogeneous magnetic field eliminated.

For phase contrast flow quantification, two complete sets of echoes must be collected, one using each bipolar gradient. Having to repeat the acquisition and collect two sets of echoes makes phase contrast acquisitions twice as long as conventional gradient echo imaging.

Subtracting two datasets is necessary to cancel out the phase errors due to field inhomogeneities. But the pairs of gradients need not be mirror-image bipolar gradients. In fact, on most commercial systems, phase contrast MR images are generated with phase images acquired with (a) flow-encoding gradients and (b) flow-compensated gradients (Figure II3-4). As with bipolar gradients, using this pair of acquisitions and subtracting one set of phase data from the other helps to reduce artifacts due to other causes of phase errors, such as field inhomogeneity.

Use of flow-compensated gradients reduces the number of acquisitions necessary to generate flow sensitivity in three directions to four. With conventional bipolar gradients, six acquisitions (one pair per direction) would be needed.

With phase contrast imaging, there is an important distinction between the direction of flow sensitivity and the plane of imaging.

For phase contrast MR angiography, there is often a need for flow sensitivity to be three-dimensional. Consequently, flow-compensated gradients are acquired in all three directions, and the entire sequence requires collection of four sets of data. The imaging plane or slab can be two- or three-dimensional.

Flow quantification acquisitions are typically 2D and set up to measure either in-plane flow or through-plane flow. Through-plane flow images are obtained by applying the velocity-encoding gradients in the slice-select direction (Figure II3-2), so that flow-sensitivity is in the through-plane direction, while the imaging plane is perpendicular to the direction of flow. Alternatively, the flow sensitivity desired may be in the plane of the image. In such cases, flow-encoding gradients are applied in both the frequencyand phase-encoding directions.

Three-dimensional phase–contrast flow quantification, with flow sensitivity in all three directions, is rarely used to image cardiovascular subjects because of the long acquisition times and complex image post-processing.