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Physics Facts - Fazed by phase?

Puzzled by the meaning of "phase encoding" in MRI. Hallmarq's Nick Bolas explains

Magnetic resonance imaging is based on a peculiar property of the nucleus of the hydrogen atom, which can be made to absorb and transmit radio waves when placed in a strong magnetic field. The exact radio frequency is precisely proportional to the exact magnetic field, so by making the magnetic field stronger on one side of an object, the position of hydrogen-containing material (e.g. water or fat) can be determined from the frequency of the received radio signal. This was a revolutionary idea when first recognized by Paul Lauterbur in 1978 (1), and is the basis of all of today's MRI. But the frequency only tells you the position in one direction (the direction of the field gradient), and a one-dimensional image is not much use for diagnosis. How is the idea extended to two dimensions?

Suppose you apply a second field gradient for just a short time, then turn it off. Nuclei in the stronger part of the gradient will precess a little faster for a moment, then go back to the same speed as all the others. If you could see the direction of their magnetic fields as a clock face, they've just switched to daylight savings - now going around at the same speed, but slightly further in advance. A change in phase, but not in frequency.

The phase can't be measured directly, because there is no fixed reference point (imagine a clock with hands on a plain face - it would be tough to tell the time). But, if a series of progressively stronger short gradient pulses is applied, some signals (those that see a strong field change when the gradient is on) will jump in a series of big phase steps, while those seeing only a small field change during the gradient pulse will only creep in a series of small steps (2).

Now we have the fundamentals for a two-dimensional image. After an initial pulse to start the nucleus precessing, a gradient is applied in one direction for a short time, then turned off. A constant gradient is then applied in another direction, and the signal is digitized and stored. The process is repeated many times, with the phase encode gradient stepped in strength each time but the frequency encode gradient always remaining the same.

The math used to convert from a series of big or small jumps in phase to distance in space is exactly the same as that used to convert from a series of big or small jumps between successive digitized points in time (i.e. the frequency).

The signals from all the repeated steps add together, so the 256 or so phase encode steps all contribute to improving the signal:noise ratio of the final image. But because a certain time (TR) has to be allowed between pulses for the nuclei to relax (with time constant T1) the most basic form of MR image sequence is quite slow. The object being imaged, or a part of it, may also move during the time between pulses, making MRI particularly sensitive to motion (e.g. respiration, heartbeat and blood flow). Special techniques are needed to reduce these problems.

The same idea can be used a second time, to create two phase encode and one frequency encode directions (3D imaging) or even a third time to phase encode all three directions of space and leave the frequency axis for chemical shift (spectroscopic imaging).

1) Lauterbur PC. Image formation by induced local interactions: examples employing nuclear magnetic resonance. Nature 1973; 242: 190–1.
2) Edelstein WA, Hutchison JMS, Johnson G, Redpath T. Spin warp NMR imaging and applications to human whole-body imaging. Phys Med Biol 1980; 25(4): 751–6.