# Magnetic Resonance Imaging

We will model the nucleus of an atom as a spinning charged conducting ball, so that it has a "nuclear magnetic moment" which we write as

M = g m N m s.

Here g is the "gyromagnetic ratio", a dimensionless quantity measured for each atom; m N is the "nuclear magneton" (with a value of n p e h / 4 p m and units of A m 2, where n p is the number of protons and m is total mass of the nucleus); and m s is the spin, which is quantized.

Now for another of physics's deep mysteries. We have seen that electric charge and spin are quantized: they only come in discrete units. As we will see in the next chapter, energy is quantized as well. The energy and spin of an atom or a nucleus can be changed by the absorption or emission of a small "quantum" of "electromagnetic radiation" called a "photon". It is electromagnetic because it is associated with changing electric and magnetic fields (see the next chapter). The quantum of energy it carries is associated with a "frequency", which specifies how often the fields return to their original configuration:

DE = h n,

where n is the frequency, measured in "Hertz" ("cycles" per second).

If we immerse a nucleus in a magnetic field, its spin (and therefore its magnetic moment) gives it potential energy. Equating

DU = - DM B

= - g m N B D m s

= h n,

we see that if m s changes from (for instance) - 1/2 to 1/2, a photon of frequency n is emitted. Similarly, we can "flip" the spin from 1/2 to - 1/2 by radiating the nucleus with photons of frequency n. We can use this radiation to image objects by the following procedure:

1. Orient all the nuclear spins in the object (ie., a patient's body) in parallel with a strong magnetic field.

2. Flip the spins of the nucleii we are interested in locating in the other direction with a strong pulse of radiation of exactly the right frequency.

3. "Listen" for the electromagnetic signal (the radiated photons) when the spins relax to their original state.

See a quicktime animation (43K) showing the incoming pulse flipping a nuclear spin, and the subsequent re-emission when the spin relaxes to its low-energy state.

The frequency will identify the isotope. For example:

Isotopem sgm N (10 - 27)AbundanceSensitivityn (MHz)
1H1/2, -1/25.5855.0599.98=142.57
2H1, 0, -1.8572.524.02*.00963.264
13C1/2, -1/21.4052.3291.11*.01594.939
19F1/2, -1/25.2572.391probe.83418.97
31P1/2, -1/22.2632.442probe.06648.341

(here B = 1 T; probes are artificially inserted; * are often enriched)

Since atomic electrons also have spin, they interact with the spins of the nucleii (the electron spins make them the source of minute magnetic fields). Hence they also contribute to this effect, causing a "chemical shift" (or "electron shift") in the frequency. This allows the identification of molecular species, since the molecular bond configurations (and hence the states of the electrons) are different for every species. To create an image, we use a B gradient. By creating a different value of B at every point in the object, each point will be associated with a different frequency. Empirically, we find that different molecules (tissues) have different "relaxation times" (diseased, ie., tumorous, tissue has a longer relaxation time than normal tissue). So by listening at intervals, we can identify diseased tissue.

Magnetic Resonance Imaging (MRI, or as it was known before the word "nuclear" took on dangerous connotations, "Nuclear Magnetic Resonance") is extremely important as the only (so far as we know) nondestructive medical imaging technique. It allows not only diagnostic imaging, but by "tagging" biological molecules with probes and using time-lapse imaging, some metabolic processes can be tracked. Recently a technique known as "Functional Magnetic Resonance Imaging" (FMRI) or "Blood Oxygenation Level Dependent Imaging" (BOLD) has made use of the fact that there is a significant difference in the magnetic susceptibility (m) of oxygenated and deoxygenated hemoglobin. By looking for locally oxygenated hemoglobin, which can be triggered by nerve cell activity, it is possible to obtain a real-time image of (for example) the processing of auditory input by the brain. This technique is currently being used to study basic brain function, and will soon be applied clinically as well.

The following are some resources related to MRI:

The next chapter is on atomic physics.

If you have stumbled on this page, and the equations look funny (or you just want to know where you are!), see the College Physics for Students of Biology and Chemistry home page.