Protons have spin, which, for our purposes here, can be viewed as a charge distribution in the nucleus revolving around an axis. Associated with the resulting current is amagnetic dipole momentcollinear with the axis of the spin. In elements with an odd number of protons, such as hydrogen, the nucleus itself will have a net magnetic moment. The objective inmagnetic resonance imaging(MRI) is to determine the density of such elements in a volume of interest within the body. The basic idea is to use strong magnetic fields to force the individual spinning nuclei to emit signals that, while too weak to be detected alone, are detectable in the aggregate. The signals are generated by the precession that results when the axes of the magnetic dipole moments are first aligned and then perturbed.
In much of MRI, it is the distribution of hydrogen in water molecules that is the object of interest, although the imaging of phosphorus to study energy transfer in biological processing is also important. There is ongo- ing work using tracers containing fluorine, to target specific areas of the body and avoid background resonance. Because the magnetic properties of blood change when the blood is oxygenated, increased activity in parts of the brain can be imaged throughfunctionalMRI (fMRI). Non-radioactive isotopes of gadolinium are often injected as contrast agents because of their ability to modify certain parameters called the T1 relaxation times.
2.4.1
Alignment
In the absence of an external magnetic field, the axes of these magnetic dipole moments have random orientation, dictated mainly by thermal ef- fects. When an external magnetic field is introduced, it induces a small fraction, about one in 105, of the dipole moments to begin to align their
axes with that of the external magnetic field. Only because the number of protons per unit of volume is so large do we get a significant number of moments aligned in this way. A strong external magnetic field, about 20,000 times that of the earth’s, is required to produce enough alignment to generate a detectable signal.
2.4.2
Precession
When the axes of the aligned magnetic dipole moments are perturbed, they begin to precess, like a spinning top, around the axis of the external magnetic field, at the Larmor frequency, which is proportional to the in- tensity of the external magnetic field. If the magnetic field intensity varies spatially, then so does the Larmor frequency. Each precessing magnetic dipole moment generates a signal; taken together, they contain informa- tion about the density of the element at the various locations within the body. As we shall see, when the external magnetic field is appropriately chosen, a Fourier relationship can be established between the information extracted from the received signal and this density function.
2.4.3
Slice Isolation
When the external magnetic field is the static field, then the Larmor frequency is the same everywhere. If, instead, we impose an external mag- netic field that varies spatially, then the Larmor frequency is also spatially varying. This external field is now said to include agradient field.
2.4.4
Tipping
When a magnetic dipole moment is given a component out of its axis of alignment, it begins to precess around its axis of alignment, with frequency equal to its Larmor frequency. To create this off-axis component, we apply aradio-frequency field(rf field) for a short time. The effect of imposing this rf field is to tip the aligned magnetic dipole moment axes away from the axis of alignment, initiating precession. The dipoles that have been tipped ninety degrees out of their axis of alignment generate the strongest signal.
2.4.5
Imaging
The information we seek about the proton density function is contained within the received signal. By carefully adding gradient fields to the exter- nal field, we can make the Larmor frequency spatially varying, so that each frequency component of the received signal contains a piece of the information we seek. The proton density function is then obtained through Fourier transformations. Fourier-transform estimation and extrapolation techniques play a major role in this rapidly expanding field [157].
2.4.6
The Line-Integral Approach
By appropriately selecting the gradient field and the radio-frequency field, it is possible to create a situation in which the received signal comes primarily from dipoles along a given line in a preselected plane. Performing an FFT of the received signal gives us line integrals of the density func- tion along lines in that plane. In this way, we obtain the three-dimensional Radon transform of the desired density function. The Central Slice Theo- rem for this case tells us that, in theory, we have the Fourier transform of the density function.
2.4.7
Phase Encoding
In the line-integral approach, the line-integral data is used to obtain values of the Fourier transform of the density function along lines through the origin in Fourier space. It would be more convenient for the FFT if we have Fourier-transform values on the points of a rectangular grid. We can obtain this by selecting the gradient fields to achievephase encoding.
2.4.8
A New Application
A recent article [262] in The Boston Globe describes a new application of MRI, as a guide for the administration of ultra-sound to kill tumors and perform bloodless surgery. In MRI-guided focused ultra-sound, the sound waves are focused to heat up the regions to be destroyed and real-time MRI imaging shows the doctor where this region is located and if the sound waves are having the desired effect. The use of this technique in other areas is also being studied: to open up the blood-brain barrier to permit chemo-therapy for brain cancers; to cure hand tremors, chronic pain, and some effects of stroke, epilepsy, and Parkinson’s disease; and to remove uterine fibroids.