Magnetic resonance imaging exploits the existence of induced nuclear magnetism in the patient. Materials with an odd number of protons or neutrons possess a weak but observable nuclear magnetic moment. Most commonly protons (1H) are imaged, although carbon (13C), phosphorous (31P), sodium (23Na), and fluorine (19F) are also of significant interest. The nuclear moments are normally randomly oriented, but they align when placed in a strong magnetic field. Typical field strengths for imaging range between 0.2 and 1.5 T, although spectroscopic and functional imaging work is often performed with higher field strengths. The nuclear magnetization is very weak; the ratio of the induced magnetization to the applied fields is only 4 × 10−9. The collection of nuclear moments is often referred to as magnetization or spins.
The static nuclear moment is far too weak to be measured when it is aligned with the strong static magnetic field. Physicists in the 1940s developed resonance techniques that permit this weak moment to be measured. The key idea is to measure the moment while it oscillates in a plane perpendicular to the static field . First one must tip the moment away from the static field. When perpendicular to the static field, the moment feels a torque proportional to the strength of the static magnetic field. The torque always points perpendicular to the magnetization and causes the spins to oscillate or precess in a plane perpendicular to the static field. The frequency of the rotation ω0 is proportional to the field:
ω0 = −γ B0
where γ , the gyromagnetic ratio, is a constant specific to the nucleus, and B0 is the magnetic field strength. The direction of B0 defines the z-axis. The precession frequency is called the Larmor frequency. The negative sign indicates the direction of the precession.
Since the precessing moments constitute a time-varying flax, they produce a measurable voltage in a loop antenna arranged to receive the x and y components of induction. It is remarkable that in MRI we are able to directly measure induction from the precessing nuclear moments of water protons.
Recall that to observe this precession, we first need to tip the magnetization away from the static field. This is accomplished with a weak rotating radiofrequency (RF) field. It can be shown that a rotating RF field introduces a fictitious field in the z direction of strength ω/γ . By tuning the frequency of the RF field to ω0, we effectively delete the B0 field. The RF slowly nutates the magnetization away from the z-axis. The Larmor relation still holds in this “rotating frame,” so the frequency of the nutation is γ B1, where B1 is the amplitude of the RF field. Since the coils receive x and y (transverse) components of induction, the signal is maximized by tipping the spins completely into the transverse plane. This is accomplished by a π/2 RF pulse, which requires γ B1τ = π/2, where τ is the duration of the RF pulse. Another useful RF pulse rotates spins by π radians. This can be used to invert spins. It also can be used to refocus transverse spins that have dephased due to B0 field inhomogeneity. This is called a spin echo and is widely used in imaging.
NMR has been used for decades in chemistry. A complex molecule is placed in a strong, highly uniform magnetic field. Electronic shielding produces microscopic field variations within the molecule so that geometrically isolated nuclei rotate about distinct fields. Each distinct magnetic environment produces a peak in the spectra of the received signal. The relative size of the spectral peaks gives the ratio of nuclei in each magnetic environment. Hence the NMR spectrum is extremely useful for elucidating molecular structure.
The NMR signal from a human is due predominantly to water protons. Since these protons exist in identical magnetic environments, they all resonate at the same frequency. Hence the NMR signal is simply proportional to the volume of the water. They key innovation for MRI is to impose spatial variations on the magnetic field to distinguish spins by their location. Applying a magnetic field gradient causes each region of the volume to oscillate at a distinct frequency. The most effective nonuniform field is a linear gradient where the field and the resulting frequencies vary linearly with distance along the object being studied. Fourier analysis of the signal obtains a map of the spatial distribution of spins. This argument is formalized below, where we derive the powerful k-space analysis of MRI.
