The basic principle underlying magnetic resonance imaging is a measurement of signals induced by the nuclear magnetic resonance (NMR) effect. The NMR phenomenon was initially discovered by Purcell and Bloch in 1946 using a technique which is today known as continuous-wave NMR [6, 7]. In 1950, Hahn demonstrated that the NMR effect can also be observed with a modified experimental method based on finite radio frequency pulses [8]. The concept of this pulsed NMR experiment is exploited in the MRI technique to obtain a signal from the sample.
Figure 2.1 shows a schematic illustration of the experimental apparatus. The NMR effect can only be observed for nuclei with nonzero spin quantum number such as 1H,
13C, 19F, or 31P. These particles have a magnetic moment which is related to their angular momentum by
µ=γJ , (2.1)
where the gyromagnetic ratioγ is a nucleus-dependent property. The dominant nucleus in MRI applications is the proton in hydrogen, and most MRI techniques focus only on resonance effects of protons. To measure nuclear resonance, the sample is placed in a
Figure 2.1: Schematic illustration of the apparatus for the pulsed NMR experiment.
static magnetic field which is aligned in z-direction B0 =B0ez. For a short duration, an alternating electromagnetic field is applied perpendicular to the static field. The electromagnetic field is generated with an inductive coil surrounding the sample, which is driven by an oscillator with an angular frequency matching theLarmor frequency
ω0 =γ B0 . (2.2)
For protons exposed to a magnetic field strength of B0 = 2.89 T, which was used in the present work, it corresponds to an angular frequency of ω0 = 7.74·108rad/s or, respectively,f0 =ω0/2π = 123.2 MHz. Because this frequency belongs to the spectrum used for radio transmission, the temporary generation of the electromagnetic field is called a radio frequency (RF) pulse.
Immediately after application of the RF pulse, a small voltage can be detected at the terminals of the induction coil. The received signal oscillates with the same Larmor frequency ω0, and its envelope decays exponentially in the order of milliseconds. It is induced by the magnetic moment of resonating protons as a consequence of the perturbation orexcitation with the RF pulse.
2.1.1 Quantum Mechanical Description
The description of the NMR phenomenon requires a quantum mechanical analysis as the effect originates from nuclear spin interaction with the magnetic field. Because the derivation of the NMR theory is described extensively in the literature [9, 10], only the main results are discussed here.
The interaction with the static magnetic field produces an energy −µ·B, which leads
to a discrete number of eigenvalues of the Hamiltonian. For a proton with spin quantum number j = 12, these energy levels are
E↑ =−γ~
2B0 E↓ =γ~
2B0 . (2.3)
The difference between both states corresponds to the energy of an oscillation at Larmor frequency
∆E =γ~B0 =~ω0 (2.4)
and explains why the resonance phenomenon occurs only at this discrete frequency.
Analysis of the expectation value of the magnetic moment hµi reveals that the z-component remains fixed in the static field, while the transversal z-component rotates in the xy-plane at angular frequency ω0. Hence, the expectation vector acts like a precessing gyroscope. If an additional time-varying field with frequency ω0 is created in the xy-plane
B =B0+B1(t) = B0 + sin(ω0t)ex+ cos(ω0t)ey , (2.5) then the expectation vectorhµiis tilted with respect to the z-axis. In the NMR exper-iment, this additional field is generated by the RF pulse. Derivation of the expectation vector with respect to time yields
dhµi
dt =hµi ×γB, (2.6)
which obeys the classical equation of motion of a gyroscope and holds true for the static and time-dependent magnetic field.
Equation (2.6) shows that the expectation vector for a single proton can take an arbi-trary orientation – regardless of the spin quantization. However, because the expecta-tion value is of a statistical nature, observing the value would require a high number of measurements. On the other hand, in practice a high number of protons is ex-cited at the same time. Therefore, it is convenient to introduce thebulk magnetization M = P
iµi, which sums over all magnetic moments within a macroscopic volume of the sample. Assuming that the protons do not interact, Eq. (2.6) is also valid for the bulk magnetization. Since many “independent” protons are observed simultaneously in the experiment, the measured bulk magnetization corresponds to the expectation value of the magnetic moment of a single proton. This justifies to describe the NMR experiment based on Eq. (2.6).
Finally, although the protons tend to reach the lower energy stateE↑when exposing the sample to the static field, in practice both statesE↑andE↓ are occupied to some extent
due to energy absorption from thermal contact. According to statistical physics, the population follows a Boltzmann distribution. In thermal equilibrium, the probability of finding a proton in either state is given by the Boltzmann factors
p(E↑) = e−E↑/kT
e−E↑/kT +e−E↓/kT p(E↓) = e−E↓/kT
e−E↑/kT +e−E↓/kT , (2.7) where k is the Boltzmann constant and T is the temperature. The Boltzmann factors can be used to derive the population difference for a macroscopic volume with proton densityρ. It yields that the bulk magnetization points along the positive z-direction in thermal equilibriumM =M0 ez, while the magnitude is given by
M0 =ρ γ2~2
4kT B0 . (2.8)
2.1.2 Relaxation Effects
The signal detected in the pulsed NMR experiment declines rapidly after the RF exci-tation. Early NMR experiments showed that there are actually two different relaxation mechanisms affecting the transverse bulk magnetization [11]. The first mechanism is called spin-lattice or longitudinal relaxation and describes an exponential recovery of the longitudinal magnetizationMz after a preceding excitation. The effect is attributed to energy exchange between the protons and their environment, returning the excited system to its thermal equilibrium state.
The second mechanism, the spin-spin or transverse relaxation, corresponds to a de-phasing of the moments inside a macroscopic volume. Because the bulk magnetization averages over all moments in the volume, its amplitude decays from destructive interfer-ence. The effect originates from frequency fluctuations caused by proton interactions.
To account for these two mechanisms, Bloch extended Eq. (2.6) by respective relaxation terms, yielding the Bloch equation
which allows to describe the evolution of the magnetization observed in NMR exper-iments [11]. In contrast to Eq. (2.6), it is written for the bulk magnetization as the relaxation terms are entirely based on empirical findings, without considering the physi-cal mechanisms on a detailed level. The recovery rate of the longitudinal magnetization is given by the T1 relaxation time, while the transversal relaxation is characterized by the T2 relaxation time. Both time constants are properties of the individual sample
material. The T2 relaxation time is always shorter than the T1 relaxation time, and, therefore, the detected signal decays with the T2 relaxation time. However, when per-forming fast repetitive excitations with incomplete recovery of the longitudinal magne-tization, the signal amplitude becomes dependent on the T1 relaxation time. Therefore, both relaxation times can be estimated from NMR experiments.
2.1.3 Conclusions for MRI
The analysis of spectroscopic NMR experiments usually requires detailed knowledge of nuclear spin physics and spin interactions. In contrast, most NMR-based imaging techniques can be properly explained with only the following four statements that sum-marize the results from the previous sections.
(i) The bulk magnetization of the protons aligns in the positive z-direction when in-serting the sample to the magnetic field.
(ii) It can be tipped to the xy-plane using a RF pulse at Larmor frequencyω0, generated by a coil perpendicular to the static field. The flip angle is determined by the amplitude and duration of the RF pulse.
(iii) The tipped magnetization acts like a gyroscope and precesses at Larmor frequency ω0 in the xy-plane. It induces an alternating voltage in the coil proportional to its transversal component.
(iv) The transversal component decays exponentially with the T2 relaxation time, while the longitudinal component relaxes with the T1 relaxation time.