Lecture 13 — February 25, 2021 Prof

Home , Bloch equations, Larmor precession

MATH 262/CME 372: Applied Fourier Analysis and Winter 2021 Elements of Modern Signal Processing Lecture 13 — February 25, 2021 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long, Edited by E. Bates

1 Outline

Agenda: Magnetic resonance imaging (MRI)

1. Bird’s eye view: physics at play, MRI machines

2. Nuclear magnetic resonance (NMR)

3. Bloch phenomenological equations

4. Relaxation times

A nice introductory video to MRI can be found here

http://www.youtube.com/watch?v=Ok9ILIYzmaY

Last Time: We surveyed two techniques for eﬃciently computing non-uniform discrete Fourier transforms. We considered two distinct tasks: Type I : evaluating the Fourier transform at uniform frequencies from non-uniform data; and Type II : evaluating the transform at non-uniform frequencies from uniform data. Finally, we mentioned that Taylor approximations could be employed to simultaneously consider non-uniform grids in both domains, the so-called Type III problem.

2 Magnetic resonance imaging

After studying computerized tomography (CT) and its connections to the Fourier transform, we now turn to discussing the principles behind magnetic resonance imaging (MRI). As we saw with X-ray tomography, the general idea of medical imaging is to measure the outcome of an interaction between the human body and a known energy source. In MRI, one measures the response of water molecules in the human body when subjected to low-energy electromagnetic pulses against a strong magnetic ﬁeld. This makes MR an attractive imaging modality to practitioners, as there is no evidence that exposure to strong magnetic ﬁelds is harmful. Of course, in order to understand how electromagnetic pulses ultimately lead to an image, we need to construct a mathematical model of the relevant physics. Once we have this model, we will understand what is necessary to recover the quantity of interest from the observed outcome.

1 One interesting aspect about MRI is that it builds on several discoveries made throughout the last century. After the early development of quantum mechanics, there was interest in observing experimentally the predictions of the theory. Of particular interest was the prediction that atomic nucleii have an intrinsic magnetic moment due to their spin. Isidor Isaac Rabi discovered the phenomenon of nuclear magnetic resonance and developed an extension of the Stern-Gerlach method to measure the magnetic moment of atomic nucleii. In 1944 he was awarded the Nobel Prize in Physics “for his resonance method for recording the magnetic properties of atomic nuclei.” Later, Felix Bloch1 and Edward Purcell extended this method for ﬂuids and solids, and they were awarded the Nobel Prize in Physics in 1952 “for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith.” It was much later when these methods were adapted to develop an imaging method. For this work, Peter Mansﬁeld and Paul Lauterbur were awarded the Nobel Prize in Physiology or Medicine in 2003 “for their discoveries concerning magnetic resonance imaging”. Isidor Rabi had to have an MRI scan performed in his later years. “I saw myself in that machine,” he remarked, “I never thought my work would come to this.”

3 A bird’s eye view

We begin our exploration of MR with a high-level description of the physics at play, in the process drawing a picture of how MRI machines work. In the later sections, we will go more in depth mathematically. There we will see the entry of quantum and statistical mechanics in describing the actual nuclear magnetic resonance. This will eventually lead us to discuss the relevant diﬀerential equations. Let us develop the ideas in progression:

1. Proton spin The development of quantum mechanics led to the discovery of the spin. The spin is a quantum-mechanical property of elementary particles that corresponds to an intrinsic notion of angular momentum. Although it has no classical-mechanical analogue, the spin of a charged particle induces a magnetic moment in the same way a loop of electric current has a magnetic moment. This magnetic moment allows the particle to interact with external magnetic ﬁelds. As a consequence of the spin of protons, atomic nucleii also have spin. In particular, for elements with odd atomic number, the resulting nuclear spin is nonzero2. Throughout these notes, we will focus our attention to hydrogen, as its nucleus is a single proton. The quantum 1 1 spin number is either + 2 or 2 , and the sign corresponds the orientation of the magnetic moment. −

2. External magnetic ﬁeld MRI machines are engineered to create very strong magnetic ﬁelds3. When the magnetic

1Felix Bloch was a member of the Department of Physics at Stanford University. For more details about this, please visit http://www.stanford.edu/dept/physics/history/. 2This is because, roughly speaking, for elements with even atomic number, the spins of the protons in the nucleus cancel each other out. 3The intensity is usually between 1.5T and 5T. For comparison, the strength of the Earth’s magnetic ﬁeld is roughly 5 × 10−5T.

2 (a) spin-up + 1 (b) spin-down 1 2 − 2 Figure 1: A hydrogen nucleus with one of two quantum spin numbers. In each illustration we see the axis of rotation in black, which is the direction of the magnetic moment, as well as the direction of spin in blue.

moment of the proton is placed in such a field, it experiences torque. Note this is a consequence of the proton’s spin, because it was the spin that gave the magnetic moment. The induced angular momentum causes the proton to precess (i.e. spin about an axis that is itself rotating with respect to some fixed axis). The axis of revolution is the direction of the magnetic field, while the axis of spin is again the direction of the magnetic moment. The precession is called Larmor precession, and its frequency (that is, the rate of revolution) is known as Larmor frequency. When we consider not just one proton but many, we shall see that there is a slight surplus of those whose spins are aligned with the magnetic field, meaning they are in the so-called low-energy state. The protons whose spins are anti-aligned with the magnetic field are in a high-energy state. This surplus gives a net magnetization in the direction of the external magnetic field.

(a) low-energy state (b) high-energy state

Figure 2: The hydrogen nucleus is now subjected to a strong magnetic ﬁeld, indicated by the vertical red arrow. The direction of revolution is dictated by the orientation of the magnetic moment (using the right-hand rule) and is shown in green.

3 3. Radio frequency field Statistical mechanics provides the means of determining the number of protons present from a measurement of the net magnetization. And if we detect varying proton densities4, we can distinguish the different materials from which they were measured. The trouble is that the rather weak net magnetization is in the direction of the much stronger external magnetic field. Consequently, it would be extremely difficult to distinguish such a small perturbation of an already strong field, especially considering the noise inherent in the measurement process. The approach of MRI machines then is to perturb the bulk magnetization in a direction perpendicular to the external field. This is done by pulsing the system with electromagnetic waves at its resonant frequency, which is exactly Larmor frequency! A key feature of the Larmor frequency is that it falls in the radio-frequency (RF) band. This is extremely convenient for two reasons5. First, an MRI machine does not need to generate high-frequency (so high-energy) pulses. And second, we are very experienced at controlling electromagnetic fields in the RF band. After all, our radio stations are doing it everyday.

(a) closer to high-energy state (b) closer to low-energy state

Figure 3: When a pulse of RF waves (depicted by the turquoise and purple sinusoids) at resonant frequency is introduced to the external magnetic ﬁeld, magnetic moments align with the pulse. Those protons in the low-energy state are brought to a higher energy state, and vice versa. The magnetic moments continue to precess about the vertical magnetic ﬁeld, but at a greater radius.

4. On-off excitation As depicted in Fig. 3, the RF pulse brings magnetic moments into alignment with the pulse, transverse to the original external field. This is been accomplished by the RF pulse at Larmor frequency producing torque on the magnetic moments. Now the direction of the net magnetization is perpendicular to the magnetic field. Furthermore, since there was a surplus of nuclei in the low-energy state, the new alignment yields a net gain in energy. That is, the system is placed into a state of excitation. When the excitation pulse is turned off, the magnetic moments tend to decay toward their initial configurations. Energy is thus released as radiation that can be recorded to measure proton densities and ultimately produce an image.

4It is customary to refer to proton density, even though we are considering hydrogen nuclei. 5It is also inconvenient because you cannot listen to the radio during an MRI scan.

4 5. Gradient field Suppose we can, indeed, accomplish all that we have described thus far. There still remains a critical obstruction. While we may have measured the energy released with on-off excitation, we do not know from where it came. Constructing an image from completely aggregate measurements is thus impossible. The solution is to use a gradient magnetic field, that is, one whose strength varies with location. The Larmor frequency naturally varies with the intensity of the magnetic field, so when the strength of the magnetic field varies with location, so too does the resonant frequency. Consequently, an RF pulse at a particular frequency will excite protons only at those locations exposed to the correct magnetic field intensity. In MRI machines, 2D cross-sections are exposed to a constant intensity so that these cross-sections can be isolated for imaging. When combined, they provide a 3D recording.

Figure 4: A mock illustration of the role of a spatially varying magnetic field. In the graphic, B0 represents the external magnetic field, a notation we will follow in the upcoming sections. A stronger magnetic field implies a higher Larmor frequency. The frequencies written here approximately correspond to a 1.5T field. In MRI machines, the variable-strength magnetic field is created by gradient coils. The quality of the gradient system affects scanning speed and image resolution.

4 Nuclear spin and Larmor precession

Now equipped with a basic understanding, we make a second pass through the topics of the previous section. In particular, we will see the mathematics governing, or at least modeling, the physics we have discussed. To start, we have seen that the interaction between proton spin and the external magnetic ﬁeld is the basis of MRI, and here we provide an overview its quantum mechanical description. We denote by I the nuclear spin operator, which in this case has quantum spin 1 number 2 . The angular momentum becomes J = ~ I. It is known that a charged particle with nonzero angular momentum has an associated magnetic moment, given by

µ = γ J, where γ is the gyromagnetic ratio (in this case, of the proton). Nuclear magnetic resonance (NMR) describes the interaction between the spin of the atomic nucleii and an external magnetic

5 field. Consider a homogeneous magnetic field B0, which we call the main field. In the absence of any other effects, the interaction between the magnetic moment and the main field is the only one that determines the energy of the proton:

E = µ, B0 . −h i That is, the energy is the component of the magnetic moment that is parallel to the main ﬁeld. This component is usually called the longitudinal component, and we denote it by µk. Since, roughly speaking, measuring the energy amounts to measuring the spin along the direction of the main ﬁeld, the longitudinal component becomes quantized. Measuring this quantity has only two possible outcomes, namely

1 B0 1 B0 µk = ~γ (spin-up) or µk = ~γ (spin-down). 2 B0 −2 B0 k k k k This implies the two possible outcomes for measuring the energy are 1 1 E = ~γ B0 (low-energy) or E = ~γ B0 (high-energy). −2 k k 2 k k Therefore, in the presence of an external field the nucleii become polarized. The two possible outcomes for measuring µk are said to be parallel or anti-parallel with the main field, corresponding to spins of + 1 and 1 , respectively. 2 − 2 On the other hand, the component of the magnetic moment orthogonal to the main field is not quantized. We understand its behavior by analyzing the average magnetization µ . It expe- h i riences a rotation about B0, the Larmor precession, due to the torque exerted on the magnetic moment by the main field: d µ = γ µ B0. (1) dth i h i ×

As we will see later, the solutions to this equation leave the longitudinal component of µ un- h i changed as µ rotates around B0 at frequency γ B0 , the Larmor frequency. This gives the interpretationh thati the magnetic moment precessesk aroundk the main ﬁeld. The direction of spin determines whether the magnetic moment has parallel or anti-parallel orientation (see Fig. 5). What happens when we consider a collection of hydrogen atoms? If we neglect the interactions between the atoms, we can study this as a system of particles, each of which again has one of two energy states: 1 1 E− = ~γ B0 (parallel) or E+ = + ~γ B0 (anti-parallel). −2 k k 2 k k

Suppose we have N hydrogen atoms. Let ∆N be the difference between the number of nuclei with spin parallel to the main field (with energy E−) and the number of nuclei with spin anti-parallel to the main field (with energy E+). Then, from statistical mechanics, we have ~γ ∆N N B0 , ≈ 2kBT k k where kB is the Boltzmann constant and T is the absolute temperature. Since ∆N > 0, there is a slightly greater number of spins aligned with the main field, which creates a net magnetization. The main idea of MR is to measure this net magnetization around a small neighborhood of a point in order to detect the number of hydrogen atoms in that region.

6 B0 B0 µ µ h i h ki

µ µ h ki h i

(a) Parallel (b) Anti-parallel

Figure 5: Schematic representation of the Larmor precession. When subjected to an external magnetic ﬁeld, the average magnetic moment of the nucleus precesses in a parallel or anti-parallel orientation. Note the longitudinal component remains constant, which is consistent with the quantization of the corresponding component of the angular momentum. The precession occurs at the Larmor frequency.

5 Bulk magnetization

A model that accounts for every spin in the sample would quickly become intractable. Further- more, it is reasonable to predict that fine-scale effects will disappear as we aggregate the behavior of each individual atom. Consequently, we will model the bulk magnetization instead. The bulk magnetization is the average magnetization over small neighborhoods due to the aggregated po- larization of each nucleus in that neighborhood. It corresponds to a mesoscopic description, as we take neighborhoods large enough that we can work with average effects and neglect quantization, but small enough that we can still localize the magnetization on a small region of space. Note that once the spins are polarized, they experience Larmor precession at the same frequency. Meanwhile, their phases are out of sync, we assume in a uniformly distributed fashion. This means that if we aggregate the contributions of each nucleus, the transverse component of the magnetization will be zero in expectation. Consequently, the average magnetization in a small neighborhood will be essentially parallel to the main field (see Fig. 6). As we wish to model aggregate behavior, we define ρ = ρ(x) to be the proton density on a small 3 neighborhood of a point x R . The bulk magnetization is then given by ∈ c −3 M0(x) = ρ(x)B0(x), c 1.284 10 K/T, (2) T ≈ × the density of the spin magnetic moments at x. Notice that M0 is a scalar multiple of B0 at any point, reflecting our observation that the average transverse magnetization is 0. As a consequence, since M0 obeys (1), it does not change with time. That is, it corresponds to an equilibrium state.

If we know B0(x) (presumably because we have generated it), then M0 encodes the proton density | | at each point in space. We can recover ρ(x) from M0(x) , and so our goal should be to measure the bulk magnetization. | | This is where our physical obstructions enter:

7 B0

µ µ h i h i (a) (b)

Figure 6: (a) In the absence of external fields, on any small region the nucleii are not polarized. (b) In the presence of an external field B0, the nucleii polarize, and their average magnetization may precess in a parallel or anti-parallel orientation with respect to the main field. Since there is a bias towards lower energies, there is a net longitudinal magnetization. Since the phases of the precessions are random, the average magnetization in the transverse plane is zero.

(1) The bulk magnetization is very small relative to the main field. From (2) we deduce that M0 will always be a small fraction of B0 , and thus a small fraction of the total field. For | | | | −6 instance, in a field of intensity 1T at room temperature, M0 10 B0 . | | ≈ | | (2) And second, how does one measure bulk magnetization only near a single point (or, as in MRI machines, along a single plane)?

Concern (1) is resolved by perturbing the bulk magnetization in a direction transverse to B0. Graphically, M0 is rotated from the z-axis (the direction of B0) to the xy-plane, where its presence can actually be detected. Of course, we need to understand how the bulk magnetization is perturbed, and what its behavior will be after it is perturbed. Furthermore, since these perturba- tions are non-constant, we will need to account for dependence on time. We have mentioned that the correct perturbation is an electromagnetic pulse at Larmor frequency. Now, for hydrogen the gyromagnetic ratio is γ 42.5764 MHz/T. Consequently, its Larmor frequency at 1.5T is about 64MHz. Indeed, this is within≈ radio-frequency band. In the next section we introduce the model used to understand the interactions between the bulk magnetization and RF pulses. As for concern (2), the use of gradient magnetic ﬁelds enables spatial encoding of resonant frequency. This translates to the spatial resolution needed to produce an image from excitation readouts.

6 Nuclear induction and Bloch equations

Nuclear induction refers to the change in electric potential caused by the reorientation of magnetic moments. It is this voltage difference that allows for the measurement of the bulk magnetization. Bloch’s equations describe how the magnetic moment changes in time due to the interaction with an external magnetic field. In the setting we have set forth, the magnetic field to be considered is of the form

B(t) = B0 + ∆B(t) , |{z} |{z} | {z } external ﬁeld main ﬁeld perturbation

8 where we assume ∆B(t) B0 at any t > 0. This consideration is motivated physically rather than mathematically:k A strongk k perturbationk may cause the system to reach a different equilibrium state once it is removed, in which case our mathematical model may no longer be representative of the physical system. In vector form, the Bloch equations are written d 1 1 M(x, t) = γ M B(x, t) (Mk M0) M⊥. (3) dt × − T1 − − T2 It is instructive to compare (3) to (1). The torque experienced by the magnetic moment from the magnetic field is still accounted for, but the two additional first-order terms now capture the relaxation of the magnetization. Indeed, if they are omitted and B = B0 is taken to be time- independent, then (3) reduces to a constant precession about B0(x) at Larmor frequency.

The components M⊥ and Mk represent the components orthogonal and parallel to the main ﬁeld B0, that is, the components in the xy-plane and along the z-axis, respectively. Once more, they are called the transverse and longitudinal components of the magnetization. M0 is the equilibrium magnetization, which is parallel to the z-axis. The constants T1 and T2 are the relaxation times. In particular, T1 is the spin-lattice relaxation time and models the decay of the longitudinal component of the magnetization to equilibrium. On the other hand, T2 is the spin-spin relaxation time, and it models the decay of the transverse component of the magnetization to equilibrium. Part of what makes Bloch’s work so intriguing is that these equations are phenomenological. That is, they have not been deduced from ﬁrst principles, but rather constructed in a way that describes experimental results accurately. In the next lecture we will turn to solving them.