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 frequen- cies 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.