A magnetic field can be generated in different ways. The most familiar form of a magnet is a bar magnet. It is usually made of iron, and one end of the magnet acts as the north pole while the other is the south pole. Alternatively, a magnetic field can be generated by passing an electrical current through a wire. Current flowing through a loop of wire generates a magnetic field perpendicular to its plane. If the wire is configured as a solenoid—that is, it is coiled tightly in a cylinder (Figure I1-2)—then the magnetic field induced along each loop of the solenoid is added to the fields induced along the other loops, and consequently a substantial magnetic field can be generated along the length of the cylinder. If the current passing through the solenoid is steady, then the magnetic field will be steady, with a north pole at one end of the coil and a south pole at the other end. The strength of the magnetic field is directly proportional to the amount of current passing through the wire and is referred to at B₀ (“B zero” or “B naught”), typically measured in units of tesla or gauss, where 1 tesla (T) = 10,000 gauss (G).

A vector is depicted as an arrow with a magnitude (or length) and angle (or orientation) with respect to some set of reference axes. Vectors can be constrained to lie within a plane (in which case they are called 2D) or they can point in any direction in space (in which case they are then called 3D). Any vector can be expressed as the sum of its x and y (for 2D) or x, y, and z components (for 3D). For example, a vector M in the xz plane with a magnitude M and angle α (alpha) relative to the z axis can be thought of as the sum of the x component, M_x, and the z component, M_z (Figure I1-4).

Similarly, a 3D vector M can be expressed as the sum of its x, y, and z components: M = M_x + M_y + M_z. In MR physics, the sum of M_x + M_y can be denoted as M_xy. Thus, M can be expressed as the sum of its z and xy components, M = M_z + M_xy (Figure I1-5). The angle that M makes with the z axis is usually referred to as α, while a second angle, φ (phi), describes the angle between M_xy and the x axis.

To determine the relationship between the magnitude M, lengths of the components M_x and M_z or M_xy and M_z, and the angles α and φ, a review of sine and cosine functions is needed.

Most MRI uses a magnet field formed by a large coil of superconducting wire, within which the subject is placed. The magnetic field directed along the main axis, or bore, of the magnet usually defines the z axis (Figure I1-7). Magnetization along the bore is referred to as longitudinal magnetization, while that perpendicular to the bore is called transverse magnetization. Because of the symmetry of magnetization in the transverse plane, the x and y directions are interchangeable. By convention, 3D magnetization vectors M are therefore considered in terms of their z (longitudinal) and xy (transverse) components: M_xy + M_z. The angle of the vector M with respect to the z axis, α, is referred to as the flip angle.

For clarification, although most MRI units are positioned so that the the bore of the magnet, the z axis, is horizontal, most textbook drawings of magnetization vectors depict the z axis pointing vertically (Figure I1-7).

In MRI, the protons across the imaging region do not usually have identical magnetic moments. Their lengths and
orientations are frequently different, depending on factors such as where they are in the magnetic field and in what types of tissues they are found. To determine the net magnetization of the entire population of protons, different subpopulations must be considered separately and their contributions to net magnetization summed. To determine net magnetization, the reader should feel comfortable with vector addition, which is reviewed in the following Background Reading.

There are at least two different ways to perform vector addition. One is to overlay each vector from head to tail (Figures I1-4 and I1-5). For example, as shown in Figure I1-4, the vector M can be formed by the sum of M_x and M_z, laid from head to tail. Similarly, in three dimensions, the magnetization vector is the sum of M_xy and M_z (Figure I1-5).

Alternatively, vectors can be summed by adding each of their orthogonal components, M_x, M_y, and M_z (Figure I1-8). For example, consider two vectors in the xz plane, M1 and M2. To add these two vectors, first break each vector down into its x and z components, M1_x, M1_z, M2_x, M2_z. Then sum the x components and z components separately: Mtotal_x = M1_x + M2_x. Mtotal_z = M1_z + M2_z. The resulting vector Mtotal is simply the sum of the component totals, Mtotal = Mtotal_x + Mtotal_z.

Because vector addition is best understood by examples, the reader is referred to Figure I1-9. As illustrated in the figure, one key concept in vector addition is that vectors pointing in opposite directions cancel each other
out. Therefore, a large population of vectors that are randomly oriented, pointing in all directions, tends to sum to zero, because for every vector, there will likely be a vector pointed in the opposite direction.

Outside of the MR environment, the magnetic moments of protons in the body are oriented in a completely random fashion, and therefore net magnetization is zero.

What happens when these protons are placed in a strong external magnetic field?

Two aspects of the behavior of the magnetic moments can be observed. First, the magnetic moments tend to align with the external magnetic field. They behave like tiny bar magnets exposed to a strong magnetic field and are inclined to align with it. This creates a net magnetization.

Second, a magnetic moment placed in an external field is induced to rotate around the axis of the magnetic field. To understand this movement, known as precession, consider the analogy of a wobbling gyroscope (Figure I1-10). A spinning gyroscope oriented vertically spins without any wobbling. However, once tipped off axis, gravity causes the spinning gyroscope to start to wobble. The wobbling has its own frequency (f in Figure I1-10), which can be considered independent of the frequency of spinning. The precession of protons’ magnetic moments is akin to this wobbling. In the case of the gyroscope, the wobbling frequency depends on gravity, while for protons the precessional frequency depends on the strength of the external magnetic field, B₀. The precessional frequency is measured either in hertz (cycles per second, in which case it is denoted by f) or in radians per second (in which case it is denoted by ω, omega), where ω = 2πf and π (pi) is approximately 3.14.

When exposed to the B₀ magnetic field, all magnetic moments precess (or wobble) around the axis of the magnetic field. Although the magnetic moments precess at about the same frequency, they do not precess together (Figure I1-11), that is, they lack phase coherence. As a result, the transverse or M_xz components are randomly oriented. Based on the concepts of vector addition (Figure I1-9) and the very large number of protons in the tissue, transverse magnetization cancels out and is zero. Consequently, the net magnetization, M, is entirely longitudinal in the direction of B₀ and stationary.

If the magnetic moments gain phase coherence, then the net magnetization precesses around the z axis at a frequency f (Figure I1-11). In other words, net magnetization acquires a transverse component that rotates in the xy plane with a frequency f.

While vectors are useful for showing the state of magnetization at a given point in time, other tools are better for depicting the behavior of magnetization vectors over time, such as precessing magnetic moments. Sinusoidal curves can illustrate the behavior of a single magnetic moment, populations of coherent moments, as well as populations of magnetic moments that are gradually losing phase coherence. The relationship between precessing vectors and their depiction as sinusoidal curves is detailed in the following Background Reading on precessing vectors, angular frequency, and sinusoidal curves.

One of the most important concepts of MR physics is the Larmor equation, which describes the relationship between the precessional frequency and the external magnetic field to which the protons are exposed.

The Larmor equation says that the angular frequency at which the magnetic moments precess around the axis of the magnetic field, called the Larmor frequency, is proportional to the strength of the magnetic field. The higher the field strength, the faster the precession,

f = γB

where f = precessional frequency, B is the magnetic field strength, and γ (gamma) is the gyromagnetic ratio. γ depends on the type of nucleus being imaged. For hydrogen protons, γ ≅ 42.6 megahertz per tesla (MHz/T; recall that a frequency denoted f is measured in cycles per second, or hertz, and 1 MHz = 1 million cycles per second). Compared to other nuclei, hydrogen protons have a large gyromagnetic ratio. This, in addition to their natural abundance, makes hydrogen ideal for MR imaging because small changes in the magnetic field cause large, and readily measurable, changes in signal. For a 1.5 T magnet, the Larmor frequency is

f = 42.6 MHz/T × 1.5 T = 63.9 MHz ≈ 64 MHz

or about 64 million cycles per second. If the magnetic field were uniformly and perfectly 1.5 T throughout, then all protons in the magnet’s bore would precess around the z axis at the same frequency of about 64 MHz.

As will be discussed later, no MRI system has a perfectly homogeneous magnetic field, B₀, throughout its bore. What then is the effect of heterogeneity in the magnetic field? The answer lies simply in understanding the Larmor equation: Protons that are exposed to slightly different magnetic fields will precess at slightly different frequencies. More specifically, protons that are exposed to fields less than 1.5 T will precess slightly more slowly than those exposed to 1.5 T, while those exposed to fields greater than 1.5 T will precess more quickly.

In MRI, the effects of magnetic field differences on the precessing of protons are characterized in terms of frequency and phase. Frequency is the rate of change of phase. It is described simply as the number of cycles (or millions of cycles) per second. By the Larmor equation, frequency changes directly in proportion to the strength of the magnetic field.

What about phase? Phase refers to the orientation of one magnetization vector relative to some reference. It can be thought of as the angle between two vectors or between a vector and a reference axis, for example the angle φ between a vector M_xy and the x axis (Figure I1-13). If two protons start in the same position but precess at different frequencies, then with time the protons will be out of phase; that is, there will be a phase difference between them. The greater the frequency difference between two protons, the greater the phase difference over a given time. For a given difference in frequencies, the more time elapsed, the greater the phase difference separating the two protons. However, phase shift is uniquely defined only across a range of 360°. When a phase shift exceeds 360° (say, 370°), it becomes indistinguishable from a phase shift 360° less (say, 10°). Nevertheless, the cumulative phase difference between two protons precessing at two frequencies is proportional to the difference in frequencies and to the duration of time that has passed (Figure I1-16).

Precessional frequencies can be deliberately made to vary with position in the magnetic field by applying a magnetic field gradient along a given direction. A magnetic field gradient refers to a magnetic field that varies in a certain direction (Figure I1-17). In the setting of a gradient, the precessional frequencies of protons vary spatially in a predictable fashion. For example, with a linear magnetic field gradient in the z direction (Figure I1-17), the field toward the subject’s feet is slightly weaker than 1.5 T, while the field toward the head is slightly stronger. The protons at the feet therefore precess more slowly than the middle of the body, and those in the head precess more quickly. For a given linear magnetic field gradient, the range of precessional frequencies across the field of view, Δf, or the bandwidth, can be predicted using the Larmor equation,

Δf = γΔB

where ΔB is the range of magnetic field strengths across relevant region.

Spatial variation of the external magnetic field causes a corresponding variation in Larmor frequencies for the magnetic moments exposed to the gradient. What would the effect of deliberately varying precessional frequencies have on the net magnetization? Based on the previous Background Reading (Figure I1-15), the cumulative signal from protons exposed to a linear magnetic field gradient will resemble a sinc function (Figure I1-18).

So far, the discussion about the Larmor frequency has assumed that all hydrogen nuclei are alike, but they are not. The molecular environment experienced by a given hydrogen proton is not exactly equal to B₀

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