Chapter M MRI Contents Introduction to nuclear magnetic resonance imaging (NMR / MRI)....................... M.2 Physics overview................................................... M.2 Spins...................................................... M.2 Magnetic moments............................................... M.4 Main field B.................................................. M.4 RF field..................................................... M.8 Relaxation.................................................... M. Field gradients................................................. M.2 Imaging overview................................................ M.4 Selective excitation preview........................................... M.4 Physics details.................................................... M.5 Bloch equation................................................. M.5 Signal equation................................................. M.9 Image formation................................................... M.2 Example : 2D Projection-reconstruction MR sequence............................ M.2 Example 2: Spin-warp or 2DFT imaging.................................... M.23 Fast imaging methods.............................................. M.24 Sampling and resolution issues for 2DFT sequence............................... M.25 Excitation and coils................................................. M.27 Excitation...................................................... M.29 Non-selective excitation............................................. M.3 Selective excitation............................................... M.3 Refocusing................................................... M.32 Relaxation effects during excitation pulses................................... M.34 Imaging considerations............................................... M.35 Off-resonance effects.............................................. M.35 Spin echos................................................... M.36 Fat/water separation............................................... M.4 Signal equation (revisited)........................................... M.4 Summary....................................................... M.42 M.
c J. Fessler, October 3, 29, 7:39 (student version) M.2 Introduction to nuclear magnetic resonance imaging (NMR / MRI) NMR principle discovered independently by Bloch and Purcell in 946, led to Nobel Prize in physics [ 4]. First MR images in early 7 s by Lauterbur [5]. Advantages of MRI Nonionizing radiation Good soft-tissue contrast High spatial resolution Flexible user control - many acquisition parameters allow optimization for specific imaging situations Negligible attenuation (although RF coil sensitivity is now a useful mechanism for fast scans) Image in arbitrary plane (or volume) Potential for chemically specified imaging Flow imaging Disadvantages of MRI High cost Complicated siting due to large magnetic fields Low sensitivity Little signal from bone Fairly long scan duration Naturally there is ongoing work to reduce or minimize the disadvantages of MRI (especially scan duration). For example, more recently real-time NMR images have been produced [6]. Physics overview As usual, we begin by describing the general physical phenomena that underly the MR imaging modality [7]. Spins Nuclei with an odd number of protons or neutrons posses an angular momentum vector J, often simply called spin. From a classical physics perspective, we can think of such nuclei as spinning charged spheres, called spins, that act as tiny dipoles. In medical imaging, the most abundant spin is H, a single proton, in water molecules. There are over stable nuclei that have an odd number of protons or neutrons. Other particularly important nuclei in biology and medicine: 2 H, 7 Li, 3 C, 9 F, 23 Na, 3 P, 27 I Associated with each spin is a nuclear magnetic dipole moment vector, or simply magnetic moment vector, µ = (µ X, µ Y, µ Z ) = µ X i + µ Y j + µ Z k that is co-linear with direction of the spin s angular momentum vector. MRI involves the interaction of these spins with several types of magnetic fields. The following three of these fields are designed and controlled by us. Main field: B. This field is (usually) aligned with the scanner axis (the z axis by convention) and is generated by superconducting coils or large permanent magnets. We select this (static) field strength when we design the system, e.g.,.5t MR scanner RF field: B (t), an amplitude-modulated pulse transmitted through a coil. This field is usually (approximately) perpendicular to B. There are many different ways to design this excitation field. Field gradients: G(t). These cause spatial variations in the relative strength of the z-oriented magnetic field. Field gradients are also user-controlled, and are what separate MRI from NMR. Paul Lauterbur and Peter Mansfield earned the 23 Nobel Prize in Physiology or Medicine by developing, analyzing, and refining the use of gradient fields G(t). In addition to the above human-controlled fields, the presence of orbiting electrons near a nucleus also change the local apparent magnetic field. This effect is called chemical shift and will be described in more detail later. It can be both a source of image
c J. Fessler, October 3, 29, 7:39 (student version) M.3 artifacts and a source of information for distinguishing different types of tissue. Fields The three user-controlled fields described above each correspond to specific instrumentation components, as illustrated in the following diagram, taken from [8]. A typical superconducting magnet is based on niobium-titanium (NbTi) alloys, requiring a liquid helium cryogenic system that cools to about 4.2 K. Block diagram Image acquisition: µ magnetic moment B main field M(r) equilibrium magnetization B RF excitation M(r, t) M X (r, t)+ı M Y (r, t) transverse magnetization G(r, t) field gradients s(t) MR signal Image reconstruction: s(t) MR signal ifft2() or something else Spatial coordinates in 3D are denoted by r = (x, y, z). ˆf(r)
c J. Fessler, October 3, 29, 7:39 (student version) M.4 Magnetic moments The magnetic moment vector µ = (µ X, µ Y, µ Z ) is related to the angular momentum vector J by µ = γ J. The nuclei-dependent constant γ is called the gyromagnetic ratio. Often we will work with the related quantity γ = γ 2π which has units Hz / Tesla. The magnitude of the magnetic moment is a fixed value determined by quantum physics: µ = µ 2 X + µ2 Y + µ2 Z = h γ I(I + ). (M.) γ = γ 2π is called the gyromagnetic ratio. For H, γ = 42.48 MHz/Tesla. h = 6.6 34 J sec denotes Planck s constant. I denotes the spin quantum number Hereafter we restrict attention to nuclei with two energy states, such as H, for which I = /2. Thus µ = h γ 3/2. Note that 3/2 is the distance from the center to the corner of the unit cube. Although the µ is fixed, normally the magnetic moment of any spin has a rapidly fluctuating random orientation. Main field B Now suppose we apply a static and spatially uniform magnetic field B (r) = B along some direction. By convention, the direction is taken to be z, i.e., B = = B Z k, where k = denotes the unit vector along z. B Z The main field strength B Z, technically the magnetic flux density, is about Tesla = 4 Gauss = Wb/m 2 = kg / (s 2 A) typically. (The earth s magnetic field is about.5 Gauss). Microscopic view The static main field along k causes Zeeman splitting: the longitudinal component µ Z of the magnetic moment is quantized: for H, or more generally µ Z = m Z h γ where m Z { I,...,I, I}. µ Z = ± h γ, (M.2) 2 Define the transverse component of the magnetic moment as that lying in the transverse plane, i.e., the x, y plane: µ XY µ X i + µ Y j. By comparing (M.) and (M.2), we see that (for H): µ XY = µ 2 µ Z 2 = h γ/ 2 = 2 µ Z. More generally, µ XY = h γ I(I + ) m 2 Z. The angle θ between the magnetic moment µ and the applied field direction k is ( arccos(µ Z / µ ) = arccos ±/ ) 3 {54.7, 8 54.7 }. However, the orientation of the transverse component µ XY of the magnetic moment is random, i.e., arctan(µ Y /µ X ) is uniformly distributed over [, 2π).
c J. Fessler, October 3, 29, 7:39 (student version) M.5 We can think of the two possibilities for the z-component µ Z as being parallel or anti-parallel to k. The parallel state has slightly lower energy than the anti-parallel state. According to quantum theory, E = µ B = µ Z B Z = m z hγ B Z. The energy difference between the two states is E = E E + = h γ B Z. The energy difference is sometimes illustrated by Zeeman diagram: B = z Higher Energy Anti-Parallel B > Lower Energy Parallel At absolute zero, all nuclei would occupy the lower energy (parallel) state. However, thermal agitation greatly exceeds E. The (equilibrium) ratio of anti-parallel (N ) to parallel (N + ) nuclei is governed by Boltzmann distribution, for which: N = N + + N is the total number of spins k b =.38 23 J/ K (Boltzmann constant) T temperature in degrees Kelvin. We will see that the signal recorded in MRI is proportional to E[N + ] N = e E+/(kbT) e E+/(k bt) + e E /(k bt), E[N + ] E[N ] = e E/(k bt). e E+/(k bt) e E /(kbt) E[N + N ] = N e E+/(k bt) + e N E /(k bt) e E+/(k bt) + e = N e E/(kbT) E /(k bt) e E/(k bt) + N E k b T = N h γ B Z k b T. (M.3) Unfortunately, at room temperature, the difference in occupancy is only a few parts per million! Thus NMR is a fairly insensitive modality. (Fortunately there is a lot of H, but imaging Na or P is much harder.) How can one increase sensitivity (signal strength)? Decrease temperature (not clinically feasible, but fine for traditional spectroscopy). Increase field strength B Z, hence today s trend towards 3T systems (and beyond). However, there are imaging tradeoffs such as increased T and increased field inhomogeneity, and practical tradeoffs such as more complicated siting. Do other things that affect the proportionality factor! This is where much MRI research occurs, e.g., using multiple receive coils.
c J. Fessler, October 3, 29, 7:39 (student version) M.6 Magnetization: Macroscopic view The net effect of the magnetic moments of numerous spins is local magnetization of the sample. In MRI we are interested in manipulating and imaging this magnetization. We will denote the magnetization by M(r, t) if it is varying with time, or M (r) if it is in an equilibrium, The units of magnetization are Amperes per meter (A/m). The equilibrium spatial distribution of magnetization is essentially proportional to the local density ρ(r) of H, i.e.: M (r) = = M Z (r) k, M Z (r) = ρ(r)(h γ) 2 M Z (r) 4k b T B Z, where ρ(r) denotes the spin density: the number of spins per unit volume. Using (M.3), we can also write M Z (r) = ρ(r)h γ 4 E[N + N ]/N. Images proportional to ρ(r) are of some interest, but barely scratch the surface of the potential of NMR imaging. Why is the magnetization along k? Because the transverse component µ XY of each spin has a random orientation. Lower Energy State (Parallel) z B > M Net Magnetization of Material Higher Energy State (Anti-Parallel) At 37 C = 3 K, for a single proton in H 2 O, M Z = 3.25 3 B Z A / m. Microscopic view: If B =, the spins are equally likely to occupy either energy state, and the dipole orientations are random. Macroscopic view: If B = then the net magnetization is zero: M (r) =. Precession Microscopic view: The transverse component of µ experiences a torque due to the applied main field B, causing it precess about about the direction of the applied field, i.e., about k or z, called nuclear precession. (Analogy: spinning top, gravitational pull.) The rate of this precession is given by the Larmor equation: ω = γ B Z or f = γ B Z. For a typical.5t main field, f 63 MHz. This frequency is in the RF regime (lowest point on FM dial is about 88MHz). A convenient way to describe this precession mathematically is to use complex notation: µ XY (t) µ X (t) + ıµ Y (t) = µ XY ()e ıωt, where µ XY () is a random initial phase. (In addition to precession there are also random fluctuations due to thermal agitation.) The direction of precession is clockwise if viewed against the direction of B, i.e., if we view the x, y plane from above. This corresponds to a left hand rule: with left thumb pointing along applied field B, precession follows the direction of the fingers. However, this precession in the transverse plane is not observable (without perturbing the system in some way to be described shortly) because neighboring spins have different random phases so the net magnetization in the transverse plane is zero. Macroscopic view: The phase of the precessing spins is random, so the net transverse magnetization is zero. Because there are more spins in the parallel state, the net magnetization M is oriented along z (in equilibrium):
c J. Fessler, October 3, 29, 7:39 (student version) M.7 Units Here is a quick check on the units of magnetization, usinghttp://en.wikipedia.org/wiki/tesla_(unit) in part. Newton: force to give kg mass a m/s 2 acceleration. N = kg m / s 2 Joule: energy required for force of Newton over meter. J = N m = kg m 2 / s 2 Ampere: current in two infinitesimal wires m apart that produces a force of 2 2 N per meters of length. N A m Gauss: Maxwell / cm 2 (CGS unit) Tesla = Wb/m 2 = kg / (s 2 A) (SI unit). Thus A m N m 2 J m 3 Now, from Boltzmann distribution it is clear that E/(k b T) is unitless, where E = h γ B Z. Rewriting: ( ) h γ M Z (r) = ρ(r) }{{} 4 ( γh) BZ, }{{} k b T spins / m 3 J / T } {{ } unitless so M Z has units: J / T / m 3 = (kg m 2 / s 2 ) / (kg / (s 2 A)) / m 3 = A / m.
c J. Fessler, October 3, 29, 7:39 (student version) M.8 RF field The collection of precessing spins is resonant in the sense that one can induce oscillations by applying energy at the appropriate wavelength/frequency/energy. Note that E = h γ B Z = hf, which is exactly the formula for the energy in the quanta of an EM field with frequency f. So a RF field tuned to f will resonate with spins. For excitation we apply an RF field perpendicular to the main field B, usually as an amplitude modulated sinusoid: cos(ω t + φ ) B (t) = B a (t) sin(ω t + φ ), (M.4) where a (t) denotes a unitless pulse envelope that we can design under computer control. We normalize such that max t a(t) =. The duration of a (t) is usually very short ( ms), so this is called a RF pulse. The field B (t) is circularly polarized, generated using quadrature RF coils. It will also be convenient to represent this RF field in complex notation: B (t) B,X (t)+ı B,Y (t) = B a (t)e ı(ωt+φ) = b (t)e ıωt, where b (t) B a (t)e ıφ. The RF field strength is B (t) = B (t) = B, and typically is a fraction of a Gauss. The above model is an idealization. In practice there are several departures that need not concern us now. Even if the RF coil were to produce a component in the z direction, this component would be negligible relative to B Z. Ideally B (t) would be spatially homogeneous, i.e., independent of r. In practice any coil design has a nonuniform field pattern, and this can be a complication in some types of MR scans, particularly at higher field strengths where the wavelengths are shorter or when multiple coils are used. Ideally the RF frequency ω exactly equals the resonant frequency of the spins. In practice due to nonuniformity of the main field strength B Z, the resonant frequency varies spatially so the RF frequency never matches the Larmor frequency everywhere. The effect of such off resonance is somewhat complicated [7, p. 87]. Microscopic view: The added RF energy can cause some of the spins to occupy the higher energy state. What might happen to the longitudinal magnetization??? How much can it change??? The RF pulse causes the transverse components of the magnetic moments of some of the spins to become in phase. This creates a nonzero transverse magnetization component. The largest possible magnitude is M XY = M Z. The exact mechanisms are beyond the scope of this course. Fortunately, when E k b T, as in NMR, classical and quantum mechanics agree, so from now on we take a classical viewpoint and only consider the net magnetization vector M(t). Macroscopic view: The magnetic moments and hence the bulk magnetization precess about the overall magnetic field B(t) = B + B (t), which is time-varying. Trying to imagine such precession directly is challenging. To simplify, we will later consider in detail a rotating frame that rotates at frequency ω, having coordinates x, y, z. In this frame, the precessing cone due to B is frozen. If φ = above, then the RF field is along the x axis in the rotating frame, and the magnetization will precess around the x axis (in this rotating frame) following the left hand rule. Due to Larmor equation, the frequency of this precession is f = γ B (t), which is much slower than f because B B Z.
c J. Fessler, October 3, 29, 7:39 (student version) M.9 The following figures illustrate how the magnetization evolves during RF excitation. We will analyze this in detail later. Laboratory Frame Rotating Frame z z θ = 9 B x M M y x y If the RF strength and duration is just right, then one can tip the magnetization into the transverse plane. (A 9 flip angle). Usually this type of excitation yields the largest signal (for a single excitation). To describe the transverse component of the magnetization, we will sometimes use the vector notation M XY (r, t) M X (r, t) i + M X (r, t) j but more often using the convenient complex mathematical notation: M XY (r, t) M X (r, t)+ı M X (r, t). Often we will use M Z (r, ) and M Z (r, + ) to denote the longitudinal magnetization immediately before and immediately after an RF excitation pulse at time t = respectively. This notation reflects the fact that RF excitation pulses are very short. We will similarly define M XY (r, ) and M XY (r, + ). For an on-resonance RF excitation pulse with flip angle α about the x axis, the effect of the RF pulse when applied to an object at equilibrium is ( M Z r, + ) ( = cos(α) M Z r, ) = cos(α) M Z ( M XY r, + ) ( = ı sin(α) M Z r, ) = ı sin(α) M Z. In particular, if α = π/2, then as illustrated in the figure above ( M Z r, + ) = ( M XY r, + ) = ı M Z.
c J. Fessler, October 3, 29, 7:39 (student version) M. Relaxation After we turn off the RF signal, the magnetization will continue to precess in the x y plane, but the system will begin to return to the equilibrium state. After an RF pulse (at t = ) with a 9 tip angle, the following occur. The longitudinal component, along the z-axis, recovers to its equilibrium value as follows (for t > ): ( M Z (r, t) = M Z (r) e t/t(r)) ( + M Z r, + ) e t/t(r). T is the spin-lattice time constant; exchange of energy between spins and surrounding lattice of electrons etc. T typically from - msec. T values lengthen with increasing B Z The magnitude of the transverse component decays as follows (for t > ): M XY (r, t) = MXY ( r, + ) e t/t 2(r). T 2 is the spin-spin time constant; loss of phase coherence due to interactions between spins. (In practice the signal decays with a faster time constant T 2 because B Z is nonuniform.) T 2 typically from - msec Transverse relaxation is caused by loss of phase coherence of the spins due to interactions between nearby microscopic dipoles causing a broadening of the spins resonant frequencies. The details require quantum mechanics. T 2 values are largely independent of B Z Relative Magnetization M z (t) / M Longitudinal Relaxation.5.5 T msec M (t) = M ( e t/t ) + M () e t/t z z.5.5 2 2.5 3 Time [sec] Relative Magnetization M(t) / M() Transverse Relaxation.8.6 M(t) = M() e t/t 2.4 T 2 4 msec.2.5.5 Time [sec] 2 2.5 3 The values of T and T 2 can vary considerably between different materials, providing very valuable contrast information.
c J. Fessler, October 3, 29, 7:39 (student version) M. Interestingly, because of the differences between T and T 2 relaxation times, the length of the magnetization vector does not stay the same during relaxation. 9 Excitation and Relaxation msec msec Mz msec Mxy Signal preview As the system returns to equilibrium state, it emits energy that can be measured in the form of RF oscillations. We can think of the body as a collection of tiny precessing magnetic dipoles. By Faraday s law, moving magnetic dipoles induce a voltage across an RF coil that is oriented to detect x y magnetization changes. The RF signal we measure is the superposition of the individual signals from all those oscillators. The RF signal is a function of the key parameters: T, T 2, and ρ (the spin density), all of which are functions of spatial location r. We would like to image the spatial distribution of one or more of those parameters from the RF signal. The signal described thus far is called free induction decay (FID) ( free because RF is off; we can also have steady-state systems).
c J. Fessler, October 3, 29, 7:39 (student version) M.2 Localization preview The wavelength of RF is in meters, so we cannot easily selectively excite or view a desired slice or voxel using RF and B Z alone. As in ultrasound, MRI uses a sequence of excitation/reception steps. For each excitation/reception, we record a D RF signal. Because we want to form a 2D (or 3D) image, we need a sequence of excitation/reception steps. Unlike ultrasound, in MRI one cannot form a useful image until after all data have been collected. In ultrasound, location (range) is relate to time. In X-ray imaging, localization is defined via collimation and small source size (so detected photon position corresponds to a particular ray though the object). In MRI, location is related to temporal frequency. Note that at typical RF wavelengths in MRI, there is relatively little attenuation in passing through the body (or through many organic materials, which is why radios work inside houses). See [9, Fig..]. Field gradients If we applied only a perfectly uniform field B, then all spins would oscillate at the same frequency, so there would be no spatial localization ability. (The RF receiver responds to the whole volume (almost) equally.) The clever solution for localization is to use special coils to induce field gradients. Example. By using suitable coils, we can induce an x gradient: B(r) = B(x, y, z) = = (B Z +xg X ) k. B Z +xg X Note that the field is still oriented along the z direction! But now the field strength is space-variant. G X usually less than G/cm. Because field-strength now depends on location, so does the resonant frequency, via the Larmor equation. So spatial location has been encoded in temporal frequency! This was the key innovation needed to move from NMR to MRI. Illustration of field gradients 2 No Gradient y 2 2 2 Z Gradient 2 y 2 2 2 Y Gradient 2 y 2 2 2 z [cm]
c J. Fessler, October 3, 29, 7:39 (student version) M.3 Chemical shift There is one more field that influences the spins. Orbital motion of electrons surrounding a nuclei perturbs the local magnetic field: B eff = B Z ( σ), where σ depends on local environment. Because of the Larmor relationship, this field shift causes a shift in the resonant frequency: ω eff = ω ( σ). All molecules may have this property, so to normalize things we define water to be a reference, and relate other chemical species to it in parts per million (ppm): chemical shift = δ = ω S ω R 6 = ω ( σ S ) ω ( σ R ) 6 = σ R σ S 6 (σ R σ S ) 6 ω R ω ( σ R ) σ R Fat (lipids with many CH 2 groups) has resonant frequency about δ = 3.5 ppm lower than water. Example. Change in z position of fat due to chemical shift: Misc z = ω γ G Z = γω γ G Z = δγ B Z γ G Z = δ B Z = 3.5 6.5 4 G =.525 mm. G Z G/cm Other topics that could be mentioned: nonlinear field gradients, magnetization transfer. Magnetic susceptibility diamagnetic materials decrease the field strength, paramagnetic materials slightly increase the field strength, ferromagnetic materials strongly increase the field strength. Because carbon and molecular oxygen (O 2 ) are diamagnetic, the field in a body is slightly lower than in air. The transitions between air and soft tissue cause magnetic field inhomogeneities that can degrade image quality if left uncorrected [].
c J. Fessler, October 3, 29, 7:39 (student version) M.4 Imaging overview The object magnetization M(r, t) is the quantity we wish to image; we do so by manipulating the applied magnetic field B(r, t). An MR imaging pulse sequence typically alternates between two phases of manipulation. Excitation: apply RF to establish a pattern of transverse magnetization Readout: turn off RF transmitter and collect an RF signal that is related to that pattern, usually varying the field gradients G. Unfortunately, a single D RF signal in general will not contain enough information to describe a 2D or 3D magnetization pattern. Therefore, the sequence is repeated multiple times. In the simplest case to consider, the excitation part is identical for each repetition, and between excitations one allows the magnetization to return to steady state. Thus, following each excitation there is some pattern M(r, ), where we let t = denote the time following excitation here. We would like to make an image of the transverse component of this pattern, i.e., a picture of m(x, y) z+ Z/2 z Z/2 M X (x, y, z, )+ı M Y (x, y, z, )dz. Typically we simply display the magnitude of this complex quantity, i.e., ( 2 ( z+ Z/2 2 z+ Z/2 m(x, y) = M X (x, y, z, )dz) + M Y (x, y, z, )dz). z Z/2 z Z/2 For each readout portion we vary G(t) so that we collect a signal related to different aspects of the magnetization so that eventually we collect enough information to make a picture of m(x, y). In some MR applications, important information is encoded in the phase of m(x, y). p. 39 Selective excitation preview If only the main field B Z is present, then the whole volume is resonant at f, so all spins are tipped by an applied RF field that oscillates at that frequency. This is called non-selective or hard excitation. Exciting the whole volume is useful for 3D imaging, but 3D imaging can be time consuming, as we will see later from our k-space analysis. Therefore, usually we would like to excite just a single plane (or thin slab) to make a manageable 2D problem. This is called selective or soft excitation []. Solution: apply a linear field gradient, e.g., along z direction (called slice selection gradient) to make traditional transaxial slices: B(r) = B(x, y, z) = B Z +z G Z = (B Z +z G Z ) k. Because MRI scanners have coils that produce field gradients in all three directions, the slice orientation is arbitrarily selectable. With a slice selection gradient enabled, the Larmor frequency is space variant: f (r) = f (x, y, z) = γ(b Z +z G Z ) (i.e., varies with slice position z). Thus to excite just plane z, in principle we would apply an RF signal whose spectrum is concentrated at f = γ(b Z +z G Z ). Such an RF signal would excite only those spins in slice z. What would be the corresponding RF signal? An infinite duration sinusoid, which is not practical. Sinc pulse Usually we content ourselves with exciting and imaging one or more thin slabs of thickness z. (A slab of finite thickness is necessary) to have enough spins excited to get a measurable RF signal.) Then for a z-gradient G Z, the desired RF spectrum is, where f = γ G Z z. Thus the corresponding RF signal in the time domain would be e ıωt sinc(t f). In practice, rect( f f f time-limited approximations are used, so the slice profile is not perfectly rectangular. (Essentially a filter design problem.) Note the relationship between spatial location (in this case z) and temporal frequency due to Larmor equation. This is key to localization in MRI.