Syllabus

Syllabus: “SPIN GYMNASTICS” Walter Kucharczyk MD
Introduction To understand MRI, it is first necessary to understand the physics of proton Nuclear Magnetic Resonance (NMR). The most important entity relevant to biologic MRI is the nucleus of the hydrogen atom in water. While other protons occur within biological molecules, water represents the most important site for MRI due to the concentration of protons in water and the dynamical properties of water. The proton is a fundamental nuclear particle which exhibits charge, mass and spin. While the first of these two concepts is familiar, the notion of spin is not as well appreciated. As the name suggests, it can be thought of as a rotation of the nucleus about its axis which in conjunction with the charge of the nucleus, gives the proton a magnetic property similar to a small bar magnet. However, in addition to the magnetic property of the nucleus, the spin together with the mass of the proton, gives it a property referred to as angular momentum. The combined effect of the spin, charge, mass and angular momentum are the ingredients which are responsible for NMR. Specifically, when a proton is placed in an applied magnetic field, it will precess or wobble. This precession is similar to that of a spinning gyroscope when placed in the earth’s gravitational field. In this case, the gyroscope appears to wobble about its axis at a specific frequency dictated by the strength of the gravitation field and the rotation characteristics of the gyroscope. In a similar manner, the proton’s precessional frequency, also known as the Larmor frequency, is dictated by the fundamental properties of the proton and is proportional to the strength of the magnetic field. For example, at a field strength of 1 Tesla, the Larmor frequency is 43.57 MHz. Doubling the magnetic field strength to 2 Tesla would increase the Larmor frequency to 87.14 MHZ. The scaling factor between Larmor frequency and magnetic field is known as the gyromagnetic ratio. Detection and Excitation A proton has a specific resonance frequency for a fixed magnetic field. The collective magnetic properties of the protons can be represented as a vector usually referred to as the “bulk magnetization” vector, which precesses about the magnetic field Bo at a frequency proportional to Bo. In order to detect this magnetization, a coil of wire is used which is connected to a sensitive amplifier which is in turn tuned to the Larmor frequency. The rotating magnetic field from the magnetization will induce a tiny NMR signal in the coil, which oscillates at the Larmor frequency. Only the time varying part of the magnetization is capable of inducing a signal in the coil and as such only the rotating component of the magnetization in the x-y plane is detectable by this method. This component of the magnetization is referred to as the “transverse” component as opposed to the “longitudinal” component parallel to the Bo field. This also means that the orientation of the receiver coil must be such that its axis lies in the transverse plane, so that the changing magnetic field of the transverse component can couple with the coil and induce a signal. Normally, the magnetization is aligned parallel to the Bo field (along the z axis) and as such cannot precess. In order to generate an NMR signal, the magnetization is tipped away from this equilibrium alignment so that a component of the magnetization lies in the transverse plane where it is free to precess. To achieve this, the spins are exposed to an alternating “B1” magnetic field which is tuned to the Larmor frequency. As the Larmor frequencies are typically in the MHz range, these pulses are referred to as radio frequency or “RF” pulses. Unlike the Bo field, the direction of the B1 field is in the transverse plane. By virtue of this alternating applied magnetic field, the spins can progressively absorb energy and by tipped away from the longitudinal axis to create a component into the transverse plane. The longer the duration of the applied field, or the greater its field strength, the greater the tip angle which can be achieved. By careful choice of the duration and strength of this applied B1 field, the magnetization can be tipped to any angle relative to the Z-axis. The Rotating Frame of Reference The motions of the magnetization vector are complex as they rotate out of alignment with the Z-axis and precess about the Z-axis during excitation and relaxation. In order to simplify our picture of these motions, it is common to view the spin system from a special frame of reference which itself rotates about the Z-axis. To appreciate this concept, imagine a turntable that revolves about the Z-axis at the Larmor frequency carrying a small camera is mounted on a turntable which rotates at the same Larmor frequency. The magnetization, turntable and the camera are seen to rotate about the Z-axis in synchrony. However, if the magnetization vector is viewed from the point of the view of the rotating camera, the magnetization vector would appear stationary with a fixed alignment relative to the Z without any precession. From this new point of view, the precessional motion of the spins can be simplified. Furthermore, if the spin precesses at a frequency which is slightly slower or faster than the frequency of the rotating frame of reference, the spin will precess in this frame at a frequency which corresponds to the difference between the spin precession frequency and that of the rotating frame. Thus, the spin can appear to precess in either direction with its frequency dictated by this difference frequency. Spin Dephasing and Transverse Signal Decay In order to appreciate the contrast mechanisms in MRI, it is necessary to understand how the magnetization of the multitude of spins in the object combines to produce the NMR signal. Recall, that vectors which are parallel or “in-phase”, add in proportion to their individual lengths. However, if the vectors are not aligned, or “out-of-phase”, their sum will not be purely additive but will reflect both their magnitudes and orientations. If spins with identical frequencies are matched to the rotating frame, the magnetization of each spin will add in phase and thus be the numeric sum of their individual magnetization. However, in a situation in which each spin experiences a slightly different magnetic field, then each will have a slightly different precession frequency. Assuming that the spins are initially aligned, this difference in frequency will cause the phase angle of each component of the magnetization to progressively drift out of alignment. Thus with increasing time, the sum of the individual magnetization vectors will decrease due to this progressive de-phasing resulting in a decay of the NMR signal. The time constant for this signal decay can be characterized as that time needed to reduce the signal to 1/e or 37% of its maximum value and is referred to as T2*. At this point, it would be natural to ask what mechanisms cause the spins to experience different magnetic fields throughout the tissue. First, we should categorize these field variations into two groups; those which are fixed in time as distinct from those which change with time. Fixed inhomogeneities of the magnetic field could result from the design of the magnet used to form the Bo field and as such are uninteresting from a biological point of view. However, even with a perfect magnet, the tissues being imaged can distort the fields as a result of their magnetic properties. Changes in tissue magnetic “susceptibility” can generate small field gradients on the order of a few parts per million than can vary throughout the tissue. As these inhomogeneities are fixed, the phase angle of individual spins will grow at a constant rate. In contrast to fixed field variations, there are other mechanisms that can generate time varying field inhomogeneities. These can arise from the fact that the protons themselves are slightly magnetic which can exchange magnetization or due to diffusive movement of spins in microscopic magnetic field inhomogeneities. This decay mechanism is referred to as spin-spin relaxation and has a decay time or T2. The key point is that spin-spin relaxation arises from these time varying magnetic spin interactions. As such, the rate of spin dephasing arising from spin-spin interactions is not constant. In any NMR experiment, spins will experience spin dephasing from both fixed and time varying field changes and both of these factors contribute to the decay time constant T2*. However, using a special combination of excitation pulses, known as a spin-echo pulse sequence, it is possible to measure the signal decay time constant arising only from the time varying changes in the magnetic field to quantify T2. Spin-Spin Relaxation and T2 Weighted MRI The spin-echo pulse sequence uses two RF pulses. The first pulse tips the spins by 90 degrees to force the total magnetization onto the transverse plane. Immediately after this pulse, the spins are in-phase and the NMR signal is maximized. Shortly thereafter, the spins undergo dephasing and the signal decays. At some arbitrary time (TE/2) after the first RF pulse, a second RF pulse is delivered which rotates all the spins by an additional 180 degrees. This moves the spins into an arrangement which mirrors their positions just prior to the 180o pulse. At a later time TE seconds after the 90 degree RF pulse, we find that the signal reappears to form an ‘echo’. The reason for this ‘spin-echo’ can be understood by considering the phase of a single spin. During the first TE/2 seconds, a spin will accumulate a phase angle of theta degrees (relative to the positive Y axis). After the second RF pulse, the phase of this same spin is now 180-theta degrees. This indicates that the phase of the spin is exactly theta degrees from the negative Y-axis. Thus, assuming that the spin continues to accumulate phase at the same rate, the phase will align with the negative Y-axis in an additional TE/2 seconds after the 180o pulse, or TE seconds after the original 90 degree RF pulse. This argument holds true for all the spins in the system, so that all the spins will re-align along the negative Y-axis to form a spin-echo at a time TE. If the rate of dephasing for all spins were constant during the TE interval, the magnitude of the transverse magnetization at TE would be identical to the magnetization immediately after the first 900 RF pulse. However, the time varying nature of spin dephasing outlined above, will create small variations in the phase angle accumulation for each spin before and after the 180 degree RF pulse. The net result is that the spins will not perfectly align at time TE resulting is a small net spin dephasing and signal decay. The longer the TE interval, the greater the opportunity for spin dephasing, resulting in loss of signal. It should be clear, that only the signal loss associated with the time varying component of spin dephasing will contribute to the echo amplitude and thus allows us to measure T2 independently of static field in homogeneities. As a result, T2 will always be longer than T2* as the latter time constant suffers from both static and time varying dephasing while T2 only experiences time varying dephasing. In the case of a spin-echo MR pulse sequence, the 90o and 180o pulses are used to form the spin-echo from which the MR image is formed and it is the magnitude of this echo which determines the brightness in the MR image. If we have two tissues of differing T2 values, the echo amplitudes for each tissue at time TE will differ depending on their respective T2 values. Thus in a T2 weighted MR image, CSF will be brighter than either grey or white matter. The Spin-Lattice Relaxation Time and T1 Weighted Imaging In the preceding section, the process of spins dephasing and causing loss of detected signal was described. However, it is important to note that spin dephasing is a loss of the arrangement of spin orientation after the initial excitation pulse. In this case, the energy which has been deposited by the RF excitation pulse has not been dissipated but only lost to detection. However, if we wait longer, this energy will slowly leave the spin system and be distributed throughout the sample or the “lattice”. The time required for this dissipation is referred to as the spin-lattice or T1 time constant and tend to be longer than the T2 time constant of a given tissue. As a result, after the spins have undergone signal loss from dephasing, the magnetization slowly grows along the z-axis until the equilibrium magnetization is finally reached. The time needed to reach 63% of the equilibrium longitudinal magnetization is referred to as the T1 time constant. T1values for biological tissues generally increase with Larmor frequency whereas T2 times are relatively constant. T1 tend to be 5 to 10 times longer than T2 for these tissues at 1.5 Tesla. Images reflecting T1 are commonly made in MRI and the details of image contrast are complex and beyond to scope of this lecture. As we will see, spin-echo MR images are made with multiple repetitions of 90 and 180 degree pulse combination followed by the detected spin-echo. This triplet of pulses is repeated multiple times every TR seconds in order to gather enough data to form an MR image. Thus it can be seen that the extent of recovery of equilibrium magnetization during the TR interval will vary depending on the tissue T1 value. A tissue with a longer T1 will recover less and contribute less to the spin-echo relative to a tissue with a shorter T1 relaxation time. As such, the short T1 tissue will appear brighter than the longer T1 tissue. While we have made this point in the context of spin-echo MRI, it is true for all MRI techniques requiring multiple excitations to collect MRI data. Summary of Relaxation Mechanisms for Proton MRI There are two main relaxation mechanisms of interest in MRI; namely, spin-lattice (T1) and spin-spin(T2) relaxation. Spin-spin relaxation is an example of a relaxation mechanism which is associated with a loss of spin order or phase and is seen as a transverse decay of signal after the initial spin excitation. The spin-spin time constant T2 is measured with a spin-echo experiment. A related time constant T2* is also seen as a signal decay from spin dephasing and is the time constant observed by simply observing the signal decay directly. In this case T2* is shorter than T2 as it includes dephasing mechanisms from both constant and time varying magnetic field inhomogeneities throughout the tissue. In contrast, T1 is the time needed for the spin system to dissipate the energy which was deposited in the tissue by the initial RF excitation pulse. In liquids, this energy dissipation is slow by comparison to the time to cause the spin system to dephase, thus T1 times are much longer than either T2 or T2*. From this discussion, it is clear that T2 can never be larger than T1. Image contrast arising from either T2 or T1 relaxation mechanism is complex and is modulated by the timing of pulses sequences and size of the flip angles resulting from the excitation pulses. Changing the TE times of spin-echo experiments will alter the T2 weighting while altering the TR interval or flip angle will control the T1 weighting in MR images. Image Formation MRI is unique as a medical imaging method in terms of the relation between the detected signals and the final image. As in any digital imaging method, the challenge of MRI is to define the intensity of the MRI signal for an array of pixels corresponding to differing points throughout the anatomy. However, unlike all other imaging methods in current use in medical imaging, the signal detecting device (receiver coils) cannot be collimated to restrict the signal to a specific location as is done in x-ray imaging, ultrasound or radionuclide imaging. Rather, the MR imaging task is unique, as the detected signals originate from the entire object rather than a single point within it. Thus in the following sections, the mechanisms used to achieve a MR images based on the fact that spins precess at a frequency proportional to the surrounding magnetic field are described. The goal is to determine the brightness of each pixels located in a three dimensional co-ordinate system based (X,Y and Z) and use three related techniques to achieve this based on selective excitation, frequency and phase encoding. To achieve this, the magnetic field in the magnet filed is distorted using magnetic field gradients to provide spatial encoding. Magnetic Field Gradients Great care is used to build the magnets for MRI so as to achieve a highly homogeneous magnetic field within the magnet bore. However, in order to create MR images, these fields must be distorted in a precise and controlled manner though the applications of magnetic field gradients. To illustrate this more fully consider the object in the presence of a gradient in the X direction. A gradient in the X direction means that the field changes only in the X direction and is constant for any point in a Z-Y plane. Furthermore, the field on this plane either increases/decreases with the +ve X position and decreases/increases with the -ve X location. Similarly, a gradient in Z means that the field is proportional only the Z location and is constant in the X-Y plane. Finally, the Y gradient causes the field to change only in the Y direction and is constant within a Z-X plane. In addition, to these gradients having a direction such as X, Y or Z, they can also have a magnitude. In this case, the meaning of the magnitude of the gradient refers to the rate with which the field changes per unit distance. Typical gradients can have values of 10 mT/m, meaning that the field changed 10 mT (i.e. 10-2 Tesla) for every meter of distance moved in object. Thus in comparison to the size of the applied magnetic field (~1 Tesla), we see that these gradients represent very small perturbations (~1%) to the overall field. As these gradients have both magnitude and direction they can be represented as vectors and can add to generate gradients in any direction by the simultaneous application of component X, Y and Z gradients. Selective Excitation The task of defining the 3D distribution of image brightness generally starts with “selective excitation”. As the name implies, this process creates a slab of tissue which is excited so that transverse magnetization is restricted to a specific plane of prescribed location and thickness. The technique involves the combination of NMR resonance, magnetic field gradients and a band limited RF excitation pulse. As an analogy to selective excitation, consider an “excitation” tuning fork set to ring at 440 Hz (the note A). Further consider that nearby is an array of tuning forks which range in frequency from the notes F through C with the tuning forks arranged in a line. When the excitation tuning fork is struck, it moves air at a frequency of 440 Hz. This creates an oscillating pressure wave that propagates until all the tuning fork are bathed with oscillating air molecules. However, only the tuning fork tuned to 440 Hz can absorb energy in synchrony with the moving air and begin to ring or resonate. However, the other tuning forks, cannot resonate with the moving air and remain silent. If the excitation tuning fork were dampened, the A note in the tuning fork array would continue to ring and emit its own sound. However the tuning forks were arranged in a linear array from F to C and since A was used to excite the array, the middle tuning fork must have undergoing excitation. By this means a specific location in space can be excited by the choice of the excitation tuning fork. It follows that using an excitation tuning fork of higher or lower frequency will move the excited region to right or left. This simple analogy is perfectly adaptable to NMR selective excitation. In this case, the spins can absorb energy only if the RF frequency is matched to the Larmor frequency. To create an arrangement similar to the line of tuning forks, a magnetic field gradient is used. In this situation, the magnetic field changes along the axis of the object. In this case, the near end of the cylinder experiences a smaller field and lower Larmor frequency than the opposite end. Consider an object to be composed of different slabs of varying Larmor frequency. Applying the RF excitation to match a frequency for the centre slab will rotate magnetization in this slab to the transverse plane where it will continue to precess. By controlling the range of frequencies used in the excitation pulse, the width of the slice can be controlled while controlling the centre frequency of the pulse, the location of the slice an be controlled. By this means, it is possible create precessing magnetization in a slab at a specific location with a specific slice thickness. The task which remains, is to define the brightness of the spins within this slab. Fourier Magnetic Resonance Imaging MR images can be constructed from so-called “K-space” (Fourier space) data. The next section will indicate how the MR imaging system generates the image signals in the form of the required K-space data. Image Space and K-Space An MR image has coordinates X and Y while the K-space data has coordinates Kx and Ky. The units of X and Y are in units of distance (i.e. centimeters) while the units of Kx and Ky are in units of 1/distance (i.e. centimeters-1). K-space dimensions are somewhat unfamiliar as they are expressed in reciprocal distances. How the MR Imager Encodes Spatial Information This section demonstrates how the MR imaging system generates data directly in the K-space domain, recognizing that in the K-space domain the image data is represented as “spatial frequencies” (or “stripes”) across the entire Field of View (FOV) of varying orientation, spatial frequency and amplitude. This section describes how the MR imaging system generates these stripes and how it determines their correct amplitude, so that when added together, they form the final image. Motions of Spins in a Gradient It is helpful to change the representation of the magnetization that was discussed in the preceding sections from a vector to something simpler. Specifically, rather than drawing a rotating vector which induces a signal in the coil, the magnetization is represented by a sphere. The sphere will rotate on its axis with one side of the sphere colored black, while the other side is white. As the magnetization rotates, the sphere revolves about its axis showing a progressive change from the white side to the black side. By looking at the progression of the color of the sphere, the progression of the phase of the spin as it evolves in varying magnetic environments is visible. Consider aa array of such black and white balls. After the excitation is created, the magnetization of each spin is in phase, and as such, all the balls are in the same orientation, showing the white face of each ball. Consider the application of a gradient in the x-direction and the resulting magnetic field deviation. When the gradient is applied, the balls experience slightly different magnetic fields. On the extreme left side, the magnetic field deviation is negative and causes the balls to rotate in a clockwise manner in the rotation frame. Towards the centre of the array, the rotation rate of the balls decreases until the centre ball is reached where the magnetic field deviation is zero. Continuing further to the right, the field is increasing gradually and results in a counter clockwise rotation rate of gradually increasing frequency. At a later time in the gradient evolution, we see that the balls are arranged to show varying degrees of black and white and a vertical stripe pattern emerges. The density of the stripe pattern (ie the spatial frequency), increases with the gradient evolution or the area under the gradient waveform. By orienting the gradient horizontally (along the X-axis), the stripe pattern is vertical. To create a horizontal stripe pattern, a gradient is applied along the Y-axis and a progression of stripe density evolution with increasing gradient application evolves in the Y-direction. Thus by application of a gradient in either the horizontal or vertical directions, stripes in the vertical and horizontal directions can be generated respectively. With increasing exposure to these gradients, the spatial frequency of the stripe pattern increases in proportion to the area under the gradient-time plot. Stripe Patterns of Arbitrary Orientation and Spatial Frequency In order to generate a stripe pattern in an arbitrary direction, consider the use of two gradients in sequence. A plot of the gradient in Gy and Gx as a function of time is shown. As shown, the amplitude of the Gy gradient waveform is incremented in steps (with fixed duration) after which a fixed Gx gradient waveform is applied. Also consider a plot where a trajectory is traced which corresponds to the area of the gradient waveforms as they evolve. In this plot the horizontal axis is the area of the Gx gradient (red area) and the vertical axis is the area under the Gy gradient (green area). Recall that the spatial frequency of the stripe pattern increases with increasing exposure to a gradient and is represented by the area of the under the gradient waveform. Thus the area of the Gx gradient corresponds to the spatial frequency Kx while the area under the Gy gradient corresponds to the value of Ky. Thus, as the Gy gradient is incremented in amplitude, the point on this plot moves progressively along the Ky axis. After each Gy gradient application, the evolution of the Gx gradient causes that point to progress along the Kx direction. Thus by the combined application of the Gx and Gy gradients all points in the K-space plot can be achieved. Determining the K-Space Amplitude of the Stripe Pattern Thus the application of gradients can create stripe patterns of varying orientation and spatial frequency. The only remaining issue to illustrate, is how the MR imaging system determines the correct amplitude for each spatial frequency needed to correctly encode the object. This is done by measuring the time dependent magnetization of the object during the application of the Gx gradient. Here the object being imaged is shown as a transverse head image made up of our tiny magnetization spheres. During the application of the gradient, the spheres generate the stripe patterns that has been discussed above and generate an NMR signal which is induced in the RF coil. This signal is sampled periodically during the application of the Gx gradient to created a detected NMR signal. The amplitude of this signal corresponds to the desired K-space amplitude for each point in the K-space plot. After repeated applications of all the Gy and Gx gradients, the full K-space representation is complete. Final MRI Pulse Sequence In summary, during the “excitation” RF pulse, a slice is selected in the presence of a Gz gradient. Then an incremented Gy gradient is used to precede the Gx gradient waveform. The NMR signal or echo is “sampled” during the application of the Gx gradient. The sampled data is then applied to the corresponding trajectory in K-space to dictate the brightness of the K space data. If N by N pixels in the x and y direction are needed in the final image, then the signal must be sampled N times for each of N incremented Gy gradient waveforms. This indicates that N separate echoes are needed in order to collect the N incremented Gy gradient applications. The timing between the successive Gy gradients is TR seconds and is the parameter used to control the T1 weighting of the image as discussed above. Similarly, the time between the selective excitation pulse and the peak of the echo formation is the TE time and is used to determine the amount of T2* weighting in an image. Conclusions and Final Comments This lecture and these materials represent a very basic introduction to the physics of MRI. Many excellent texts have been written which delve deeper into the subject and discuss in greater detail the nature of MRI contrast, other imaging methods, and pulses sequences and common artifacts seen in MRI. References: 1) Plewes DB, Kucharczyk W. Physics of MRI: a primer. J Magn Reson Imaging. 2012 May;35(5):1038-54.

Acknowledgements

Professor Don Plewes

References

No reference found.