, where J is the angular momentum quantum number, and is Planck's constant divided by 2p
. The absolute value squared of angular momentum is given by
Magnetic Resonance Imaging
S.A. Marshall
Introduction
Magnetic resonance imaging (MRI) is an outgrowth of various mapping procedures that have been applied to the reconstruction of images. Over the past several years these procedures have been greatly refined, the principal motivation for their refinements being their use in medicine as diagnostic tools. Examples of such mappings include medical diagnostics using x-rays and magnetic resonance, astrophysics, aids to navigation, and the detection and characterization of macroscopic defects in solids. Perhaps the most successful and highly developed application in the area of medicine has been computerized x-ray tomograph imaging, or CT. This is because CT is a diagnostic technique which is noninvasive and has minimal impact on the subject being imaged.
An image may be defined as one that allows for a faithful representation of an object and provides information on one or more aspects of the object. Consider an image as an object representation in, say, two dimensions whose image reconstruction data are sensitive to some property of the object. The object representation is the result of a one-to-one mapping operation which generates a point in image space for every point in object space. For practical reasons, every object point should generate a unique image point and vice versa. Furthermore, the corresponding object and image points should be unique and continuous, and the mapping should preserve angles between points.
An ordinary photograph possesses virtually all of the qualities that one might expect of an image. The photograph provides the viewer with a sense of the object and produces analog information that can be retrieved and subsequently subjected to manipulation. If the object is three-dimensional, the photograph can provide a perspective which can give the viewer a basis for interpreting the image. Another useful property of a photograph is its exposure or photographic density. Every point on a photograph's two-dimensional surface represents an exposure spot whose density will to some degree be faithful to the object it is meant to represent. Another aspect of the photograph is its set of emulsion grains and their locations. The surface of a photograph consists of a permanent two-dimensional record of the density of silver crystals at the various image points. This record represents analog information which can be converted to digital information by decomposing it into volume cells and assigning to each such cell a photographic density. Each cell may be given an appropriate set of identifying numbers, such as the x and y surface coordinates, and a photographic density number z. These numbers can then be assigned to each cell and processed to reconstruct an image.
Each cell of the image may be thought of as a picture element called a pixel. A set of numbers (x, y, z) can be assigned to each pixel which refer to a column number, a row number, and a brightness number. From these data, and with the help of a computer and an imaging algorithm, one can reconstruct an image which can then be viewed on a computer's screen or presented as an x-y graphic plot. Refinements may then be made to enhance the presentation of this raw image. For example, if the image were to be that of the sun's disk, the z-axis of the image could be color-enhanced with various shades to represent regions with different temperatures.
There are many applications for which the surface of an object as seen in an ordinary photograph is of limited value. If the regions of interest are located within an object's interior, then the surface photograph would not be very useful. A familiar example is the need for a medically noninvasive diagnostic tool for investigating regions of the body that lie beneath the skin, such as a broken bone, a foreign body, or a malformed internal organ. In many cases, x-rays could provide the medically required information. Other modes that provide images of internal regions of a body include ultrasound, g -rays, magnetic resonance, and autography. These techniques produce a single sheet of information whose exposed area has pixel densities that are proportional to integrated radiation attenuation or photon counts for the cases of x-ray, g -ray, or autograph images or to reflected sound wave intensities for ultrasound images.
Computerized tomography (CT) is a procedure which can make use of any of the radiations referred to above. However, the terminology is generally associated with x-ray imaging techniques. Tomography refers to a slice or sheet image of the object. If the formalisms of CT could be used with absorptions of radiations other than x-rays, it could be applied to the production of images constructed from such data.
Some very general ideas have been presented concerning the relationships between objects, representations, and images. These notions will now be incorporated into a discussion on the reconstruction of images using the techniques of CT and then applied to MRI.
Computerized Tomography
Consider an object whose image is to be reconstructed by means of CT. This is accomplished by taking appropriate two-dimensional slice data of the object, accumulating these data in a computer's memory, and then constructing an image through the use of some imaging algorithm.
As an example, consider an object of some cross-sectional shape having a well-defined boundary whose body exhibits some distribution of x-ray absorbance. Let us assume that the object is located above the x-y plane near the origin and that the x-rays uniformly illuminate the object from above. An x-ray detector with an adjustable iris is positioned beneath the object and the source as in Figure 1. The detector is scanned over the surface of the object, thereby generating a triplet of numbers (x, y, z) which may be stored in the computer's memory. Of this triplet, z and y represent the horizontal and vertical positions of the detector, while z represents the intensity of the transmitted x-ray beam. Thus, at each location of the detector, the accumulated data can be converted to represent x-ray absorbance. The stored information is then subjected to an imaging algorithm which assigns to every picture element a brightness corresponding to the object's x-ray absorbance.
The resolution of the image will be a function of the iris's area, the collimation and intensity of the x-ray beam, the wavelength of the radiation, and the sensitivity of the detector. Within certain limits, these parameters may be selected so as to provide the desired image resolution. For example, higher resolution may be achieved by reducing the area of the detector's iris and choosing a finer scan of the detector. However, as the iris's area is reduced, the signal intensity at the detector is diminished. As the iris size is reduced further, the desired signals will eventually be overwhelmed by detector noise. Another parameter of limited variability is radiation wavelength. At shorter wavelengths (higher photon energies), x-rays tend to penetrate more deeply into body tissues and thereby yield more diagnostic information. However, this strategy is limited since the exposure of biological tissues to shorter x-ray wavelengths can result in cell damage.
Finally, computer memory places another practical limit on the degree of image resolution. Every picture element or pixel requires the storage of information such as its coordinates and x-ray absorbance number. Each image slice requires a large number of pixels, which may place heavy demand upon computer memory. As a result, the degree to which an image can be resolved will depend on the size of the computer's memory.
Another method of image production is reconstruction from rays. Consider an object in the x-y plane whose center is located above the origin of the coordinate axes. An x-ray source located on a graduated circle is capable of rotating full circle about the center of coordinates. An x-ray detector is positioned opposite the x-ray source and follows the source as it is rotated about the object as in Figure 2. Such a system produces two pieces of image information. One is the orientation of the x-ray beam, and the second is the total x-ray absorbance. If a second detector samples the intensity of the x-ray beam before it enters the object, the difference in intensities of the x-rays can be determined without concern for either momentary fluctuations or gradual degradation of intensities in the x-ray source.
Another factor of x-ray absorbance which must be considered is the spectral character of the x-ray source. If different regions of an object have x-ray absorbencies which depend upon the wavelength l , one must use x-rays that are nearly monochromatic. Alternatively, corrections in the imaging algorithms must be made to account for the polychromatic nature of the x-ray source.
Successful procedures now exist for the reconstruction of images from x-ray absorption data. Currently, a number of highly developed computer-oriented mathematical techniques are being used to reconstruct images from observational data. Although only a few primitive examples have been discussed, several techniques exist, each of which is designed to address a specific class of problems. The choice of one technique over another will most likely depend upon the nature of the imaging problem, its complexity, and the computer resources required to execute an image.
Magnetic Resonance Spectroscopy
To understand magnetic resonance imaging, it is necessary first to become familiar with some of the concepts of magnetic resonance spectroscopy. Consider a system at the microscopic level characterized by the quantum mechanical angular momentum L = J , where J is the angular momentum quantum number, and is Planck's constant divided by 2p
. The absolute value squared of angular momentum is given by
(1)
When a particle with angular momentum carries an electric charge, a magnetic dipole moment is generated whose magnitude is given by
(2)
where g is known as the magnetogyric ratio. The energy of a magnetic moment m in the presence of a magnetic field H is E = -m × H , or its equivalent
E = - -m H cos q (3)
where q
is the angle between the magnetic moment and the magnetic field. Classically, a magnetic moment can assume any orientation with respect to the magnetic field. That is, q
can take on any value from 0 to 180. However, at the atomic or quantum level, the number of such orientations is restricted such that the projection of m
upon the magnetic field direction is limited to a finite and well-prescribed set of values (see Section 42.5). Its projection upon the magnetic field is g
m , where m is the "magnetic" quantum number whose values are restricted to the following:
m = J, (J - 1), (J - 2),¼ , -(J - 2), -(J - 1), -J (4)
The two most primitive examples of a magnetic moment are the proton and electron. Each of these particles, known as fermions, has an irreducible spin angular momentum whose quantum number is S =. The magnetic moment associated with spin is given by
(5)
For the proton, the g factor is g p = 2.675 x 108 s-1 T-1 and for the electron it is g e = - 1.759 x 1011 s-1 T-1. Thus, a proton (or an electron) in a magnetic field has only two allowed states identified by the quantum numbers m =. The energies associated with these two states are given by
(6)
This linear dependence of energy upon magnetic field is shown in Figure 3.
A classical treatment shows that the time rate of change of angular momentum is equal to the applied torque t = m x H . Hence, a magnetic dipole moment in a magnetic field will experience a time rate of change of angular momentum given by = t = m x H, and since m = g L we will find that
(7)
Since the torque acting on a system is perpendicular to both m and H , the resulting motion will be a precession of about the magnetic field. This is similar to the motion of a top spinning in a gravitational field. Furthermore, since the magnetic force acting upon a moving electrical charge is directed at right angles to its instantaneous velocity, no work is done by the magnetic force upon the dipole moment. Thus, once initiated, the precession will continue unattenuated until some dissipative mechanism is introduced to remove energy from the system.
If the system of nuclei is in thermal contact with a heat bath whose heat capacity is large compared to the system, what will be the outcome of total polarization? To answer this question, suppose that at some instant a system consisting of a very large number N of protons were polarized so that all have spin quantum numbers. The total magnetization of the system would then be, and its energy would be. This would be a configuration of minimum spin energy and maximum spin order. If this system were completely isolated from the rest of the universe, it would be characterized by a spin temperature of absolute zero since it would have no spin energy to give up nor any means of increasing its spin order. Such a system could not be in thermal equilibrium with its heat bath. Eventually, a real system would arrive at the temperature of the bath and would have a magnetic polarization corresponding to that temperature. This limiting or equilibrium polarization is given by the Boltzmann relation, where is the difference in energy between the two spin states, k is the Boltzmann constant, and T is the absolute temperature. Further, the z-component of the magnetization Mz will advance towards M0, the equilibrium value of magnetization. The differential equation governing the return to equilibrium is
(8)
whose solution is given by
(9)
where t is time and T1 is the longitudinal (spin-lattice) relaxation time, which characterizes the time it takes Mz to return to thermal equilibrium.
Another parameter which characterizes the dynamics of a spin system moving towards thermal equilibrium is the transverse or spin-spin relaxation time T2. To understand its nature, consider a system of nuclei in thermal equilibrium with its bath. Let the system be perturbed at some time t = 0 such that its spin order increases while its total energy remains unchanged. Although energy is conserved, the increase in spin order comes at the expense of an increase in transverse magnetic polarization, Mxy. As the system returns to thermal equilibrium, Mxy. goes to zero. The system returns to thermal equilibrium at a rate characterized by T2. In general T2 is shorter than or at best equal to T1.
The details of the relaxation parameters T1 and T2 are known to be quite complicated and dependent upon several variables such as temperature, nuclear motion, the concentration of magnetic species, and the external magnetic field intensity. However, the concept of a relaxation time has proven to be very useful in correlating specific interactions with the characteristics of drift towards equilibrium. Consequently, relaxation times are universally used in magnetic resonance spectroscopy.
Magnetic resonance can best be understood by using Equation 6 to find the energy difference between two states of a spin 
system of nuclei, say, protons, in the presence of a magnetic field H
. If the difference in energy 
is supplied by a radiation field whose photons have energy 
, then nuclei in the lower energy state will absorb photons and be raised to the upper energy state. The radiation field will also cause nuclei in the upper state to give up a photon of energy and thereby drop to the lower energy state. On average, the radiation field would lose no energy and the system of nuclei would remain unchanged. This assumes, however, that the upper and lower states are equally populated. In fact, at ordinary temperatures, that is not the case. Statistical mechanics shows that for a quantum system of spin particles, the relation between the populations n+ and n- corresponding to the states is given by
(10)
Thus, the lower energy state will be more populated to the extent determined by the exponential term known as the Boltzmann factor. If the energy difference D E is small compared to kT, then the following approximation holds:
(11)
Since the probability of induced absorption is equal to the probability of induced emission, the result is a net absorption of photons because 
.
If the radiation field is characterized by a single frequency w 0 or by a very narrow band of frequencies w 0 ± d w , where d w is small compared to w 0 but larger than the width of the resonance absorption, then one can observe an absorption line by choosing the magnetic field such that. On the other hand, if the radiation frequency is held at w 0 while the magnetic field is scanned through H0, then a resonance can be observed in the magnetic field domain. This is referred to as CW or continuous wave spectroscopy.
Another kind of spectroscopy that is more appropriate to magnetic resonance imaging is the pulsed variety. To understand pulse magnetic resonance spectroscopy, consider a system of protons in the presence of a steady magnetic field H0 along the z-axis. At ordinary temperatures, the protons will be slightly polarized and have a small but finite net magnetization Mz along the direction of the magnetic field, while the total nuclear magnetization M will precess about the z-axis. Because of this precession, the transverse magnetizations Mz and My will have average values that vanish. Now suppose a radiation field of frequency w 0 = g H0 is impressed upon this system of protons having an x-component whose time dependence is Hx = Hx0 cos(w 0t). Such a linearly polarized field may be expressed as the sum of two counter-rotating circularly polarized fields Hr and H, where r is a field that rotates about the z-axis in a counter clockwise manner, while rotates about the z-axis in a clockwise manner. The component He, which rotates in the sense opposite to that of precession, will on average, or over a rotation of 2p radians, receive as much torque from M as it imparts. Thus its influence on M may be ignored. The only component of Hx that affects the magnetization vector is Hr, which rotates in the same sense as does the precession of the magnetization vector. In the rotating frame, Hr remains in phase with M, thereby imparting a constant torque upon the latter. This added torque causes M to undergo a continuous increase in its polar angle q . The increase in q will continue as long as the system is exposed to this oscillating field. This additional angular motion of M is referred to as nutation. As the tip of the vector M precesses about the direction of the applied field H0, it undergoes a nodding motion (see Figure 4). If the angular frequency of the radiation field is w , one finds that the effective magnetic field fixed in the rotating reference frame is given by
(12)
where 
is a unit vector lying along the x-axis in the rotating frame and k is a unit vector along the z-axis fixed in the lab frame. When w
is equal to the Larmor angular velocity w
0, the term in brackets will vanish since w
0 = g
H
0. Furthermore, when w
= w
0, the effective field H
eff is equal to H
r.
Now suppose that an oscillating field H r of angular frequency w 0 is applied, for a short time d t, to a collection of protons in a magnetic field H 0. In the rotating frame, the vector M will dip through an angle given by q = g Hrd t. If the strength of Hr and the time of its duration d t are such that g Hrd t = p , then M will rotate into the negative z-axis. Such an orientation represents a state of maximum magnetic energy. Consequently, the system will return to thermal equilibrium in a manner characterized by the longitudinal relaxation time, T1 (according to Eq. 9).
Next, suppose that the duration time of the pulse is halved so that g Hrd t = p /2. At the end of this pulse, all of the nuclear magnetic moments will be aligned parallel to the yâ-axis of the rotating frame. This is not a state of thermal equilibrium; hence the system will immediately move towards thermal equilibrium by giving up its yâ magnetization. It does so in a time characterized by T2, the transverse relaxation time, assuming this time is short compared to T1. The transverse magnetization of the system returns to its state of equilibrium according to the relation
(13)
The manner in which the system returns to thermal equilibrium is a randomization process caused by inhomogeneities in the external field and the effect of neighboring magnetic moments. The magnetization, initially polarized along yâ, gradually randomizes in the xâ-yâ plane, and "fans out" in this plane as in Figure 5. The time dependence of this process is given by Equation 13, and is referred to as free induction decay.
If the time lapsed is long compared to the relaxation time T2, the transverse magnetization will vanish. However, if a second pulse is applied to the system, a 180o pulse, the magnetic moments will later refocus to yield the original transverse magnetization and a second magnetic resonance signal referred to as a spin echo will result. The spin echo signal can be viewed as one which is a synthesis of a free induction that decays as time flows in the negative direction and a second free induction that decays as time flows in the forward direction. The two are joined together at the zero of time at which their maxima coincide.
To recapitulate, a spin echo signal can be generated as the system of nuclei is exposed to a pair of radio frequency magnetic field pulses polarized along the rotating x-axis. The duration of these pulses must be such that g Hrd t is 90o for the first pulse and 180o for the second. The time between the two pulses should be several transverse relaxation times to insure that the transverse magnetization has time to partially randomize. Yet, it must be short compared to the longitudinal relaxation time to avoid the loss of magnetization as the system tends to return to thermal equilibrium. Figure 6 shows the consequences of a 90o pulse followed by a 180o pulse.
There is a fundamental difference between magnetic resonance absorption spectroscopy and the two forms of pulsed spectroscopy. In absorption spectroscopy, information is gathered in the frequency or magnetic field domain, while in pulsed spectroscopy, information is gathered in the time domain. Both techniques are equivalent in that the resonance absorption shape function can be obtained from the free induction shape function using a Fourier transformation. Pulsed spectroscopy has the advantage of being very fast compared to absorption spectroscopy. For example, a single 90° pulse is sufficient to produce a signal which can be captured by a suitable signal processing device. This signal may then be subjected to an appropriate Fourier transformation to provide the absorption shape function. The entire process may be conducted very rapidly using a computer to first control the pulse sequencing and then to perform the various mathematical manipulations.
At the heart of magnetic resonance imaging is the magnetic resonance absorption shape function, or simply the absorption function, which is used to drive an imaging algorithm. In practice, an integrated density function is required for image reconstruction. This function is directly proportional to the integrated magnetic resonance absorption. Image reconstruction is obtained by establishing a set of procedures for encoding the coordinates of object volume elements with magnetic field strength values. One uses gradient magnetic fields (spatially varying fields) to produce this encoding. A procedure is established for encoding each point of a planar sheet with a unique magnetic field strength. Once this is accomplished, an image reconstruction algorithm may be used to produce an image.
To better understand the advantage of pulsed spectroscopy over frequency domain spectroscopy, consider the following experiment. Suppose the entire x-y plane is subjected to a uniform magnetic field, and an rf pulse is applied to the system which rotates the magnetization by 90o. All magnetic moments will now be rotated from the z-axis to the rotating yâ-axis, and free induction will proceed along with its decay. After some interval of time (short compared to T2), let a gradient magnetic field be applied along the x-axis. Then each volume element of the object will have its magnetic moments precess at a frequency characterized by the local magnetic field identified with the space coordinates of the object. The result is a single complex signal which is the superposition of all free induction decay signals along the direction of the field gradient. This signal may then be subjected to a Fourier transformation which will yield the entire distribution of precession frequencies within the slice along with the signal intensity associated with each frequency. This information may then be used as input data to an imaging algorithm.
One of the earliest magnetic resonance images reported in the scientific literature is a view of two 1-mm diameter water-filled capillary tubes shown in Figure 7. By comparison, Figure 8 shows state-of-the-art magnetic resonance images of a human head. As mentioned earlier, MRI is a diagnostic technique that is considered to be noninvasive. Photons associated with the rf signals have energies of about 10-7 eV. Since molecular bond strengths are much larger (of the order of 1 eV), the rf photons cause little cellular damage. In comparison, x-rays or g -rays have energies ranging from 104 to 106 eV and have the potential of causing considerable cellular damage. This is the main advantage that MRI has over other imaging techniques in medical diagnostics.
Suggested Readings
Carrington, A., and A.D. McLachlan, Introduction to Magnetic Resonance With Applications to Chemistry and Chemical Physics, New York, Harper & Row, 1967.
Herman, G.T., Image Reconstruction from Projections. The Fundamentals of Computerized Tomography, New York, Academic Press, Inc., 1980.
Keller, P.J., Basic Principles of Magnetic Resonance Imaging, Milwaukee, General Electric Company, 1988.
Mansfield, P., and P.G. Morris, NMR Imaging in Biomedicine, New York, Academic Press, Inc., 1982.
Slichter, C.P., Principles of Magnetic Resonance, New York, Harper & Row, 1963.
Questions
Problems
Figure Captions
Figure 1
One method of obtaining an image is by using a rotatable x-ray source that uniformly illuminates a stationary object. A x-ray detector placed beneath the object takes transmission intensity measurements at each of the indicated grid points.
Figure 2
Another method of obtaining images used in x-ray computerized tomography, CT. The x-ray source and detector scan the object by rotating over a full circle. The integrated x-ray intensity beam is obtained for each of a uniformly spaced set of discrete orientations.
Figure 3
A schematic representation of the energy dependence on magnetic field of a two-state quantum mechanical system corresponding to S = 1/2.
Figure 4
The behavior of a magnetic moment in a uniform magnetic field. If no alternating magnetic field is applied to the system, a pure precession takes place. When an alternating field of angular velocity w
0 is applied, a nutation results in the motion of the moment. Eventually, the magnetization will tip into the xâ-yâ plane.
Figure 5
A rotation of the magnetization vector from the laboratory z-axis into the rotating frame's yâ-axis. After completing this rotation, all the magnetic moments in the system will be phase. As time evolves, this system of moments fans out in the manner discussed in the text.
Figure 6
A schematic representation sequence of two pulses, one of 90°
duration followed by another of 180°
duration. The consequences of this sequence of pulses is the generation of a free induction decay followed by a spin echo.
Figure 7
An early magnetic resonance image of two water-filled capillary tubes 1 mm in diameter with their centers separated by about 3 mm. (a) This shows the method of projections used in producing the image. (b) The reconstructed image. (From P.C. Lauterbur, Nature, 242:190, 1973.)
Figure 8
State-of-the-art magnetic resonance images taken of the human head. (a) is a sagittal view and (b) is a coronal view. The slice images are of 1 cm thickness and the result of a 128 x 128 point reconstruction. (Photo Researchers, Inc.)