Magnetic Resonance Imaging

Magnetic Resonance Imaging (5 May Version)

Purpose: The goal of this experiment is to introduce the fundamental principles and practice of magnetic resonance imaging (MRI), a form of NMR spectroscopy. A two-dimensional (2D) image of the stem or bud from a plant will be obtained and the utility of NMR relaxation as a means of enhancing contrast will be examined. This experiment was developed by Cyrus Maher, Class of 2006, as his senior thesis project in chemistry.

References

* denotes those references which should be read as preparation for the experiment.

1)* S. L. Smith, "Nuclear Magnetic Resonance Imaging", Anal. Chem., 57, 595A-608A (1985).

2)* H. Friebolin, Basic One- and Two-Dimensional NMR Spectroscopy, 4th. ed., Wiley-VCH, Weinheim, 2005. (Examine the material on MRI, relaxation, and the spin-echo exoperiment.)

3) P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy, OxfordUniv. Press, Oxford, 1991. (This monograph is one of the 3 great treatises in NMR.)

4) Z.-P. Liang and P. C. Lauterbur, Principles of Magnetic Resonance Imaging, IEEE Press, New York, 2000. (Liang and Nobel Laureate Lauterbur provide a careful mathematical treatment of MRI. The text was written to accompany a course at the University of Illinois.)

5) K. R. Brownstein and C. E. Tarr, "Importance of Classical Diffusion in NMR Studies of Water in Biological Cells", Phys. Rev. A, 19, 2446-2453 (1979).

Introduction and Basic Theory

In the following sections, vectors are printed in bold face and operators inItalics.

NMR spectroscopy is based on the magnetic properties of the atomic nucleus. Any nucleus with non-zero spin (s>0) is a magnetic dipole with a magnetic dipole moment  = gNS where the nuclear magneton, N = eh/4πmH = 0.50504 x 10-26 J/T = 0.50504 x 10-23 erg/Gauss. In the case of the proton which will be the spin examined in this experiment, s, the spin quantum number, is ½ and g, a quantum mechanical correction factor to classical physics, is 5.585486. When a single nucleus with spin (spin angular momentum) is placed in a magnetic field with strength B, the Hamiltonian energy operator is given by

H = -·B = -gNS·B (1)

If the magnetic field is entirely along the z axis, i.e. B = B0k, the Hamiltonian reduces to

H = -gNB0Sz (2)

Recall that the Sz operator yields the z component of the spin angular moment and its eigenvalue in units of h/2π is given by the quantum number m. m is +½and -½for a proton. Therefore, the magnetic energy of the nucleus is given by

E = -gNB0m (3)

In an NMR experiment, the spins are excited by a pulse of radiofrequency radiation. The selection rules for the excitation are Δs = 0 and Δm = 1. Therefore, the frequency of the transition is given by

ν = ΔE/h = gNB0/h (4)

NMR spectroscopists often measure frequency as angular frequency ω in radian/s sec where ω = 2πν.

ω = [gN(2π/h)]B0 = γB0 (5)

γ is called the gyromagnetic ratio; its value in cgs-esu units for the proton is 26753 radianGauss-1s-1. In a conventional NMR experiment, the instrument is adjusted for a homogeneous magnetic field. That is, B0 is constant across the sample and all protons with the same chemical shift exhibit the same NMR frequency. In the case of our instrument, B0 = 9.3974 Tesla or 93974 Gauss soω = 2.5473 x 109 radian/s and ν = 400.13 MHz.

Imaging via NMR spectroscopy, i.e. MRI, is achieved by the use of gradients. Their application with be first explained via imaging along the z axis. In MRI the magnetic field generated by the superconducting magnet, B0, is supplemented by a gradient, a linear variation in the magnetic field.

B = B0 + Gzz where Gz = (B/z) (6)

A gradient is generated by passing a current through a special coil whereby Gz = ЖI. In the case of our spectrometer, the current supply for the gradients consists of 3 channels. Each channel can produce upon computer control a current in the range -10 A to 10 A. The conversion factor Ж, the so-called coil constant, is 5.56 Gauss/cm-A. It follows from equations (5) and (6) that the NMR frequency ω is now a function of position.

ω = ω(z) = γB0 + γGzz (7)

Recall that a spectrum is a graph of proton density versus frequency. In the case of no gradient, the case in conventional NMR spectroscopy, the spectrum will consist of a sharp line centered at ω = γB0. However, with MRI, the spectrum becomes a continuous function. The Fourier transform of the signal yields proton density, ρ, as a function of frequency. Since the frequency depends linearly on position, the spectrum, a graph of ρ versus ω, is easily converted with the aid of equation (7) to an image, a plot of ρ versus z.

Multi-dimensional MRI is achieved by the use of a triple-axis probe, one with coils that can independently create gradients along 3 orthogonal directions. Our new probe which can only detect and excite protons has this functionality. Our other probes, while quite versatile for chemical purposes, can only apply a gradient along the z or axial direction and only permit one-dimensional imaging.

Experiments which utilize a triple-axis probe for 3D imaging are now based on multi-dimensional spectroscopy. Richard Ernst, our Robbins Lecturer in 1998, received the Nobel Prize in Chemistry for the development of Fourier-transform NMR and multi-dimensional NMR. In order to understand Ernst's contribution, consider the simplest 2D experiment. Ernst divides the experiment into preparation (P), evolution (E), and detection (D) periods. During the preparation period, a coherent oscillation of the magnetic dipoles is created by the application of a radiofrequency pulse or sequence of pulses. Following excitation, the magnetization is allowed to oscillate for a time period t1, the evolution period. Finally, the receiver and the A/D converter are gated on and the magnetization is measured TD(2) times during the detection period, t2. The experiment is completed TD(1) times. With each pass, the length of the evolution period is incremented. Upon completion of the experiment, the signal is a function of two independent time scales, t1 and t2. A two-dimensional Fourier transform yields a spectrum as a function of two angular frequencies, ω1 and ω2.

A 2D experiment can be converted into 3D imaging with the following modifications.

a) A selective radiofrequency pulse is used during the preparation period along with a gradient along the z axis. Selectivity is understood by referring to the Heisenberg Uncertainty Principle. If a wave form is applied for a very short period, Δt, there is a large spread in the wave form's frequency, Δν. That is, (Δt)(Δν) ~1. This would the case of a hard pulse, e.g. 5-10 μs, where the radiation covers the entire spectrum. Conversely, if Δt, the pulse length, is sufficiently long, the radiation seen in the probe covers a very narrow range of frequencies. We use in our MRI methods pulse lengths of 1.5 to 3.0 ms. The selectivity of the preparation pulse is further enhanced by adjusting the shape of the pulse. Hence, the radiation is centered at a frequency ω' and covers the range ω' - (Δω/2) < ω < ω' + (Δω/2). Because of equation (7), the selective pulse only excites protons in a narrow slice along the z axis, z' - (Δz/2) < z < z' + (Δz/2), and only these protons contribute to the signal during the detection period. For obvious reasons, the gradient applied during the preparation period is called the Slice gradient.

b) During the evolution period, a second gradient is applied along the y axis. Hence, the oscillation frequency of the magnetization varies along the y axis and information about the structure of the specimen along the y axis is encoded in the signal. This information is not detected directly as the signal is only detected during the detection period. However, the state of the magnetization at the start of the detection period depends on its state or phase at the end of the evolution period. In the vector model of spin angular momentum, the angular momentum vector precesses about the z axis at an angular frequency defined by equation (7). At the end of the evolution period t1, the phase of the signal is given by φ = ω(y)t1 where the notation for the angular frequency, ω(y), emphasizes the application of a y gradient, often called the Phase gradient.

c) Finally during the preparation period, a third gradient, the Read gradient, is applied along the x axis. Therefore information on the position along the x axis is directly encoded in the signal collected during t2. A double Fourier transformation is followed by a transformation of the axes from a frequency scale to a position scale. The result is a plot of proton density in the x-y plane in a slice perpendicular to the z axis.

For technical reasons. two important modifications are made in the experiment. A full discussion including several details not discussed here is provided in the comprehensive treatises by Callaghan and Liang & Lauterbur. A 180 selective pulse is inserted in the middle of the evolution period. The pulse sequence becomes

(90)-(TE/2)-(180)-(TE/2). TE is called the echo period. This is the famous Carr-Purcell spin echo sequence. It is discussed in Friebolin and is employed to correct for the loss of signal resulting from dephasing of the spins caused by magnet inhomogeneity and the application of the Slice gradient. After application of the first pulse, the magnetization rapidly decays. However, after the second pulse, it recovers as an echo and peaks in amplitude at a time TE after the first pulse. One can apply a sequence of 180 pulses and delays, This will in fact be one of our approaches. Secondly, rather than varying t1 as a constant Gy as originally proposed by Ernst, most MRI units hold t1 constant and increment the gradient Gy. Note in either case, position and frequency information during the evolution period is encoded into the detected signal via the phase at the end of the evolution period.

In the execution of the modified Carr-Purcell pulse sequence, one generates a trail of echoes. You will observe that the intensity of the echoes decreases exponentially as a function of the total echo time, nTE where n = 1, 2, etc. That is,

I = I0exp(-nTE/T2) (8).

In equation (8), T2 is the time constant for the decay of the signal. This decay results from the dephasing of the spins contributing to the signal. In other words, T2, often called the transverse or spin-spin relaxation time, is the time constant for the loss of coherence and is due to natural, irreversible processes. Relaxation is the term applied to the molecular basis for the loss of coherence. If the mechanism responsible for relaxation is known, values of T2 can provide additional information about the sample. Relaxation is also used as a tool to optimize contrast as different populations of water can relax at different rates. At the macroscopic level, relaxation is also manifested in two other ways: the rate of return to equilibrium with a time constant T1 and the rate of energy change between dipoles. The energy exchange can be measured via the nuclear Overhauser effect (nOe).

Relaxation is a spectroscopic phenomenon that affects line shapes, intensities, and in MRI contrast. At the molecular level, relaxation is caused by any process that produces a randomly varying magnetic field. In bulk water the most important mechanism for relaxation is the interaction between magnetic dipoles. However, in this experiment the water is confined in biological structures such as cells and a different relaxation mechanism which was developed by Brownstein and Tarr dominates. When more than one mechanism, e.g. dipole-dipole (DD) and Brownstein-Tarr (BT), contributes to relaxation, the net, experimental value of T2 is given by equation (9).

1/T2 = 1/T2,DD + 1/T2,BT (9).

You will show that T2,DD, i.e. T2 from the dipole-dipole mechanism, is orders of magnitude greater than the experimental value of T2. It follows from equation (9) that the experimental value of T2 is dominated by T2,BB and therefore T2 T2,BT. This is an important result as it enables you to use the Brownstein-Tarr mechanism to interpret the data.

In order to draw the conclusion made above, one must be able to estimate T2,DD from the dipole-dipole relaxation mechanism. To this end, consider the rotation of a water molecule, a process described by a random-walk model. The time scale for this random-walk is set by the rotational correlation time, c. As the molecule rotates, one hydrogen spin, a magnetic dipole, perceives a vector change in the magnetic field due to the movement of the other spin, also a dipole. In the case of water, the correlation time for this rotational modulation of the magnetic field is in the range of picoseconds. One can estimate the correlation time for water by applying hydrodynamic theory at the molecular level and assuming that the water molecule is spherical. This assumption of spherical symmetry works very well for globular proteins. The result derived originally by Einstein and Debye is

c = V/kBT (10)

where V is the molecular volume, a quantity generated by Spartan, and  is the viscosity of the solvent. With an estimate of c in hand, T2 is calculated via equation (10). Note that cgs units are used in equation (11). The details for this celebrated result can be found in Callaghan.

1/T1 = 1/T2 = (3/2)[h2γ4/4π2rHH6]c (11)

In equation (11), rHH is the proton-proton distance in water. A good estimate of rHH and V can be obtained via molecular modeling, e.g. a Hartree-Fock calcuation in Spartan with a 3-21G* basis set. The sixth-power dependence on the relation between the molecular rotation and observables is the basis for the application of the nuclear Overhauser effect to the determination of protein structure.

From the perspective of the physical chemist, relaxation and the value of T2,DD are important as they provide a window on molecular mobility on the picosecond time scale. If protons on a molecule, e.g. a protein in bulk water, have different mobilities, their T2's will also be different. Note that a decrease in mobility, e.g. a longer correlation time, leads to adecrease in T2. If one lengthens the echo time, the signal will be dominated by the more mobile proton. If the dipole-dipole model dominates, the structure containing this water would stand out in the MRI image. However, there is a cost for this enhancement. As one lengthens the echo time, the signal for all waters will decrease. Obtaining good contrast will require an increase in the number of scans in order to restore the lost signal-to-noise.

As part of the calculations for the experiment, you will calculate T2,DD and show that it is ca. 3 s. In contrast, you will find that the experimental value of T2 is ca. 20-30 ms. Consequently, a mechanism other than dipole-dipole, e.g. the Brownstein-Tarr mechanism, makes the dominant contribution to T2 and the value of T2 provides information about the size of the space in which the water molecule is confined. Translational diffusion is a random-walk process described quantitatively by the relationship

rrms = <r20.5 = [6Dt]0.5 (12)

where rrms is root-mean-square distance and D is the translational diffusion constant. During a period t, 68% of the molecules will travel up to rrms from the position at t = 0. For the MRI experiment, t equals the total echo time and one finds that rrms is of the order of magnitude of the cell dimensions. That is, during the execution of the pulse sequence, a typical water molecule has a high probability of diffusing to the cell wall. The encounter between the water molecule and the cell wall abruptly changes the phase of its Larmor precession and is the origin of relaxation. After working out the mathematical details, Brownstein and Tarr showed that T2 is directly proportional to the size of the enclosure. The constant of proportionality, which depends on the detailed geometry of the enclosure and the details of the interaction between water and the cell wall, cannot be determined in the MRI experiment. Therefore, we can only draw qualitative conclusions from the value of T2. Water populations in different spaces will exhibit different values of T2. Water in a smaller space will on the average require less time to diffuse to the cell wall and will have a smaller T2.

Experimental Procedure

You will use Version 2.1.1 of ParaVision to acquire and process your images. ParaVision is Bruker's MRI software. You will be using the same software as that found on the large medical instruments.We have modified the Chemistry Department's NMR spectrometer so that it can perform authentic MRI. To this end, we replaced our single-channel gradient amplifier with a triple-channel gradient amplifier and purchased a proton only, triple-axis gradient probe. We retained the original 9 Tesla magnet which has a bore size of only 52 mm. This narrow bore must accommodate the probe with its compact electronics and the NMR tube. You will insert your biological specimen in the same 5.00 mm OD, 4.22 mm ID NMR tube that you used in organic chemistry and the barrier experiment. Hence the diameter of your specimen must not exceed 4.2 mm. Furthermore, the receiver/transmitter coil can only excite protons over a maximum range of 16 mm and an effective range of 11 mm. Furthermore, in contrast to the large medical units which use water-cooled gradient coils and large currents (up to 200 A!!), our gradient coils are air cooled. Hence, we must be conservative in the currents used and the length of the periods when gradients are applied. As a result of these constraints, we are limited to the imaging of small specimens with axial symmetry and must devote 30 minutes or more to the acquisition of a good image. However, we can achieve far superior resolution, 0.05 mm or better, with our arrangement. Resolution on a medical unit is 1 mm or less.

In this section, boldface marks xwinnmr, ParaVision, and Unix commands.Italics will be used to highlight xwinnmr and ParaVision buttons.

A) Preliminary steps done in xwinnmr.

1) Secure a biological specimen and load it into a 5 mm NMR tube. The specimen should contain mobile water. Its outer diameter, OD, should not exceed the inner diameter, ID, of the NMR tube. Plant materials have been found to work well. A desiccated fly would not work as most of the hydrogen is tied up in relatively immobile macromolecules that give very broad signals. In a hydrated animal specimen, both water and fat molecules which differ in chemical shift by ca. 3 ppm, are present and contribute significantly to the signal. Unless the contribution of the fat to the fid is suppressed, its presence limits resolution in the MRI image. Special techniques are required to image solids. Something that wiggles will give a very poor image since long acquisition times are required on our spectrometer.