Fundamentals of NMR Imaging

In this section, we will describe three building blocks of NMR imaging phase encoding, frequency encoding and slice selection. All three are related to the signal by the fourth equation  

Let us now derive the equations that relate the spatial information to the signal behavior. As we have seen previously, a spin at position r possesses a Larmor frequency co(r) = y B(r) = v( Bo + g r). It is convenient to subtract the reference value, given by the average field, oi0 = v B0. so that we obtain the frequency difference relative to an (arbitrarily chosen) position r= 0  

In this equation, we have made the replacement k = (1/2 ir)yg8 in order to introduce the Fourier conjugate variable to r. This is because formally Eq. (1.6) is a Fourier transformation. What we really want to know is the shape of the sample, p(r), which we can derive by applying the inverse Fourier transformation to the signal function  

However, in order to be able to apply the inverse Fourier transformation, we need to know the dependence of the signal not only for a particular value of k (one gradient pulse), but as a continuous function. In practice, it is the Fast Fourier Transform (FFT) that is performed rather than the full, analytical Fourier Transform, so that the sampling of k-space at discrete, equidistant steps (typically 32, 64, 128) is being performed. 

The recipe for the first building block of NMR imaging, the phase encoding, thus goes like this apply a phase gradient of effective area k acquire the signal S(k) repeat for a number of different equidistant values of k perform the inverse 

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These examples demonstrate the principal feasibility of imaging rigid polymers while addressing questions of general interest. Yet the experimental efforts involved are substantial and the samples need to be kept small. Therefore, NMR imaging of rigid solids appears to be restricted to selected fundamental investigations. 

In the opening chapter, An Introduction to Solution, Solid-State, and Imaging NMR Spectroscopy Leslie Butler (Louisiana State University) introduces the fundamental structure parameters in the NMR experiment through a discussion on solution state H NMR. The shielding of nuclei by core and valence electrons, gives rise to those structure-pregnant numbers called chemical shifts, d, that have been accrued and correlated since the earliest days of NMR. Scalar coupling,... 

Medical Imaging. An area in which applied mathematics has become fundamentally important is the field of medical imaging, especially as it applies to magnetic resonance imaging (MRI). The MRI technique developed from nuclear magnetic resonance (NMR) analysis commonly used in analytical chemistry to determine molecular structures. In NMR, measurements are obtained of the absorption of specific radio frequencies by molecules held within a magnetic field. The strength of each absorption and specific patterns of absorptions are characteristic of the structure of the particular molecule and so can be used to determine unequivocally the exact molecular structure of a material. 

It should be noted that NMR studies are performed on proteins in their native solution state. Since the protein molecules are moving around in solution, the NMR technique measures scalar quantities (torsion angles and interproton distances) this is fundamentally different from x-ray crystallography in which the static crystal lattice allows a vector image of the molecule to be obtained. Since NMR studies are performed on proteins in solution, the technique can be used to probe intricate details of the dynamics of the protein. 

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