Über die Autorin bzw. den Autor

Siegfried Stapf received his PhD in Physics at the University of Ulm, Germany, in 1996. Following a postdoctoral stay at the University of Nottingham, UK, he currently holds a position as Hochschuldozent at the RWTH Aachen, Germany. His main research interests cover the fields of molecular dynamics and order of confined fluids and soft matter, as well as transport phenomena and structure/dynamics relations in complex media investigated with advanced Nuclear Magnetic Resonance Imaging techniques.

Song-I Han received her Doctoral Degree in Natural Sciences (Dr.rer.nat) from Aachen University of Technology, Germany, in 2001. She was awarded with the first Raymond Andrew Prize of the Ampere Society for an outstanding PhD thesis in magnetic resonance imaging. She pursued her postdoctoral studies at the University of California, Berkeley under the sponsorship of the Feodor Lynen Fellowship of the Alexander von Humboldt Foundation. Dr. Han joined as an Assistant Professor the Department of Chemistry and Biochemistry at the University of California, Santa Barbara in 2004. Her research expertise lies in magnetic resonance flow imaging methodologies and her research objectives are technique developments for orders of magnitude faster and more sensitive NMR spectroscopy and imaging.

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NMR Imaging in Chemical Engineering

John Wiley & Sons

Chapter One

Siegfried Stapf and Song-I Han

1.1

A Brief Comment

Whenever one wants to present an experimental technique for a certain application, one is faced with the question of how much depth the introduction should go into. This is particularly true for NMR, which has developed into an area of research of literally arbitrary complexity, while the basics of NMR imaging can be grasped sufficiently well with little background knowledge. Because this book focuses on applications of NMRI to chemical engineering problems, we will provide the reader with only the basic tools of NMR imaging that are necessary to appreciate the full potential and flexibility of the method as such, in fact its beauty. Much as the admiration for a masterpiece of music does not necessarily require the possession of the skills to reproduce it, but it benefits from a certain understanding of the structure and the context of the composition. Owing to the limited space that is available, however, it must remain beyond the scope of this Introduction to give a complete account of even the simplest relationships. Those readers who are already familiar with the basics of NMR may glance over this Introduction. Laypersons will obtain a crude overview but are directed to a limited number of standard textbooks given at the end of this chapter, a list which is not intended to be complete but should give a starting point for learning more about NMR and is kept short intentionally. Of the large number of NMR textbooks available, only relatively few concentrate on imaging techniques, and the majority of these are aimed at the medical researcher. In fact, medical imaging is in a very advanced state as far as applications are concerned, and it is worthwhile looking across the boundaries to get an overview of advances in, e.g., fast imaging, contrast factors, motion suppression or data processing techniques. For a more thorough understanding, the reader is referred to the lists of references at the end of each of the 29 chapters in this book, which together give an extensive account of the state-of-the-art of non-medical NMR imaging.

1.2 The Very Basics of NMR

If one dissects the abbreviation NMRI into its individual words, the essential features of the method are all covered: we are exploiting the interaction of the Nuclear magnetic properties with external static Magnetic fields that leads to Resonance phenomena with oscillating magnetic fields in the radiofrequency regime in order to obtain an Image of an object. The use of the term "nuclear" has become so unpopular in medical sciences that the abbreviation "MRI" is more common nowadays; although the "N" just states that the experiment is performed on the atomic nuclei, which make up all matter, including our body - to distinguish it from techniques that use the response of the electron. We should merely keep in mind that we are dealing with a quantum mechanical phenomenon. However, most of the systems that are relevant to NMR imaging are weakly coupled homonuclear systems of solution or soft materials where motional averaging and secular approximations are valid for simplified classical description to be sufficient. If one wants to exploit, e.g., the dipolar coupled spin network in ordered soft materials or large molecular assemblies for contrast imaging, a more detailed knowledge of the quantum mechanical relationships becomes inevitable.

We attempt to describe NMR Imaging in a simplified manner using only three essential equations that explain why we see a signal and what it looks like. The first equation describes the nuclear spin magnetization, thus the strength of the NMR signal (and indeed much more):

[M.sub.0] = N [[gamma].sup.2] [[??].sup.2] l(l + 1)/3[k.sub.B] T [B.sub.0] (1.1)

This equation is called the Curie law and relates the equilibrium magnetization [M.sub.0] to the strength of the magnetic field [B.sub.0]. The constants have the following meaning: I is the nuclear spin quantum number (see below), [gamma] is the gyromagnetic ratio specific for a given isotope, [??] is Planck's constant, [k.sub.B] is Boltzmann's constant, N is the number of nuclei and T is the temperature.

What it essentially tells us is that the magnetization increases linearly with the number of nuclei and the magnetic field, and with the square of the gyromagnetic ratio [gamma]. [M.sub.0] is the quantity which after all translates into the NMR signal that we measure, so it should be as large as possible. In order to obtain maximum magnetization, one therefore wants to use a very strong magnetic field (although the ease with which weak fields can be generated possesses a significant attraction, see Chapters 2.2 and 2.4), and take advantage of a nucleus with a large [gamma]. Of all stable isotopes, the nucleus of the hydrogen atom, [sup.1]H, has the largest [gamma]. Furthermore, it is contained in all organic matter and the [sup.1]H isotope has almost 100% natural abundance, which is the reason why the vast majority of imaging experiments are done with [sup.1]H nuclei. However, several advanced applications are presented in this book that exploit the additional information provided by other nuclei such as [sup.2]H (Chapter 2.8), [sup.7]Li (Chapter 3.4), [sup.23]Na (Chapters 3.4 and 5.4), [sup.35]Cl (Chapter 3.4), the metals [sup.27]Al, [sup.51]V and [sup.55]Mn (Chapter 5.4) or the monatomic gas [sup.129]Xe (Chapters 2.6 and 5.3).

The origin of the Curie law is found in the nuclear equivalent of the magnetization, the spin. It resembles a compass needle that is located at the core of the nucleus; brought into a magnetic field, its energy state will depend on its orientation relative to the field direction. The spin is a quantum property, i.e., it can only assume quantized (half-integer or integer) values, given by the total spin quantum number: I = 0, +1/2, +1, .... The main difference from a real compass needle is that when a spin is brought into the magnetic field, all the different orientations it can assume correspond to only a limited number of discrete energy levels, quantized between -I and +I in half-integer steps, so that 2I + 1 possibilities result. One often symbolizes this effect by an arrow of a constant length that is oriented at well defined angles relative to [B.sub.0], but this is merely a crutch for visualizing the quantum mechanical property through a classical one. Independent of how one imagines the spin, one needs to keep in mind that only by bringing the spins into an external magnetic field can the different orientations of the spins differ in energy, which - as stated above - takes place in a non-continuous but quantized fashion, with energy differences of [DELTA]E = [??][[omega].sub.0], where [[omega].sub.0] is the Larmor frequency. Its meaning will be discussed shortly. Keep in mind that not one but an ensemble of spins are present, which are distributed between the discrete energy levels. According to thermodynamics, the lower energy states are more likely to be populated - the average number of spins found in the different energy level states is given by the Boltzmann distribution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we have restricted ourselves to the case of two energy levels as are found for I = 1/2: -1/2, +1/2 (Figure 1.1). It describes the [sup.1]H nucleus and many other isotopes that are important for imaging. The Curie law originates from the Boltzmann distribution. If we insert typical values for the magnetic field strength (say, 10 tesla) and room temperature (T = 298 K), we end up with a tiny fraction of 0.007% in population difference for the [sup.1]H nuclei. This difference is what provides the NMR signal. If we add up all (quantum mechanical) spins to a (classical) bulk magnetization, most of them cancel out, but only this very small population difference determines the actual value of [M.sub.0]. For this reason NMR is traditionally regarded as an insensitive method. Although advanced techniques have made sample volumes down to nanoliters and concentrations down to micromolar detectable under certain circumstances, NMR spectroscopy and imaging can still not compete in terms of sensitivity with techniques such as fluorescence spectroscopy or atomic force microscopy, which are basically capable of detecting single atoms or molecules. However, the power of NMR lies in its unique ability to encode a cornucopia of parameters, such as chemical structure, molecular structure, alignment and other physical properties, interaction between atoms and molecules, incoherent dynamics (fluctuation, rotation, diffusion) and coherent flow (translation) of the sample into the complex NMR signal, instead of simply measuring signal amplitudes of carriers that can be referred to distance or position information.

The longitudinal magnetization [M.sub.0], which we have just defined, is an equilibrium state for which a direct measurement would be of limited use for our purposes. The principle of NMR techniques, however, is not to measure this equilibrium quantity, but the response, thus the change of non-equilibrium transverse magnetization with time, which induces a voltage in a receiver coil enclosing the sample. When subjecting the spin system to an oscillating radiofrequency (rf) field, resonance phenomena can be utilized in such a way that, at the end of the irradiation, the magnetization M is manipulated to be oriented perpendicular to the magnetic field [B.sub.0], i.e., out of its equilibrium. In this state, the spin ensemble's net magnetization is precessing about [B.sub.0], and this precession takes place in a coherent manner ("in phase") among the spins in the ensemble as long as this coherence is not destroyed by natural or artificial influences. It turns out that controlling the time evolution of the coherence by means of a series of rf field and static magnetic field gradient pulses (pulse sequence) leaves a wealth of possibilities to encode information into the signal and extract it again at the time of acquisition. The time-dependence of M in the magnetic field, describing this precession motion, follows the second essential equation:

dM/dt = [gamma] M B (1.2)

The time dependence of the magnetization vector, M(t), is thus related to the cross-product of M and B. Keep in mind also that the magnetic field can be time-dependent. We have replaced [B.sub.0] by B to indicate that the magnetic field can consist of different contributions. In particular, the rf field that interacts with the spins in the sample is a time-dependent magnetic field, [B.sub.rf], it is precisely this field that, when taken into account during the application of the rf pulse, results in the magnetization being rotated out of its equilibrium orientation. To a first approximation, however, we consider the magnetic field to be static. We can then solve Eq. (1.2) and obtain:

M.sub.x = [M.sub.0] sin [[omega].sub.0]t

[M.sub.y] = [M.sub.0] cos [[omega].sub.0]t [M.sub.+] = [M.sub.0] exp (i[[omega].sub.0]t)

The quantity of interest is the precession of the components perpendicular to [B.sub.0] that are measured in the experiment by induced voltage in the coil, which is subsequently amplified and demodulated. We can write them either as individual components [M.sub.x], [M.sub.y], or by a vector [M.sub.+], which combines both of them. In the static field, the precession about [B.sub.0] occurs with the Larmor frequency [[omega].sub.0] = [gamma] [B.sub.0]. If we neglect those processes which dampen the amplitude of the rotating transverse magnetization as precession proceeds, this already describes the frequency that we pick up with our receiver coil, and it is the third and perhaps the most important of our three fundamental equations of NMR:

[omega] = [gamma]|B| (1.3)

Now what do we learn from this? Given that the gyromagnetic ratio [gamma] is known for all nuclei with a very high precision, the measurement of the signal frequency [[omega].sub.0] allows us to determine the actual value of the magnetic field precisely! Indeed this is what NMR is basically doing, with one remarkable exception: B is not just the externally applied field, but it is the magnetic field at the local position of the nucleus itself, which may vary from one nucleus to the other. A large part of the toolbox of NMR is built around this simple dependence. There are two regimes that can be distinguished, providing totally different information about our system.

The microscopic regime is given by the immediate vicinity of the nucleus. It is surrounded by electrons the motion of which - just like the motion of any electric charge - induces a magnetic field that shields the nucleus from the external field, resulting in the nucleus specific local magnetic field. The single electron of the hydrogen atom shields a fraction of some [10.sup.-5] of the external field. Because the shielding depends on the actual charge distribution which, in turn, is a consequence of the molecular environment of the hydrogen atom, a particular moiety can be identified by the shielding effect it has on the nucleus. From [omega] = [gamma] B we see that the resonance frequency of all nuclei varies proportionally with the field strength. The difference relative to a standard sample is called the chemical shift and is measured in unitless numbers, given as ppm (parts per million, [10.sup.-6]). Comparing all proton-containing chemical substances, the total range of [sup.1]H nuclei resonance frequencies covers about 12 ppm. For instance, in an external magnetic field of 9.4 tesla, i.e., at a resonance frequency of about 400 MHz, the maximum difference in frequencies observed is only about 4800 Hz. To distinguish subtle differences between the molecules, the resolution of a good spectrometer must be much better, often values of [10.sup.-10] (i.e., 0.4 Hz in our example) can be reached.

The macroscopic regime is the one that is directly accessible by the magnet designer. By introducing, on purpose, an inhomogeneity to the magnetic field by means of additional coils, B and therefore [omega] are made functions of the position. Of course, this only makes sense if each value of the field B occurs only once in the entire sample volume, so that an unambiguous assignment of the position is possible. The most obvious solution is the generation of a linear field dependence B(r) = [B.sub.0] + gr for which inversion of the frequency into position is directly applicable. The following chapter will address this relationship.

(Continues...)

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