# User:Bci2/2DFT NMRI

2D-FT Nuclear Magnetic resonance imaging (2D-FT NMRI), or Two-dimensional Fourier transform magnetic resonance imaging (NMRI), is primarily a non--invasive imaging technique most commonly used in biomedical research and medical radiology/nuclear medicine/MRI to visualize structures and functions of the living systems and single cells. For example it can provides fairly detailed images of a human body in any selected cross-sectional plane, such as longitudinal, transversal, sagital, etc. NMRI provides much greater contrast especially for the different soft tissues of the body than computed tomography (CT) as its most sensitive option observes the nuclear spin distribution and dynamics of highly mobile molecules that contain the naturally abundant, stable hydrogen isotope 1H as in plasma water molecules, blood, disolved metabolites and fats. This approach makes it most useful in cardiovascular, oncological (cancer), neurological (brain), musculoskeletal, and cartilage imaging. Unlike CT, it uses no ionizing radiation, and also unlike nuclear imaging it does not employ any radioactive isotopes. Some of the first MRI images reported were published in 1973[1] and the first study performed on a human took place on July 3, 1977.[2] Earlier papers were also published by Peter Mansfield[3] in UK (Nobel Laureate in 2003), and R. Damadian in the USA, (together with an approved patent for magnetic imaging). Unpublished high-resolution' (50 micron resolution) images of other living systems, such as hydrated wheat grains, were obtained and communicated in UK in 1977-1979, and were subsequently confirmed by articles published in Nature.

Advanced clinical diagnostics and biomedical research NMR Imaging instrument.

## NMRI Principle

Certain nuclei such as 1H nuclei, or fermions' have spin-1/2, because there are two spin states, referred to as "up" and "down" states. The nuclear magnetic resonance absorption phenomenon occurs when samples containing such nuclear spins are placed in a static magnetic field and a very short radiofrequency pulse is applied with a center, or carrier, frequency matching that of the transition between the up and down states of the spin-1/2 1H nuclei that were polarized by the static magnetic field. [4] Very low field schemes have also been recently reported.[5]

## Chemical Shifts

NMR is a very useful family of techniques for chemical and biochemical research because of the chemical shift; this effect consists in a frequency shift of the nuclear magnetic resonance for specific chemical groups or atoms as a result of the partial shielding of the corresponding nuclei from the applied, static external magnetic field by the electron orbitals (or molecular orbitals) surrounding such nuclei present in the chemical groups. Thus, the higher the electron density surounding a specific nucleus the larger the chemical shift will be. The resulting magnetic field at the nucleus is thus lower than the applied external magnetic field and the resonance frequencies observed as a result of such shielding are lower than the value that would be observed in the absence of any electronic orbital shielding. Furthermore, in order to obtain a chemical shift value independent of the strength of the applied magnetic field and allow for the direct comparison of spectra obtained at different magnetic field values, the chemical shift is defined by the ratio of the strength of the local magnetic field value at the observed (electron orbital-shielded) nucleus by the external magnetic field strength, Hloc/ H0. The first NMR observations of the chemical shift, with the correct physical chemistry interpretation, were reported for 19F containing compounds in the early 1950's by Herbert S. Gutowsky and Charles P. Slichter from the University of Illinois at Urbana (USA).

## NMR Imaging Principles

A number of methods have been devised for combining magnetic field gradients and radiofrequency pulsed excitation to obtain an image. Two major maethods involve either 2D -FT or 3D-FT[6] reconstruction from projections, somewhat similar to Computed Tomography, with the exception of the image interpretation that in the former case must include dynamic and relaxation/contrast enhancement information as well. Other schemes involve building the NMR image either point-by-point or line-by-line. Some schemes use instead gradients in the rf field rather than in the static magnetic field. The majority of NMR images routinely obtained are either by the Two-Dimensional Fourier Transform (2D-FT) technique (with slice selection), or by the Three-Dimensional Fourier Transform (3D--FT) techniques that are however much more time consuming at present. 2D-FT NMRI is sometime called in common parlance a "spin-warp". An NMR image corresponds to a spectrum consisting of a number of spatial frequencies' at different locations in the sample investigated, or in a patient.[7] A two–dimensional Fourier transformation of such a "real" image may be considered as a representation of such "real waves" by a matrix of spatial frequencies known as the k–space. We shall see next in some mathematical detail how the 2D-FT computation works to obtain 2D-FT NMR images.

## Two-dimensional Fourier transform imaging and spectroscopy

A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both involving standard', one-dimensional Fourier transforms. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original function that was transformed at the first stage), but a Fourier transform in a second variable-- which is shifted' in value-- relative to that involved in the result of the first Fourier transform. Such 2D-FT analysis is a very powerful method for both NMRI and two-dimensional nuclear magnetic resonance spectroscopy (2D-FT NMRS)[8] that allows the three-dimensional reconstruction of polymer and biopolymer structures at atomic resolution]].[9] for molecular weights (Mw) of dissolved biopolymers in aqueous solutions (for example) up to about 50,000 Mw. For larger biopolymers or polymers, more complex methods have been developed to obtain limited structural resolution needed for partial 3D-reconstructions of higher molecular structures, e.g. for up 900,000 Mw or even oriented microcrystals in aqueous suspensions or single crystals; such methods have also been reported for in vivo 2D-FT NMR spectroscopic studies of algae, bacteria, yeast and certain mammalian cells, including human ones. The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR hyperspectral imaging (2D-FT NIR-HS), or in MRI imaging for research and clinical, diagnostic applications in Medicine. In the latter case, 2D-FT NIR-HS has recently allowed the identification of single, malignant cancer cells surrounded by healthy human breast tissue at about 1 micron resolution, well-beyond the resolution obtainable 2D-FT NMRI for such systems in the limited time available for such diagnostic investigations (and also in magnetic fields up to the FDA approved magnetic field strength H0 of 4.7 T, as shown in the top image of the state-of-the-art NMRI instrument). A more precise mathematical definition of the double' (2D) Fourier transform involved in both 2D NMRI and 2D-FT NMRS is specified next, and a precise example follows this generally accepted definition.

### 2D-FT Definition

A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables, ${\displaystyle f(x_{1},x_{2})}$, carried first in the first variable ${\displaystyle x_{1}}$, followed by the Fourier transform in the second variable ${\displaystyle x_{2}}$ of the resulting function ${\displaystyle F(s_{1},x_{2})}$. Note that in the case of both 2D-FT NMRI and 2D-FT NMRS the two independent variables in this definition are in the time domain, whereas the results of the two successive Fourier transforms have, of course, frequencies as the independent variable in the NMRS, and ultimately spatial coordinates for both 2D NMRI and 2D-FT NMRS following computer structural recontructions based on special algorithms that are different from FT or 2D-FT. Moreover, such structural algorithms are different for 2D NMRI and 2D-FT NMRS: in the former case they involve macroscopic, or anatomical structure detrmination, whereas in the latter case of 2D-FT NMRS the atomic structure reconstruction algorithms are based on the quantum theory of a microphysical (quantum) process such as nuclear Overhauser enhancement NOE, or specific magnetic dipole-dipole interactions[10] between neighbor nuclei.

### Example 1

A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals :${\displaystyle s(t_{1},t_{2})}$ yielding a real 2D-FT NMR spectrum' (collection of 1D FT-NMR spectra) represented by a matrix S whose elements are

${\displaystyle S(\nu _{1},\nu _{2})={\textbf {Re}}\int \int cos(\nu _{1}t_{1})exp^{(-i\nu _{2}t_{2})}s(t_{1},t_{2})dt_{1}dt_{2}}$

where :${\displaystyle \nu _{1}}$ and :${\displaystyle \nu _{2}}$ denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively, in the 2D NMR experiments. Next, the \emph{covariance matrix} is calculated in the frequency domain according to the following equation

${\displaystyle C(\nu _{2}',\nu _{2})=S^{T}S=\sum _{\nu ^{1}}[S(\nu _{1},\nu _{2}')S(\nu _{1},\nu _{2})],}$ with :${\displaystyle \nu _{2},\nu _{2}'}$ taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies :${\displaystyle \nu _{1}}$.

### Example 2

Atomic Structure from 2D-FT STEM Images of electron distributions in a high-temperature cuprate superconductor paracrystal' reveal both the domains (or location') and the local symmetry of the 'pseudo-gap' in the electron-pair correlation band responsible for the high--temperature superconductivity effect (obtained at Cornell University). So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an additional, earlier Nobel prize for 2D-FT of X-ray data (CAT scans'); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.[11]

## Brief explanation of NMRI diagnostic uses in Pathology

As an example, a diseased tissue such as a malign tumor, can be detected by 2D-FT NMRI because the hydrogen nuclei of molecules in different tissues return to their equilibrium spin state at different relaxation rates, and also because of the manner in which a malign tumor spreads and grows rapidly along the blood vessels adjacent to the tumor, also inducing further vascularization to occur. By changing the pulse delays in the RF pulse sequence employed, and/or the RF pulse sequence itself, one may obtain a relaxation--based contrast', or contrast enhancement between different types of body tissue, such as normal vs. diseased tissue cells for example. Excluded from such diagnostic observations by NMRI are all patients with ferromagnetic metal implants, (e.g., cochlear implants), and all cardiac pacemaker patients who cannot undergo any NMRI scan because of the very intense magnetic and RF fields employed in NMRI which would strongly interfere with the correct functioning of such pacemakers. It is, however, conceivable that future developments may also include along with the NMRI diagnostic treatments with special techniques involving applied magnetic fields and very high frequency RF. Already, surgery with special tools is being experimented on in the presence of NMR imaging of subjects.Thus, NMRI is used to image almost every part of the body, and is especially useful for diagnosis in neurological conditions, disorders of the muscles and joints, for evaluating tumors, such as in lung or skin cancers, abnormalities in the heart (especially in children with hereditary disorders), blood vessels, CAD, atherosclerosis and cardiac infarcts (courtesy of Dr. Robert R. Edelman)

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## Reference list

1. Lauterbur, P.C., Nobel Laureate in 2003 (1973). "Image Formation by Induced Local Interactions: Examples of Employing Nuclear Magnetic Resonance". Nature 242: 190–1. doi:10.1038/242190a0.
2. [http://www.howstuffworks.com/mri.htm/printable Howstuffworks "How MRI Works"
3. Peter Mansfield. 2003.Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI
4. Antoine Abragam. 1968. Principles of Nuclear Magnetic Resonance., 895 pp., Cambridge University Press: Cambridge, UK.
5. Raftery D (2006). "MRI without the magnet". Proc Natl Acad Sci USA. 103 (34): 12657–8. PMC 1568902. PMID 16912110. doi:10.1073/pnas.0605625103. Unknown parameter |month= ignored (|date= suggested) (help)
6. Wu Y, Chesler DA, Glimcher MJ; et al. (1999). "Multinuclear solid-state three-dimensional MRI of bone and synthetic calcium phosphates". Proc. Natl. Acad. Sci. U.S.A. 96 (4): 1574–8. PMC 15521. PMID 9990066. doi:10.1073/pnas.96.4.1574. Unknown parameter |month= ignored (|date= suggested) (help)
7. *Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh (1999). Magnetic resonance imaging: physical principles and sequence design. New York: J. Wiley & Sons. ISBN 0-471-35128-8.
8. Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy.Nobel Lecture, on December 9, 1992.
9. http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance#Nuclear_spin_and_magnets Kurt Wutrich in 1982-1986 : 2D-FT NMR of solutions
10. Charles P. Slichter.1996. Principles of Magnetic Resonance. Springer: Berlin and New York, Third Edition., 651pp. ISBN 0-387-50157-6.
11. Protein structure determination in solution by NMR spectroscopy Wuthrich K. J Biol Chem. 1990 December 25;265(36):22059-62.

## References

• Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. Nobel Lecture, on December 9, 1992.
• Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst's Nobel Laureate Lecture delivered on December 2, 1992, "A new approach to measure two-dimensional (2D) spectra." has been proposed by Jean Jeener at an Ampere Summer School in Basko Polje, Yugoslavia, 1971 (Jean Jeneer,1971}). He suggested a 2D Fourier transform experiment consisting of two $\pi/2$ pulses with a variable time $t_1$ between the pulses and the time variable $t_2$ measuring the time elapsed after the second pulse as shown in Fig. 6 that expands the principles of Fig. 1. Measuring the response $s(t_1,t_2)$ of the two-pulse sequence and Fourier-transformation with respect to both time variables produces a two-dimensional spectrum $S(O_1,O_2)$ of the desired form. This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$ experiments that can also easily be expanded to multidimensional spectroscopy.
• Dudley, Robert, L (1993). "High-Field NMR Instrumentation". Ch. 10 in Physical Chemistry of Food Processes (New York: Van Nostrand-Reinhold) 2: 421–30. ISBN 0-442-00582-2.
• Baianu, I.C.; Kumosinski, Thomas (1993). "NMR Principles and Applications to Structure and Hydration,". Ch.9 in Physical Chemistry of Food Processes (New York: Van Nostrand-Reinhold) 2: 338–420. ISBN 0-442-00582-2. Unknown parameter |month= ignored (|date=` suggested) (help)
• Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh (1999). Magnetic resonance imaging: physical principles and sequence design. New York: J. Wiley & Sons. ISBN 0-471-35128-8.