User:Bci2/2D-FT NMR

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 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 higly 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 and the first study performed on a human took place on July 3, 1977. Earlier papers were also published by Peter Mansfield in UK (Nobel Laureate in 2007, 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 Unpublished `high-resolution' (50 micron resolution) images of other living systems, such as hydrated wheat grains, were obtained and communicated in UK in 1977-1979, and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.

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.

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 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. 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
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 three-dimensional reconstruction of polymer and biopolymer structures by two-dimensional Nuclear Magnetic Resonance  (Kurt Wutrich 1986: 2D-FT NMR of solutions) for molecular weights (Mw) of the dissolved polymers up to about 50,000 Mw. For larger biopolymers or polymers, more complex methods have been developed to obtain the desired resolution needed for the 3D-reconstruction of higher molecular structures, e.g. for 900,000 Mw, methods that can also be utilized in vivo. The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR hyperspectral imaging, or in MRI imaging for research and clinical, diagnostic applications in Medicine. A more precise mathematical definition of the `double' Fourier transform involved is specified next, and a precise example follows the definition. A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables, $$f(x_1, x_2)$$, carried first in the first variable $$x_1$$, followed by the Fourier transform in the second variable $$x_2$$ of the resulting function $$F(s_1, x_2)$$.

Example 1
A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals :$$s(t_1, t_2)$$ yielding a real 2D-FT NMR `spectrum' (collection of 1D FT-NMR spectra) represented by a matrix S whose elements are
 * $$S(\nu_1,\nu_2) = \textbf{Re} \int \int cos(\nu_1 t_1)exp^{(-i\nu_2 t_2)} s(t_1, t_2)dt_1 dt_2$$

where :$$\nu_1$$ and :$$\nu_2$$ 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
 * $$ C(\nu_2', \nu_2) = S^T S = \sum_{\nu^1}[S(\nu_1,\nu_2')S(\nu_1,\nu_2)],$$ with :$$\nu_2, \nu_2'$$ taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies :$$\nu_1$$.

Example 2
2D-FT STEM Images (obtained at Cornell University) 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 (maybe a possible, next Nobel if and only if the mathematical physics treatment is also developed to include also such results). 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.

Brief explanation of NMRI Diagnostic Uses in Pathology
As an example, a diseased tissue, such as that inside tumors, can be detected because the hydrogen nuclei of molecules in different tissues return to their equilibrium spin state at different relaxation rates. By changing the pulse delays in the RF pulse sequence employed, and or the pulse sequence itself one may obtain a `relaxation-based contrast' 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 some metal implants, 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. It is 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 especiallyuseful 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 and atherosclerosis.