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'''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 (vision)|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 <sup>1</sup>H as in plasma water molecules, blood, disolved metabolites and fats. This approach makes it most useful in  [[cardiovascular magnetic resonance|cardiovascular]],  [[oncology|oncological]] (cancer), [[neurology|neurological]] (brain), [[human musculoskeletal system|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<ref name=lauterbur>{{ cite journal |author=Lauterbur, P.C., Nobel Laureate in 2003 | title=Image Formation by Induced Local Interactions: Examples of Employing Nuclear Magnetic Resonance |journal=Nature |year=1973 |volume=242 |pages=190–1 |doi=10.1038/242190a0 }}</ref> and the first study performed on a human took place on July 3, 1977.<ref>[http://www.howstuffworks.com/mri.htm/printable Howstuffworks "How MRI Works"</ref> Earlier papers were also
published by Peter Mansfield<ref>Peter Mansfield. 2003.[http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdf Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI]</ref> 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 Unpublished `high-resolution' (50 micron resolution) images of other living systems, such as hydrated wheat grains, were obtained and communicated in UK in 1977-1979, and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''. 
[[Image:Modern 3T MRI.JPG|thumb|564px|right|Advanced clinical diagnostics and biomedical research NMR Imaging instrument.]]

== NMRI Principle==
Certain nuclei such as <sup>1</sup>H 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 <sup>1</sup>H nuclei that were polarized by the static magnetic field. <ref>Antoine Abragam. 1968. ''Principles of Nuclear Magnetic Resonance.'', 895 pp., Cambridge University Press: Cambridge, UK.</ref>
Very low field schemes have also been recently reported.<ref name= Raftery>{{cite journal
|author=Raftery D |title=MRI without the magnet |journal=Proc Natl Acad Sci USA. |volume=103 |issue=34 |pages=12657–8 |year=2006 |month=August |pmid=16912110 |pmc=1568902 |doi=10.1073/pnas.0605625103
}}</ref>

==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 orbital]]s (or [[molecular orbital]]s) 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, H<sub>loc</sub>/ H<sub>0</sub>.
The first NMR observations of the chemical shift, with the correct physical chemistry interpretation, were reported for <sup>19</sup>F 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<ref name=Wu1999>{{cite journal
|author=Wu Y, Chesler DA, Glimcher MJ, ''et al'' |title=Multinuclear solid-state three-dimensional MRI of bone and synthetic calcium phosphates |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=96 |issue=4 |pages=1574–8 |year=1999 |month=February |pmid=9990066 |pmc=15521 |doi=10.1073/pnas.96.4.1574 |url=http://www.pnas.org/cgi/pmidlookup?view=long&pmid=9990066
}}</ref> reconstruction from projections, somewhat similar to [[Computed tomography|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 [http://www.math.cuhk.edu.hk/course/mat2071a/lec1_08.ppt 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.<ref>*{{cite book |author=Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh |title=Magnetic resonance imaging: physical principles and sequence design |publisher=J. Wiley & Sons |location=New York |year=1999 |isbn=0-471-35128-8 }}
</ref> 
A two–dimensional [[Fourier transform]]ation of such a "real" image may be considered as a representation of such "real waves" by a matrix of [[spatial frequency|spatial frequencies]] known as the [[k-space (MRI)|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)<ref>Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy.[http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf  Nobel Lecture], on December 9, 1992.</ref> that allows the three-dimensional reconstruction of polymer and biopolymer structures at atomic resolution]].<ref>
http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance#Nuclear_spin_and_magnets  Kurt Wutrich in 1982-1986 : 2D-FT NMR of solutions</ref> 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 [[Near infrared spectroscopy|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 '''H'''<sub>0</sub> 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, <math>f(x_1, x_2)</math>, carried first in the first variable <math>x_1</math>, followed by the Fourier transform in the second variable <math>x_2</math> of the resulting function <math>F(s_1, x_2)</math>.                                     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<ref>Charles P. Slichter.1996. ''Principles of Magnetic Resonance.'' Springer: Berlin and New York, Third Edition., 651pp. ISBN 0-387-50157-6.
</ref> between neighbor nuclei.

=== Example 1===
A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals :<math>s(t_1, t_2)</math>  yielding a real 2D-FT NMR `spectrum' (collection of 1D FT-NMR spectra) represented by a matrix '''S''' whose elements are
:<math>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</math>
where :<math>\nu_1</math> and :<math>\nu_2</math> 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
:<math> C(\nu_2', \nu_2) = S^T S = \sum_{\nu^1}[S(\nu_1,\nu_2')S(\nu_1,\nu_2)],</math> with :<math>\nu_2, \nu_2'</math> taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies :<math>\nu_1</math>.
=== Example 2 ===
[http://www.physorg.com/news129395045.html 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  [http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf Chemistry, Physiology and Medicine] received very significant recognition.<ref>[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=2266107&query_hl=33&itool=pubmed_docsum Protein structure determination in solution by NMR spectroscopy] Wuthrich K. J Biol Chem. 1990 [[December 25]];265(36):22059-62.</ref>

==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 implant]]s), 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 [http://www.mr-tip.com/serv1.php?type=img&img=Cardiac%20Infarct%20Short%20Axis%20Cine%204 cardiac infarcts] (courtesy of Dr. Robert R. Edelman)

==See also==

{{Col-begin}}
{{Col-1-of-3}}
*[[Nuclear magnetic resonance]] (NMR)
*[[Medical imaging]]
*[[Protein nuclear magnetic resonance spectroscopy]]
*[[Kurt Wüthrich]]
*[[Chemical shift]]
*[[Computed tomography]] (CT)
*[[Fourier transform spectroscopy]](FTS)
*[[Richard R. Ernst]]
{{Col-2-of-3}}
*[[Magnetic resonance microscopy]]
*[[Solid-state NMR]]
*[[Herbert S. Gutowsky]]
*[[John S. Waugh]]
*[[Charles P. Slichter]]
*[[Near infrared spectroscopy|FT-NIRS]] (NIR)
*[[Magnetic resonance elastography]]
{{Col-3-of-3}}
*[[Relaxation]]
*[[Earth's field NMR]] (EFNMR)
*[[Robinson oscillator]]
{{Commons}}
{{Col-end}}

==Reference list==
{{Reflist}}
{{Reflist|2}}
 <references/>

==References==
 <references/>

*Antoine Abragam. 1968. ''Principles of Nuclear Magnetic Resonance.'', 895 pp., Cambridge University Press: Cambridge, UK.
*Charles P. Slichter.1996. ''Principles of Magnetic Resonance.'' Springer: Berlin and New York, Third Edition., 651pp. ISBN 0-387-50157-6.
*Kurt Wüthrich. 1986, ''NMR of Proteins and Nucleic Acids''., J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore. ([http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrich-lecture.pdf Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules]
*[http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=2266107&query_hl=33&itool=pubmed_docsum Protein structure determination in solution by NMR spectroscopy] Wuthrich K. J Biol Chem. 1990 [[December 25]];265(36):22059-62

*2D-FT NMRI Instrument image: [http://upload.wikimedia.org/wikipedia/en/b/bf/HWB-NMRv900.jpg  A JPG color image of a 2D-FT NMRI `monster' Instrument]. 

*Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. [http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf  Nobel Lecture], on December 9, 1992.

*Peter Mansfield. 2003.[http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdf Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI]

*D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. [http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf PDF of 2D-FT Imaging Applications to NMRI in Medical Research.] Worcester Polytechnic Institute. (Includes many 2D-FT NMR images of human brains.)

*Paul Lauterbur. 2003.[http://nobelprize.org/nobel_prizes/medicine/laureates/2003/    Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.]

*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.''

*{{cite journal
|author=Dudley, Robert, L |title=High-Field NMR Instrumentation|journal= Ch. 10 in Physical Chemistry of Food Processes |volume=2 |pages=421-30|year=1993 ||publisher= Van Nostrand-Reinhold |location=New York |isbn=0-442-00582-2}}

*{{cite journal
|author=Baianu, I.C.; Kumosinski, Thomas |title=NMR Principles and Applications to Structure and Hydration,|journal= Ch.9 in Physical Chemistry of Food Processes |volume=2 |publisher= Van Nostrand-Reinhold |location=New York ||pages=338-420| year=1993 |month=August|isbn=0-442-00582-2}}
 
*{{cite book |author=Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh |title=Magnetic resonance imaging: physical principles and sequence design |publisher=J. Wiley & Sons |location=New York |year=1999 |isbn=0-471-35128-8 }}

*{{cite journal
|author=Raftery D |title=MRI without the magnet |journal=Proc Natl Acad Sci USA. |volume=103 |issue=34 |pages=12657–8 |year=2006 |month=August |pmid=16912110 |pmc=1568902 |doi=10.1073/pnas.0605625103}}

*{{cite journal
|author=Wu Y, Chesler DA, Glimcher MJ, ''et al'' |title=Multinuclear solid-state three-dimensional MRI of bone and synthetic calcium phosphates |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=96 |issue=4 |pages=1574–8 |year=1999 |month=February |pmid=9990066 |pmc=15521 |doi=10.1073/pnas.96.4.1574 |url=http://www.pnas.org/cgi/pmidlookup?view=long&pmid=9990066
}}

==External links==
*[http://www.mr-tip.com/exam_gifs/cardiac_infarct_short_axis_cine_6.gif Cardiac Infarct or "heart attack" Imaged in Real Time by 2D-FT NMRI]
*[http://www.patiencys.com/MRI/ 3D Animation Movie about MRI Exam]
*[http://www.e-mri.org Interactive Flash Animation on MRI] - ''Online Magnetic Resonance Imaging physics and technique course''
*[http://www.ismrm.org International Society for Magnetic Resonance in Medicine]
*[http://www.simplyphysics.com/flying_objects.html Danger of objects flying into the scanner]

== Related Wikipedia websites ==
*[[Medical imaging]]
*[[Computed tomography]]
*[[Magnetic resonance microscopy]]
*[[Fourier transform spectroscopy]]
*[[Near infrared spectroscopy|FT-NIRS]]
*[[Magnetic resonance elastography]]
{{Commons}}
*[[Nuclear magnetic resonance]] (NMR)
*[[Chemical shift]]
*[[Relaxation]]
*[[Robinson oscillator]]
*[[Earth's field NMR]] (EFNMR)
*[[Rabi cycle]]

''This article incorporates material by the original author  from [http://planetphysics.org/encyclopedia/2DFTImaging.html 2D-FT MR- Imaging and related Nobel awards] 
on [http://planetphysics.org/ PlanetPhysics], which is licensed under the [[Wikipedia:Text of the GNU Free Documentation License|GFDL]].''



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