Algebraic geometry and analytic geometry - Revision history

132.73.204.166: /* Formal statement of GAGA */

2014-11-10T12:29:03Z

Formal statement of GAGA

← Older revision		Revision as of 14:29, 10 November 2014
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84.52.131.50: /* Formal statement of GAGA */ The added less formal discussion renders the invitation to tidy the entry up obsolete

2014-02-20T22:45:12Z

Formal statement of GAGA: The added less formal discussion renders the invitation to tidy the entry up obsolete

← Older revision		Revision as of 00:45, 21 February 2014
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	~~[[Image:MRI T2 Brain axial image.jpg\|thumb\|Standard MRI: no neural tracts~~ can ~~be seen in this high resolution T2-weighted axial image of the brain]]~~		Gabrielle Straub is what for you can call me although it's not the a large number of feminine of names. Fish keeping is what I actually do every week. Managing people is my time frame job now. My house is now inside of South Carolina. Go to my site to find out more: http://prometeu.net<br><br>Feel free to surf to my blog post: [http://prometeu.net how to hack clash of clans]
	~~In [[neuroscience]],~~ '~~''tractography''' is a [[3D modeling]] technique used to visually represent [[neural tract]]~~s using data collected by [[diffusion tensor imaging]] (DTI).<ref name=Filler2009b>{{cite journal \|title=The History, Development and Impact of Computed Imaging in Neurological Diagnosis and Neurosurgery: CT, MRI, and DTI \|year=2010 \|last1=Filler \|first1=Aaron \|journal= [http://www.ispub.com/journal/the_internet_journal_of_neurosurgery/volume_7_number_1_39/article/the-history-development-and-impact-of-computed-imaging-in-neurological-diagnosis-and-neurosurgery-ct-mri-and-dti.html Internet Journal of Neurosurgery]\|volume=7\|issue=1}}</ref> It uses special techniques of [[magnetic resonance imaging]] (MRI), and computer-based [[image analysis]]. The results are presented in two- and three-dimensional images.

	In addition to the long tracts that connect the [[brain]] to the rest of the body, there are complicated [[biological neural networks\|neural networks]] formed by short connections among different [[Cerebral cortex\|cortical]] and [[subcortical]] regions. The existence of these bundles has been revealed by [[histochemistry]] and [[biology\|biological]] techniques on [[post-mortem]] specimens. [[Brain tracts]] are not identifiable by direct exam, [[computed tomography\|CT]], or [[MRI]] scans. This difficulty explains the paucity of their description in [[neuroanatomy]] atlases and the poor understanding of their functions.

	~~The MRI sequences used look at the symmetry of brain water diffusion. Bundles of fiber tracts make~~ the ~~water diffuse asymmetrically in~~ a ~~[[tensor]], the major axis parallel to the direction of the fibers. The asymmetry here is called [[anisotropy]]. There is a direct relationship between the~~ number of ~~fibers and the degree~~ of ~~anisotropy.~~

	~~[[Image:DTI Brain Tractographic Image Set~~.~~jpg\|600 px]]~~

	~~Figure legend:~~
	Diffusion tensor imaging (DTI) data have been used to seed various tractographic assessments of this patient's brain. These are seen in superior (A), posterior (B), and lateral views (C&D). The seeds have been used to develop arcuate and superior longitudinal fasciculi in (A) and (B), for brainstem, and corona radiata in (C), and as combined data sets in (D). Some of the two dimensional projections of the tractographic result are also shown. The data set may be rotated continuously into various planes to better appreciate the structure. Color has been assigned based on the dominant direction of the fibers. There is asymmetry in the tractographic fiber volume between the right and left arcuate fasciculus (Raf & Laf) (smaller on the left) and between the right and left superior longitudinal fasciculus (Rslf & Lslf) (smaller on the right). ~~Also seen are Tapetum (Ta), Left corona radiata (Lcr) and Left middle cerebellar peduncle (Lmcp).~~

	~~== Mathematics ==~~

	Using [[Diffusion MRI\|diffusion tensor MRI]], one can measure the apparent diffusion coefficient at each voxel in the image, and after multilinear regression across multiple images, the whole diffusion tensor can be reconstructed.<ref>{{cite journal \|author=Basser, P., et al. \|title=''In Vivo'' Fiber Tractography Using DT-MRI Data. \|journal=Magnetic Resonance in Medicine \|volume=44 \|pages=625–632 \|year=2000 \|doi=10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O}}</ref>

	~~Suppose there~~ is ~~a fiber tract of interest in the sample~~. ~~Following the [[Frenet–Serret formulas]], we can formulate the space-path of the fiber tract as a parametrized curve:~~

	~~:<math> \frac{d{\mathbf r}(s)}{ds}={\mathbf T}(s) </math>~~

	~~where <math> {\mathbf T}(s) </math>~~ is ~~the tangent vector~~ of ~~the curve~~. The reconstructed diffusion tensor <math> D </math> can be treated as a matrix, and we can easily compute its [[eigenvectors]] <math> \lambda_1,\lambda_2,\lambda_3 </math> and eigenvectors <math> {\mathbf u}_1,{\mathbf u}_2,{\mathbf u}_3 </math>. By equating the eigenvector corresponding to ~~the largest eigenvalue with the direction of the curve~~:

	:~~<math> \frac{d{\mathbf r}(s)}{ds}={\mathbf u}_1({\mathbf r}(s)) </math>~~

	~~we can solve for <math> {\mathbf r}(s) <~~/~~math> given the data for <math> {\mathbf u}_1(s) <~~/~~math>. This can be done using numerical integration, e.g., using [[Runge-Kutta]], and by interpolating the principal eigenvectors~~.

	~~== MRI technique ==~~

	<~~!-- Image with unknown copyright status removed: [[Image:Fallon_Petrovic_DTI_lat3.jpg\|right\|Image of a streamlined DTI scan of the whole human brain seen from the side: Fallon&Petrovic UC Irvine]] --~~>
	~~[[Image:DTI-sagittal-fibers.jpg\|thumb\|240px\|Tractographic reconstruction of neural connections via Diffusion tensor imaging (DTI).]]~~
	Tractography is performed using data from diffusion tensor imaging. Diffusion MRI, a method to produce images of the molecular diffusion process in tissues, was introduced in 1985, notably for its potential for neuroimaging.<~~ref~~>Le Bihan D. and Breton,E. ''Imagerie de diffusion ''in vivo'' par résonance magnétique nucléaire''. C.R.Acad.Sc.Paris T.301, Série II:1109-1112, 1985</ref><ref>Le Bihan D., Breton E., Lallemand D., Grenier P., Cabanis E., Laval-Jeantet M. ''MR Imaging of Intravoxel Incoherent Motions: Application to Diffusion and Perfusion in Neurologic Disorders'', Radiology, 161,401-407, 1986.</ref> A improvement of diffusion MRI, called Diffusion Tensor MRI or DTI <ref>Basser PJ, Mattiello J, Le Bihan D. ''MR Diffusion Tensor Spectroscopy and Imaging''. ''Biophys. J''. 66:259-267, 1994</ref><ref>Basser PJ, Mattiello J, Le Bihan D. ''Estimation of the effective self-diffusion tensor from the NMR spin echo''. J Magn Reson B. 1994 Mar;103(3):247-54.</ref> allows to fully characterize molecular diffusion in the 3 dimensions of space. Free diffusion occurs equally in all directions. This is termed "[[isotropic]]" diffusion. If the water diffuses in a medium with barriers, the diffusion will be uneven, which is termed "[[anisotropic]]" diffusion. In such a case, the relative mobility of the [[molecules]] from the origin has a shape different from a [[sphere]]. This shape is often modeled as an [[ellipsoid]], and the technique is then called [[diffusion tensor imaging]]. Barriers can be many things—cell membranes, axons, myelin, etc.; but in white matter the principal barrier is the [[myelin]] sheath of [[axons]]. Bundles of axons provide a barrier to perpendicular diffusion and a path for parallel diffusion along the orientation of the fibers.

	Anisotropic diffusion is expected to be increased in areas of high mature axonal order. Conditions where the myelin or the structure of the axon are disrupted, such as [[Physical trauma\|trauma]], [[tumors]], and [[inflammation]] reduce anisotropy, as the barriers are affected by destruction or disorganization.

	Anisotropy is measured in several ways. One way is by a ratio called "[[fractional anisotropy]]" (FA). An anisotropy of "0" corresponds to a perfect sphere, whereas 1 is an ideal linear diffusion. Well-defined tracts have FA larger than 0.20. Few regions have FA larger than 0.90. The number gives information of how aspherical the diffusion is but says nothing of the direction.

	~~Each anisotropy is linked to an orientation of the predominant axis (predominant direction of the diffusion). Post-processing programs are able to extract this directional information.~~

	This additional information is difficult to represent on 2D grey-scaled images. To overcome this problem a color code is introduced . Basic colors can tell the observer how the fibers are oriented in a 3D-coordinate system: This is termed an "anisotropic map". The software could encode the colors in this way:

	*Red indicates directions in the X axis: right to left or left to right.
	*Green indicates directions in the Y axis: ~~[[Posterior (anatomy)\|posterior]] to anterior or from [[anterior]] to posterior.~~
	*Blue indicates directions in the Z axis: foot-to-head direction or vice versa

	~~Notice that the technique is unable to discriminate the "positive" or "negative" direction in the same axis.~~

	~~==See also==~~
	* [[Connectome]]
	* [[Diffusion MRI]]
	* [[Connectogram]]

	~~==References==~~
	~~<references/>~~

	~~==External links==~~
	* [http://~~www~~.~~mitk.org/DiffusionImaging MITK Diffusion: Free software for the processing of diffusion-weighted MR data including tractography]~~
	* [http://www.brainlab.com/product/item/fibertracking-and-functional-software Brainlab FiberTracking and Functional Software]
	* [http://www-sop.inria.fr/asclepios/software/MedINRIA/index.php MedINRIA]: A [[freeware]] (non-commercial use only) for DTI tractography
	* [http://www.slicer.org 3D Slicer: Free software with DTI tractography calculation and visualization (BSD License)]
	* [http://trackvis.org TrackVis: DTI tractography visualization software]
	* [http://dipy.org Dipy: Diffusion Imaging in Python]
	* [http://www.visielab.ua.ac.be/publications/improved-analysis-brain-connectivity-using-high-angular-resolution-diffusion-imaging PhD thesis on Diffusion tractography: Improved analysis of ~~brain connectivity using high angular resolution diffusion imaging]~~

	~~[[Category:Magnetic resonance imaging]~~]

129.206.26.233: /* Formal statement of GAGA */

2014-01-02T16:18:51Z

Formal statement of GAGA

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en>TakuyaMurata: /* Moishezon manifolds */ lk

2012-08-31T11:16:09Z

Moishezon manifolds: lk

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