Diffusion Tensor

Imaging OptimizationResearch by Ken Stephenson

Dr. Nathan Yanasek

Dr. Joseph Hauger

This Is Your Brain

Graduate Seminar Presentation Revisited 2

• Necessitation of a

passive way to explore

the brain.

• Dissection will kill you.

This Is Your Brain

On MRI

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• Magnetic fields are

harmless compared to a

scalpel.

• Intensity of diffusion

shows tissue density.

• Great diagnostic tool.

• But there’s more…

This Is Your Brain

On DTI

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• Same apparatus and data.

• Intensity AND directionality

of diffusion shows nerve

pathways.

• More information.

• Potentially a better diagnostic

tool.

This Is Your Brain

On DTI

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• Successive layers can be

scanned and compiled

to yield even more

information.

• An entirely new

diagnostic tool is born.

This Is Your Brain

On DTI Tractography

http://youtu.be/ZpAzY5-tDWE?t=40sGraduate Seminar Presentation Revisited 6

• 3D mapping of

the neuronal

pathways.

• Great

possibilities for

diagnosis.

*

The Theory

𝐴 𝒙 = 𝐴0𝑒−𝑏 𝒙𝑇𝐃 𝒙

𝛿 𝒙 =1

𝑏𝑙𝑛

𝐴0𝐴 𝒙

= 𝒙𝑇𝐃 𝒙

𝛅 = 𝐗𝐓𝐃 𝐗

𝛅 = 𝐗 𝐃 𝐃 = 𝐗−1𝛅

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• Gaussian Diffusion

model.

• Solve for the tensor

requires an over-

determined system.

• Fitting scheme

required… but

which one?

Pretty Picture

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• A different

tensor for each

voxel.

• 512x512 voxels

per scan.

*

Signal Attenuation

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𝐴 𝒙 = 𝐴0𝑒−𝑏 𝒙𝑇𝐃 𝒙

• Application of

magnetic field causes

Larmor precession

• Removing the field,

the regions mix and

relax.

Fitting Scheme

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𝐃 = 𝐗−1𝛅

• Pseudo-Inverse.

• Exact analogy to

what is seen in

introductory level

science.

• Don’t believe me?

Let’s prove it!

𝐗−1 = 𝐗𝑇 𝐗 𝐗𝑇 −1

*

Fitting the Diffusion Tensor

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• Quadric Surface.

• More dimensions

but still linear in

each dimension.

• The simply line is an

exact analogy.

Least Squares Fit Overview

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• Looking at

the 2D case.

• Intuitive by

comparison.

Least Squares Fit Overview

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• Definition of error.

• Best slope value.

Least Squares Fit Overview

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• Best intercept value.

• Combining the equations.

Least Squares Fit Overview

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• Do I smell Linear

Algebra?

• Boom! QED

• Now, what were we doing

with this again?

𝐗 𝐃 = 𝛅 𝐃 = 𝐗−1𝛅

*

Least Squares Fit

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• Symmetry

assumed.

• Weighting schemes

also possible.

Gaussian Distribution

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• Average, most

probable, and true

value.

• Symmetry is your

friend.

• But how do you get a

negative signal?

Rician Distribution

Graduate Seminar Presentation Revisited 18

• Not symmetric.

• Least-Square finds

the average.

• Student grades are

also not usually

normally distributed!

• What do we do?

*

NFC

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• Nature of the

pseudo-inverse.

• Nature of the noise.

• Perhaps a different

model?

“I need more mathematics.”

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Credits

• In the order presented

• http://www.huffingtonpost.com/cynthia-germanotta/a-kinder-braver-mental-health-world_b_3381158.html

• http://radiopaedia.org/images/153877

• http://www.auntminnie.com/index.aspx?sec=ser&sub=def&pag=dis&ItemID=90628

• http://dti-tk.sourceforge.net/pmwiki/pmwiki.php?=Documentation.TsaAdvanced

• http://e-mri.blogspot.com/p/diffusion-tensor-imaging-dti-is.html

• http://www.nstarlab.nl/content/stimulating-brain-tms-during-fmri-scanning

• http://image.diku.dk/shark/sphinx_pages/build/html/rest_sources/tutorials/algorithms/linearRegression.html

• http://ufert.se/user-acquisition/mobile-game-analytics/when-why-and-how-you-should-use-linear-regression/

• http://www.bioen.utah.edu/wiki/index.php?title=Noise_modeling_and_Probability_Theory

• http://en.wikipedia.org/wiki/Rice_distribution

• http://www.google.com/imgres?imgurl=&imgrefurl=http%3A%2F%2Fmastibite.com%2Fwallpaper%2FWallpapers%2FDevil%2Ffunny-devil.jpg.php&h=0&w=0&sz=1&tbnid=YhodTdMEBX-oLM&tbnh=194&tbnw=259&zoom=1&docid=ubDWqTnYHiBJ4M&ei=JdEoUq_jNPDk4APHzYHoAQ&ved=0CAEQsCU

• http://foglobe.com/albert-einstein.html

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