diffusion tensor imaging optimization - usc...
TRANSCRIPT
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
Graduate Seminar Presentation Revisited 3
• 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
Graduate Seminar Presentation Revisited 4
• 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
Graduate Seminar Presentation Revisited 5
• 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
Graduate Seminar Presentation Revisited 7
• Gaussian Diffusion
model.
• Solve for the tensor
requires an over-
determined system.
• Fitting scheme
required… but
which one?
Pretty Picture
Graduate Seminar Presentation Revisited 8
• A different
tensor for each
voxel.
• 512x512 voxels
per scan.
*
Signal Attenuation
Graduate Seminar Presentation Revisited 9
𝐴 𝒙 = 𝐴0𝑒−𝑏 𝒙𝑇𝐃 𝒙
• Application of
magnetic field causes
Larmor precession
• Removing the field,
the regions mix and
relax.
Fitting Scheme
Graduate Seminar Presentation Revisited 10
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
Graduate Seminar Presentation Revisited 11
• Quadric Surface.
• More dimensions
but still linear in
each dimension.
• The simply line is an
exact analogy.
Least Squares Fit Overview
Graduate Seminar Presentation Revisited 12
• Looking at
the 2D case.
• Intuitive by
comparison.
Least Squares Fit Overview
Graduate Seminar Presentation Revisited 13
• Definition of error.
• Best slope value.
Least Squares Fit Overview
Graduate Seminar Presentation Revisited 14
• Best intercept value.
• Combining the equations.
Least Squares Fit Overview
Graduate Seminar Presentation Revisited 15
• Do I smell Linear
Algebra?
• Boom! QED
• Now, what were we doing
with this again?
𝐗 𝐃 = 𝛅 𝐃 = 𝐗−1𝛅
*
Least Squares Fit
Graduate Seminar Presentation Revisited 16
• Symmetry
assumed.
• Weighting schemes
also possible.
Gaussian Distribution
Graduate Seminar Presentation Revisited 17
• 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
Graduate Seminar Presentation Revisited 19
• Nature of the
pseudo-inverse.
• Nature of the noise.
• Perhaps a different
model?
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
Graduate Seminar Presentation Revisited 21