The Project Molecular Diffusion in MRI

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The Project Molecular Diffusion in MRI

• Technical application of tracking fiber
  (Tractografía)
• investigator Martha Liliana Mora V.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• State of the art.
• Physical phenomenon DTI.
• Introduction.
• Methods.
• 4.1 Algorithm RF Inhomogeneity Correction
  Algorithm in MRI.
• 4.2 Algorithm Registration.
• 4.3 Algorithm Diffusion Isotropy
• 4.4 Algorithm Diffusion Anisotropy Tensor MRI.
• 4.5 Algorithm DTI - Higher Resolution.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• 5. Implementation of Methods. (proof in the
  software)
• 6 .The Objective of this visit Brigham and
  womens hospital Harvard Medical School.
• 6.1 Collaboration with the group of BWH's work in
  publications.
• 6.2 Training in the acquisition, processing,
  analysis, and application of Diffusion Tensor
  Imaging.
• 6.3 Future works with the group of BWH. HMS.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• State of the art.
• Stejskal, E. O., and Tanner, J.E. Spin-diffusion
  measurements spin echoes in the presence of a
  time-dependent field gradient. J. Chem. Phys. 42,
  288-92. (1965).
• Difusión en imagen de resonancia magnética. It
  was introduced for LeBihan en 1985. Art. Le
  Bihan D, Breton E. Imagerie de diffusion in vivo
  par résonance magnétique nucléaire. CR Acad Sci
  Paris 19853011109-1112.
• Difusion Tensor by Basser et al (Mattiello J. Le
  Bihan) Diffusion tensor echo-planar imaging of
  human brain. In proceedings of the SMRM,
  Estimation of the effective self-diffusion tensor
  from the NMR spin echo. J. Magn. Reson 1994
• Diffusion Tensor Imaging Concepts and
  Applications (Denis Le Bihan, Jean Francois
  Mangin, Cyril Poupon, Chris A. Clark. Journal of
  Magnetic Resonance Imaging (2001).

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• State of the art.
• Diffusion Tensor Imaging Image Acquisition and
  Processing Tools. Surgical Planning Laboratory,
  Technical Report 354. Martha E. Shenton, Ph.D.,
  Marek Kubicki, M.D., Ph.D., Robert W. McCarley,
  M.D.
• An Analysis Tools for Quantification of Diffusion
  Tensor MRI Data. Hae-Jeong Park, Martha E.
  Shenton, Carl-Fredrik Westin. Division of Nuclear
  Medicine, Dept. of Diagnostic Radiology, Yonsei
  University, Colege of Medicine, Shinchon-dong,
  Seodaemun-gu, Seoul 120-749, Korea. Laboratory of
  Mathematics in Imaging, Dept. of Radiology,
  Brigham and Womens Hospital Harvard Medical
  School Boston USA.
• DTI and MTR abnormalities in schizophrenia
  Analysis of white matter integrity.
• M. Kubicki et al. Neuroimagen 25 (2005)
  1109-1118.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• State of the art.
• P. Perona and J. Malik. Scale-space and edge
  detection using anisotropic diffusion. IEEE
  Transactions on Pattern Analysis and Machine
  Intelligence, 12(7)629-639,July 1990.
• J. Weickert. Theoretical foundations of
  anisotropic diffusion in image processing.
  Computing Supplement, 11221-236, 1996.
• J. Weickert. A review of nonlinear diffusion
  ltering. Scale-Space Theory in Computer Vision,
  Lecture Notes in Comp. Science (Springer,
  Berlin), 12523-28,1997. Invited Paper.
• L. Alvarez, P.L. Lions, and J.M. Morel. Image
  selective smoothing and edge detection by
  nonlinear diffusion (II). SIAM Journal of
  Numerical Analysis, 29845-866,1992.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• State of the art.
• P. Abry and A. Aldroubi. Designing
  multiresolution analysis-type wavelets and their
  fast algorithms. J. Fourier Anal. Appl., to
  appear.
• S. Mallat. Multiresolution approximations and
  wavelet orthonormal bases of . Trans. Am.
  Math Soc., 315(1) 69-97, 1989.
• S. Mallat. A theory for multiresolution signal
  decomposition The wavelet representation. IEEE
  Trans. Signal Proc., II(7) 674-693, 1989.

emailmartha.mora_at_urjc.es
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DIFUSIÓN
Difusión Restringida (ANISOTROPíA)
Difusión Libre (ISOTROPÍA)
emailmartha.mora_at_urjc.es
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DIFUSIÓN
It does not obtain direction
equal loss of sign.
emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.

4.1 Algorithm RF Inhomogeneity Correction
Algorithm in MRI. Publication Juan A.
Hernandez, Martha L. Mora, Emanuele Schiavi, and
Pablo Toharia. ISBMDA 2004, LNCS 3337, pp. 18,
2004. Publisher Springer-Verlag Berlin
Heidelberg 2004
emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Registration criteria.
• Quantitative measure of a good match.
• Focus on intensity based measures.
• Spatial transform type
• Allowable mapping from one image to another.
• Optimization algorithm used
• Optimize transform parameters w.r.t to measure.
• Image interpolation method
• - Value of image at non-grid position

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Registration Framework
• Generic framework for building intensity based
  registration
• algorithms.
• Each functionality encapsulated as components.
• Components are inter-changeable allowing a
  combinatorial
• variety of registration methods.
• Components are generic
• Can be used outside the registration framework

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.2 Registration Framework Components

Registration Framework
Transform Parameters
Fixed image
Image Similarity Metric
Moving image
Cost Function Optimizer
Image Interpolator
Resample Image Filter
Resampled image
Transform
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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Transfrom
• Encapsulates the mapping of points and vectors
  from an input space to an output space.
• Provides a variety of transforms from simple
  translation, rotation and scaling to general
  affine and kernel transforms.
• Forward versus inverse mapping
• Parameters and Jacobians

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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Forward and Inverse Mappings
• Relationship between points of two images
  can be expressed in two ways
• Forward
• Pixel of input image mapped onto the output image
• Inverse
• Output pixels are mapped back onto the input
  image
• Encapsulates the mapping of points and vectors
  from an input space to an output space.
• Provides a variety of transforms from simple
  translation, rotation and scaling to general
  affine and kernel transforms.
• Forward versus inverse mapping
• Parameters and Jacobians

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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Translation Transform
• Maps all points by adding a constant
  vector
• Parameters
• i- parameter represent the translation in
  the i-dimension.
• Jacobian in 2D

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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Euler2D Tranform
• Represents a rotation and translation in 2D
• Parameters
• Jacobian in 2D

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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Image Registration Classification.
• Euler3D Tranform
• Represents 3D rotation and translation
• - Rotation about each coordinate axis.
• Parameters

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Molecular Diffusion in MRI

• Methods.
• 4.3 Algorithm Diffusion Isotropy Image
  derivates.
• The derivative of the image with respect to
  the variable is written
• For vector-valued images , we have
  and
• The derivation of a scalar image with
  respect to its spatial coordinates is
• called the image gradient and is noted by
• 
  

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Molecular Diffusion in MRI

• Methods.
• 4.3 Algorithm Diffusion Isotropy
• It is for 2D images (p 2) and 3D volumes (p
  3)
• when p 3

when p 2
emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• 
  

Methods. 4.3 Algorithm Diffusion Isotropy
This equation used in the physics to describe
solid flows, this one is known as the equation of
the diffusion. Koenderink noticed in that the
solution at a particular time t is the
convolution of the original image with a
normalized 2D Gaussian kernel of variance

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.3 Algorithm Diffusion Isotropy
• With
• 
  and
• Is a normalized 2D Guassian kernel of variance
  
• Perona Malik. The idea is built on the fact
  that the heat equation can be written in a
  divergence form

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.3 Algorithm Diffusion Isotropy - divergence
  based PDE
• Other authors proposed to use a function
  depending on the convolved
• gradient norm
  rather than simply considering
• where
• 
  is a normalized 2D Gaussian kernel of
  variance
• A major generalization of divergence-based
  equations has been recently proposed by
• Weickert.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.3 Algorithm Diffusion Isotropy - divergence
  based PDE
• A major generalization of divergence-based
  equations has been recently proposed by Weickert.
  
• he considered image pixels as chemical
  concentrations diffusing with respect to some
  physical laws (Fick Law and continuity equations)
  and proposed a very generic equation

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.4 Algorithm Diffusion Anisotropy Tensor MRI.
• This is justied by the fact that spectral
  elements of diffusion tensors
  are the important data that provide signicant
  structural informations
• For DT-MRI images, the diagonal matrix
  measures the water
  molecule velocity in the brain fibers, while the
  tensor orientation provides important clues
  to the structure and geometric organization of
  these fibers.
• Significant physiological values can also be
  computed from
• - Mean diffusivity
• - Partial anisotropy
• - Volumen ratio

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm Diffusion Anisotropy Tensor MRI.
• Regularization of the tensor diffusivities
• Different anisotropic PDE's can be used to
  regularize the tensor diffusivities,
• Depending on the considered application.
• For instance,the following diffusion schemes
  could be considered for analysis
• - Process each eigenvalue separately, with
  classical scalar regularization schemes.
• - Process the eigenvector
  using vector valued diffusion PDEs.
• - Include a-priori spectral informations inside
  the diffusion equation, in order to drive the
• diffusion process. For instance, it could be done
  like this, for DT-MRI regularization purposes
• where D is a diffusion tensor that drives the
  regularization process.

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods.
• 4.2 Algorithm DTI Multiresolution
  Approximations and Their Associated Wavelets.
• There is a class of DWT that can be implemented
  using extremely efficient algorithms. Aldroubi -
  S.Mallat.
• These types of wavelet transforms are associated
  with mathematical structures called
  multiresolution approximations of (MRA).
• A multiresolution approximation of is a set
  of spaces that are generated by
  dilating and translating a single function
  .

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Molecular Diffusion in MRI

• Methods. Multiresolution Approximations and Their
  Associated Wavelets.
• Where are
  the dilations (or reductions) and translations of
• The function called the scaling
  function. Moreover, for fixed the set
• is requered to form an
  unconditional basis of .
• If the funcctions form an
  orthogonal basis of . Then we call
• an orthogonal scaling function.
• The spaces are required to satisfy the
  additional properties

emailmartha.mora_at_urjc.es
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Molecular Diffusion in MRI

• Methods. 4.2 Multiresolution Approximations and
  Their Associated Wavelets.
• Properties (i) (iv), the scaling function
  that is used to generated the MRA cannot be
  chosen arbitrarily.
• In fact since and since
• 
  .
• Conclude that the generating function
  must be a linear combination
• of the basis
  
• This last relation is often called the two-scale
  relation or the refinement equation, and the
  sequence is the generating sequence
  which is crucial in the implementation of the DWT
  associated with multiresolutions.

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Molecular Diffusion in MRI

THANKS YOU