PPT

The Project Molecular The Project Molecular Diffusion in MRIDiffusion in MRI

Technical application of tracking Technical application of tracking fiber (Tractografía)fiber (Tractografía)

investigator: Martha Liliana Mora V.investigator: Martha Liliana Mora V.

e:mail:[email protected]

Molecular Diffusion in MRIMolecular Diffusion in MRI

1.1. State of the art.State of the art.

2.2. Physical phenomenon DTI.Physical phenomenon DTI.

3.3. Introduction.Introduction.

4.4. Methods.Methods.

4.1 Algorithm RF Inhomogeneity Correction Algorithm in MRI.4.1 Algorithm RF Inhomogeneity Correction Algorithm in MRI. 4.2 Algorithm Registration.4.2 Algorithm Registration.

4.3 Algorithm Diffusion Isotropy4.3 Algorithm Diffusion Isotropy

4.4 Algorithm Diffusion Anisotropy Tensor MRI.4.4 Algorithm Diffusion Anisotropy Tensor MRI.

4.5 Algorithm DTI - Higher Resolution.4.5 Algorithm DTI - Higher Resolution.


5. Implementation of Methods. (proof in the software)5. Implementation of Methods. (proof in the software)

6 .The Objective of this visit Brigham and women´s 6 .The Objective of this visit Brigham and women´s hospital – Harvard Medical School.hospital – Harvard Medical School.

6.16.1 Collaboration with the group of BWH's work in publications.Collaboration with the group of BWH's work in publications.

6.2 Training in the acquisition, processing, analysis, and application of 6.2 Training in the acquisition, processing, analysis, and application of Diffusion Tensor Imaging.Diffusion Tensor Imaging.

6.3 6.3 Future works with the group of BWH. – HMS.Future works with the group of BWH. – HMS.



1.1. State of the artState of the art..

Stejskal, E. O., and Tanner, J.E. Spin-diffusion measurements: spin echoes Stejskal, E. O., and Tanner, J.E. Spin-diffusion measurements: spin echoes in the presence of a time-dependent field gradientin the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288-92. . J. Chem. Phys. 42, 288-92. (1965).(1965).

Difusión en imagen de resonancia magnética.Difusión en imagen de resonancia magnética. It was introduced for LeBihan en It was introduced for LeBihan en 1985. Art. – Le Bihan D, Breton E. Imagerie de diffusion in vivo par résonance 1985. Art. – Le Bihan D, Breton E. Imagerie de diffusion in vivo par résonance magnétique nucléaire. CR Acad Sci Paris 1985;301:1109-1112.magnétique nucléaire. CR Acad Sci Paris 1985;301:1109-1112.

Difusion Tensor by Basser et al (Mattiello J. Le Bihan) Diffusion tensor echo-Difusion Tensor by Basser et al (Mattiello J. Le Bihan) Diffusion tensor echo-planar imaging of human brain. In proceedings of the SMRMplanar imaging of human brain. In proceedings of the SMRM, Estimation of the , Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson 1994;effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson 1994;

Diffusion Tensor Imaging: Concepts and Applications;Diffusion Tensor Imaging: Concepts and Applications; (Denis Le Bihan, Jean (Denis Le Bihan, Jean Francois Mangin, Cyril Poupon, Chris A. Clark. Journal of Magnetic Resonance Francois Mangin, Cyril Poupon, Chris A. Clark. Journal of Magnetic Resonance Imaging (2001).Imaging (2001).



1.1. State of the art.State of the art. Diffusion Tensor Imaging – Image Acquisition and Processing Tools.Diffusion Tensor Imaging – Image Acquisition and Processing Tools. Surgical Surgical

Planning Laboratory, Technical Report # 354. Martha E. Shenton, Ph.D., Marek Planning Laboratory, Technical Report # 354. Martha E. Shenton, Ph.D., Marek Kubicki, M.D., Ph.D., Robert W. McCarley, M.D. Kubicki, M.D., Ph.D., Robert W. McCarley, M.D.

An Analysis Tools for Quantification of Diffusion Tensor MRI DataAn Analysis Tools for Quantification of Diffusion Tensor MRI Data. Hae-Jeong . Hae-Jeong Park, Martha E. Shenton, Carl-Fredrik Westin. Division of Nuclear Medicine, Dept. Park, Martha E. Shenton, Carl-Fredrik Westin. Division of Nuclear Medicine, Dept. of Diagnostic Radiology, Yonsei University, Colege of Medicine, Shinchon-dong, of Diagnostic Radiology, Yonsei University, Colege of Medicine, Shinchon-dong, Seodaemun-gu, Seoul 120-749, Korea. Laboratory of Mathematics in Imaging, Seodaemun-gu, Seoul 120-749, Korea. Laboratory of Mathematics in Imaging, Dept. of Radiology, Brigham and Women’s Hospital Harvard Medical School Dept. of Radiology, Brigham and Women’s Hospital Harvard Medical School Boston – USA.Boston – USA.

DTI and MTR abnormalities in schizophrenia: Analysis of white matter DTI and MTR abnormalities in schizophrenia: Analysis of white matter integrity.integrity.

M. Kubicki et al. Neuroimagen 25 (2005) 1109-1118. M. Kubicki et al. Neuroimagen 25 (2005) 1109-1118.

Molecular Diffusion in MRI


1.1. State of the art.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, 11:221-236, 1996.

J. Weickert. A review of nonlinear diffusion ltering. Scale-Space Theory in Computer Vision, Lecture Notes in Comp. Science (Springer, Berlin), 1252:3-28,1997. 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, 29:845-866,1992.



1.1. State of the art.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.



)(2 L

DIFUSIÓN

Difusión Restringida(ANISOTROPíA)

Difusión Libre(ISOTROPÍA)

trrD ˆˆ6

1

)(ˆ rCDJ


DIFUSIÓNDIFUSIÓN

bDeDA )(It does not obtain direction

equal loss of sign.

3)( 2 t

TtGb


Methods.Methods.


4.1 Algorithm RF Inhomogeneity Correction Algorithm in MRI.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. 1–8, 2004. Publisher: Springer-Verlag Berlin Heidelberg 2004


Molecular Diffusion in MRIMolecular Diffusion in MRIMethods.Methods.4.2 Algorithm Image Registration Classification.4.2 Algorithm Image Registration Classification.

Registration criteria.Registration criteria.- Quantitative measure of a “good match”.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




Methods.Methods.4.2 Algorithm Image Registration Classification.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

Methods.Methods.

4.2 4.2 Registration Framework Components


Registration Framework

Image Similarity

Metric

Cost Function

Optimizer

Transform

Image Interpolator

Resample Image

Filter

Fixedimage

Movingimage

TransformParameters

Resampled image

Methods.Methods.4.2 Algorithm Image Registration Classification.4.2 Algorithm Image Registration Classification.TransfromTransfrom

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


Molecular Diffusion in MRIMolecular Diffusion in MRI Methods.Methods. 4.2 Algorithm Image Registration Classification.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


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:

txx '

tp

10

01J



Euler2D Tranform:

Represents a rotation and translation in 2D

Parameters:

Jacobian in 2D:

y

x

t

t

y

x

y

x

cossin

sincos

'

'

yx ttp ,,

10)(sin)(cos

01)(cossin

yx

yxJ



Euler3D Tranform: Represents 3D rotation and translation - Rotation about each coordinate axis.

Parameters:

txRRRx

txRRRx

yxz

xyz

'

'

zyxzyx tttp ,,,,,


Methods.Methods.

4.3 Algorithm Diffusion Isotropy – Image derivates.4.3 Algorithm Diffusion Isotropy – Image derivates.

• The derivative of the image with respect to the variable is written The derivative of the image with respect to the variable is written

• For vector-valued images , we have andFor vector-valued images , we have and

The derivation of a scalar image with respect to its spatial coordinates is The derivation of a scalar image with respect to its spatial coordinates is called the image gradient and is noted by :called the image gradient and is noted by :

I

a

II a

I na RxI )(

T

n

aaa a

IIII

,....,, 21

I x

I


Methods.Methods.

4.3 Algorithm Diffusion Isotropy 4.3 Algorithm Diffusion Isotropy

It is for 2D images (p = 2) and 3D volumes (p = 3) :

when p = 3when p = 3


Tyx III , when p = 2when p = 2

Tzyx IIII ,,



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

It

I

noisyI G t2

GII noisyt * dvduvuGvyuxIyxI noisyt ,,),(


Methods.Methods.

4.3 Algorithm Diffusion Isotropy 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 :


2

22

2 2exp

2

1

yx

G

t2

IdivIt

I


Molecular Diffusion in MRIMethods.

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.

GIg *GI * I

IGIgdivt

I

*

2

22

2 2exp

2

1

yx

G


Molecular Diffusion in MRIMethods.

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:

IDdivt

I



Methods.Methods.4.4 Algorithm Diffusion Anisotropy Tensor MRI.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:

TUTUT

321 ,, diagT

U

T

321 Tr

2

322

21

232

231

221

2

FA

321321 /27 VR



Methods.Methods.

4.2 Algorithm Diffusion Anisotropy Tensor MRI.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 PDE´s.

- 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.

ldiagD

l

l

lil VRFADdivt

,....,,



Methods.Methods.

4.2 Algorithm DTI – Multiresolution Approximations and Their Associated 4.2 Algorithm DTI – Multiresolution Approximations and Their Associated Wavelets.Wavelets.

• There is a class of DWT that can be implemented using extremely efficient There is a class of DWT that can be implemented using extremely efficient algorithms. [Aldroubi - S.Mallat]. algorithms. [Aldroubi - S.Mallat].

• These types of wavelet transforms are associated with mathematical These types of wavelet transforms are associated with mathematical structures called multiresolution approximations of (MRA).structures called multiresolution approximations of (MRA).

• A multiresolution approximation of is a set of spaces that are A multiresolution approximation of is a set of spaces that are generated by dilating and translating a single function .generated by dilating and translating a single function .


2L

2L

zjjV

t

zk

jjj lctkjkcV 2),(,)(


Methods. Methods. Multiresolution Approximations and Their Associated Wavelets.Multiresolution Approximations and Their Associated Wavelets.

Where are the dilations (or reductions) and translations ofWhere are the dilations (or reductions) and translations of

The function called the scaling function. Moreover, for fixed the set The function called the scaling function. Moreover, for fixed the set

is requered to form an unconditional basis of .is requered to form an unconditional basis of .

If the funcctions form an orthogonal basis of . Then we call If the funcctions form an orthogonal basis of . Then we call an orthogonal scaling function.an orthogonal scaling function.

• The spaces are required to satisfy the additional properties:


ktt jjkj 22 2/

,

t j

Zktkj ),(, jV

Zktkj ),(, jV )(t

jV

....;)......( 101 VVVi

);()( 20 RLofsubspaceclosedaisVii

;)( 2LVUClosViii jzj

.0)( jzjVViv


Methods. 4.2 Methods. 4.2 Multiresolution Approximations and Their Associated Wavelets.Multiresolution Approximations and Their Associated Wavelets.

Properties (i) – (iv), the scaling function that is used to generated the MRA cannot Properties (i) – (iv), the scaling function that is used to generated the MRA cannot be chosen arbitrarily. be chosen arbitrarily.

In fact since and since In fact since and since

.. Conclude that the generating function must be a linear combinationConclude that the generating function must be a linear combination

of the basis :of the basis :

This last relation is often called the two-scale relation or the refinement equation, and 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 the sequence is the generating sequence which is crucial in the implementation of the DWT associated with multiresolutions.of the DWT associated with multiresolutions.

)(t

01 VV 01

2/1 )()2/(2 VtandVt

)2/(2 2/1 t

0Vofzkkt

zk

ktkht )(22/ 2/1

)(kh


PPT

Health & Medicine

algorithm diffusion

project molecular diffusion

mri methods

algorithm registration

mri state

spindiffusion measurements

image processing

optimization algorithm