# Patent

## Title
Method of analyzing diffusion-weighted magnetic resonance image

## Abstract

A method of analyzing a diffusion-weighted magnetic resonance image is adapted to analyze an orientation distribution function ψ0 of nerve fibers at a vector in space. The method includes the following steps: (A) finding an orientation {circumflex over (μ)}a with the largest probability in the ODF ψ0; (B) maximizing a single fiber ODF ψ′; (C) analyzing the maximized single fiber ODF ψ′ to obtain quantitative information; (D) deducting the maximized single fiber ODF ψ′ obtained in step (B) from the entire ODF ψ0; and (E) stopping computation if the result obtained in step (D) is determined to be smaller than a predetermined value, and repeating steps (A), (B), (C), and (D) if otherwise.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
This application claims priority of Taiwanese Application No. 096125046, filed on Jul. 10, 2007.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to a method of analyzing information, more particularly to a method of qualitatively and quantitatively analyzing a diffusion-weighted magnetic resonance image.
2. Description of the Related Art
Magnetic resonance imaging (MRI) is a non-invasive imaging method for examining organs and tissues of the human body. It works on the principle that magnetic fields can change the direction of magnetic spin of hydrogen atoms in a living body, and excite resonance of hydrogen atom nuclei, and it uses a computer to analyze electromagnetic signals to create images. Since about 70% of the human body is water (H2O), which contains a large amount of hydrogen atom nuclei, MRI has been widely applied to clinical imaging.
Diffusion-weighted MR images (hereinafter referred to as DWI), which are capable of revealing detailed structures of tissue fibers, are realized through applying magnetic gradient pulses and receiving signals responding to diffusion of water molecules. Since diffusion of water molecules in tissues is mainly influenced by tissue structures, diffusion characteristics of water molecules can be used to estimate detailed structures of tissue fibers.
To analyze DWI, diffusion tensor imaging (DTI) techniques utilizing tensor analysis have been developed. Living organs are composed of non-homogeneous tissues, where the diffusion of water molecules exhibits an anisotropic pattern. Conventional diffusion coefficients can hardly reflect the diffusion characteristics of the water molecules clearly. Generally, magnetic gradients are applied in six specific directions to construct six diffusion-weighted images, along with one null image (imaging without magnetic gradient), which are analyzed to thereby obtain a tensor matrix. In DTI, a symmetric three-dimensional tensor matrix is used to describe the diffusion strength, and a tissue structure is inferred and constructed through eigen analysis.
A current method of improvement, known as high angular resolution diffusion imaging (HARDI), is to apply magnetic gradients in more directions during sampling. However, for the cerebral tissues of a human brain, the nerve fibers are largely intertwined, and it is difficult to discern the orientations of the fibers at fiber intersections. This manifests the fact that better analysis results cannot be obtained due to insufficient degree of freedom of the DTI matrix.
By further calculating HARDI information, an orientation distribution function (ODF) of a fiber at any one point in space can be obtained. However, techniques for qualitative and quantitative analysis of ODF are required if the number and orientations of the fibers also need to be known.
A current technical accomplishment is primarily concerned with finding local maximum values in ODFs to serve as orientations of the fibers. However, for ODFs at fiber intersections, if an included angle between two intersecting fibers is relatively small or if there are less branching fibers, such a method cannot be used to clearly analyze less conspicuous fibers. Therefore, with respect to qualitative and quantitative techniques for analyzing ODFs, further researches and developments are required.
SUMMARY OF THE INVENTION
Therefore, the object of the present invention is to provide a method of qualitatively and quantitatively analyzing diffusion-weighted magnetic resonance images in order to reveal intravoxel fibers.
Accordingly, a method of analyzing a diffusion-weighted magnetic resonance image is adapted to analyze an orientation distribution function (ODF) ψ0 of a nerve fiber at a voxel, and comprises the following steps:
(A) finding an orientation {circumflex over (μ)}a with the largest probability in the entire ODF ψ0;
(B) finding a maximized single fiber ODF ψ′ along the orientation {circumflex over (μ)}a;
(C) analyzing the ODF ψ′ found in step (B) to obtain quantitative information;
(D) deducting the ODF ψ′ obtained in step (B) from the entire ODF ψ0; and
(E) stopping computation if the result obtained in step (D) is determined to be smaller than a predetermined value, and repeating steps (A), (B), (C), and (D) if otherwise.



BRIEF DESCRIPTION OF THE DRAWINGS
Other features and advantages of the present invention will become apparent in the following detailed description of the preferred embodiment with reference to the accompanying drawings, of which:
FIG. 1 is a flowchart of the preferred embodiment of a method of analyzing a diffusion-weighted magnetic resonance image according to the invention;
FIG. 2 is a schematic diagram showing a single fiber ODF ψ′;
FIG. 3 is a schematic diagram depicting three fibers that can be resolved from the entire ODF ψ0;
FIG. 4 illustrates the ODF found in a cortical spinal track; and
FIG. 5 illustrates exemplary analytical results of an ODF using the preferred embodiment.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Orientations of intersecting or crossing fibers in the DWI are difficult to be resolved. In the preferred embodiment, an ODF ψ0 of a voxel in space is first obtained by analyzing high angular resolution diffusion images (HARDIs). The ODF ψ0 will be such as that shown in FIG. 4 when presented in three dimensions. The purpose of analysis in this embodiment is to analyze the function ψ0 of the voxel, and to resolve the orientation and strength of principal fibers, as well as the orientations and strengths of other fibers in descending order of principality (refer to FIG. 3).
As shown in FIG. 1, the method according to this embodiment includes the following steps:
In step S1, a largest probability distribution orientation {circumflex over (μ)}a of the entire ODF ψ0 is found to serve as the orientation of the principal fiber, {circumflex over (μ)}a=arg max ψ0({circumflex over (μ)}).
In step S2, a maximized single fiber distribution function ψ′ along the orientation {circumflex over (μ)}a is found by analyzing the entire ODF ψ′ using the following steps:
In sub-step S21, a correlation between any two orientations {circumflex over (μ)}i, {circumflex over (μ)}j is set to be an absolute value of a dot product (inner product) thereof, thereby obtaining a correlation matrix

C=<Cij>, cij=|{circumflex over (μ)}i·{circumflex over (μ)}j|.

In sub-step S22, referring to FIG. 2, with the maximum probability orientation being {circumflex over (μ)}a, the single fiber ODF ψ′ is maximized, which satisfies the following three conditions:
1. If cia>cja, then ψ′({circumflex over (μ)}i)>ψ′({circumflex over (μ)}j). In other words, the larger the included angle, the smaller will be the correlation Cai, and the distribution probability of the orientation {circumflex over (μ)}i.
2. If cia=cja, then ψ′({circumflex over (μ)}i)=ψ′({circumflex over (μ)}j). In other words, since the single fiber distribution is axially symmetric, all orientations that form the same included angle with the orientation {circumflex over (μ)}a, i.e., orientations with equivalent correlations, should have equivalent distribution probabilities.
3. For any orientation {circumflex over (μ)}i, ψ′({circumflex over (μ)}i)≦ψ0({circumflex over (μ)}i) has to be satisfied. That is, the distribution probability of the single fiber ODF ψ′ should not be greater than the probability of the entire ODF ψ0.
In step S3, the maximized single fiber ODF ψ′ is further analyzed to obtain quantitative information. Content of the function ψ′ includes principal fiber orientations and probability information.
In step S4, with reference to FIG. 5, the single fiber ODF ψ′ is deducted from the entire ODF ψ0, i.e., the most evident fiber distribution is deducted therefrom to obtain the remaining ODF ψ1.
In step S5, a determination is made as to whether sum of the remaining ODF ψ1 is smaller than a predetermined value. In the negative, this indicates that there are remaining fibers, and ψ0 is substituted by ψ1 and steps S1 to S4 are repeated. In the affirmative, this indicates that there are no other fiber orientations with obvious distribution probability, and iterative computation should be stopped. For example, with reference to FIG. 5, supposing the total sum of an updated function ψ1 obtained in step S4 during the third iterative computation is smaller than the predetermined value, this indicates that there is a negligible remainder after deducting various fiber ODFs sequentially from the entire ODF ψ0. After several iterations, ψ0 may have three sets of fibers ψ′(1), ψ′(2), ψ′(3) (see FIG. 3).
In the above embodiment, the method of this invention is exemplified by merely performing iterative computations of a voxel.
In sum, the method of analyzing diffusion-weighted magnetic resonance images according to the present invention exploits the anisotropic and asymmetric characteristics of ODFs to resolve crossing fibers. Thus, qualitative and quantitative analysis of ODFs can be accurately conducted.
While the present invention has been described in connection with what is considered the most practical and preferred embodiment, it is understood that this invention is not limited to the disclosed embodiment but is intended to cover various arrangements included within the spirit and scope of the broadest interpretation so as to encompass all such modifications and equivalent arrangements.

## Claims

1. A method of analyzing diffusion-weighted magnetic resonance images, said method being adapted to analyze an ODF ψ0 of nerve fibers at a voxel in space, and comprising the following steps:
(A) finding an orientation {circumflex over (μ)}a with the largest probability in the ODF ψ0;
(B) maximizing a single fiber ODF ψ′;
(C) analyzing the maximized single fiber ODF ψ′ to obtain quantitative information;
(D) deducting the maximized single fiber ODF ψ′ obtained in step (B) from the entire ODF ψ0; and
(E) stopping computation if the result obtained in step (D) satisfies a termination condition, and substituting the result obtained in step (D) for the orientation distribution function ψ0 in step (A), and repeating steps (A) to (D) if otherwise.

2. The method as claimed in claim 1, wherein, the distribution function ψ′ is found in step (B) using the following sub-steps: (B1) setting a correlation between any two orientations {circumflex over (μ)}i, {circumflex over (μ)}j, in space to be an absolute value of a dot product thereof so as to obtain a correlation matrix C=<cij>, cij=|{circumflex over (μ)}i·{circumflex over (μ)}j|; and (B2) finding a maximized single fiber ODF which satisfies the following three conditions: if cia>cja, then ψ′({circumflex over (μ)}i)>ψ′({circumflex over (μ)}j); if cia=cja, then ψ′({circumflex over (μ)}i)=ψ′({circumflex over (μ)}j); and for any orientation {circumflex over (μ)}i, ψ′({circumflex over (μ)}i)≦ψ0({circumflex over (μ)}i) has to be satisfied.

3. The method as claimed in claim 1, wherein steps (A) to (D) are repeated a number of times until the total sum of the remaining ODF of the result obtained in step (D) is smaller than a predetermined value.

4. The method as claimed in claim 1, wherein the orientation distribution function ψ0 is obtained by analyzing high angular resolution diffusion-weighted images.

5. The method as claimed in claim 1, wherein the termination condition in step (E) is whether the result obtained in step (D) is smaller than a predetermined value, computation being stopped if the result is determined to be smaller than the predetermined value, steps (A), (B), (C), and (D) being repeated if otherwise.