CN110533758B - Brain fiber asymmetric reconstruction method based on hydrodynamic differential equation - Google Patents

Abstract

A brain fiber asymmetric reconstruction method based on a hydrodynamic differential equation, which regards fiber bundles as flow bundles and models fiber reconstruction by introducing related concepts in hydrodynamic; describing the continuity of the fiber within the voxel in terms of a divergence concept in the fluid mechanics; describing the spatial consistency of the fiber bundles by extending the concept of divergence between voxels; and calculating the microstructure reconstruction of the asymmetric brain fiber through the optimization of the two constraint conditions. The experiment is respectively compared with the current popular fiber reconstruction method in the aspects of simulated magnetic resonance data and actual clinical data, and the experimental result proves that the method provided by the invention improves the accuracy of fiber reconstruction.

Description

Brain fiber asymmetric reconstruction method based on hydrodynamic differential equation

Technical neighborhood

The invention relates to medical imaging and neuroanatomical neighborhood under computer graphics, in particular to a brain fiber microstructure reconstruction method.

Background

The brain is a comprehensive organ for controlling the complex functional activities of human beings such as logic thinking, learning and memory, movement, emotion and the like, and the exploration of the working mechanism of the human brain is the front edge and hot spot of the current scientific research. The brain white matter fiber reconstruction technology is a unique non-invasive method for imaging the fiber tissue microstructure in the living brain at present by imaging the brain fiber space microstructure information with anatomical significance, and has become an important technical means for brain science researches such as brain cognitive mechanism exploration, nerve disease pathological analysis, brain operation navigation and the like.

Reconstruction of brain fiber microstructure is a fundamental step in brain fiber imaging, providing accurate fiber direction estimation for fiber bundle tracking. The traditional reconstruction method usually only depends on the monosomic information to reconstruct the microstructure of the voxels, and the model type is basically a centrosymmetric model, so that the accuracy of fiber direction reconstruction is limited. It is a hot spot of research to propose new, more accurate brain fiber microstructure reconstruction models.

Disclosure of Invention

In order to solve the problems of low dependence and low precision of the existing brain fiber microstructure reconstruction method on single element information and central symmetry of a model, the invention provides an asymmetric brain fiber microstructure reconstruction method based on a hydrodynamic differential equation by combining neighborhood information of voxels.

The technical scheme adopted for solving the technical problems is as follows:

a method for asymmetrically reconstructing brain fibers based on a hydrodynamic differential equation, comprising the steps of:

step one, asymmetric fiber trajectory distribution (fiber trajectory distribution, FTD) function:

considering the fiber bundles as a stream, the stream consists of a series of sets of streamlines s= { S _i I=1, ····, n, any point in space (x, y, z) is the flow field at that point, the fiber direction at any point within a voxel is represented by a flow field:

FTD is represented as a flow field distribution throughout the voxel, approximated by a set of ternary quadratic polynomials:

υ(x,y,z)＝AC(x,y,z) (2)

wherein the coefficient matrix a is defined as follows:

c (x, y, z) = [ x ] denoted by C ² ,y ² ,z ² ,xy,xz,yz,x,y,z,1] ^T ；

Combining the spatial continuity of the fiber bundles on the neighborhood, wherein the process is as follows:

2.1 continuity constraints for intra-voxel FTD

Assuming that the diffusion displacement of water molecules in the same fiber bundle remains continuous, using the continuous incompressible fluid theory, the spatial continuity of the fiber trajectory is described by introducing the concept of divergence of the fiber flow on the diffusion tensor vector field:

when the FTD does not belong to the start or end region of the nerve fiber bundle, divΩ satisfies the following formula:

divΩ＝0 (5)

the simultaneous formulas (2), (3), (4) and (5) are obtained:

2.2 spatial continuity constraints between FTD voxels while equation (6) ensures that the FTD within a voxel meets continuity, but does not indicate inter-voxel fiber bundle continuity, the same fiber bundle should be consistent between adjacent voxels, i.e., the corresponding FTD should meet inter-voxel continuity, an FTD inter-voxel consistency function is proposed to characterize fiber bundle continuity between adjacent voxels, assuming that the voxel is a unit cube, N _c ＝(c ⁰ ,c ¹ ,…,c ⁵ ) Representing six voxels adjacent to the center voxel c, denoted by A _c ＝(A ⁰ ,A ¹ ,…,A ⁵ ) FTD coefficients representing neighborhood voxels, for adjacent voxels c and c traversed by the stream ⁱ The continuity function divΓ (x, y, z) of any point of its intersection is defined as follows:

wherein Γ is the adjacent voxel connecting surface through which the flow passes, S is the flow, and v ⁱ C and c are respectively ⁱ FTD of (c) ⁱ Mapping to the same coordinate system as c:

wherein a is _jk And

a and A respectively ⁱ The continuity function div ψ over the entire plane Γ is the area integral of divΓ (x, y, z):

each voxel has six adjacent voxels, and the continuity function among the voxels is the sum of the continuity functions of six connecting surfaces:

the combined formula (2) -formula (9) is obtained:

wherein a is _jk Elements in FTD coefficient matrix a representing the center voxel,

FTD coefficient matrix A representing the ith voxel of the neighborhood ⁱ Elements of (a) and (b);

step three, calculating FTD

Calculating the FTD by minimizing intra-voxel and inter-voxel continuity functions aims at making the fiber trajectory distribution most closely to the fiber direction distribution function (fiber orientation distribution, FOD), and the coefficient matrix a in the FTD can be calculated by optimizing the cost function as follows:

where Φ (v (x, y, z)) is the probability of FOD at point (x, y, z), we take the 26 neighborhood c= [ C ] of the central voxel for simplicity of calculation ₁ ,c ₂ ,…,c ₂₆ ]Peak p= [ P ] of middle FOD ₁ ,p ₂ ,…,p ₂₆ ]As an approximation of Φ (v (x, y, z)), the formula (11) is simplified as follows:

after the flow field coefficient A is obtained, the FTD of the voxel is obtained.

The beneficial effects of the invention are as follows: the accuracy of the fiber reconstruction is improved.

Drawings

FIG. 1 is a schematic diagram of a center voxel and a neighborhood voxel.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

Referring to fig. 1, a method for asymmetrically reconstructing brain fibers based on a hydrodynamic differential equation includes the steps of:

step one, asymmetric fiber trajectory distribution (fiber trajectory distribution, FTD) function:

considering the fiber bundles as a stream, the stream consists of a series of sets of streamlines s= { S _i I=1, ····, n, any point in space (x, y, z) is the flow field at that point, the fiber direction at any point within a voxel is represented by a flow field:

FTD is represented as a flow field distribution throughout the voxel, approximated by a set of ternary quadratic polynomials:

υ(x,y,z)＝AC(x,y,z) (2)

wherein the coefficient matrix a is defined as follows:

c (x, y, z) = [ x ] denoted by C ² ,y ² ,z ² ,xy,xz,yz,x,y,z,1] ^T ；

Combining the spatial continuity of the fiber bundles on the neighborhood,

since during dMRI data acquisition, the brain nerve fibers can be seen as unchanged, corresponding to related concepts in fluid mechanics, i.e. the stream is seen as an incompressible constant stream; the process is as follows:

2.1 continuity constraints for intra-voxel FTD

Assuming that the diffusion displacement of water molecules in the same fiber bundle remains continuous, using the continuous incompressible fluid theory, the spatial continuity of the fiber trajectory is described by introducing the concept of divergence of the fiber flow on the diffusion tensor vector field:

when the FTD does not belong to the start or end region of the nerve fiber bundle, divΩ satisfies the following formula:

divΩ＝0 (5)

the simultaneous formulas (2), (3), (4) and (5) are obtained:

2.2 spatial continuity constraints between FTD voxels while equation (6) ensures that the FTD within a voxel meets continuity, but does not indicate inter-voxel fiber bundle continuity, the same fiber bundle should be consistent between adjacent voxels, i.e., the corresponding FTD should meet inter-voxel continuity, an FTD inter-voxel consistency function is proposed to characterize fiber bundle continuity between adjacent voxels, assuming that the voxel is a unit cube, N _c ＝(c ⁰ ,c ¹ ,…,c ⁵ ) Representing six voxels adjacent to the center voxel c, denoted by A _c ＝(A ⁰ ,A ¹ ,…,A ⁵ ) FTD coefficients representing neighborhood voxels, for adjacent voxels c and c traversed by the stream ⁱ The continuity function divΓ (x, y, z) of any point of its intersection) The definition is as follows:

wherein Γ is the adjacent voxel connecting surface through which the flow passes, S is the flow, and v ⁱ C and c are respectively ⁱ FTD of (c) ⁱ Mapping to the same coordinate system as c:

wherein a is _jk And

a and A respectively ⁱ The continuity function div ψ over the entire plane Γ is the area integral of divΓ (x, y, z):

each voxel has six adjacent voxels, and the continuity function among the voxels is the sum of the continuity functions of six connecting surfaces:

the combined formula (2) -formula (9) is obtained:

wherein a is _jk Elements in FTD coefficient matrix a representing the center voxel,

FTD coefficient matrix A representing the ith voxel of the neighborhood ⁱ Elements of (a) and (b);

step three, calculating FTD

Calculating the FTD by minimizing intra-voxel and inter-voxel continuity functions aims at making the fiber trajectory distribution most closely to the fiber direction distribution function (fiber orientation distribution, FOD), and the coefficient matrix a in the FTD can be calculated by optimizing the cost function as follows:

where Φ (v (x, y, z)) is the probability of FOD at point (x, y, z), we take the 26 neighborhood c= [ C ] of the central voxel for simplicity of calculation ₁ ,c ₂ ,…,c ₂₆ ]Peak p= [ P ] of middle FOD ₁ ,p ₂ ,…,p ₂₆ ]As an approximation of Φ (v (x, y, z)), the formula (11) is simplified as follows:

after the flow field coefficient A is obtained, the FTD of the voxel is obtained.

Claims (1)

1. A method for asymmetric reconstruction of brain fibers based on hydrodynamic differential equations, the method comprising the steps of:

step one, an asymmetric fiber track distribution FTD function:

considering the fiber bundles as a stream, the stream consists of a series of sets of streamlines s= { S _i I=1, ····, n, any point in space (x, y, z) is the flow field at that point, the fiber direction at any point within a voxel is represented by a flow field:

FTD is represented as a flow field distribution throughout the voxel, approximated by a set of ternary quadratic polynomials:

υ(x,y,z)＝AC(x,y,z) (2)

wherein the coefficient matrix a is defined as follows:

c (x, y, z) = [ x ] denoted by C ² ,y ² ,z ² ,xy,xz,yz,x,y,z,1] ^T ；

Combining the spatial continuity of the fiber bundles on the neighborhood, wherein the process is as follows:

2.1 continuity constraints for intra-voxel FTD

Assuming that the diffusion displacement of water molecules in the same fiber bundle remains continuous, using the continuous incompressible fluid theory, the spatial continuity of the fiber trajectory is described by introducing the concept of divergence of the fiber flow on the diffusion tensor vector field:

when the FTD does not belong to the start or end region of the nerve fiber bundle, divΩ satisfies the following formula:

divΩ＝0 (5)

the simultaneous formulas (2), (3), (4) and (5) are obtained:

2.2 spatial continuity constraints between FTD voxels

Although formula (6) can ensure that the FTD in the voxels meets the continuity, but can not represent the fiber bundle continuity among the voxels, the same fiber bundle is consistent among the adjacent voxels, namely the corresponding FTD should meet the continuity among the voxels, a consistency function among the FTD voxels is proposed to characterize the continuity of the fiber bundle among the adjacent voxels, and the voxels are assumed to be unit cubes, and N is _c ＝(c ⁰ ,c ¹ ,…,c ⁵ ) Representing that the central voxels c are adjacentSix voxels, denoted A _c ＝(A ⁰ ,A ¹ ,…,A ⁵ ) FTD coefficients representing neighborhood voxels, for adjacent voxels c and c traversed by the stream ⁱ The continuity function divΓ (x, y, z) of any point of its intersection is defined as follows:

wherein Γ is the adjacent voxel connecting surface through which the flow passes, S is the flow, and v ⁱ C and c are respectively ⁱ FTD of (c) ⁱ Mapping to the same coordinate system as c:

wherein a is _jk And

a and A respectively ⁱ I=0, …,5; j=1, 2,3; k=0, …,9, the continuity function div ψ over the whole plane Γ is the area integral of divΓ (x, y, z):

each voxel has six adjacent voxels, and the continuity function among the voxels is the sum of the continuity functions of six connecting surfaces:

the combined formula (2) -formula (9) is obtained:

wherein a is _jk Elements in FTD coefficient matrix a representing the center voxel,

FTD coefficient matrix A representing the ith voxel of the neighborhood ⁱ Elements of (a) and (b);

step three, calculating FTD

Calculating FTD by minimizing intra-voxel and inter-voxel continuity functions aims at making the fiber track distribution most closely to the fiber direction distribution function FOD, and the coefficient matrix a in FTD is calculated by optimizing the cost function:

where Φ (v (x, y, z)) is the probability that FOD is at point (x, y, z), the 26 neighborhood c= [ C ] of the center voxel is taken ₁ ,c ₂ ,…,c ₂₆ ]Peak p= [ P ] of middle FOD ₁ ,p ₂ ,…,p ₂₆ ]As an approximation of Φ (v (x, y, z)), the formula (11) is simplified as follows:

after the flow field coefficient A is obtained, the FTD of the voxel is obtained.

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