CN108346150A - A kind of cortex thickness method of estimation based on atlas analysis - Google Patents

Abstract

Cortical thickness estimation from magnetic resonance imaging (MRI) is an important technique for studying neurodegenerative diseases of the brain. The present invention proposes a method for estimating the thickness of the cerebral cortex based on atlas analysis, which can effectively capture the geometric changes of the cerebral cortex. First, the method approximates the shape of the cerebral cortex in MRI by constructing a tetrahedral mesh model and reflects its intrinsic geometric properties; The distribution of the internal temperature field; finally, construct the heat transfer probability representation system represented by the eigenvalue and eigenvector of the Laplace-Beltramin operator through the mechanism of map analysis, and track the maximum heat transfer probability to determine the gradient field line in the cerebral cortex temperature field The direction and length of the cortex are finally obtained to obtain the thickness information of the cerebral cortex.

Description

A method for estimating cerebral cortex thickness based on atlas analysis

technical field

The invention belongs to the field of computer application technology, relates to MRI cerebral cortex morphology change detection technology, and is used for early diagnosis of neurodegenerative diseases such as AD.

Background technique

Alzheimer's disease (AD) has become a neurodegenerative disease that seriously endangers the health of the elderly and seriously threatens human health. In order to develop effective treatments to hinder the progression of the disease, it is necessary to accurately diagnose the mild cognitive impairment (MCI) population that is easily transformed into AD at an early stage, and reveal the development trend of the disease according to its evolutionary trajectory. Since the appearance of AD symptoms starts in a local area, it is manifested as a thinning of the thickness of the cerebral cortex in the local area, and then this phenomenon spreads to other parts of the brain. Therefore, many researchers regard the thickness change of cerebral cortex as the auxiliary diagnosis basis of AD disease. For this reason, it has been an important research topic in this field to extract the thickness information of the cerebral cortex in a local area by using a suitable detection method.

Magnetic resonance imaging (MRI) technology can display the distribution of physical parameters (such as density) of the human brain in space, and obtain tomographic images and three-dimensional volume images in any direction, so it has been widely used in the early diagnosis of neurodegenerative diseases such as AD and reveal its evolution. This technology combined with computer graphics technology can quantify and evaluate the small morphological changes of the cerebral cortex, and is an analytical method for clinically revealing the geometric structure changes of the cerebral cortex without damage. However, the quantitative analysis of MRI cortical morphological changes still faces the following core problems:

1) How to use geometric morphometric properties to generate a tetrahedral mesh model of the cerebral cortex that maintains the original MRI image data. Reduce the fitting error caused by approximating MRI image data with cube grids in the past;

2) How to improve the accuracy and robustness of cortical thickness calculation on the basis of the original Lagrangian method with the help of the essential geometric characteristics of the cerebral cortex embodied in the map theory. Reduce the calculation error caused by relying solely on the Lagrangian method to solve the thickness information.

Contents of the invention

The purpose of the present invention is to provide a method for estimating the thickness of cerebral cortex based on the atlas analysis theory, which is characterized by constructing the tetrahedral grid representation form of the cerebral cortex nuclear magnetic resonance (MRI) image and calculating the heat conduction law in the harmonic energy field. To capture geometric information in the MRI cerebral cortex, the specific steps are described as follows:

The first step: Cube grid generation: On the basis of maintaining the boundary shape, generate a background grid composed of cube units; use the fast marching method based on the vertex connection relationship to calculate the spatial position attributes of the background cube grid points with symbolic features , through the probability distribution of the spatial position of the grid points near the edge surface, the algorithm for adaptively adjusting the grid side length is designed to achieve the best degree of fitting to the cerebral cortex;

Step 2: Tetrahedral meshing: Tetrahedral meshing is carried out on the edge face and internal vertex background mesh respectively in the form of pyramid splitting and body-centered cubic lattice, and then the four faces after splitting The volume grid is used to cut and organize the zero value surface;

Step 3: Tetrahedral grid optimization: Using the three-dimensional deformation description theory based on the spatial displacement gradient tensor, design measurement strategies for inductive tetrahedral unit micro-deformation and grid optimization such as elastic modulus, smooth modulus, and fidelity modulus ; Construct a minimum harmonic energy model and solution algorithm that takes into account both external shape preservation and internal structure optimization of the cerebral cortex, and optimize the tetrahedral grid structure;

Step 4: Construct the Laplace-Beltramin operator: the harmonic energy expression defined on the Riemannian manifold M start with Respectively represent the grid vertex positions on the Riemannian manifold M, It is a function expression defined on the vertices of the mesh; combining the vector expressions of each surface of the tetrahedron unit and the coordinate expression of the center of gravity of any point in the body, the geometry between the vertices of the tetrahedron with the side length and the corresponding dihedral angle variable is derived Constraint relationship ; Combining with the differential equation of heat conduction in three-dimensional space, under the given Dirichlet condition (fixing the temperature of the inner and outer surfaces of the cerebral cortex), the Laplace-Beltramin operator is constructed The discrete expression of;

Step 5: Calculating the internal temperature field distribution of the cerebral cortex: first construct the Dirichlet boundary condition of the cerebral cortex; define the entire tetrahedral mesh as a limited solution space under the condition of thermal equilibrium, and use the finite element method to solve the heat conduction equation: ,in is the Laplace operator, the subscript M is the entire tetrahedral grid space, is the spatial position and time As the temperature function of the variable, the above partial differential equation is transformed into a linear system to solve the temperature field distribution inside the cerebral cortex;

Step 6: Calculate the thickness of the cerebral cortex: by studying the physical meaning of the Brownian motion probability density function on the Riemannian manifold of the cerebral cortex, construct the eigenvalue and eigenvector representation of the Laplace-Beltramin operator determined by the local geometric structure of the cerebral cortex The heat transfer probability characterization system; and calculate the gradient curve of heat conduction based on the maximum heat transfer probability between two adjacent isothermal layers, and finally obtain the thickness information of the inner and outer surfaces of the cerebral cortex.

This method can effectively capture the geometric changes of the cerebral cortex, and can assist in the early diagnosis of mild cognitive impairment (MCI) who are easily transformed into AD symptoms, and reveal the development trend of the disease according to its evolution trajectory.

Description of drawings

Figure 1. Cube grid generation results. Figure 1(a) is the original sphere image, and Figure 1(b) is the result after filling with cube voxels while maintaining the edge shape of the sphere.

Figure 2 Three templates for reconstructing a new tetrahedron in the remaining geometry after cutting off the vertex indicated by "+". The large circles with thick lines indicate that the vertex is located in the inner, outer and edge of the cerebral cortex in the initial state, respectively.

Fig. 3 (a) is an image of the cerebral cortex obtained based on the tetrahedral mesh generation algorithm of the present invention; Fig. 3 (b) is a cross-sectional view of the tetrahedral mesh of the cerebral cortex, which is filled with 154,908 tetrahedrons; Fig. 3 (c ) is the distribution range of dihedral angles of all tetrahedrons in the generated tetrahedral grid; Figure 3(d) is the distribution range of quality coefficients of all tetrahedrons in the generated tetrahedral grid.

Figure 4 Schematic diagram of tetrahedron unit. Usually in a tetrahedral unit, called edge with edges and dihedral The positional relationship is relative, for the edge length.

Figure 5(a) is a composite image, the whole is a cubic structure with a spherical cavity in the middle. After filling tetrahedrons between the outer surface of the cube and the inner surface of the sphere, a tetrahedral grid structure as shown in the figure is formed. Figure 5(b) is the isothermal surface cut out from the tetrahedral mesh (from left to right, from top to down, respectively 0.4 degrees -0.9 degrees). Figure 5(c) is the temperature flow field lines between the outer surface and the inner surface. Figure 5(d) is the thickness information from the outer surface to the inner surface obtained according to the length of the temperature flow field line.

Fig. 6 is using the method of the present invention and the FreeSurfer method (Fischl, B., Dale, A.M., 2000. Measuring the thickness of the human cerebral cortex from magnetic resonance images. Proc. Natl. Acad. Sci. USA 97, 11050-11055.) The projection map of the significant difference between the three groups of cerebral cortex thickness of Alzheimer's disease patients (AD), mild cognitive impairment patients (MCI) and normal people (CTL) under statistical probability. Among them, Fig. 6(a) and Fig. 6(b) are the significant difference mapping diagrams of AD and CTL groups obtained by the method of the present invention and the FreeSurfer method respectively; Fig. 6(c) and Fig. 6(d) are respectively The mapping map of the significant difference between the MCI and CTL groups obtained by the method and the FreeSurfer method. Here, the smaller the p value, the more significant the difference in thickness between groups.

Detailed ways

The purpose of the present invention is to provide a method for estimating the thickness of cerebral cortex based on the atlas analysis theory, which is characterized by constructing the tetrahedral grid representation form of the cerebral cortex nuclear magnetic resonance (MRI) image and calculating the heat conduction law in the harmonic energy field. To capture geometric information in the MRI cerebral cortex, the specific steps are described as follows:

Step 1: Cube grid generation: Firstly, the cube grid is used to fill the MRI binary image, and the spatial attribute position of the grid vertices is determined by the distance function from the point to the edge As determined, the calculation of the distance function is obtained by the fast marching method based on the connection relationship of vertices, and the vertices are marked ( ) is the edge surface, by calculating the ratio of the number of vertices on the edge to the number of all vertices on the edge surface, the size of the side length of the background grid is adaptively adjusted; for example, the result of filling a sphere with cubic voxel units is shown in the figure 1 shown;

Step 2: Tetrahedral meshing: Use the form of pyramid and body-centered cubic lattice to divide the edge surface and internal vertices into tetrahedrons. Negative checksum and linear scale function zero-value surface (all of The surface connected by points) is cut and sorted. For some cases where the original tetrahedron structure collapses due to cutting, it is necessary to summarize some templates through the previous research work to reconstruct the new tetrahedron, as shown in Figure 2;

The third step: Tetrahedral grid optimization: use the harmonic energy function G ( U ) containing the vertex displacement vector to minimize the spatial position of each vertex, where U=xX , U is the displacement vector, x is the coordinate vector after fine-tuning, X is the original coordinate vector, G ( U ) is composed of penalty term, smooth term and fidelity term; according to the finite strain theory, the partial differential of the displacement vector relative to the spatial coordinate system is called the spatial displacement gradient tensor, namely: , where F is called the deformation gradient tensor, is the gradient operator, then the Green deformation tensor is , the eigenvalues of C form three physical quantities describing the degree of deformation: stretching amount, shearing amount, and volume; on this basis, the stretching amount penalty and shearing amount generated under large topological shape changes are generated Penalty and volume penalty item; the threshold condition for starting the penalty is mainly to measure the relationship between the stretching amount and the volume amount, the shear amount and the volume amount, and rely on the penalty item to maintain the generation of the tetrahedral mesh during each iteration Quality; the smooth item uses the difference between the position vector and the displacement vector of each tetrahedron as the input of the smooth function, and then optimizes the position of each vertex on the surface of the area and the internal tetrahedron by minimizing the finite differential method, eliminating the vertex coordinates Ups and downs; the fidelity item only acts on the vertices on the boundary during the algorithm process, that is, if the point on the boundary after iteration deviates too far from the original voxel position, a penalty will be imposed; the minimization of the harmonic energy function uses the Sobolev gradient algorithm Finally, a tetrahedral grid that maintains the topological shape of the boundary of the MRI image data and the overall smoothness and stability is generated; Figure 3 shows the grid image of the cerebral cortex obtained by using the tetrahedral grid generation algorithm in this paper. It can be seen from Figure 3 that due to the implementation of the network The grid edge preservation and structure optimization strategy ensures that the grid conforms to the fine topology of the original image, and at the same time improves the generation quality of each tetrahedron in the grid, providing accuracy for subsequent numerical calculations such as Laplace-Beltramin operators guarantee;

Step 4: Construct the Laplace-Beltramin operator: define the generated tetrahedral mesh as a limited solution area containing internal and edge vertices, and then calculate the local rigidity matrix of the tetrahedral mesh according to formula (1) :

(1)

in, is the vertex number in the tetrahedral mesh, is the discrete harmonic energy factor, and its calculation method is as follows: In a specific tetrahedral grid, assuming that the edge is shared by n tetrahedrons, then in each tetrahedron, the edge The opposite edges and dihedral angles are denoted as and , as shown in Figure 4, then in all n tetrahedrons, the edge All the opposite edges and dihedral angles can be written as ,but

(2)

Finally, use the formula (3) to get the Laplace-Beltramin operator under the Dirichlet boundary condition:

(3)

where D is a diagonal matrix defined as ;

Step 5: Calculating the internal temperature field distribution of the cerebral cortex: first assign the temperature of each vertex on the outer surface of the cerebral cortex to 1, and assign the temperature of each vertex on the inner surface to 0, so as to construct the Dirichlet boundary condition of the cerebral cortex; through the formula ( 3) The obtained Laplace-Beltramin operator As a functional operator to solve the stable diffusion distribution of heat in the harmonic energy field inside the cortex; use the following formula to solve the temperature field distribution inside the cortex:

(4)

in, Yes A vector representing n vertices on the inner surface of the cortex, Yes The vector of m represents the vertices of the inner and outer surfaces of the cerebral cortex; for example, in the composite image 5(a), after solving the temperature field between the inner and outer surfaces using formula (4), each isotherm cut out from the tetrahedral mesh noodle;

Step 6: Calculating the thickness of the cerebral cortex: First, according to the map analysis theory, construct the thermonuclear diffusion factor:

(5)

in, and represent the i-th eigenvalue and eigenvector of the Laplace-Beltramin operator, respectively, Indicates the probability density function of unit heat from x along the propagation path to y within the conduction time t, and its maximum probability density propagation path is the gradient direction of heat propagation; then according to the isothermal layer obtained in the fifth step, from the outer surface of the cerebral cortex Starting from a vertex x of , use the formula (5) to calculate the heat transfer probability density from x to each point in the next isothermal layer , search for the y point of the next isothermal layer corresponding to the maximum probability density, connect x and y is the heat propagation gradient direction of the two adjacent isothermal layers; then start from y, search for the maximum Heat probability density propagation path, and so on, until the inner surface, connect the maximum heat probability density propagation path between all isothermal layers, you can get the local thickness value from a point on the outer surface of the cerebral cortex to the inner surface, traverse all the outer surface At the vertex, the global thickness value of the cerebral cortex can be obtained by using the calculation method in the sixth step. Figure 5(c) is the heat transfer gradient curve calculated according to the maximum heat probability density using the map analysis theory; Figure 5(d) is the thickness information mapped to the outer surface according to the length of the temperature flow field line from the outer surface to the inner surface.

Claims (1)

1. A method for estimating the thickness of a cerebral cortex based on a spectrum analysis theory is characterized in that geometric information in the cerebral cortex of a brain cortex Magnetic Resonance (MRI) is captured by constructing a tetrahedral mesh representation form of the MRI image of the cerebral cortex and calculating a heat conduction rule in a harmonic energy field, and the specific steps are described as follows:

the first step is as follows: and (3) generating a cubic grid: generating a background grid consisting of cubic units on the basis of keeping the shape of the boundary; calculating the spatial position attribute of the grid point of the background cube with the symbolic feature by adopting a fast marching method based on the vertex connection relation, and designing an operation rule for adaptively adjusting the side length of the grid through the probability distribution of the spatial position of the grid point near the edge surface so as to achieve the optimal degree of fitting the cerebral cortex;

a second step of tetrahedral mesh division, which is to respectively perform tetrahedral division on the background meshes at the edge surface and the internal vertex by adopting a pyramid division form and a body-centered cubic lattice division form, and then perform zero surface cutting and finishing on the divided tetrahedral meshes;

the third step: and (3) tetrahedral mesh optimization: designing measurement strategies such as elastic modulus, smooth modulus, fidelity modulus and the like for sensing the micro deformation of the tetrahedral unit and optimizing the grid by using a three-dimensional deformation description theory based on the spatial displacement gradient tensor; constructing a minimum harmonic energy model and a solving algorithm which take account of the preservation of the external morphology of the cerebral cortex and the optimization of the internal structure, and optimizing a tetrahedral mesh structure;

the fourth step: constructing a Laplace-Beltramin operator: by expression of harmonic energy defined on Riemannian manifold MTo start with, whereinRespectively representing the mesh vertex positions on the riemann manifold M,is a function expression defined on the mesh vertex; combining vector expressions of all surfaces of the tetrahedron units and barycentric coordinate expressions of any point in the body, and deducing a geometric constraint relation between tetrahedron vertexes with side length and corresponding two-side angle variables(ii) a Combining a three-dimensional space heat conduction differential equation, and constructing a Laplace-Beltramin operator under a given Dirichlet condition (temperature of the inner surface and the outer surface of a fixed cerebral cortex)The discrete expression of (a);

the fifth step: calculating the internal temperature field distribution of the cerebral cortex: firstly, constructing Dirichlet boundary conditions of a cerebral cortex; defining the whole tetrahedral mesh as a finite solution space under the condition of thermal balance, and solving a heat conduction equation by using a finite element method:whereinAs Laplace operator, subscriptMIs the whole tetrahedral mesh space and is a tetrahedral mesh,is in spatial positionAnd timeConverting the partial differential equation into a linear system to solve the temperature field distribution in the cerebral cortex as a temperature function of a variable;

and a sixth step: calculating the thickness of the cerebral cortex: constructing a heat transfer probability characterization system which is determined by a local geometric structure of the cerebral cortex and is characterized by a characteristic value and a characteristic vector of a Laplace-Beltramin operator by researching the physical significance of a Brownian motion probability density function on the riemann manifold of the cerebral cortex; and calculating a gradient curve of heat conduction between two adjacent isothermal layers based on the maximum heat transfer probability.