CN111612746A - Dynamic detection method of functional brain network central node based on graph theory - Google Patents

Dynamic detection method of functional brain network central node based on graph theory Download PDF

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CN111612746A
CN111612746A CN202010364805.XA CN202010364805A CN111612746A CN 111612746 A CN111612746 A CN 111612746A CN 202010364805 A CN202010364805 A CN 202010364805A CN 111612746 A CN111612746 A CN 111612746A
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颜成钢
陈安琪
朱嘉凯
孙垚棋
张继勇
张勇东
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Hangzhou Dianzi University
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Abstract

The invention discloses a dynamic detection method of a functional brain network central node based on graph theory. The invention is improved on the basis of a multivariable central pivot node detection method, so that the central pivot node which is more reliable and accords with the cognitive activity of neuroscience can be detected. First, the blood oxygen signal is divided into several segments with time as dimension by using the sliding window technique. And detecting the pivot node in the corresponding time window in the sliding window of each period of time, thereby obtaining a change track of the pivot node along with the movement of the sliding window. Finally, the change track is used as a constraint to act on a multivariate detection method, so that more reliable and accurate central node can be dynamically detected.

Description

Dynamic detection method of functional brain network central node based on graph theory
Technical Field
The invention relates to the field of neuroscience brain network research, in particular to a dynamic detection method of a functional brain network central node based on graph theory.
Background
Resting state magnetic resonance imaging (fMRI) provides a non-invasive method to measure changes in brain blood oxygenation. In the resting state, the subject does not perform any specific task, and the subject spontaneously develops neural activity, the fluctuations of which are related to the changes in the blood oxygen concentration signal. The blood oxygen concentration signal is used to calculate the connectivity between brain regions, thereby constructing a functional brain network.
The brain network can be divided into a plurality of modules, some of which are responsible for vision, some of which are responsible for hearing and the like, and the modular structure enables people to distinguish different roles and statuses of brain area nodes more carefully. For example, some nodes are important in the module in which they are located, but not necessarily important for the entire network, and are called regional core nodes (provisonalhub), while other nodes, although having limited functionality in their own module, are connected to different modules and maintain connectivity of the entire network, and are called hub nodes (connectorhuub). The central node is connected with different functional modules in a brain network, and plays a very important role in connection in the brain due to the high centrality of the central node, such as information integration and participation in various cognitive activities, and the characteristics show that the central node is more easily influenced on the brain when suffering from disease attack. Recently, a consensus has been reached in the neuroimaging field that the brain's functional network changes throughout the scan time, even in an unprocessed environment. Many studies have shown that dynamic patterns are more relevant to certain cranial nerve diseases. Therefore, if we can dynamically detect central nodes more accurately, we can help us to analyze and understand the pathological mechanism of these diseases better, and can also be used for assisting the early diagnosis and treatment of diseases.
At present, methods for detecting a central node are designed for a static functional brain network, and are difficult to capture dynamic changes of the central node along with time, so that the detection result does not have time consistency and the change of the central node cannot be guaranteed to be consistent with cognitive changes presented in the functional brain network.
Disclosure of Invention
The invention provides a dynamic detection method for a functional brain network central node based on graph theory. Aiming at the defects of the static detection method, the multivariate central node detection method is improved, so that the central node which is more reliable and accords with the cognitive activity of neuroscience can be detected.
First, the blood oxygen signal is divided into several segments with time as dimension by using the sliding window technique. In the sliding window of each period of time, the hub node in the corresponding time window is detected, so as to obtain a variation track (as shown in fig. 1) of the hub node moving along with the sliding window. Finally, the change track is used as a constraint to act on a multivariate detection method, so that more reliable and accurate central node can be dynamically detected.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1, constructing a functional brain network in each sliding window by using Pearson correlation;
basic data format and preliminary knowledge, G ═ V, W, to represent a functional brain network, where V represents the set of N brain region nodes,
Figure BDA0002476193790000021
is an N × N adjacency matrix wijRepresenting the elements of the ith row and the jth column in the adjacent matrix W, and particularly representing the association degree of the ith brain area node and the jth brain area node; next, a Laplace matrix L ═ D-W is calculated, where D is a diagonal element of
Figure BDA0002476193790000022
The degree matrix of (c).
In graph theory, graph G is formed by a set of orthogonal vectors Φ obtained by eigen-decomposing a laplacian matrix L, i.e. L ═ ΦTΛ Φ, diagonal matrix Λ ═ diag [ λ [ ]1,λ2,...,λN]And λ1,λ2,...,λNRepresenting the eigenvalues sorted in ascending order.
The eigenvalues correspond to the orthogonal vectors one by one and are eigenvalues of a Laplace matrix L;
if all brain region nodes are connected in a functional brain network without separate parts, then the minimum eigenvalue λ is1Is zero. The number of zero eigenvalues equals the number of subgraphs. I.e. when lambda1And λ2The number of subgraphs is 2 when all values are zero.
Step 2: improved multivariate central node detection
2-1. Each brain region node v in a functional brain networkiAre all associated with a binary flag s in a selection vectoriIs associated with where s i0 denotes the brain region node viBeing a hub node, si1 means not a hub node. And judging whether the central node is the expected central node or not by removing the damage degree of the selected brain area node to the functional brain network, wherein the damage degree is judged by the quantity condition of zero eigenvalues of the residual functional brain network after the brain area node is removed.
The judgment criteria are as follows: and if the number of the zero eigenvalues of the residual functional brain networks is increased more than that of other nodes after the selected nodes are deleted, the node is taken as a candidate central node.
2-2. since multiple pivot nodes need to be selected at one time, the pair selection vector s ═ s1,s2,…,sn]Is an NP-hard problem. According to the KyFan theorem, the problem is converted into the sum of the minimum K characteristic values before minimization,
if there are K central nodes in the N brain area nodes of the functional brain network, the calculation is as follows:
Figure BDA0002476193790000031
then further derivation yields the final objective function:
Figure BDA0002476193790000032
wherein λ isiRepresenting the ith characteristic value;
Figure BDA0002476193790000033
representing a matrix;
2-3. place each element of the selection vector S on the diagonal of the diagonal matrix S. Wherein L iss=D-STWS represents the laplace matrix of the remaining functional brain network, with subscript s being the marker for differentiation;
Figure BDA0002476193790000036
a K-dimensional matrix representing nodes of each brain region in the remaining functional brain network, and is subjected to FTF ═ I orthogonal constraint, so the solution needs to be optimized for F and S:
and F, optimizing: fixing the diagonal matrix S to obtain a closed form solution of F, the closed form solution being LsK orthogonal vectors corresponding to the first K minimum eigenvalues of (1).
And (4) optimizing S: fixing F, the objective function of the optimized diagonal matrix S becomes:
Figure BDA0002476193790000034
wherein
Figure BDA0002476193790000035
Is an element aij=wij||fi-fj||2N × N, and fiAnd fjRespectively representing the ith row vector and the jth row vector of the F matrix.
2-4, because the optimized objective function is not strictly convex, and s is a binary selection vector which is not beneficial to optimization, an auxiliary vector is introduced
Figure BDA0002476193790000041
And simplifying the optimized objective function as follows:
Figure BDA0002476193790000042
where P is another diagonal matrix derived from the auxiliary vector P. Intuitively, the selection vector s is the result of the binarization of the auxiliary vector p.
Also in equation (4), P and S are solved alternately:
and F, optimizing: fixing the diagonal matrix S to obtain a closed form solution of F, the closed form solution being LsK orthogonal vectors corresponding to the first K minimum eigenvalues of (1).
Optimizing S and P: and F is fixed, and the diagonal matrix S and the auxiliary matrix P are optimized and solved by utilizing an augmented Lagrange multiplier method.
And step 3: dynamic detection of functional brain network hub nodes
The algorithm framework for backbone node detection versus dynamic function network of the present invention is shown in FIG. 2. Specifically, the following is shown:
and 3-1, segmenting the whole blood oxygen signal into T groups of overlapped sliding windows. At each time t, each brain region node v is estimatediCorresponding pi,piIs the ith element in the auxiliary vector p.
3-2. connection of piForm a track
Figure BDA0002476193790000043
Wherein
Figure BDA0002476193790000044
Is a continuous function pi(τ) at a time point τtDiscrete sampling of (2). Thus a continuous function pi(τ) the value of τ at any other point in time can be calculated using Radial Basis Functions (RBFs):
Figure BDA0002476193790000045
where the parameter sigma is used to control the strength of the trajectory smoothing,
Figure BDA0002476193790000046
representing the weight of the auxiliary vector of the ith brain region node at time t, i.e. the weight
Figure BDA0002476193790000047
Represents piThe weight at time t; tau-tautRepresenting the time difference.
3-3. given the trajectory
Figure BDA0002476193790000048
Set of radial basis functions
Figure BDA0002476193790000049
Calculated by the following method:
Figure BDA00024761937900000410
wherein the parameter λ is used to control the continuous function pi(τ) intensity of time dependence.
3-4. initializing a diagonal matrix S consisting of the selection vectors S for each sliding window using the improved diagonal matrix P consisting of auxiliary vectors, as indicated by the arrows in fig. 2.
And 3-5, substituting the improved diagonal matrix P formed by the auxiliary vectors into the step 2-3, and repeating the optimization solution until the target converges.
The invention has the following beneficial effects:
the output of the method of the invention is the central nodes which change along with the change of the brain network, and the central nodes are more accurate than the traditional method at each time point, and the correlation information between the time points is also kept.
Drawings
FIG. 1 is an overall framework of a hub node dynamic detection method;
FIG. 2 is an algorithm framework for a hub node dynamic detection method;
FIG. 3 is an analysis graph of the stability of the conventional method and the present method;
FIG. 4 is a graph of the visualization of the detection of normal human pivot nodes by the conventional method and the present method;
FIG. 5 is a graph of the visualization of the central node detection of a obsessive-compulsive patient using the conventional method and the present method;
Detailed Description
The present invention will be described in further detail below with reference to the accompanying drawings and experiments.
The invention utilizes the sliding window technology to divide the blood oxygen signal into several segments by taking time as a dimension. In the sliding window of each period of time, the hub node in the corresponding time window is detected, so as to obtain a variation track (as shown in fig. 1) of the hub node moving along with the sliding window. Finally, the change track is used as a constraint to act on a multivariate detection method, so that more reliable and accurate central node can be dynamically detected.
The embodiment of the invention mainly comprises the following steps:
step (1) data of 63 normal persons and 62 obsessive-compulsive patients were selected for the experiment, each subject had T1 weighted magnetic resonance image (specific parameters TR 8 ms, TE 1.7ms, flip angle 20 °, resolution 1.0 × 1.0 × 1.0.0 mm2) And resting state functional magnetic resonance data (specific parameters are TR 2s, TE 60ms, flip angle 90 degrees, resolution 3.0 × 3.0.0 3.0 × 4.0.0 mm2) Each subject contained 230 detection time points.
Step (2) all these experimental data were registered into an AAL template, divided into 116 brain regions. And calculating the correlation among the brain areas to obtain a corresponding functional brain network connection matrix W as the input of the experiment.
And (3) setting experimental parameters.
The size of the sliding window is set to 10% of the size of the whole time point; the detection number of the pivot nodes is set to be 12; the parameter σ of the radial basis RBF is set to 0.7; the optimal lambda parameter of the objective function (2) is found to be 0.6 by the grid search method.
And (4): finally, the objective function is aligned
Figure BDA0002476193790000061
And sequentially optimizing F and S until convergence, and obtaining a final result.
And (3) analyzing an experimental result:
for each individual, counting the number of the pivot nodes detected in each sliding window, constructing a corresponding histogram, and finally calculating a corresponding entropy value.
The lower the entropy shows that the change of the pivot node in the whole detection process is less, and as shown in fig. 3, the entropy values are all above the diagonal, which shows that the entropy values detected by the method are lower and more stable.
We further visualize the change results of the pivot nodes in a period of time, as shown in fig. 4 and fig. 5, the pivot nodes detected by our method have no change within 3 TRs (about 6 seconds), while the pivot nodes detected by the conventional method all have a jump at the second TR moment and return to the previous state at the next moment, which is unusual and not in accordance with the law of cognitive change. Thus, it can be seen that our method is more stable and reliable than the conventional method.

Claims (5)

1. A dynamic detection method for a functional brain network central node based on graph theory is characterized in that a sliding window technology is utilized to averagely divide blood oxygen signals into a plurality of segments by taking time as a dimension; detecting a pivot node in the corresponding time window in the sliding window of each period of time, thereby obtaining a change track of the pivot node along with the movement of the sliding window; and finally, the change track is used as a constraint to act on a multivariate detection method, so that more reliable and accurate central node can be dynamically detected.
2. The dynamic detection method for a graph theory-based functional brain network hub node according to claim 1, comprising the steps of:
step 1, constructing a functional brain network in each sliding window by using Pearson correlation;
a functional brain network is denoted by G ═ (V, W), where V denotes the set of N brain region nodes,
Figure FDA0002476193780000011
is an N × N adjacency matrix, wijRepresenting the elements of the ith row and the jth column in the adjacent matrix W, and particularly representing the association degree of the ith brain area node and the jth brain area node; next, a Laplace matrix L ═ D-W is calculated, where D is a diagonal element of
Figure FDA0002476193780000012
A degree matrix of (c);
in graph theory, graph G is formed by a set of orthogonal vectors Φ obtained by eigen-decomposing a laplacian matrix L, i.e. L ═ ΦTΛ Φ, diagonal matrix Λ ═ diag [ λ [ ]1,λ2,...,λN]And λ1,λ2,...,λNRepresenting eigenvalues sorted in ascending order;
step 2: improved multivariate central node detection
2-1. Each brain region node v in a functional brain networkiAre all associated with a binary flag s in a selection vectoriIs associated with where si denotes the brain region node viBeing a hub node, si1 means not a hub node; judging whether the central node is the expected central node or not by removing the damage degree of the selected brain area node to the functional brain network, wherein the damage degree is judged by the quantity condition of zero eigenvalues of the residual functional brain network after removing the brain area node;
2-2. since multiple pivot nodes need to be selected at one time, the pair selection vector s ═ s1,s2,…,sn]Is an NP-hard problem; converting the problem into the sum of K minimum eigenvalues before minimization according to the Ky Fan theorem;
if there are K central nodes in the N brain area nodes of the functional brain network, the calculation is as follows:
Figure FDA0002476193780000021
then further derivation yields the final objective function:
Figure FDA0002476193780000022
wherein λ isiRepresenting the ith characteristic value;
Figure FDA0002476193780000023
representing a matrix;
2-3, putting each element of the selection vector S on the diagonal of the diagonal matrix S; wherein L iss=D-STWS represents the laplace matrix of the remaining functional brain network, with subscript s being the marker for differentiation;
Figure FDA0002476193780000024
a K-dimensional matrix representing nodes of each brain region in the remaining functional brain network, and is subjected to FTF ═ I orthogonal constraint, so the solution needs to be optimized for F and S:
and F, optimizing: fixing the diagonal matrix S to obtain a closed form solution of F, the closed form solution being LsK orthogonal vectors corresponding to the first K minimum eigenvalues;
and (4) optimizing S: fixing F, the objective function of the optimized diagonal matrix S becomes:
Figure FDA0002476193780000025
wherein
Figure FDA0002476193780000026
Is an element aij=wij||fi-fj||2N × N, and fiAnd fjRespectively representing ith row vector and jth row vector of the F matrix;
2-4, because the optimized objective function is not strictly convex, and s is a binary selection vector which is not beneficial to optimization, an auxiliary vector is introduced
Figure FDA0002476193780000027
And simplifying the optimized objective function as follows:
Figure FDA0002476193780000028
where P is another diagonal matrix derived from the auxiliary vector P; intuitively, the selection vector s is the binarization result of the auxiliary vector p;
also in equation (4), P and S are solved alternately:
and F, optimizing: fixing the diagonal matrix S to obtain a closed form solution of F, the closed form solution being LsK orthogonal vectors corresponding to the first K minimum eigenvalues;
optimizing S and P: fixing F, and optimally solving a diagonal matrix S and an auxiliary matrix P by using an augmented Lagrange multiplier method;
and step 3: dynamic detection of functional brain network hub nodes
3-1, segmenting the whole blood oxygen signal into T groups of overlapped sliding windows; at each time t, each brain region node v is estimatediCorresponding pi,piIs the ith element in the auxiliary vector p;
3-2. connection of piForm a track
Figure FDA0002476193780000031
Wherein
Figure FDA0002476193780000032
Is a continuous function pi(τ) at a point in time { τt-discrete sampling of { right left over }; thus a continuous function pi(τ) the value of τ at any other point in time can be calculated using Radial Basis Functions (RBFs):
Figure FDA0002476193780000033
where the parameter sigma is used to control the strength of the trajectory smoothing,
Figure FDA0002476193780000034
representing the weight of the auxiliary vector of the ith brain region node at time t, i.e. the weight
Figure FDA0002476193780000035
Represents piThe weight at time t; tau-tautRepresents a time difference;
3-3. given the trajectory
Figure FDA0002476193780000036
Set of radial basis functions
Figure FDA0002476193780000037
Calculated by the following method:
Figure FDA0002476193780000038
wherein the parameter λ is used to control the continuous function pi(τ) intensity of the time dependence;
3-4, initializing a diagonal matrix S consisting of selection vectors S of each sliding window by using the diagonal matrix P consisting of the improved auxiliary vectors, as shown by an arrow in FIG. 2;
and 3-5, substituting the improved diagonal matrix P formed by the auxiliary vectors into the step 2-3, and repeating the optimization solution until the target converges.
3. The method according to claim 2, wherein the eigenvalues are in one-to-one correspondence with orthogonal vectors and are eigenvalues of a laplacian matrix L.
4. A method according to claim 2 or 3, wherein if all brain area nodes are connected in a functional brain network without separate parts, the minimum eigenvalue λ is1Is zero; the number of zero eigenvalues equals the number of subgraphs; i.e. when lambda1And λ2All have values ofAt zero, the number of subgraphs is 2.
5. The method according to claim 4, wherein the criterion is as follows: and if the number of the zero eigenvalues of the residual functional brain networks is increased more than that of other nodes after the selected nodes are deleted, the node is taken as a candidate central node.
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