Space factor function principal component method for brain function structure recognition

Abstract

The invention discloses a space factor function principal component method for brain function structure identification. The method comprises the following steps: S1, inputting original data; s2, extracting function data by utilizing a factor process; s3, decomposing the factor load into a smooth function of a space coordinate and an additionally determined piecewise constant matrix; s4, applying the principal component analysis of the function to the potential process to process the correlation on the variables, and obtaining the final form of the function; determining a block structure of the brain; a regression model about the cognitive function is established as a covariable and is used for analyzing the influence of the ROI volume on the cognitive function. According to the method, the internal correlation, the spatial correlation and the segmentation smoothness of the ROI volume curve can be effectively captured, and the low-dimensional scalar features easy to operate are extracted from the ROI volume curve for subsequent regression analysis, so that the prediction precision of cognitive competence is improved.

Description

TECHNICAL FIELD

[0001] The present application relates to a new spatial factor guided functional principal component analysis method for processing spatially dependent high-dimensional functional data to identify brain functional structure. BACKGROUND

[0002] Alzheimer's disease (AD) is a prevalent and irreversible brain disorder in the elderly population, characterized by memory loss, reduced thinking ability, language skills, and reasoning ability, as well as a series of behavioral deficits. Given the irreversible nature of AD and the current situation that it cannot be completely cured, early prediction of cognitive decline is crucial for detecting changes in the disease at the initial stage. Changes in brain volume, particularly the gradual atrophy of specific structures, are significant neuro-related features associated with AD, and are particularly important for understanding the pathological significance. As AD worsens, neuronal damage and death lead to changes in brain volume, with changes in brain regions related to cognitive function and memory being particularly evident. The volume of the region of interest (ROI) can be directly used as a feature to predict cognitive decline in patients with Alzheimer's disease, which helps early disease detection and the adoption of appropriate intervention measures. In addition, studying the volume changes of specific ROIs can provide insight into the pathophysiology of AD, reveal potential mechanisms of disease progression, and help discover potential biomarkers for monitoring disease progression.

[0003] When assessing the impact of ROI volume changes in AD, using scalar volume measures provides a direct approach. However, volume density curves offer more advantages compared to scalar volumes. First, density curves can more comprehensively characterize the changes in brain tissue associated with AD. Second, density curves provide more observations for analysis, making them more sensitive to AD-related changes. Analyzing volume density curves faces two major challenges. First, we need to identify the high intrinsic and spatial correlation between ROIs. A reasonable assumption is that ROIs closer in distance tend to be more similar. However, the brain is composed of spatially continuous regions, each containing multiple ROIs. These regions can differ in cortical thickness, neuron density, connectivity patterns, and functional expression. Therefore, the second challenge is to maintain the differences between different regions while considering spatial correlation.

[0004] Current methods for processing high-dimensional functional data struggle to balance the correlation between different functions and the temporal correlation within a single function, and none of them can take advantage of additional spatial coordinate information. When further introducing spatial coordinate information, most existing methods can only use the information of the adjacency matrix, thus failing to fully utilize the coordinate information. Some methods can fully utilize the coordinate information, but they all assume that all functional curve measurements change continuously with spatial coordinates, ignoring the differences between different brain regions. SUMMARY

[0005] The present application aims at overcoming the deficiencies of the prior art, and provides a spatial factor function principal component method for brain function structure recognition.

[0006] The present application aims at overcoming the deficiencies of the prior art, and provides a spatial factor function principal component method for brain function structure recognition.

[0007] S1, original data input: data is represented as , is the brain local volume measurement of the first th ROI on the quantile , , is the number of ROIs;

[0008] S2, extracting dimensional function data by using a factor process , and the specific form of the factor model is:

[0009] ;

[0010] wherein, is a latent hidden process vector of dimensions, and the dimension of the latent hidden process vector is far less than the dimension of the function data; is a determined loading matrix; is a measurement error vector, each component of which has a mean of 0, a variance of , and is independent of ; S3, decomposing the factor load into a smooth function of spatial coordinates and an additional determined piecewise constant matrix; assuming that the three-dimensional coordinates of the center of the first

[0011] th ROI are , the factor model is modified, and the specific form is:

[0012] ;

[0013] wherein is a matrix composed of the values of the smooth function on the first three spatial coordinates: is used for processing non-smooth information, and if the first th and the second th ROIs belong to the same region, then​ ;

[0014] S4. Applying principal component analysis of functions to latent processes To handle variables Correlation on;

[0015] right Each component Using Karhunen-Lo Expand ve, use truncation to the beginning Karhunen-Lovation of basis functions The truncated form of ve is:

[0016] ;

[0017] The following notation is given: , ; It is the covariance function The corresponding number One orthogonal characteristic function; , ; It is the first one with a mean of 0 and decreasing variance. The scores of each orthogonal principal component;

[0018] The specific form of principal component analysis is as follows:

[0019] ;

[0020] The final form of the function is:

[0021] ;

[0022] use Determine the compartmentalized structure of the brain; A regression model of cognitive function was established using ROI volume as a covariate to analyze the impact of ROI volume on cognitive function.

[0023] The beneficial effects of this invention are:

[0024] 1) Significantly improved prediction accuracy for density curves. In actual ADNI data analysis, compared with traditional high-dimensional function data and spatiotemporal data modeling methods, the out-of-sample prediction accuracy of this invention is the highest. The improved prediction performance is due to multiple factors: this invention combines the inherent characteristics of the ROI volume curve with its quantiles. The correlation on the surface effectively re-represents high-dimensional functional data with spatial correlation. This model decomposes the load into a smooth function of spatial coordinates. and piecewise constant matrix , capable of handling multiple regions with piecewise smooth characteristics, taking into account the discontinuities and significant spatial correlation therein.

[0025] 2) has interpretability. The load matrix, piecewise constant matrix and score matrix in the model have explicit interpretability. The load matrix and piecewise constant matrix depict the characteristics of ROI measurement piecewise smoothness, and the resulting score matrix can be directly used to construct a regression modeling task to predict MMSE scores (used to measure cognitive function). Based on specific conversion, the model can identify ROIs related to cognitive decline.

[0026] 3) The model has high scalability and can be used to fit various high-dimensional function data. The present application is not limited to analyzing ROI density curves. When spatial information cannot be obtained, the factor load decomposition process can be omitted, and only the initial factor model is used. When the data has spatial correlation but no piecewise structure, the piecewise constant matrix can be omitted; when the data has other structures, additional penalty terms can be introduced in the calculation to consider the actual situation. BRIEF DESCRIPTION OF DRAWINGS

[0027] Figure 1 is a flowchart of the spatial factor guided function principal component analysis method of the present application;

[0028] Figure 2 is the spatial distribution of ROIs in 8 regions of the brain;

[0029] Figure 3 is a regression coefficient function graph. DETAILED DESCRIPTION

[0030] The present application first develops a spatial factor function principal component (SF-FPCA) method for brain function structure recognition, which effectively captures the inherent spatial correlation and piecewise smoothness of the ROI volume curve. The basic concept of SF-FPCA is to model the intrinsic correlation of ROI measurements using factor processes. Then apply univariate FPCA to each component of these factor processes to handle the correlation on the variable and further extract features. In addition, by decomposing the load into a smooth function of spatial coordinates and a piecewise constant matrix, SF-FPCA incorporates spatial coordinates to describe the relationship between the correlation between ROIs and spatial information. The technical solutions of the present application are further described below in conjunction with the drawings.

[0031] As Figure 1 shown, a spatial factor function principal component method for brain function structure recognition of the present application is used to process magnetic resonance imaging data, including the following steps:

[0032] S1. Raw Data Input: Brain MRI images are first standardized using N4 bias correction and other methods, followed by log-Jacobian transformation to obtain data from the brain. The volume density curves of each ROI were obtained, and finally, through logarithmic quantile density transformation, the volume density curves were converted into logarithmic quantile functions. The resulting data is represented as follows: This is used as the input data for the present invention; in the ADNI data, It is the first ROI at the quantile Measurement of local brain volume. , The number of ROIs;

[0033] S2, Extracting using factorial processes dimensional function data This refers to the intrinsic correlation between different ROI density curves; drawing on the ideas of the classic factor model, it is assumed that... The correlation between them stems from the sharing of underlying processes. The specific form of the factor model is as follows:

[0034] ;

[0035] in, yes A latent process vector of dimension 1, and its dimension 2... Much smaller than Dimensions ; It is a definite load matrix; The mean of each component is 0, and the variance is... And with Independent measurement error vectors;

[0036] S3 decomposes the factor loadings into smooth functions of spatial coordinates and additionally determined piecewise constant matrices; the model mentioned in S2 only captures the correlations between ROIs and does not integrate spatial coordinates, thus ignoring potential spatial information. To address this limitation, assume the... The three-dimensional coordinates of the center of each ROI are A modification to the factor model is proposed; its specific form is as follows:

[0037] ;

[0038] in It is a smooth function In the 1st~ The matrix formed by the values ​​taken at each spatial coordinate: for The value at the first spatial coordinate, Dimensions And so on. Dimensions ;matrix This approach is used to handle additional non-smooth information. It explains the inter-variable and spatial correlations between ROIs through a low-rank and smooth structure. (Matrix) The differences between different regions are explained by a piecewise constant structure. Specifically, if the first... and If ROIs belong to the same region, then... The number and composition of these regions are flexible and not predetermined. The piecewise constant structure means that the value is the same constant within each region. Therefore, the load is smooth overall within each region. However, on different regions... The different values ​​of reflect the spatial differences between different regions. Through the decomposition of the load matrix, this scheme captures spatial correlation (through a smooth function of spatial coordinates). ) and highlighting the differences between different regions (through piecewise constants) A balance was achieved between them.

[0039] S4. Applying Functional Principal Component Analysis (FPCA) to Latent Processes To handle variables Correlation on;

[0040] right Each component Using Karhunen-Lo The ve expansion method is a common approach in principal component analysis of functions, and its specific form is as follows:

[0041] ;

[0042] in, It is the covariance function The corresponding number One orthogonal characteristic function; It is the first one with a mean of 0 and decreasing variance. The scores of each orthogonal principal component;

[0043] In practice, truncation to the beginning is usually used. Karhunen-Lovation of basis functions The truncated form of ve is:

[0044] ;

[0045] The following notation is given: , ; , The specific form of principal component analysis is as follows:

[0046] ;

[0047] The final form of the function is:

[0048] ;

[0049] use Determine the compartmentalized structure of the brain; A regression model of cognitive function was established using ROI volume as a covariate to analyze the impact of ROI volume on cognitive function.

[0050] This method is called space-factor-guided functional principal component analysis (SF-FPCA).

[0051] To achieve the identifiability of this scheme, the following processing steps are also included: introducing information about The smoothness assumption is that it can be approximated by the truncated basis function expansion; specifically, it is:

[0052] ;

[0053] in The representative coefficient matrix has dimensions τ×q. j ; It is a pre-defined length. basis functions; given symbols ( It is a length of The function vector, ~ These are the function vectors. exist ~ The value of , With the dimension being τ×q, the final solution is as follows:

[0054] ;

[0055] In the actual processing of brain magnetic resonance imaging data, the data that can be obtained is the first... 10 subjects ROI volume curve exist quantile points The observed values ​​on the above will be the first ROI volume curves for each subject At the l-th observation point The observed values ​​on are denoted as , In order to obtain and To estimate the value, an iterative algorithm is proposed, the specific process of which is as follows (where we use the superscript (r) to denote the update value of a component in the r-th iteration): First, the method for calculating the initial value is given: Let the total sample size be... , represent exist The covariance matrix at the location is related to The matrix after integration. Let... , initial value yes Sample approximation form The former The eigenvectors corresponding to the eigenvalues ​​are multiplied by... It is represented by the following notation:

[0056] ;

[0057] in Represents a symmetric matrix The former The matrix formed by the eigenvectors corresponding to the largest eigenvalues. Representing the The ROI volume curve of each subject at the l-th observation point The observed values ​​on (similar definitions follow) This represents the number of observation points.

[0058] After that, through Each column pair Return, and then from Separation and The specific form is as follows:

[0059] ;

[0060] After obtaining and After initializing the values, according to , obtain The initial values ​​are as follows:

[0061] ;

[0062] Given an initial value, the following iterative steps are given: At the... During the step, update using the following least squares loss. :

[0063] ;

[0064] Obtained through the ADMM algorithm Update.

[0065] To account for the load-segmented structure, the following is introduced: :

[0066] ;

[0067] in and These are pre-defined adjustment parameters. The Frobenius norm of a vector, the first term Used to determine The group is used as the benchmark; in the second item It is about The decreasing function reflects the characteristic that ROIs that are close to each other are more likely to belong to the same region;

[0068] After obtaining and Then, according to the plan By combining the model's identifiability conditions, we can obtain... The update is as follows:

[0069] ;

[0070] Then update according to the following formula :

[0071] ;

[0072] Last consideration Update: Estimating unknown characteristic functions using spline techniques ; Approximating via splines, It has the following approximate form:

[0073] ;

[0074] in yes They are orthogonal basis function vectors, each They are all orthogonal spline coefficient matrices. It is the FPCA coefficient. ;

[0075] Define the following symbols:

[0076] , ,

[0077] ;

[0078] Updated with the ideas of the classic factor model and as follows:

[0079] ;

[0080] ;

[0081] in, yes The Next update That is to The components that need to be updated are replaced with the most recently updated values, and then the resulting matrix is ​​formed.

[0082] Last Update ,in yes The row components;

[0083] Repeat the above iterative steps until... , Convergence. (The result is...) and The estimate.

[0084] Obtained After the estimation, if , will the and the Each ROI is divided into the same region; based on this method, in this embodiment, out of all 95 ROIs, 89 ROIs are divided into three blocks, containing 26, 29, and 34 ROIs respectively. In addition, 2 ROIs are grouped together, and the remaining 4 ROIs are each assigned to an independent region. To visualize the locations of these different regions, a graph is drawn. Figure 2 The image shown. Figure 2This diagram shows the spatial distribution of Regions of Interest (ROIs) in eight brain regions. (a) Piece 1: ROIs are mainly located in the middle, deep, and posterior regions of the brain; (b) Piece 2: ROIs are mainly located in the anterior and basal parts of the cerebral cortex; (c) Piece 3: ROIs are mainly located in the upper and lateral parts of the cerebral cortex; (d) Piece 4: ROIs located in the right preorbital cortex and insula; (e) Piece 5: ROIs located in the right posterior cingulate cortex; (f) Piece 6: ROIs located in the right transverse temporal region; (g) Piece 7: ROIs located in the right supraorbital region; (h) Piece 8: ROIs located in the right middle temporal region. In the diagram, ROIs in the left brain are marked in red, and ROIs in the right brain are marked in blue. This diagram illustrates the spatial distribution of ROIs, which helps to better understand their location in brain anatomy. Figure 3 As shown in (d), some ROIs in the right brain are located in the three major regions (i.e. Figure 3 Aside from (a)-(c)), this pattern was not observed in the left hemisphere. This difference highlights the distinction between the left and right hemispheres. Similar differences have been found in subsequent studies of cognitive function.

[0085] Obtained After estimating the ROI volume, it was used as a covariate to establish a linear regression model with MMSE as the response variable, to explore the impact of ROI volume on cognitive function. If using... Indicates the return to the middle and The corresponding regression coefficients, then This represents a measure of the impact of the ROI on MMSE. After appropriate transformation, we obtain... ,in represent The regression coefficient function. Through this transformation, MMSE is correlated with the extracted features. The regression relationship between them can be transformed into MMSE and the original covariates. The regression relationship. If The The different signs of the component functions indicate that X j (t) may have a positive effect in some regions and a negative effect in others, which poses a challenge to assessing its overall impact. Therefore, if The The point-by-point confidence interval of the i-th component function does not wrap around the x-axis, assuming that the i-th component function... Atrophy of regional areas of interest (ROIs) is a significant risk factor for Alzheimer's disease (AD). Based on this approach, 36 ROIs were identified that affect cognitive function, and all of them had a positive impact. Of these 36 ROIs, 21 are located in the left hemisphere, 12 in the right hemisphere, and the remaining 3 ROIs are located in the brainstem, cerebellar lobules IV and VI-VII, respectively. Figure 3In the table, (a) shows the mean curve and confidence band of the regression coefficient function of the left hippocampus; (b)-(c) show the mean curve and confidence band of the regression coefficient function of the left and right hippocampal paragyri, respectively; and (d) shows the estimation comparison of the left and right hippocampal paragyri (red for the left and blue for the right).

[0086] like Figure 3 As shown in (a), atrophy of the left hippocampus is associated with cognitive decline, but no similar effect was found in the right hippocampus. Furthermore, as... Figure 3 As shown in (b) and (c), atrophy of the parahippocampal gyrus is associated with cognitive decline, and the left and right parahippocampal gyruses exhibit similar mechanisms of influence. Figure 3 As shown in (d), the influence on the left side is greater than that on the right side.

[0087] Analysis of the hippocampus and parahippocampal gyrus suggests that the mechanisms of influence between the left and right hemispheres may be similar. However, in reality, including the parahippocampal gyrus, we only identified a total of five brain regions where the left and right halves had similar effects on MMSE. Furthermore, within these five regions, it was impossible to determine which hemisphere had a greater influence. Besides these five pairs of ROIs, the trends of the coefficient functions of the corresponding ROIs for the left and right hemispheres were even completely different. These observations indicate that there are differences in the influence of the left and right hemispheres on cognitive function.

[0088] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A spatial factor function principal component method for brain functional structure recognition, characterized in that, Includes the following steps: S1. Raw data input: Data is represented as follows , It is the first ROI at the quantile Measurement of local brain volume. , The number of ROIs; S2, Extracting using factorial processes dimensional function data The specific form of the factor model is as follows: ； in, yes A latent process vector of dimension 1. Much smaller than Dimensions ; It is a definite load matrix; The mean of each component is 0, and the variance is... And with Independent measurement error vectors; S3. Decompose the factor loadings into smooth functions of spatial coordinates and additionally determined piecewise constant matrices; assume the first... The three-dimensional coordinates of the center of each ROI are The factor model is modified as follows: ； in It is a smooth function In the 1st~ The matrix formed by the values ​​taken at each spatial coordinate: Used to process non-smooth information, if the... and If ROIs belong to the same region, then... ; S4. Applying principal component analysis of functions to latent processes To handle variables Correlation on; right Each component Using Karhunen-Lo Expand ve, use truncation to the beginning Karhunen-Lovation of basis functions The truncated form of ve is: ； The following notation is given: , ; It is the covariance function The corresponding number One orthogonal characteristic function; , ; It is the first one with a mean of 0 and decreasing variance. The scores of each orthogonal principal component; The specific form of principal component analysis is as follows: ； The final form of the function is: ； use Determine the compartmentalized structure of the brain; A regression model of cognitive function was established using ROI volume as a covariate to analyze the impact of ROI volume on cognitive function.

2. The spatial factor function principal component method for brain functional structure recognition according to claim 1, characterized in that, It also includes the following processing steps: introducing information about The smoothness assumption is made, assuming that it can be approximated by the truncated basis function expansion; the final solution is as follows: ； Representative coefficient matrix; yes exist ~ The value of , These are pre-defined basis functions; In order to obtain and To estimate the result, an iterative algorithm is proposed, the specific process of which is as follows: First, the method for calculating the initial values ​​is given: Let the total sample size be... ,remember ; initial value Represented as: ； in Represents a symmetric matrix The former The matrix formed by the eigenvectors corresponding to the largest eigenvalues. Representing the The ROI volume curve of each subject at the l-th quantile Measurement of local brain volume. The number of quantiles; Afterwards, through Each column pair Return, and then from Separation and The specific form is as follows: ； After obtaining and After initializing the values, according to , obtain The initial values ​​are as follows: ； Given an initial value, the following iterative steps are given: At the... During the step, update using least squares loss. ; After obtaining and After that, I obtained The update is as follows: ； Then update according to the following formula : ； Last consideration Update: Estimating unknown characteristic functions using spline techniques ; Approximating via splines, It has the following approximate form: ； in yes They are orthogonal basis function vectors, each They are all orthogonal spline coefficient matrices. yes ; Define the following symbols: ， ， ； Updated with the ideas of the classic factor model and as follows: ； ； in, yes The This is the second update; Last Update ,in yes The row components; Repeat the iteration until , Convergence, obtain and The estimate.

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