CN107194911B - Minimum nuclear error analysis method based on diffusion MRI microstructure imaging - Google Patents

Minimum nuclear error analysis method based on diffusion MRI microstructure imaging Download PDF

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CN107194911B
CN107194911B CN201710251439.5A CN201710251439A CN107194911B CN 107194911 B CN107194911 B CN 107194911B CN 201710251439 A CN201710251439 A CN 201710251439A CN 107194911 B CN107194911 B CN 107194911B
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冯远静
金丽玲
潘一源
周思琪
吴烨
曾庆润
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Abstract

A minimum nuclear error analysis method based on diffusion MRI microstructure imaging comprises the following steps: (1) establishing a diffusion tissue model, wherein the microstructure model comprises three characteristics of a microstructure: linear structural anisotropy (LSA, D)l) Anisotropy of planar Structure (PSA, D)p) Isotropic (SSI, D) with spherical structures) (ii) a Each feature model in each voxel is described as a hybrid anisotropic/isotropic model; (2) calculating a feature scalar, the amount of blocked and limited diffusion being captured by calculating the difference between the estimated microstructure dimensions; (3) the minimum kernel error analysis method generally represents; (4) and optimizing an algorithm, introducing an auxiliary matrix variable Z, and enabling S-theta W to be Y, wherein the minimization problem is solved by utilizing an enhanced Lagrangian multiplier. The invention provides a diffusion MRI microstructure imaging-based minimum nuclear error analysis method for stably and efficiently estimating a fiber structure with a small intersection angle by combining a diffusion tensor imaging technology.

Description

Minimum nuclear error analysis method based on diffusion MRI microstructure imaging
Technical Field
The invention relates to the field of medical imaging and neuroanatomy under computer graphics, in particular to a minimum nuclear error analysis method based on diffusion MRI microstructure imaging.
Background
With the development of the era and the progress of medical imaging technology, diffusion tensor imaging technology has greater and greater influence in neuroscience research, and the existence of advanced neuroimaging technology is indispensable in the era; diffusion tensor imaging is an emerging method of describing brain structures; at present, the diffusion tensor imaging technology is widely applied to auxiliary means of psychiatric diseases and diagnosis, and even can be used for making preoperative surgical plans, so that the contribution of the diffusion tensor imaging technology in the medical field has no alternative advantages; therefore, the research on the diffusion tensor imaging technology has great significance for brain science.
Diffusion Tensor Imaging (DTI) is the most widely used method in the clinic, in which fiber tracking algorithms form anatomically significant fiber space microstructures; since the fiber tracking algorithm based on the DTI model cannot correctly reflect the real connection condition of fibers in the brain, methods such as High Angular Resolution Diffusion Imaging (HARDI) and the like are proposed to solve the problem of crossing of a plurality of fibers.
Disclosure of Invention
In order to overcome the defect that the fiber structure with a small intersection angle cannot be stably and efficiently estimated in the prior art, the invention provides a diffusion MRI microstructure imaging-based minimum nuclear error analysis method for stably and efficiently estimating the fiber structure with the small intersection angle by combining a diffusion tensor imaging technology.
In order to solve the technical problems, the invention provides the following technical scheme:
a minimum nuclear error analysis method based on diffusion MRI microstructure imaging comprises the following steps:
(1) establishing a diffusion tissue model
The microstructure model contains three features of the microstructure: linear structural anisotropy (LSA, D)l) Anisotropy of planar Structure (PSA, D)p) Isotropic (SSI, D) with spherical structures) (ii) a Each feature model in each voxel is described as a hybrid anisotropic/isotropic model:
wherein S (0) represents the baseline signal, DiRepresenting the subset i diffusion tensor, b gradient direction, g gradient direction, m maximum number of subsets having intersections with m cardinal directions in a voxel, fiRepresenting the volume fraction of the subset i;
linear combinatorial mixing of three microstructure models:
Figure BDA0001272187960000022
wherein S (b, g) represents a diffusion signal, wl、wp、wisoAnd wanisoRepresents the non-negative mixing fraction of each microstructure, where wi0oTable isotropy, wanisoApparent anisotropy, M1，Mp，MsThe weight corresponding to each microstructure is represented,
Figure BDA0001272187960000023
Figure BDA0001272187960000024
wiso+waniso＝1；
diffusion model D obtained by learning different features from dMRI datasetl，DpAnd Ds，DlDenotes an anisotropy parallel to the main direction, wherein the eigenvalues {0,0, λlSatisfy the Condition FAl1 and
Figure BDA0001272187960000025
Dprepresenting anisotropy perpendicular to the main direction, eigenvalues { lambdap,λp0 satisfies the condition
Figure BDA0001272187960000026
And MDP＝2RAP；DsIndicating isotropy in each direction, wherein the eigenvalues { λ }s,λs,λsSatisfy the Condition FAs0 and RAs＝0‘’
(2) Computing a feature scalar
The amount of blocked and limited diffusion is captured by calculating the difference between the estimated microstructure dimensions;
Figure BDA0001272187960000027
wherein v islpThe relative difference between LSA and PSA, v, is describedpisoIndicates the relative difference between PSA and SSI, vlpanThe relative difference between the anisotropy measured for LSA and PSA is shown;
(3) the minimum kernel error analysis method generally shows
The measurement process is expressed by the general formula:
S＝ΘW+η
whereinIs a coefficient of the fraction of diffuse tissue,
Figure BDA0001272187960000032
is a vector representation of the dMRI signal measured in voxels, where
Figure BDA0001272187960000033
i∈{1,…ngEta, represents the captured noise,
Figure BDA0001272187960000034
the method specifically models an observation matrix of a convolution operator by utilizing LAD, PAD and SID;
calculating a coefficient W by minimizing the mean square error between S and the measurement, W being estimated as the solution to the optimization problem;
each fiber bundle is considered to be the same path in the direction of access within the consistency term; evaluating regression coefficients by a model kernel norm minimization problem and obtaining a regular matrix regression model by modifying weight sparsity:
Figure BDA0001272187960000035
wherein
Figure BDA0001272187960000036
Andis the set of measured signals and coefficients that determine the relative weight of each subset along each cardinal direction, N being the number of subsets; lambda is a weighting factor defined by a user, the weighting factor is different from 0 to 1, sparsity between data fitting is controlled, and lambda is set according to a general penalty level; gamma is the sum of the diagonal except for the last term of 0A diagonal matrix of 1, wherein:
Figure BDA0001272187960000038
where σ is the noise standard deviation measured from the background signal;
(4) optimization of algorithms
Introducing an auxiliary matrix variable Z, and enabling S-theta W to be Y, wherein the minimization problem is solved by utilizing an enhanced Lagrange multiplier;
Figure BDA0001272187960000039
where μ is a penalty parameter and Z is an array of Lagrangian multipliers; ADMM can be iterated as follows: by passingIteration W, which is a typical problem for weighted versions of LARS; by Yk+1＝argminYLμ(Y, W, Z) iterating Y; the optimal solution can be calculated by a singular value threshold algorithm; finally, in the standard ADMM method, the Lagrangian multiplier is represented by Zk+1＝Zk+ μ (S- Θ W-Y) iteration;
in the acceleration case, the initial residual is unchanged
Figure BDA0001272187960000041
Derivation of dual residuals
Figure BDA0001272187960000042
Termination criteria are as follows:
Figure BDA0001272187960000043
wherein
Figure BDA0001272187960000044
Wherein q, p, σabs、σrelTo adjust the parameters.
The invention has the beneficial effects that: a fiber structure with a small intersection angle is stably and efficiently estimated.
Detailed Description
The present invention is further explained below.
A minimum nuclear error analysis method based on diffusion MRI microstructure imaging comprises the following steps:
(1) establishing a diffusion tissue model:
NEMI proposes a new microstructure model, containing three features of the microstructure: linear structural anisotropy (LSA, D)l) Anisotropy of planar Structure (PSA, D)p) Isotropic (SSI, D) with spherical structures) (ii) a In general, each feature model in each voxel can be described as a hybrid anisotropic/isotropic model:
Figure BDA0001272187960000046
wherein S (0) represents the baseline signal, DiRepresenting the subset i diffusion tensor, b gradient direction, g gradient direction, m maximum number of subsets having intersections with m cardinal directions in a voxel, fiRepresenting the volume fraction of the subset i;
linear combinatorial mixing of three microstructure models:
Figure BDA0001272187960000047
wherein S (b, g) represents a diffusion signal, wl、wp、wisoAnd wanisoRepresents the non-negative mixing fraction of each microstructure, where wisoTable isotropy, wanisoApparent anisotropy, Ml，Mp，MsThe weight corresponding to each microstructure is represented,
Figure BDA0001272187960000052
wiso+waniso＝1。
diffusion model D obtained by learning different features from dMRI datasetl，DpAnd Ds. In particular, DlDenotes an anisotropy parallel to the main direction, wherein the eigenvalues {0,0, λlSatisfy the Condition FAl1 and
Figure BDA0001272187960000053
Figure BDA0001272187960000054
similarly, DpRepresenting anisotropy perpendicular to the main direction, eigenvalues { lambdap,λp0 satisfies the conditionAnd MDP＝2RAP；DsIndicating isotropy in each direction, wherein the eigenvalues { λ }s,λs,λsSatisfy the Condition FAS0 and RAS＝0。
(2) Calculating a feature scalar:
diffusion in tissue is often limited or impeded and is related to structural information of each part, rather than being independent between each voxel; the amount of blocked and restricted diffusion can be captured by calculating the difference between the estimated microstructure dimensions.
Figure BDA0001272187960000056
Wherein v islpThe relative difference between LSA and PSA, v, is describedpisoIndicates the relative difference between PSA and SSI, vlpanThe relative difference between the anisotropy measured for LSA and PSA is shown.
(3) The minimum kernel error analysis method generally represents the form:
the measurement process can be expressed by the general formula:
S＝ΘW+η
wherein
Figure BDA0001272187960000057
Is a coefficient of the fraction of diffuse tissue,
Figure BDA0001272187960000058
is a vector representation of the dMRI signal measured in voxels, where
Figure BDA0001272187960000059
i∈{1,…ngEta, represents the captured noise,
Figure BDA00012721879600000510
is an observation matrix that explicitly models the convolution operator using LAD, PAD and SID.
The coefficient W is typically calculated by minimizing the mean square error between S and the measurement. In this case, W is estimated as the solution to the optimization problem; unlike most sparse reconstructions, sparse constraints are not imposed on all diffuse tissue scores; otherwise, it would be difficult to separate the SID from the dMRI measurement.
Each fiber bundle may be considered to be the same path in the direction of access within the consistency term; under the influence of observation or requirements, in many applications, the residual signal Θ W-S on the optimal solution is typically low rank, the regression coefficients can be evaluated by a model kernel norm minimization problem, and a regular matrix regression model is obtained with modified weight sparseness:
Figure BDA0001272187960000061
wherein
Figure BDA0001272187960000062
And
Figure BDA0001272187960000063
is a set of measured signals and is determined perThe coefficient of the relative weight of each subset along each basic direction, and N is the number of the subsets; λ is a user-defined weighting factor, varying from 0 to 1, controlling the sparsity between data fits, and should be set according to a universal penalty level. Γ is a diagonal matrix with diagonals all 1 except for the last term of 0, where:
Figure BDA0001272187960000064
where σ is the standard deviation of the noise measured from the background signal.
(4) And (3) algorithm optimization:
the NEMI problem is solved by an alternating direction method using multipliers, introducing an auxiliary matrix variable Z, and letting S- Θ W ═ Y, the minimization problem is solved with an enhanced lagrange multiplier.
Figure BDA0001272187960000065
Where μ is a penalty parameter and Z is an array of Lagrangian multipliers; ADMM can be iterated as follows: by passingIteration W, which is a typical problem for weighted versions of LARS; by Yk+1＝arg minYLμ(Y, W, Z) iterating Y; the optimal solution can be calculated by a singular value threshold algorithm; finally, in the standard ADMM method, the Lagrangian multiplier is represented by Zk+1＝ZkAnd + mu (S-theta W-Y) iteration.
In the acceleration case, the initial residual is unchanged
Figure BDA0001272187960000067
Derivation of dual residuals
Figure BDA0001272187960000068
Termination criteria are as follows:
Figure BDA0001272187960000069
wherein
Figure BDA0001272187960000072
Wherein q, p, σabs、σrelIn order to adjust the parameters, manual setting is required.
This example evaluates the framework by creating a new minimum nuclear error problem that yields all structural information in order to isolate the parameter framework that extracts the diffusion signals between each water molecule.

Claims (1)

1. A minimum nuclear error analysis method based on diffusion MRI microstructure imaging is characterized in that: the method comprises the following steps:
(1) establishing a diffusion tissue model
The microstructure model contains three features of the microstructure: linear structural anisotropy (LSA, D)ι) Anisotropy of planar Structure (PSA, D)p) Isotropic (SSI, D) with spherical structures) (ii) a Each feature model in each voxel is described as a hybrid anisotropic/isotropic model:
wherein S (0) represents the baseline signal, DiRepresenting the subset i diffusion tensor, b gradient direction, g gradient direction, m maximum number of subsets having intersections with m cardinal directions in a voxel, fiRepresenting the volume fraction of the subset i;
linear combinatorial mixing of three microstructure models:
Figure FDA0002189100890000012
wherein S (b, g) represents a diffusion signal, wl、wp、wisoAnd wanisoRepresents the non-negative mixing fraction of each microstructure, where wisoTable isotropy, wanisoApparent anisotropy, Ml，Mp，MsThe weight corresponding to each microstructure is represented,
Figure FDA0002189100890000013
Figure FDA0002189100890000014
wiso+waniso＝1；
diffusion model D obtained by learning different features from dMRI datasetl，DpAnd Ds，DlDenotes an anisotropy parallel to the main direction, wherein the eigenvalues {0,0, λlSatisfy the Condition FAt1 and
Figure FDA0002189100890000015
Dprepresenting anisotropy perpendicular to the main direction, eigenvalues { lambdap，λp0 satisfies the condition
Figure FDA0002189100890000016
And MDP＝2RAP；DsIndicating isotropy in each direction, wherein the eigenvalues { λ }s，λs，λsSatisfy the Condition FAS0 and RAS＝0‘’
(2) Computing a feature scalar
The amount of blocked and limited diffusion is captured by calculating the difference between the estimated microstructure dimensions;
Figure FDA0002189100890000017
wherein v islpThe relative difference between LSA and PSA, v, is describedpisoIndicates the relative difference between PSA and SSI, vlpanShowing the difference between the anisotropy of the LSA and PSA measurementsThe relative difference of (a);
(3) the minimum kernel error analysis method generally shows
The measurement process is expressed by the general formula:
S＝ΘW+η
wherein
Figure FDA0002189100890000018
Is a coefficient of the fraction of diffuse tissue,
Figure FDA0002189100890000019
is a vector representation of the dMRI signal measured in voxels, where
Figure FDA00021891008900000110
Eta represents the noise of the acquisition and,
Figure FDA00021891008900000111
is an observation matrix for explicitly modeling convolution operators by using LSA, PSA and SSI;
calculating a coefficient W by minimizing the mean square error between S and the measurement, W being estimated as the solution to the optimization problem;
each fiber bundle is considered to be the same path in the direction of access within the consistency term; evaluating regression coefficients by a model kernel norm minimization problem and obtaining a regular matrix regression model by modifying weight sparsity:
Figure FDA0002189100890000021
whereinAnd
Figure FDA0002189100890000023
is the set of measured signals and coefficients that determine the relative weight of each subset along each cardinal direction, N being the number of subsets; λ is a user-defined weighting factor, from 0 to 1And controlling sparsity between data fits, wherein lambda is set according to a general penalty level; Γ is a diagonal matrix with diagonals all 1 except for the last term of 0, where:
where σ is the noise standard deviation measured from the background signal;
(4) optimization of algorithms
Introducing an auxiliary matrix variable Z, and enabling S-theta W to be Y, wherein the minimization problem is solved by utilizing an enhanced Lagrange multiplier;
Figure FDA0002189100890000025
where μ is a penalty parameter and Z is an array of Lagrangian multipliers; ADMM can be iterated as follows: by passing
Figure FDA0002189100890000026
Iteration W, which is a typical problem for weighted versions of LARS; by Yk+1＝arg minYLμ(Y, W, Z) iterating Y; the optimal solution can be calculated by a singular value threshold algorithm; finally, in the standard ADMM method, the Lagrangian multiplier is represented by Zk+1＝Zk+ μ (S- Θ W-Y) iteration;
in the acceleration case, the initial residual is unchangedDerivation of dual residuals
Figure FDA0002189100890000028
Termination criteria are as follows:
Figure FDA0002189100890000029
wherein
Figure FDA00021891008900000210
Figure FDA00021891008900000211
Wherein q, p, σabs、σrelTo adjust the parameters.
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