CN114638908B - Method for reconstructing magnetoacoustic magnetic particle concentration image under saturated magnetization state - Google Patents
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Abstract
The invention discloses a magnetoacoustic magnetic particle concentration image reconstruction method under a saturation magnetization state, which comprises the following steps: based on the simulation model, acquiring sound pressure data at the ultrasonic transducer and gradient magnetic field data at the reconstruction region; constructing a sound pressure matrix by utilizing sound pressure data acquired by each ultrasonic transducer, and constructing a system matrix by utilizing gradient magnetic field data at the reconstruction region, wherein the system matrix is used for describing that a direct corresponding relation exists between sound pressure and the concentration partial derivative of the magnetic nano particles; and then obtaining the concentration partial derivative distribution of the magnetic nano particles by using an LSQR method, so as to obtain the concentration distribution of the magnetic nano particles. The invention uses the LSQR-trapezoid formula method, can quickly, stably and high-quality reconstruct the concentration distribution, has clear image boundary and has good application prospect.
Description
Technical Field
The invention belongs to the technical field of concentration distribution image reconstruction, and particularly relates to a magneto-acoustic magnetic particle concentration image reconstruction method under a saturated magnetization state based on a least square QR decomposition method-trapezoidal formula method.
Background
Magnetic Nanoparticles (MNPs) are widely used in the biomedical field due to their low toxicity, good biocompatibility, magnetic responsiveness and controllability under the action of an applied magnetic field, including: magnetic hyperthermia, drug delivery, targeted therapy, gene therapy, and the like. Magnetic particle imaging (MAGNETIC PARTICLE IMAGING, MPI) was the imaging method that was first applied to magnetic nanoparticles for medical diagnostics, gleich B et al reported the MPI imaging method on Nature for the first time in 2005. However, the spatial resolution is affected by theory and equipment factors, and in order to further improve the spatial resolution, induction magnetoacoustic particle concentration imaging (Magneto-Acoustic Concentration Tomography WITH MAGNETIC Induction MACT-MI) is proposed for the first time in 2020, shi Xiaoyu and the like at 1-5mm at present, the method naturally solves the problem of electromagnetic interference between a driving coil and a detecting coil, combines the advantages of electromagnetic technology and ultrasonic technology, and has the advantages of non-invasiveness, good contrast, high sensitivity, high spatial resolution and the like.
For the research of MACT-MI inverse problem, the magnetic nanoparticle concentration image reconstruction method of magneto-acoustic coupling (patent number: 201911020966.0) applied by 2019, 12, 25, xu Zhengyang and the like discloses a magnetic nanoparticle concentration image reconstruction method of magneto-acoustic coupling, a time inversion method is used for carrying out sound source reconstruction, a sound pressure derivation process is involved, the influence of data noise on a reconstruction result is amplified, the algorithm stability is poor, and the reconstruction result has boundary singularities. 2021, hu Yu et al propose a magnetoacoustic magnetic particle concentration imaging method based on TSVD (truncated singular value decomposition), which has a good imaging effect when the coefficient matrix is a small sparse matrix, but has an unsatisfactory imaging effect on a large sparse coefficient matrix, and the theoretical formula is only suitable for solving the condition of uniform concentration distribution, so that the application range is narrow, and further research is needed for reconstructing the inverse problem of MACT-MI.
Disclosure of Invention
The invention aims to solve the technical problem of providing a magnetoacoustic magnetic particle concentration image reconstruction method in a saturated magnetization state aiming at the defects of the prior art. The preparation method can reconstruct concentration distribution rapidly, stably and high-quality by utilizing an LSQR-trapezoid formula method, has clear image boundary and has good application prospect.
In order to solve the technical problems, the invention adopts the following technical scheme: the method for reconstructing the magnetoacoustic magnetic particle concentration image in the saturated magnetization state is characterized by comprising the following steps of:
Step one, setting a simulation initial condition of magnetoacoustic magnetic particles in a saturated magnetization state, and acquiring sound pressure data at an ultrasonic transducer and gradient magnetic field data at a reconstruction region based on a preset simulation model; the reconstruction region is formed by taking a magnetic nanoparticle group as a center, selecting a Wmm multiplied by Wmm region as a reconstruction region, and carrying out finite element division on the reconstruction region to divide the reconstruction region into M multiplied by M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each grid;
constructing a sound pressure matrix by utilizing sound pressure data acquired by each ultrasonic transducer, and constructing a system matrix by utilizing gradient magnetic field data at the reconstruction area, wherein the system matrix is used for describing that a direct corresponding relation exists between sound pressure and the concentration partial derivative of the magnetic nano particles;
thirdly, acquiring concentration partial derivative distribution of the magnetic nano particles by using the sound pressure matrix and the system matrix constructed in the second step through an LSQR method;
And step four, acquiring magnetic nanoparticle concentration distribution by utilizing magnetic nanoparticle concentration partial derivative distribution, so as to acquire a magnetoacoustic magnetic particle concentration reconstruction image in a saturated magnetization state.
Preferably, in the first step, setting a simulation initial condition of magnetoacoustic magnetic particles in a saturated magnetization state, and acquiring sound pressure data at an ultrasonic transducer and gradient magnetic field data at a reconstruction region based on a preset simulation model; selecting a Wmm×Wmm region as a reconstruction region by taking a magnetic nanoparticle group as a center, and carrying out finite element division on the reconstruction region to divide the reconstruction region into M×M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each grid; comprising the following steps:
Initializing magnetic nanoparticle parameters, simulating conditions of a Helmholtz coil and a Maxwell coil current, selecting an EMG308 model as a magnetic nanoparticle, placing a magnetic nanoparticle group at the center of the coil, respectively introducing current into the Helmholtz coil and the Maxwell coil, providing a static magnetic field B sat by the Helmholtz coil to enable the magnetic nanoparticle to reach a saturated state, and providing a uniform gradient magnetic field B g by the Maxwell coil; and drawing a circle with a fixed scanning radius by taking the magnetic nanoparticle group as the center, discretely arranging a plurality of ultrasonic transducers on the circle, acquiring sound pressure data of a plurality of time points received by each ultrasonic transducer, and acquiring an original sound field p (r, t) at each ultrasonic transducer.
Preferably, a circle is drawn with a fixed scanning radius by taking a magnetic nanoparticle group as the center, 165 ultrasonic transducers are discretely arranged on the circle, and sound pressure data of 470 time points received by each ultrasonic transducer are obtained; selecting a 50mm multiplied by 50mm area as a reconstruction area by taking the magnetic nanoparticle group as the center, and carrying out finite element division on the reconstruction area to divide the reconstruction area into 250 multiplied by 250 grids.
Preferably, the sound pressure data acquired at each ultrasonic transducer is used to construct a sound pressure matrix, and the gradient magnetic field data at the reconstruction region is used to construct a system matrix, where the system matrix is used to describe that a direct correspondence exists between sound pressure and a concentration partial derivative of the magnetic nanoparticles, and the method includes:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
In the formula (1), f (r', t) is the magnetic force received by the magnetic nano particles at the moment t; r' is the position of the sound source point, and t is time; n (r') is the concentration of the magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle; The magnitude of the gradient magnetic field at the position of the sound source point at the moment t; e z denotes a unit vector in the z direction;
The sound source and sound pressure relationship is as follows:
In the formula (2), p (r, t) is the sound pressure at any point at the time t, and the unit is Pa; r is any point position; c s is the speed of sound in the biological tissue in m/s; is a sound source item; t is time; r 'is the position of the source point, r= |r-R' |; omega is the entire study area;
due to the unidirectional nature of the magnetic force, the sound source term can be expressed as
Bringing formula (3) into formula (2) to obtain
Since in calculating the problem of temporal spatial distribution,Can be processed according to constant, so that
The above equation can be used to obtain a direct correspondence between sound pressure and partial concentration derivative, and describing the correspondence through the system matrix a can be abstracted to obtain a matrix relation:
Ax＝b (6)
Wherein A is a system matrix, x is a concentration partial derivative, b is a sound pressure matrix, and sound pressure data in the sound pressure matrix b corresponds to p (r, t); and combining gradient magnetic field data according to the formula (5) and the formula (6), thereby obtaining a system matrix A.
Preferably, in the third step, the concentration partial derivative distribution of the magnetic nanoparticles is obtained by using the sound pressure matrix and the system matrix constructed in the second step by using an LSQR method, including:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally obtaining concentration partial derivative distribution;
The solving process comprises the following steps:
Let orthonormal matrix U k＝[u1,u2,…,uk](uj∈Rmi) and Double diagonal matrix of
Wherein (. Alpha. 1,α2,…,αk∈R;β2,β3,…,βk+1. Epsilon.R).
The iterative process is as follows: the iterative process is as follows:
Step 301, condition initialization
β1＝||b||;α1＝||ATu1||2;β1u1＝b；
Step 302 diagonalizing a coefficient matrix
βj+1＝||Avj-αjuj||2;αj+1＝||ATuj+1-βj+1vj||2;
(j＝1,2,...,k)。
Step 303, calculating QR decomposition intermediate variables
Step 304, updating the values of x and intermediate variable w
Step 305, judging iteration conditions
If ||ax k -b||ε is satisfied, the iteration is terminated, where epsilon is the allowable error, epsilon=0.01 is set;
for j=1, 2..k, steps 302-305 are repeated.
Preferably, in the fourth step, the magnetic nanoparticle concentration distribution is obtained by using the magnetic nanoparticle concentration partial derivative distribution, so as to obtain a magnetoacoustic magnetic particle concentration reconstruction image, including:
The concentration partial derivative is known to satisfy the following relationship with the concentration distribution:
Assuming the side length of the grid is l, l=0.2 mm, the concentration partial derivative x can be expressed as:
The concentration profile N (r') can be expressed as
The trapezoidal formula method with simple calculation and higher precision is introduced for solving, and the method can be obtained
Wherein, c is more than or equal to 0 and less than or equal to 249,1 and d is more than or equal to 250, and the concentration distribution can be obtained by combining the boundary conditions of x (c, d).
Compared with the prior art, the invention has the following advantages:
1. The invention aims to solve the problems of low reconstruction speed, poor reconstruction quality, boundary singularity and the like in the existing magnetoacoustic magnetic particle concentration imaging method. The invention solves the concentration distribution of magnetoacoustic magnetic particles based on the LSQR-trapezoidal formula method, can quickly and stably reconstruct the concentration distribution with high quality, has clear image boundary and has no singularity.
2. According to the invention, a calculation area is subjected to grid division of a specified grid according to a finite element method, sound field data and magnetic field data are discretized, a system matrix reflecting the relation between sound pressure and concentration partial derivative is constructed, the concentration partial derivative distribution is reconstructed by using an LSQR method, and a concentration distribution image is reconstructed by using a trapezoidal formula method.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of a method for reconstructing a magnetoacoustic magnetic particle concentration image in a saturated magnetization state based on a least squares QR decomposition method-trapezoidal formula method.
Fig. 2 is a simulation model.
Fig. 3 is a Helmholtz coil current.
Fig. 4 is a schematic diagram of sound pressure data and magnetic field data acquisition.
Fig. 5 is a gradient concentration model.
Fig. 6a is a diagram of simulation results.
Fig. 6b is a graph of target concentration.
Detailed Description
Example 1
Magneto-acoustic magnetic particle concentration imaging method
As shown in fig. 1, the calculation region is subjected to grid division of a predetermined grid according to a finite element method, sound field data and magnetic field data are discretized, a system matrix reflecting the relationship between sound pressure and concentration partial derivative is constructed, the concentration partial derivative distribution is reconstructed by using an LSQR method, and a concentration distribution image is reconstructed by using a trapezoidal formula method. As shown in fig. 1, the present invention is implemented by the following technical scheme: the invention provides a magnetic sound magnetic particle concentration image reconstruction method under a saturated magnetization state based on an LSQR-trapezoidal formula method, which specifically comprises the following steps:
S1, setting a simulation initial condition of magnetoacoustic magnetic particles in a saturated magnetization state, and acquiring sound pressure data at an ultrasonic transducer and gradient magnetic field data at a reconstruction region based on a preset simulation model;
Further, the simulation model building mode of the magnetoacoustic magnetic particles in the saturated magnetization state is as follows: simulation study is carried out by establishing a two-dimensional axisymmetric model by means of COMSOL multiphysics5.6, as shown in fig. 2, the coil material is copper, the coil is placed by taking the z axis as a central axis, the radius of the coil is r=150mm, the origin is the coil center, and the middle cylindrical area is the study area.
1. Simulation conditions
(1) Parameters of magnetic nanoparticles
From EMG 308 (Ferrotec (USA) Corporation), the specifications of which are shown in Table 1.
TABLE 1 EMG 308 Specification
(2) Excitation current
And a constant current with the magnitude of 50A in the anticlockwise direction is simultaneously introduced into the upper coil and the lower coil of the Helmholtz coil, the strength of the generated magnetic field is about 1.93 multiplied by 10 5 A/m and is larger than the saturated magnetic field strength of the EMG308, and the magnetic nano particles reach a saturated state. The upper coil of the Maxwell coil is supplied with a counterclockwise current, the lower coil is supplied with a clockwise current with the same magnitude, and the coils are supplied with currents with the time characteristics shown in fig. 3.
Specifically, initializing simulation conditions of magnetic nanoparticle parameters, a Helmholtz coil and a Maxwell coil current, selecting an EMG308 model as a magnetic nanoparticle, placing a magnetic nanoparticle group at the center of the coil, respectively introducing current into the Helmholtz coil and the Maxwell coil, providing a static magnetic field B sat by the Helmholtz coil to enable the magnetic nanoparticle to reach a saturated state, and providing a uniform gradient magnetic field B g by the Maxwell coil; as shown in fig. 4, a circle is drawn with a fixed scanning radius by taking a magnetic nanoparticle group as the center, 165 ultrasonic transducers are discretely arranged on the circle, sound pressure data of 470 time points received by each ultrasonic transducer are obtained, and an original sound field p (r, t) at each ultrasonic transducer is obtained; selecting a 50mm multiplied by 50mm region as a reconstruction region by taking a magnetic nanoparticle group as a center, carrying out finite element division on the reconstruction region, dividing the reconstruction region into 250 multiplied by 250 grids, and extracting gradient magnetic field data at each grid.
S2, constructing a sound pressure matrix by utilizing sound pressure data acquired by each ultrasonic transducer, and constructing a system matrix by utilizing gradient magnetic field data at the reconstruction area, wherein the system matrix is used for describing that a direct corresponding relation exists between sound pressure and the concentration partial derivative of the magnetic nano particles;
Constructing a sound pressure matrix by using sound pressure data acquired at each ultrasonic transducer, and constructing a system matrix by using gradient magnetic field data at the reconstruction region, wherein the system matrix is used for describing that a direct corresponding relation exists between sound pressure and a concentration partial derivative of magnetic nano particles, and comprises the following steps:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
In the formula (1), f (r', t) is the magnetic force received by the magnetic nano particles at the moment t; r' is the position of the sound source point, and t is time; n (r') is the concentration of the magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle; The magnitude of the gradient magnetic field at the position of the sound source point at the moment t; e z denotes a unit vector in the z direction;
The sound source and sound pressure relationship is as follows:
In the formula (2), p (r, t) is the sound pressure at any point at the time t, and the unit is Pa; r is any point position; c s is the speed of sound in the biological tissue in m/s; is a sound source item; t is time; r 'is the position of the source point, r= |r-R' |; omega is the entire study area;
due to the unidirectional nature of the magnetic force, the sound source term can be expressed as
Bringing formula (3) into formula (2) to obtain
Since in calculating the problem of temporal spatial distribution,Can be processed according to constant, so that
The above equation can be used to obtain a direct correspondence between sound pressure and partial concentration derivative, and describing the correspondence through the system matrix a can be abstracted to obtain a matrix relation:
Ax＝b (6)
wherein A is a system matrix, x is a concentration partial derivative, b is a sound pressure matrix, and sound pressure data in the sound pressure matrix b is from p (r, t).
Combining gradient magnetic field data according to the formulas (5) and (6) to obtain a system matrix A, wherein the size of the system matrix A is 77550 multiplied by 62500; sound pressure matrix b is constructed using sound pressure data at the ultrasonic transducer, the matrix size being 77550 x 1.
S3, acquiring concentration partial derivative distribution of the magnetic nano particles by using an LSQR method by utilizing the sound pressure matrix and the system matrix constructed in the step two;
further, obtaining the concentration partial derivative distribution of the magnetic nanoparticles includes:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally obtaining concentration partial derivative distribution;
The solving process comprises the following steps:
Let orthonormal matrix U k＝[u1,u2,…,uk](uj∈Rmi) and Double diagonal matrix of
Wherein (. Alpha. 1,α2,…,αk∈R;β2,β3,…,βk+1. Epsilon.R).
The iterative process is as follows: the iterative process is as follows:
Step 301, condition initialization
β1＝||b||;α1＝||ATu1||2;β1u1＝b；
Step 302 diagonalizing a coefficient matrix
βj+1＝||Avj-αjuj||2;αj+1＝||ATuj+1-βj+1vj||2;
(j＝1,2,...,k)。
Step 303, calculating QR decomposition intermediate variables
Step 304, updating the values of x and intermediate variable w
Step 305, judging iteration conditions
If ||ax k -b||ε is satisfied, the iteration is terminated, where epsilon is the allowable error, epsilon=0.01 is set;
for j=1, 2..k, steps 302-305 are repeated.
S4, acquiring magnetic nanoparticle concentration distribution by utilizing magnetic nanoparticle concentration partial derivative distribution, so as to acquire a magnetoacoustic magnetic particle concentration reconstruction image in a saturated magnetization state.
Further, the method for obtaining the magnetic nanoparticle concentration distribution by using the magnetic nanoparticle concentration partial derivative distribution to obtain the magnetoacoustic magnetic particle concentration reconstruction image in the saturated magnetization state comprises the following steps:
The concentration partial derivative is known to satisfy the following relationship with the concentration distribution:
Assuming the side length of the grid is l, l=0.2 mm, the concentration partial derivative x can be expressed as:
The concentration profile N (r') can be expressed as
The trapezoidal formula method with simple calculation and higher precision is introduced for solving, and the method can be obtained
Wherein, c is more than or equal to 0 and less than or equal to 249,1 and d is more than or equal to 250, and the concentration distribution can be obtained by combining the boundary conditions of x (c, d).
The concentration distribution can be rebuilt through the steps, the concentration gradient model is established in consideration of the fact that MNPs are in dispersion gradient in biological tissues, as shown in fig. 5, a peripheral circular area is a bionic area, an inner area is an imitated MNPs area, EMG 308 is selected for MNPs, the rebuilt result and a concentration rebuilt target are shown in fig. 6a and 6b, the concentration distribution can be rebuilt with high quality by the method provided by the invention, no singularity exists, and the applicability of the invention is further evaluated, the correlation coefficient CC and the relative error RE are introduced, and the target concentration distribution and the rebuilt concentration distribution are compared by adopting the following formula:
Wherein N n is a target nanoparticle concentration value, N n,r is a reconstructed nanoparticle concentration value, As an average value of the concentration of the target nanoparticles,Is the average value of the concentration of the target nano particles.
Through evaluation of the invention, the correlation coefficient and the relative error of the invention are 0.9816 and 0.2761 respectively, and the reconstruction accuracy is higher.
The above description is only of the preferred embodiments of the present invention, and is not intended to limit the present invention. Any simple modification, variation and equivalent variation of the above embodiments according to the technical substance of the invention still fall within the scope of the technical solution of the invention.
Claims (6)
1. The method for reconstructing the magnetoacoustic magnetic particle concentration image in the saturated magnetization state is characterized by comprising the following steps of:
Step one, setting a simulation initial condition of magnetoacoustic magnetic particles in a saturated magnetization state, and acquiring sound pressure data at an ultrasonic transducer and gradient magnetic field data at a reconstruction region based on a preset simulation model; the reconstruction region is formed by taking a region with the length of Wmm multiplied by Wmm as a center of a magnetic nanoparticle group as a reconstruction region, and carrying out finite element division on the reconstruction region to divide the reconstruction region into M multiplied by M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each grid;
constructing a sound pressure matrix by utilizing sound pressure data acquired by each ultrasonic transducer, and constructing a system matrix by utilizing gradient magnetic field data at the reconstruction area, wherein the system matrix is used for describing that a direct corresponding relation exists between sound pressure and the concentration partial derivative of the magnetic nano particles;
thirdly, acquiring concentration partial derivative distribution of the magnetic nano particles by using the sound pressure matrix and the system matrix constructed in the second step through an LSQR method;
And step four, acquiring magnetic nanoparticle concentration distribution by utilizing magnetic nanoparticle concentration partial derivative distribution, so as to acquire a magnetoacoustic magnetic particle concentration reconstruction image in a saturated magnetization state.
2. The method for reconstructing a magnetoacoustic magnetic particle concentration image in a saturated magnetization state according to claim 1, wherein in the first step, a simulation initial condition of magnetoacoustic magnetic particles in a saturated magnetization state is set, and sound pressure data at an ultrasonic transducer and gradient magnetic field data at a reconstruction region are obtained based on a preset simulation model; selecting an Nmm multiplied by Nmm region as a reconstruction region by taking a magnetic nanoparticle group as a center, and carrying out finite element division on the reconstruction region to divide the reconstruction region into M multiplied by M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each grid; comprising the following steps:
Initializing magnetic nanoparticle parameters, simulating conditions of a Helmholtz coil and a Maxwell coil current, selecting an EMG308 model as a magnetic nanoparticle, placing a magnetic nanoparticle group at the center of the coil, respectively introducing current into the Helmholtz coil and the Maxwell coil, providing a static magnetic field B sat by the Helmholtz coil to enable the magnetic nanoparticle to reach a saturated state, and providing a uniform gradient magnetic field B g by the Maxwell coil; and drawing a circle with a fixed scanning radius by taking the magnetic nanoparticle group as the center, discretely arranging a plurality of ultrasonic transducers on the circle, acquiring sound pressure data of a plurality of time points received by each ultrasonic transducer, and acquiring an original sound field p (r, t) at each ultrasonic transducer.
3. The method for reconstructing the magnetic acoustic magnetic particle concentration image in the saturated magnetization state according to claim 1, wherein a circle is drawn with a fixed scanning radius by taking a magnetic nanoparticle group as a center, 165 ultrasonic transducers are discretely arranged on the circle, and sound pressure data of 470 time points received by each ultrasonic transducer are obtained; selecting a 50mm multiplied by 50mm area as a reconstruction area by taking the magnetic nanoparticle group as the center, and carrying out finite element division on the reconstruction area to divide the reconstruction area into 250 multiplied by 250 grids.
4. The method for reconstructing a magnetic-acoustic-magnetic particle concentration image in a saturated magnetization state according to claim 2, wherein constructing a sound pressure matrix from sound pressure data acquired at each ultrasonic transducer, constructing a system matrix from gradient magnetic field data at the reconstruction region, the system matrix describing a direct correspondence between sound pressure and a concentration partial derivative of magnetic nanoparticles, comprises:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
In the formula (1), f (r', t) is the magnetic force received by the magnetic nano particles at the moment t; r' is the position of the sound source point, and t is time; n (r') is the concentration of the magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle; The magnitude of the gradient magnetic field at the position of the sound source point at the moment t; e z denotes a unit vector in the z direction;
The sound source and sound pressure relationship is as follows:
In the formula (2), p (r, t) is the sound pressure at any point at the time t, and the unit is Pa; r is any point position; c s is the speed of sound in the biological tissue in m/s; f (r, t) is the sound source term; t is time; r 'is the position of the source point, r= |r-R' |; omega is the entire study area;
due to the unidirectional nature of the magnetic force, the sound source term can be expressed as
Bringing formula (3) into formula (2) to obtain
Since in calculating the problem of temporal spatial distribution,Can be processed according to constant, so that
The above equation can be used to obtain a direct correspondence between sound pressure and partial concentration derivative, and describing the correspondence through the system matrix a can be abstracted to obtain a matrix relation:
Ax＝b (6)
wherein A is a system matrix, x is a concentration partial derivative, b is a sound pressure matrix, and sound pressure data in the sound pressure matrix corresponds to p (r, t); and combining gradient magnetic field data according to the formula (5) and the formula (6), thereby obtaining a system matrix A.
5. The method for reconstructing a magnetic acoustic magnetic particle concentration image in a saturated magnetization state according to claim 2, wherein in the third step, the concentration partial derivative distribution of the magnetic nanoparticles is obtained by using the sound pressure matrix and the system matrix constructed in the second step by using the LSQR method, and the method comprises the following steps:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally obtaining concentration partial derivative distribution;
The solving process comprises the following steps:
Let orthonormal matrix U k＝[u1,u2,…,uk](uj∈Rmi) and Double diagonal matrix of
Wherein (α 1,α2,…,αk∈R;β2,β3,…,βk+1 ε R);
the iterative process is as follows: the iterative process is as follows:
Step 301, condition initialization
β1＝||b||;α1＝||ATu1||2;β1u1＝b；
α1v1＝ATu1;w1＝v1;x0＝0；
Step 302 diagonalizing a coefficient matrix
βj+1＝||Avj-αjuj||2;αj+1＝||ATuj+1-βj+1vj||2;
(j＝1,2,...,k)；
Step 303, calculating QR decomposition intermediate variables
Step 304, updating the values of x and intermediate variable w
Step 305, judging iteration conditions
If ||ax k -b||ε is satisfied, the iteration is terminated, where epsilon is the allowable error, epsilon=0.01 is set;
for j=1, 2..k, steps 302-305 are repeated.
6. A method for reconstructing a magnetoacoustic magnetic particle concentration image in a saturated magnetization state according to claim 3, wherein in step four, the magnetic nanoparticle concentration distribution is obtained by using the magnetic nanoparticle concentration partial derivative distribution, thereby obtaining a magnetoacoustic magnetic particle concentration reconstructed image, comprising:
The concentration partial derivative is known to satisfy the following relationship with the concentration distribution:
Assuming the side length of the grid is l, l=0.2 mm, the concentration partial derivative x can be expressed as:
The concentration profile N (r') can be expressed as
The trapezoidal formula method with simple calculation and higher precision is introduced for solving, and the method can be obtained
Wherein, c is more than or equal to 0 and less than or equal to 249,1 and d is more than or equal to 250, and the concentration distribution can be obtained by combining the boundary conditions of x (c, d).
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