CN114638908A - Magneto-acoustic-magnetic particle concentration image reconstruction method under saturation magnetization state - Google Patents

Magneto-acoustic-magnetic particle concentration image reconstruction method under saturation magnetization state Download PDF

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CN114638908A
CN114638908A CN202210225528.3A CN202210225528A CN114638908A CN 114638908 A CN114638908 A CN 114638908A CN 202210225528 A CN202210225528 A CN 202210225528A CN 114638908 A CN114638908 A CN 114638908A
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闫孝姮
许洪
陈伟华
曹媛
李军
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Liaoning Technical University
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Abstract

The invention discloses a method for reconstructing a magnetoacoustic magnetic particle concentration image in a saturation magnetization state, which comprises the following steps: based on the simulation model, obtaining sound pressure data at the ultrasonic transducer and gradient magnetic field data at the reconstruction region; 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 a direct corresponding relation between the sound pressure and a concentration partial derivative of the magnetic nanoparticles; then the two are used for obtaining the concentration partial derivative distribution of the magnetic nano particles by an LSQR method, and the concentration distribution of the magnetic nano particles can be obtained. The invention uses the LSQR-trapezoidal formula method, can reconstruct the concentration distribution quickly, stably and with high quality, has clear image boundary and has good application prospect.

Description

Magneto-acoustic-magnetic particle concentration image reconstruction method under saturation magnetization state
Technical Field
The invention belongs to the technical field of concentration distribution image reconstruction, and particularly relates to a method for reconstructing a magnetoacoustic particle concentration image in a saturation 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 (MPI) is the earliest Imaging method to apply Magnetic nanoparticles to medical diagnosis, and in 2005, Gleich B et al reported MPI Imaging methods for the first time in Nature. However, the spatial resolution is influenced by theory and equipment factors, currently 1-5mm, in order to further improve the spatial resolution, in 2020, schargi and the like firstly propose inductive magnetoacoustic particle Concentration imaging (Magnetic-Acoustic Concentration with Magnetic index MACT-MI), and the method naturally overcomes the problem of electromagnetic interference between a driving coil and a detecting coil, integrates the advantages of an electromagnetic technology and an ultrasonic technology, and has the advantages of no wound, good contrast, high sensitivity, high spatial resolution and the like.
For the research on the MACT-MI inverse problem, a magnetoacoustic coupled magnetic nanoparticle concentration image reconstruction method (patent number: 201911020966.0) applied by Sun-grant, Inc. in 2019, 12/25/2019 discloses a magnetoacoustic coupled magnetic nanoparticle concentration image reconstruction method, wherein a time reversal method is used for 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 singularity. In 2021, Huyu et al proposed a magnetic acoustic 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 when dealing with a large sparse coefficient matrix, and the theoretical formula thereof is only suitable for solving the situation of uniform concentration distribution, and has a narrow application range, so that the inverse problem reconstruction of MACT-MI requires further research.
Disclosure of Invention
The technical problem to be solved by the present invention is to provide a method for reconstructing a magnetoacoustic particle density image in a saturation magnetization state, aiming at the above-mentioned deficiencies of the prior art. The preparation method utilizes an LSQR-trapezoidal formula method, can reconstruct the concentration distribution rapidly, stably and in high quality, has clear image boundary and has good application prospect.
In order to solve the technical problems, the invention adopts the technical scheme that: a magneto-acoustic particle concentration image reconstruction method in a saturation magnetization state is characterized by comprising the following steps:
firstly, setting a simulation initial condition of magnetic-acoustic-magnetic particles in a saturation 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 characterized in that a region Wmm multiplied by Wmm is selected as a reconstruction region by taking a magnetic nanoparticle swarm as a center, and the reconstruction region is subjected to finite element division and is divided into M multiplied by M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each of the meshes;
secondly, 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 a direct corresponding relation between the sound pressure and a concentration partial derivative of the magnetic nano particles;
thirdly, acquiring the concentration partial derivative distribution of the magnetic nanoparticles by using the sound pressure matrix and the system matrix constructed in the second step through an LSQR (least square resonance) method;
and step four, acquiring the concentration distribution of the magnetic nanoparticles by utilizing the partial derivative distribution of the concentration of the magnetic nanoparticles, thereby acquiring a magnetic acoustic magnetic particle concentration reconstruction image in a saturation magnetization state.
Preferably, the first step sets a simulation initial condition of the magnetoacoustic-magnetic particles in a saturation magnetization state, and acquires sound pressure data at the ultrasonic transducer and gradient magnetic field data at the reconstruction region based on a preset simulation model; selecting a Wmm multiplied by Wmm area as a reconstruction area by taking the magnetic nanoparticle swarm as a center, and carrying out finite element division on the reconstruction area to divide the reconstruction area into M multiplied by M grids; the gradient magnetic field data at the reconstruction region comprises gradient magnetic field data at each grid; the method comprises the following steps:
initializing simulation conditions of magnetic nanoparticle parameters and currents of a Helmholtz coil and a Maxwell coil, selecting EMG308 as magnetic nanoparticles, placing magnetic nanoparticle groups at the central position of the coil, respectively introducing currents into the Helmholtz coil and the Maxwell coil, and providing a static magnetic field B by the Helmholtz coilsatThe magnetic nano particles are saturated, and a uniform gradient magnetic field B is provided by a Maxwell coilg(ii) a Drawing a circle by using the fixed scanning radius with the magnetic nanoparticle group as a 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 by a fixed scanning radius with the magnetic nanoparticle group as a center, 165 ultrasonic transducers are discretely arranged on the circle, and 470 sound pressure data of time points received by each ultrasonic transducer are acquired; selecting a 50mm multiplied by 50mm area as a reconstruction area by taking the magnetic nanoparticle group as a 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 matrix is constructed by using sound pressure data acquired at each ultrasonic transducer, and the system matrix is constructed by using gradient magnetic field data at the reconstruction region, and the system matrix is used for describing that a direct corresponding relation exists between the sound pressure and the concentration partial derivative of the magnetic nano particles, and the method comprises the following steps:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
Figure BDA0003539072200000031
In the formula (1), f (r', t) is the magnetic force applied to the magnetic nanoparticles at the time t; r' is the position of the sound source point, and t is time; n (r') is the concentration of magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle;
Figure BDA0003539072200000032
the gradient magnetic field is the position of the sound source point at the time t; e.g. of a cylinderzRepresents a unit vector in the z direction;
the sound source and sound pressure relationship is as follows:
Figure BDA0003539072200000033
in the formula (2), p (r, t) is the sound pressure at any point of time t, and the unit is Pa; r is an arbitrary point position; c. CsIs the speed of sound in biological tissue, in m/s;
Figure BDA0003539072200000034
is a sound source item; t is time; r 'is the position of the sound source point, and R is | R-R' |; Ω is the entire study area;
due to the unidirectionality of the magnetic force, the sound source term can be expressed as
Figure BDA0003539072200000041
Bringing formula (3) into formula (2) to obtain
Figure BDA0003539072200000042
Since in computing the temporal spatial distribution problem,
Figure BDA0003539072200000043
can be processed according to constant value, so that
Figure BDA0003539072200000044
From the above equation, there is a direct correspondence between the sound pressure and the concentration partial derivative, and the correspondence is described by a system matrix a, so that a matrix relation equation can be abstracted:
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); the system matrix a is obtained according to equations (5) and (6) in combination with the gradient magnetic field data.
Preferably, in the third step, the LSQR method is used to obtain the partial derivative distribution of the concentration of the magnetic nanoparticles by using the sound pressure matrix and the system matrix constructed in the second step, and the method includes:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally solving the concentration partial derivative distribution;
the solving process comprises the following steps:
let the orthonormal matrix Uk＝[u1,u2,…,uk](uj∈Rmi) And
Figure BDA0003539072200000045
the double diagonal matrix is
Figure BDA0003539072200000046
Wherein (alpha)1,α2,…,αk∈R；β2,β3,…,βk+1∈R)。
The iterative process is as follows: the iterative process is as follows:
step 301, condition initialization
β1＝||b||；α1＝||ATu1||2；β1u1＝b；
Figure BDA0003539072200000051
Step 302, diagonalizing the coefficient matrix
βj+1＝||Avj-αjuj||2；αj+1＝||ATuj+1-βj+1vj||2；
Figure BDA0003539072200000052
(j＝1,2,...,k)。
Step 303, calculating QR decomposition intermediate variables
Figure BDA0003539072200000053
Figure BDA0003539072200000054
Step 304, updating the values of x and the intermediate variable w
Figure BDA0003539072200000055
Step 305, judging iteration conditions
If | | Ax is satisfiedk-b | ≦ epsilon and terminating the iteration, where epsilon is the allowed error, set epsilon 0.01;
repeat steps 302-305 for j 1, 2.
Preferably, in the fourth step, the magnetic nanoparticle concentration distribution is acquired by using the magnetic nanoparticle concentration partial derivative distribution, so as to acquire a magnetoacoustic magnetic particle concentration reconstruction image, including:
the partial derivative of the concentration and the concentration distribution are known to satisfy the following relationship:
Figure BDA0003539072200000056
assuming that the side length of the grid is l, l is 0.2mm, the concentration partial derivative x can be expressed as:
Figure BDA0003539072200000057
the concentration profile N (r') can be expressed as
Figure BDA0003539072200000061
The solution is obtained by introducing a trapezoidal formula method with simple calculation and higher precision
Figure BDA0003539072200000062
Wherein c is more than or equal to 0 and less than or equal to 249, d is more than or equal to 1 and less 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 slow reconstruction speed, poor reconstruction quality, boundary singularity and the like in the existing magnetic-acoustic-magnetic particle concentration imaging method. The invention solves the concentration distribution of the magnetoacoustic-magnetic particles based on the LSQR-trapezoidal formula method, can reconstruct the concentration distribution rapidly and stably with high quality, and has clear image boundary and no singularity.
2. The method comprises the steps of carrying out specified grid division on a calculation region according to a finite element method, discretizing sound field data and magnetic field data, constructing a system matrix reflecting the relation between sound pressure and concentration partial derivative, reconstructing concentration partial derivative distribution by using an LSQR method, and reconstructing by using a trapezoidal formula method to obtain a concentration distribution image.
The technical solution of the present invention is further described in detail by the accompanying drawings and examples.
Drawings
FIG. 1 is a flow chart of a method for reconstructing a magnetoacoustic particle density image in a saturation magnetization state based on a least square QR decomposition method-trapezoidal formula method.
FIG. 2 is a simulation model.
Figure 3 shows the 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 graph of simulation results.
FIG. 6b is a target concentration graph.
Detailed Description
Example 1
Magnetic-acoustic-magnetic particle concentration imaging method
As shown in fig. 1, a 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 relation between sound pressure and a concentration partial derivative is constructed, concentration partial derivative distribution is reconstructed by an LSQR method, and a concentration distribution image is reconstructed by a trapezoidal formula method. As shown in fig. 1, the present invention is realized by the following technical solutions: the invention provides a method for reconstructing a magnetoacoustic particle concentration image in a saturation magnetization state based on an LSQR-trapezoidal formula method, which specifically comprises the following steps:
s1, setting initial simulation conditions of the magnetic-acoustic-magnetic particles in a saturation magnetization state, and acquiring sound pressure data at the ultrasonic transducer and gradient magnetic field data at a reconstruction region based on a preset simulation model;
further, the simulation model building method of the magnetoacoustic-magnetic particles in the saturation magnetization state is as follows: a two-dimensional axisymmetric model is established by means of COMSOL multiphysics5.6 for simulation study, as shown in fig. 2, a coil material is copper, the coil material is placed with a z-axis as a central axis, the radius of the coil is r equal to 150mm, an origin is taken as a coil center, and a middle cylindrical region is taken as a study region.
1. Simulation conditions
(1) Parameters of magnetic nanoparticles
Obtained from EMG308 (Ferrotec (USA) Corporation) and having the specifications shown in Table 1.
TABLE 1 EMG308 Specifications
Figure BDA0003539072200000071
(2) Excitation current
Constant current with 50A in the counterclockwise direction is simultaneously introduced into the upper and lower coils of the Helmholtz coil, and the generated magnetic field intensity is about 1.93 multiplied by 105A/m is larger than the saturation magnetic field intensity of EMG308, and the magnetic nano particles reachTo a saturated state. The current in the counterclockwise direction is fed into the upper coil of the Maxwell coil, the current in the clockwise direction with the same magnitude is fed into the lower coil, and the currents with the time characteristics shown in fig. 3 are fed into the coils.
Specifically, initializing simulation conditions of magnetic nanoparticle parameters and currents of a Helmholtz coil and a Maxwell coil, selecting EMG308 as a model as magnetic nanoparticles, placing magnetic nanoparticle groups at the center of the coil, respectively introducing currents into the Helmholtz coil and the Maxwell coil, and providing a static magnetic field B by the Helmholtz coilsatThe magnetic nano particles are saturated, and a uniform gradient magnetic field B is provided by a Maxwell coilg(ii) a As shown in fig. 4, a circle is drawn with a fixed scanning radius centering on the magnetic nanoparticle group, 165 ultrasonic transducers are discretely arranged on the circle, 470 sound pressure data of 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 area as a reconstruction area by taking the magnetic nanoparticle group as a center, carrying out finite element division on the reconstruction area, dividing the reconstruction area into 250 multiplied by 250 grids, and extracting gradient magnetic field data at each grid.
S2, 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 a direct corresponding relation between the sound pressure and a concentration partial derivative of the magnetic nanoparticles;
the method comprises the following steps of constructing a sound pressure matrix by using sound pressure data acquired at each ultrasonic transducer, constructing a system matrix by using gradient magnetic field data at a reconstruction region, wherein the system matrix is used for describing a direct corresponding relation between the sound pressure and a concentration partial derivative of magnetic nanoparticles, and comprises the following steps:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
Figure BDA0003539072200000081
In the formula (1), f (r', t) is the magnetic force applied to the magnetic nanoparticles at the time t; r' is the sound of interestThe position of the source point, t being the time; n (r') is the concentration of magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle;
Figure BDA0003539072200000091
the gradient magnetic field is the position of the sound source point at the time t; e.g. of the typezRepresents a unit vector in the z direction;
the sound source and sound pressure relationship is as follows:
Figure BDA0003539072200000092
in the formula (2), p (r, t) is the sound pressure of any point at the time t, and the unit is Pa; r is an arbitrary point position; c. CsIs the speed of sound in biological tissue, in m/s;
Figure BDA0003539072200000093
is a sound source item; t is time; r 'is the position of the sound source point, and R is | R-R' |; Ω is the entire study area;
due to the unidirectionality of the magnetic force, the sound source term can be expressed as
Figure BDA0003539072200000094
Bringing formula (3) into formula (2) to obtain
Figure BDA0003539072200000095
Since in computing the temporal spatial distribution problem,
Figure BDA0003539072200000096
can be processed according to constant value, so that
Figure BDA0003539072200000097
From the above equation, there is a direct correspondence between the sound pressure and the concentration partial derivative, and the correspondence is described by a system matrix a, so that a matrix relation equation can be abstracted:
Ax＝b (6)
where a is the system matrix, x is the partial derivative of concentration, b is the sound pressure matrix, and the sound pressure data in the sound pressure matrix b is from p (r, t).
Combining the gradient magnetic field data according to equations (5) and (6) to obtain a system matrix A, wherein the size of the system matrix is 77550X 62500; the sound pressure data at the ultrasound transducer is used to construct a sound pressure matrix b, which has a size of 77550 x 1.
S3, obtaining the concentration partial derivative distribution of the magnetic nanoparticles by using the sound pressure matrix and the system matrix constructed in the second step and using an LSQR method;
further, obtaining the partial derivative distribution of the concentration of the magnetic nanoparticles comprises:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally solving the concentration partial derivative distribution;
the solving process comprises the following steps:
let the orthonormal matrix Uk＝[u1,u2,…,uk](uj∈Rmi) And
Figure BDA0003539072200000101
the double diagonal matrix is
Figure BDA0003539072200000102
Wherein (alpha)1,α2,…,αk∈R；β2,β3,…,βk+1∈R)。
The iterative process is as follows: the iterative process is as follows:
step 301, condition initialization
β1＝||b||；α1＝||ATu1||2；β1u1＝b；
Figure BDA0003539072200000103
Step 302, diagonalizing the coefficient matrix
βj+1＝||Avj-αjuj||2；αj+1＝||ATuj+1-βj+1vj||2；
Figure BDA0003539072200000104
(j＝1,2,...,k)。
Step 303, calculating QR decomposition intermediate variables
Figure BDA0003539072200000105
Figure BDA0003539072200000106
Step 304, updating the values of x and the intermediate variable w
Figure BDA0003539072200000111
Step 305, determining iteration conditions
If | | Ax is satisfiedk-b | ≦ epsilon and terminating the iteration, where epsilon is the allowed error, set epsilon 0.01;
repeat steps 302-305 for j 1, 2.
S4, obtaining the magnetic nanoparticle concentration distribution by utilizing the magnetic nanoparticle concentration partial derivative distribution, thereby obtaining a magnetic acoustic magnetic particle concentration reconstruction image in a saturation magnetization state.
Further, acquiring the magnetic nanoparticle concentration distribution by using the magnetic nanoparticle concentration partial derivative distribution so as to obtain a magnetoacoustic magnetic particle concentration reconstruction image in a saturation magnetization state, wherein the acquiring step comprises the following steps:
the relation between the known concentration partial derivative and the concentration distribution satisfies the following relation:
Figure BDA0003539072200000112
assuming that the side length of the grid is l, l is 0.2mm, the concentration partial derivative x can be expressed as:
Figure BDA0003539072200000113
the concentration profile N (r') can be expressed as
Figure BDA0003539072200000114
The solution is obtained by introducing a trapezoidal formula method with simple calculation and higher precision
Figure BDA0003539072200000115
Wherein c is more than or equal to 0 and less than or equal to 249, d is more than or equal to 1 and less 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 reconstructed through the steps, a concentration gradient model is established by considering that MNPs are dispersed and gradually changed in biological tissues, as shown in figure 5, a peripheral circular area is a bionic area, an internal area is an MNPs imitation area, the MNPs adopt EMG308, and the reconstruction result and the concentration reconstruction target are shown in figures 6a and 6b, the method provided by the invention can reconstruct the concentration distribution with high quality, has no singularity, introduces a correlation coefficient CC and a relative error RE for further evaluating the applicability of the invention, and compares the target concentration distribution with the reconstructed concentration distribution, and specifically adopts the following formula:
Figure BDA0003539072200000121
Figure BDA0003539072200000122
in the formula NnIs the concentration value of the target nanoparticle, Nn,rIn order to reconstruct the concentration values of the nanoparticles,
Figure BDA0003539072200000123
the average value of the concentration of the target nano-particles,
Figure BDA0003539072200000124
mean values of target nanoparticle concentrations.
Through the evaluation of the invention, the correlation coefficients and the relative errors of the invention are respectively 0.9816 and 0.2761, and the reconstruction precision is high.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the present invention in any way. Any simple modification, change and equivalent changes of the above embodiments according to the technical essence of the invention are still within the protection scope of the technical solution of the invention.

Claims (6)

1. A magneto-acoustic particle concentration image reconstruction method in a saturation magnetization state is characterized by comprising the following steps:
step one, setting a simulation initial condition of magnetic-acoustic magnetic particles in a saturation 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 characterized in that a region Wmm multiplied by Wmm is selected as a reconstruction region by taking a magnetic nanoparticle swarm as a center, and the reconstruction region is subjected to finite element division and is divided into M multiplied by M grids; the gradient magnetic field data at the reconstruction region includes gradient magnetic field data at each of the meshes;
secondly, 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 a direct corresponding relation between the sound pressure and a concentration partial derivative of the magnetic nano particles;
thirdly, acquiring the concentration partial derivative distribution of the magnetic nanoparticles by using the sound pressure matrix and the system matrix constructed in the second step through an LSQR (least square resonance) method;
and step four, acquiring the concentration distribution of the magnetic nanoparticles by utilizing the partial derivative distribution of the concentration of the magnetic nanoparticles, thereby acquiring a magnetic acoustic magnetic particle concentration reconstruction image in a saturation magnetization state.
2. The method for reconstructing the image of the concentration of the magnetoacoustic particles under the saturation magnetization state according to claim 1, wherein in the first step, initial simulation conditions of the magnetoacoustic particles under the saturation magnetization state are set, and based on a preset simulation model, sound pressure data at the ultrasonic transducer and gradient magnetic field data at the reconstruction region are obtained; selecting an Nmm multiplied by Nmm area as a reconstruction area by taking the magnetic nanoparticle swarm as a center, and carrying out finite element division on the reconstruction area to divide the reconstruction area into M multiplied by M grids; the gradient magnetic field data at the reconstruction region comprises gradient magnetic field data at each grid; the method comprises the following steps:
initializing simulation conditions of magnetic nanoparticle parameters and currents of a Helmholtz coil and a Maxwell coil, selecting EMG308 as magnetic nanoparticles, placing magnetic nanoparticle groups at the central position of the coil, respectively introducing currents into the Helmholtz coil and the Maxwell coil, and providing a static magnetic field B by the Helmholtz coilsatThe magnetic nano particles are saturated, and a uniform gradient magnetic field B is provided by a Maxwell coilg(ii) a Drawing a circle by using the fixed scanning radius with the magnetic nanoparticle group as a 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 an image of a concentration of magnetic, acoustic, and magnetic particles under a saturation magnetization state according to claim 1, wherein a circle is drawn with a fixed scanning radius centering on the group of magnetic nanoparticles, 165 ultrasonic transducers are discretely disposed on the circle, and sound pressure data of 470 time points received by each ultrasonic transducer is obtained; selecting a 50mm multiplied by 50mm area as a reconstruction area by taking the magnetic nanoparticle group as a 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 an image of the concentration of magnetoacoustic particles under a saturation magnetization state according to claim 2, wherein a sound pressure matrix is constructed by using sound pressure data acquired at each ultrasonic transducer, and a system matrix is constructed by using gradient magnetic field data at the reconstruction region, the system matrix being used for describing a direct correspondence relationship between the sound pressure and partial derivatives of the concentration of the magnetic nanoparticles, and the method comprises:
the magnetic force to which the magnetic nanoparticles are subjected is expressed as
Figure FDA0003539072190000021
In the formula (1), f (r', t) is the magnetic force applied to the magnetic nanoparticles at the time t; r' is the position of the sound source point, and t is time; n (r') is the concentration of magnetic nanoparticles; m represents the magnetic moment of the magnetic nanoparticle;
Figure FDA0003539072190000022
the gradient magnetic field is the position of the sound source point at the time t; e.g. of the typezRepresents a unit vector in the z direction;
the sound source is related to the sound pressure by the following equation:
Figure FDA0003539072190000023
in the formula (2), p (r, t) is the sound pressure at any point of time t, and the unit is Pa; r is an arbitrary point position; c. CsIs the speed of sound in biological tissue, in m/s; · (r, t) is sound source item; t is time; r 'is the position of the sound source point, and R is | R-R' |; Ω is the entire study area;
due to the unidirectionality of the magnetic force, the sound source term can be expressed as
Figure FDA0003539072190000024
Bringing formula (3) into formula (2) to obtain
Figure FDA0003539072190000031
Since in computing the temporal spatial distribution problem,
Figure FDA0003539072190000032
can be processed according to constant value, so that
Figure FDA0003539072190000033
From the above equation, there is a direct correspondence between the sound pressure and the concentration partial derivative, and the correspondence is described by a system matrix a, so that a matrix relation equation can be abstracted:
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); the system matrix a is obtained according to equations (5) and (6) in combination with the gradient magnetic field data.
5. The method for reconstructing the image of the concentration of the magnetoacoustic-magnetic particles under the saturation magnetization state according to claim 2, wherein the step three of obtaining the partial derivative distribution of the concentration of the magnetic nanoparticles by using the LSQR method by using the sound pressure matrix and the system matrix constructed in the step two comprises:
introducing an LSQR method, solving a system matrix serving as a large sparse coefficient matrix, and finally solving the concentration partial derivative distribution;
the solving process comprises the following steps:
let the orthonormal matrix Uk＝[u1,u2,…,uk](uj∈Rmi) And
Figure FDA0003539072190000034
the double diagonal matrix is
Figure FDA0003539072190000035
Wherein (alpha)1,α2,…,αk∈R；β2,β3,…,βk+1∈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；
Figure FDA0003539072190000041
Step 302, diagonalizing the coefficient matrix
βj+1＝||Avj-αjuj||2；αj+1＝||ATuj+1-βj+1vj||2；
Figure FDA0003539072190000042
(j＝1,2,...,k)。
Step 303, calculating QR decomposition intermediate variables
Figure FDA0003539072190000043
Figure FDA0003539072190000044
Step 304, updating the values of x and the intermediate variable w
Figure FDA0003539072190000045
Step 305, determining iteration conditions
If | | | Ax is satisfiedk-b | ≦ epsilon and terminating the iteration, where epsilon is the allowed error, set epsilon 0.01;
repeat steps 302-305 for j 1, 2.
6. The method for reconstructing an image of the concentration of magnetoacoustic magnetic particles under a saturation magnetization state according to claim 3, wherein the step four of obtaining the reconstructed image of the concentration of magnetoacoustic magnetic particles by using the magnetic nanoparticle concentration partial derivative distribution comprises:
the relation between the known concentration partial derivative and the concentration distribution satisfies the following relation:
Figure FDA0003539072190000046
assuming that the side length of the grid is l, l is 0.2mm, the concentration partial derivative x can be expressed as:
Figure FDA0003539072190000047
the concentration profile N (r') can be expressed as
Figure FDA0003539072190000051
The solution is obtained by introducing a trapezoidal formula method with simple calculation and higher precision
Figure FDA0003539072190000052
Wherein c is more than or equal to 0 and less than or equal to 249, d is more than or equal to 1 and less 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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