Photoacoustic image reconstruction method enhanced by ultrasound tomography

Abstract

The present invention relates to the technical field of medical image reconstruction, and discloses a method for photoacoustic image reconstruction enhanced by ultrasonic tomography, comprising: each array element of the transducer emits an ultrasonic signal in turn, and other array elements receive the ultrasonic signal, and the sound velocity information of the tissue is reconstructed through ultrasonic tomography technology; a pulsed laser is used to irradiate the tissue, and the transducer is used to receive the photoacoustic signal emitted by the tissue; a linear equation group is established to describe the relationship between the initial pressure distribution and the photoacoustic signal, and the equation group is iteratively solved to reconstruct the initial pressure distribution. The present invention uses the same device to realize dual-modal imaging of ultrasonic tomography and photoacoustic tomography, and obtains a sound velocity distribution diagram and a photoacoustic original signal respectively. By introducing the sound velocity information of ultrasonic tomography into the reconstruction of the photoacoustic image, the reconstruction quality of the photoacoustic image is effectively improved. Experimental and simulation results show that the method has high operability and application effectiveness.

Description

Technical Field

[0001] The present invention relates to the technical field of medical image reconstruction, and in particular to a photoacoustic image reconstruction method enhanced by ultrasonic tomography. Background Art

[0002] Photoacoustic computed tomography (PACT) is an emerging non-invasive biomedical imaging modality with the advantages of high spatiotemporal resolution, rich contrast, and no ionizing radiation. In PACT, after the laser pulse irradiates the tissue, the tissue absorbs light and generates a transient pressure distribution (called the initial pressure distribution). The initial pressure distribution propagates in the medium, thereby generating an acoustic wave signal (called the photoacoustic signal). The ultrasound receiver captures the acoustic wave signal and converts it into an electrical signal, and then generates an image of the tissue through signal processing and image reconstruction algorithms.

[0003] Based on the implementation principle of photoacoustic tomography, image reconstruction is very critical in the imaging process and directly affects the image quality of the imaging system. At present, the mainstream photoacoustic image reconstruction algorithms are all based on the assumption that the imaging medium is uniform and has a single sound velocity. However, in the actual imaging environment, the existence of acoustic heterogeneity makes the assumption invalid, which ultimately leads to inaccurate image reconstruction. Under this limitation, iterative reconstruction (IR) algorithm, as a numerical algorithm that can couple the imaging system model and consider the medium inhomogeneity, can provide higher quality image reconstruction in actual imaging.

[0004] The prior art has established an iterative forward numerical model of photoacoustic in inhomogeneous media, obtained the system matrix in inhomogeneous media through discretization, and verified the effectiveness of the iterative reconstruction algorithm in inhomogeneous media through simulation. In addition, the prior art also analyzes the significance of the conjugate transpose operator of the system matrix in photoacoustic tomography, and provides a theoretical tool for considering medium inhomogeneity in the iterative reconstruction algorithm. These studies have shown that whether acoustic heterogeneity is considered in the reconstruction algorithm of photoacoustic tomography has an important impact on the reconstruction quality of the photoacoustic image. However, there are currently few studies that apply the iterative reconstruction algorithm in inhomogeneous media to actual experiments to effectively improve the reconstruction quality of photoacoustic images in experimental data.

[0005] Since the inhomogeneity of the medium needs to be considered in the iterative reconstruction algorithm of photoacoustic tomography, it is very important to accurately obtain the sound velocity information of the medium. Ultrasound computed tomography (USCT) is an imaging technology similar to X-ray computed tomography. It detects the sound wave signal that penetrates the tissue and inverts the acoustic properties of the tissue, such as the sound velocity information. Summary of the invention

[0006] In order to solve the above technical problems, the present invention provides a method for photoacoustic image reconstruction enhanced by ultrasonic tomography. First, the effectiveness of the method in the present invention is verified by simulation in inhomogeneous media of different degrees. Secondly, dual-modal imaging of ultrasonic tomography and photoacoustic tomography is realized by the same set of devices to obtain the sound velocity distribution and photoacoustic signal respectively. Finally, by coupling the medium sound velocity information reconstructed by ultrasonic tomography to the iterative reconstruction of photoacoustic tomography, the photoacoustic image quality of leaf vein phantoms and living mice is improved. The results show that the iterative reconstruction of photoacoustic images enhanced by ultrasonic tomography has application effectiveness and experimental operability, can effectively improve the quality of photoacoustic images, and has important research significance and application value.

[0007] The present invention adopts the following technical solution:

[0008] A method for photoacoustic image reconstruction enhanced by ultrasonic tomography comprises the following steps:

[0009] Step 1: After the pulsed laser irradiates the tissue, the photoacoustic signal emitted by the tissue is detected by the transducer;

[0010] Step 2: establish a linear equation group to describe the relationship between the initial pressure distribution and the detected photoacoustic signal, and obtain the reconstructed initial pressure distribution, i.e., the photoacoustic image, by iteratively solving the linear equation group, so that the error between the photoacoustic signal estimated according to the reconstructed initial pressure distribution and the photoacoustic signal actually detected by the transducer is minimized, and output the initial pressure distribution at this time as the optimal solution, specifically including: according to the physical process of imaging, the linear equation group is expressed as:

[0011] ; (1)

[0012] in, is the system matrix, which is the mapping between the initial pressure distribution and the photoacoustic signal. is the vector representation of the photoacoustic image; the solution of equation (1) is transformed into the following problem:

[0013] ; (5)

[0014] in, Represents auxiliary variables; is the vector representation of the photoacoustic signal; is the regularization term; is the regularization penalty coefficient; represents the L2 norm; is the penalty coefficient; the rotation iteration algorithm is used to solve equation (5), and the sound velocity information of the tissue is incorporated into the solution process through the k-Wave toolkit.

[0015] Furthermore, the photoacoustic image to be reconstructed Divide into pixel regions, the grayscale value in each pixel region is considered constant, and the vector representation of the photoacoustic image is , express Length, for No. Elements, superscript represents transposition; the transducer comprises array elements, and the sampling length of each array element is , then the vector representation of the detected photoacoustic signal is , is the total sampling length of all elements of the transducer, ; for The elements.

[0016] Furthermore, the process of solving the sound velocity information of the tissue specifically includes:

[0017] Based on the transducer used in step 1, reconstructing the sound velocity information of the tissue by ultrasonic tomography technology;

[0018] A single array element of the transducer sends out an ultrasonic signal, and other array elements of the transducer receive the ultrasonic signal;

[0019] By repeating the process of sending and receiving ultrasonic signals for each array element and extracting the propagation time of the ultrasonic signal from sending to receiving, the sound velocity information of the tissue is reconstructed through the inversion reconstruction algorithm.

[0020] Furthermore, the k-Wave toolkit is used to incorporate the tissue sound velocity information into the solution process, specifically including: the operation of formula (1) represents the forward model, and the operation of formula (5) represents the reverse model; the k-Wave toolkit of MATLAB can incorporate the tissue sound velocity information into the implementation process of the forward model and the reverse model.

[0021] Furthermore, the use of a rotation iteration algorithm to solve equation (5) specifically includes the following steps:

[0022] S231, fixed is a known quantity, Perform optimization solution:

[0023] ;

[0024] S232, fixed is a known quantity, Perform optimization solution:

[0025] ;

[0026] S233, Update :

[0027] ;

[0028] Repeat steps S231 to S233 until Reaching the preset threshold .

[0029] Compared with the prior art, the beneficial technical effects of the present invention are:

[0030] The modular solution framework of the iterative reconstruction algorithm under inhomogeneous media proposed in the present invention transforms the direct solution of the objective function into multiple independent sub-problems for research. The modular solution framework of the iterative reconstruction algorithm has the stability of modular solution and can standardize the reconstruction process; at the same time, it has the flexibility of easy modification of regularization terms and can apply different denoising methods according to actual problems.

[0031] The ultrasonic tomography-enhanced photoacoustic image reconstruction proposed in the present invention can effectively improve the reconstruction quality of photoacoustic images. The present invention can realize dual-modal imaging of ultrasonic tomography and photoacoustic tomography through the same set of devices, couple the sound velocity information obtained by ultrasonic tomography reconstruction into the photoacoustic iterative reconstruction algorithm, and realize the improvement of the quality of photoacoustic image reconstruction. At the same time, the ultrasonic tomography-enhanced photoacoustic image reconstruction is realized through simulation verification, leaf vein phantom and mouse in vivo experimental results.

[0032] The present invention focuses on the iterative reconstruction algorithm in photoacoustic tomography and proposes a photoacoustic image reconstruction method enhanced by ultrasonic tomography. The main contents include a modular solution framework of the iterative reconstruction algorithm in inhomogeneous media and a photoacoustic image reconstruction scheme enhanced by ultrasonic tomography. The method proposed in the present invention effectively solves the limitation of sound velocity heterogeneity in the actual photoacoustic imaging environment, can improve the reconstruction quality of the actual photoacoustic image, and has important research significance and application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a schematic diagram of the principle of the photoacoustic image reconstruction method enhanced by ultrasonic tomography of the present invention;

[0034] Figure 2 It is a schematic diagram of iterative simulation reconstruction results in a simple inhomogeneous medium according to an embodiment of the present invention;

[0035] Figure 3 A schematic diagram of iterative simulation reconstruction results in a complex inhomogeneous medium according to an embodiment of the present invention;

[0036] Figure 4 This is a schematic diagram of the reconstruction results of the leaf vein phantom iterative experiment under the non-uniform medium in one embodiment of the present invention;

[0037] Figure 5 This is a schematic diagram of the reconstruction results of an iterative experiment on a mouse in vivo in a non-uniform medium according to an embodiment of the present invention. DETAILED DESCRIPTION

[0038] A preferred embodiment of the present invention is described in detail below with reference to the accompanying drawings.

[0039] The main purpose of the present invention is to propose a method for photoacoustic image reconstruction enhanced by ultrasonic tomography, which can effectively improve the quality of photoacoustic images in actual imaging environments.

[0040] In order to achieve the above-mentioned object, the present invention includes a modular solution framework of an iterative reconstruction algorithm and a photoacoustic image reconstruction scheme for ultrasonic tomography enhancement.

[0041] 1. Modular solution framework of iterative reconstruction algorithm:

[0042] Here, the principle of the iterative reconstruction algorithm is first given. The iterative reconstruction algorithm updates the initial pressure distribution in an iterative manner so that the error between the photoacoustic signal estimated according to the initial pressure distribution and the photoacoustic signal collected by the actual transducer is minimized, and the optimal solution of the initial pressure distribution at this time is output, that is, the reconstructed photoacoustic image is obtained.

[0043] The basic idea of ​​the iterative reconstruction algorithm is to establish a linear equation system to describe the relationship between the initial pressure distribution and the detected photoacoustic signal, and to obtain the optimal initial pressure distribution by iteratively solving the linear equation system. , which is divided into pixel area, and the gray value in each pixel area is considered constant, then the vector representation of the photoacoustic image is .

[0044] The transducer for receiving the photoacoustic signal comprises array elements, and the sampling length of a single array element is , then the vector representation of the photoacoustic signal is ,in, is the total sampling length of all elements of the transducer, According to the physical process of imaging, the forward model can be expressed as:

[0045] ; (1)

[0046] in, is the system matrix. The forward process of the photoacoustic imaging system is the mapping between the initial pressure distribution and the photoacoustic signal, and its accuracy directly affects the quality of the reconstructed image of the imaging photoacoustic imaging system.

[0047] In the actual iterative reconstruction algorithm, the problem is transformed into a least squares problem:

[0048] ; (2)

[0049] in, represents the L2 norm, It means that when equation (3) is minimized Since the solution process is ill-conditioned, a regular term is generally added to optimize the solution process, and the problem can be transformed into:

[0050] ; (3)

[0051] in, is the regularization term; the first part on the right side of formula (3) represents the fidelity term, and the second part represents the penalty term.

[0052] Next, rewrite equation (3) into the following equivalent form:

[0053] ; (4)

[0054] in, is the system matrix; It is a photoacoustic signal; is the regularization term; is the regularization penalty coefficient, which is used to control the influence of the regularization term.

[0055] Introducing a penalty coefficient and an auxiliary variable , this auxiliary variable is used to In Approximation is made, so formula (4) is transformed into:

[0056] ; (5)

[0057] Among them, when When , the solution of equation (5) converges to the solution of equation (4).

[0058] The operation of formula (1) represents the forward model, and the operation of formula (5) represents the reverse model. The implementation process of these two models can take into account the sound velocity information of the inhomogeneous medium, which can be implemented through the k-Wave toolkit of MATLAB.

[0059] The present invention adopts a rotation iteration algorithm to solve equation (5), which specifically includes the following steps:

[0060] S231: Fixed is a known quantity, Perform optimization solution:

[0061] ; (6)

[0062] S232: Fixed is a known quantity, Perform optimization solution:

[0063] ; (7)

[0064] S233: Update :

[0065] ; (8)

[0066] Repeat steps S231 to S233 until Reach the preset .

[0067] The above shows the modular solution framework of the iterative reconstruction algorithm in photoacoustic tomography proposed in the present invention. The target equation to be solved is equivalently transformed from equation (4) to equation (5), and the solution of equation (5) is divided into several separate sub-problems for study, which shows that the proposed framework has the stability of modular solution. It can be seen from the steps of solving equation (5) that when different regularization terms are used, the solution method of step S232 will not change because it does not contain regularization terms; at the same time, the solution method of step S231 will be different with the selection of different regularization terms, but the solution of step S231 does not have a system matrix. The problem of solving this problem becomes a simple denoising problem. In the field of image processing, there has been in-depth research on the denoising problem of images, and different regularization terms can be modified for denoising and solving. Therefore, this framework has the flexibility to easily modify regularization terms. In addition, for the solution of step S232, the currently widely used deep learning method can be selected for denoising, which improves the scalability of the iterative reconstruction algorithm.

[0068] 2. Ultrasonic tomography enhanced photoacoustic image reconstruction scheme:

[0069] To achieve photoacoustic image reconstruction in inhomogeneous media, it is necessary to know the sound velocity information of the medium in advance (i.e., the sound velocity distribution of the medium). Therefore, how to accurately obtain the sound velocity information of the medium is very important in the iterative reconstruction algorithm in inhomogeneous media.

[0070] Ultrasonic tomography is a high-resolution imaging technology that uses ultrasound to transmit through tissue to reconstruct tissue characteristics (such as sound velocity, attenuation, and other information). In the present invention, dual-modal imaging of ultrasonic tomography and photoacoustic tomography is realized based on the same imaging device. Specifically, the sound velocity information of the tissue is first reconstructed by ultrasonic tomography, and then the sound velocity information is coupled to the photoacoustic iterative reconstruction algorithm under inhomogeneous media, and finally the enhancement process of photoacoustic image reconstruction is realized. Figure 1 The principle process of using ultrasound tomography to enhance photoacoustic image reconstruction is demonstrated. Figure 1 It can be seen that the use of a single sound velocity to reconstruct photoacoustic images in inhomogeneous media will lead to inaccurate reconstruction; by coupling the medium sound velocity information reconstructed by ultrasonic tomography in the iterative reconstruction algorithm of photoacoustic tomography, the reconstruction quality of photoacoustic images can be improved and accurate reconstruction can be achieved, such as Figure 1 As shown in the white dashed box.

[0071] The present invention is further illustrated by the following two examples.

[0072] Example 1

[0073] The effectiveness of the iterative reconstruction algorithm for image reconstruction in inhomogeneous media of the present invention is illustrated by iterative simulation reconstruction in inhomogeneous media of different degrees. The simulation object of this embodiment is a vascular phantom, the transducer is a ring array transducer containing 256 array elements, the radius is 25 mm, the sampling rate is set to 62.5 MS / s (MS / s means megasamples per second), the reconstruction grid size is 201×201, the grid spacing is 0.1 mm, and the medium is set to a non-uniform sound velocity distribution. The photoacoustic signal is obtained by the k-Wave toolkit, no noise interference is added, and the Tikhonov regularization term is added to the photoacoustic image reconstruction process.

[0074] Figure 2 The results of photoacoustic image reconstruction in simple inhomogeneous media are demonstrated. Figure 2 (a) shows the relative distribution of the sound source and the transducer; Figure 2 (b) shows the real image of the vascular phantom; Figure 2 (c) shows the sound velocity distribution in the reconstructed area, which consists of two patches of uneven sound velocity; Figure 2 (d) and Figure 2 (e) shows the photoacoustic image reconstruction results using a single sound speed of 1505 m / s and 1510 m / s respectively; Figure 2 (f) in the figure shows the coupling Figure 2 (c) is the result of photoacoustic image reconstruction after the sound velocity distribution.

[0075] Figure 3 The results of photoacoustic image reconstruction in complex inhomogeneous media are demonstrated. Figure 3 (a) shows the relative distribution of the sound source and the transducer; Figure 3 (b) shows the real image of the vascular phantom; Figure 3 (c) shows the sound velocity distribution in the reconstructed area, which is composed of multiple patches of uneven sound velocity; Figure 3 (d) and Figure 3 (e) shows the photoacoustic image reconstruction results using a single sound speed of 1505 m / s and 1510 m / s respectively; Figure 3 (f) in the figure shows the coupling Figure 3 (c) is the result of photoacoustic image reconstruction after the sound velocity distribution.

[0076] This embodiment shows the iterative simulation reconstruction results under simple inhomogeneous media and complex inhomogeneous media, and compares them with the photoacoustic image reconstruction results without considering the inhomogeneity of the medium. The results show that when the medium is a simple inhomogeneous medium, it is possible to select a suitable single sound speed to obtain relatively good results; when the medium is a complex inhomogeneous medium, it is difficult to select a suitable single sound speed to obtain relatively good results. The iterative reconstruction algorithm for the sound speed information of the coupling medium can reconstruct the object to be measured well, effectively improve the reconstruction quality of the photoacoustic image, and verify the effectiveness of the iterative reconstruction algorithm for inhomogeneous media in the present invention.

[0077] Example 2

[0078] The practical application of the photoacoustic image reconstruction method enhanced by ultrasonic tomography of the present invention is illustrated by iterative experimental reconstruction in two groups of inhomogeneous media.

[0079] First, the phantom iterative experimental reconstruction results of inhomogeneous media are presented. The experimental object is a leaf vein phantom, the sampling rate is set to 62.5 MS / s, the reconstruction grid size is 301×301, and the grid spacing is 0.1 mm. The photoacoustic signal is acquired by the experimental transducer, and the total variation regularization term is added to the photoacoustic image reconstruction process. Figure 4 The results of photoacoustic image reconstruction of leaf vein phantom experiment are shown. Figure 4 (a) shows the experimental setup. During the experiment, the phantom was placed in room temperature water for imaging. The phantom was made of two materials: high-speed gelatin inside and agar outside with a speed slightly higher than that of water. Figure 4 As shown in the upper right corner of (a), the middle layer of the phantom is the leaf vein phantom, which serves as the target of photoacoustic tomography. Figure 4(b) shows the sound velocity distribution reconstructed by ultrasound tomography, which is consistent with the experimental phantom, which is composed of high-sound-velocity gelatin, low-sound-velocity agar, and slightly lower-sound-velocity water. Figure 4 (c) shows the result of photoacoustic image reconstruction using a single sound velocity of 1490 m / s; Figure 4 (d) shows that the coupling Figure 4 The photoacoustic image reconstruction result after the sound velocity distribution in (b). Figure 4 (c) and Figure 4 As can be seen from the black circles and arrows in (d), the reconstruction result of the sound velocity distribution of the coupling medium is significantly better than the reconstruction result using a single sound velocity, which improves the quality of the photoacoustic image.

[0080] Then, the reconstruction results of the in vivo iterative experiment in inhomogeneous media were shown. The experimental subject was a mouse, the sampling rate was set to 62.5 MS / s, the reconstruction grid size was 251×301, and the grid spacing was 0.1 mm. The photoacoustic signal was acquired by the experimental transducer, and the total variation regularization term was added to the photoacoustic image reconstruction process. Figure 5 The results of photoacoustic image reconstruction of mice in vivo experiments after coupling the sound velocity distribution reconstructed by ultrasound tomography are shown. Specifically, the sound velocity distribution within the imaging range is reconstructed by ultrasound tomography, and then the photoacoustic image reconstruction of the sound velocity distribution of the coupling medium is performed. Figure 5 (a) to Figure 5 (c) Figure 5 (d) to Figure 5 (f) shows the analysis results of the mouse liver cross section, kidney and spleen cross section, respectively. Figure 5 The numbers 1 to 4 indicate the liver, spleen, left kidney, and right kidney respectively. Figure 5 (a) and Figure 5 (d) in the figure shows the sound velocity distribution of two sections reconstructed by ultrasound tomography, including the liver, spleen, kidney, normal tissue, and coupling medium water. Figure 5 (b) and Figure 5 (e) shows the photoacoustic image reconstruction results of two sections using a single sound velocity of 1507 m / s. It can be seen that Figure 5 The results show that some vascular bifurcations are inaccurately reconstructed, especially at the edge of the mouse at the boundary between the two media, as indicated by the black arrows. Figure 5 (c) and Figure 5 (f) in the figure shows the photoacoustic image reconstruction results of the sound velocity distribution of the coupling medium. From the results, the coupling of the photoacoustic image reconstruction based on the sound velocity measured by the ultrasonic tomography experiment makes the blood vessels shown by the black arrows recover from bifurcation to a focused state, achieving more accurate photoacoustic image reconstruction and improving the quality of photoacoustic images.

[0081] It is obvious to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting, and the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that all changes falling within the meaning and scope of the equivalent elements of the claims are included in the present invention, and any reference numerals in the claims should not be regarded as limiting the claims involved.

[0082] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A method for photoacoustic image reconstruction enhanced by ultrasonic tomography, characterized in that: The following steps are involved: Step 1: After the pulsed laser irradiates the tissue, the photoacoustic signal emitted by the tissue is detected by the transducer; Step 2: establish a linear equation group to describe the relationship between the initial pressure distribution and the detected photoacoustic signal, and obtain the reconstructed initial pressure distribution, i.e., the photoacoustic image, by iteratively solving the linear equation group, so that the error between the photoacoustic signal estimated according to the reconstructed initial pressure distribution and the photoacoustic signal actually detected by the transducer is minimized, and output the initial pressure distribution at this time as the optimal solution, specifically including: according to the physical process of imaging, the linear equation group is expressed as: ； （1） in, is the system matrix, which is the mapping between the initial pressure distribution and the photoacoustic signal. is the vector representation of the photoacoustic image; the solution of equation (1) is transformed into the following problem: ； （5） in, Represents auxiliary variables; is the vector representation of the photoacoustic signal; is the regularization term; is the regularization penalty coefficient; represents the L2 norm; is the penalty coefficient; the rotation iteration algorithm is used to solve equation (5), and the sound velocity information of the tissue is incorporated into the solution process through the k-Wave toolkit; The photoacoustic image to be reconstructed Divide into pixel regions, the grayscale value in each pixel region is considered constant, and the vector representation of the photoacoustic image is , express Length, for No. Elements, superscript represents transposition; the transducer comprises array elements, and the sampling length of each array element is , then the vector representation of the detected photoacoustic signal is , is the total sampling length of all elements of the transducer, ; for The elements; The method of solving equation (5) by using a rotating iteration algorithm specifically includes the following steps: S231, fixed is a known quantity, Perform optimization solution: ； S232, fixed is a known quantity, Perform optimization solution: ； S233, Update : ； Repeat steps S231 to S233 until Reaching the preset threshold .

2. The method for photoacoustic image reconstruction enhanced by ultrasonic tomography according to claim 1, characterized in that: The process of solving the sound velocity information of the tissue specifically includes: Based on the transducer used in step 1, reconstructing the sound velocity information of the tissue by ultrasonic tomography technology; A single array element of the transducer sends out an ultrasonic signal, and other array elements of the transducer receive the ultrasonic signal; By repeating the process of sending and receiving ultrasonic signals for each array element and extracting the propagation time of the ultrasonic signal from sending to receiving, the sound velocity information of the tissue is reconstructed through the inversion reconstruction algorithm.

3. The method for photoacoustic image reconstruction enhanced by ultrasonic tomography according to claim 1, characterized in that: The k-Wave toolkit is used to incorporate the tissue sound velocity information into the solution process, specifically including: the operation of formula (1) represents the forward model, and the operation of formula (5) represents the reverse model; the k-Wave toolkit of MATLAB can incorporate the tissue sound velocity information in the implementation process of the forward model and the reverse model.

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