Reconstruction of acoustic inhomogeneous property in Magnetoacoustic Tomography with Magnetic Induction

Abstract

Magnetoacoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid imaging modality proposed to reconstruct the electrical impedance property in biological tissue with high spatial resolution. The tissue in most parts of human body has inhomogeneous acoustic properties, which will significantly affect the propagations of ultrasound waves. And then it leads to potential distortion and blurring of the acoustic source and conductivity images in the reconstruction. The purpose of this paper is to reconstruct both the acoustic source and conductivity distribution in an acoustically heterogeneous medium. And a new algorithm is presented based on the generalized finite element method (GFEM) and inhomogeneous time reversal method. We established an acoustic inhomogeneous model, in which different regions of the model the acoustic signals spread with different speeds. The numerical simulation experiments showed feasibility of the new method. Then we applied the proposed algorithm to reconstruct the conductivity of a gel phantom with our MAT-MI system. The experimental results indicate that the acoustic inhomogeneity of tissues in terms of speed variation can be reconstructed in the process of MAT-MI imaging.

Keywords

MAT-MI acoustic inhomogeneity time reversal method

1. Introduction

The wealth of tissue information is precious in clinic, including acoustic properties (acoustic speed and intensity) and electrical properties (conductivity and permittivity). Therefore, tissue properties reconstruction is very attractive in biomedical imaging. It has been reported that electrical properties of biological tissue are sensitive to physiological and pathological conditions. A hybrid imaging modality named Magnetoacoustic Tomography with Magnetic Induction was introduced with theoretical and experimental investigations [1, 2]. It is proposed to avoid the “shielding effect” in Magnetoacoustic Tomography (MAT) and Hall Effect Imaging (HEI) associated with the usage of surface electrodes. It was reported that the spatial resolution of MAT-MI is better than 1 mm [3], which is higher than Electrical Impedance Tomography (EIT) and Magnetic Induction Tomography (MIT). Although Magnetic Resonance Electrical Impedance Tomography (MREIT) can achieve high spatial resolution, it is currently limited by its requirement of high level current injection. However, MAT-MI requires much less uniformity and magnetic field stability than MRI.

In MAT-MI, a sample is located in a static field and eddy currents are induced in the sample by a time-varying magnetic field. Acoustic vibration is generated due to the Lorentz force and the sample will emit acoustic waves. MAT-MI utilizes the measurements of the acoustic pressure around the sample to reconstruct the conductivity distribution, as shown in Fig. 1.

Figure 1.

Illustration of MAT-MI.

At present, most of reconstruction algorithms for MAT-MI make an assumption that the tissue is acoustically homogeneous. The tissue in most parts of human body has inhomogeneous acoustic properties, which will significantly affect the propagations of ultrasound waves. And then it leads to potential distortion and blurring of the acoustic source and conductivity images in the reconstruction. In reference [4], an extra ultrasonic transmission tomography (UTT) method is used to reconstruct the distribution of an acoustic speed in MAT-MI. Because of the introduction of a new modality it makes the process more complicated. The authors adopted the finite difference time domain (FDTD) method based on time reversal acoustic theory, and used the redundancy of data detection to reconstruct the distributions of acoustic properties in reference [5]. In their scanning processing of MAT-MI, each rotational position in rotational trace has a translate processing in order to get redundant data. Thus it caused the scanning time lengthened.

In this study, we proposed an inhomogeneous time reversal algorithm to quantitatively measure the distribution of acoustic speed in the tissue. The sound speed is used to reconstruct its source. Therefore, we reconstructed images without using additional information from other imaging methods and shorten the scan time. In this paper, firstly, the generalized finite element method is used to model and compute the forward problem of MAT-MI. Secondly, the inhomogeneous time reversal reconstruction algorithm is proposed to solve the inverse problem of MAT-MI with inhomogeneity. Finally, both the acoustic source and conductivity distribution in an acoustically heterogeneous medium are reconstructed. Using the proposed method, the problems of sources with variable acoustic speeds and conductivity distribution in MAT-MI are able to be solved and reconstructed.

2. Methods

2.1 Generalized finite element method and wave equation of MAT-MI

We use generalized finite element method [6] to model the inhomogeneity of the tissue and solve inhomogeneity medium, which is different to the traditional finite element method (FEM) [7, 8, 9]. In GFEM, the node is generalized, so it can have more than two or three generalized degrees of freedom. At each generalized node, we can take a polynomial to define a generalized type of nodal shape function $\vec{u}_{i}$ and the interpolation function $\vec{\varphi}_{i}$ . The field variable $\text{U}^{h}$ can be written as a summation

${U}^{h}=\sum\limits_{i=1}^{N}{\vec{u}_{i}\vec{\varphi}_{i}}$ (1)

where $N$ is the number of generalized node. The generalized finite element can be guaranteed to be highly accurate and easily applied to the acoustic wave equation.

The wave equation governing the pressure distribution is given as

$\nabla^{2}p(r,t)-\frac{1}{c_{\text{s}}^{2}}\frac{\partial^{2}p(r,t)}{\partial t% ^{2}}=\nabla\cdot(J\times B_{0})$ (2)

where $p$ , $r$ , $t$ and $c_{s}$ represent the pressure, the spatial point, time and the acoustic speed in the media, respectively. The MAT-MI acoustic source cover unit volume can be further expanded as

$\nabla\cdot(J\times B_{0})=\sigma\left(-\frac{\partial B_{1}}{\partial t}% \right)\cdot B_{0}+(\nabla\sigma\times E)\cdot B_{0}$ (3)

2.2 Inhomogeneous time reversal method

Time reversal algorithm [10, 11] is one of the methods for imaging reconstruction. It can be used to solve the MAT-MI inverse problem. This method is a direct reconstruction algorithm in which the source $\psi(r)$ is computed directly from the measurement $\tilde{p}(r,k)$ at the transducers. The time reversal algorithm can be written as

$\psi(r)=\frac{1}{\pi c_{0}^{2}}\oint\limits_{\partial\Gamma}{dS_{d}}\int_{-% \infty}^{\infty}{dk\tilde{p}}(r_{d},k)\tilde{n}\cdot\nabla_{d}G^{\ast}(r_{d},r)$ (4)

where $r_{d}$ is the location of transducers, $G^{\ast}(r_{d},r)$ is the complex conjugate of Green’s function, $\partial\Gamma$ is a closed surface where the transducers are located and $c_{0}$ is the background speed of sound, $\tilde{n}$ is the unit normal vector of $\partial\Gamma$ pointing to the source. In time domain, this equation is applicable to both acoustically homogeneous and inhomogeneous media.

2.3 Green’s function

From Green’s function identity

$\int\limits_{V}(\phi\nabla^{2}\psi-\psi\nabla^{2}\varphi)dV=\oint_{S}({\varphi% \nabla\psi-\psi\nabla\varphi)\cdot}dS$ (5)

Substituting $\psi={G}^{\ast}(r_{d},r)$ and $\phi=\tilde{p}(r_{d},k)$ into the Green’s identity, the left hand side of the equation becomes

$\displaystyle\int\limits_{V}{\text{\{}\tilde{p}\nabla^{2}G^{\ast}-G^{\ast}% \nabla^{2}\tilde{p}\text{\}}}dV=\int\limits_{V}{\text{\{}\tilde{p}(-\delta(r)-% k^{2}G^{\ast})-G^{\ast}(-\psi(r)-k^{2}\tilde{p})\text{\}}}dV=-\int\limits_{V}{% \text{\{}\tilde{p}\delta(r)+k^{2}\tilde{p}G^{\ast}-G^{\ast}\psi(r)-k^{2}G^{% \ast}\tilde{p}\text{\}}}dV=0$ (6)

where the source $\psi$ is computed directly form the measure $\tilde{p}$ at the transducers, $\phi$ represents the sound pressure of the k layer, $\delta(r)$ is a dielectric constant. It is assumed that the density $\rho(r)$ is homogeneous and hence the Helmholtz equation $(\nabla^{2}+k^{2})\tilde{p}=-\psi$ and $(\nabla^{2}+{k}^{2})G=-\delta(r-{r}^{\prime})$ . This equation is another form of time-reversal which can be implemented more easily since there is no derivative for the Green’s function.

2.4 Fast Fourier Transformation

To implement the inhomogeneous time reversal equation with Fast Fourier Transform (FFT), a Cartesian grid mesh is created and the volume of each element is denoted as $\Delta{V}$ .

At the transducer locations, a quantity can be computed by using

$\phi(r)=\frac{\partial\tilde{p}^{\ast}(r,k)}{\partial n}\frac{\Delta S}{\Delta V}$ (7)

where $\Delta S$ is a small surface area and its summation is the total surface area enclosing the entire volume, and $\phi(r)$ is zero elsewhere. $\phi(r)$ acts as an acoustic source and the acoustic wave $\varphi(r,k)$ can be generated as

$\varphi(r,k)=\int_{V}{\phi(r)G(r,r^{\prime})dV^{\prime}}$ (8)

Then the computed $\varphi(r,k)$ is related to the $\psi(r)$ in equation

$\psi(r)=\frac{1}{\pi c_{0}^{2}}\int_{-\infty}^{\infty}{dk\varphi^{*}(r,k)}$ (9)

where $*$ represents conjugate, $\varphi^{*}(r,k)$ is the conjugate calculation of sound pressure. The main difficulty in solving acoustic nonuniform media is that the nonuniformity is unknown. However, if the acoustic characteristics of a portion of the medium can be known in advance, the shape and size of the nonuniform medium can be estimated under conditions known to have acoustic parameters.

The main idea of the inhomogeneous time reversal consists three steps. Firstly, the homogeneous time reversal is applied to reconstruct an initial image. Secondly, the initial image is segmented into different regions where the acoustic properties are known. Thirdly, each region with the acoustic parameters is assigned and a map of inhomogeneity is formed.

Figure 2.

Generalized finite element model.

3. Numerical and experimental results

In the simulation, we established a concentric spherical GFEM model by approximating biological tissue to evaluate the performance of inhomogeneous time reversal method, as shown in Figs 2 and 3. In our numerical simulation, the GFEM is used to solve the distribution of acoustic sources in acoustic inhomogeneous material at first, and the acoustic pressure signal is then calculated. Finally, the acoustic source is reconstructed by using the time reversal algorithm.

Figure 3.

Concentric spherical model.

Figure 4.

Acoustic source [mP].

Figure 5.

Computer simulation and algorithm evaluation. (a), (b) are eddy current distribution in $x$ and $y$ axis direction, respectively [A/m ${}^{2}$ ]. (c) is the acoustic pressure signal of both in homogeneous (colored in green) and inhomogeneous (colored in red) media. (d) is reconstruction of acoustic speed distribution. (e) is obtained with the inhomogeneity map estimated from the homogeneous time reversal. (f) is solved with the inhomogeneity map estimated from the inhomogeneous time reversal. (g) is the reconstruction of conductivity distribution. (h) is line profiles of the reconstructed acoustic source at $y=$ 0.

Figure 6.

MAT-MI device.

Figure 7.

Phantom.

Figure 8.

Acoustic pressure.

Figure 3 shows a two-layer concentric spherical model used in this numerical simulation. The outer sphere has a radius of 6 mm and the speed of sound in it is 1700 m/s. The inner sphere has a radius of 3 mm and the speed of sound in it is 2000 m/s. Space sound speed is 1500 m/s. The acoustic source distribution is shown in Fig. 4. The feasibility of the algorithm is evaluated with computer simulation, as shown in Fig. 5. Figure 5a, b is eddy current distribution in the $x$ and $y$ axis respectively. From Fig. 5c, one can see the distribution of the acoustic pressure signals. The vibration source has large peaks at the boundaries, with each peak extending approximately 30 mm in the spatial dimension. This peak size is related to the numerical calculation grid. In the continuous case, it would reduce to a pulsed function. When changes occur in the inner conductivity value or inner radius, the changes can be clearly observed in the estimated pressure plots at the transducer position. Sound pressure signal of acoustic homogeneous model is detected with four distinct peaks, corresponding to the four boundaries of the simulation model. However, sound pressure signal of acoustic inhomogeneous model is detected with six distinct peaks, corresponding to both the four boundaries of the model and the sound speed variation. Since the sound speeds both in outer layer and inner layer are larger than that within the space, the distance between the third and fourth peaks (from left to right) is smaller than the distance between the second and third peaks (from left to right) in the homogeneous model. In the inhomogeneous model it can be seen in the presence of a relatively weak sound pressure signal and this signal acts as a boundary sound pressure generated reflected sound wave. The acoustic speed distribution is reconstructed as shown in Fig. 5d. The reconstructed acoustic speed distribution agrees very well with the parameters in the two-layer concentric spherical model as shown in Fig. 3. Figure 5e is obtained with the inhomogeneity map estimated from the homogeneous time reversal. While Fig. 5f is reconstructed with the inhomogeneity map estimated from the inhomogeneous time reversal. Figure 5g is the reconstruction of conductivity distribution. Figure 5h is the comparison curve among the target, the reconstructed acoustic sources both with the homogeneous time reversal and with the inhomogeneous time reversal. As shown in Fig. 5e, f and h, the acoustic source distribution is successfully reconstructed. And the correlation coefficient between the target image and the inhomogeneous reconstructed image is 99% and the relative error is less than 2%. The correlation coefficient between the target image and the homogeneous reconstructed image is 90% and the relative error is 3.5%. These results show that a much better performance can be obtained by using the proposed inhomogeneous time reversal algorithm when compared to the results obtained by using the conventional homogeneous time reversal algorithm in MAT-MI image reconstruction.

Figure 9.

Acoustic source reconstruction.

The experimental system photos are shown in Fig. 6. In Fig. 6, the static magnetic field is generated by two symmetrically placed permanent magnets. The magnetic field in the region is relatively uniform with a flux density of about 0.5 T. The excitation field which lasts 2 $\mu$ s and a center frequency of 0.5 MHz is generated by the signal generator and amplified by the power amplifier with a transistor switch driving a pair of Helmholtz coils. A submerged ultrasonic transducer is selected to measure the acoustic signal. The transducer scans the sample in a circle with the radius of 40 cm and scanning step of 1.25 ${}^{\circ}$ . The received ultrasonic signal is amplified by 50 dB gain and sampled at 5 MHz. Signal averaging was performed to improve the SNR. There is one small cylinder like gels with 0% salinity embedded in the center of a big cylinder like gel with 15% salinity, as shown in Fig. 7. The ultrasonic signal is recorded as shown in Fig. 8. By using the detected acoustic pressure data the distribution of the acoustic source is reconstructed with the inhomogeneous time reversal algorithm as shown in Fig. 9. In terms of the shape and the size of the conductivity boundaries, the reconstructed image agrees very well with the cross section of the gel phantom.

4. Conclusions and discussions

The feasibility of applying the proposed MAT-MI method for inverse computation of the inhomogeneity of tissues is studied in the paper. In the previous work, the reconstruction methods of MAT-MI were tested in soft tissues, which are acoustically homogeneous. However, most biological tissues contain parts with different acoustic properties. Therefore, a new approach for MAT-MI was developed to reconstruct the acoustic sources in an acoustically inhomogeneous medium. Firstly, the numerical results indicate that the inhomogeneity map estimated with the inhomogeneous time reversal is very close to the ground truth. Secondly, the results of inhomogeneous time reversal show drastic improvements over the homogeneous time reversal method, Finally, both the acoustic source and conductivity distribution in an acoustically heterogeneous medium are reconstructed in this paper. The results demonstrate the effectiveness of the proposed algorithm for acoustic inhomogeneous image reconstruction, and show that the proposed algorithm may become an important MAT-MI method for acoustics non-uniform tissues.

MAT-MI is a non-invasive impedance imaging approach with high spatial resolution and accuracy. In present study, our imaging model is still a bit simple, and this model may not tenable enough in some applications. In future, we will consider more complex real models, such as real head model from CT or MRI images. It could be of importance to construct a complex real model with acoustic inhomogeneity taken into account, which will contribute to achieve higher resolution in MAT-MI imaging.

In conclusion, the feasibility of imaging electrical properties and acoustic properties of the object by means of the MAT-MI approach is demonstrated with the present simulation and experiment study. With a simple concentric spherical model, the present simulation results suggest that the inhomogeneous time reversal promises to provide acoustic speed distribution and conductivity distribution with high contrast resolution in MAT-MI imaging.

Footnotes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant No. 51677053, No. 51737003, No. 51377045, No. 51077040 and No. 51607056, the Scientific and Technological Research Projects of Hebei Province China under Grant No. E2015202292, No. E2015202050, No. E2017202190, No. 15272002 and No. 15275704, the Scientific and Technological Research Project of Higher Education under Grant No. ZD2017020, No. BJ2016013, No. QN2016044 and the High Level Talent Support Program under Grant No. C2015005012.

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