An improved method for quantitative recovery of conductivity using tomographically measured thermoacoustic data

Abstract

Noninvasive extraction of tissue conductivity distribution is important in brain imaging and cancer detection. Here we present an improved method that can accurately image tissue conductivity using tomographically measured microwave-induced thermoacoustic data. Our reconstruction algorithm is first tested using simulations, and then validated using tissue phantom experiments where saline-containing tubes are used as target(s) with various target sizes, positions and conductivities. The average error of reconstruction for the simulations is reduced from 4.87% to 1.38% compared with the previous algorithm. The experimental results obtained suggest that accurate quantitative thermoacoustic imaging would provide a potential tool for precision medicine.

Keywords

Thermoacoustic imaging conductivity distribution quantitative thermoacoustic tomography method

1 Intoduction

Tissue electromagnetic (EM) properties such as conductivity can be used as a fingerprint to identify diseased tissue [1 –3]. For example, malignant breast or brain tumor has higher conductivity than benign or normal tissue [4 –7]. Microwave-induced thermoacoustic tomography (TAT) based on EM properties is a high resolution hybrid imaging method combining the advantages of both microwave and ultrasound imaging, and thus far has been developed [8 –16] for breast cancer diagnosis [9 –11], brain imaging [12], angiography [13 , 17], and joint imaging [15]. When microwave irradiate to a target, the target absorbs microwave energy and expands, producing a mechanical wave that travels outward. By detecting this mechanical wave, a TAT image containing the EM properties information is formed. However, because of the highly inhomogeneous distribution of microwave field in tissue [18, 19], the energy loss reconstructed by traditional TAT methods cannot quantitatively characterize the distribution of EM properties such as conductivity. Quantitative thermoacoustic tomography (qTAT) is an extension of TAT, which can quantitatively recover tissue conductivity distribution and remove the influence of the inhomogeneous field distribution on the recovered images [8 , 20– 22].

In our previous work [8, 22], we showed that conductivity distribution could be recovered by a finite element-based reconstruction algorithm coupled with the Helmholtz equation. However, an assumption of a uniform EM field distribution was made. In addition, calibration was required to estimate the microwave strength and boundary parameters. In this study, we describe an improved qTAT method to remove the uniform EM field assumption and calibration process, the two limitations associated with our previous qTAT method, using an adaptive minimization strategy and the actual inhomogeneous EM field distribution from the antenna waveguide used. To further optimize our new method, deionized water coupled with a dielectric matching layer was used to reduce the influence of microwave scattering.

2 Materials and methods

2.1 Quantitative thermoacoustic tomography method

Propagation of thermoacoustic signals in tissue can be described by the following thermoacoustic wave equation [22]: $\nabla^{2} p (r, t) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} p (r, t) = - \frac{β σ}{C_{p}} \frac{\partial}{\partial t} {| E (r, t) |}^{2}$ (1) where p is the pressure wave, t is the time, r is the spatial variable, c is the acoustic velocity, C_p is the specific heat capacity at constant pressure, and β is the thermal coefficient of volume expansion, σ is the electrical conductivity, and E is the electric field.

The energy loss distribution s ( r ) = σ|E (r) |², can be recovered by our finite element-based qTAT algorithm [22]. In brief, Eq. (1) was firstly discretized according to finite element method, and then an iterated scheme was implemented based on Tikhnove regularization method, and finally the energy loss density was iteratively calculated. The conductivity distribution can be obtained [22] by substituting s(r) into the following Helmholtz equation: $\nabla \cdot (\frac{1}{μ_{r}} \nabla E_{s}) + k^{2} ɛ_{r} E_{s} = 0$ (2) where μ _r is the relative permeability, E_s is the incident electric field, k is the wave number, $ɛ_{r} = ɛ + i \frac{σ}{ω}$ is the effective dielectric constant, ɛ is the relative dielectric constant, and ω is the angular frequency.

2.2 Adaptive quantitative thermoacoustic tomography (a-qTAT) method

In TAT, a waveguide is typically used as the antenna to deliver the microwave to the object via coupling medium (e.g., oil). In a waveguide, the electric field in TE₁₀ mode can be written as: $E_{x} = E_{z} = 0$ (3) $E_{y} = - \frac{i ω μ a}{π} A_{10} sin (\frac{π x}{a}) e^{- i β z}$ (4) where E_x,E_y,E_z are the electric field components in the x, y and z directions, respectively, a is width of waveguide, A₁₀ is the amplitude constant in TE₁₀ mode, and β is the propagation constant. Because E_y distribution is symmetric, the propagation of microwave can be approximated to a two-dimensional plane with y = 0. Therefore, the incident electric field E_s can be calculated through the electric field distribution of the waveguide.

In our previous work, the uniformly distributed plane wave model (UDPW), which assumed uniform energy distribution, was used in the conductivity reconstruction. The distribution can be considered as homogeneous in small reconstruction area, but for a large areas, UDPW does not accurately recover the distribution. In this work, the conductivity reconstruction is based on the two-dimensional (2D) waveguide model (2DWT) which describes microwave radiation more accurately than UDPW. The electric field (E-field) distributions given by the two models are compared and shown in Fig. 1. Here, the E-field distributions obtained by CST^® simulation software are regarded as the actual ones. We calculated the E-field distributions in x-y plane (see Fig. 1 (b) and 1 (c)), while the E-field distributions given by the UDPW and the 2DWT are shown in Fig. 1 (b) and 1 (c), respectively. To quantitatively compare the two models, the E-field distribution profiles at y = 0 are shown in Fig. 1 (d). From Fig. 1, we can see that the 2DWT model describes the electric field distribution more accurately than the UDPW model.

Fig.1

Comparison of the uniformly distributed plane wave model (UDPW) and the 2D waveguide transmission model (2DWT). Electric field distributions given by the actual one (a), UDPW (b), 2DWT (c), and profiles (d) along a transect crossing the center in (a), (b) and (c), respectively.

To eliminate the calibration procedure used in our previous qTAT algorithm, the following adaptive method is adopted here: (a) the energy loss distribution s _r ( r ) is reconstructed using thermoacoustic signals; (b) estimate the possible range of conductivity σ_e and relative dielectric constant ɛ _e values of the object; (c) a series of energy loss distributions, s _ij (r), are calculated by substituting all the possible conductivity and relative dielectric constant values into Eq. (2) with E_s. where i represents the serial number of different relative dielectric constants and j represents the serial number of different conductivity; (d) calculate error map: err_ij = ∥ s_ij ( r )-s_r ( r ) ∥ ₂; (e) Steps (a)-(d) are repeated until a minimum is reached where conductivity distribution σ_c is obtained at the minimum. In this calculation, an average ɛ _e value of target is used. Furthermore, for experimental datas, some pre-processing including detrend, resample and threshold was performed before the back-projection imaging reconstruction.

2.3 Phantom experiments system

For the phantom experiments, a 1 GHz microwave generator (peak power 1– 4 kW, frequency 0.95– 1.2 GHz, pulse duration 0.25– 50μs, and repetition rate 20– 25000 Hz; PG5KB, Epsco USA) was used to generate microwave pulses, which were delivered to the object through a home-made waveguide filled with castor oil (relative permittivity ɛ = 4.3) coupled with a composite matching layer (ɛ = 15.5, loss tangent <0.001, thickness 18.5 mm; TP-1/2, Taizhou Wangling Insulating Materials Factory, Taizhou, China) [23]. Given the pulse width of our microwave generator, the spatial resolution is not higher than 0.19 mm [24]. A customized ultrasonic transducer (central frequency: 2.55 MHz, active-element diameter: 25.4 mm; 3dB bandwidth:100%; IMASONIC SAS, France) was used for thermoacoustic signal collection, and the signal was amplified by a home-made low noise amplifier (0.2– 2MHz, 60dB), and digitized by a 50 MHz sampling rate acquisition card (PCI4732, VIDTS. Inc., China). In this study, limited by the waveguide position (see Fig. 2), the transducer could be rotated only over 320° around the phantom in the x-y plane, and, an interval of 2° was used here to collect the signals.

Fig.2

Schematic of the experimental setup.

3 Simulations and experiments results

We conducted several simulations and phantom experiments to test and validate the proposed a-qTAT approach. In the simulations, a single target with a diameter of 3 mm, a conductivity of 8.18 S/m and a relative dielectric constant of 62 was embedded in a background having 40 mm in diameter, a conductivity of 0.001 S/m and a relative dielectric constant of 78. The exact conductivity map and simulation results obtained using both qTAT and a-qTAT are presented in Fig. 3 (a)– (c), respectively. To better compare the differences among the images Fig. 3, image profiles across the target center are shown in Fig. 3 (d). Furthermore, the err _ij during the calculation is shown in Fig. 3 (e). In err _ij map£¬the difference between the simulated and the exact absorption distributions of the target under a combination of conductivity and relative permittivity is described. From this map, the relative error is minimized when conductivity is 8 S/m and the relative permittivity is 63. Therefore, an average is ɛ _e = 63 of target was used in the iterative calculation of σ _c in this case. This iterative process was detailed elsewhere [22].

Fig.3

Simulations using a-qTAT and qTAT. The exact conductivity map (a), reconstructed conductivity distribution by qTAT (b) and a-qTAT (c);(d): conductivity profiles along a transect crossing the background, (e): err _ij during calculation.

In the first experiment, saline-containing tubes were used as the target(s) where the conductivity and relative dielectric constant were varied by changing the concentration of salt in the target(s). Given the concentration of added salt, the conductivity and relative dielectric constant of the targets could then be calculated through the Debye equation [25]. In the first experiment, a target with a diameter of 3 mm, a conductivity of 8.18 S/m and a relative dielectric constant of 62 at 1 GHz was imaged. A single target was placed at different positions, and the resulting images at all the positions were then fused in a single image, shown in Fig. 4(a). It can be seen that the target is quantitatively recovered at these different positions without the impact of the uneven field distribution. In addition, the typical time domain and spectrum of thermoacoustic signals for this case is shown in Fig. 4(b) and (c), respectively.

Fig.4

Fused conductivity image for a single target located at different positions (a), typical time domain and spectrum of thermoacoustic signal (b-c).

In the second experiment, two cases were examined. In the first case, NaCl solution with a conductivity of 8.18 S/m and a relative dielectric constant of 62 on 1 GHz was filled in a plastic tube (target) with a diameter of 5.2, 4.5 and 3.0 mm (see Table 1). In the second case, NaCl solutions with a conductivity of 8.18, 12.57 and 10.45 S/m and a relative dielectric constant of 62, 52 and 57 on 1 GHz were, respectively, filled in a plastic pipe with a diameter of 5.2, 4.5 and 3.0 mm (see Table 2). The conductivity images reconstructed by a-qTAT are shown in Figs. 5 (a) and 6 (a), respectively, for cases 1 and 2. Furthermore, the profiles along lines crossing the center of each target (Figs. 5a and 6a) are shown in Fig. 5 (b)– 5(d) and Fig. 6 (b)– 6(d), respectively. We immediately note that the size and conductivity value of the target(s) are well reconstructed for both cases.

Table 1

The parameters used for case 1

	Diameter(mm)	Position	ɛ	σ(S/m)
Background			78.4	5.55×10^–6
Target 1	5.2	Top	62	8.18
Target 2	4.5	Middle	62	8.18
Target 3	3.0	Bottom	62 8.18

Table 2

The parameters for case 2

	Diameter(mm)	Position	ɛ	σ(S/m)
Background			78.4	5.55×10^–6
Target 1	5.2	Top	62	8.18
Target 2	4.5	Middle	52	12.57
Target 3	3.0	Bottom	57	10.45

4 Discussion and conclusion

Quantitatively, from Fig. 3, we found that the average error in reconstruction for a-qTAT is only 1.38% compared to an error of 4.87% for qTAT in simulations. Meanwhile, we note that, the images recovered by qTAT show pitting/artifacts, which is corrected in the new method. In simulation, the conductivity σ _e and relative dielectric constant ɛ _e used in adaptive estimation process is 1, 2, 3, \dots , 15 S/m and 51, 53, 55, \dots , 99, respectively. From the calculation of err _ij (shown in Fig. 2(e)), the err _ij reaches the minimum at an optimal value of ɛ _e . Therefore, ɛ _e = 63 was selected as the average relative dielectric constant in the target for conductivity reconstruction. In addition, by comparing Fig. 4 with Figs. 5 and 6, we also noted that no hollow regions in the image shown in Fig. 4 are larger than those in Figs. 5 and 6 (in the phantoms with a diameter of 3 mm, but hollow region in the phantoms with a diameter of 4.5 and 5.2 mm vs. 3 mm in diameter). The hollow region are due to the limited bandwidths [26] of the transducer and amplifier used in our imaging system. We plan to optimize the hardware to solve this problem. In addition, we will consider the optimization of the adaptive process to improve its computational speed. Finally, we will explore the applications of quantitative thermoacoustic imaging technique in breast cancer detection and brain imaging.

Fig.5

Reconstructed conductivity image (a) and profiles (b-d) for case 1.

Fig.6

Reconstructed conductivity image (a) and profiles (b-d) for case 2.

In conclusion, we have presented an adaptive qTAT method that can provide significantly improved accuracy for recovery of conductivity over previous qTAT methods. We plan to further evaluate its utility through in vivo experiments in future.

Footnotes

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (61701076 and 11674047).

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