A one-step method for quantitative microwave-induced thermoacoustic tomography

Abstract

BACKGROUND:

Electrical conductivity directly correlates with tissue functional information such as blood and water contents, and quantitative extraction of tissue conductivity is of significant importance for disease detection and diagnosis using microwave-induced thermoacoustic tomography (TAT).

OBJECTIVE:

The existing quantitative TAT (qTAT) approaches capable of extracting tissue conductivity require two steps for the recovery of conductivity. Such two steps approaches depend on an accurate knowledge of the microwave energy loss distribution in tissue and offer a slow computational convergence rate. The purpose of this study is to develop a new algorithm to reconstruct tissue conductivity with higher reconstruction accuracy and greater computational efficiency.

METHODS:

We propose an improved qTAT method for direct recovery of tissue conductivity from thermoacoustic data measured along the boundary with only one step without the dependence of microwave energy loss information. The feasibility of our one-step qTAT method is validated in both simulated and tissue-mimicking phantom experiments with single-target and multi-target configurations with different contrast levels.

RESULTS:

Compared with the previous two-step methods, our one-step qTAT method improves the accuracy of conductivity recovery with approximately one-fold reduction in the mean absolute error (MAE) and root mean square error (RMSE) with p-values greater than 0.05. In addition, the convergence rate is improved by more than two folds for the one-step method.

CONCLUSIONS:

The study demonstrates that new method can quantitatively reconstruct conductivity of tissue more accurately and efficiently over the existing qTAT methods, leading to potentially enhanced accuracy for disease detection and diagnosis.

Keywords

Conductivity distribution thermoacoustic tomography quantitative reconstruction method

1 Introduction

Microwave-induced thermoacoustic tomography (TAT), as a hybrid medical imaging modality, can be used to detect breast cancer [1 –3], and brain [4], joint [5, 6], and vascular diseases [7, 8], etc., because it takes the advantages of high resolution ultrasound imaging and high contract microwave imaging [8, 9]. When a microwave signal is irradiated to tissue, the tissue absorbs microwave energy and then expands to produce thermoelastic ultrasonic signals outward, which can be detected by ultrasonic transducers distributed around the tissue. Using these detected signals, a TAT image containing the tissue electrical properties [10], such as conductivity, can be formed.

It has been documented that the biological tissue conductivity that directly correlates with biological tissue structural and functional information such as water content and hemoglobin concentration [11]. Benign or normal tissue has lower conductivity than malignant breast or brain tumors [12, 13]. These physiological parameters are important for accurate diagnostic decision-making. While most of the conventional TAT methods are qualitative [14 –16], which can reconstruct the absorbed microwave energy density, etc., and cannot quantitatively characterize the distribution of electromagnetic properties such as conductivity. In addition, with the wide application of deep learning, the deep-learning-based TAT has been proposed successively [17 –20], and although the deep learning-based TAT method [20] can quantitatively reconstruct the dielectric properties of biological samples, it relies on a large number of training sets for quantitative reconstruction. Our group first proposed and developed a two-step algorithm for quantitative TAT (qTAT) reconstruction of conductivity [21], and further experimentally validated the feasibility of this qTAT in breast cancer [22, 23], which showed that it is possible to recover biological tissue conductivity by a finite element-based iterative inversion algorithm coupled with Helmholtz equation. However, the accuracy of conductivity reconstruction obtained by using a fitting approach in the second step of the original qTAT algorithm needs further improvement. Toward this end, we described a two-step regularized Newton method (RNM) [24]. However, the RNM still has a limitation of a slow convergence rate, which means that the reconstruction requires more iterations to get close to the true solution. In practical or clinical applications, the slow convergence means computationally costly. Higher accuracy in reconstruction means better accuracy of disease diagnosis. However, both the existing quantitative algorithms require two steps, and the reconstruction of conductivity depends on the accuracy of the microwave energy loss recovered in the first step.

In this study, we propose a novel one-step quantitative reconstruction algorithm to overcome the limitations associated with the previous two-step methods. Both simulated and phantom experiments are used to test and validate our improved algorithm.

2 Materials and Methods

2.1 Image reconstruction algorithm

Our reconstruction method requires only one step to directly obtain the conductivity from an initial guess of conductivity by minimizing an object function, where the initial guess of conductivity is to set an arbitrary initial value of conductivity, e.g. σ₀ = 0.001S/m. The object function is a weighted sum of the squared difference between observed and calculated acoustic field data around the boundary locations given by $Min : F = \sum_{i = 1}^{M} {(p_{i}^{o} - p_{i}^{c})}^{2}$ (1) where $p^{o} = {(p_{1}^{o}, p_{2}^{o}, p_{3}^{o}, \dots, p_{M}^{o})}^{T}$ and $p^{c} = {(p_{1}^{c}, p_{2}^{c}, p_{3}^{c}, \dots, p_{M}^{c})}^{T}$ are the measured and computed acoustic fields for i _{= 1,2,3,⋯,M} boundary locations. p^c are computed based on the finite-element solution to the following thermoacoustic wave equation in Equation (2) is written as $\nabla^{2} p (r, t) = \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} p (r, t) - \frac{β_{e} σ}{C_{p}} \frac{\partial}{\partial t} | E_{s} (r, t) |^{2}$ (2) where p_(r,t) is the acoustic pressure at position r and time t, c is the acoustic speed, β_e is the volume expansion coefficient, c_p is the specific heat capacity at constant pressure, σ is the conductivity. Following Jiang et al. [21], E_s is the electric field, which is obtained by the following Helmholtz equation $\nabla \cdot (\frac{1}{μ_{r}} \nabla E_{s}) + K^{2} ɛ_{r} E_{s} = 0$ (3) where μ_r is the relative permeability, k is the wave number, ɛ_r is the effective dielectric constant.

The following matrix equation capable of inverse solution of σ_a is stated as ${\begin{array}{l} (J^{T} J + λ^{'} I) Δ x = J^{T} (p^{o} - p^{c}) \\ σ_{a} = σ_{0} + ξ Δ x \end{array}$ (4) where Δx is the update vector for σ_a , ξ is computed from a backtracking line search [25, 26] and I is the identity matrix, and λ′ is the regularization parameter determined by combined Marquardt and Tikhonov regularization schemes. We have found that when $λ^{'} = \frac{0.5 \times trace (J^{T} J) \times er r^{2}}{N}$ , the penalty term varies with error, and the reconstruction algorithm obtains the best results, where trace(.) represents the trace of the matrix, the error function is determined as $err = \frac{{∥ p^{o} - p_{i}^{c} ∥}_{2}}{{∥ p^{o} - p_{1}^{c} ∥}_{2}}$ (5)

J is the Jacobian matrix formed by $\frac{\partial p}{\partial σ}$ at the boundary measurement, which is equivalent to the product of the sensitivity of pressure to energy density $(\frac{\partial p}{\partial φ})$ and the sensitivity of energy density to conductivity $(\frac{\partial φ}{\partial σ})$ , So, Jacobian matrix J are written as $\frac{\partial p_{i}}{\partial σ_{j}} = \sum_{k = 1}^{N} (\frac{\partial p_{i}}{\partial φ_{k}} \frac{\partial φ_{k}}{\partial σ_{j}}) (i = 1, 2, \dots, M; j = 1, 2, \dots, N; k = 1, 2, \dots, N)$ (6) in which N is nodal number of the finite element mesh in the entire solution domain, the sensitivities of $\frac{\partial p}{\partial φ}$ in each iteration can be computed by the finite element solution Equation (2) [21], and the absorbed energy density φ = σ|E_s|², the derivatives $\frac{\partial φ}{\partial σ}$ in Equation (6) are further stated as $\frac{\partial φ_{k}}{\partial σ_{j}} = {\begin{matrix} {| E_{s, k} |}^{2} + σ_{k} \cdot 2 | E_{s, k} | \cdot (\frac{\partial | E_{s, k} |}{\partial σ_{j}}) (k = j) \\ σ_{k} \cdot 2 | E_{s, k} | \cdot (\frac{\partial | E_{s, k} |}{\partial σ_{j}}) (k \neq j) \end{matrix}$ (7)

Electric field E_s in Equation (7) can be calculated by the finite element solution to Equation (3), and the finite element discretized form of Equation (3) is stated as [21] $A E_{s} = B$ (8)

Based on Equation (8), the sensitivity $\frac{\partial E_{s}}{\partial σ}$ in Equation (7) is obtained from the following equation, which is used in adjoint methods [27, 28] $A \frac{\partial E_{s}}{\partial σ} = \frac{\partial B}{\partial σ} - \frac{\partial A}{\partial σ} E_{s}$ (9)

Therefore, it is possible to quantitatively reconstruct the conductivity distribution by solving Equations (2) and (4) based on the finite element method in just one step, thus minimizing the object function given in Equation (1). A flowchart of the one-step quantitative reconstruction conductivity algorithm is shown in Fig. 1.

Fig. 1

Flowchart of this proposed study.

2.2 Experimental system

In this work, a single transducer scanning thermoacoustic imaging system (see Fig. 2) was used for the phantom experiments. The pulsed microwaves were generated by an S-band microwave generator, a microwave source with a center frequency of 3 GHz and a pulse width of 550 ns. In the experiments, the microwave energy was radiated through a standard horn antenna (aperture: 114×114 mm², gain: 10 dB) to a sample immersed in coupling agent-transformer oil, which absorbed the microwave energy and generated a thermoacoustic signal. The generated thermoacoustic signal was detected by an ultrasonic transducer (center frequency: 2.25 MHz, crystal diameter: 12 mm), and after passing through a low-noise amplifier (bandwidth: 260 KHz-2.2 MHz, amplification: 58 dB), it was acquired by an acquisition card with a sampling rate of 50 MHz. In the experiments, the ultrasound transducer was scanned laterally over a 360° range to acquire the thermoacoustic signal at 180 positions for image reconstructions.

Fig. 2

Schematic of the experimental setup.

3 Simulated and experimental results

We conducted several simulation and phantom experiments to validate the proposed one-step qTAT reconstruction method. The quantitative reconstruction was performed in simulation and phantom experiments using a finite element mesh of 1584 nodes and 2986 elements for the simulations and of 4459 nodes and 8736 elements for the phantom experiments.

Other key known parameters involved in the simulations are as follows: relative permeability μ_r = 1.0 H/m,vacuum dielectric constant ɛ₀ = 8.854 × 10^-12 F/m,relative dielectric constant ɛ_r = 71 F/m, initial conductivity σ₀ = 0.001 S/m,microwave frequency f = 3.0 × 10⁹ Hz and the 40 mm diameter circular background had a conductivity of σ = 0.1 S/m. The results for a single circular target with a diameter of 10 mm and a conductivity of 0.2 S/m are shown in Fig. 3 where Fig. 3(a) provides the exact conductivity image, and Figs. 3(b)-3(c) present the reconstructed distributions of conductivity images by RNM and one-step qTAT methods, respectively. The conductivity property profiles along a transect crossing the center of target, shown in Fig. 3(d). To better compare the reconstruction results of the two methods, Fig. 3(e) presents the relative error between the reconstructed and exact conductivity using the two methods at the same iteration step. We can see that the error of the one-step qTAT method decreases rapidly compared to the RNM, where the inversion process becomes stable after 7 iterations by one-step qTAT method, but 14 iterations by the two-step RNM for this case. Figures 3(a)-3(c) show that the reconstructed images using the two methods agree with the initial conductivity distribution image. The simulation experiments aim to illustrate that the convergence rate for the one-step qTAT method is better than the RNM method.

Fig. 3

Simulations using RNM and one-step method. The exact conductivity image (a), reconstructed conductivity image using RNM (b), one-step (c), conductivity property profiles along a transect crossing the center of target by the two reconstruction methods (d), and relative error vs iteration steps (e).

Figure 4 gives reconstructed conductivity images by RNM and one-step qTAT method, where three circular targets having a 10 mm diameter with different contrast levels (0.2, 0.3, 0.4 S/m) were embedded in a background. Figures 4(a)-4(c) show the exact and the recovered conductivity images with different methods, and Figs. 4(d)-4(e) present the conductivity property profiles along transects crossing the center of each target. To better compare the reconstruction results of the two methods, the relative error between the reconstructed and exact conductivity using the two methods at the same iteration step is given in Fig. 4(f), which clearly shows that the error of the one-step qTAT method decreases rapidly compared to the RNM, where the one-step qTAT method becomes stable in the inversion process after 8 iterations, but 16 iterations for the RNM in this case.

Fig. 4

Simulations using RNM and one-step method. The exact conductivity image (a), reconstructed conductivity image using RNM (b), one-step (c), conductivity property profiles along the transects crossing the center of each target by the two reconstruction methods (d-e), and relative error vs iteration steps (f).

In the phantom experiments, small saline tubes (each with 3 mm in diameter) containing different concentrations were used as targets to verify the ability of one-step qTAT method to reconstruct conductivity more accurately and efficiently, where the conductivity and relative dielectric constant were varied by changing the concentration of salt in the target through the Debye model [29].

In the first phantom experiment, a single target having 2% 3% or 4% concentration NaCl solution (the concentration of saline with different salt contents) was placed at different locations in the background. Quantitative conductivity values recovered from the first phantom experiment using RNM and one-step qTAT method are shown in Table 1. The recovered conductivity images and their profiles for cases #I to #III with 2% concentration, cases #IV to #VI with 3% concentration and cases #VII to IX with 4% concentration single target using the two reconstruction methods, are given in Fig. 5. Figures 5(a), 5(d) and 5(g) show the conductivity images using RNM, while Figs. 5(b), 5(e) and 5(h) present the conductivity images using the one-step qTAT method. While we see that the reconstructed images of the two methods are almost indistinguishable visually, however, the recovered conductivity property profiles across the target center are clearly different (as shown in Figs. 5(c), 5(f) and 5(i)) with more accurately reconstructed target conductivity using one-step qTAT method. To better illustrate this finding, we used two quantitative measures, mean absolute error (MAE) and root mean square error (RMSE), to quantitatively evaluate the conductivity reconstructions as shown in Tables 2 and 3, where the ground truth of conductivity of the brine tube used to compute MAE and RMSE is calculated by Debye model.

Table 1

Comparison of the reconstructed conductivity values of a single target obtained using the two reconstruction methods

Case#	Brine	Conductivity	RNM			One-step
	concentration	(S/m)	conductivity(S/m)			conductivity(S/m)
			Max	Mean	SD	Max	Mean	SD
I	2%	3.34	3.42	3.00	1.62	3.46	3.03	1.63
II	2%	3.34	3.02	2.84	1.53	3.25	3.05	1.64
III	2%	3.34	3.29	3.04	1.59	3.36	3.10	1.62
IV	3%	4.81	4.58	4.01	2.24	4.68	4.09	2.28
V	3%	4.81	4.57	3.86	2.26	4.89	4.09	2.40
VI	3%	4.81	4.20	3.82	2.14	4.55	4.13	2.31
VII	4%	6.21	6.10	5.95	3.37	6.23	6.07	3.44
VIII	4%	6.21	6.31	6.11	3.39	6.30	6.09	3.38
IX	4%	6.21	6.28	5.98	3.42	6.34	6.03	3.44

Fig. 5

Reconstructed conductivity images and profiles for the single target case #I (a-c), case #IV (d-f), and case #VII (g-i) by RNM (left column) and the one-step method (middle column); Conductivity property profiles along the transects crossing the center of each target (right column).

Table 2

MAE and RMSE calculations for the two methods using the max values of the single target phantom data

Algorithms	MAE			RMSE
	Mean	SD	SEM	Mean	SD	SEM
RNM	0.2011	0.1405	0.0811	0.2295	0.1564	0.0903
One-step	0.1045	0.0453	0.0261	0.1178	0.0488	0.0282

Table 3

MAE and RMSE calculations for the two methods using the mean values of the single target phantom data

Algorithms	MAE			RMSE
	Mean	SD	SEM	Mean	SD	SEM
RNM	0.4967	0.3723	0.2149	0.5051	0.3680	0.2125
One-step	0.3778	0.2925	0.1689	0.3791	0.2916	0.1683

Furthermore, Fig. 6 gives the relative error between the reconstructed and exact conductivity using the two methods in the same iteration step, which shows that the one-step qTAT method has a better convergence rate than the RNM. Figures 6(a), 6(b) and 6(c) present a comparison of the convergence rate for the two methods in reconstructing the conductivity of NaCl solutions with concentrations of 2% 3% and 4% respectively. Statistical data for the reconstructed conductivity for each concentration are shown in Fig. 7 and are analyzed using mean±SEM (standard error of the mean). Figures 7(a)-7(b) show the statistical analysis of the max and mean values of the recovered conductivity for each concentration.

Fig. 6

Convergence rates for the two methods for a single target with 2% (a), 3% (b) and 4% (c) concentrations.

Fig. 7

Statistics analysis of the recovered conductivity for each concentration with the max values (a), with the mean values (b). Error bars represent SEM.

In the second phantom experiment, three targets having 2% 3% and 4% concentration NaCl solution was placed in the background. The conductivity images recovered by RNM and one-step qTAT method are shown in Figs. 8(a) and 8(b), respectively. Figures 8(c), 8(d) and 8(e) give comparative profiles along the transects crossing the center of each target, and the mean values of conductivity in the reconstructed target region are presented in Table 4. In addition, the relative error between the reconstructed and exact conductivity using the two methods in the same iteration step is shown in Fig. 8(f). From Fig. 8, we immediately notice that the conductivity values of the three targets are well reconstructed by using one-step qTAT method.

Fig. 8

Reconstructed conductivity images by the RNM (a) and the one-step method (b); (c-e): conductivity property profiles along the transects crossing the center of each target, (f): relative error vs iteration steps.

Table 4

Comparison of the reconstructed conductivity values for multiple targets obtained using the two methods

Background	Brine concentration	Conductivity (S/m)	RNMconductivity(S/m)	One-stepconductivity(S/m)
			Mean	Mean
Target 1	2%	3.34	2.75	3.00
Target 2	4%	6.21	6.19	6.10
Target 3	3%	4.81	3.61	4.51

In the third phantom experiment, the accuracy of conductivity reconstruction was compared and analyzed between the fitting method used in previous state-of-the-art studies in this field and the one-step qTAT method used here. The recovered conductivity images and their profiles for cases #I with 1.18% and 2.48% concentrations, cases #II with 1.18% and 3.98% concentrations dual-target using the two reconstruction methods, are given in Fig. 9. The conductivity images recovered by the fitting and one-step qTAT method are shown in Figs. 9(a), 9(e) and 9(b), 9(f), respectively. Figures 9(c)-9(d) and 9(g)-9(h) give comparative profiles along the transects crossing the center of each target, and the mean values of conductivity in the reconstructed target region are presented in Table 5.

Fig. 9

Reconstructed conductivity images for the dual-target case #I (a-b), case #II (e-f) by fitting method (first line) and the one-step method (second line); Conductivity property profiles along the transects crossing the center of each target (third and fourth line).

4 Discussion and conclusions

From the results presented in Figs. 3 and 4, while we see that both methods can accurately reconstruct the conductivity of the simulated target, the convergence rate for the one-step qTAT method is ∼2 times faster than the RNM for both cases with single and multiple targets. Where the RNM algorithm requires 14–16 iterations to converge, whereas the one-step qTAT method needs only 7-8 iterations to converge. Similarly, the RNM algorithm requires more iterations than the one-step qTAT method to reach convergence using the phantom experimental data, resulting in a difference of 3 minutes in computational time.

By comparing the results shown in Fig. 5 and Table 1, due to the inhomogeneous microwave field distribution of thermoacoustic imaging, the same target absorbs microwaves inconsistently at different locations, resulting in differences in the reconstructed conductivity. Two methods are used to calculate separately, although the standard deviation of conductivity (SD) on the reconstructed target region using one-step qTAT method is lower than that of the RNM algorithm, it only means that the reconstructed conductivity values in the target region by the one-step qTAT method deviates more from the mean value than the RNM algorithm, which does not affect the quantitative judgment of the whole target region.

Based on the max and mean values of the reconstructed target region, we note that accuracy of the conductivity reconstruction at different locations is significantly improved with the one-step qTAT method, being closer to the exact value. Together with the data shown in Tables 2-3, we analyzed two quantitative measures, MAE and RMSE to shown an improvement of more than one-fold using the one-step qTAT method. By setting the significance level p to 0.05, p = 0.0573 > 0.05 for MAE and p = 0.0527 > 0.05 for RMSE the calculation is based on the max values of conductivity in the reconstructed region, and p = 0.1548 > 0.05 for MAE and p = 0.1532 > 0.05 for RMSE is calculated based on the mean values of conductivity in the reconstructed region. The results show that no significant errors arise using the new one-step qTAT method. And as shown by the results in Fig. 7, which use the one-sided student’s test method (n = 3 for both salt concentration 2%3% and 4%), we obtain p value of less than 1% which means that the results presented here are statistically significant by using the one-step qTAT method. Further, we investigated the robustness and accuracy of the one-step qTAT method in reconstructing multiple targets with different conductivities. Additionally, the ability of the two methods to reconstruct the conductivity is analyzed by quantitative comparison through the index of relative error, which is calculated as shown in Equation (10) below. $Err = \frac{| σ^{R} - σ^{T} |}{σ^{T}} \times 100 %$ (10) where σ^R is the reconstructed value of conductivity, and σ^T is the true value of the target conductivity. The results shown in Fig. 8 and Table 4 indicate that the relative reconstruction errors based on the one-step qTAT method for the target having 2% 3% and 4% saline concentrations are 10.18% 6.24% and 1.77% respectively, while the relative errors based on the RNM method for the target with 2% 3% and 4% saline concentrations are 17.66% 24.95% and 0.32% respectively. From Figs. 6 and 8(f), we see that for both single and multiple targets, the one-step qTAT method provides a stable solution in the inversion process after 8 iterations, while the RNM requires significantly more iterations to become stable, especially for the multi-target case.

In addition, the results shown in Fig. 9 and Table 5 suggest that the one-step qTAT method is more accurate than the fitting method in terms of both the reconstruction of conductivity images and conductivity property profiles. The contrast of the two targets is 1:1.97 (Figs. 9(a)-9(b)) and 1:2.96 (Figs. 9(e)-9(f)) corresponding to Case #I and Case #II, respectively. Combined with Table 5, we obtain that for case #I, the contrast of the two targets reconstructed by the one-step and fitting methods is 1:1.78 and 1:2.44, respectively. In particular, for Case #II, the contrast of the two targets reconstructed by the one-step method is 1:1.65, while the contrast of the two targets reconstructed by the fitting method is 1:1.09. The results demonstrate that the one-step qTAT method presented here can quantitatively reconstruct the conductivity more accurately.

Table 5

Comparison of the reconstructed conductivity values for multiple targets obtained using the two methods

Case#	Location	Brine concentration	Conductivity (S/m)	Fittingconductivity(S/m)	One-stepconductivity(S/m)
				Mean	Mean
I	left	1.18%	2.06	1.53	2.17
right	2.48%	4.05	3.73	3.87
II	left	1.18%	2.06	5.13	3.36
right	3.98%	6.09	5.57	5.56

In conclusion, we have demonstrated that it is possible to directly reconstruct conductivity using the thermoacoustic wave equation coupled with the Helmholtz equation for electromagnetic field in just one step. The described one-step qTAT method can significantly improve the conductivity reconstruction accuracy and convergence rate compared to the RNM. We plan to evaluate the utility of our improved method to reconstruct tissue conductivity in vivo in the near future. As for the conductivity reconstruction of blood vessels with smaller diameters, we need to refer to our group’s dual-modality experimental system to complete the thermoacoustic data acquisition using the polarization effect, and optimize the one-step qTAT to complete the conductivity reconstruction according to the actual experimental environment, which will be reported subsequently.

Footnotes

Acknowledgments

This research was supported in part by Chongqing University of Posts and Telecommunications Doctoral Innovative Talents Program (BYJS202215, BYJS202117) and Chongqing Municipal Education Commission Science and Technology Research Program Youth Project (KJQN202000607).

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