Evaluation of reconstruction algorithms for a stationary digital breast tomosynthesis system using a carbon nanotube X-ray source array

Abstract

Breast cancer is the most frequently diagnosed cancer in women worldwide. Digital breast tomosynthesis (DBT), which is based on limited-angle tomography, was developed to solve tissue overlapping problems associated with traditional breast mammography. However, due to the problems associated with tube movement during the process of data acquisition, stationary DBT (s-DBT) was developed to allow the X-ray source array to stay stationary during the DBT scanning process. In this work, we evaluate four widely used and investigated DBT image reconstruction algorithms, including the commercial Feldkamp-Davis-Kress algorithm (FBP), the simultaneous iterative reconstruction technique (SIRT), the simultaneous algebraic reconstruction technique (SART) and the total variation regularized SART (SART-TV) for an s-DBT imaging system that we set up in our own laboratory for studies using a semi-elliptical digital phantom and a rubber breast phantom to determine the most superior algorithm for s-DBT image reconstruction among the four algorithms. Several quantitative indexes for image quality assessment, including the peak signal-noise ratio (PSNR), the root mean square error (RMSE) and the structural similarity (SSIM), are used to determine the best algorithm for the imaging system that we set up. Image resolutions are measured via the calculation of the contrast-to-noise ratio (CNR) and artefact spread function (ASF). The experimental results show that the SART-TV algorithm gives reconstructed images with the highest PSNR and SSIM values and the lowest RMSE values in terms of image accuracy and similarity, along with the highest CNR values calculated for the selected features and the best ASF curves in terms of image resolution in the horizontal and vertical directions. Thus, the SART-TV algorithm is proven to be the best algorithm for use in s-DBT image reconstruction for the specific imaging task in our study.

Keywords

Stationary digital breast tomosynthesis (s-DBT)carbon nanotube (CNT)image reconstruction image accuracy image resolution

1 Introduction

Breast cancer is the most common type of cancer in women. Despite tremendous progress in medicine, the incidence and mortality rates of female breast cancer remain high both worldwide and in China [1]. Regardless of race or ethnicity, the incidence appears to be increasing [2]. Digital mammography is currently the gold standard for the early detection of breast cancer. However, limitations, such as high false-positive and false-negative rates, have been well documented [3]. The widely used technique of digital breast tomosynthesis (DBT) can mitigate these limitations and overcome the overlapping phenomena associated with mammography [4].

Digital tomosynthesis is a 3D imaging modality that employs a form of limited-angle tomography to produce sectional images [5]. The common design of a DBT system involves a rotating anode with a single X-ray source. Two rotation methods, step-and-shoot motion and continuous motion, in which the X-ray tube moves along an arc and acquires 2D projections within a limited angle from a single scan [3]. The motion of the X-ray source during the process of image correction can cause remarkable motion blurring, as analyzed by Yang et al. [6]. Such methods are also associated with relatively high scanning times since the reduction of the scanning time is limited by the rotational speed of the unique source [7]. To overcome these shortcomings of traditional DBT systems, stationary digital breast tomosynthesis (s-DBT) systems have been proposed in recent years [6, 7]. Quan et al. presented a multi-beam system that uses X-ray sources in linear arrays arranged in a square geometric pattern [8]. Qian et al. reported an s-DBT system that uses a multi-beam field emission X-ray (MBFEX) source based on a carbon nanotube (CNT) [9]. Neither the distributed X-ray sources nor the panel detectors rotate during one scanning cycle, and the object phantom remains stationary. Artefacts caused by mechanical rotation can be reduced by using electronic scanning systems to obtain images from different angles. A simplified imaging system and improved temporal efficiency are additional reported advantages of s-DBT systems compared to conventional DBT systems [6].

Among current reconstruction techniques in digital tomosynthesis imaging, both analytical reconstruction and iterative reconstruction are being investigated. The most widely known classical analytical algorithm is filtered back projection (FBP), which is normally used in transmission computed tomography (CT) [10]. As in CT, when the FBP method is applied to cone-beam geometry involving highly incomplete frequency sampling, reconstruction errors are introduced [11]. For 3D imaging, the Feldkamp-Davis-Kress (FDK) algorithm, which was modified from the 2D FBP algorithm presented by Feldkamp et al., is widely used [12]. Many works on the utilization of the FDK method for DBT imaging have been reported, including works aiming at better filter designs and algorithm evaluations [13 –15]. The commercial DBT imaging systems manufactured by Hologic Selenia and Siemens use the FDK method for image reconstruction [16, 17]. In addition to the analytical methods, iterative reconstruction (IR) has been shown to provide a good balance between improved image quality and low- and high-frequency features [18]. The algebraic reconstruction technique (ART) is the earliest proposed iterative algorithm for image reconstruction for CT [19]. It updates the object image under a sequential mode of calculation, which guarantees rapid convergence but causes serious image noise. By contrast, the simultaneous iterative reconstruction technique (SIRT) applies batch-mode updating of the image intensity, namely, updating the image based on the integral projection data set in one iteration, which requires more updating epochs for convergence and provides smoother images [20]. To combine the advantages of both the ART and SIRT algorithms, Andersen and Kak proposed the simultaneous algebraic reconstruction technique (SART) for CT image reconstruction. The SART method also uses batch-mode updating but with smaller batches, considering the projection data of one projection view in one updating step to reconstruct the object [21]. The SIRT and the SART have been reported to be employed in the task of DBT image reconstruction [22, 23] and commercial or prototype DBT systems [17] and were studied in the current work.

In addition to the simple usage of fidelity-driven IR methods, total variation (TV) regularization can be introduced to improve the resultant image quality by denoising the reconstructed image concomitantly with the reconstruction iterations [24]. The TV minimization method is proven to be capable of edge-reserved image denoising. It was first proposed as an image denoising technique by Rudin et al. and was applied for limited-viewing angle CT image reconstruction in work by Velikina et al. [25, 26]. The commercial system manufactured by IMS Giotto involves TV regularization in its image reconstructions [17].

In this work, we evaluated the applicability of four widely used algorithms for transmission computed tomography, namely, FDK, SIRT, SART and the TV regularized SART (SART-TV), for DBT imaging based on the s-DBT system that we set up in our laboratory. The s-DBT system that we used has 9 electronically controlled CNT X-ray sources in an arc array across a 40° angular span that can be instantaneously switched on or off. Several well-known quantitative assessment criteria were applied to evaluate the algorithms used in terms of image reconstruction accuracy, image structural similarity and image in-plane and vertical resolutions. By quantitatively evaluating the reconstructed images given by the four discussed algorithms, we determined the best one for the task of image reconstruction in s-DBT imaging.

The remaining sections of the paper are organized as follows. Section 2 describes the detailed geometry information, imaging model and algorithms used in the current study. The evaluation methods are also outlined in this section. In Section 3, we discuss on the experimental results of the simulative and physical experiments, and Section 4 gives the conclusion of the study in this article.

2 Materials and methods

2.1 s-DBT system

The s-DBT system, as shown in Fig. 1, was constructed with a CNT X-ray source array. In total, 9 individual carbon nanotube (CNT) X-ray sources were distributed in an arc array, covering a 40° angle with equiangular spacing of 4.84° from the isocenter. Each X-ray source consists of a CNT cathode, a gate, a single focusing electrode and an anode set at 28 kV voltage and 20 mA current. The effective exposure time is 50 ms. An electronic system was implemented to control and switch the sources. The total size of the stationary detector (AXS 2430, Analogic Corporation) is 24×30 cm, and the detector has 2816×3584 uniformly arrayed detector units. This detector is a premium large-field-of-view detector offering a state-of-the-art tomosynthesis performance. It has a high modulation-transfer function (MTF) of 90% at 0.1 lp/mm, which allows the acquisition of high-quality images. The distance from the central CNT X-ray source to the stationary detector was 650 mm, and the isocenter of the source array was located on the bearing platform of the detector. Between the bearing platform and the image receiving surface was an air gap if 10 mm depth. The schematic geometric structure and the actual s-DBT system are shown in Fig. 1.

Fig. 1

(Left) Schematic geometric structure of the studied s-DBT system. (Right) The rubber breast phantom.

2.2 Imaging experiments

2.2.1 Simulative experiment

A semi-elliptical digital breast phantom was used for the simulative experiment in this study. The digital phantom contained 50 image slices, with the size and thickness of every slice being 256×256 and 1 mm. The horizontal pixel size of the phantom was designed to be 0.5 mm×0.5 mm. As is shown in Fig. 2, several features were introduced in different slices of the phantom to represent different abnormal tissues in human breasts, including the ellipsoids representing cancer tissues in slices 10 and 25 and the small spheres representing microcalcifications in slice 40. The geometric parameters and conditions for the simulative experiment, including the source-to-detector distance, the source-to-isocenter distance and the source distribution, were set to be the same as those of the established s-DBT system described in the last section. Simulated noise in the projection data was generated through a similar method to that described in Hu et al.’s work [27, 28], and the simulated incident X-ray intensity was set to 5e⁴ to represent a low tube current X-ray emission. The detector channel size in the simulative experiment was set to 2 mm×2 mm, and the size of the projection image was 100×200. We fully reconstructed the 50 slices of the images with the four discussed algorithms.

Fig. 2

Reconstructed images in the simulative experiment.

2.2.2 Physical experiment

A rubber breast phantom was used for the physical experiment in this study. The investigated phantom is shown in Fig. 1. As indicated by the arrows, two types of insertions represented two typical types of lesions in a human breast. The white inclusions marked by the red arrows are of high contrast and simulate microcalcifications, and the black inclusions marked by the white arrows are of relatively low contrast and simulate breast tumors. There were, in total, 11 inserted “breast tumors” and 2 inserted “microcalcification tissues” in the breast phantom. The thickness of the breast phantom was approximately 50 mm, so the output reconstructed three-dimensional image contained 50 slices, with the thickness of every slice being 1 mm. The original projection images were cut to the size of 2800×3500, and 2×2 binning was applied to obtain projection images with the size of 1400×1750 to accelerate the reconstructions. Therefore, the pixel dimensions of the finally used projection data were 0.17 mm×0.17 mm. The horizontal size of the output image was set to 256×512, with the pixel dimensions being 0.35 mm×0.35 mm.

2.3 Image reconstruction

In this work, we evaluated the four widely investigated and used tomographic image reconstruction algorithms, namely, FDK, SIRT, SART and SART-TV for the s-DBT system described in the last section. For the implementation of the standard FDK method, a ramp filter and a window filter in the frequency domain of the projection data was applied. After projection filtering the inverse Fourier transform, negative values appeared in the space domain of the projection data; thus, a non-negative operation was conducted on the reconstructed image.

The core of the SIRT method involves using the whole set of projection data to update the object image. The implementation of this algorithm in this study was actually based on a least squared cost-dominated optimization. The updating step is $x^{n + 1} = x^{n} + λ^{n} \cdot G^{*} (y - G x^{n})$ (1) where yis the projection data, Gis the system matrix characterizing the forward projection of the s-DBT system, and G^* represents the back-projection operation. λⁿis an adaptive updating step size that is important for the stable running of SIRT and was calculated by [29]. $λ^{n} = \frac{{∥ G^{*} (y - G x^{n}) ∥}_{2}^{2}}{{∥ G G^{*} (y - G x^{n}) ∥}_{2}^{2}} \cdot \sqrt{\frac{N_{slice}}{M_{view}}}$ (2) with N_slice being the number of pixels of one image slice and M_view being the number of pixels of one view of projection data.

The SART algorithm was first proposed by Andersen and Kak for tomographic image reconstruction [21]. In this work, we applied the generalized version of the image updating rule given by Jiang and Wang [30], which shows $x_{j}^{n + 1} = x_{j}^{n} + \frac{ω}{G_{+, i}} \cdot \sum_{i = 1}^{M} \frac{G_{i, j}}{G_{i, +}} (y_{i} - {(G x^{n})}_{i}) .$ (3)

In the above formula, G_+,i is the sum of the elements of the jth column of G, and similarly, G_i,+is the sum of the elements of the ith column of G. M denotes the total number of projection data values, and ω is a relaxation parameter controlling the stability of the SART algorithm, which was set to 0.8 and 0.3 for the simulative and physical experiments, respectively.

The implementation of the SART-TV method in this work includes a three-dimensional TV minimization step after every fidelity update of the SART algorithm. We applied the TV minimization method described in Ertas et al.’s work [24]. The relaxation parameter ω is same to that in the SART reconstruction, and the TV minimization step size β was set to 15 for the simulative experiment and 3 for the physical experiment for the best reconstruction performance.

We employed an open-source toolbox for the DBT imaging investigation, which was programmed in MATLAB 2019a (MathWorks; Natick, MA, USA), to conduct basic operations such as forward and backward projection and FDK reconstruction [31]. We executed enough iterations for the iterative reconstruction algorithms—namely, 40 for the simulative experiments and 35 for the physical experiments—to attain convergence.

2.4 Evaluation methods

To assess the performances of the FDK, SIRT, SART and SART-TV algorithms based on the s-DBT system that we developed, several quantitative criteria were used. The imaging accuracy, similarity to the ground truth, and resolution of the reconstructed images were of interest in the evaluation. The RMSE indicates the numerical difference between the obtained image and the original image, which represents the ground truth. The PSNR, which is a measure of the imaging error, is related to the RMSE, as it approaches infinity as the RMSE approaches zero [32, 33]. These metrics were calculated as follows: $RMSE = \sqrt{\frac{\sum_{i = 1}^{N} {({x_{i}}^{recon} - x_{i}^{true})}^{2}}{N}}$ (4) and $PSNR = 10 lo g_{10} \frac{ma x^{2} (x^{true})}{\frac{1}{N - 1} \sum_{i = 1}^{N} {({x_{i}}^{recon} - x_{i}^{true})}^{2}}$ (5) where x^reconand x^trueindicate the reconstructed image and the ground-truth phantom, respectively, and Nis the total number of pixels in the region of interest (ROI) under consideration.

The SSIM between the reconstructed image and the ground-truth phantom is calculated as follows [34]: $SSIM = l (x^{recon}, x^{true}) \times c (x^{recon}, x^{true}) \times s (x^{recon}, x^{true})$ (6) with l, c, and sbeing the luminance index, contrast index and structural index, respectively, which were calculated by $l (x^{recon}, x^{true}) = \frac{2 μ_{recon} μ_{true} + C_{1}}{{μ_{recon}}^{2} + {μ_{true}}^{2} + C_{1}}$ (7) $c (x^{recon}, x^{true}) = \frac{2 σ_{recon} σ_{true} + C_{2}}{{σ_{recon}}^{2} + {σ_{true}}^{2} + C_{2}}$ (8) and $s (x^{recon}, x^{true}) = \frac{σ_{recon - true} + C_{3}}{σ_{recon} σ_{true} + C_{3}}$ (9) respectively. In the above three formulas, C₁, C₁ and C₁are three small positive constants, μ_recon and μ_trueare the mean values of the calculated map M^reconand the ground truth-map M^true, respectively, and σ_recon and σ_trueare the standard deviations calculated from μ_reconand μ_true, respectively. σ_recon-trueis the covariance between x^reconand x^true. The calculation of the standard deviation is given by $σ = \frac{1}{N - 1} \sum_{j = 1}^{N} (x_{j} - μ) .$ (10)

And the covariance is defined by $σ_{recon - true} = \frac{1}{N - 1} \sum_{j = 1}^{N} ({x_{i}}^{recon} - μ_{recon}) (x_{i}^{true} - μ_{true}) .$ (11)

To jointly evaluate the contrast resolution and the effect of noise reduction in the reconstructed image, the contrast-to-noise ratio (CNR) was calculated. The CNR was calculated by [35]. $CNR = \frac{μ_{f} - μ_{bg}}{\sqrt{σ_{f} + σ_{bg}}},$ (12) where μ_f and μ_bg are the mean intensities of the image features and the image background, respectively, and σ_f and σ_bg are the standard deviations. In addition to the contrast resolution in the horizontal plane, the resolution in the z-direction was also evaluated through the calculation of the artefact spread function (ASF) of the simulated calcifications in the reconstructed images [15]. The formula for the calculated ASF is given by $ASF (z) = \frac{μ_{a} (z) - μ_{bg} (z)}{μ_{f} (z_{0}) - μ_{bg} (z_{0})},$ (13) where μ_a(z) denotes the mean intensity of the off-plane artefact located at height z and μ_bg(z) denotes the mean intensity of image background in the same slice. μ_f(z₀) and μ_bg(z₀) are the mean intensities of the in-plane feature and background, respectively, with the feature being located at height z₀. We note that for the simulative experiments, the z₀ of a feature was known, and for the physical experiments dealing with the rubber breast phantom, z₀ was determined by visual inspection conducted slice by slice.

3 Results and discussion

3.1 Simulative semi-elliptical phantom study

3.1.1 Reconstruction accuracy

Figure 2 shows the 10th, 25th and 40th slices of the reconstructed three-dimensional simulative DBT image via the FDK, SIRT, SART and SART-TV algorithms. From visual inspection, the SART-TV method gives the best results in terms of image quality and similarity to the ground-truth phantom. Figure 3 shows the calculated PSNR and RMSE values of every slices of the reconstructed images. Since the results from the FDK method are not comparable to those of the three IR methods employed, the quantitative values of the FDK reconstructed image slices are not shown. For every slice of the reconstructed image, the SART-TV method gave the highest PSNR value and the lowest RMSE value compared to SIRT and SART, from which one can conclude that from the aspect of image accuracy, the SART-TV algorithm has a superior performance.

Fig. 3

(Left) PSNR values of the reconstructed image slices. (Right) RMSE values of the reconstructed image slices.

3.1.2 Structural similarity

Figure 4 gives the calculated SSIM values of every slice of the reconstructed DBT images. Again, the results from the FDK method are not shown. The SART-TV algorithm gives the resultant image with highest structural similarity to the ground-truth phantom, as indicated by the highest SSIM values for every 2D slice.

Fig. 4

SSIM values of the reconstructed image slices.

3.1.3 Contrast recolution

The calculated CNR values of two selected features representing breast tumors, which were respectively located in the 10th and 25th slices, are given in Fig. 5. The regions of the image features and backgrounds used for the calculation of the CNR values are marked by the red wireframes in the phantom slices shown on the right. It is remarkable that the SART-TV gave the highest CNRs for both selected features, indicating that it reconstructs the DBT image with the highest contrast resolution. The low CNR values of the FDK-reconstructed image may be attributed to the uncontrolled image noise, since the corresponding features shown in Fig. 2 seem the most detectable compared to the results of the IR algorithms.

Fig. 5

CNR values of the reconstructed features in the 10th and 25th slices (simulative experiment).

3.1.4 Vertical resolution

In this study, the vertical resolutions of the reconstructed images were evaluated by the calculation of an ASF curve based on the simulated microcalcification tissue, as shown in Fig. 6. The discussed image feature located in the 40th slice of the simulative volume and the region of the background are marked in the phantom image on the right. The ASF curves from the 30th slice to the 50th slice for all the discussed algorithms are shown. The performances in terms of vertical resolution of the three IR algorithms are not remarkably discrepant and are superior to that of the FDK method since the former ASF curves are narrower, indicating that off-plane artefacts were better suppressed by the iterative reconstruction methods.

Fig. 6

ASF curves of the reconstructed feature for the different algorithms (simulative experiment).

3.2 Physical breast phantom study

3.2.1 Visual inspection

Figure 7 shows the 24th, 28th and 38th slices of the reconstructed DBT images from the four discussed algorithms in the physical rubber phantom experiment. The natures of the four algorithms are better illustrated. The FDK method gives acceptable results but with significant image noise. By contrast, the SIRT method gives smoother images with a lower level of image noise but flatted image features. The results given by SART seem interposed between the FDK and SIRT, with less noise compared to FDK and visibly clearer image features compared to SIRT. However, the SART-TV again seems the most superior among the four algorithms, since it gives results with remarkably controlled image noise and sufficiently identifiable features.

Fig. 7

Reconstructed image slices (24th, 28th and 38th) of the physical experiment.

3.2.2 Contrast resolution

In our physical rubber phantom study, we selected three features representing breast tumor tissues located at different heights. The corresponding slices in the reconstructed image are approximately the 24th, 28th, and 38th slices. Figure 8 shows the CNR values calculated from the marked regions of features and backgrounds. One can determine that the level of image noise plays an important role in the resultant CNR measurements since the CNR value of the SIRT is remarkably higher than that of the SART. The SART-TV-reconstructed image features have the highest CNR values, illustrating that the TV-regularized algorithm has the most superior performance in terms of image contrast resolution and noise reduction.

Fig. 8

CNR values of the reconstructed features in the 24th, 28th and 38th slices (simulative experiment).

3.2.3 Vertical resolution

Figure 9 shows the ASF curves calculated from the two selected features representing two microcalcification tissues located in different slices (the 23rd and 30th) of the reconstructed images. The selected image features and the related regions of the background are marked in the two images on the right. Different from the results given by the simulative study, the FDK method seems to have narrower ASF curves; however, the results indicate that it introduces abnormal off-plane artefacts, which may lead to imprecise feature vertical localization. Compared to SIRT, the SART and SART-TV algorithms give reconstructed three-dimensional images with higher vertical resolutions, as stated by the calculated ASF curves.

Fig. 9

ASF curves of the reconstructed features from different algorithms (physical experiment). (Up) Feature located in the 23rd slice. (Down) Feature located in the 30th slice.

4 Conclusions

In this study, we evaluated four image reconstruction algorithms, namely, FDK, SIRT, SART and SART-TV, for an s-DBT imaging system set up with carbon nanotube X-ray sources and an AXS-2430 detector manufactured by Analogic Corporation using different assessment criteria in terms of the reconstruction accuracy, image structural similarity to ground truth and contrast resolution and vertical resolution of the image features. A simulative experiment using a semi-elliptical digital phantom and a physical experiment using a rubber breast phantom with embedded features representing breast tumors and microcalcification tissues were conducted. The simulative experimental results show that in term of the image reconstruction accuracy, the SART-TV method gives resultant images with highest PSNR values and lowest RMSE values with respect to the ground-truth phantom. The best structural similarity to the ground truth of the SART-TV-reconstructed image is also manifested by the higher SSIM values. Additionally, the SART-TV method also has the best performance in terms of the horizontal contrast resolution and vertical resolution, as illustrated by the highest CNR values calculated from selected image features and the best ASF curves compared to FDK, SIRT and SART. Thus, in summary, we conclude that SART-TV has the best performance in the specific s-DBT imaging task in this study for image reconstruction among the four discussed algorithms.

Conflic of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Footnotes

Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their constructive comments and suggestions. This work was supported by the Guangdong Special Support Program of China (2017TQ04R395), Shenzhen Basic Research Program of China (JCYJ20170413162354654), the National Natural Science Foundation of China (81871441), the Shenzhen International Cooperation Research Project of China (GJHZ20180928115824168), the Guangdong International Science and Technology Cooperation Project of China (2018A050506064), and the Natural Science Foundation of Guangdong Province in China (2020A1515010733).

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