Tomosynthesis is a technique used in general X-ray radiography and mammography systems to obtain three-dimensional images. In particular, this technology has been adopted for mammography systems, in conjunction with 2D mammography [1]. Tomosynthesis has been reported to outperform general X-ray imaging in detecting nodular shadows in the lung field and fracture lines [2,3]. Moreover, in terms of patient exposure, it can acquire images with a significantly lower radiation dose than computed tomography (CT). Compared to simple X-ray or mammography images, special indices are used to evaluate image quality since tomosynthesis acquires a three-dimensional image. For example, artifact spread function and focal spot motion have been used for this purpose [4,5]. The European Reference Organization for Quality Assured Breast Screening and Diagnostic Services (EUREF) has published its guideline on breast tomosynthesis image quality [6]. The Z-resolution, which corresponds to the slice thickness in CT images, is an important indicator of the image quality of tomosynthesis. The Z-resolution is not arbitrarily determined during image reconstruction, as with CT images but depends on the alignment during imaging and the parameters used for image reconstruction [7]. According to EUREF guidelines, the Z-resolution is measured using a micrometallic sphere (bead) with a diameter of 1.0 mm. However, there is no explanation on the reason for choosing 1.0 mm as bead diameter. As other reports used beads of various diameters, there is no unified view on bead diameter [8,9]. Furthermore, no studies have investigated the relationship between bead diameter and Z-resolution. The bead diameter has the greatest influence on the Z-resolution measurement results. Since finding beads with very small diameter can be challenging, in this study, we simulated tomosynthesis images of a bead of different diameters and measured the effect of bead diameter on Z-resolution measurement.

1. Simulated bead image

Fig. 1 shows images of the bead created during the simulation. The bead images were C language-based and created using Visual Studio 2022 (Microsoft Corp., Redmond, WA, USA). The matrix size, pixel value, pixel size, and output of the base image were set to 1,024×1,024×1,024 pixels, 0, 0.1 mm, and 16-bit, respectively. A uniform area of 800×800×500 pixels³ was set at the center of the base image to support the simulated bead. The pixel value was set to 60. The bead diameters were 0.025, 0.05, 0.1, 0.2, 0.3, 0.7, and 1.3 mm (Fig. 2). The bead’s center pixel value was set to 200. These pixel values were chosen such that, when reconstructed in the tomosynthesis image, the maximum available pixel values were not reached. The following is a description of the 0.025 and 0.05 mm renderings. For a pixel size set to 0.1 mm, the 0.025 and 0.05 mm bead images were within 1 pixel, even if the images were enlarged by projection and could not be distinguished. Therefore, the pixel value of the 0.025 mm bead image was set to 100, half the pixel value of the 0.05-mm bead image.

Figure 1. Simulated bead images. The pixel values of the base image and supporter were set to 0 and 60, respectively.

Figure 2. Each simulated bead image. The pixel value of the bead image was 200; however, for a diameter of 0.025 mm, it was half (100) than that of the other diameters.

2. Projection

A simulated bead image was created by projecting the created bead image onto a virtual detector. Fig. 3 shows a schematic of the relationship between the subject and the virtual detector to create the virtual projection. In the coordinates (f_x, f_y, f_z) of the virtual X-ray tube focus, f_y=0 because the X-ray tube moves along the x-axis. The distance from the focal point to the detector center is D; thus, f_z=D. Therefore, the coordinates of the focal point are given by Eq. (1).

Figure 3. Schematic of the virtual projection for the projected image. The number of pixels of the focus-shift value and the minimum D value employed are 1,000 and 500 pixels, respectively. Therefore, the X-ray tube angular range is ±45°.

where dx is the distance moved per unit time by the focus. If the coordinates to be virtually projected onto the detector are (x_d, y_d, and z_d), the distance between these coordinates and the focus can be calculated using the three-square theorem as follows:

The unit direction vector (a_x, a_y, a_z) from the focal point to the projection position is given by:

Therefore, the coordinates (x_d, y_d, z_d) of the projected image can be expressed using these equations. The minimum values of the focus-shifted distance and D were set to 1000 and 500 pixels, respectively. Therefore, the angular range of the X-ray tube was set as ±45°; 74 projected images were acquired. Fig. 4 shows the procedure followed to obtain the projected image.

Figure 4. Simulated projected images. The number of projected images was 74.

3. Reconstruction

As the projected image is formed from a limited angle, artifacts from outside the field of view are generated when reconstructing the image [10]. Therefore, before reconstructing the tomosynthesis image, a Fourier transform was performed on the projected image and a low-pass filter was applied to reduce the artifacts in the high-frequency region. In this study, a low-pass filter was applied to the projected image before the reconstruction (Fig. 5). The tomosynthesis image was reconstructed by applying the filtered back-projection (FBP) method to the projected image, which suppresses high frequencies (Fig. 6a). The reconstruction interval was set at 1.0 mm.

Figure 5. Low-pass filter shapes applied to the filtered back-projection.

Figure 6. Simulated (a) and real (b) tomosynthesis images of the bead. The numbers below the figures correspond to the simulated diameter.

4. Real bead image

We conducted an evaluation to confirm that the Z-resolution obtained from the tomosynthesis image generated by the simulation did not deviate from the Z-resolution obtained from the real bead image obtained using general X-ray equipment (RADspeed Pro EDGE package; Shimadzu Corp., Kyoto, Japan). The pixel size of the indirect-conversion flat panel detector mounted on this equipment was 0.15 mm. The acquisition alignment is shown in Fig. 7. A bead with a diameter of 0.3 mm was placed on a 200-mm acrylic plate just below the focal spot. The source-image receptor distance was 1,100 mm. The exposure conditions were 70 kV, 2.5 mAs, and 12 ms per exposure. The angular range of the X-ray tube during imaging was set to ±30° and that of the simulation to ±45°; therefore, the angular range differed between the simulation and the real bead image. However, this equipment had an angular range of ±30° centered on the table surface; therefore, the apparent angular range with respect to the bead was approximately ±40°. The tomosynthesis image of the real bead was obtained by reconstructing the acquired 60 projected images using the FBP method (Fig. 6b). The low-pass filter used in the reconstruction was the closest to the filter shape shown in Fig. 5; the reconstruction interval was set to 1.0 mm.

Figure 7. Schematic of the acquisition for the real projected image using general X-ray equipment. The angular range of the real bead image (±30°) differed from the simulated bead image (±45°). However, the apparent angular range to the bead increases as the X-ray tube moves around the table surface.

5. Z-resolution

The Z-resolution measurement method is illustrated in Fig. 8. A region of interest was placed at the center coordinates of the bead image in the reconstructed tomosynthesis image; thus, the maximum pixel value along the z-axis was obtained. The slice sensitivity profile (SSP) along the z-axis was obtained by plotting the acquired pixel values and the full width at half maximum (FWHM) of the SSP peak was defined as the Z-resolution [7].

Figure 8. Schematic of the Z-resolution measurement method. A region of interest was placed at the center of the bead image, and slice sensitivity profile was acquired by plotting the maximum pixel values of the bead images. FWHM, full width at half maximum.

6. Frequency response

The frequency response along the z-axis was calculated from the simulated bead image. The acquired SSP was converted to a power value by the 1D Fourier transform and normalized to a spatial frequency of 0 cycles/mm to obtain the frequency response. However, as the sampling interval was 1.0 mm (slice interval), the frequency response included the influence of areasing.

1. Comparison between simulated and real bead images

The acquired SSP from a simulated 0.3-mm bead and that by a real 0.3-mm bead are shown in Fig. 9. The Z-resolutions calculated from each SSP were 2.33 and 2.68 mm for the simulated and real bead images, respectively.

Figure 9. Acquired slice sensitivity profile of the simulated and real bead image. The Z-resolutions of the simulated and real bead image were 2.33 and 2.68 mm, respectively.

2. Z-resolution calculation

The measured SSP values are shown in Fig. 10. The SSPs obtained with bead diameters of 0.025, 0.05, and 0.1 mm had approximately the same shape, so did those of bead diameters 0.2 and 0.3 mm. However, the larger the diameter of the virtual bead, the larger the SSP width. The FWHM was calculated from these SSPs to provide the Z-resolution as shown in Fig. 11. The Z-resolution of beads of 0.025, 0.05, and 0.1 mm diameter was approximately equal (1.6 mm), whereas that for beads with 0.2 and 0.3 mm diameter was larger than for 0.1 mm and nearly equal between the two. For larger diameters, the Z-resolution increased with increasing diameter.

Figure 10. Acquired slice sensitivity profile (SSP) of each simulated bead image. The shape of SSP of 0.025 mm was equivalent to these of the 0.05 and 0.1 mm.

Figure 11. Results of the Z-resolution of each simulated bead diameter. There were no change of the Z-resolution value of the diameter of 0.025, 0.05, and 0.1 mm.

3. Results of the frequency response

The calculated frequency response is shown in Fig. 12. Similar to the Z-resolution, the frequency response was smaller for larger bead diameters. The results obtained for beads of 0.025, 0.05, and 0.1 mm diameter were like those of the Z-resolution. However, the frequency response results for beads with 0.2 and 0.3 mm diameters were slightly different. Fig. 13 shows the bead images (0.025, 0.05 and 0.1 mm) in the x-z plane. The distribution of pixel values was similar for the three images.

Figure 12. Results of the frequency response of each simulated bead diameter. The frequency responses of 0.2 and 0.3 mm were diverged comparison with the results of the Z-resolution.

Figure 13. The x-z plane view of simulated bead of (a) 0.025 mm, (b) 0.05 mm, and (c) 0.1 mm.

In a previous report, we measured the Z-resolution using the ball point at the tip of a ballpoint pen [7]. However, ballpoint pens of very small diameters are not commercially available. In this study, we created a bead image with a small diameter via simulation and measured the Z-resolution, something that could not be evaluated in a real image. This allowed us to clarify the relationship between bead diameter and Z-resolution. Moreover, the simulation allowed us to measure the Z-resolution more consistently than with real images, without being affected by other factors. For example, the pixel values at the base of the SSP were slightly different between the lower and upper slices of the peak in real images (Fig. 9). Therefore, it was difficult to determine the base pixel values, resulting in errors in FWHM measurement. However, in the SSP acquired by the simulation, the base pixel values were equivalent and FWHM measurement error was less likely to occur (Fig. 10).

To confirm that the results of the simulation bead image did not deviate from a real bead image, a real 0.3-mm bead image was acquired and the Z-resolution of the two images was compared. Despite the slightly different bead alignment, including the angular range of the X-ray tube, the two SSP shapes and the Z-resolution were similar. Therefore, the findings obtained from this simulation can be applied to real images as well.

In addition, the Z-resolution increased with increasing bead diameter. Therefore, the Z-resolution depends on bead diameter, so bead diameter should be considered when measuring the Z-resolution. Notably, for small diameters (0.025, 0.05, and 0.1 mm) there was no change in Z-resolution. Therefore, it was confirmed that using beads with a diameter smaller than the pixel size does not affect the measurement of Z-resolution. In addition, EUREF guidelines recommend a 1.0 mm diameter bead, which may overestimate the Z-resolution. Therefore, according to our results, this guideline needs to be revised to measure the true Z-resolution of tomosynthesis images.

The z-axis frequency response obtained from the SSP was similar to that of the Z-resolution. The frequency responses diverged for beads with 0.2 and 0.3 mm diameters despite the absence of significant differences in Z-resolution. The SSP showed an identical peak shape; however, the pixel value change to the base was slightly different. This probably explains the divergence between the two. The difference between 0.1 and 0.2 mm was larger than that between 0.2 and 0.3 mm diameters. This was also true for the Z-resolution, which was larger, that is, 1.60 mm for a 0.1 mm diameter and 2.23 mm for a 0.2 mm diameter, respectively. A possible reason for this finding may be that the 0.1 mm diameter bead image could be represented by 2 pixels, whereas the 0.2 mm diameter bead image required 3 pixels.

In this study, we investigated the relationship between bead diameter and Z-resolution in tomosynthesis the using simulation-generated bead images. The Z-resolution increased with larger bead diameters. Furthermore, the Z-resolution was not affected for bead diameters smaller than the pixel size. Based on these findings, when performing true Z-resolution measurements, the bead diameter should be carefully chosen.

The authors have nothing to disclose.

All relevant data are within the paper.