2.1.

Neural Network Material Decomposition

The energy-dependent x-ray attenuation coefficient of a material can be represented as a linear combination of basis functions, which, for example, may represent the attenuation coefficients of two basis materials.¹⁹ The line integral through an object with energy-dependent x-ray attenuation distribution $\mu (x,y,z,E)$ measured along a ray path $\overrightarrow{r}$ by an x-ray transmission measurement can be represented as

This study used an NN to estimate the equivalent thicknesses through two basis materials, $t_1$ and $t_2$ from the measured photon-counting spectral measurements. A feedforward NN with a single hidden layer, as shown in Fig. 1, was implemented as in previous work^13,14 using the Python Scikit-Learn module.²⁰ A network with one hidden layer was used for this regression problem, as theoretically a single-layer network is sufficient to approximate any function. The inputs to the NN are the log normalized energy-bin counts data $p_k$, which are calculated from the number of counts detected in energy bin $k$ in the presence object, $n_k$, and in the absence of the object, $n_{k,0}$:

Fig. 1

Overview of proposed NN material decomposition estimator. The NN consists of an input layer (log normalized energy bin counts data), an adjustable hidden layer, and an output layer (basis material thicknesses). The number of nodes in the hidden layer was determined using a leave-one-out cross validation study.

The energy bin data are input to the NN after log normalization so that the NN does not need to learn the approximate exponential relationship between the basis material thicknesses and the counts measurements. The number of elements in the hidden layer was investigated as described in Sec. 2.4. The NN outputs the estimated thicknesses of the two basis materials.

The NN used a sigmoid activation function, an L2 regularization parameter of 0.001, and the limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm for network training.²¹ The sigmoid activation was selected because it is commonly used for shallow network architectures for which computation time is not a factor.²² The NN was trained with log-normalized transmission measurements through known thicknesses of the basis materials, as will be described in Sec. 2.3. During training, the NN learns the functional relationship between the basis material thicknesses and the energy-bin counts measurements. Pulse pileup effects are expected to change this functional relationship. This study trained an NN for a specific tube current setting corresponding to specific incident count rate through air and investigated the ability of the NN to learn the relationship between log-normalized energy-bin measurements and basis material thicknesses at that incident count-rate level. Once trained, the NN estimates the basis material pathlengths for an input set of log normalized energy-bin data measurements acquired at the same count-rate level used for training. Throughout this work, count-rate level refers to the counts per second per detector element incident on the detector through air.

2.2.

Photon-Counting X-Ray Acquisition

Photon-counting spectral data were acquired on a bench-top spectral CT system with a CdTe photon-counting detector (DxRay, Northridge, California) and microfocus x-ray tube (L9181-02, Hamamatsu, Hamamatsu City, Japan). The detector consisted of an array of $4\times 64$, $1.4\mathrm{mm}\times 1.0\mathrm{mm}$ elements with four comparator channels per element and a maximum count rate of 10⁶ counts per second per element as stated by the manufacturer. Data were acquired with a tube voltage of 90 kV and with four detected energy bins at thresholds of [20–45], [45–55], [55–65], [65–90] keV. To investigate the effects of pulse pileup, acquisitions were performed at five tube current settings of 0.02, 0.03, 0.06, 0.09, and 0.12 mA, with the detector placed 72 cm from the x-ray source. The acquisition time was adjusted for each tube current setting so that the tube current-time product was 0.456 mAs for all experiments, ensuring that the same number of photons were incident the detector for all experiments so that changes in noise were due to pileup conditions rather than differences in the number of photons reaching the detector.

The tube current settings used in this study were approximately three orders of magnitude lower than what is used in clinical CT systems. The count rate capabilities of the bench-top detector were approximately three orders of magnitude lower than what is expected to be required for clinical photon-counting CT systems.²³ To generalize the results of this study to the wide range of photon-counting detector capabilities, we report the studied cases as a percentage of the maximum detector count rate, so that the results are reported relative to the level of pileup instead of a detector-specific count-rate level. Figure 2 shows the detected count rate versus the incident count rate through air for the five investigated tube current settings, which corresponded to 9%, 14%, 27%, 40%, and 54% of the maximum detector count rate, demonstrating the level of pileup loss at each count-rate level and energy bin.

Fig. 2

Detected count rate plotted against the count rate incident on the detector in air for (a) each energy bin and (b) the sum of all energy bins plotted for one example detector element. Each marker represents one of the five count-rate levels investigated in this study, corresponding to 9%, 14%, 27%, 40%, and 54% of the maximum detector count rate (10⁶). The ideal linear relationship between the detected and incident count rate, which would occur in the absence of pileup effects, is plotted for comparison. The ideal relationship was estimated by linear regression using the origin and the two lowest count rate data points. The plotted relationships are specific to the studied detector.

2.3.

Neural Network Training

Calibration data for NN training were obtained by acquiring transmission measurements through a step wedge phantom consisting of 25 combinations of PMMA (0 to 4, 2.54 cm slabs) and aluminum (0 to 4, 0.64 cm slabs). Calibration data were acquired at each of the five count-rate levels shown in Fig. 2. The ground truth aluminum thicknesses were multiplied by four during network training to encourage similar levels of bias between the estimated PMMA and aluminum thicknesses. When the network was used for decomposition after training, the aluminum estimates were divided by four to compensate for this scaling. An NN was trained for each detector element and for each incident count-rate level. As proposed in our previous work, a transfer learning technique was used during training to reduce variations in bias across detector elements which can cause ring artifacts in reconstructed CT images.¹⁴ For each count-rate level, a network was first trained using the calibration energy-bin counts data from all detector elements. This produced an initial set of network weights that fit the average detector response. These weights were then used to initialize a network for each detector element. The element-specific networks were further trained using the calibration data from that specific detector element.

2.4.

Selection of Number of Hidden Layer Elements

A leave-one-out cross-validation study was performed to select the number of nodes in the hidden layer and to investigate whether pulse pileup effects alter the optimal network configuration. The number of hidden-layer nodes was varied between 5 and 100. During each cross-validation run, the network was trained on 24 calibration measurements, with one calibration measurement left out for testing. The trained network was then used to predict the basis material thicknesses of the test calibration measurement. The calibration points at the extreme corners of the calibration space (0-cm PMMA, 0-cm aluminum) and (10.16-cm PMMA, 2.54-cm aluminum) were not used as testing datasets to prevent the need for the network to extrapolate beyond the calibrated space. The root mean squared error (RMSE) between the true basis material thicknesses and the thicknesses estimated by the NN was calculated across all cross-validation test data. The number of hidden elements for subsequent studies was selected based on the resulting RMSE, after which an NN was trained for each count-rate level using all 25 calibration measurements.

2.5.

NN Evaluation Using Test Material X-Ray Transmission Measurements

To quantify the performance of the NN material decomposition across different pileup conditions, photon-counting x-ray transmission measurements were acquired through each of 3.81-cm-thick Teflon (8545K26, McMaster-Carr, Elmhurst, Illinois), 5.08-cm thick Delrin (8545K26, McMaster-Carr, Elmhurst, Illinois), and 2.54-cm-thick neoprene (9013K12, McMaster-Carr, Elmhurst, Illinois) using the acquisition protocol and count-rate levels described in Sec. 2.2. The transmission measurements were repeated five times for each test material and count-rate level. The log normalized energy-bin counts data at each count-rate level were input to the corresponding network trained for each detector element and count-rate level, resulting in the estimated basis material thicknesses for each count-rate level, detector element, and for each of the five trials.

The two basis material thicknesses returned by the NN estimator represent a point in two-dimensional (2D) material decomposition space. The 2D Euclidean distance, $D$, between the true basis material thickness coordinates, $t_{\mathrm{PMMA}}$ and $t_{\mathrm{Al}}$, and the basis material thicknesses estimated by the NN, ${\widehat{t}}_{\mathrm{PMMA}}$, and ${\widehat{t}}_{\mathrm{Al}}$, was calculated as a metric of material decomposition accuracy, as expressed in the equation below, which is also proportional to the RMSE. The ground truth material decomposition coefficients were estimated by simulating the polyenergetic photon-counting transmission measurements using the attenuation coefficient functions provided by the NIST XCOM database²³ followed by maximum likelihood decomposition. The uncertainty of the NIST attenuation functions is estimated at 1% to 2%.²⁴ This distance metric gives more weight to error in the basis material that has a greater contribution to the overall attenuation.

The median, lower quartile, and upper quartile of the distance metric were calculated across the five trials and all detector elements for each count-rate level and for each of the Teflon, Delrin, and neoprene test materials. The ground truth decomposition values contain uncertainty due to potential impurities in the materials, uncertainty of NIST attenuation coefficients, and uncertainty introduced during material decomposition. As an example for the 3.81-cm Teflon test slab, introducing $-2\%$ bias in the density and $-2\%$ bias in the NIST attenuation coefficients causes 0.2-cm distance error in the material decomposition estimates. To further evaluate NN performance without these uncertainties, the distance error metric was calculated for three combinations of known thicknesses of PMMA and aluminum using results of the leave-one-out study. The tested PMMA and aluminum combinations were selected as those with composition most similar to each of the three test materials, as described in Table 1.

Table 1

Ground truth basis material thicknesses for the three test materials and for the three combinations of known thicknesses of PMMA and aluminum used to evaluate NN performance. The combinations of PMMA and aluminum were selected to be similar to the compositions of the test materials.

2.6.

NN Evaluation Using Photon-Counting CT Acquisitions

To more fully evaluate the NN performance across incident count-rate level, photon-counting CT data were acquired of a 6.35-cm-diameter PMMA phantom with 1.9-mm-diameter cylindrical inserts of LDPE (8754K46, McMaster-Carr, Elmhurst, Illinois), PMMA (8573K13, McMaster-Carr, Elmhurst, Illinois), and Teflon (8546K15, McMaster-Carr, Elmhurst, Illinois). CT data were acquired at 120 views at 14%, 27%, and 40% of the maximum detector count rate, with the scan time scaled at each tube current setting to maintain a constant tube-current-time-product across count-rate levels. At each view, the detector was translated to two positions to encompass the phantom field of view. The trained NN for each count-rate was applied to decompose the CT data into PMMA and aluminum basis sinograms, which were then reconstructed into basis images using filtered backprojection.

NN performance was evaluated by extracting ROIs within the Teflon, LDPE, and PMMA inserts in each of the resulting PMMA and aluminum basis images at each count-rate level. The median, lower quartile, and upper quartile of the distance metric were calculated within each ROI and count-rate level. The standard deviation of basis map values within the ROIs was also calculated.

3.1.

Selection of Number of Hidden Layer Elements

Figure 3 shows the RMSE resulting from the leave-one-out cross-validation study of the number of hidden-layer nodes. The RMSE, calculated by averaging all validation trials and detector elements, is plotted for each tested number of nodes and for each studied count-rate level. The results demonstrate that a hidden layer with 50 to 60 nodes generally provided low validation error for all count-rate levels, while 30 nodes provided lowest RMSE for the lowest count-rate level and 70 nodes provided lowest RMSE for the two higher count-rate levels. The lowest count-rate level (9% of maximum count rate) resulted in the lowest RMSE for all network architectures. The RMSE was similar across the other count-rate levels when 50 or 60 nodes were used. Based on these results, the number of hidden layer nodes in the NN was 30, 30, 40, 70, and 70 for the respective count-rate levels of 9%, 14%, 27%, 40%, and 54% in the subsequent study. A study was also performed using 50 hidden-layer nodes for each count-rate level, with results presented in the Appendix.

Fig. 3

The results of the leave-one-out cross validation study are plotted for each of the studied detected count-rate levels. The detected count-rate levels are represented as percent of the maximum detector count rate. The RMSE of the validation test points, averaged across the validation trials and across all detector elements, are plotted for the different number of nodes in the hidden layer.

3.2.

NN Evaluation Using Test Material X-Ray Transmission Measurements

Figure 4 shows the NN material decomposition estimates in the 2-D material decomposition space for each detector element, count-rate level, and trial. Results are presented for the test combinations of calibration materials (top row) and the test material slabs (bottom row). The ground truth material decomposition values are also plotted.

Fig. 4

Material decomposition values estimated by the NN displayed in the 2-D material decomposition space. (a) The results of decomposing the known combinations of (PMMA, aluminum) thicknesses in the leave-one-out study. (b) Results from the test materials. Results are plotted for each test material and detector element, and, for the test materials, each of the five trials. Each marker type represents the detected count-rate level specified as the percent of the maximum detector count rate. The ground truth material decomposition coefficients are also plotted for each case.

Figure 5 first plots the material decomposition distance error against detected count-rate level for an NN that was trained with data from only the lowest count-rate level (9%) and then applied to decompose data from all count-rate levels. While this approach is not recommended, these results demonstrate the magnitude of potential error due to pileup, which increases with increasing count-rate level. Figure 5 also plots the distance error that results from NNs trained specifically for each count-rate level, demonstrating the ability of the trained NNs to reduce error due to pileup. For example, the error in the Teflon material decomposition was 0.12 cm for the 54% count-rate level when using an NN trained specifically with data acquired at that count-rate level, compared with an error of 7.1 cm when the NN was trained with data from the 9% count-rate level.

Fig. 5

The top row displays the distance error plotted against detected count-rate level for an NN trained with data from the lowest detected count-rate level and then applied to data acquired at varying count rates for (a) (PMMA, Al) test combinations and (b) test materials. The bottom row displays the distance error resulting from NNs trained specifically for each count-rate level for (c) (PMMA, Al) test combinations and (b) test materials. The plotted points represent the median distance error across all detector elements and trials, while the error bars represent the upper and lower quartiles.

As seen in Fig. 4, the incident count rate affected the relative contributions of basis material thicknesses estimated by the NNs. For example, PMMA had a larger contribution to the Teflon decomposition at low count rates than high count rates, while the aluminum contribution demonstrated the opposite trend. However, the overall distance error for the Teflon case remained fairly constant with incident count-rate level, with error of 0.33 cm at the 9% count-rate level and 0.12 at the 54% count-rate level. All NNs overestimated the PMMA contribution of neoprene while underestimating the aluminum contribution. The neoprene results demonstrated the highest overall error, with the error increasing with count-rate level. The neoprene error at the 9% count-rate level was 0.65 cm, compared with 1.14 cm at the 54% level for the NNs specifically trained at each count-rate level. While the error increased with count rate for the neoprene decomposition, the count-rate-specific NN considerably reduced the potential error due to pileup when compared with the 5.2-cm error for the 54% count-rate level when using an NN trained at 9% count-rate level. The test combination of 0.0-cm PMMA and 0.635-cm aluminum, which was close to neoprene in the decomposition space, did not show the trend of increasing error with increasing count rate. There was no clear trend between count-rate level and material decomposition error for Delrin, Teflon, or for the tested combinations of PMMA and aluminum. For materials other than neoprene, the differences in basis material decomposition were within the estimated 0.2-cm uncertainty of the ground truth values.

Figure 6 shows the standard deviation of the basis material thicknesses estimated by the NN for each test material and count-rate level. To reduce the effects of detector element-to-element variations on the quantification of standard deviation, the standard deviation was calculated across the five trials and then averaged across detector elements. As seen in Fig. 6, the standard deviation of the NN estimates did not show a clear trend with count-rate level. The 40% count-rate level demonstrated the highest standard deviation, which may suggest that using 70 nodes in the hidden layer may not be optimal and may cause overfitting. The appendix evaluates the performance of NNs with 50 nodes in the hidden layer for each count-rate level. The results in the appendix demonstrate that using 50 nodes may be beneficial for the 40% and 54% count-rate levels.

Fig. 6

Standard deviation of the PMMA and aluminum basis material thicknesses estimated by a NN trained for each count-rate level. The standard deviation was calculated across the five trials and then averaged across detector elements for each test material.

3.3.

NN Evaluation Using Photon-Counting CT Acquisitions

The PMMA and aluminum basis images reconstructed from the basis sinograms estimated by the NNs are shown in Fig. 7 for incident count rates of 14%, 27%, and 40% of the maximum detector count rate. The number of hidden nodes used in the NN for the 14% and 27% count-rate levels was 30 and 40, respectively, as in the results presented in Sec. 3.2. As described in the Appendix, an NN architecture with 50 hidden-layer nodes was determined to be advantageous for the 40% count-rate level and was used for this study. The images in Fig. 7 demonstrate similar gray-level values despite the different pileup effects. Figure 8 shows the median distance error for ROIs within the Teflon, PMMA, and LDPE rods of the phantom. The quantitative error for all three rods decreased with increasing count-rate. For example, in the Teflon rod, the material decomposition error was 0.089, 0.066, 0.054 at count-rate levels of 14%, 27%, and 40%, respectively. The noise level in the basis images varied with count-rate level but did not show the trend of increased noise with count-rate level that would be expected due to dead time losses. For example, the standard deviation, which takes into account both noise and ring artifacts, was 0.012, 0.017, and 0.006 in the Teflon ROI of the aluminum basis image for the 14%, 27%, and 40% count-rate levels, respectively, and 0.054, 0.064, 0.043 in the Teflon ROI of the PMMA basis image.

Fig. 7

PMMA and aluminum basis images reconstructed from the basis sinograms that were estimated by the trained NN for each studied flux level. The rods, from brightest to darkest, are Teflon, PMMA, LDPE, and air. The 50-keV virtual monoenergetic image formed by the basis images is also shown for each flux level. The flux level is reported as the percentage of the maximum detector count rate.

Fig. 8

Euclidean distance error metric estimated in the Teflon, PMMA, and LDPE regions of the basis images shown in Fig. 7 for the three studied count-rate levels. The data points represent the median error in the ROI. The error bars represent the upper and lower quartile of the error in the ROI. The basis coefficients in the CT basis images represent the contribution of each basis material and are thus unitless.