Beam profile assessment in spectral CT scanners

Abstract In this paper, we present a method that uses a combination of experimental and modeled data to assess properties of x‐ray beam measured using a small‐animal spectral scanner. The spatial properties of the beam profile are characterized by beam profile shape, the angular offset along the rotational axis, and the photon count difference between experimental and modeled data at the central beam axis. Temporal stability of the beam profile is assessed by measuring intra‐ and interscan count variations. The beam profile assessment method was evaluated on several spectral CT scanners equipped with Medipix3RX‐based detectors. On a well‐calibrated spectral CT scanner, we measured an integral count error of 0.5%, intrascan count variation of 0.1%, and an interscan count variation of less than 1%. The angular offset of the beam center ranged from 0.8° to 1.6° for the studied spectral CT scanners. We also demonstrate the capability of this method to identify poor performance of the system through analyzing the deviation of the experimental beam profile from the model. This technique can, therefore, aid in monitoring the system performance to obtain a robust spectral CT; providing the reliable quantitative images. Furthermore, the accurate offset parameters of a spectral scanner provided by this method allow us to incorporate a more realistic form of the photon distribution in the polychromatic‐based image reconstruction models. Both improvements of the reliability of the system and accuracy of the volume reconstruction result in a better discrimination and quantification of the imaged materials.

performance. 1,2 Characterizing the beamlines is also essential in particle accelerators for them to be operated with optimal output. 3 The polychromaticity of the x-ray beams used in the computed tomography (CT) scanners necessitates the accurate modeling of beam profile in these machines. CT scanners are used for diagnostic imaging (kilovoltage range) and image-guided radiotherapy (megavoltage range). Many methods have been published for beam profile measurements of such systems. Among them, the work published by Malts et al. can be referred to. They presented a method of characterizing the spatial variation in the intensity and energy of the incident beam in diagnostic and treatment cone beam CT. 4 Beam profile characterization is also a prerequisite to optimize performance of spectral CT scanners operating on the basis of photon-counting detectors. The optimal performance of the spectral CT scanner is achievable when the energy and position of the incident photon are measured accurately. 5 Beam profile assessment methods examining various properties of the beam profile can be used to identify the parameters that prevent accurate measurement of photon energy and position. The properties of the beam such as photon intensity and its angular distribution not only needs to be characterized at initial installation, but beam profiles also need to be regularly assessed for identifying the distortion caused by either deterioration of the x-ray tube performance during its lifetime or instabilities of other scanner components. In this study, aforementioned properties are characterized for the beam profiles measured using MARS small-animal spectral CT manufactured by MARS Bioimaging Ltd., New Zealand.
The MARS spectral CT scanners use Medipix photon counting detectors to provide 3D tomographic images with both high spatial and high spectral resolution. The energy resolved information enables simultaneous discrimination and quantification of different materials based on their spectral signatures. 6,7 The spectral imaging allows the extraction of functional and anatomical features of the tissues via tracing biomarkers and pharmaceuticals in a low dose and noninvasive way. 8,9 MARS imaging has been used in various preclinical applications such as characterizing the composition of excised vulnerable atherosclerosis plaques in arteries, 10 functional imaging of arthritic cartilage, 11 and targeting cancerous cells using nanoparticles. 12 The x-ray tube used in the spectral CT provides a cone shape photon distribution which typically varies over the imaging field. 4 Furthermore, x-ray tube manufacturing and alignment variation of the beam direction with detector plane in a spectral CT scanner also makes the photon beam profile specific to that system. For instance, relative geometric offsets due to tube anode orientation may spatially shift the recorded beam profile. To identify such a variation, the beam profile of each spectral CT scanner needs to be characterized. The information obtained from beam profile characterization can then be used to calibrate the image reconstruction models. Providing the more realistic form of the photon distribution to the forward model allows better image reconstruction, and as a consequence better material identification and quantification. Performing spectral reconstruction with an inaccurate characterization of the x-ray beam has the potential to cause significant material misspecification. 13 Reconstruction problems can also arise when random fluctuations occur in the beam profile due to instability of CT scanner components, such as the x-ray tube and detector. Fluctuations in the beam profile are more likely when spectral data are acquired over a long exposure time. Relatively long exposure time is required because photon counting detectors can optimally operate at low photon flux. 14 The use of low photon flux ensures maintaining spatial and spectral fidelity of the images in two aspects. 5,15 Firstly, the small pixel size of the photon counting detectors such as Medipix3RX favors the use of x-ray tubes with small focal spot sizes (e.g., 50 lm) to maximize spatial resolution. 5,14,16 Striking the smaller area of the anode target by electrons, in turn, generates a lower photon flux. 14,17 Secondly, due to limited pulse resolution time of such detectors, the energy information of a high flux beam cannot be resolved correctly. The energy of coincident photons is accumulated and registered at a higher energy of each initial photon. This pulse pile-up effect results in the loss of spectral information. 16,18 To minimize the occurrence of this effect, incident photon flux needs to be reduced.
Acquiring data with longer exposure time, while using the low photon flux, provides sufficient counts; resulting in a higher signal to noise ratio in reconstructed images. However, the detector performance may degrade due to increasing ASIC temperature and as a consequence, charge loss occurs due to detector polarization during long acquisition time. Therefore, the beam profile stability needs to be monitored in such a system to ensure that there is no count drift during imaging. In response to this need, we have developed a beam profile assessment and characterization method. The method enables quantification of the temporal and spatial properties of beam profiles and assessment through comparison with modeled beam profiles.
In this paper, we introduce a parametrized semi-analytic source model and the experimental requirements. We then explain the procedure of developing the beam profile assessment method, and present method evaluation results obtained from one well-calibrated and two poorly calibrated MARS spectral CTs.

| MATERIALS AND METHODS
A workflow diagram of the beam profile assessment method is depicted in Fig. 1. An experimental beam profile is provided to the method. Then a modeled beam profile is prepared from a semi-analytic source model based on the equivalent spatial parameters of the measurement. Measured and modeled data are then preprocessed to reach the same level of conformity to be comparable with each other. In the next step, several properties of the beam profile are measured. In the comparison step, the measured properties are compared with the modeled beam profile. If a significant discrepancy is identified, it indicates potential issues with calibration or components of the systems.

2.A | Modeling the beam profile
The beam profile assessment technique requires the use of an x-ray source model that describes the spatial variation in the x-ray beam away from the central axis. For this purpose, we have utilized a parameterized semi-analytic source model fitted to the x-ray tube with a 50 lm focal spot and 20°anode angle. 19 The general formula of this source model, S hu EV , is presented by eq. (1).
where, S hu EV provides the spectral components of the x-ray spectra as a function of energy, E, tube voltage, V, and angular distribution of h and u. h is the camera translation which is along the scanner rotational axis and u expresses the anode-cathode direction that is orthogonal to the rotational axis and the beam direction as demonstrated in Fig. 2. S 00 provides the x-ray spectrum at the beam center for a given tube voltage A, B, and C are coefficients, which depend on x-ray energy (keV) and tube voltage (kVp). n h and n u represent the beam offsets along h and u with respect to the center. This source model currently can be used for the x-ray tubes with the voltage range of 30-120 kVp, and angular photon distribution within h = AE 17 and u = AE 5.5°. Further details can be found in Ref. [20,21].
To obtain a modeled beam profile, the first step is to extract a spatial photon distribution from the source model based on the tube voltage, filtration, and geometric features of the scanned data. The magnified beam profile shown in Fig. 3 is an example of a modeled photon distribution in a typical field of view fitted to scan a small object size like mouse. It should be noted that the x-ray photon distribution across the rotational axis (h) is analyzed in this study and the count variation along u is assumed to be negligible (i.e., 0.06% in a typical field of view).
The second step is to correct the source model output for the factors that modulate the incident photons as a result of detector properties. The beam profile assessment algorithm adjusts the incident counts for two major detector effects according to eq. (2): where, l E is linear attenuation coefficient (mm À1 ) of the sensor layer which varies with energy, E. The thickness of the sensor layer, t, which is 2 mm for the Medipix3RX detector was used in this study.
Another phenomenon which distorts the spectral performance of the detector is coincident photon pile-up that happens when the photons arrive in a time domain less than the dead time of the 1. An overview of beam profile assessment method.

2.B | Measurement of the beam profile
The spectral scanner used in this study was MARS small-bore CT In Fig. 4, each frame is divided into five groups for the demonstration purpose, but to obtain more points across the beam profile, the average of each four rows of pixels is typically used. This grouping provides 32 data points at each camera position. Thus, the beam profile resolution of a five-camera position scan is extended up to 160 points along h.
c. Temporal beam profile is measured to monitor the variation in the beam profile over time at a given position. In the small-bore spectral CT setups with single-chip camera, the dataset of each camera position is consecutively collected before the next one.
To obtain temporal information of the recorded counts, the data-  | 291 e. Regression is applied to the measured data using a second-degree polynomial curve fitting to extract the beam profile. This fitted curve is expected to follow the parabolic shape as the modeled beam profile. The measured beam profile obtained from this step is then normalized with respect to the peak of the parabola to assess some properties like its shape as explained in the next section.

2.C | Beam profile properties
Several properties of the beam profile have been determined in this study to efficiently characterize the spatial beam distribution in a spectral CT scanner. The reliability of the beam profile's properties is also assessed by comparing them with properties of the modeled beam profile.
Beam profile shapes are assessed using the concavity and latus rectum of the beam profile parabola. To determine the sign of concavity, a simple test is to calculate the second derivative of the measured beam profile. The latus rectum of a parabola is the chord that passes through the focus, which is perpendicular to the major axis transversing the curves at two points. 24 The measured beam profiles are expected to be concave down similar to the modeled beam profile. The average of the latus rectums in all temporal beam profiles is calculated and compared with the latus rectum of the molded beam profile. In addition, the variation in the temporal beam profiles is considered to assess the beam profile. Interscan count variation can be measured by comparing count drift between different scans at the beam center. The interscan variability arises from changes in the system state such as increasing the ASIC temperature or detector polarization due to heavy use of the CT system.

2.D | Method evaluation
To validate the efficiency of the beam profile assessment method, a MARS spectral scanner was used which completely passed a series of QA tests to check the stability of every component of the scanner such as high voltage power supply, x-ray tube, and detector, as well as performing several geometrical alignment tests. This system, therefore, was considered as a well-calibrated system. Three datasets including 720 flat-field frames in each were collected by a single-chip CdTe-Medipix3RX at every camera position. The camera was translated to five positions and the distance from the center of the camera position to the x-ray source was set to 187 mm. Each single exposure was performed by an 80 kVp x-ray beam with the intensity of 30 lA during 120 ms. A 3.1 mm aluminum sheet was also used to filter the x-ray beam in addition to an intrinsic filter of 1.8 mm aluminum.
Furthermore, the efficiency of the beam profile assessment method was evaluated by performing this method to the poorly calibrated CT systems. For these series of measurements, five camera positions were chosen and 720 flat-field frames were collected at each position. The source to detector distance was set at 270 mm. In every flat-field measurement, the camera was exposed by a 120 kVp x-ray beam with the intensity of 20 lA during 180 ms. The output spectrum was filtered by 0.375 mm brass.

| RESULTS
This section reports the results of the beam profile assessment method performed on several MARS spectral CTs with different levels of calibration quality.

3.A | Beam profile assessment for a well-calibrated system
The assessment beam profile method was applied to all three datasets acquired from a MARS spectral scanner considered as a wellcalibrated scanner. The photon distribution along h from one of these datasets is demonstrated in Fig. 7a. We checked for bias in this dataset by inspecting the ratio of the measured noise (i.e., variance/mean) to the expected noise (i.e., 1= ffiffiffi n p where n is photon flux across the number of frames for each pixel). The histogram of this ratio for a group of counts is presented in Fig. 7b. The bell-shaped histogram with an average of one indicates a Poisson distribution.
Next, a quadratic function was fitted to this dataset, as shown in There is 0.03°difference between the average latus rectum of the temporal beam profiles and model, which is within the experimental uncertainty. Hence, the shapes of these temporal beam profiles are well-matched with the model as shown in Fig. 8b.
Second, the angular offset of this measurement along h is 0.8°AE 0.07°, as shown in Fig. 8a. The solid red curve shows the measured beam profile after applying the angular offset adjustment.
The standard deviation value (AE0.07°) is approximately one-tenth of the angular offset, which is low enough to accept the angular variation in the temporal beam profiles.
Third, there is an intrascan count variation in this measurement as shown in Fig. 8b. The deviation of the beam profile in the last time interval with respect to the first one is around 0.1%, which is negligible for this scan. It is evident that the beam profile is quite stable on this CT scanner.
Fourth, the magnitude of the measured beam profile was compared with the model at the beam center. In Fig. 9, the blue curve shows the beam profile of this dataset plotted against the model.
The difference between the integral counts of this dataset and model is around 0.5% at h ¼ 0.

| 293
Finally, the beam profiles of two other datasets collected by the same spectral CT are also plotted in Fig. 9. The interscan variation between these three datasets is just above 0.3% due to statistical error. Low interscan variation indicates that this CT system can reliably perform the same scan.    Table 1.

| DISCUSSION
The results demonstrated that the beam profile assessment technique can efficiently be used to monitor the performance of the spectral CT scanner. This method could precisely exploit the parameters varying between different beam profiles.

Measured beam profile in T 2
Measured beam profile in T 3 F I G . 1 0 . A beam profile with a minor defect due to intrascan variation at the end of the scan (i.e., most positive h value). Although this dataset has relatively large angular offset, the variation in the angular offsets between all temporal beam profiles is within the acceptable range. The quadratic polynomial function fitted to the first set of experiments was evaluated by RMSE values, which was less than 1% for a series of stable beam profiles presented in this study. The results of the beam profile assessment in the MARS spectral CT with the same setup showed that even if the beam profile shape was deformed, a quadratic curve fitting could still express the actual shape of the measured profile. As a worst-case scenario, it can be referred to an anode defect occurrence in which the beam profile may not follow the quadratic trend. It is noteworthy that the anode defect is very unlikely to happen for the low power x-ray tubes 20 used in a spectral micro-CT system like MARS as the amount of heat formed at the anode is a fraction of heat generated from the anode surface of the high-power x-ray tube used 25 in the conventional CT scanner.
However, the performance of the x-ray tube in a spectral CT needs to be tested regularly to provide ongoing assurance of the system prior to performing a scan. This test can be performed systematically using the beam profile assessment method presented by this work.
The beam profile shapes in the poorly calibrated system may suf- The angular offset along h , which was measured from partially stable beam profiles that varied from 0.8°to 1.6°. As previously noted, the main reason for beam profile offset is unavoidable tolerance of the x-ray tube during manufacturing. Another possible reason is flex of the scanner components. Unlike the angular offset variation between a series of spectral CT systems, the angular offset of the equivalent temporal beam profiles should be identical within an acceptable uncertainty. The large angular offset (Fig. 11) is another evidence of poor geometric calibration of the scanner.
The angular offset along u was not measured in this study because the horizontal dimension of a typical field of view is small in the MARS small-bore scanners. In the case of using a wider horizontal field of view, u offset would need to be considered. It may change the amount of vertical angular offsets and the skewness of the beam profile that requires further investigation.
To analyze the intrascan variation in the integral counts, differences between temporal beam profiles are measured. The concept of a temporal beam profile is proposed, based on count sampling at each camera position for different time intervals. It is expected that the number of counts at each position should remain the same with reasonable uncertainty, provided that scanner components are working in a steady state during data acquisition. Therefore, any inconsistencies in these beam profiles would indicate intrascan count variation.
The location of the intrascan variation also provides some clues about the origin of variation. The deviation of the temporal beam profiles at the beginning of the scan could be due to including the x-ray tube warm-up time in the acquisition time. Temporal deviations that appear at the end of the scan show a degradation in recorded counts, probably resulting from a gradual rise in ASIC temperature or detector polarization during data acquisition (Fig. 10). If the beam profiles in different time domains behave chaotically (Fig. 11), it is evidence of transient distortion occurring across the entire the scan. In general, intrascan count variation can increase the variation in other beam profile properties. For instance, the large tolerance of angular offset shown in Fig. 11 is due to large intrascan count variation.
The results of assessing the integral counts at the beam center in a well-calibrated MARS CT indicated that the measured counts and those calculated from the model are well-agreed as shown in Fig. 7a. The minor difference (<0.5%) is due to not correcting the source model for other potential detector effects such as incomplete charge collection, cadmium fluorescence, and charge sharing. 15,[26][27][28] In the poorly calibrated CT system, a large difference (25-30%) was observed between the experimental and modeled counts. Possible reasons are inaccuracies in the geometric calibration, such as the source to detector distance, and filter thickness. If the scan is performed under incorrect setup parameters, detector may operates in the nonlinear dynamic range; resulting in an unstable beam profile.
The beam profile assessment in a series of scans performed by the same spectral CT scanner revealed that there is no significant interscan variation in integral counts at the beam center in a well-calibrated system (0.3% in Fig. 9). The interscan count variation analysis can be used as a part of quality assurance (QA) test for measuring repeatability error in each spectral CT scanner. A scanner fails the assessment test when a large interscan variation is observed between the scans performed iteratively on the same day.
In human spectral CT with a larger field of view, multichip detectors are used, which require more accurate and faster troubleshooting. Using a multichip detector, the entire beam profile can potentially be acquired at a single exposure. Because of this, the overall trends of spatial and temporal beam profiles are formed by the beam profile of each individual chip in the detector array. Providing correct beam profiles for all detectors in an array is essential, particularly when they are operated in a helical scan. 17 Translation of the beam profile assessment method from the single-chip detector to the multichip detector array can be performed by analyzing the response of each chip. Stitching the beam profiles measured by all detectors together would provide higher resolution of the overall beam profile as more spatial points are available using the multichip detector.
Although the output of this technique indicates the beam profile fluctuation in a single-chip camera, it does not address the main source of this fluctuation. It is expected that in a multichip camera, we can differentiate between an unstable x-ray tube and a faulty detector array. This is because of simultaneously obtaining the correlated spectral signal in a multichip camera, over a larger h direction by different chips. In addition, further investigation is required to precisely determine an uncertainty range for each property of the beam profile.

| CONCLUSION
The method presented in this paper qualitatively and quantitatively assesses various beam profiles, which can assist in improving spectral CT performance in two ways. Firstly, the method can identify the presence of various calibration issues in a spectral CT scanner.
It offers a simple and fast check of the beam profile during manufacturing. It also aids in reliably performing quality assessment at different stages from manufacturing through to the final product.
Secondly, the accurate offset parameters of the beam profile provided by this work can also be used for additional geometric calibration of the x-ray source model. The use of optimized x-ray source model in the spectral reconstruction techniques will improve the accuracy of material identification and quantification.

ACKNOWLEDGMENTS
The authors thank the University of Otago, University of Canterbury, MARS Bioimaging Ltd., Medipix collaborations at CERN, National Heart Foundation, and Maurice and Phyllis Paykel Trust for supporting this research.

CONF LICT OF I NTEREST
No conflicts of interest.