Development and tuning of models for accurate simulation of CT spatial resolution using CatSim

Abstract

Objective

We sought to systematically evaluate CatSim’s ability to accurately simulate the spatial resolution produced by a typical 64-detector-row clinical CT scanner in the projection and image domains, over the range of clinically used X-ray techniques.

Approach

Using a 64-detector-row clinical scanner, we scanned two phantoms designed to evaluate spatial resolution in the projection and image domains. These empirical scans were performed over the standard clinically used range of X-ray techniques (kV, and mA). We extracted projection data from the scanner, and we reconstructed images. For the CatSim simulations, we developed digital phantoms to represent the phantoms used in the empirical scans. We developed a new, realistic model for the X-ray source focal spot, and we empirically tuned a published model for the X-ray detector temporal response. We applied these phantoms and models to simulate scans equivalent to the empirical scans, and we reconstructed the simulated projections using the same methods used for the empirical scans. For the empirical and simulated scans, we qualitatively and quantitatively compared the projection-domain and image-domain point-spread functions (PSFs) as well as the image-domain modulation transfer functions (MTFs). We reported four quantitative metrics and the percent error between the empirical and simulated results.

Main Results

Qualitatively, the PSFs matched well in both the projection and image domains. Quantitatively, all four metrics generally agreed well, with most of the average errors substantially less than 5% for all X-ray techniques. Although the errors tended to increase with decreasing kV, we found that the CatSim simulations agreed with the empirical scans within limits required for the anticipated applications of CatSim.

Significance

The new focal spot model and the new detector temporal response model are significant contributions to CatSim because they enabled achieving the desired level of agreement between empirical and simulated results. With these new models and this validation, CatSim users can be confident that the spatial resolution represented by simulations faithfully represents results that would be obtained by a real scanner, within reasonable, known limits. Furthermore, users of CatSim can vary parameters including but not limited to system geometry, focal spot size/shape and detector parameters, beyond the values available in physical scanners, and be confident in the results. Therefore, CatSim can be used to explore new hardware designs as well as new scanning and reconstruction methods, thus enabling acceleration of improved CT scan capabilities.

Introduction

X-ray-based imaging modalities are crucial tools for clinical screening, diagnosis, treatment planning, and response monitoring of a wide range of diseases. Development of improvements to these imaging methods can be enabled and accelerated by simulation tools. The X-ray-based Cancer Imaging Simulation Toolkit (XCIST) is currently being developed in the context of cancer imaging but can more broadly be applied to develop improved imaging systems for many medical-imaging needs. (Wu et al., 2022) XCIST includes CatSim (De Man et al., 2007) as the core simulator, as well as numerical phantoms (Segars et al., 2008) and image-reconstruction tools. XCIST is an open-source toolkit accessible to academic and commercial researchers; rigorous validation of its critical features will enable its wide application by scientists and engineers for whom such sophisticated tools may not be currently available.

Spatial resolution is one of the most important system performance characteristics to model correctly. The models that contribute to the spatial resolution in the simulated images include the focal spot (FS) (Papadakis et al., 2021), X-ray transport process, detector, X-ray detection process, system architecture, scan trajectory, and reconstruction. Many of these are straightforward to model and have been validated previously (Wu et al., 2022). However, two new, critical models that contribute to spatial resolution were developed (Figure 1A) and evaluated for this work, including a realistic FS model and a detector temporal response model (Duclos et al., 2003). These models are significant new contributions to CatSim. Using these new models, we evaluated agreement (Figure 1B) between empirical results from a commonly used 64-row clinical CT scanner and simulated results obtained when modeling the same scanner using CatSim within the XCIST framework. This evaluation is also a significant contribution to CatSim because it is important for users to understand the regime over which CatSim has been validated, in terms of spatial position within the scan field of view (FOV), orientation (e.g., lateral versus longitudinal and radial versus azimuthal), and the quantitative limits of CatSim’s ability to predict the performance of a real scanner.

Figure 1. Workflow for this paper.

A) New model development. B) Evaluation of simulation results versus empirical results.

Methods

We compared the spatial resolution obtained empirically with a real scanner with the resolution obtained when simulating the same scanner. We use the term “empirical” to denote methods and results obtained using hardware (the real scanner), and the terms “simulation” or “simulated” to denote methods and results obtained using software (CatSim). We intentionally do not use the term “experimental” to denote empirical work because we find that that leads to confusion, since experiments are performed in both hardware and software.

We evaluated the spatial resolution obtained in both the projection domain (PD) and image domain (ID) using previously developed approaches.(De Man, 2001). Importantly, we did not intend to assess the absolute performance of the modeled system, because that is affected by multiple factors that are controlled but not optimized in this work. Rather, we focus on agreement between empirical and simulated results.

We reported four primary metrics; we introduce these and their abbreviations here, for clarity. In each domain, we report the full width at half maximum (FWHM) of the point spread function (PSF), designated PSF50_proj and PSF50_image. We also report the image-domain modulation transfer function (MTF) at 50% and 10% modulation, designated MTF50 and MTF10. We appended each of these abbreviations with _emp and _sim for empirical and simulation, respectively. Each of these metrics are reported in various orientations and positions; we state these conditions in figure axes and figure and table captions.

Parameters used for simulations reflected the nominal empirical configuration, with additional models used to simulate imprecise and/or non-ideal effects such as focal spot (FS) size/shape/contour and detector scintillator temporal response. We performed experiments to evaluate the resolution in the projection domain and in reconstructed images. Data in empirical projections were converted to post-log p-values, but all standard corrections for non-ideal effects (e.g., temporal response, low signal) were disabled. All simulated data were computed using CatSim within the XCIST package, publicly available from (Wu et al., 2022).

The concept of sub-sampling is especially important for simulation of spatial resolution. This is described in detail in (Wu et al., 2022). In summary, for each projection, X-ray attenuation line integrals are calculated for multiple user-specified samples of the focal spot (lateral and longitudinal subsamples, i.e., width and length, respectively), detector cells (lateral and longitudinal subsamples, i.e., column and row, respectively) and view angle (angular sub-samples). Sampling parameters are specified below.

The key hardware, simulation models, and parameters used are listed below.

CT system

Empirical results were obtained using the CT scanner and scan protocols described in Table 1. For the simulation experiments, the system model precisely represented this system and scan parameters.

Table 1.

CT system parameters

Scanner model: Lightspeed VCT® (GE HealthCare, Chicago, IL)

X-ray tube potentials: 80 kV, 100 kV, 120 kV, and 140 kV

Source-to-isocenter distance (SID): 541 mm

Source-to-detector distance (SDD): 949 mm

Detector column size: 1.0239 mm

Detector row size: 1.098782 mm

Detector column count: 888

Detector row count: 64

Rotation time: 1 second

Number of views per rotation: 984

Focal spots: small and large

X-ray tube currents: 100 mA and 300 mA for the small FS and 350 mA and 600 mA for the large FS

Bowtie filter: large

Patient table: not used (phantoms were suspended in air)

Phantoms

The phantoms for the simulation experiments were analytic phantoms that precisely represented the physical phantoms used for each empirical experiment, described below.

Phantom for projection domain experiments:

We constructed the phantom by inserting two 0.7-mm-diameter stainless steel (SS) wires into a 450-mm-diameter foam cylinder. The wires were located approximately 218 mm from the axis of rotation (AOR); one wire was approximately parallel to the AOR and the other was approximately 45° from the AOR (Figure 2A). We reconstructed images only to measure the distance of the wires from the AOR; we then created a virtual phantom for simulations using those wire positions. A tilted object can be used to evaluate longitudinal (z) resolution in CT (Polacin et al., 1994). In our implementation of this concept, as the z resolution decreases, the wire PSF will become more blurred along the direction in which the wire is tilted, since the longitudinal convolution with the slice sensitivity profile will translate into an in-plane convolution with a similar profile, magnified by the tangent of the tilt angle. This z-resolution blur effect is superimposed on the blur effects associated with in-plane spatial resolution.

Figure 2. Phantoms.

(A) For the projection-domain experiments. (B) For the image-domain experiments.

Phantom for image domain experiments:

We also constructed this phantom using a 450-mm-diameter foam cylinder. We drilled nineteen ~10-mm holes in the foam at positions ranging from ~0 to ~225 mm from the phantom center, in ~12.5-mm increments (Figure 2B). We fastened the ends of 0.1-mm tungsten wires at the faces of the foam, attempting to keep the wires taut and centered in the hole. We scanned the phantom, measured the wire positions in the images, and created a virtual phantom for the simulations using the measured wire positions.

No attempt was made to exclude the impact of the finite wire thickness from the PSF and MTF metrics since our goal was not to evaluate the absolute system spatial resolution but rather show the close correspondence between simulations and empirical results.

Reconstruction

XCIST’s equiangular Feldkamp-Davis-Kress (FDK) (Feldkamp et al., 1984) routine was used for all reconstructions. For empirical experiments, the projection data were obtained from the scanner for reconstruction using the XCIST FDK routine.

Focal spot simulation model

When using CatSim, the FS used for a given simulation is selectable from a point source, uniform or Gaussian models, or realistic FS models. Any of these models except the point source can be parameterized and then subsampled per CatSim’s configuration parameters. The realistic models were developed from 2D FS photographs made using a precision pinhole and X-ray camera (Bookstein and Steck, 1971); the resulting FS images were then processed by averaging and background subtraction. For these simulations, we used the VCT small and VCT large models, parameterized and sampled as shown in Table 2. The model measures the FS size from the FS image in each direction (width and length) at the user-specified threshold. The model then scales the FS image to the user-specified size in each direction, and samples the scaled FS image to the specified number of samples using the pixels in the FS image at 2% of the maximum intensity as the start and end of the sampled area. Therefore, the FS size and the number of samples is user specified but the sample spacing (mm) is determined by the model. Using this model, one can simulate the performance of a system with various specified FS sizes, for example to evaluate hypothetical system performance with hypothetical FS sizes. However, in this work we specified the estimated true FS sizes as measured from the FS images, so that we could compare empirical versus simulated results. The FS images used by the model, and images representing the FS model, as parameterized in Table 2, are shown in Figure 3. To assist CatSim users, a more detailed discussion of the FS model is provided in Supplemental Information S1.

Table 2.

Focal spot parameters used for the reported simulations.

		Small	Large
FS image pixel size (mm) (width and length)		0.00372
FS size threshold (width and length)		0.5
FS size (mm) (a)	Width	0.897	1.247
	Length	0.565	1.009
FS model number of samples	Width	15
	Length (b)	15
	Length (c)	4
FS model sample spacing (mm) (a)	Width	0.100	0.121
	Length (b)	0.073	0.108
	Length (c)	0.274	0.404

^(a)Determined from the FS image by the FS model’s algorithm, then specified for experiments herein

^(b)For tilted wire experiments

^(c)For all other experiments

Figure 3. Focal spots.

(A) Small FS. (B) Large FS. 2D images and 3D contour plots are shown. The 2D images are scaled to reflect the sample spacing, so the focal spot aspect ratio is correct, as shown by the 1-mm scale bars. The 3D contour plots use a square pixel, so the number of samples is correctly represented but the sample spacing is not to scale.

Detector simulation model

CatSim supports arbitrary detector geometries and multiple detector physics effects, which are specified using configuration parameters. The temporal characteristics of the HiLight detector used in the VCT scanner have been reported in detail (Hsieh, Gurmen and King, 2009); we used the reported multi-exponential temporal response function in our simulations:

$$h\left(t\right)=\sum _{n=1}^N\frac{{\alpha }_n}{{\tau }_n}U\left(t\right)e^{\frac{-t}{{\tau }_n}}$$

For this work, we used a bi-exponential model (i.e., N = 2); we therefore required two scaling coefficients (α₁ and α₂) and two time constants (τ₁ and τ₂). We used the published values for the coefficients and time constants except that we adjusted the primary time constant to achieve better temporal agreement between empirical and simulated results. We used the nominal detector geometry specified in Table 1, and other detector parameters as specified in Table 3. The simulation parameters for detector fill factor and crosstalk are nominal values but do not necessarily precisely represent the detector’s true physical properties. Note that the detector sample spacing is the detector cell (DC) aperture size (the DC size with the fill factor applied) divided by the number of samples.

Table 3.

Detector parameters used for reported simulations.

Number of detector rows	PD sims	4
	ID sims	64
Cell fill factor (column and row)		0.9
Number of samples	Column	15
	Row (a)	4
	Row (b)	15
X-ray crosstalk	Column	0.025
	Row	0.020
Optical crosstalk	Column	0.045
	Row	0.040
Temporal response	α1	0.930
	α2	0.070
	τ1 (ms)	0.9
	τ2 (ms)	6.0

^(a)For PD experiments

^(b)For ID experiments

System simulation model

Each projection includes multiple simulated line integrals of attenuation. The spectrum model is sampled at a user-specified number of energy bins; the attenuation coefficients for each material are averaged over the energy ranges for each energy bin. For this work, we used kV/5 (the tube potential/5) energy bins. To estimate the effect of gantry rotation during one projection, multiple sub-projections are formed at sub-view angles, the number of which is user-specified. For this work, we used 15 view samples. At each sub-view, line integrals are calculated from each FS sample position to each DC sample position, at each sampled energy. Therefore, the total number of line integrals simulated is the product of the numbers of: energy bins, views, view samples, FS samples, DCs, and DC samples. For this work, the number of line integrals calculated for the simplest simulated scan (PSF50_proj at 80 kV) was 80/5 energy bins × 984 views × 15 view samples × 15 FS width samples × 4 FS length samples × 888 detector columns × 4 detector rows × 15 DC column samples × 4 DC row samples = 3×10¹³ line integrals. Running single-threaded CatSim on a Linux workstation with 256 AMD EPYC 7702 processors running at 1.564 GHz, this simulation ran in about 15 minutes. The most complex simulations (using 64 detector rows and 15 longitudinal samples for the FS and DCs) required calculation of 2×10¹⁶ line integrals, and processing time scaled accordingly. Multi-threaded CatSim is also available in the current implementation of XCIST, which may substantially reduce computation time depending on the hardware.

Analysis – projection-domain experiments

Each projection from a rotating scan contains a 2D image of the 0.7-mm wire. After taking the negative log of the projections, for each of the 984 projections, we calculated the full width at half the maximum (FWHM) of the wire PSF at the second of four simulated detector rows to determine PSF50_proj.

Note that, when scanned (Figure 4), the wire is sometimes close to the source (position S), close to the detector (position D), or tangential to the trajectory of the wire relative to the source and detector (position T). At each projection, we normalized the measured PSF50_proj by the magnification factor (SDD/source-to-object distance) of the wire at that position, and converted from detector columns to mm. We next plotted the PSF50_proj values for all projections and applied a third-order spline fit to the data (Figure 5). Finally, for each projection, we calculated the difference between the fitted curves for empirical and simulated data. We performed this analysis for both the wire parallel to the AOR and the tilted wire, and analyzed the results for view ranges corresponding to the S, D, and T positions, as well as for all views. Our box-and-whisker plots are produced using Microsoft Excel version 2302. Per Microsoft, the lines in the boxes of these plots represent the mean (average) rather than the more conventional median.

Figure 4. Critical wire positions for the projection-domain experiments.

The PSF of the wire was analyzed at 984 views angles during a complete 360° rotation, but the most interesting view angles are when the wire is closest to the source (S), the detector (D), and tangential to the wire trajectory (T).

Figure 5. PSF50_proj.

The projection-domain PSF is shown for 984 projections during a complete 360° rotation, for both empirical and simulated data. The approximate positions S, D, and T are indicated. The dotted vertical lines designate the boundaries of the view ranges that were used for analysis of the three wire positions.

Analysis – image-domain experiments

We used XCIST’s FDK reconstruction with the Bone kernel to reconstruct a 10-mm ROI around each wire, then we corrected for wires potentially non-parallel to the AOR by fitting the images’ maximum value pixels to a line and shifting the images accordingly, and we averaged the shifted images (Figure 6). We applied a 2D weighting function to mask out the foam. We determined the line from the AOR to each wire, and measured the PSF50 along that line for the radial response (PSF50_img_rad) and perpendicular to that line for the azimuthal response (PSF50_img_azim). We then applied a 2D FFT to the image and, for each of these orientations, we averaged pixel values over a ±15° range and measured the MTF at 50% and 10% modulation (MTF50 and MTF10). When analyzing the empirical scans, we found that five wires were misaligned in the holes and the data were unusable; these wires (indicated with yellow circles) were excluded for both empirical and simulated experiments.

Figure 6. Image-domain analysis.

One slice of a reconstructed image is shown (background) from the empirical scan of the 19-wire phantom. Five wires (yellow circles) were excluded due to wire alignment issues. Five examples of high-resolution region-of-interest reconstructions are shown (inset images); these are displayed at W/L = 900/−600 HU. The red line denotes the radius from the AOR to a selected wire; the red and orange regions in the corresponding inset image designate the angular ranges from which the radial and azimuthal PSF curves were determined.

In total, six image-domain parameters were reported: PSF50_img_rad, PSF50_img_azim, MTF50_rad, MTF50_azim, MTF10_rad, and MTF10_azim. To evaluate trends in each parameter’s agreement between empirical and simulated results, we analyzed the percent errors for each X-ray technique and orientation.

Perturbation experiments

Spatial resolution is determined by several primary system characteristics, including FS size, DC size, and view sampling. We have previously shown the image-domain effect of varying these parameters using an anthropomoric phantom. (Wu et al., 2022) In addition, there are a few nonideal effects that impact spatial resolution, including the temporal detector response mentioned above and the crosstalk between detector cells. To evaluate and demonstrate the impact of these, and to further explore the effect of FS size, we performed four experiments in the projection domain in which we varied these parameters, as shown in Table 4. Each parameter was varied independently while the others were held at their nominal values, except that the four crosstalk parameters were varied simultaneously. We processed the data as described above, under “Analysis - projection-domain experiments.”

Table 4.

Parameters and values used for the perturbation experiments.

Perturbed parameters		Decrease (a)	Nominal	Increase (b)
τ1 (ms)		0.500	1.000	1.500
X-ray crosstalk	Column	0	0.025	0.050
	Row	0	0.020	0.040
Optical crosstalk	Column	0	0.045	0.090
	Row	0	0.040	0.080
FS size (mm)	Width	0	0.897	1.794
	Length	0.283	0.565	0.848

Simulation details

All configuration files and code required to perform the simulations reported herein are available to XCIST/CatSim users. (Zhang, 2023)

Results

Projection domain

Figure 7 shows selected PD PSFs at 120 kV, 300 mA (small FS size) using the nominal simulation parameters with the wire at the critical positions S, D, and T. The general characteristics of the empirical PSFs were reproduced by the simulation. The PSFs at positions S and D showed asymmetry in opposite directions. The S position had a broad PSF and had the largest apparent area under the curve. The T position produced the narrowest PSF with the smallest apparent area. The simulation results consistently produce about 20% to 30% higher amplitude PSFs compared to the empirical results.

Figure 7. PD PSFs at critical wire positions.

PSFs are shown at (A) position S, (B) position D and (C) position T. Vertical axes are equal and horizontal axis widths are equal.

Figure 8 shows the corresponding results for PSF50_proj for all projections, using both the phantom with the wire parallel to and tilted from the AOR. In general, agreement is good, although there are few regions in the empirical data where noise causes the fitted curves to differ locally from the simulated data.

Figure 8. Projection-domain PSF50.

PSF50_proj is plotted versus projection number for (A) the phantom with the wire parallel to the AOR and (B) the phantom with the wire tilted from the AOR, for 120 kV, 300 mA.

Figure 9 shows the results of the pertubation experiments. Note that these are normalized by the magnification of the wire at each view angle, so the plotted PSF50 is the perceived size at the wire. For all experiments, the PSF was at local maxima when the wire was at positions S and D, and at local minima when the wire was at the T positions. When the primary time constant (τ₁) was varied by 50% from the nominal value, PSF50 was constant at the T positions, but varied by about 28% at positions S and D. When the crosstalk was increased from 0 to twice the nominal value, PSF50 was nearly constant at position S, but steadily increased as the wire moved to position D. When the FS length was varied by 50% from the nominal value, PSF50 was nearly constant at positions S and D, but varied by about −20% and +40% at the T positions, for the −50% and +50% change in FS length, respectively. When the FS width was varied from 0 to twice the nominal value, PSF50 was nearly constant at position D, but steadily increased as the wire moved to position S.

Figure 9. Perturbation experiment results.

(A) The primary detector response time constant is varied by +/−50%. Critical positions S, D, and T are shown (see Figure 4). (B) All four crosstalk parameters are simultaneously varied from 0 to twice their nominal values. (C) The FS length is varied by +/−50%. (D) The FS width is varied from 0 mm to twice it’s nominal width.

The remaining simulation results reported all used the nominal parameter values.

Figure 10 shows the results for both the wire parallel to and tilted from the AOR for view ranges corresponding to the S, D, and T positions, as well as for all views. For the S and D positions, we note a small negative trend with increasing kV and mA. Also for the S and D positions, the average errors and the central quartiles are all well within ±5%. For the T position, there are dependencies with kV and mA that are more complex. The 80-kV, 100-mA average errors are consistently within 0 to about +5%, i.e., the simulations are predicting a larger PSF than the empirical data, and the central quartiles are relatively small, within a few percent. However, for all other techniques, the bias is negative, and there are larger average errors, approaching 10% for a few low-kV techniques, but there is a trend toward reduced errors with increasing kV. When all views are included, the T-position data dominates the distributions, but the average errors are all within 5%.

Figure 10. Projection-domain PSF50.

All vertical axes are error, simulation versus empirical results.

Image domain

Figure 11 shows exemplary ID PSF images for a wire near image center and for a wire near the edge of the scan field of view (SFOV), both using 120 kV and 300 mA (small FS). The PSF images show ringing artifacts for the center position, and azimuthal blur for the edge position. Figure 11 shows profiles through these same PSFs. For the center wire, the radial and azimuthal PSFs were nearly identical, so only the radial result is shown. The PSFs in both directions are shown for the edge wire. Note that the image of the wire in the center is very bright (about 6000 HU) but the image of the wire near the edge is blurry and therefore not as bright (about −200 HU) (also see Figure 6). Agreement for PSF50_img between empirical and simulation is good (within 5%). The pulse amplitude is consistently about 10% higher for the simulated versus empirical results. The shapes of the PSFs qualitively agree between the empirical and simulated results; note the ringing near −1000 HU in Figure 11(A) and the asymetry in Figure 11(C).

Figure 11. Exemplary image-domain PSFs.

A) Near image center, the radial PSF (shown) is nearly identical to the azimuthal PSF (not shown). B) Radial PSF and C) azimuthal PSF near the edge of the image FOV. The horizontal axes are equal, but the vertical axes differ. Data were acquired at 120 kV, 300 mA (small FS).

Figure 12 shows the three image-domain metrics that we chose (PSF50, MTF50, and MTF10), each for the radial and azimuthal orientations, plotted versus distance from the AOR. Agreement between empirical and simulation results is very good, including aspects in which the results are surprising. For example, PSF50 is larger for the centermost wire than one might expect considering the trend of the wires at about 25 mm from the AOR to about 80 mm from the AOR. Also, when the radial and azimuthal PSF50 curves begin to diverge at about 100 mm from the AOR, there is a distinct drop in PSF50 that is accurately predicted by the simulation. Surprisingly, the radial and azimuthal MTF50 curves cross, but this is also predicted by the simulation. Finally, the curve for MTF10 in the radial orientation shows a clear discontinuity at about 80 mm from the AOR, which is predicted by the simulation.

Figure 12. Exemplary image-domain spatial resolution versus distance from the AOR.

(A) PSF50, (B) MTF50, and (C) MTF10 are shown for the radial and azimuthal directions. Data were acquired at 120 kV, 300 mA (small FS).

Figure 13 shows the MTF curves for the centermost wire and for the outermost wire, as scanned at 120 kV and 300 mA (small FS). Generally, the empirical and simulation results agree well, but there are a few areas of notable disagreement. For the radial orientation (Figure 13A), there is disagreement of up to about 12% for the edge wire at low frequencies (< 5 lp/cm), and disagreement of up to about 2% for the center wire at intermediate frequencies (about 5 lp/cm to 10 lp/cm). For the azimuthal orientation (Figure 13B), there is a difference of about 10% in the “ringing” in the curve for the edge wire between about 5 lp/cm and 10 lp/cm.

Figure 13. Exemplary MTF curves.

MTF in the (A) radial and (B) azimuthal directions are shown for a wire near the center and a wire near the edge of the image FOV. Data were acquired at 120 kV, 300 mA (small FS).

Figure 14 shows the results for PSF50, MTF50, and MTF10 for both the radial and azimuthal orientations. In general, errors are within ±5% with no substantial trend or bias, with some exceptions. For the radial PSF50, the average error in the 80-kV, 600-mA data is about −6%, with about 5% quartile size; however, with increasing kV, the average errors trend toward zero. For the radial MTF50, the average errors have a negative bias of about −5%, and for a few techniques the average errors exceed −5%. For the azimuthal MTF50, the average error in the 80-kV, 600-mA data is about +7%, with about 3% quartile size; however, with increasing kV, the average errors trend toward zero. To better understand the nature of the error distribution, we plotted the errors versus the distance of the wires from the AOR (Figure S2 in the Supplemental Materials). We found that for the three conditions that include average errors exceeding 5%, errors are within ±5% in the center 25-cm diameter of the scan field-of-view (SFOV), but there are trends toward the perimeter of the SFOV that lead to larger errors.

Figure 14. Image-domain PSF50, MTF50, and MTF10.

All vertical axes are error, simulation versus empirical results.

Discussion

Simulation parameter choices.

We chose the FS and detector sampling that we used because we are confident that these choices will provide sufficient sampling for our purposes, which was to compare the simulated results with the empirical ground truth. We did not explore sampling strategies to find the optimal strategy for the minimum computation time required to achieve acceptable agreement. We chose the key detector temporal response parameter τ1 to achieve good qualitative agreement in the PD PSF shapes and good average quantitative agreement in the four quantitative parameters that we used (PSF50_proj, PSF50_imag, MTF50_imag, and MTF10_imag). We used detector crosstalk parameters that we have used in the past and seem to produce reasonable agreement, although spatial resolution is not strongly dependent on these parameters, as discussed in the next paragraph. After obtaining and evaluating the reported results, we did not adjust any parameters in an attempt to improve agreement.

Projection-domain PSFs.

The shape of the PD PSFs agreed well, especially after we adjusted the value of τ1 to 90% of the published value. The asymmetry of the S-position and D-position PSFs agreed, thus validating several CatSim models. The breadth of the PSF at the S and D positions is largest due to a combination of the gantry rotational blur and the detector temporal response. This effect is magnified for the S position because the wire position as projected on the detector is highly magnified due to the wire’s close proximity to the source. When the wire is at the T position, the absence of rotational blur (and absence of the asymmetry that results from the blurring) is because there is only a small lateral displacement in the projection of the wire over several views; the resulting PSF with the wire in this position most closely represents the optics related to the focal spot and detector feature sizes and is minimally affected by rotational or temporal effects.

We noted that the simulations consistently produce about 20% to 30% higher amplitude PSFs compared to the empirical results. We speculate that this is due to imperfect models of the wire material and of the X-ray tube spectrum, which are outside the scope of this paper and do not impact the resolution modeling accuracy.

Perturbation experiments.

These experiments are intended to illustrate the effects of varying different design parameters. First, changing the detector temporal response has a strong impact on PSF50_proj at the S and D wire positions and minimal effect at the T positions for the reasons discussed in the previous paragraph. Second, because crosstalk is a detector effect, it has the largest impact on PSF50_proj when the wire is at the D position, a smaller effect at the T positions, and essentially no effect when the wire is at the S position. The effect of the FS length is, as one might expect, negligible at the S and D positions, because at those wire positions, the detector “sees” only a straight-on view of the FS; therefore, only the FS width is relevant. However, at the T positions, the detector “sees” the FS from an oblique perspective, which means that a diagonal view of the FS, defined by both its width and length, forms the projection of the wire. Therefore, as the FS length is varied, PSF50_proj varies proportionately. Finally, the FS width most strongly affects PSF50_proj when the wire is closest to the FS (at the S position) and the effect is smaller at the T position and nearly negligible at the D position.

Projection-domain PSF50 with different wire angulation.

The disagreements between empirical and simulated results are generally small (average errors when all views are included are less than 5%). The S-position results suggest that our FS model is accurate, with the disagreement generally increasing with decreasing kV. S-position results are dominated by the FS model. Our new model was made from an average of FS photo made at 80 kV, 100 kV, 120 kV, and 140 kV. Although FS size does depend on kV and mA to a small extent, we chose not to include those dependencies in our FS model in the interest of simplicity. The D-position results show that our detector and detection models are quite accurate, although there is a negative trend with increasing kV that suggests a small detection-physics effect that is not accounted for in our detection model. Clearly, the results are strongly dominated by errors near the T position. As discussed above, the detector response model contributes strongly to simulated PSFs at the T position. On average, our model produces good agreement at the T position, but some techniques produce positive errors while other techniques produce negative errors. This suggests that some benefit might be gained from a more complex detector temporal response model, i.e., a model that includes kV and mA dependencies. However, because the errors when all views are included are acceptable for most anticipated applications of CatSim, we accept the results as a compromise between ideal accuracy and unwieldy complexity.

We hypothesize that a secondary source of errors is noise in the empirical data, causing the fitted curve to become less smooth than expected – in principle, these curves should be distorted sinusoidal functions, without local variations as seen in the empirical results. We believe that more averaging and perhaps more complex fitting would produce a smoother curve and even better agreement. However, we feel that these results show sufficient agreement to demonstrate the validity of CatSim’s models. Quantitatively, the disagreement between empirical and simulated tilted-wire results was acceptable for most anticipated applications of CatSim. We note that the maximum disagreements generally occurred where our spline fitting permitted local distortions in the sinusoidal PSF-versus-view-angle functions; we hypothesize that this is non-physical and closer agreement would be achieved by using a more sophisticated fitting function. We also note that the disagreement in results generally increased with decreasing kV. Our FS model was made from a FS photo made at 120 kV. Although FS size does depend on kV to some extent, we chose not to include this dependence in our FS model in the interest of simplicity and because the effect is small. Note that the PSF50 of the tilted wire is always larger than PSF50 of the straight wire since it includes the additional blur due to the finite slice thickness. Since the simulated and empirical results are in good agreement for both straight and tilted wire, we conclude that the z-resolution is also modeled well.

Image-domain results at various wire positions and in radial and azimuthal directions.

As mentioned in the Results section, there were several notable and sometimes unanticipated features in the empirical ID PSF shapes and in the curves showing each metric’s values versus position; these were very well predicted by the simulations. These qualitative observations provide good confidence in the validity of CatSim’s models, especially when the characteristics were unanticipated. Specifically, we are uncertain about the root cause of the discontinuity in the PSF50 and MTF50 curves between 80mm and 100mm, but we hypothesize that they are induced by some subtle aliasing artifacts that impact the measurements. The observations regarding disagreements in the PSF and MTF curves can be explained from the projection-domain results, but, from our experience, the errors in image-domain results would not have a perceivable effect on the spatial resolution in clinical images. Quantitatively, agreement in results for all ID metrics was again within reasonable limits, and disagreement again increased with decreasing kV, for the reason given above.

Limitations.

As discussed above, our results show that CatSim could benefit from more complex, kV- and mA-dependent models for the focal spot, detection physics, and detector temporal response. However, in the interest of simplicity, we believe that the current models are sufficient for most anticipated applications of CatSim. The curve-fitting method that we used for the PSF50_proj experiment permitted the results to deviate from the expected distorted-sinusoidal shape; however, the results still showed sufficiently close agreement to support confident conclusions. The construction of the ID phantom was imperfect, leading to the exclusion of five of the nineteen wires from the results. However, the curves still contained sufficient information to have confidence in the results.

Conclusion

This study cumulatively evaluated all the models used by CatSim that have an impact on spatial resolution in the context of a typical 64-detector-row clinical CT scanner. These models include two new models required to accurately simulate spatial resolution: a realistic model for the X-ray source’s focal spot and an empirically tuned model for the X-ray detector’s temporal response. The results qualitatively and quantitatively validate CatSim’s ability to simulate a real scanner’s spatial resolution within reasonable limits.