Quantitative accuracy of CT numbers: Theoretical analyses and experimental studies

Abstract

Purpose

The CT number accuracy, that is, CT number bias, plays an important role in clinical diagnosis. When strategies to reduce radiation dose are discussed, it is important to make sure that the CT number bias is controlled within an acceptable range. The purpose of this paper was to investigate the dependence of CT number bias on radiation dose level and on image contrast (i.e., the difference in CT number between the ROI and the background) in Computed Tomography (CT).

Methods

A lesion‐background model was introduced to theoretically study how the CT number bias changes with radiation exposure level and with CT number contrast when a simple linear reconstruction algorithm such as filtered backprojection (FBP) is used. The theoretical results were validated with experimental studies using a benchtop CT system equipped with a photon‐counting detector (XC‐HYDRA FX50, XCounter AB, Sweden) and a clinical diagnostic MDCT scanner (Discovery CT750 HD, GE Healthcare, Waukesha, WI, USA) equipped with an energy‐integrating detector. The Catphan phantom (Catphan 600, the Phantom Laboratory, Salem, NY, USA) was scanned at different mAs levels and 50 scans were performed for each mAs. The bias of CT number was evaluated for each combination of mAs and ROIs with different contrast levels. An anthropomorphic phantom (ATOM 10‐year‐old phantom, Model 706, CIRS Inc. Norfolk, VA, USA) with much more heterogeneous object content was used to test the applicability of the theory to the more general image object cases.

Results

Both theoretical and experimental studies showed that the CT number bias is inversely proportional to the radiation exposure level yet linearly dependent on the CT number contrast between the lesion and the background, that is, $\text{Bias}({\widehat{\mu }}_1^{\mathrm{FBP}})=\frac{\alpha }{\mathit{mAs}}(1+\beta \mathrm{\Delta }HU)$.

Conclusions

The quantitative accuracy of CT numbers can be problematic and thus needs some extra attention when radiation dose is reduced. In this work, we showed that the bias of the FBP reconstruction increases as mAs is reduced; both positive and negative bias can be observed depending on the contrast difference between a targeted ROI and its surrounding background tissues.

1. Introduction

Given the widespread use of computed tomography (CT) in diagnostic imaging, image‐guided interventions and image‐guided radiation therapy, the potential biological risks associated with the use of ionizing x‐ray radiation in CT has become a public concern. To address this concern, efforts from clinical societies, medical physics communities, industries, and federal funding agencies have been taken¹ to develop hardware, software, and optimized scanning protocols to reduce radiation dose while maintaining the needed image quality for clinical tasks. Hardware technologies involve modifications to the current systems for dose‐efficient data acquisition, including new detector design to improve quantum efficiency and reduce electronic noise,² tube current/tube potential modulation,^{3, 4, 5, 6} and dynamic bowtie filter design.^{7, 8, 9} In conjunction with these hardware developments, software methods have also been developed to reconstruct low noise images from noisy data acquired at reduced radiation dose levels. These methods include a wide range of iterative image reconstruction algorithms (see review articles Refs. [10] and [11] and references therein) and a variety of denoizing methods in either the raw detector counts domain before log‐transform^{12, 13, 14} or in sinogram space after the log‐transform.^{15, 16, 17, 18} Although the technical details of these software methods vary from one to another, the basic objectives are essentially the same: eliminate structured noise streaks and reduce the overall noise level while maximally preserving the edges of the image object. In addition to these technological developments, it is imperative that medical physicists work with clinicians to optimize CT scanning protocols to have the lowest radiation dose level achievable while maintaining the necessary image quality required for a specific clinical imaging task.

To assess the performance of these low‐dose strategies, quantitative image quality metrics such as noise variance, noise power spectrum (NPS), modulation transfer function (MTF), and some task‐based approaches are usually employed.^{19, 20, 21, 22, 23, 24, 25} These quantitative image quality assessment methods primarily focus on the performance of noise reduction, the capability of edge preservation, and the joint impact on a given imaging task such as low‐contrast lesion detection.

However, less attention has been paid to the accuracy of the reconstructed CT numbers. In layman's terms, the accuracy of CT number concerns the answer to the following question: An image object is repeatedly scanned at two radiation dose levels, one low exposure level and one high exposure level, but at the same tube potential, after image reconstruction, if one measures the ensemble average of the CT numbers at a given region of interest (ROI), would the mean CT number be the same at these two radiation dose levels? If the mean CT numbers remain the same, that is, difference of the mean CT numbers is zero, then the CT number is considered as unbiased. Otherwise, the CT number is said to be biased. In clinical applications, the accuracy of CT number plays an important role. For instance, the relative change of CT numbers is often used as a quantitative metric by radiologists in their decision‐making process. The accuracy of CT number is also critically important for attenuation corrections in PET/CT²⁶ and in radiation therapy treatment planning.²⁷

When radiation dose is reduced, it has been shown^{28, 29} that potential bias can be introduced in the reconstructed image and the CT number accuracy could be degraded, particularly for reconstruction methods where the postlog line integrals are used as input projection data. When a log‐transform is used to generate line integral projection data, due to the convexity of the negative logarithm function, − ln (X), one can immediately conclude that E[− ln (X)]≥− ln (E[X]) from the well‐known Jensen's inequality.^{29, 30} This results in a nonzero bias in log‐transformed projection data, that is, the mean value of the log‐transformed projection data is different from the projection data acquired at a high dose level (when electronic noise from the readout electronics are neglected, the ensemble mean of the repeatedly measured counts at low exposure level is equivalent to the counts at a corresponding high exposure level). The bias in log‐transformed projection data approximately follows an inverse relationship with the counts, namely, the bias of log‐transformed data increases as the radiation dose level is reduced. As a direct consequence of this bias in projection data, a positive bias is associated with the CT number in the FBP reconstructed image as shown by other investigators,²⁹ using a single‐component scalar CT model. However, there are many scientific questions that remain to be addressed. These questions include: (a) When the background material around the lesion is considered, how is the CT number bias quantitatively related to the contrast, that is, the CT number difference between the lesion and the background? (b) Is the measured CT number always positively biased? or can the CT number bias be negatively biased? (c) In clinical practice, when a detection task is given with certain acceptable bias threshold value, what would be the lowest dose limit to keep the bias below the threshold? The purpose of the paper was to address the first two technical questions and to provide insights into the technical aspects of the third question. Specifically, to answer these questions, a lesion‐background model similar to that in Ref. [31] was introduced to develop a theoretical model to derive the quantitative relationship between CT number bias, exposure level, and the CT number contrast. After that, experimental studies were performed to validate the theoretical results.

2. Theoretical model

In this section, a theoretical model was developed to study the CT number bias and how it is related to the external parameter of exposure level and the internal parameter of CT number contrast.

2.A. Lesion‐background model

To simplify the theoretical analysis, let us assume that there is a square‐shaped lesion with linear attenuation coefficient μ ₁ and linear dimension l ₁ embedded into a background with linear attenuation coefficient μ ₂ and a linear dimension l ₂ (Fig. 1).

Figure 1.

Lesion‐background model used for theoretical analysis. The lesion (inner square) has a linear attenuation coefficient of μ ₁ with length l ₁; background (outer square) has a linear attenuation coefficient of μ ₂ with length l ₂. b _1i and b _2i are the number of incident photons, ${\overline{y}}_{1i}$ and ${\overline{y}}_{1i}$ are the expected detected photons. [Color figure can be viewed at wileyonlinelibrary.com]

Assuming that an ideal photon‐counting detector is used in measurement and each measurement is independent and follows Poisson statistics, that is,

$$P(y_{1i})=e^{-{\overline{y}}_{1i}}\frac{{\overline{y}}_{1i}^{y_{1i}}}{y_{1i}!},$$ (1)

$$P(y_{2i})=e^{-{\overline{y}}_{2i}}\frac{{\overline{y}}_{2i}^{y_{2i}}}{y_{2i}!}.$$ (2)

Here, the expected photon counts ${\overline{y}}_{1i}$ and ${\overline{y}}_{2i}$ can be written as

$${\overline{y}}_{1i}=b_{1i}{e}^{-[{\mu }_1{l}_1+{\mu }_2\left(l_2-l_1\right)]},$$ (3)

$${\overline{y}}_{2i}=b_{2i}{e}^{-{\mu }_2{l}_2},$$ (4)

where b _1i and b _2i are numbers of incident x‐ray photons and are assumed to be known constants. For the measured dataset $\{y_{1i},y_{2i}\}$ (i = 1,2,···,N), the joint probability of the measurements is then given by

$$P(\{y_{1i},y_{2i}\})=\prod _{i=1}^{N}P(y_{1i})P(y_{2i}).$$ (5)

If the lengths of both the lesion and background are known, then the image reconstruction task is to obtain the estimates ${\widehat{\mu }}_1$ and ${\widehat{\mu }}_2$ from a series of independent measurements $y_{1i},y_{2i},(i=1,2,\dots ,N)$ at different exposure levels (i.e., b _1i and b _2i may be different from one measurement to another). This can be accomplished via the so‐called maximum likelihood estimation (MLE) method. In this method, the following negative log‐likelihood function, $L({\mu }_1,{\mu }_2)=-lnP(\{y_{1i},y_{2i}\})$, is minimized to estimate ${\widehat{\mu }}_1$ and ${\widehat{\mu }}_2$ from the measured dataset (see Appendix A for details):

$$\begin{array}{cc}\hfill L({\mu }_1,{\mu }_2)=& \sum _{i=1}^N{b}_{1i}{e}^{-[{\mu }_1{l}_1+{\mu }_2\left(l_2-l_1\right)]}+b_{2i}{e}^{-{\mu }_2{l}_2}\hfill \\ & +y_{1i}[{\mu }_1{l}_1+{\mu }_2(l_2-l_1)]+y_{2i}{\mu }_2{l}_2.\hfill \end{array}$$ (6)

To simplify the mathematical calculations, the quadratic approximation to the negative log‐likelihood function is often used to approximate the above negative log‐likelihood function (see Appendix A for details):

$$L^Q\left({\mu }_1,{\mu }_2\right)=\frac{1}{2}\sum _{i=1}^N{y}_{1i}{\left[{\mu }_1{l}_1+{\mu }_2\left(l_2-l_1\right)-ln\frac{b_{1i}}{y_{1i}}\right]}^2+\frac{1}{2}\sum _{i=1}^N{y}_{2i}{\left({\mu }_2{l}_2-ln\frac{b_{2i}}{y_{2i}}\right)}^2.$$ (7)

By setting the partial derivatives of the function L ^Q with respect to μ ₁ and μ ₂ to zero, one obtains the following weighted least squares estimator as follows:

$${\widehat{\mu }}_1^{\mathrm{WLS}}=\frac{1}{l_1}\frac{\sum _{i=1}^N{y}_{1i}ln\frac{b_{1i}}{y_{1i}}}{\sum _{i=1}^N{y}_{1i}}-\frac{l_2-l_1}{l_1{l}_2}\frac{\sum _{i=1}^N{y}_{2i}ln\frac{b_{2i}}{y_{2i}}}{\sum _{i=1}^N{y}_{2i}},$$ (8)

$${\widehat{\mu }}_2^{\mathrm{WLS}}=\frac{1}{l_2}\frac{\sum _{i=1}^N{y}_{2i}ln\frac{b_{2i}}{y_{2i}}}{\sum _{i=1}^N{y}_{2i}}.$$ (9)

When the weighting of each line integral measurement is set to one (i.e., $y_{1i}/\sum _{i=1}^N{y}_{1i}\to \frac{1}{N}$ and $y_{2i}/\sum _{i=1}^N{y}_{2i}\to \frac{1}{N}$), one obtains the following FBP equivalent estimator:

$${\widehat{\mu }}_1^{\mathrm{FBP}}=\frac{1}{N{l}_1}\sum _{i=1}^{N}ln\frac{b_{1i}}{y_{1i}}-\frac{l_2-l_1}{N{l}_1{l}_2}\sum _{i=1}^{N}ln\frac{b_{2i}}{y_{2i}},$$ (10)

$${\widehat{\mu }}_2^{\mathrm{FBP}}=\frac{1}{N{l}_2}\sum _{i=1}^{N}ln\frac{b_{2i}}{y_{2i}}.$$ (11)

Using the above derived estimators ${\widehat{\mu }}_1^{\mathrm{FBP}}$ and ${\widehat{\mu }}_2^{\mathrm{FBP}}$, we can discuss the CT number bias in the next subsection.

2.B. Estimation of the CT number bias in reconstruction

Bias of the estimator is defined as

$$\text{Bias}[\widehat{\mu }]=E[\widehat{\mu }]-\mu .$$ (12)

As discussed in Refs. [28] and [29], direct calculation of $E[\widehat{\mu }]$ may be difficult because $\widehat{\mu }$ is a nonlinear function. To approximately calculate the mean, one can perform the Taylor expansion of a general function $\widehat{\mu }=g(Y)$ around its mean value $\overline{Y}$. For ${\widehat{\mu }}_1$, Y = {y ₁₁,…,y _1N ,y ₂₁,…,y _2N }. For ${\widehat{\mu }}_2$, Y = {y ₂₁,y ₂₂,…,y _2N }. Corresponding mean values are ${\overline{y}}_{1i}=b_{1i}{e}^{-[{\mu }_1{l}_1+{\mu }_2(l_2-l_1)]}$ and ${\overline{y}}_{2i}=b_{2i}{e}^{-{\mu }_2{l}_2}$. As a result, under the independent measurement assumption, one obtains the following approximation of $E[\widehat{\mu }]$:

$$E\left[{\widehat{\mu }}_1\right]-{\mu }_1\approx \frac{1}{2}\sum _{i=1}^N\frac{{\partial }^2}{\partial y_{1i}^2}g\left(\overline{Y}\right)\text{Var}\left(y_{1i}\right)+\frac{1}{2}\sum _{i=1}^N\frac{{\partial }^2}{\partial y_{2i}^2}g\left(\overline{Y}\right)\text{Var}\left(y_{2i}\right),$$ (13)

$$E\left[{\widehat{\mu }}_2\right]-{\mu }_2\approx \frac{1}{2}\sum _{i=1}^N\frac{{\partial }^2}{\partial y_{2i}^2}g\left(\overline{Y}\right)\text{Var}\left(y_{2i}\right).$$ (14)

Using Eqs. (13) and (14), biases of the FBP reconstruction can be calculated as follows:

$$\text{Bias}\left({\widehat{\mu }}_1^{\mathrm{FBP}}\right)=\frac{e^{{\mu }_2{l}_2}}{2l_2}\frac{1}{N}\sum _{i=1}^N\frac{1}{b_{2i}}\left[1+\frac{l_2}{l_1}\left(e^{\mathrm{\Delta }\mu l_1}\frac{\sum _{i=1}^N\frac{1}{b_{1i}}}{\sum _{i=1}^N\frac{1}{b_{2i}}}-1\right)\right],$$ (15)

$$\text{Bias}\left({\widehat{\mu }}_2^{\mathrm{FBP}}\right)=\frac{e^{{\mu }_2{l}_2}}{2l_2}\frac{1}{N}\sum _{i=1}^N\frac{1}{b_{2i}},$$ (16)

where Δμ = μ ₁−μ ₂. When $\mathrm{\Delta }\mu l_1$ is a small number, $e^{\mathrm{\Delta }\mu l_1}\approx 1+\mathrm{\Delta }\mu l_1$, the bias of the lesion can be rewritten as

$$\text{Bias}\left({\widehat{\mu }}_1^{\mathrm{FBP}}\right)=\frac{e^{{\mu }_2{l}_2}}{2l_2}\frac{1}{N}\sum _{i=1}^N\frac{1}{b_{2i}}\left[\mathrm{\Delta }\mu l_2\frac{\sum _{i=1}^N\frac{1}{b_{1i}}}{\sum _{i=1}^N\frac{1}{b_{2i}}}+\frac{l_2}{l_1}\frac{\sum _{i=1}^N\frac{1}{b_{1i}}}{\sum _{i=1}^N\frac{1}{b_{2i}}}-\frac{l_2}{l_1}+1\right].$$ (17)

If the number of incident photons for each measurement remains constant, that is, b _1i = b _2i = b,(i = 1,2,…,N), then the results can be simplified as:

$$\text{Bias}\left({\widehat{\mu }}_1^{\mathrm{FBP}}\right)=\frac{e^{{\mu }_2{l}_2}}{2l_{2}b}\left(1+\mathrm{\Delta }\mu l_2\right),$$ (18)

$$\text{Bias}\left({\widehat{\mu }}_2^{\mathrm{FBP}}\right)=\frac{e^{{\mu }_2{l}_2}}{2l_{2}b}.$$ (19)

As can be seen from Eq. (18), the bias of the lesion depends on both the background and the contrast. For a given system and tube potential, the entrance photon number b is proportional to the mAs, that is, b = k × mAs, where k is a system‐dependent scaling factor. Based on the relationship between μ and HU, we have $\mathrm{\Delta }\mu =\mathrm{\Delta }HU\frac{{\mu }_w}{1000}$, where μ _w is the liner attenuation coefficient of water, and the bias of the lesion can be rewritten as

$$\text{Bias}\left({{\widehat{\mu }}_1}^{\mathrm{FBP}}\right)=\frac{\alpha }{\mathit{mAs}}\left(1+\beta \mathrm{\Delta }HU\right),$$ (20)

where $\alpha =\frac{e^{{\mu }_2{l}_2}}{2l_{2}k}$, $\beta =\frac{{\mu }_w{l}_2}{1000}$ are two constants determined by both the system and the background of the phantom. In most clinical cases, the background material in an object is approximately water equivalent. In this case ΔHU is simply the CT number of the lesion. Eq. (20) is the main result of the theoretical analysis, which states that for a given system and a fixed tube potential, the bias of the reconstruction follows an inverse relationship with mAs and a linear relationship with the CT number contrast ΔHU. Note that the result shown in Eq. (20) was derived under the uniform x‐ray fluence condition as shown in the argument before Eq. (18). However, as shown in Eq. (17), the inverse relationship with mAs and linear dependence on image contrast is valid for much more general cases. In the following section, experimental studies were performed to validate these theoretical results.

3. Experimental validation

In this section, we present validation methods using two different imaging systems. The first is a benchtop CT imaging system equipped with a photon‐counting detector which is consistent with the photon‐counting assumption used in the theoretical analysis. Second, a diagnostic CT scanner was used to check the applicability of the derived theoretical result to the clinical CT systems that are equipped with energy‐integrating detectors.

3.A. Phantoms used in validation studies

A Catphan phantom (Catphan 600, the Phantom Laboratory, Salem, NY, USA) was scanned on both systems validate the theoretical results. The central axial slice was aligned to the CTP 404 section of the phantom, and images of this section were used for the bias analysis.

To further test the applicability of the derived theoretical model for the nonuniform fluence case, an anthropomorphic pediatric phantom (ATOM 10‐year‐old phantom, Model 706, CIRS Inc. Norfolk, VA, USA) with heterogeneous object content was scanned on the photon‐counting CT system. Similar data analyses were performed to study the CT number accuracy.

3.B. Photon‐counting CT system

The benchtop CT system in this work is equipped with a photon‐counting detector (XC‐HYDRA FX50, XCounter AB, Sweden) as shown in Fig. 2.

Figure 2.

The benchtop photon‐counting CT system. [Color figure can be viewed at wileyonlinelibrary.com]

The detector has a native pixel size of 100 μm and covers an area of 510 mm × 6 mm. The system has a medical grade rotating‐anode x‐ray tube with a 1.0‐mm nominal focal spot (G1582, Varian Medical Systems, Salt Lake City, UT, USA). Axial beam collimation was applied to reduce scatter. For the Catphan phantom scan, the x‐ray beam was filtered by a 0.5 mm copper sheet and a custom‐made bowtie filter. For the CT acquisition, the object was rotated at 15^∘/s, and 720 views were acquired over 360^∘ for each scan. A 2 × 2 binning of the detector pixels was used. Image reconstruction was performed using the Feldkamp–Davis–Kress (FDK) algorithm³² with a ramp kernel. Reconstructed voxel size was 0.2 × 0.2 × 0.2 mm³. Tube potential was fixed at 80 kVp, and independent scans were acquired at eight different mAs levels ranging from 69.6 to 367.2 mAs. Corresponding CTDI_vol values were measured, ranging from 1.4 to 7.2 mGy. For the pediatric phantom scan, the x‐ray beam was filtered by a 0.2‐mm copper sheet and the bowtie filter was not used. The object was rotated at 7.5^∘/s, and 1440 views were acquired. Tube potential was 80 kVp, four different mAs levels were used, ranging from 24 to 576 mAs, with corresponding CTDI_vol values ranging from 0.7 to 17.8 mGy.

3.C. Diagnostic MDCT scanner

The second system used in validation studies was a 64‐slice diagnostic MDCT scanner (Discovery CT750 HD, GE Healthcare, Waukesha, WI, USA). This system was operated in axial mode with a detector coverage of 20 × 0.625 mm at isocenter and a Head bowtie filter. The tube potential was set at 80 kVp and independent scans of the Catphan phantom were acquired at eleven different mAs levels ranging from 4 to 48 mAs, with reported CTDI_vol values ranging from 0.3 to 3.5 mGy. Images with pixel size of 0.43 × 0.43 mm² and slice thickness of 0.625 mm were reconstructed with the conventional FBP method using the Standard kernel.

3.D. Workflow of the data analysis

As discussed in the previous section, in order to experimentally measure the reconstruction bias, the surrogate of the ground truth μ, hereafter referred to as the reference, can be determined by the following steps: perform the average of the raw counts of the 50 scans from the highest mAs level; perform the logarithm transform to generate the line integral data; and perform FDK reconstruction. For each mAs level, each of the 50 repeated scans was reconstructed first and then averaged to obtain $E[\widehat{\mu }]$; bias is then calculated by a further subtraction of the reference. Figure 3 shows the workflow of the experimental data analysis.

Figure 3.

Workflow of the experimental analysis.

4. Experimental results

4.A. Results of the Catphan phantom from the photon‐counting CT system

Figure 4 (a) shows an example of the reconstructed image from the benchtop photon‐counting CT system with the lowest exposure level and (b) shows the corresponding bias map. The reconstructed reference of the benchtop CT system and ROIs used for the analysis are shown in (c). The mean value within each ROI was used for the determination of bias.

Figure 4.

(a). A reconstructed image from the benchtop CT system with 69.6 mAs exposure, L/W: 100/2000. (b). The bias image for the 69.6 mAs exposure, L/W: 20/100. (c). The reconstructed reference and ROIs used for the analysis, L/W: 100/2000. [Color figure can be viewed at wileyonlinelibrary.com]

4.A.1. Dependence of CT number bias on exposure level

To validate the mAs dependence of the reconstruction bias, the biases of four representative ROIs were plotted, as shown in Fig. 5. Each data point represents the measured CT number for a given ROI and mAs level. Error bars in the plot represent the standard error of the mean (SEM). Based on the linear fit of the data, the experimental results have a good agreement with the theoretical prediction, that is, bias is inversely proportional to mAs.

Figure 5.

Bias as a function of mAs ⁻¹.

4.A.2. Dependence of CT number bias on contrast

To validate the contrast dependence of the reconstruction bias, the measured bias was plotted as a function of contrast (ΔHU) for four representative mAs levels. Experimental results shown in Fig. 6 have a good agreement with the theoretical result which predicts a linear dependence of the bias on the CT number contrast.

Figure 6.

Bias as a function of contrast ΔHU.

4.A.3. Dependence of CT number bias on exposure level and contrast

To jointly validate the mAs and contrast dependence of the reconstruction bias, experimental data were fitted according to the theoretical expression in Eq. (20). Figure 7 shows the fitting result, where fitting parameters α and β are determined. Results show good agreement with the theoretical model, as indicated by a small root mean square error (2 HU) of the fitting results.

Figure 7.

Bias as a function of contrast and mAs. Fitting parameters: α = 1559, β = 0.0031. Root mean square error of the fit is 2.0 HU; R ² = 0.990. [Color figure can be viewed at wileyonlinelibrary.com]

4.B. Results of the Catphan phantom from the diagnostic CT scanner

Experimental results from the diagnostic CT scanner are shown in Figs. 8, 9, 10. Similar to the results from the benchtop photon‐counting CT, they are in good agreement with the theoretical predictions. Larger deviations can be attributed to the use of the energy‐integrating detector. The theoretical derivations assumed a photon‐counting detector was used and the noise is dominated by Poisson noise. In energy‐integrating detectors, when the dose is reduced, the contributions of the electronic noise become more significant. As a result, the experimental results slightly deviated from the theoretical predictions.

Figure 8.

Bias as a function of mAs ⁻¹ for the diagnostic scanner.

Figure 9.

Bias as a function of contrast ΔHU for the diagnostic scanner.

Figure 10.

Bias as a function of contrast and mAs. Fitting parameters: α = 64.82, β = 0.0036. Root mean square error of the fit is 6.7 HU; R ² = 0.861. [Color figure can be viewed at wileyonlinelibrary.com]

Once the parameters α and β were determined, the iso‐bias contour map could be generated, as shown in Fig. 11. Using the obtained iso‐bias map, one can determine the lowest possible mAs level such that an acceptable bias for a given contrast level is not exceeded.

Figure 11.

Iso‐bias map. [Color figure can be viewed at wileyonlinelibrary.com]

4.C. Results of the pediatric phantom from the photon‐counting CT system

Figure 12 (a) shows an example of the reconstructed image from the photon‐counting CT system with 24 mAs exposure and (b) shows the corresponding bias map. The reconstructed reference and ROIs used for the analysis are shown in (c). Three types of ROIs with different tissue equivalent plugs were used for analysis: bone (ROI 1), soft tissue (ROI 2), and lung tissue (ROI 3). The soft tissue region in the center was considered as the background.

Figure 12.

(a). A reconstructed image from the benchtop CT system with 24 mAs exposure, L/W: ‐100/4000. (b). The bias image for the 24 mAs exposure, L/W: 80/150. (c). The reconstructed reference and ROIs used for the analysis, L/W: ‐100/2000. [Color figure can be viewed at wileyonlinelibrary.com]

As can be seen from the results shown in Figs. 13 and 14, the mAs and contrast dependence of the reconstruction bias derived from the simple model still apply.

Figure 13.

Bias as a function of mAs ⁻¹ for the pediatric phantom.

Figure 14.

Bias as a function of contrast ΔHU for the pediatric phantom.

5. Discussion

The lesion‐background model and theoretical results of the bias were validated experimentally. Based on the theoretical results, several implications can be given as follows. First, the magnitude of the bias increases as dose is reduced. While most CT reconstruction and postprocessing algorithms focus on the noise reduction performance, additional attention should be paid to the issue of bias since we pursue low‐dose CT scans to produce not only high‐quality images but also accurate reconstructions. Second, from Eq. (20), both positive bias and negative bias can appear in the same reconstruction of the phantom. If the lesion has a positive contrast, then the bias is always positive; however, if the lesion has a negative contrast, the bias can be either negative or positive, depending on the term (1+βΔHU). These findings are supported by the experimental results shown in Figs. 5 and 5.

Some limitations of the paper are worth mentioning: First, although the biases of CT number were theoretically and experimentally studied using phantoms, further studies are needed to understand how the parameters α and β in Eq. (20) depend on the shape, size, and attenuation properties of the image object. A quantitative understanding of these potential properties is the key in translating the conclusion in this paper to clinical human subject studies. Potentially, these two parameters may be estimated experimentally in phantom studies. For example, one may use an anthropomorphic phantom to perform calibration studies to estimate these two parameters. After these two parameters are fixed, an iso‐bias map can be generated to guide the prescription of scanning parameters for human subject studies. Second, the clinical impact of CT number bias is dependent on the specific diagnostic imaging tasks. When CT numbers are used as a quantitative biomarker in clinical disease diagnosis, the accuracy of CT number is crucially important. However, in some other clinical tasks, the diagnosis may primarily count on the visual conspicuity level of a lesion. In this case, the accuracy of CT number may not be as critical as the previous one. In fact, one may even argue that the bias could potentially enhance the contrast and thus might be beneficial in some clinical context. This can be seen by subtracting Eq. (19) from Eq. (18), which shows that the relative bias will always lead to an increase in the contrast. Therefore, a quantitative understanding of the potential CT number bias is needed for an accurate understanding of pros and cons in clinical context. Third, note that Eq. (20) was derived assuming an uniform incident fluence, such that the fluence on the detector is relatively uniform. The condition corresponds to the case of a round object with the use of an appropriate bowtie filter. If the bowtie filter is not applied, l ₂ in Eq. (20) has to be interpreted as the mean path length traveling through the voxel being reconstructed. So the voxel close to the center of a round object will have higher bias compared to the voxel on the peripheral region. Fourth, for a very high‐contrast large lesion, $e^{\mathrm{\Delta }\mu l_1}\approx 1+\mathrm{\Delta }\mu l_1$ may not apply, therefore the linear dependence on the contrast may become inaccurate. This can be seen from the bias map shown in Fig. 12(b), where the spine region appears brighter compared to other bone ROIs due to a larger bias amplitude. Last but not least, we did not consider the case of extremely low exposure levels in which a significant amount of the detector readout counts can be zero or even negative (for energy‐integrating detectors due to the confounding electronic noise). Since the log‐transform ln (x) does not allow zero or negative arguments, these zero or negative counts must be corrected before calculating the line integral. For example, these zero or negative counts may be replaced by an arbitrarily chosen small positive number, or interpolated from neighboring detector pixels, or processed using other low signal correction methods. Despite the methods used to correct these zero or negative counts, these low signal correction operations will inevitably introduce bias to the postlog line integral data. The impact of these low signal correction schemes on quantitative CT number accuracy is an interesting topic but it is not addressed in this paper. In other words, with advanced low signal correction methods being used to handle zero counts or negative counts, the mAs and contrast dependence shown in Eq. (20) may need to be further modified. In this work, the lowest mAs was selected so that the number of the detector pixels with either zero counts or negative counts was small.

As shown in this paper, the quantitative CT number accuracy depends on the choice of statistical weighting schemes in image reconstruction. With the introduction of the variety of iterative image reconstruction algorithms to enable low‐dose CT, it would be interesting to investigate the impact of these image reconstruction methods on quantitative CT number accuracy. Since the potential CT number bias would depend on the specific methods to handle the statistical weight of measured data, and information about these methods may require access to the manufacturer's proprietary information, the topic of CT number bias with iterative image reconstruction methods was not discussed in this paper.

6. Conclusion

The CT number bias in FBP reconstruction was theoretically analyzed based on a lesion‐background model. Analytical results have been derived to characterize the dependence of bias on mAs and CT number contrast. Theoretical results were validated by experimental studies. The results demonstrated that reconstruction bias is linearly dependent on the CT number contrast and inversely proportional to the exposure level.

Appendix A.

Based on Eqs. (3), (4), (5), the negative log‐likelihood function for the proposed CT reconstruction problem can be calculated as follows:

$$\begin{array}{cc}\hfill L({\mu }_1,{\mu }_2)& =-lnP(\{y_{1i},y_{2i}\})\hfill \\ & =-ln\prod _{i=1}^{N}P(y_{1i})P(y_{2i})\hfill \\ & =-ln\prod _{i=1}^N{e}^{-{\overline{y}}_{1i}}\frac{{\overline{y}}_{1i}^{y_{1i}}}{y_{1i}!}{e}^{-{\overline{y}}_{2i}}\frac{{\overline{y}}_{2i}^{y_{2i}}}{y_{2i}!}\hfill \\ & =\sum _{i=1}^N{\overline{y}}_{1i}-y_{1i}ln{\overline{y}}_{1i}+{\overline{y}}_{2i}-y_{2i}ln{\overline{y}}_{2i}\hfill \\ & =\sum _{i=1}^N{b}_{1i}{e}^{-[{\mu }_1{l}_1+{\mu }_2\left(l_2-l_1\right)]}+b_{2i}{e}^{-{\mu }_2{l}_2}\hfill \\ & +y_{1i}[{\mu }_1{l}_1+{\mu }_2(l_2-l_1)]+y_{2i}{\mu }_2{l}_2.\hfill \end{array}$$ (21)

Note that the irrelevant constants $ln(y_{1i}!)$ and $ln(y_{2i}!)$ were discarded in the last two equalities.

Finding the solutions that minimize the above nonquadratic cost function is usually very difficult. Therefore, a quadratic approximation is often used to simplify the analysis.³³ For convenience, the above negative log‐likelihood function can be written as follows:

$$L({\mu }_1,{\mu }_2)=\sum _{i=1}^N{h}_{1i}[{\mu }_1{l}_1+{\mu }_2\left(l_2-l_1\right)]+h_{2i}\left({\mu }_2{l}_2\right),$$ (22)

where the functions h _1i (θ) and h _2i (θ) are defined as follows:

$$h_{1i}(\theta )=b_{1i}{e}^{-\theta }+y_{1i}\theta ,$$ (23)

$$h_{2i}(\theta )=b_{2i}{e}^{-\theta }+y_{2i}\theta .$$ (24)

Applying the second‐order Taylor expansion to h _1i (θ) around estimates ${\widehat{\theta }}_{1i}$ and h _2i (θ) around ${\widehat{\theta }}_{2i}$, we have:

$$h_{1i}(\theta )\approx h_{1i}({\widehat{\theta }}_{1i})+{\dot{h}}_{1i}({\widehat{\theta }}_{1i})(\theta -{\widehat{\theta }}_{1i})+\frac{{\ddot{h}}_{1i}({\widehat{\theta }}_{1i})}{2}{(\theta -{\widehat{\theta }}_{1i})}^2,$$ (25)

$$h_{2i}(\theta )\approx h_{2i}({\widehat{\theta }}_{2i})+{\dot{h}}_{2i}({\widehat{\theta }}_{2i})(\theta -{\widehat{\theta }}_{2i})+\frac{{\ddot{h}}_{2i}({\widehat{\theta }}_{2i})}{2}{(\theta -{\widehat{\theta }}_{2i})}^2,$$ (26)

where ${\dot{h}}_i$ and ${\ddot{h}}_i$ represent first‐ and second‐order derivatives of h _i . The estimators are obtained as follows:

$${\widehat{\theta }}_{1i}=ln\frac{b_{1i}}{y_{1i}},$$ (27)

$${\widehat{\theta }}_{2i}=ln\frac{b_{2i}}{y_{2i}}.$$ (28)

Substituting Eqs. (27), (28) into Eqs. (25) and (26) and dropping the irrelevant constants, we have:

$$h_{1i}(\theta )\approx \frac{y_{1i}}{2}{\left(\theta -ln\frac{b_{1i}}{y_{1i}}\right)}^2,$$ (29)

$$h_{2i}(\theta )\approx \frac{y_{2i}}{2}{\left(\theta -ln\frac{b_{2i}}{y_{2i}}\right)}^2.$$ (30)

Finally, the formula in Eq. (7) can be obtained using Eqs. (22),(29), and (30).