Realistic simulation of reduced-dose CT with noise modeling and sinogram synthesis using DICOM CT images Chang Won Kim and Jong Hyo Kim Citation: Medical Physics 41, 011901 (2014); doi: 10.1118/1.4830431 View online: http://dx.doi.org/10.1118/1.4830431 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/41/1?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Quantum noise properties of CT images with anatomical textured backgrounds across reconstruction algorithms: FBP and SAFIRE Med. Phys. 41, 091908 (2014); 10.1118/1.4893497 CT head-scan dosimetry in an anthropomorphic phantom and associated measurement of ACR accreditationphantom imaging metrics under clinically representative scan conditions Med. Phys. 40, 081917 (2013); 10.1118/1.4815964 Prediction of human observer performance in a 2-alternative forced choice low-contrast detection task using channelized Hotelling observer: Impact of radiation dose and reconstruction algorithms Med. Phys. 40, 041908 (2013); 10.1118/1.4794498 Noise reduction in low-dose cone beam CT by incorporating prior volumetric image information Med. Phys. 39, 2569 (2012); 10.1118/1.3702592 Three-dimensional anisotropic adaptive filtering of projection data for noise reduction in cone beam CT Med. Phys. 38, 5896 (2011); 10.1118/1.3633901

Realistic simulation of reduced-dose CT with noise modeling and sinogram synthesis using DICOM CT images Chang Won Kim Interdisciplinary Program of Bioengineering Major Seoul National University College of Engineering, San 56-1, Silim-dong, Gwanak-gu, Seoul 152-742, South Korea and Institute of Radiation Medicine, Seoul National University College of Medicine, 28, Yongon-dong, Chongno-gu, Seoul 110-744, South Korea

Jong Hyo Kima) Department of Radiology, Institute of Radiation Medicine, Seoul National University College of Medicine, 28, Yongon-dong, Chongno-gu, Seoul, 110-744, Korea; Department of Transdisciplinary Studies, Graduate School of Convergence Science and Technology, Seoul National University, Suwon, Gyeonggi-do, 443-270, Korea; and Advanced Institutes of Convergence Technology, Seoul National University, Suwon, Gyeonggi-do, 443-270, Korea

(Received 12 May 2013; revised 12 October 2013; accepted for publication 1 November 2013; published 4 December 2013) Purpose: Reducing the patient dose while maintaining the diagnostic image quality during CT exams is the subject of a growing number of studies, in which simulations of reduced-dose CT with patient data have been used as an effective technique when exploring the potential of various dose reduction techniques. Difficulties in accessing raw sinogram data, however, have restricted the use of this technique to a limited number of institutions. Here, we present a novel reduced-dose CT simulation technique which provides realistic low-dose images without the requirement of raw sinogram data. Methods: Two key characteristics of CT systems, the noise equivalent quanta (NEQ) and the algorithmic modulation transfer function (MTF), were measured for various combinations of object attenuation and tube currents by analyzing the noise power spectrum (NPS) of CT images obtained with a set of phantoms. Those measurements were used to develop a comprehensive CT noise model covering the reduced x-ray photon flux, object attenuation, system noise, and bow-tie filter, which was then employed to generate a simulated noise sinogram for the reduced-dose condition with the use of a synthetic sinogram generated from a reference CT image. The simulated noise sinogram was filtered with the algorithmic MTF and back-projected to create a noise CT image, which was then added to the reference CT image, finally providing a simulated reduced-dose CT image. The simulation performance was evaluated in terms of the degree of NPS similarity, the noise magnitude, the bow-tie filter effect, and the streak noise pattern at photon starvation sites with the set of phantom images. Results: The simulation results showed good agreement with actual low-dose CT images in terms of their visual appearance and in a quantitative evaluation test. The magnitude and shape of the NPS curves of the simulated low-dose images agreed well with those of real low-dose images, showing discrepancies of less than +/−3.2% in terms of the noise power at the peak height and +/−1.2% in terms of the spatial frequency at the peak height. The magnitudes of the noise measured for 12 different combinations the phantom size, tube current, and reconstruction kernel for the simulated and real low-dose images were very similar, with differences of 0.1 to 4.7%. The p value for a statistical testing of the difference in the noise magnitude ranged from 0.99 to 0.11, showing that there was no difference statistically between the noise magnitudes of the real and simulated low-dose images using our method. The strength and pattern of the streak noise in an anthropomorphic phantom was also consistent with expectations. Conclusions: A novel reduced-dose CT simulation technique was developed which uses only CT images while not requiring raw sinogram data. Our method can provide realistic simulation results under reduced-dose conditions both in terms of the noise magnitude and the textual appearance. This technique has the potential to promote clinical research for patient dose reductions. © 2014 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4830431] Key words: noise model, NPS, NEQ, algorithmic MTF, reduced-dose simulation, filtered backprojection 1. INTRODUCTION Computed tomography is being increasingly used in healthcare applications given its rapid technological advancement leading to rapid and high-resolution imaging of the three011901-1

Med. Phys. 41 (1), January 2014

dimensional details of anatomical structures at a relatively low cost. More than 62 million CT examinations were performed in the United States in 2007, and this rate is increasing by around 10% each year.1 On the other hand, also increasing are concerns about radiation exposure associated with CT

0094-2405/2014/41(1)/011901/16/$30.00

© 2014 Am. Assoc. Phys. Med.

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C. Kim and J. Kim: Realistic simulation of reduced-dose CT using DICOM CT images

examinations, in keeping with the elevated level of awareness and growing reports on the risk of ionized radiation to public health. A report from the National Research Council of the National Academies in 2006 states that even a low level of exposure of ionizing radiation such as that from a CT examination could lead to the development of cancer. Moreover, Brenner et al. reported that 29 000 cancer cases reported in the US in 2007 may have been associated with CT examinations.2 Therefore, the demand for the optimization of CT protocols according to the as low as reasonably achievable (ALARA) principle has been raised, leading to a variety of studies to realize this principle in clinical settings. Reduced-dose CT simulation has been proposed as an effective research tool to aid those studies in clinical environments. This technique enables the generation of CT images equivalent to any reduced dose condition via the addition of appropriate levels of noise to a given image or raw data, thereby circumventing the ethical burden involved in repeating CT examinations with difference exposure conditions for the same patients.3, 4 The images obtained with this technique can be effectively utilized for evaluating image quality and lesion detectability levels according to varying dose levels. Reduced-dose CT simulation techniques can be classified into two categories: those based on raw sinograms and those based on CT images. The first type (the raw sinogram method) was introduced by Mayo et al.,5 who added random Gaussian noise to raw sinogram data and reconstructed the data to generate simulated low-dose CT images, thus demonstrating the feasibility of a reduced-dose CT simulation technique. Frush et al. improved this technique by adjusting the magnitude of the Gaussian random noise at each pixel on the raw sinogram data according to the mAs condition and the attenuation of the target object.6 The basic idea of using the raw sinogram data was retained and applied to clinical CT image simulations,7–15 which was then advanced by Massoumzadeh et al. to include the effect of a bow-tie filter and the system noise of the CT equipment.16 Recently Zabic et al. extended the noise model to reflect the noise covariances between the noises of generated low-dose and reference scan data as well as the photon starvation and ring artifact appearing at an ultra-low-dose level.28 The raw sinogram-based method uses a reconstruction system provided by the CT vendor and thus provides realistic images with different reconstruction kernels which are indistinguishable from actual scanned CT images obtained under the same reduced-dose condition. However, the raw sinogram data are neither available at general academic institutions unless special care is taken nor are their formats known, thus hindering the widespread use of this data for academic research purposes. On the other hand, the CT imagebased method is advantageous in that image data are much readily available. Moreover, the standard DICOM format is used, which may allow researchers to apply this method to wider areas of study. The CT image-based method was introduced by Britten et al.,17 who, in a rather simple method, added spatially correlated noise to the CT image that was used. After the initial trial, generating images or patterns with added noise was attempted by many researchers.18–21 More recently, Adam et al. proposed a technique that used synthetic Medical Physics, Vol. 41, No. 1, January 2014

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sinograms, though this was found to be limited in its application to clinical studies due to the fact that it requires CT images in a special condition while also not guaranteeing the realism of the simulated images in terms of how they resemble actual CT scan images.20, 21 The limitations of conventional image-based methods arise from the lack of an appropriate noise model in the simulation process due to the difficulties in extracting the system parameters and reflecting the physical processes involved in generating the unique noise patterns of low-dose CT images. Therefore, for a wider application range of the reduced-dose simulation method in clinical research, required is the development of a sophisticated image-based method which provides realistic simulated images comparable to real CT scans while employing a noise model that faithfully reflects the CT physics of noise generation and the characteristics of vendorspecific filtered back-projection kernels. This study presents a novel image-based reduced-dose CT simulation method in which a noise model is developed based on measurements of the key characteristics of the CT system, with the model then used to generate reduced dose-equivalent CT noise images by means of sinogram synthesis and filtered back-projection procedures, finally providing realistic simulated reduced-dose CT images. In Sec. 2, we discuss the theory of the noise model. Section 3 introduces the procedures of the analysis of the noise characteristics. Section 4 evaluates the performance of the proposed method. Finally, a discussion and concluding comments are given in Secs. 5 and 6, respectively.

2. THEORY OF NOISE MODELING The statistical properties of the noise in a CT image originate from random fluctuations inherent in the detection of a finite number of x-ray photons and its treatment in the reconstruction process. This can be effectively characterized by the noise power spectrum (NPS). Equation (1), which was derived by Wagner et al.22 and Hanson,23 describes the NPS of CT as a function of the radial frequency, NPS(f ) =

πf MTF2a lg (f ), NEQ

(1)

where NEQ refers to the noise equivalent density per image. Note that NEQ defined here is different from the conventional definition referring to the effective number of detected photons per detector pixel. Also noteworthy is that Eq. (1) relates the NPS measured on a CT image to the two key system characteristics, the NEQ and algorithmic MTF, which determine the magnitude and textural pattern of the noise, respectively. While NEQ is a monotonic increasing function of the dose level, NPS is inversely related to the dose. Considering the stochastic nature of noise, the NPS of axially scanned CT images can be treated as the sum of two separate NPSs of higher doses, which can be stated as NPSlow (f ) = NPShigh (f ) + NPSadded (f ).

(2)

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Therefore, if we regard the attenuation coefficient function μdose (x, y) of a CT image at a given dose as the sum of the noise-free component I(x, y) and the additive noise ηdose (x, y) as μdose (x, y) = I (x, y) + ηdose (x, y),

(3)

μlow (x, y) of a low-dose scan CT can be said to be equivalent to adding a noise image having an appropriate NPS to its higher dose image, in this case μlow (x, y) = I (x, y) + ηlow (x, y) = I (x, y) + ηhigh (x, y) + ηadded (x, y) = μhigh (x, y) + ηadded (x, y),

(4)

where the NPSs of ηlow (x, y), ηhigh (x, y), and ηadded (x, y) follow the relationship described in Eq. (2). Therefore, the problem of simulating reduced-dose CT is equivalent to generating a CT noise image following the NPS with an appropriate magnitude and a spectral shape, which, in turn, is reduced to determining the NEQadded at the desired dose and MTFalg values of the CT system. In Secs. 2.A and 2.B, we discuss a noise model in CT which is based on an extended expression of NPS as a continuous function of the radiation dose and the attenuation of scanned objects as well as the radial frequency in relation to simulating reduced-dose CT.

2.A. Analysis

This section discusses NPS analysis to extract the necessary key characteristics, NEQ and algorithmic MTF, to develop a continuous function of NPS for the radiation dose along with the attenuation of scanned objects as well as the radial frequency.   2  1  −2πi(xfx +yfy )  η(x, y)e (5) NPS(f ) =   . S S Equation (5) describes how the NPS is measured from a CT image, where S is the area of measurement ROI, and η(x, y) is the noise image of a uniform object with an intensity scale converted to linear attenuation coefficients. Usually, a water or acrylic phantom with a diameter of 20–25 cm is used for the measurement of the NPS. As discussed above, the magnitude of NPS is inversely related to NEQ, which is mainly determined by the dose and the attenuation of the scanned object. Accordingly, NPS can be rewritten as   2  1  −2πi(xfx +yfy )  η(x, y)dose,A e NPS(f )dose,A =  , S S

(6) where A is the attenuation of the object along the ray path at the object center. From the definition of MTF, the value of MTFalg converges to unity at the frequency range near zero: lim MTFalg (f ) f →0

= 1. Under this condition, NPS is controlled only by the ramp Medical Physics, Vol. 41, No. 1, January 2014

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component of the filter’s frequency response, which ensures the acquiring of NEQ by taking the inverse of the slope of NPS24 as π NEQdose,A = . (7) NPSdose,A (f ) lim |f | f →0

Equation (7) ensures an array of NEQ values for a dose range and object attenuations using a set of phantoms with different diameters and scan conditions with varying dose levels. Here, we consider the contribution of electronic system noise to NEQ. The mean number of detected photon sd at a CT detector can be described as the multiplication of the incident flux Q0 with the transmittance of an object e−A , sd = Q0 · e−A ,

(8)

where A is the attenuation coefficient of the object along the beam path. As the electronic system noise can be regarded to follow a zero-mean normal distribution while also remaining independent of the quantum noise following a Poisson distribution, the noise at the detector can be modeled to have variance of σd2 = sd + σe2 .

(9)

Here, we introduce NEQd , the noise-equivalent quanta per detector pixel. Recalling that the noise-equivalent quanta of a detector pixel is the square of the signal-to-noise ratio,  2 sd sd 2 = . (10) NEQd = σd sd + σe2 In order to convert the NEQd value of the detector pixel domain to the NEQ value of the reconstructed image domain, as in Eq. (1), we need two scale factors: one is the inverse of the detector pixel size (1/a) to give a normalized value per unit length, and the other is the number of views m to take into account the integration effect over one rotation. Applying the scale factors to Eq. (10) gives NEQdose,A =

sd 2 m . sd + σe2 a

(11)

At this point, we include the dose-dependent property in the electronic system noise σe2 . Referring to a previous study16 in which the electronic system noise was modeled as a weighted sum of the thermal noise and dose-dependent noise terms, σe2 (mAs) = α · c · mAs + β,

(12)

where α and β are respectively the weight factors for the dosedependent noise and thermal noise terms, c means collimation, and where mAs is the tube current time product representing the dose level. Putting Eqs. (8) and (12) into Eq. (11) gives a model of NEQ in which the effects of the dose and the object’s attenuation are fully incorporated: NEQ(Q0 , A) =

Q0 ·

m (Q0 · e−A )2 . + α · c · mAs + β a

e−A

(13)

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Letting Q0 = γ · c · mAs in Eq. (13) finally gives a complete model of NEQ, which is expressed as a function of mAs and A: NEQ(mAs,A) =

m (γ · c · mAs · e−A )2 . γ · c · mAs · e−A + α · c · mAs + β a

(14)

Equating Eqs. (13) and (7) for a number of measurements and applying a curve fitting technique, one can easily obtain the values for the three parameters in Eq. (13). Thus, we obtained a noise model that can predict the magnitude of NEQ at any arbitrary dose level and object attenuation. Now, we turn to the spectral property of the noise, which determines the noise texture. Rearranging Eq. (1) and replacing Eq. (7) with NEQ gives ⎤1/2

⎡ MTFa lg (f ) =

NPS(f ) |f | ⎢ ⎣ lim NPS(f ) f →0 |f |

⎥ ⎦

.

(15)

frequency f: NPS(mAs, A, f ) =

πf MTF2a lg (f ) NEQ(mAs, A)

=

πf a γ · c · mAs · e−A + α · c · mAs + β MTF2a lg (f ). m (γ · c · mAs · e−A )2 (16)

2.B. Synthesis

This section describes how an appropriate CT noise image can be generated using the key characteristics of a CT system to simulate reduced-dose CT. From Eq. (1) and Eq. (2), NPSadded (f ) = NPSlow (f ) − NPShigh (f )

1 1 MTF2a lg (f ) − = πf NEQlow NEQhigh = πf

Finally, putting Eqs. (13) and (14) into Eq. (1) gives a complete noise model of NPS, which is a function of the dose and object’s attenuation as well as the radial

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1 MTF2a lg (f ). NEQadded

(17)

If we let rmAs = mAslow /mAshigh and apply the noise model of Eq. (14), the above equation becomes

1 1 1 − = NEQadded NEQ(rmAs · mAshigh , A) NEQ(mAshigh , A) a = m



γ · c · mAshigh · e−A + α · c · mAshigh + β γ · c · rmAs · mAshigh · e−A + α · c · rmAs · mAshigh + β − (γ · c · rmAs · mAshigh · e−A )2 (γ · c · mAshigh · e−A )2

. (18)

2

Recalling that NEQd is (SNR) at the detector pixels and that it is scaled by a/m, the variance of the noise to be added at the detector pixels is determined via



1 1 2 −A −1 +β · 2 −1 . (19) σadded = c · mAshigh · (γ · e + α) · rmAs rmAs At this point, we adapt our noise model to the detector response of a real-world CT system. As the transmission e−A of a scanned object is a function of the detector pixel and gantry angle, the magnitude of the added noise is as follows: 



1 1 −1 +β · 2 −1 . (20) σadded (ξ, θ ) = c · mAshigh · (γ · e−A(ξ,θ) + α) · rmAs rmAs

Here, ξ is the detector pixel index and θ is the gantry angle step. In the theoretical derivation, we only consider the parallel beam geometry for convenience. The attenuation sinogram of a scanned object A(ξ , θ ) can be obtained by taking the Radon transform of the noise-free attenuation coefficient function such that ∞ ∞ I (x, y) · δ(x cos θ + y sin θ − ξ )dxdy. A(ξ, θ ) = −∞ −∞

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(21)

At this point, we take the effect of the bow-tie filter into account when calculating the added noise. As the property of the bow-tie filter does not depend on the gantry angle, we can let the transmission of the bow-tie filter be a function of the detector pixel, as B(ξ ). When we define the linear sinogram of a high dose sd, high (ξ , θ ) as the response function of the detector pixel and gantry angle at a high-dose scan, then sd,high (ξ, θ ) = γ · c · mAshigh · B(ξ ) · e−A(ξ,θ) .

(22)

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The linear sinogram of the reduced dose can then be considered as virtually equal to adding white Gaussian noise with a magnitude determined by Eq. (20) to the linear sinogram of a high dose, sd,low (ξ, θ ) = γ · c · mAshigh · B(ξ ) · e−A(ξ,θ) + σadded (ξ, θ ) · W GN.

(23)

When we compute the added noise in the attenuation sinogram, Aadded (ξ, θ ) = − ln(sd,low (ξ, θ )/(γ · c · mAshigh · B(ξ ))) −A(ξ, θ ).

(24)

Finally, we obtain the added CT noise image ηadded (x, y) by taking the inverse Radon transform of the noise sinogram after filtering with MTFalg , as determined by Eq. (14), π  ∞ A˜ added (f, θ )|f |MTFa lg (f ) ηadded (x, y) = 0

−∞



× ej 2πf (x cos θ+y sin θ) df dθ,

(25)

where A˜ added (f, θ ) is the 1D Fourier transform of Aadded (ξ , θ ). 3. METHODS AND MATERIALS 3.A. Overall procedure

The procedures of the proposed method were performed in two phases: an analysis phase to extract the key characteristics of the CT system from the NPS measurements, and a synthesis phase to generate a synthetic noise CT image using the extracted characteristics. The procedures are summarized in Fig. 1. The analysis phase begins with obtaining CT scans for a set of phantoms designed to be used for extracting the parameters of the noise model in this study. NPS calculations were carried out using the CT images for the phantom set, followed by an estimation of a NEQ set and the algorithmic MTF from the NPS data set. Then, a curve fitting technique was applied to the NEQ estimates, thereby determining the required parameters of the noise model. In the synthesis phase, a CT study of a high dose was retrieved from the PACS server, from which a synthetic attenuation sinogram was generated. Next, a matrix of white Gaussian noise was generated while varying the magnitude determined according to the noise model obtained in the analysis phase to create a noise sinogram, which was filtered and backprojected to produce a noise CT image. Finally, the noise image was added to the original CT image to provide a simulated reduced-dose CT image. 3.B. Scanner, scan condition, and computing environment

The 16-row CT scanner (Somatom Sensation 16, Siemens, DE) located at Seoul National University Hospital (SNUH) Medical Physics, Vol. 41, No. 1, January 2014

F IG . 1. Summary of the analysis and synthesis procedures.

was used in this study. A cylindrical water phantom with a 20 cm diameter and a 20 cm length, a tapered cylindrical acrylic phantom consisting of five disks each with a 2 cm thickness and different diameters (35, 32.5, 30, 20, and 15 cm), and an anthropomorphic chest phantom were used to measure the parameters and for the performance evaluation of the developed simulator. To determine the algorithmic MTF, the cylindrical water phantom was scanned at 120 kVp 30 mAs with 16 × 1.5 mm beam collimation in axial scan mode and was reconstructed with eight different kernels (B10f, B20f, B30f, B40f, B50f, B60f, B70f, and B80f) to have a 250 mm FOV. For the NEQ measurement, the tapered cylindrical acrylic phantom was scanned at 120 KVp at different current levels (30, 40, 50, 100, and 200 mAs) with beam collimation of 16 × 0.75 mm, after which it was reconstructed with two kernels (B30f and B50f). An anthropomorphic chest phantom was scanned under four exposure conditions (20, 40, 60, and 200 mAs at 120 kVp, 16 × 1.0 mm beam collimation) and was reconstructed with the B60f kernel for use in the performance evaluation. All data analyses and simulations were carried out with software programmed with MATLAB (MathWorks, Natick, MA) on a personal computer equipped with 2.66 GHz Q8400 Quad-Core CPU and 8 GB RAM. 3.C. Calculation of NPS and determination of the algorithmic MTF

The NPS of CT was used as an information source in extracting the system characteristics of CT in this study. Therefore, special care was taken to prevent the introduction of undesirable artifacts into the NPS calculations. It is known that noise in CT images consists not only of stochastic quantum noise but also structured noise.22, 24 Usually, the structured

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noise arises from scatter, dark currents, the nonuniform detector gain, beam hardening, shading, and other unknown factors. This structured noise is nonstochastic and can be classified as an artifact that exhibits a smooth trend depending on the pixel locations, which appear as a sharp peak at a low frequency on NPS if not treated appropriately.24 Therefore, the effective removal of this structured noise is critical for an accurate estimation of NEQ and MTFalg values, as those characteristics strongly rely on the slope of NPS in the lowfrequency range. In this study, we removed this structural noise using a de-trending technique in which a polynomial fitted image Cp (x, y) was subtracted from the original CT image C(x, y), NPS2d (u, v) =

2 xy  F T {C(x, y) − Cp (x, y)} . (26) Nx Ny

An ROI of 128 × 128 pixels was taken at image center, and a 2D surface function was obtained by applying a surface fitting technique to the ROI data with polynomial function.30–32 The low frequency peaks of NPS were successfully removed after applying this detrending technique. From the detrended NPS2d , a series of 1D NPS( f )θi was obtained by taking an interpolated sampling of NPS2d (u, v) along the radial direction at varying angles of θ i over 0 ∼ π , and NPS1d ( f ) was calculated as the average of NPS( f )θi : NPS1d (f ) =

Nθ 1  NPS(f )θi . Nθ i=1

(27)

This procedure was repeated for 40 images to produce a mean NPS curve, which was used for the subsequent analysis. To obtain a cleaner NPS curve free of irregular fluctuations, additional curve smoothing was realized by applying iterative 1D diffusion to the calculated NPS curve.

F IG . 2. Comparison of the measured values and fitted curves of the NEQd model.

tom sizes (15, 20, 25 cm). For these smaller phantoms, we assumed that the detector response was dominated by the high flux of the incident photons and that the system noise could be neglected. The remaining data were used with the obtained γ for the nonlinear curve fitting of NEQ to extract the other two parameters via Eq. (14). The three parameters thus obtained were 2620 for γ , 0.014 for α, and 3 for β. Figure 2 compares the 20 NEQd measurements versus the fitted curve of the NEQd model on log scale for the five phantom diameters. Over the range of experimental conditions, the fitted NEQd model agrees well with the measurements, including the linear part in the small phantom range as well as the part that is slightly bent down in the large phantom range.

3.D. Measurement of NEQ and system noise

NEQ measurements were carried out using CT images of the tapered acrylic phantom. A total of 20 smoothed NPS curves were calculated using the same procedure described in Sec. 3.C for a combination of four different mAs values and five phantom sizes. The intensity scale of NPS in HU was then converted to attenuation coefficients with an effective energy level of 87.4 keV for the 120 kVp sample and the corresponding water attenuation coefficient of 0.18/cm in reference to a previous study.24 From each NPS curve, the NEQdose,A estimate was derived using Eq. (7) followed by conversion to the detector level NEQd by scaling with the detector pixel pitch divided by the number of projections per rotation (a/m). Then, a curve fitting technique was applied to obtain the three parameters of the noise model (γ , α, β) using those 20 NEQd estimate data. MATLAB was used for the curve fitting based on the “robust nonlinear fitting function,” which uses a least-square fitting method with iterative reweighting. In order to improve the reliability of the curve fitting process, the detector gain parameter γ was obtained first using a subset of data which corresponds to relatively small phanMedical Physics, Vol. 41, No. 1, January 2014

3.E. Generation of a synthetic sinogram

First, a DICOM high-dose CT image was obtained from the PACS server and the intensity values in HU were converted to attenuation coefficients by applying the conversion scale, as outlined in Sec. 3.D. The 2D attenuation coefficient function μhigh (x, y) thus generated includes the noise component ηhigh (x, y) as well as noise-free function I(x, y). If the noisy attenuation coefficient function μhigh (x, y) is used in synthetic sonogram generation, the noise covariance between the newly generated reduced-dose noise and the noise component ηhigh (x, y) causes an elevation of noise level on reduceddose image.28 Therefore, in order to reduce the noise covariance effect, an image denoising technique was applied to the μhigh (x, y) before generation of synthetic sinogram. A total variation (TV) minimization-base de-noising technique was chosen considering its edge preserving performance,29 inf

u∈BV ( )

F (μ) = T V (μ) + λ μ − μ2L2( ) ,

(28)

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011901-7

of D was sourced from the DICOM header, and the values of Np (1160) and Nd (672), the total number of projections and the number of detector pixels, respectively, were obtained from the literature.16 The synthetic attenuation sinogram was then converted to produce a synthetic linear sinogram of a high dose as sd,high (di , gj ) = γ · c · mAs · B(di ) · e−A(di ,gj ) ,

(31)

where B(di ) is the bow-tie filter profile function, which will be described in the Appendix. 3.F. Generation of a noise image

White Gaussian noise (WGN) with a zero mean and unity variance was generated with the MATLAB randn function and multiplied by the noise magnitude of the added noise with Eq. (20), which was then incorporated into the equation used to generate the simulated linear sinogram of a low dose as ssim,low (di , gj ) = ss.high (di , gj ) F IG . 3. Geometry of the fan beam projection scheme used for the generation of the synthetic sinogram in this study.

where T V (μ) ≡



|∇μ|dx, μ is input attenuation image, and



μ is updated denoised image by TV-minimization iteration. The parameterλ was set to 0.1. Then, a synthetic attenuation sinogram was generated through a fan-beam projection operation. The geometry of the fan beam projection scheme used in this study is shown in Fig. 3. In order to realize the line integral along the beam path, the coordinate of each point along the beam path needed to be represented as a function of the distance from the source point S to the current point t. Thus, if we let beam angle φ = θ + σ , the coordinate of the beam path can be expressed as x(t) = −D · sin σ + t · sin φ y(t) = −D · cos σ − t · cos φ,

(29)

Nr 

[μ(x(nt), y(nt))]

n=0

= t

Nr 

[μ(−D · sin(gj ) + nt · sin(di + gj ),

n=0

−D · cos(gj ) − nt · cos(di + gj ))],

(30)

where di , i = −Nd /2,. . . ,0,. . . , Nd /2-1 denotes the arc angle step and gj , j = 1,. . . ,Np is the gantry angle step. The value Medical Physics, Vol. 41, No. 1, January 2014

(32)

The simulated linear low-dose sinogram was then converted back to produce the simulated attenuation low-dose sinogram (Asim,low ):

ssim,low (di , gj ) Asim,low (di , gj ) = − ln . (33) γ · c · mAs · B(di ) This was used to produce the attenuation sinogram of the added noise (Aadded ): Aadded (di , gj ) = Asim,low (di , gj ) − A(di , gj ).

(34)

Then, a fan-beam filtered back-projection procedure was applied to the attenuation sinogram of the added noise to produce a noise CT image. Unlike parallel-beam geometry, filtered back-projection in fan-beam geometry requires an additional procedure25 including weighting and modification of the attenuation profiles and kernel functions via μ added (x, y)

where θ and σ are correspondingly the gantry and arc angles and where D is the distance from the source to the object center. The synthetic attenuation sinogram A(di , gj ) is then obtained as the sum of the sampled attenuation coefficients at each point along the ray path steps scaled by the sampling interval t as A(di , gj ) = t

+ σadded (di , gj ) · W GN (0, 1).

⎡ ×⎣

Np 1 2π  = Np i=1 L2 (x, y, gi )



Nd /2−1

⎤

ˆ j )⎦ . (35) Aˆ added (gi , d (x, y, gi ) − dj ) · h(d

j =−Nd /2

In Eq. (35), Aˆ added is the modified attenuation profile, hˆ is the modified discrete kernel filter function, and the angle d gives the location of a ray within a given fan-beam at the given gantry step angle. The reconstructed linear attenuation efficient of the added noise was then converted to HU to generate the added noise image (ηadded ), which was finally added to the original high-dose CT image (Chigh ), resulting in a simulated reduced-dose CT image. Figure 4 illustrates the synthesis procedure with an example CT image selected from the acrylic phantom image set. A reference high-dose image for the 15 cm diameter section with a 20 cm FOV taken with 100 mAs is shown in Fig. 4(a). A separate real 40 mAs image is shown in Fig. 4(b) for comparison. The synthetic attenuation sinogram generated with the reference image and the

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(a)

(b)

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parison of NPSs between real and simulated low-dose images using the water phantom, while the magnitude of the noise was statistically assessed to test if there was a difference between the real and simulated low-dose images using the acrylic phantom for two typical reconstruction kernels (B30f and B50f). The effect of bow-tie filter was assessed by comparing the noise trend around the periphery of the acrylic phantom for the simulated low-dose images with and without a bow-tie filter in the noise model in reference to the real low-dose image. The streak noise pattern by photon starvation was evaluated by visually comparing the noise patterns radiating from the high-attenuation sites of the real and simulated low-dose chest images of the humanoid phantom. 4.A.1. NPS comparison with the water phantom

(c)

(d)

(e)

(f)

Example images of the real and simulated low-dose scans used for the NPS calculations are compared side-by-side in Fig. 5 for the B30f and B50f reconstruction kernels. The granule size and the magnitude of the noise on the simulated lowdose image appear to agree closely with the actual low-dose image in both reconstruction kernels. Compared in Fig. 6 are the NPS curves calculated from CT images of real and simulated low-dose scans under 30, 60, and 120 mAs conditions with the water phantom for the two different reconstruction kernels (B30f and B50f). The mean and standard deviations of the NPS curves were calculated from 20 CT images in each set. Note that the NPS curves of the simulated low-dose images are in good agreement with those of the real low-dose images in terms of the spectral shapes and magnitudes for all dose conditions and reconstruction kernels.

F IG . 4. Intermediate results in the low-dose simulation procedure according to the proposed method. Display window of CT images is [−50 400], that of synthetic attenuation sinogram is [0 10], and that of added noise attenuation sinogram is [0 0.1]: (a) real CT image of 200 mAs (Ihigh ), (b) real CT image of 40 mAs (Ilow ), (c) synthetic attenuation sinogram As (d,g) of (a), (d) added noise attenuation sinogram Aadded (g,d) generated with (c), (e) added noise image (Nadded ), and (f) simulated 40 mAs image with (a).

simulated attenuation sinogram of the added noise are shown in Figs. 4(c) and 4(d), respectively. Clearly seen is the random pattern of the white Gaussian noise in Fig. 4(c), which is transformed into a radial streak pattern in the peripheral area of the reconstructed noise image in Fig. 4(e). The final simulated 40 mAs image in Fig. 4(f) is comparable to the real 40 mAs image in terms of the noise magnitude and textural appearance as well.

(a)

(b)

(c)

(d)

4. RESULTS 4.A. Validation of noise model

Validation of the proposed low-dose simulation method was carried out in terms of the spectral shape and magnitude of the noise, the effect of the bow-tie filter, and the noise pattern caused by photon starvation using different phantom data sets. The validity of spectral shape was assessed by a comMedical Physics, Vol. 41, No. 1, January 2014

F IG . 5. Real CT images of 30 mAs for (a) the B30f and (c) the B50f kernels, and simulated 30 mAs image from 240 mAs for (b) the B30f and (d) the B50f kernels.

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for the B30f kernel and from 0.00% to +6.50% for the B50f kernel. 4.A.2. Statistical validation of the noise magnitude with the acrylic phantom

The validity of the noise magnitude was assessed statistically using the acrylic phantom images of three different diameters (25 cm, 30 cm, and 35 cm) scanned with two tube current settings (50 mAs and 100 mAs) and reconstructed with two different kernels (B30f and B50f). For the simulation, the 200 mAs image was used as a reference. The magnitude of the noise was determined as the standard deviation within an ROI of 50 × 50 pixels placed at the image center for each setting, which was repeated for 20 CT images in each set to produce the mean and standard deviation of the noise magnitude. A statistical t-test was performed to determine if there was a difference in the noise magnitude between the real and simulated low-dose images in each set. Table I compares the measured noise magnitudes in the real and simulated lowdose sets, showing their percent differences and the results of the statistical t-testing. In each set, the noise magnitudes of the simulated low-dose image matched closely to those of the real images within a difference range of 0.35 ± 3.51 HU. In addition, the statistical t-test results show that the simulated images were not different statistically from the real low-dose images in terms of the noise magnitude.

(a)

4.A.3. Effect of the bow-tie filter on noise magnitude profile

(b)

F IG . 6. Graphs comparing the averaged NPS curves of real and simulated CT scans at three different dose levels (30, 60, and 120 mAs) for (a) the B30f and (b) B50f kernels.

Percent differences of the noise power at peak heights of the NPS curves ranged from −0.57% to +2.09% for the B30f kernel and from +0.70% to +5.56% for the B50f kernel. Also, the percent differences of the spatial frequency at the peak height of the NPS curves ranged from −2.21% to +2.99%

The appropriateness of the bow-tie filter function used in our noise model was assessed by examining the noise magnitude profile of the acrylic phantom images for the 30 cm diameter section. The noise magnitude was determined as the pixel-wise standard deviation calculated by using 40 images of the same slice location. Shown in Figs. 7(f)–7(h) are heat maps of the noise magnitudes obtained for the 50 mAs real CT scan and the simulated 50 mAs images with and without the bow-tie filter function, respectively. The simulated low-dose images were generated

TABLE I. Comparison of the noise magnitude and t-test values for 12 different combinations of the phantom size, tube current, and reconstruction kernel. Reconstruction kernel B30f

Phantom diameter [cm]

Tube current [mAs]

Noise level in HU (real CT image)

Noise level in HU (simulated image)

Difference [HU]

p-value

25

50 100 50 100 50 100 50 100 50 100 50 100

53.10 ± 1.86 38.09 ± 1.98 93.46 ± 2.92 64.92 ± 1.38 173.31 ± 7.27 115.63 ± 3.97 148.52 ± 3.78 107.34 ± 3.95 261.10 ± 5.61 180.47 ± 2.29 467.68 ± 22.37 323.07 ± 9.24

53.56 ± 1.98 37.63 ± 1.55 93.47 ± 2.45 65.02 ± 1.10 173.94 ± 6.68 116.43 ± 3.48 148.63 ± 3.99 105.36 ± 2.52 259.98 ± 5.74 181.83 ± 3.58 477.63 ± 22.31 317.63 ± 10.66

0.26 − 0.46 − 0.06 0.10 0.64 0.80 0.11 − 1.98 − 1.12 1.36 9.95 − 5.44

0.41 0.44 0.99 0.82 0.79 0.52 0.90 0.82 0.56 0.20 0.19 0.11

30 35 B50f

25 30 35

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(a)

(b)

(c)

(d)

(f)

(g)

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(e)

(h)

F IG . 7. Evaluation results of the effect of the bow-tie filter function on the noise magnitude: (a) An example phantom image of the 50 mAs scan. (c)–(e) Magnified views of (a) a simulated low-dose image with the bow-tile filter function, and the same without the bow-tile filter function, respectively. (f)–(h) show noise heat maps of (c)–(e) as determined with the pixel-wise standard deviation calculated using 40 images of the same slice location. (b) Line profiles of (f)–(h).

from 200 mAs images. Compared in Fig. 7(b) are the line profiles of those three noise heat maps. The decrease of the noise magnitudes seen at the peripheral zone on the simulated lowdose image without the bow-tie filter function as compared to the real low-dose image is resolved to an indistinguishable level after the inclusion of the bow-tie filter function in the noise model.

4.B. Evaluation of noise streak pattern with anthropomorphic phantom

The noise streak pattern caused by focal photon starvation was evaluated with an anthropomorphic phantom, which contained highly attenuating structures mimicking human anatomy in the chest region. Figure 8(b) shows the simulated noise image generated with the phantom image in Fig. 8(a), which includes several bony structures along the lateral direction. Clearly seen is the streak noise pattern radiating from Medical Physics, Vol. 41, No. 1, January 2014

locations of highly attenuated bony structures, which is also clear in the simulated low-dose image in Fig. 8(d) compared to that shown in the real low-dose image in Fig. 8(c). In an attempt to evaluate the increased magnitude of the noise by streak artifacts, several ROIs were placed at uniform tissue sites where relatively strong streak noises were produced by nearby bony structures, and the noise magnitude at those ROIs was compared between real and simulated low-dose images for three different tube current conditions (20 mAs, 40 mAs, and 60 mAs). As shown in Table II, the magnitude of the noise at those ROI sites in the simulated low-dose image agrees well with that of the real low-dose image in all of the tube current conditions within a difference of 2%. 5. DISCUSSION This study presented a novel image-based reduced-dose CT simulation method in which a noise model was used to synthesize a reduced dose-equivalent noise image by

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(a)

(b)

(c)

(d)

F IG . 8. Results of the noise streak pattern evaluation with the anthropomorphic phantom: (a) The reference CT image of 120 kVp, 200 mAs, 1 mm slice, B60f, FOV 330, (b) synthesized noise image, (c) real 20 mAs image, and (d) simulated 20 mAs image produced by adding (b) to (a).

applying the key physical characteristics of the CT system derived from NPS measurements with a phantom study. Thus far, it has been regarded that the conventional imagebased reduced-dose CT simulation method does not provide realistic simulation results due to the lack of an appropriate noise model taking various physical and system factors into account. This has limited its use in clinical research applications despite the potential advantage of wide applicability to retrospective clinical studies. Various factors are known to be involved in the generation of unique noise patterns appearing in low-dose CT images, including a reduced number of detected quanta, system noise, the structure of the bow-tie filter, attenuation of the scanned object, and the reconstruction kernel. Hence, in order to provide a realistic result, the simulation method should be able to quantify those factors from the experimental measurements of a CT system, incorporate that into a noise model, and reflect it in the noise generation process. It is generally regarded that quantifying such physical and system factors is difficult with only reconstructed CT TABLE II. Comparison of the noise level on a humanoid phantom image.

Mean noise level of real CT image [HU] Mean noise level of simulated reduced dose image [HU] Percent difference [%]

20 mAs

40 mAs

137.57

95.64

79.72

135.39

96.75

78.68

1.58

− 1.16

1.31

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60 mAs

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image data. In fact, results are rarely available for CT noise models, as only a few papers have reported experiments using raw sinogram data,17–21 possibly due to the difficulty of accessing raw sinogram data in academic institutions. This study was undertaken to overcome such limitations and thereby enable the generation of realistic reduced-dose images of patients using only a reference CT image. Physical factors known to be involved in determining the magnitude of CT noise include the energy spectrum and the quantum influx of the x-ray source, the spectral response of the detector, system noise, object attenuation, beam hardening, and scattering. As some of these factors interact with each other in complex ways depending on the composition and geometric configuration of the patient’s body, it is very difficult and impractical to identify each factor and calculate their interactions in each patient case. Therefore, this study took the practical approach of directly modeling the NEQ of the CT system, which is a noise index representing the integrative effect of various factors, as a function of two key variables, mAs and attenuation. In this approach, we began with a series of NPS measurement for six different sizes of an acrylic disk with four steps of mAs conditions, from which NEQ estimates were obtained for a total of 24 different conditions. By applying a curve fitting method to those NEQ estimates, we could determine the parameters of the noise model and thereby predict the magnitude of noise as a function of mAs and attenuation. The NEQs predicted by our model for acrylic disks with diameters of 10, 15, 20, and 25 cm at 25 mAs were 5549.9, 1863.8, 624.5, and 207.8, which closely matched the previously documented values of 5166.7, 1822.8, 653.9, and 234.9 by Whitning et al.10 Despite the importance of NEQ modeling in analyses of CT noise, no previous study reported such modeling based on measurements made on CT images according to the knowledge of the authors. This NEQ modeling method presented here for the first time can be applied usefully in an analysis of the noise properties of CT systems in various environments. In order to provide a realistic appearance of noise textures in simulated low-dose CT, it is essential to estimate the filter kernel used in the vendor’s proprietary reconstruction system accurately, which is the determinant of the NPS shape in CT. The property of the filter kernel can be represented as its frequency domain counterpart, the algorithmic MTF. Although, the MTF of a CT system is conventionally measured with wire or bead scanning, it is the total MTF reflecting the combined effect of focal spot blurring and magnification together with the filter kernel, and it is different from the algorithm MTF purely representing the filter kernel. In this study, we measured the algorithmic MTF for the first time using the relationship between NPS and algorithmic MTF as previously presented by Wagner et al.22 The vendor proprietary algorithmic MTF of commercial CT systems, although it affects the image quality of CT to a great extent, remains unexplored in the literature. Therefore, we verified the validity of the measured algorithmic MTF by assessing its consistency among those curves derived from a set of NPSs obtained under different conditions, showing a discrepancy of less than 5%. On the other hand, a comparison with the

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total MTF measured with the wire scanning method verified the higher response of the algorithmic MTF, particularly in the mid- to high-frequency range. The proposed method provided realistic simulation results of reduced-dose CT in terms of the noise magnitude and textural appearance by applying the derived NEQ model and algorithmic MTF during the generation of noise. The results of quantitative and qualitative evaluation experiments with an acrylic phantom and an anthropomorphic phantom demonstrate that the simulated low-dose CT provided by our method is not distinguishable statistically from that of real low-dose CT scans, showing p values of 0.99 to 0.11 and percent differences in the range of −2.13 to 1.84. We did not include an observer experiment when visually assessing the similarity between the simulated reduced dose and real low-dose scan CTs in this study, as we considered this to be unnecessary. A report from a previous study showed that observers could not distinguish images with changed tube currents of less than 25%, which translates to a 12% difference in the standard deviation of the CT images.16 Therefore, the percent differences in our simulation results sufficiently lower than the known threshold ( 20, the amplified electronic noise domi-

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F IG . 13. Figures compare the profile of (a) mean of 500 images, and (b) noise variance of 500 images of the three methods shown in Figs. 12(a)–12(c).

nates over the photon starvation noise in ultra-low-dose condition, and therefore the minute error between Poisson model and additive Gaussian noise model vanishes at this condition. Rather, the noise covariance between the generated low-dose and existing high-dose noise has stronger influence on the noise statistic, which is prevented in our method by employing the forward projection of a denoised CT image and verified in experiment I.

ACKNOWLEDGMENTS

F IG . 12. Figures compare the mean of 500 ultra-low-dose CT images (10 mAs) generated with (a) direct Poisson method, (b) our method, and (c) Massoumzadeh’s method. Images are displayed with windows [−300,800]. Note that (a)–(c) are reconstruction results with the sinogram of (b) in which both photon and electronic noise components are assumed to present. Medical Physics, Vol. 41, No. 1, January 2014

The research was supported by the Converging Research Center Program through the Ministry of Science, ICT and Future Planning, Korea (2013K000423) and in part by the Interdisciplinary Research Initiatives Program by College of Engineering and College of Medicine, Seoul National University (2012).

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a) Author

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