Comparison of automated beam hardening correction (ABHC) algorithms for myocardial perfusion imaging using computed tomography

Abstract

Purpose:

Myocardial perfusion imaging using computed tomography (MPI-CT) and coronary CT angiography (CTA) have the potential to make CT an ideal noninvasive imaging gatekeeper exam for invasive coronary angiography. However, beam hardening can prevent accurate blood flow estimation in dynamic MPI-CT and can create artifacts that resemble flow deficits in single-shot MPI-CT. In this work, we compare four automatic beam hardening correction algorithms (ABHCs) applied to CT images, for their ability to produce accurate single images of contrast and accurate MPI flow maps using images from conventional CT systems, without energy sensitivity.

Methods:

Previously, we reported a method, herein called ABHC-1, where we iteratively optimized a cost function sensitive to beam hardening artifacts in MPI-CT images and used a low order polynomial correction on projections of segmentation-processed CT images. Here, we report results from two new algorithms with higher order polynomial corrections, ABHC-2 and ABH C-3 (with three and seven free parameters, respectively), having potentially better correction but likely reduced estimability. Additionally, we compared results to an algorithm reported by others in the literature (ABHC-NH). Comparisons were made on a digital static phantom with simulated water, bone, and iodine regions; on a digital dynamic anthropomorphic phantom, with simulated blood flow; and on preclinical porcine experiments. We obtained CT images on a prototype spectral detector CT (Philips Healthcare) scanner that provided both conventional and virtual keV images, allowing us to quantitatively compare corrected CT images to virtual keV images. To test these methods’ parameter optimization sensitivity to noise, we evaluated results on images obtained using different mAs.

Results:

In images of the static phantom, ABHC-2 reduced beam hardening artifacts better than our previous ABHC-1 algorithm, giving artifacts smaller than 1.8 HU, even in the presence of high noise which should affect parameter optimization. Taken together, the quality of static phantom results ordered ABHC-2> ABHC-3> ABHC-1>> ABHC-NH. In an anthropomorphic MPI-CT simulator with homogeneous myocardial blood flow of 100 ml·min⁻¹·100 g⁻¹, blood flow estimation results were 122 ± 24 (FBP), 135 ± 24 (ABHC-NH), 104 ± 14 (ABHC-1), 100 ± 12 (ABHC-2), and 108 ± 18 (ABHC-3) ml·min⁻¹·100 g⁻¹, showing ABHC-2 as a clear winner. Visual and quantitative evaluations showed much improved homogeneity of myocardial flow with ABHC-2, nearly eliminating substantial artifacts in uncorrected flow maps which could be misconstrued as flow deficits. ABHC-2 performed universally better than ABHC-1, ABHC-3, and ABHC-NH in simulations with different acquisitions (varying noise and kVp values). In the presence of a simulated flow deficit, all ABHC methods retained the flow deficit, and ABHC-2 gave the most accurate flow ratio and homogeneity. ABHC-3 corrected phantom flow values were slightly better than ABHC-2, in noiseless images, suggesting that reduced quality in noisy images was due to reduced estimability. In an experiment with a pig expected to have uniform flow, ABHC-2 applied to conventional images improved flow maps to compare favorably to those from 70keV images.

Conclusion:

The automated algorithm can be used with different parametric BH correction models. ABHC-2 improved MPI-CT blood flow estimation as compared to other approaches and was robust to noisy images. In simulation and preclinical experiments, ABHC-2 gave results approaching gold standard 70 keV measurements.

1. INTRODUCTION

Myocardial perfusion imaging using computed tomography (MPI-CT) is a promising way to measure qualitative and quantitative myocardial blood flow. Numerous reports show the feasibility of cardiac CT perfusion in preclinical¹⁻⁴ and clinical studies.⁵⁻¹¹ With coronary CTA assessment of anatomy and perfusion assessment of functional flow, CT can be used as a very informative gatekeeper exam for invasive coronary angiography.^12,13 No other noninvasive exam (e.g., SPECT, PET, or MRI) can provide both anatomy and flow function. There are two principal obstacles to quantitative flow assessment. First, computational methods can give inaccurate, imprecise results leading to varying results.^14–20 However, our robust physiologic model²¹ and super-voxel SLICR method²² have gone a long way towards making more accurate myocardial blood flow assessment more accurate and precise. Second, BH artifacts can reduce the accuracy of quantitative blood flow estimation and create false positives in qualitative assessment of myocardial ischemia.²³ A dynamic MPI-CT exam is comprised of 30 to 40 volumes scanned over a series of heartbeats (cardiac cycles) as the contrast flows through the ventricles and myocardium. Each volume contains a different amount of iodine in the ventricles and main arteries, and therefore produces constantly changing BH artifacts. This makes MPI-CT a very demanding application for BH correction, and suggests the need for our study.

There are different BH correction approaches suitable for correcting the excess beam hardening due to pools of iodine as in CT perfusion imaging. Hardware approaches include prefiltration using a thin metal plate and bow-tie filter to harden the x-ray spectrum and reduce the effect of beam hardening, and energy-sensitive CT scanners (e.g., kVp switching, dual source, and spectral detector) that can apply material decomposition and create virtual keV images relatively immune to beam hardening.^2,24–27 Another approach uses phantoms with known characteristics to calibrate each scanner and correct projections before reconstruction.^28–31 This method is currently used in most CT scanners to achieve water BH correction. One can correct projections prior to image reconstruction. Several methods assume access to original projection data in addition to scanner information (e.g., tube spectra, source-to–detector geometry, and prefiltration), Methods use Poisson noise characteristics, projection consistency, or intermediate images ascribed to different materials to correct projections for improved reconstructions.^32–34 Some methods use an iterative aproproach.^{1,4,8,9,13–29} As raw projection data are unavailable in many medical imaging applications, others, including us, have postprocessed images, requiring image-based methods for BH correction. Typically, the image will be segmented to high and low attenuating materials and forward projections of the segmented images will be obtained to estimate contributions of different materials to projections.^30,31 Projections will be corrected to give improved reconstructions. Some of these methods are iterative, including iterative beam hardening correction,³⁵ dynamic iterative beam hardening correction,³⁶ and our automatic beam hardening correction (ABHC-1) method.³⁷ There have been multiple polynomial correction models proposed. They include those used by So et al,²³ empirical BHC (EBHC) proposed by Kyriakou et al³⁸ and an extension of EBHC proposed by Nett and Hsieh,³⁹ giving two, three, and seven parameters, respectively.

In our previous publication,³⁷ we introduced a calibration-free, scanner-independent, automated beam hardening correction (ABHC) algorithm suitable for MPI-CT. Briefly, using reconstructed DICOM images as input, the method optimized parameters in a low order polynomial correction algorithm according to a cost function tailored for CT cardiac per fusion. The cost function included terms sensitive to BH artifacts in CT cardiac perfusion images: cupping artifacts in the aorta and ventricle and streaks in the myocardium. Results on single images and MBF maps were good even though MPI-CT is the most demanding beam hardening correction application due to the large pool of iodine in the ventricular cavity and aorta.

Here we compare new algorithms with higher order polynomial corrections (ABHC-2 and ABHC-3) to our previous algorithm (ABHC-1) and to the method by Nett and Hsieh,³⁹ hereafter called ABHC-NH. Unlike our cost function tail ored to CT cardiac perfusion, ABHC-NH uses total variation of the corrected image as cost function. One potential concern is that methods with more parameters will suffer from reduced estimability, a problem of which we are well acquainted in flow estimation algorithms,²¹ suggesting the need for a variety of tests. Methods are evaluated on static and dynamic simulated CT digital phantoms, physical phantoms, and on pig myocardial perfusion images. We obtain CT images on a prototype spectral detector CT (SDCT, Philips Healthcare) scanner that provides both conventional and virtual keV images, allowing us to quantitatively compare corrected CT images to virtual keV images that significantly reduce beam hardening. We visually and quantitatively compare beam hardening corrections obtained with ABHC-NH, ABHC-1, ABHC-2 and ABHC-3 in both static images and MBF maps. Since ABHC-NH, ABHC-2, ABHC-3 have additional parameters as compared to ABHC-1, we are particularly concerned about robustness of corrections in noisy images.²¹ In this regard, we test methods on images acquired at different mAs values.

2. THEORY

In our ABHC algorithms, we iteratively minimized a cost function sensitive to BH artifacts by optimizing parameters in a polynomial correction model applied to projections of segmented CT images. In a previous report, we used a low order polynomial correction method, ABHC-1 (two free parameters). Here, we incorporate two higher order polynomial correction methods, ABHC-2 (three parameters) and ABHC-3 (seven parameters), with possibly better correction but with potentially reduced estimability. We will first briefly describe the polynomial corrections and then describe the enveloping ABHC parameter estimation algorithm. To apply these methods, we first segment DICOM images into component images consisting of high attenuating materials (HAM) and water images for ABHC-1 and −2 and transform input images corresponding to three HU ranges, water, middle attenuating materials (MAM), and HAM, for ABHC-NH and ABHC-3. We used a soft threshold for the segmentation as described by Kyriakou et al.³⁸ Briefly, to segment the image into three materials, we have three thresholds, T_water, T_iodine, and T_bone. Pixels with HU value below T_water are assumed to be water, pixels with HU value between T_water and T_iodine are assumed to be a linear combination of water and iodine, pixels with HU value between T_iodine, and T_bone are assumed to be a linear combination of iodine and bone, and pixels with HU above T_bone are assume to be bone.

2.A. Simple second order polynomial correction (ABHC-1)

ABHC-1 uses a 2-parameter expansion using the HAM image as a basis. Details of ABHC-1 can be found in our previous work.³⁷ The corrected image I_C is obtained by subtracting the error image I_E from the original image I. The error image I_E is found using the equation below, where R and R⁻¹ are the Radon transform and the inverse Radon transform respectively:

$$\begin{gathered} I_C=I-I_E \\ {I}_E=a\cdot I_{HAM}+b\cdot R^{-1}{\left(R\left(I_{HAM}\right)\right)}^2 \end{gathered}$$ (1)

ABHC-1 adjusts scalar parameters “a” and “b” until the cost function is minimized.

2.B. Higher order polynomial correction (ABHC-2)

We now briefly derive the correction used in ABHC-2. A CT projection along ray L is given by:

$$q(L)=-ln\int dE\Omega (L,E)e^{-\underset{0}{\overset{\infty }{\int }}dr\mu (r,E)}$$ (2)

where q(L)is the projection measurement along ray L, Ω(L,E)is the normalized spectrum along ray L, and μ(r,E)is the linear attenuation at position rat energy E. In the case of a single-material BH correction, we assume the composition:

$$\mu (r,E)=f_0(r){\psi }_0(E)$$ (3)

where ψ₀(E) = μ_en/ρ(E)is the mass-energy absorption coefficient which is the energy dependence of the material (e.g., water) and f ₀(r) = ρ(r) is the density at position r. Water precorrection that is done on every CT scanner reconstructs f₀ (r) and Eq. (2) will become:

$$\begin{gathered} q=-ln\int dE\Omega (E)e^{P_0{\psi }_0(E)} \\ {P}_0=\underset{0}{\overset{\infty }{\int }}dr{f}_0(r) \end{gathered}$$ (4)

The projection q is what we obtain from the scanner; a water corrected image that might suffer from residual BH due to other HAMs. P₀ is the water image projection. For two materials Eq. (3) becomes:

$$\begin{gathered} \mu (r,E)=f_1(r){\psi }_1(E)+f_2(r){\psi }_2(E) \\ =\left(f_1(r)+f_2(r)\right){\psi }_1(E)+f_2(r)\left({\psi }_2(E)-{\psi }_1(E)\right) \\ ={\widehat{f}}_1(r){\psi }_1(E)+f_2(r){\widehat{\psi }}_2(E) \end{gathered}$$ (5)

and Eq. (4) becomes:

$$\begin{gathered} q=-ln\int dE\Omega (E)e^{-{\widehat{P}}_1{\psi }_1(E)-P_2{\widehat{\psi }}_2(E)} \\ {\widehat{P}}_1=\underset{0}{\overset{\infty }{\int }}{\widehat{f}}_1(r)dr=\underset{0}{\overset{\infty }{\int }}\left(f_1(r)+f_2(r)\right)dr \end{gathered}$$ (6)

Combining Eqs. (4) and (6) gives:

$$\int dE\Omega (E)e^{P_{0\psi }{}_0(E)}=\int dE\Omega (E)e^{-{\widehat{P}}_1{\psi }_1(E)-P_2{\widehat{\psi }}_2(E)}$$ (7)

Note that the LHS is the projection data obtained from the scanner. To create a BH artifact-free image, ${\widehat{f}}_1(r)=f_1(r)+f_2(r)$, at a single arbitrary energy E₀, we use ψ₁(E₀) = ψ₂(E₀) = 1. Using a series expansion for ${\widehat{P}}_1$ (the corrected projection), we obtain:

$$\begin{gathered} {\widehat{P}}_1\left(P_0,P_2\right)=\sum _{i,j}{c}_{ij}{P}_0^i{P}_2^j \\ =c_{10}{P}_0+c_{01}{P}_2+c_{11}{P}_0{P}_2 \\ +c_{20}{P}_0^2+c_{02}{P}_2^2+O(3+) \end{gathered}$$ (8)

In order to determine the value of some of those parameters (c_ij), we will check a case where there are no HAMs in the image, that is, P₂ = 0 and ${\widehat{P}}_1=P_0$. In this case, the following must apply:

$$c_{10}=1and{c}_{i0}=\text{0}\forall i\ne 1$$ (9)

The final result is given by:

$${\widehat{P}}_1\left(P_0,P_2\right)=P_0+c_{01}{P}_2+c_{11}{P}_0{P}_2+c_{02}{P}_2^2+O(3+)$$ (10)

Due to the linearity of the Radon transform, the basis projections combinations presented in Eq. (10) can be filtered back projected to form basis images [Eq. (11)].

$${\widehat{f}}_1(r)=f_0(r)+c_{01}{f}_{01}(r)+c_{11}{f}_{11}(r)+c_{02}{f}_{02}(r)$$ (11)

Note that the coefficients c₀₁, c₁₁ andc₀₂ will be found by the ABHC algorithm. To compare with ABHC-1, I_c = I − I_E, where $I_c={\widehat{f}}_1(r),I=f_0(r)$, and I_E = − (c₀₁f₀₁ (r) + c₁₁ f₁₁ (r) + c₀₂f₀₂ (r)).

We summarize the correction process below. Steps are:

1. Segment input image to water image and HAM image.

2. Forward project water and HAM image to obtain P₀ and P₂, respectively. We used the Matlab⁴⁰ function “fanbeam” to perform the forward projection with the fan-beam geometry.

3. Calculate the raw data combinations P₀P₂ and $P_2^2$.

4. Reconstruct the basis images f₁₁ and f₀₂ , respectively, using FBP.

5. Construct the final image ${\widehat{f}}_1(r)$ using Eq. (11).

2.C. Higher order polynomial correction with three materials (ABHC-3)

We extended ABH C-2 to three ranges of materials which are water, HAM, and MAM. Equation (8) becomes:

$$\begin{gathered} {\widehat{P}}_1\left(P_0,P_2,P_3\right)=\sum _{i,j,k}{c}_{ijk}{P}_0^i{P}_2^j{P}_3^k \\ =P_0+c_{010}{P}_2+c_{110}{P}_0{P}_2+c_{020}{P}_2^2+c_{001}{P}_3 \\ +c_{101}{P}_0{P}_3+c_{002}{P}_3^2+c_{011}{P}_2{P}_3+\dots \end{gathered}$$ (12)

Similar to ABHC-2, the basis projections combinations presented in Eq. (12) can be filtered back projected to form basis images [Eq. (13)]. To compare with ABHC-1, I_c = I − I_E, where $I_c={\widehat{f}}_1(r)$, I = f₀(r), and I_E = −(c₀₁f₀₁(r) + c₁₁f₁₁(r) + c₀₂f₀₂(r) + c₀₀₁ f₀₀₁ (r) + c₁₀₁f₁₀₁ (r) + c₀₀₂ f₀₀₂(r) + c₀₁₁ f₀₁₁(r) + …).

$$\begin{gathered} {\widehat{f}}_1(r)=f_0(r)+c_{01}{f}_{01}(r)+c_{11}{f}_{11}(r)+c_{02}{f}_{02}(r) \\ +c_{001}{f}_{001}(r)+c_{101}{f}_{101}(r)+c_{002}{f}_{002}(r)+c_{011}{f}_{011}(r)+\dots \end{gathered}$$ (13)

Coefficients c₀₁, c₁₁, c₀₂, c₀₀₁, c₁₀₁, c₀₀₂, and c₀₁₁ will be found by the ABHC algorithm. We summarize the correction process of ABHC-3 below. Steps are:

1. Segment input image to water image, HAM image and MAM image.

2. Forward project water, HAM, and MAM image to obtain P₀, P₂ and P₃, respectively.

3. Calculate the raw data combinations P₀P₂, $P_2^2$, P₀P₃, $P_3^2$, and P₂P₃.

4. Reconstruct the basis images f₁₁, f₀₂, f₁₀₁, f₀₀₂, and f₀₁₁, respectively, using FBP.

5. Construct the final image ${\widehat{f}}_1(r)$ using Eq. (13).

2.D. Automatic beam hardening correction (ABHC)

To automatically perform beam hardening correction, we embed either one of the above correction algorithms in a parameter estimation process. In our ABHC, we iteratively optimize beam hardening correction parameters to minimize a cost function sensitive to BH artifacts (i.e., streaks and cupping). Once optimal correction parameters are found, a corrected image or image sequence is created.

We use the same cost function in our previous work³⁷:

$$\begin{gathered} \psi \left(I_C(x,y\mid \theta )\right)=\alpha \cdot T{V}_{myo}\left(\widehat{f}\left(I_C(x,y\mid \theta )\right)\right) \\ +(1-\alpha )\cdot F_{LV}\left(I_{C-LV}(x,y\mid \theta )\right) \end{gathered}$$ (14)

where ψ is the total cost of the corrected image, I_C, computed using the ABHC parameter vector θ. The coefficient α determines the relative weights of each term of the cost function. The first term, TV_myo, measures the total variation (TV) within the myocardial region. Since BH artifacts are relatively low frequency as compare to noise, we applied a Gaussian filer, $\widehat{f}$, to the corrected image to reduce image noise and streaks in the myocardial region. This ensures that the cost of TV_myo is caused by BH artifact but not overwhelming by noise. TV_myo is given by:

$$T{V}_{myo}\left(\widehat{f}\left(I_C(x,y\mid \theta )\right)\right)=\sqrt{\sum _{myo}G/A_{myo}}$$ (15)

where G is the gradient magnitude of the filtered corrected image within the myocardial region.A_myo is the area of the myocardium in pixels and is used to normalize TV_myo. The second term, F_LV, measures “cupping” in the left ventricle. The cupping phenomenon is the artificial darkening (reduction in HU) towards the middle of a homogeneous attenuating structure due to beam hardening. Peripheral HU values better reflect actual HU values than those in the middle which are artifactually reduced. (Cupping can be especially problematic as it can give erroneous arterial input functions.) We measure F_LV by segmenting left ventricle region in the corrected image, (I_C−LV (x,y|θ)). We estimate the “true” HU value of the left ventricle, LV_true, within a 4-pixel rim at the edge of the ventricle as described previously.³⁷ F_LV is the sum of the squared difference between every pixel in the left ventricle to LV_true:

$$F_{LV}\left(I_{C-LV}(x,y\mid \theta )\right)=\sqrt{\sum _{LV}{\left(I_C(x,y\mid \theta )-L{V}_{\text{true }}\right)}^2/A_{LV}}$$ (16)

where A_LV is the area of the left ventricle used to normalize the F_LV term.

2.E. Automatic beam hardening correction with total variation (ABHC-NH)

We implemented ABHC-NH, a seven parameter correction method in the literature for comparison.³⁹ Similar to ABHC-3, this method transforms images into three materials. The difference between ABHC-NH and ABHC-3 is the parameter estimation process. In ABHC-NH, the seven parameters are found by minimizing the total variation of the corrected image using simplex optimization.⁴¹

3. EXPERIMENTAL MET HODS

3.A. Digital phantoms and CT simulations

Digital phantoms were used along with a CT simulator to generate ground truth data for evaluation of the ABHC algorithms. The CT simulator has been previously reported^2,37 and models a CT scanner (Brilliance 64, Philips Healthcare), accounting for a number of scanner characteristics: cone beam source, finite width detector grid, x-ray spectrum generated by the source, x-ray pre-filtration, and x-ray spectral sensitivity of the detector. Digital phantoms are generated by the combination of 3D geometric objects, for example, ellipsoids. Line integrals are computed analytically based on the known object geometries as well as known mass-attenuation spectra from NIST.⁴² Realistic CT noise is added using Poisson noise determined by the predefined x-ray tube current and computed attenuation values. Two digital phantoms were simulated for evaluation of ABHC: a water cylinder phantom with four simulated test tubes containing different iodine concentrations, pelvic bone, and compact bone (shown later), and an anthropomorphic cardiac phantom (Fig. 1). The anthropomorphic cardiac phantom includes major structures observed in cardiac CT imaging. The cardiac phantom was used to generate both static and dynamic perfusion scans. In the dynamic perfusion scans, a series of images were acquired by dynamically changing iodine concentrations in the left and right ventricle blood pools, left and right ventricle myocardium, and descending aorta. The dynamic changes in iodine concentrations were determined from a physiologic perfusion simulator as described in previous work.⁴³

Fig. 1.

Simulated porcine phantom. (a) A porcine animal scanned on a prototype Philips SDCT scanner using a cardiac perfusion protocol (virtual 70 keV image). (b) Simulated cardiac porcine phantom, reconstructed using FBP. Major structures which affect BH artifacts are included in the simulation, such as the ribs, spine, ventricle blood pools, and descending aorta. The blood pools and myocardium enhance with iodine concentrations as determined by a physiologic perfusion simulator. W = 60/L = 360.

3.B. CT imaging

Physical phantom and in vivo images were acquired using a prototype version of the IQon spectral detector CT scanner (SDCT, Philips Healthcare). The energy-sensitive SDCT scanner enables reconstruction of both conventional, polychromatic images and virtual mono-energetic images from the same raw projection data. Virtual mono-energetic images are reconstructed from a projection-based material decomposition and have greatly suppressed BH artifacts. Physical phantom data were acquired with 120 kVp tube voltage, 100 mAs tube current, 0.27 s rotation speed, 360 degrees (full-scan) acquisition, and were reconstructed with filtered back projection at a 120 mm field-of-view, 512 × 512 matrix size, 4 cm longitudinal coverage, 2 mm contiguous slice thickness, with conventional (120 kVp) and virtual mono-energetic (70 keV) reconstructions. In vivo dynamic perfusion images were acquired using a protocol with 40 ECG-gated image frames acquired at 45% R-R cycle (end systole) with other settings being the same and images reconstructed with both conventional 120 kVp and mono-energetic 70 keV reconstructions. In both phantom and in vivo, 70 keV images were used as the ground truth.

3.C. Preclinical in vivo imaging study

An in vivo, porcine model of coronary artery stenosis was used to obtain dynamic CT myocardial perfusion imaging scans for evaluation of ABHC. This in vivo model has been reported previously for the evaluation and development of CT-MPI processing.^2,27 Female Yorkshire pigs, aged 13–15 weeks and at weight 40–50 kg, were used. An angioplasty balloon was inserted under x-ray fluoroscopic guidance into the left anterior descending coronary artery. An invasive pressure-wire⁴⁴ was placed distal to the angioplasty balloon for real-time monitoring of fractional flow reserve (FFR). The angioplasty balloon was deflated to allow healthy baseline homogeneous perfusion (FFR = 1). The angioplasty balloon was inflated to induce left ventricle myocardium with target FFR < 0.7. Animal experiments were performed under an approved IACUC protocol.

3.D. Blood flow estimation

A previously developed robust physiologic model (RPM) was used to estimate myocardial blood flow (MBF) maps from dynamic perfusion images. RPM was recently developed and demonstrated to provide stable MBF estimates over a range of clinically relevant imaging conditions and hemodynamic states.²¹ RPM is a reduced form of the Johnson-Wilson tissue homogeneity model which uses prior knowledge of contrast agent extraction and tissue physiology to constrain the extraction fraction and intravascular transit time parameters. Three free parameters are in RPM; the time delay, MBF, and decay constant. Parameter estimation was done using a gradient-based optimization algorithm with a semi-analytic implementation of convolution, a method which has been reported to provide rapid and reliable convergence to globally optimal parameters.²¹

MBF was computed from dynamic perfusion images after registration, segmentation, and noise reduction, as described previously.^2,21,37 Images were registered using a nonrigid registration algorithm with a normalized mutual information similarity metric.⁴⁵ The myocardium was segmented using a semi-automated approach implemented in a research version of the Medis QM ass software (www.medis.nl/). Image noise, which has been shown to bias model-based MBF methods,²¹ was reduced by averaging over 5 × 5 voxel kernels within the myocardial mask. Myocardial voxels outside the mask region were not used for noise reduction. Averaging only within the myocardium mask avoids blurring with the lungs and blood pools while still obtaining smoother per fusion curves for MBF estimation. After noise reduction, MBF was estimated.

3.E. Simulation experiments

Simulations using the digital water phantom and digital cardiac phantom were used to evaluate the ABHC algorithm at conventional, polychromatic imaging as compared to mono-energetic imaging. For both phantoms, BH artifacts presented data were generated using an x-ray tube voltage of 120 kVp for the digital water cylinder phantom and the digital cardiac phantom. BH artifacts presented data included a projection-based correction for water beam hardening. For both phantoms, the ground truth, BH artifacts-free data were generated using a mono-energetic x-ray source of 70 keV. All images were reconstructed using filtered back projection. In the water phantom, BH artifacts were evaluated without correction of filtered back projection images (FBP), and with correction by ABHC-NH, ABHC-1, ABHC-2 and ABHC-3, and compared to ground truth images. In the cardiac phantom, dynamic perfusion data were simulated with homogeneous myocardial blood flow of 100 ml·min⁻¹·100 g⁻¹. From the dynamic perfusion images, MBF was computed using the RPM method. The effect of beam hardening on MBF was determined on FBP images without correction and with the four corrections, and compared to simulated MBF.

3.F. In vivo experiment

The ABHC algorithms were evaluated in conventional 120 kVp images acquired with the in vivo porcine model of coronary stenosis and compared to the ground truth 70 keV images with respect to BH artifacts. In order to evaluate the ability for ABHC to reduce BH artifacts, dynamic perfusion images were captured with FFR = 1.0. In order to evaluate the ability for ABHC reduce BH artifacts effect on MBF maps, MBF values were evaluated in MBF maps produced from uncorrected and corrected conventional images, and compared to ground truth, as obtained from 70 keV virtual images.

3.G. Beam hardening artifact assessments

We quantitatively assessed beam hardening in single images and in perfusion flow maps. As known values are available in single image and blood flow simulations, we evaluated bias (difference between estimated and actual values) and precision (standard deviation of HU value or myocardial blood flow value) within the myocardium or within a ROI. In the case of pig data, we evaluated results against the fact that we expected uniform myocardial blood flow in a healthy pig.

4. RESULTS

In Figs. 2–3, we compare uncorrected images (FBP) and corrected images (ABHC-NH, ABHC-1, ABHC-2, and ABHC-3) from static, simulated CT phantoms. In Fig. 2, the four correction algorithms clearly reduce tell-tale streaking in noise-free images. Quantitatively, ABHC-2 and ABHC-3 give the best results, with a slight edge to ABHC-2 [Fig. 2(d)]. With ABHC-2, errors are typically < 1.8 HU. In Fig. 3, similar reductions in BH artifacts are seen in noisy images simulated at 100 mAs, indicating robustness of the ABHC algorithms to noise. Taken together, the quality of static phantom results ordered ABHC-2 > ABHC-3 > ABHC-1 >> ABHC-NH.

Fig. 2.

Simulated phantom with four iodine inserts, reconstructed with FBP (a), corrected with ABHC-NH (b), corrected with ABHC-1 (c), corrected with ABHC-2 (d), and corrected with ABHC-3 (e). Inserts are iodine 15 mg/mL, compact bone, iodine 13 mg/mL, and male pelvic bone from 2 o’clock to 10 o’clock. Visually, ABHC-2 gives the best result. Average measured HU value in different ROIs (d) are plotted, along with a red horizontal line represents the phantom’s background HU value without artifacts as measured from a mono-energetic image. ABHC-2 provides improved correction (within 1.8 HU of the ideal result) as compared to ABHC-NH, ABHC-1 and ABHC-3 (within 7.8, 3,1, 2.1 HU of the ideal result, respectively). Details: 120 kVp, W = 200, L = 20.

Fig. 3.

Simulated phantom with Poisson noise (140 kVp, 100 mAs, reconstructed with FBP (a), corrected with: ABHC-NH (b), ABHC-1 (c), ABHC-2 (d), and ABHC-3 (e). As in the noiseless case (Fig. 2), ABHC-1 and ABHC-3 significantly reduces BH artifacts, but was outperformed by ABHC-2. In these noisy simulations, mean values within ROIs are similar to the noise-free case in Fig 2. W = 200 L = 20.

In Figs. 4–7, we tested the robustness of our beam hardening correction approaches on myocardial blood flow under different simulated scan conditions designed to test robustness of corrections. Independent variables include kVp to affect beam hardening and signal, and mAs to affect noise in images. Figures 4 and 5 compare FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3 in simulated MPI-CT studies having a uniform myocardial blood flow 100 ml · min⁻¹·100g⁻¹) at two mAs values at the standard 120 kVp. Parameters in the digital phantom were chosen to accentuate beam hardening by creating more beam hardening than was found experimentally in animal experiments shown later. At 120 kVp and 200 mAs, BH artifacts in the FBP images cause significant inhomogeneity and under/over estimation of blood flow with an average of 122 ± 24 ml · min⁻¹ · 100g⁻¹. MBF is much more homogeneous after using ABHC-1, ABHC-2 and ABHC-3, giving average flows of 104 ± 14, 100 ± 12, and 108 ± 18 ml · min⁻¹ · 100g⁻¹, respectively. Both bias (difference between the estimated mean and actual) and imprecision (standard deviation) are significantly improved. ABHC-NH is even more biased than FBP, giving 135 ± 24 ml · min⁻¹ · 100g⁻¹. In Fig. 5, we compare flows in different ROIs along the myocardium against the actual value (red line) at both dose levels. At 200 mAs, flow ratios between lowest to highest flow ROIs for FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3 were 0.57, 0.68, 0.83, 0.84, and 0.72 and standard deviations across ROI’s were 24, 25, 15, 13 and 18 ml · min⁻¹ · 100g ⁻¹, respectively.

Fig. 4.

Blood flow estimated from digital phantom with simulated homogeneous blood flow at different mAs. The first row shows simulation with 200 mAs. The second row shows simulation with 100 mAs. From the first column to the fourth column are: FBP without correction, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3, respectively. Simulated blood flow is 100ml · min⁻¹ · 100g ⁻¹. Blood flow homogeneity is improved with ABHC-1 and ABHC-3 and even more with ABHC-2 estimation in both 200 and 100 mAs. In the condition of 120 kVp and 200 mAs, average flows over the myocardium are 122 ± 24, 135 ± 24, 104 ± 14, 100 ± 12, and 108 ± 18 ml · min⁻¹ · 100g ⁻¹ for FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3 respectively. In condition of 120kVp and 100 mAs, average flows over the myocardium are 122 ± 30, 137 ± 31, 104 ± 19, 101 ± 10, and 93 ± 18 ml ·min⁻¹ · 100g ⁻¹ for FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3 respectively. Nominal condition (120 kVp) was used.

Fig. 7.

Effect of scan conditions on beam hardening corrections. At 120 kVp, conditions include nominal dose (200 mAs) as well as increased noise at reduced mAs values, and noiseless. Also shown is the result from a 200 mAs, beam hardening-free 70 keV acquisition. Plotted are average and standard deviation of blood flow across the entire simulated myocardium. In noise-free condition, ABHC-3 outperforms other methods. In other conditions, ABHC-2 gives improved estimation as compared to all other methods and FPB without correction. The horizontal red line shows actual simulated MBF.

Fig. 5.

Blood flows in ROI’s within the myocardium. ROIs are defined on the left and results are plotted on the right for FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3. The horizontal red line represents the true simulated flow. At 200 mAs, the standard deviation across ROI’s was 24, 25, 15, 13 and 18 ml · min⁻¹ · 100g⁻¹, respectively. At 100 mAs, the standard deviation across ROIs was 30, 31, 18, 17 and 19 ml · min⁻¹ · 100g⁻¹, respectively. At 200 mAs, blood flow ratios between lowest to highest flow was 0.57, 0.68, 0.83, 0.84, and 0.72, respectively. At 100 mAs, blood flow ratios between lowest to highest flow were 0.55, 0.65, 0.8, 0.81, and 0.71, respectively. Data were obtained from the simulation experiment in Fig. 4.

In Fig. 6, we compared ABHC algorithms at two kVp values and at standard 200 mAs. All ABHC’s improved blood flow assessment with ABHC-2 producing better results. In Fig. 7, ABHC-2 gives a mean value very close to the actual value (red horizontal line) at all mAs values. FBP without correction clearly leads to overestimation of mean flow. To test whether ABHC methods over-compensate when beam hardening is not present, we simulated a 70 keV acquisition with a mono-energetic x-ray source. Both ABHC-1 and ABHC-2 gave unbiased results. ABHC-2, even with its increased number of correction parameters as compared to ABHC-1, produced robust results from noisy, low dose scans. ABHC-3 corrected phantom flow values slightly better than ABHC-2, when no noise was added, suggesting that reduced quality in the presence of noise was due to an estimation problem. The mean absolute error of mean flow value to the simulated flow across all conditions are 18.8, 25.2, 4, 1.6, and 8 ml · min⁻¹ · 100g⁻¹ for FBP, ABHC-NH, ABHC-1, ABHC-2, and ABHC-3, respectively.

Fig. 6.

Blood flow estimation from simulated flow of 100ml · min⁻¹ · 100g⁻¹ scanned at different kVps. Top row was scanned at 140 kVp, and bottom row at 80 kVp. First column was reconstructed using FBP. From the second to the fifth columns are FBP corrected with ABHC-NH, ABHC-1, 2 and 3 respectively. ABHC-1, 2 and 3 improves blood flow estimation in both 80 and 140 kVp. ABHC-2 outperforms all methods in both cases. Nominal condition (200 mAs) was used.

We performed experiments on the estimability of ABHC-3. We obtained ABHC-3 parameters from noise-free images and applied the same correction parameters to 200 mAs data. The average flow over the myocardium improved from 108 ± 18 to 102 ± 13 ml · min⁻¹ · 100g⁻¹, indicating that we did not achieve the best parameters in the relatively low noise 200 mAs data. Using ABHC-3, we initiated parameter optimization from many different star ting values and did not obtain significantly improved results. It appears that even with images of nominal noise, ABHC-3 has limited estimability.

To ensure that beam hardening correction does not remove flow deficit, we performed simulations with known flow deficits. For example, Fig. 8 shows example MBF maps from the digital phantom with a simulated flow deficit (flow ratio 0.6), clearly seen in the 70 keV case. Gross effects are seen on the map with FBP. ABHC-2 compares favorably to the 70 keV gold standard result. ABHC-NH, ABHC-1 and ABHC-3 show poorer corrections. Flow ratios computed over the entire regions and reported in the figure legend improved with corrections. Importantly, none of the ABHC corrections visually or quantitatively removed the flow deficit. In all simulations and experiments, we never found a situation where our correction method removed a flow deficit (see Discussion).

Fig. 8.

Digital phantom with flow deficit. There was a simulated flow deficit from 9 o’clock to 3 o’clock with flow 0.6 times the reference value. Shown are MBF maps generated from FBP (a), corrected with ABHC-NH (b), corrected with ABHC-1 (c), corrected with ABHC-2 (d), corrected with ABHC-3 (e), and 70 keV (f). ABHC-2 corrected the MBF map to closely match the 70 keV result, with ABHC-NH, ABHC-1 and ABHC-3 giving somewhat poorer corrections. As measured over entire regions, flow ratios, standard deviation, and standard error were (0.59, 0.45, 0.011), (0.60, 0.35, 0.008), (0.63, 0.29, 0.007), (0.62, 0.11, 0.002), (0.63, 0.13, 0.003), and (0.64, 0.09, 0.002) for FBP, ABHC-NH, ABHC-1, ABHC-2, ABHC-3, and 70 keV respectively. Nominal imaging conditions (i.e., 120 kVp and 200 mAs) with actual flow values (60 and 100 ml · min⁻¹ · 100g⁻¹) were used.

Figure 9 shows example MBF maps of healthy pig without a simulated stenosis [fractional flow reserve (FFR) = 1] generated from FBP, ABHC-NH, ABHC-1, ABHC-2, ABHC-3, and 70 keV scans. MBF maps are more homogeneous after correction with the ABHC algorithms. ABHC-2 compares somewhat more favorably than ABHC-1 and ABHC-3 to the gold standard at 70 keV. We examined relative flows around the myocardium by computing flow ratios, as given in the legend. Correction of FBP with ABHC-2 improved the flow ratio from 0.78 to 0.85, very close to the 0.86 obtained with 70 keV images. In addition, ABHC-2 corrected individual images to compare favorably with corresponding 70 keV images (not shown). We observe that beam hardening correction can reduce artifactual flow deficits which might confound clinical decision making.

Fig. 9.

Healthy (FFR = 1) pig MBF maps generated from FBP (a), corrected with ABHC-NH (b), corrected with ABHC-1 (c), corrected with ABHC-2 (d), corrected with ABHC-3 (e), and 70 keV (f). Flow ratios between average flow in an ROI located at 12 o’clock and 3 o’clock are 0.78, 0.75, 0.82, 0.85, 0.81 and 0.86 for FBP, ABHC-NH, ABHC-1, ABHC-2, ABHC-3 and 70 keV respectively. W = 360/ L = 60. ABHC-2 shows the closest flow ratio to 70 keV.

5. DISCUSSION

We have developed methods for calibration-free, automatic beam hardening correction on DICOM images from a conventional CT scanner, for the demanding application of cardiac CT perfusion. In this report, we proposed two new polynomial correction methods, ABHC-2 and ABHC-3, which consider images to be comprised of two (high and low attenuating) and three (water, iodine, and bone) materials, respectively. Corrections require optimization of three and seven parameters, respectively. We compared results to existing methods: ABHC-1, as reported by us³⁷ and ABHC-NH, as reported by Nett and Hsieh.³⁹ In general, we determined that the quality of corrections were ordered ABHC-2 > ABHC-3 > ABH C-1 >> ABHC-NH for static phantom HU measurements, dynamic phantom blood flow measurements, and porcine experiments. Results with ABHC-2 were robust across different noise (dose) levels and kVp values. Using ABHC-2 will enable accurate estimation of MBF using conventional CT scanner designs without the need for energy-sensitive CT scanners or specialized dual-energy data acquisition.

We now analyze more fully the correction algorithms. The additional free parameters in ABHC-NH, ABHC-2, and ABHC-3 as compared to our previous algorithm, ABHC-1, could reduce estimability, limiting the ability to obtain robust beam hardening correction. In general, adding more parameters produces a more complex cost function, potentially adding local minima, a phenomenon typically made more prominent with added noise. This has been shown by us for the case of model-based, MBF estimation from CT images.²¹ When we tested the ABHC methods at different mAs values, both ABH C-1 and ABHC-2 improved bias and precision over the FBP images at all of the tested dose levels (Figs. 7–9). Despite the potential for ABHC-2 to have poorer estimability under high noise conditions, we found that it outperformed ABHC-1 in all cases. However, the seven parameters methods, ABHC-NH and ABHC-3, showed poorer bias and precision compared to FBP with noisy images (Fig. 7). To reduce the effect of local minima, we evaluated different parameter initializations on ABHC-3, but results did not significantly improve. Potentially improved optimization methods, including computationally intensive methods for obtaining global optimization, would improve results, but this was beyond the scope of this report. Robustness to noise is important, as dose in dynamic CT perfusion is a concern and ABHC processing would be necessary prior to low dose MBF quantification methods such as super-voxel clustering in the “SLICR” method recently reported by us.²² At a voxel, the blood flow estimation process requires two inputs: the time-attenuation curve at the voxel and the a rterial input function. Both can be affected by BH artifact. We used an arterial input function as determined in the left ventricle. In our ABHC formulation, we included a term which acts to reduce the cupping artifact in the left ventricular cavity and aorta. ABHC-NH uses total variation averaged across the entire image rather than the cost function optimized for CT perfusion in our ABHC methods. We observed that the ABHC methods tended to restore HU values within the left ventricle and aorta better than ABHC-NH. Hence, this is likely the reason that our ABHC methods outperformed ABHC-NH in all blood flow cases tested. Importantly, ABHC algorithms are “safe.” We never observed inadvertent reduction/removal of an actual flow deficit in a large number of simulated and pig experiments. This is expected because the ABHC correction algorithms are based on physical principles using projections, rather than direct application to images. This assertion is reinforced in Fig. 7 where application of ABHC-1 and ABHC-2 to images devoid of beam hardening (70 keV images) did not degrade results, indicating that the algorithm correctly “senses and corrects” only actual beam hardening.

ABHC-2 could potentially improve clinical decisions. Currently, a 20% reduction in flow as determined by fractional flow reserve of less than 0.8 is considered hemodynamically significant and war rants intervention.⁴⁶ Using a similar threshold for flow ratio, we found instances where a false positive without correction was removed with ABHC-2 (Figs. 4, 5). In addition, we found that ABHC-2 may further reduce potential for false positives as compared to ABHC-1 (Fig. 9). It should be noted that typically physicians are interested in large 3D regions corresponding to coronary artery territories, so small regional variations in flow that are sometimes found will likely not confound the diagnosis of a territory having a flow deficit.

Experiments showed improved flow estimates at low 80 kVp as compared to high 140 kVp, both before and after beam hardening correction (Fig. 6). This observation goes against the conventional wisdom that beam hardening effects will be reduced at higher kVp values. When individual images prior to correction are examined, beam hardening is more evident in images at low kVp than at high kVp. However, because the iodine signal in the myocardium is greater at the lower kVp, the percent change of beam hardening is smaller at low kVp than at high kVp. Hence, perfusion estimates are improved. ABHC-2 improves flow estimates at both kVp values.

In conclusion, results with ABHC-2 indicate that accurate estimation of MBF is possible using DICOM images from conventional CT scanner designs. This could extend the utility of conventional CT scanners for the demanding application of myocardial perfusion.