Patient-Specific Whole-Body Attenuation Correction Maps from a CT System for Conjugate-View-Based Activity Quantification: Method Development and Evaluation

Image data

Patient images are obtained from three different clinical radionuclide therapy studies involving patients on ¹⁷⁷Lu-DOTA-octreotate with disseminated neuroendocrine tumors ⁸ and ¹¹¹In-labeled rituximab ⁹ and ¹¹¹In-labeled Zevalin™ in patients with non-Hodgkins B-cell lymphoma. ¹⁰ Whole-body X-ray scout images for 31 different patients are included in the evaluation, where some of the patients have been imaged and weighed at several occasions with intervals of between 1 week and 2 months, so that altogether 79 scout images were obtained. For 10 of the patients, both scout images and CT studies have been acquired at the same imaging occasion, and for some patients, this procedure has been repeated up to four times. Included patients are 14 males and 17 females, where males have an average weight of 77 kg (range 62–93 kg) and a body–mass index (BMI) of 23.6 kg/m² (range 18.9–26.0 kg/m²), whereas females have an average weight of 66.2 kg (range 44–90 kg) and a BMI of 20.7 kg/m² (range 15.9–30.0 kg/m²). The patients included are thus a representative population, with a broad range of body morphometries.

Images are acquired using a Discovery VH SPECT/CT system (General Electric, Milwaukee, WI), including two scintillation camera heads and a single-slice CT unit (HawkEye™). Whole-body scintillation camera images are acquired in a 256×1024 matrix size with a pixel size of 2.21×2.21 mm². Scout acquisitions are performed using a tube voltage of 140 kV, a tube current of 2.5 mA, and 0.5-mm Copper filtering, with a scanning time of ∼2 minutes for a whole-body scan. The acquired matrix has a size of 256×2000 pixels, and a pixel dimension in the vertical direction of 1.0 mm. In the lateral direction, the fan-beam acquisition geometry produces zooming by a factor that depends on the object distance from the X-ray tube, so that the pixel dimensions in the acquired matrix are not equal. In a previous work, ⁷ the lateral pixel dimension was experimentally determined for a distance that corresponds to the central depth of the patient, and was obtained to 1.163 mm. To obtain the same image dimensions as for the whole-body scintillation camera images, the scout image is linearly interpolated into an image matrix with a size of 202×904, giving pixel dimensions of 2.21×2.21 mm². The resulting image matrix is then padded with zeros, so that the final matrix dimensions become 256×1024. CT images are acquired using the same tube voltage and intensity as for the scout acquisition, with a rotation speed of 2.6 rpm. The time required is ∼10 minutes, yielding transaxial section images with an initial thickness of 1.0 cm, which are then down-sampled by the SPECT/CT system to different image dimensions, depending on the specifications given for the clinical SPECT studies that are acquired in parallel. The dimensions of the CT images are thus different for the different clinical studies from which data are obtained, and are either in a matrix size of 256×256×128 with voxel dimensions of 2.0×2.0×4.0 mm³, or in a 128×128×64 matrix with voxel dimensions of either 3.3×3.3×6.6 mm³ or 4.0×4.0×8.0 mm³. All image processing is performed offline by image export using the DICOM format and processing within LundADose software developed in IDL. ¹¹

In the further text and equations, bold symbols denote matrices. Image coordinates are denoted i, j, and l, where i runs along the patient's left–right direction; j runs along the head–foot direction; and l runs along the anterior–posterior direction. In equations where i, j, and l are not explicitly specified, all image coordinates are supposed to be included in the calculations.

Attenuation correction in the conjugate view method

The basic formalism underlying the conjugate view method can be found elsewhere. ^{1 –3} We have previously presented a pixel-based variant of this method that has been applied for quantification from whole-body scintillation camera images. ^{8,12 –14} Here, an activity image, A (MBq) is estimated using the expression: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\textbf{\textit{A}} ( i , j ) = \varepsilon^{- 1} \cdot \textbf{\textit{AC}}_{Em} ( i , j ) \cdot \left[ Sc \left( \textbf{\textit{C}}_{Em}^{ant} ( i , j ) \right) \cdot Sc \left( \textbf{\textit{C}}_{Em}^{post} ( i , j ) \right) \right] ^{1 / 2} \tag{1}\end{align*}\end{document}

where ɛ is the scintillation camera system sensitivity (cpsMBq⁻¹) measured in air, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{C}}_{Em}^{ant}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{C}}_{Em}^{post}$$\end{document} (cps) are anterior- and posterior-view count-rate images acquired for the photon energy, E_m, of the imaging radionuclide. The function Sc denotes scatter correction of the count-rate images, which in our implementation is performed using deconvolution by scatter-point spread functions determined from Monte Carlo simulation. ¹⁵ The image AC _Em is the attenuation correction map valid for E_m , as described in detail below. Before application of Equation 1, the image AC _Em is convolved with a Gaussian function with a similar spatial resolution as that of the emission images. Furthermore, the images \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{C}}_{Em}^{ant}$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{C}}_{Em}^{post}$$\end{document} are spatially registered to AC _Em (Fig. 1E) by a method tailored for whole-body images. ¹⁶ For activity quantification of organs, the activity image, A , is further processed by drawing ROIs, correcting for background and overlapping activities, and application of a source thickness correction calculated for each ROI. ¹² A perhaps more common way of implementing the conjugate view method is to delineate organ ROIs directly on the anterior–posterior count-rate images, and then perform the geometric mean calculation on the total count rate determined from each ROI. ³ Attenuation correction is then performed by multiplying this geometric mean value by an attenuation correction factor (ACF _Em ). The ACF _Em is typically determined by delineating organ ROIs in the attenuation correction map and determining the ROI mean value, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm ACF}_{Em} = \overline{\textbf{\textit{AC}}_{Em}} ( i , j \in ROI )$$\end{document} . In either of the implementation schemes, region- as well as pixel-based, an attenuation correction map is required.

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FIG. 1.

(A–C) Scout-based attenuation correction map of one male patient, with (A) the patient contour used for calculation of the weight-based calibration factor, (B) the border of the regions used for the different fractions of bony and soft tissue in the weighting image W, (C) the ROIs used for evaluation of the ACF in different organs. Image (D) shows the CT-based attenuation correction map that is spatially registered to the scout, with organ ROIs; (E) shows the activity image A (Equation 1) overlaid on the scout attenuation correction map, and (F) shows the activity image with organ ROIs typically used for activity quantification. ACF, attenuation correction factor. Color images available online at www.liebertpub.com/cbr

A general model expression of an attenuation correction map valid for an arbitrary photon energy E can be described according to the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\textbf{\textit{AC}}_E ( i , j ) = \left[ exp ( \textbf{\textit{R}}_E ( i , j ) ) \right] ^{1 / 2} \tag{2}\end{align*}\end{document}

with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\textbf{\textit{R}}_E ( i , j ) = \int \limits_0^{\textbf{\textit{L}} ( i , j )} {\bold \mu}_E ( i , j , l ) {\rm d}l = \int \limits_0^{\textbf{\textit{L}} (i , j)} ({\bold \mu} / {\bold \rho} ) _E ( i , j , l ) \cdot {\bold \rho} ( i , j , l) {\rm d}l \tag{3}\end{align*}\end{document}

In this equation, l is the continuous coordinate along the anterior–posterior projection line, and L (i, j) is the physical patient thickness (cm) at each image position (i, j). The 3D distributions μ_E and (μ/ρ)_E describe the linear attenuation coefficient (cm⁻¹), and the mass attenuation coefficient (cm²g⁻¹), respectively, at different positions in the patient for the photon energy E. The 3D distribution ρ is the mass density (g cm³) and R _E is a unitless 2D matrix describing the line integral of the various attenuation coefficients that the photons encounter along their passage through the patient.

The attenuation correction maps, or ACFs, are commonly measured using transmission imaging with and without the patient in position. ³ For the X-ray scout, a single value representative of the count rate in air, that is, without the patient in position, is obtained from the mandatory CT calibration procedure. The acquired scout image obtained from the X-ray system is a version of the image R _E in Equation 3, with E equal to the transmission photon energy, E_t , that is, R_Et . For the scout image, E_t is really the photon energy spectrum of the X-ray system, which, for our system, was previously found to be well represented by an energy of 70 keV. ⁷ However, the scout image is in relative values that need to be rescaled to obtain the quantitative image R _Et . Denoting the raw acquired scout image by S₀ , the values in S₀ range between the camera-specific values s_min and s_max (for our system equal to −1000 and 3096, respectively). Following our previous work, ⁷ an estimate of the image R _Et , that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat {\textbf{\textit{R}}}_{Et}$$\end{document} is determined from S₀ by renormalization to the image S and then scaling by an acquisition-specific calibration factor, c (unitless), according to the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\hat {\textbf{\textit{R}}}_{Et} ( i , j ) = c \cdot( \textbf{\textit{S}}_0 ( i , j ) - s_{min} ) / ( s_{max} - s_{min} )= c \cdot \textbf{\textit{S}} ( {i , j} ) \tag{4}\end{align*}\end{document}

where the minimum value in S is thus zero, and (s_max –s_min ) equals 4096 for our system. The factor c thus constitutes the link between the relative image S and the quantitative image. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat {\textbf{\textit{R}}}_{Et}$$\end{document} . In our previous work, the factor c was obtained from a system file in the SPECT/CT scanner. ⁷ For situations when this factor is not available from the system, an alternative method of obtaining the factor c is derived.

Determination of the weight-based calibration factor

As seen from Equation 3, the physical factors that govern the distribution R _E (or R _Et ) are the distributions of the mass attenuation coefficient and the mass density. The difference between the mass attenuation coefficients of various soft tissues is small, since the dominant photon interaction process is by Compton scattering. For instance, for 70 keV photons, the mass attenuation coefficient equals 0.194 g cm² for both lung and soft tissue, ^17,18 and the difference in the linear attenuation coefficient is governed by the difference in mass densities. For bony tissues, the linear attenuation coefficient is increased due to a higher probability for photoelectric absorption. However, there is a marked difference between spongiosa and cortical bone, and for energies above 70 keV, the value of the linear attenuation coefficient of spongiosa is closer to that of soft tissue than of cortical bone. ^{17 –19} Although the amount of cortical bone is in many regions small, ²⁰ it affects the image R _Et due to the relatively high value of the linear attenuation coefficient. However, the amount of cortical bone along the projection line is not available from the planar scout image, and to arrive at a quantitative estimate of R _Et , approximations are required, as derived in Appendix 1. Following these derivations, with approximations illustrated in Figure 2, the expression for the weight-based calibration factor, c_W , is obtained as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split}c_W & = \hat{m}_{TB} ( \Delta i \Delta j ) ^{- 1} \left\{ \mathop\sum_{i , j \in pat} \textbf{\textit{S}} ( i , j ) \cdot [ \rho^s \textbf{\textit{W}} ( i , j ) + \rho^b ( 1 - \textbf{\textit{W}} (i , j))] \cdot [ \mu_{Et}^s \textbf{\textit{W}} (i , j) + \mu_{Et}^b ( 1 - \textbf{\textit{W}}(i , j))] ^{-1} \right\}^{-1}\end{split} \tag{{\rm A}8} \end{align*}\end{document}

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FIG. 2.

Illustration of the approximations made in the derived equations. The physical thickness of the patient at a location corresponding to the pixel coordinates (i,j) is denoted L (i,j). It is constituted of different amounts of cortical bone, spongiosa, various soft tissues, and lung. The thickness seen by the x-rays is a “radiological thickness” denoted R _E (Equation 3). The underlying idea is that R _E can be represented by a sum of two compartments, a soft tissue-equivalent compartment and a compartment of cortical bone, where the respective contributions are given by the product of the linear attenuation coefficient and the equivalent thicknesses (Equation A5). The soft tissue- and bone-equivalent thickness \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$L^{sb}_{eq}$$\end{document} is the sum of the equivalent thickness of cortical bone, (1− W ) \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$L^{sb}_{eq}$$\end{document} , and the soft tissue-equivalent thickness, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$L^{s}_{eq} = WL^{sb}_{eq}$$\end{document} (Equations Al and A2). The equivalent thickness of cortical bone is assumed to be equal to the physical thickness, L ^b , whereas the soft tissue-equivalent thickness \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$L^{s}_{eq}$$\end{document} represents contributions from spongiosa, soft tissues, and lung (Equation Al).

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{m}_{TB}$$\end{document} (g) is the patient weight as measured from a scale; ΔiΔj is the pixel area (cm²); ρ^s and ρ^b are the mass densities (g cm³) of soft tissue and cortical bone, respectively, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Et}^s$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Et}^b$$\end{document} (cm⁻¹) are the linear attenuation coefficients for soft tissue and cortical bone for the transmission photon energy, respectively. ^17,18 The summation is performed over pixels with coordinates (i and j) that represent the patient in the image. The distribution W is a matrix of weighing factors with a maximum value of one, describing the position-dependent fraction of the soft tissue- and bone-equivalent thickness distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$L_{eq}^{sb}$$\end{document} that is represented by soft tissue (see Appendix 1).

In Equation A8, the image S is determined from the measured scout image and Equation 4, and the pixel area is determined by standard camera calibration procedures. The pixel coordinates (i and j) belonging to the area occupied by the patient are determined by applying a segmentation procedure in the image S . The patient contour is delineated by applying the automatic threshold method by Otsu, ^21,22 so that the patient contour is precisely enclosed (Fig. 1A). For patients who have a metal hip implant, which results in a very high-intensity focus in the scout image, some more elaboration is required. Due to a higher probability for photoelectric interaction in metal, it is not well approximated by body tissue, so that Equation A8 gives a poorer approximation in such cases. Also, the high-intensity focus presents a problem for the automatic segmentation procedure, reducing its ability to estimate the threshold for the patient contour. In this work, two of the evaluated patients have a hip implant, and this has been practically handled by manually delineating a region over the hip implant in the image S and replacing the pixel values by values just beside the region. Smaller metal details such as bracelets or teeth are left unprocessed. The weighing factors in W are not available from the planar scout image, and to determine this distribution, some assumptions need to be made. In this work, the values in W are set for different regions that are manually marked on the image S , as displayed in Figure 1B. The weighing factors have been previously determined by comparison of the values in a projected whole-body CT with the corresponding scout image values for one male patient. The values in W were thus determined to 0.9, 1.0, and 0.9 for the head, the upper torso, and the pelvis–legs, respectively. ¹⁴ For investigation of the sensitivity to the inclusion (or exclusion) of the partitioning distribution W in Equation A8, an expression where W (i,j) is set to unity everywhere is also given as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$c{\prime}_W$$\end{document} according to Equation A9 (Appendix 1).

Application of the calibration factor to obtain the attenuation correction map

By use of Equations 4 and 2, a scout-based attenuation correction map valid for the transmission imaging energy, AC _Et , may be obtained. In Equation 4, the factor c can thus either be the scanner system-based calibration factor, further denoted c_sys , or the weight-based calibration factor, c_W . To obtain an attenuation correction map for the emission imaging energy, AC _Em , to be applied in Equation 1, an energy scaling is required from E_t to the energy of the imaging radionuclide, E_m . This is performed by use of Equations 2, A5, and A6, so that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split} & \textbf{\textit{AC}}_{Em} (i , j) = \left[exp (\hat {\textbf{\textit{R}}}_{Em} (i , j)) \right]^{1/2} \\& = \left(exp \left[\left( \frac {{\mu_{Em}^s} \textbf{\textit{W}}(i , j) + {\mu_{Em}^b} (1-\textbf{\textit{W}}(i , j))} {\mu_{Et}^s \textbf{\textit{W}} (i , j) + \mu_{Et}^b (1- \textbf{\textit{W}} (i , j))} \right) \cdot c_x \cdot{\textbf{\textit{S}} (i , j)} \right] \right)^{1/2}\end{split} \tag{5}\end{align*}\end{document}

where c_x temporarily denotes either c_sys or c_W . The factors \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Em}^s$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Em}^b$$\end{document} are the linear attenuation coefficients for soft tissue and bone, respectively, for the energy of the emission imaging radionuclide. ¹⁸ It should be noted that since linear attenuation coefficients are applied in this equation, it is assumed that scatter correction is performed before attenuation correction, following Equation 1. When explicit scatter correction of count-rate images is not performed, a commonly used method is to apply an effective attenuation coefficient in the attenuation correction. ^3,5,6 The use of such a scheme can be incorporated in Equation 5 by substituting \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Em}^s$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Em}^b$$\end{document} for the effective attenuation coefficients, whose values must then be determined by separate measurements. The attenuation coefficients \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Et}^s$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Et}^b$$\end{document} in the denominator should in any method be the linear ones, since these are applied to convert the scout image, which is assumed to be acquired in a narrow-beam geometry, to a map of the soft-tissue equivalent thickness. At application of the attenuation correction in Equation 1, the values in the image AC _Em for locations outside the patient should have a value of one, as seen from Equations 2 and 3 where zero attenuation gives R _E (i, j)=0 and AC _E (i, j)=1. After calculation of AC _Em , the image values outside the patient contour are therefore explicitly set to one.

In Equations A8 and 5 the values used for the physical parameters, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\mu_{Et}^s , \mu_{Et}^b , \mu_{Em}^s , \mu_{Em}^b , \rho^s$$\end{document} and ρ^b are obtained from the National Institute of Standards and Technology (NIST). ¹⁸ For soft tissue, reference values for ICRU Four Component Soft Tissue ¹⁷ are used, giving for 70 keV photons \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( \mu / \rho ) _{70}^s$$\end{document} =0.1934 g cm² and ρ^s =1.0 g cm³. For cortical bone, values for Cortical Bone ICRU-44¹⁷ are used, giving for 70 keV \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( \mu / \rho ) _{70}^b$$\end{document} =0.269 g cm² and ρ^b =1.92 g cm³.

The practical steps required to implement the weight-based calibration method for obtaining a scout-based attenuation correction map are summarized in Appendix 2.

Evaluation

First, the underlying idea that the patient weight can be estimated from scout images was tested by application of Equation A7 to the 79 patient S images and using c_sys . For the same studies, the values of c_W were then determined using Equation A8 and compared to the values of c_sys .

The accuracy of the scout image-based maps, AC _Et and AC _Em , was determined by a comparison to independently obtained AC maps from CT measurements, which, unlike the scout image values, were quantitative. From a CT image, an estimate of the 3D map of the linear attenuation coefficients, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat {\mu}_E$$\end{document} (cm⁻¹) was determined as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\hat {\mu}_E ( i , j , l ) = ( \mu / \rho ) _{E} \cdot \hat{\bold \rho} ( i , j , l ) \tag{6}\end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{\bold \rho}$$\end{document} (g cm³) is an estimate of the mass density distribution, ρ, of the patient. The distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\hat{\bold \rho}$$\end{document} was obtained using a bilinear function, φ following \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}\hat {\rho} ( i , j , l ) = \phi ( I ( i , j , l )) = \begin{cases}a_1 + b_1 \cdot ( I ( i , j , l ) ) ; ( I ( i , j, l ) ) < H_1 \\ a_2 + b_2 \cdot ( I ( i , j , l ) ) ; ( I ( i , j, l ) ) \ge H_1\end{cases} \tag{7}\end{align*}\end{document}

where I(i, j, l) is the CT image values (Hounsfield units, HU), added by a value of 1000. The values of the parameters in this equation were determined by measurement of a standard calibration phantom, ²³ by determining the HU of eight tissue-equivalent inserts with various mass densities and fitting a bilinear function to the density-versus-HU data, as previously described in. ^12,24 The parameter values were a₁ =−1.57×10⁻² (g/cm³), b₁=1.03×10⁻³ (g/cm³/HU), a₂=2.94×10⁻¹ (g/cm³), b₂=7.31×10⁻⁴ (g/cm³/HU), and H₁=1025 (HU). The mass attenuation coefficients \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$( \mu / \rho ) _{E}$$\end{document} for soft tissue, skeleton spongiosa, and cortical bone were obtained from NIST ¹⁸ and ICRU Report 46¹⁹ for the energies of interest, E=E_t (70 keV) and E=E_m (208, 245, and 364 keV). A classification of the voxels in the mass density image was performed, where values <1.051 g cm³ were classified as soft tissue, values above 1.2 g cm³ as cortical bone, and intermediate values as spongiosa. The respective mass attenuation coefficient was then multiplied by the mass density value for the respective set of voxels. It should be noted that although a voxel classification was made for selecting the mass attenuation coefficient value, the mass densities as well as the linear attenuation coefficients were determined on a voxel-by-voxel basis. A CT-based image \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$R_E^{CT}$$\end{document} was then calculated by numerical integration along the anterior–posterior direction: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \textbf{\textit{R}}_E^{CT} (i , j) = \Delta l \cdot{\mathop\sum_l} \hat{\mu}_E (i , j , l) \tag{8} \end{align*}\end{document}

where Δl is the voxel dimension in the anterior–posterior direction (cm). An attenuation correction map, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$AC_E^{CT}$$\end{document} , was then calculated by insertion of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\textbf{\textit{R}}_E^{CT}$$\end{document} into Equation 2.

For evaluation, organ ROIs were applied to both the scout image-based and the CT-based AC maps (Fig. 1C, D). The ROIs were delineated using both the scout image and the planar whole-body scintillation camera images that had previously been spatially registered to the scout image-based AC map. ¹⁶ To enable the use of the same set of organ ROIs for the CT-based map, this was also registered to the scout-based map. To ensure that the ROIs were well within the organ boundary and thus representative of the organ attenuation value, the ROIs were made smaller than those normally used for activity quantification (Fig. 1C, F). The ACF in each ROI was determined as, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${\rm ACF} = \overline{\textbf{\textit{AC}}_E} ( i , j \in ROI )$$\end{document} , and the deviation between the values from the scout image-based and the CT-based AC maps was calculated for each patient study, n, as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split} Dev_n = &\left( \overline{\textbf{\textit{AC}}_E}( i , j \in ROI ) - \overline {\textbf{\textit{AC}}_E^{CT}} ( i , j \in ROI ) \right) / \\& \overline {\textbf{\textit{AC}}_E^{CT}}( i , j \in ROI ) \cdot 100 \ (\%)\end{split} \tag{9}\end{align*}\end{document}

for E=70, 208, 245, and 364 keV. For each organ, the bias and precision were calculated by taking the mean and standard deviation of the deviations (%), for the total N studies according to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}Bias = N^{- 1} \cdot \sum\nolimits_N Dev_n \tag{10}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}Precision = ( N - 1 ) ^{- 1} \cdot \left[ \mathop\sum_N ( Dev_n - Bias ) ^2 \right] ^{1 / 2} \tag{11}\end{align*}\end{document}