Attenuation correction in emission tomography using the emission data--A review.

Attenuation correction in emission tomography using the emission data—A review Yannick Berkera) and Yusheng Li Department of Radiology, University of Pennsylvania, 3620 Hamilton Walk, Philadelphia, Pennsylvania 19104

(Received 7 July 2015; revised 19 November 2015; accepted for publication 25 November 2015; published 14 January 2016) The problem of attenuation correction (AC) for quantitative positron emission tomography (PET) had been considered solved to a large extent after the commercial availability of devices combining PET with computed tomography (CT) in 2001; single photon emission computed tomography (SPECT) has seen a similar development. However, stimulated in particular by technical advances toward clinical systems combining PET and magnetic resonance imaging (MRI), research interest in alternative approaches for PET AC has grown substantially in the last years. In this comprehensive literature review, the authors first present theoretical results with relevance to simultaneous reconstruction of attenuation and activity. The authors then look back at the early history of this research area especially in PET; since this history is closely interwoven with that of similar approaches in SPECT, these will also be covered. We then review algorithmic advances in PET, including analytic and iterative algorithms. The analytic approaches are either based on the Helgason–Ludwig data consistency conditions of the Radon transform, or generalizations of John’s partial differential equation; with respect to iterative methods, we discuss maximum likelihood reconstruction of attenuation and activity (MLAA), the maximum likelihood attenuation correction factors (MLACF) algorithm, and their offspring. The description of methods is followed by a structured account of applications for simultaneous reconstruction techniques: this discussion covers organ-specific applications, applications specific to PET/MRI, applications using supplemental transmission information, and motion-aware applications. After briefly summarizing SPECT applications, we consider recent developments using emission data other than unscattered photons. In summary, developments using time-of-flight (TOF) PET emission data for AC have shown promising advances and open a wide range of applications. These techniques may both remedy deficiencies of purely MRI-based AC approaches in PET/MRI and improve standalone PET imaging. C 2016 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4938264] Key words: attenuation correction, emission data, reconstruction, PET, SPECT

NOMENCLATURE AC: Attenuation correction ACF: Attenuation-correction factors ADMM: Alternating direction method of multipliers AF: Attenuation factors ART: Algebraic reconstruction technique CBM: Continuous bed motion CE: Consistency equation ConTraSPECT: Consistency transmission SPECT CT: Computed tomography DCC: Data consistency condition DIRECT: Direct image reconstruction for TOF EM: Expectation maximization EX: Emission ERT: Exponential Radon transform FBP: Filtered backprojection 18 FDG: F-fludeoxyglucose FORE: Fourier rebinning FOV: Field of view FWHM: Full width at half maximum GMM: Gaussian mixture model IntraSPECT: Intrinsic transmission SPECT 807

Med. Phys. 43 (2), February 2016

JETT: LAC: LLF: LOR: LSO: MAP: ML-JRM: MP-JRM: MLAA: MLACF: MLRR: MLTA:

MLTR: MRI: NAC:

Joint Emission and Transmission Tomography Linear attenuation coefficient Log-likelihood function Line of response Lutetium orthosilicate Maximum a-posteriori Maximum likelihood joint image reconstruction/ motion estimation Minimal MRI prior joint image reconstruction/ motion estimation Maximum likelihood reconstruction of attenuation and activity Maximum likelihood attenuation correction factors Maximum likelihood reconstruction of activity and registration of attenuation Maximum likelihood reconstruction of the transformation between PET and CT images and activity Maximum likelihood gradient ascent for transmission tomography Magnetic resonance imaging Non-attenuation corrected

0094-2405/2016/43(2)/807/26/$30.00

© 2016 Am. Assoc. Phys. Med.

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Y. Berker and Y. Li: AC in PET and SPECT using emission data

OFOV: OSEM: OSEMIS: OSTR: PDE: PET: PSCD: RAMLA: SPECT: S2A: SNR: SOR: SSS: SMLGA:

Outside of field of view Ordered subsets expectation maximization Ordered subsets EM-IntraSPECT Ordered subsets transmission (Ref. 274) Partial differential equation Positron emission tomography Paraboloidal surrogates coordinate descent Row-action maximum likelihood algorithm Single photon emission computed tomography Scatter to attenuation Signal-to-noise ratio Surface of response Single scatter simulation Scatter based maximum likelihood gradient ascent TX: Transmission TOF: Time of flight UTE: Ultrashort echo time λ: Activity, activity map µ: Attenuation, attenuation map a P , a S , (a−1 ): Attenuation (correction) factors P √ i: Imaginary unit, i = −1 i: Line of response j: Image pixel or voxel m P, mS : Measured data p, R λ: Nonattenuated activity sinogram P: PET (s,φ): 2-D sinogram parameters s: Scatter and randoms background S: SPECT t: Time-of-flight parameter P: Projection operator P ⊤: Back-projection operator R (R µ ): (Attenuated) Radon transform R µ: Attenuation sinogram

1. INTRODUCTION The problem of attenuation correction (AC) for quantitative positron emission tomography (PET) had been considered solved to a large extent after the commercial availability of devices combining PET with computed tomography (CT) in 2001;1 single photon emission computed tomography (SPECT) has seen a similar development.2 However, stimulated in particular by technical advances toward clinical systems combining PET and magnetic resonance imaging (MRI),3,4 research interest in alternative approaches for PET AC has grown substantially in the last years. Attenuation in PET describes the loss of detected photon pairs due to photon scattering and photoelectric effects induced by the presence of dense material along lines of response (LOR). The main challenge of attenuation correction lies in finding reliable attenuation-correction factors (ACF) compensating for this loss before or during image reconstruction. ACF are often, but not necessarily, calculated from an attenuation map (µ-map) which represents the spatial distribution of linear attenuation coefficients (LAC) of objects in the PET imaging Medical Physics, Vol. 43, No. 2, February 2016

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field of view (FOV); hence, of the patient and equipment supporting or surrounding the patient. Such a µ-map, if available, may also be used to compute an estimate of the scatter contribution within the—ideally—unscattered PET emission data (photopeak scatter), for example, through Monte-Carlo or Single Scatter Simulation (SSS).5–9 In many applications, accurate µ-maps (and hence, ACF) can be challenging to obtain. Transmission measurements have been used to this end for a long time, using PET detectors in both coincidence mode [with positron-emitting sources such as 68 Ge/68Ga (Refs. 10 and 11)] and singles mode [with 64Cu,12 68 Ge/68Ga,13 or gamma sources such as 137Cs (Ref. 14)]. These approaches benefited particularly from the introduction of PET/CT, which is using an X-ray source and dedicated detectors for transmission measurements and is now considered the gold standard for AC. However, even CT-based AC has its limitations, such as metal-induced or beam-hardening artifacts, or patient motion between CT and PET acquisitions. Furthermore, conversions from Hounsfield numbers measured using polychromatic X-radiation to LAC for monochromatic radiation from radionuclides are mainly approximations.1 But even accepting these limitations, one may find that CT is undesirable or unavailable in applications of one of mainly two kinds. Examples of the first category are pediatric applications, in which radiation dose delivered to the patient is a major concern, and research applications, in particular those with serial PET scans. Here, a CT-based µ-map can in principle be acquired, but the subject would benefit from alternative approaches. Furthermore, motion-aware AC using only CT information is limited due to dose constraints. PET/MRI is an important example of the latter category with many clinical applications,15 where the strong magnetic field prohibits the presence and use of state-of-the-art rotating X-ray tubes, hence rendering transmission-based AC challenging.3,4 The situation is similar for low-cost PET scanners, where a CT may not be integrated by design. If available, such as in PET/MRI, MRI-based AC is often the next best option; however, MRI-based µ-maps may still suffer from artifacts through patient motion, truncation of the MRI FOV, metal implants, and tissue misclassification (in particular, missing bone information).16 Hence, PET-AC methods other than CT- or MRI-based ones are still urgently needed. In this paper, we review a large subset of these methods, in particular, those making use of the emission data for attenuation correction. This review is not meant as a replacement or mere update of previous works: such as those recent ones summarizing the state of the art in PET AC based on transmission measurements,1,17 based on MRI,18–20 for neurological applications,21 for PET/MRI,22–24 or even more general ones.25,26 By contrast, we specifically focus and expand on the literature for estimation of attenuation maps from PET (and SPECT) emission data, which is only briefly touched in other reviews, if at all.20–25 Estimation of attenuation information from PET emission data shares common roots with similar applications in SPECT. Due to current high research interest in PET-related applications, we focus our attenuation to the PET literature,

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but will collect the relevant SPECT literature in passing. Historical as well as methodological accounts of general image-reconstruction algorithms for PET and SPECT, which we will not cover, can be found in Refs. 27–30. This manuscript is structured as follows. We first state the problem (Sec. 2) and summarize a few general observations and theoretical results (Sec. 3), some of which may be familiar to the reader. We then trace the common roots of simultaneous reconstruction in PET and SPECT in Sec. 4, before we focus on PET imaging. We will review algorithmic approaches, divided into those based on analytic data consistency conditions (Sec. 5) and iterative ones (Sec. 6), before turning our attention to PET applications in Sec. 7. The (to date) much smaller field of algorithms and applications for SPECT will be considered in Sec. 8. Finally, in Sec. 9 we present some recent advances in extracting quantitative information from Compton-scattered coincides, and conclude our review in Sec. 10. 2. NOTATION AND PROBLEM STATEMENT The problem of attenuation correction in PET and SPECT can be briefly formulated in the following idealized, twodimensional (2-D) example; let R+0 denote the set of nonnegative real numbers. Consider two positive, real functions λ,µ : R2 → R+0 representing spatial distributions of radioactivity concentrations (λ, activity map) and linear attenuation coefficients (µ, attenuation map), respectively; both sufficiently smooth and decaying toward 0 rapidly enough such that all following integral expressions are well-defined. Lines in the plane (compare Fig. 1) are parameterized by their distance31 s ∈ R to (0,0), and their angle with the y-axis, φ ∈ [0,2π). With l a parameter along the line, we thus have x s,φ (l) = scosφ − l sinφ and ys,φ (l) = ssinφ + l cosφ. In more concise

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ˆ where uˆ = (cosφ, vector notation, this reads xs,φ (l) = suˆ + l n, sinφ)⊤ and nˆ = (−sinφ,cosφ)⊤ (the hat denoting unit vectors ˆ Then we describe both PET (P) and SPECT (S) as in n). measurements without attenuation using p : R × [0,2π) → R+0 ,  ∞ p(s,φ) = R λ(s,φ) B dlλ(xs,φ (l)), (1) −∞

where we call p the (nonattenuated) activity sinogram, and R the Radon transform or X-ray transform.32 With attenuation, p is replaced by m P, S : R × [0,2π) → R+0 :  ∞ m P, S (s,φ) = dl a P, S (s,φ,l) · λ(xs,φ (l)). (2) −∞

From this point of view, the one difference between the two modalities lies in the attenuation factors (AF), which we denote by a P, S : R × [0,2π) × R → (0,1] and which can be expressed with the help of the Beer–Lambert law. In PET, a P is independent of the position along the line l and is related to the µ-map through (  ∞ ) a P (s,φ,l) = a P (s,φ) = exp − dr µ(xs,φ (r)) (3) −∞

(again, compare Fig. 1). The corresponding, only slightly different, expression for a SPECT measurement is (  ∞ ) a S (s,φ,l) = exp − dr µ(xs,φ (r)) . (4) l

The—seemingly small—difference in the lower limit of integration has fundamental consequences. Since attenuation effects are constant along each LOR in PET, the attenuation factors a P can be separated from the integrand. Thus, Eq. (2) is simply the Radon transform of λ multiplied with a P —which in turn is given by the Radon transform of µ. Equations (2) and (3) can thus be combined to m P = a P · R λ = exp(−R µ) · R λ.

(5)

We call R µ the attenuation sinogram. In contrast to the above, m S in SPECT is given by the attenuated Radon transform. The a S are related to µ by a formulation of the more complex divergent-beam transform in Eq. (4), and there is no simple SPECT analogue to Eq. (5). One commonly abbreviates this case of Eq. (2) to m S = R µ λ.

(6)

We now briefly state the problem of inverting Eqs. (5) and (6) in two different scenarios, with and without knowledge about the attenuation.

F. 1. A line L in the plane indicated by its parameterization (s, φ), and the (s, l) coordinate system, rotated by φ with respect to (x, y). In both SPECT and PET, the measurements are related to the line integral of λ along lines L. The two modalities differ in how these measurements are influenced by photon attenuation (here indicated by the simplified object). In SPECT, from any source position (yellow diamond, ◆), a single photon is emitted in the direction of increasing l and sees attenuation along its trajectory only (orange ray). Hence, attenuation in SPECT is depth-dependent. In PET, by contrast, two photons are emitted in directions opposite each other: thus, attenuation along the totality of L (orange and green rays) influences the measured data, regardless of the source position on L. Medical Physics, Vol. 43, No. 2, February 2016

2.A. Activity reconstruction correcting for known attenuation

In PET, finding λ given (m P , a P ) or (m P , µ) is straightforward using the inverse of the Radon transform,33 which we denote R −1. The reconstruction can be efficiently implemented using filtered backprojection (FBP) or nonlinear, iterative algorithms such as maximum likelihood expectation maximization (MLEM; Ref. 34), ordered subset expectation maximization (OSEM; Ref. 35), or the row-action maximum likelihood algorithm (RAMLA; Ref. 36). Thus, we have by

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analytic inversion −1 λ = R −1 a−1 P · m P = R (exp(R µ) · m P ),

(7)

where a−1 P , the element-wise inverse of a P , is commonly referred to as ACF. Note that the sensitivity of AC toward inaccuracies in a P is higher where the attenuation factors a P are low (thus, on LORs with high attenuation) or where measured data m have high values (on LORs with high number of measured coincidences). In the extreme case, the ACF may take arbitrary values where m P = 0 and need to be known only elsewhere. Hence, a complete attenuation map is not strictly required for attenuation correction. As a further consequence, Eq. (7) allows for two different attenuation maps to be equally suited for AC as long as their forward projections (the ACF) are similar enough on lines of response with low ACF or many counts. Hence, attenuation-correction methods need to be evaluated and compared based on their performance in realistic PET data. Only if influences from the activity distribution or the reconstruction algorithms are to be eliminated, evaluation shall be based on the generated attenuation map (if available), attenuation sinogram, attenuation factors, or ACF. For SPECT, it is intuitively obvious that the nonexistence of simple ACF in SPECT complicates the situation, even when the µ-map is known. Hence, a popular early method, known as the Chang method, disregarded attenuation during image reconstruction, and applied an empirical postprocessing step in image space followed by iterative refinements.37 Alternatively, fully iterative methods such as the aforementioned MLEM, OSEM, or RAMLA are as applicable as in PET. By contrast, analytic inversion of the attenuated Radon transform has required much additional effort and was only possible after the turn of the millennium;38,39 however, the practical impact of analytic inversion in SPECT has remained limited due to difficulties integrating additional physical effects. 2.B. Simultaneous reconstruction of attenuation and activity

In both modalities, the situation is more involved when µ is unknown. Especially in SPECT, the problem of finding µ has been coined the identification problem of emission tomography40 due to the role of µ in R µ . Various approaches have been proposed over the decades to handle this case, estimating µ (as well as λ) from m P, S . Note that while the title of this paper might suggest that attenuation correction using emission data is a separate step in image reconstruction, in which µ is found first and only then used for attenuation-corrected reconstruction of λ, this is rarely the reality. Many approaches estimate both quantities simultaneously or return λ as a byproduct of finding µ. 3. GENERAL OBSERVATIONS AND THEORETICAL RESULTS The aim of this section is to summarize findings related to the problem of simultaneous reconstruction without reference to any particular algorithm or implementation. Medical Physics, Vol. 43, No. 2, February 2016

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3.A. The Radon and X-ray transforms are uniquely invertible

As mentioned in Sec. 2, the 2-D Radon transforms R and R µ are uniquely invertible,33 as is the 3-D X-ray transform.41 This holds for the class of physically relevant functions which are bounded and of compact support. An enormous collection of theoretical results for a variety of different Radon and X-ray transforms has been published by Markoe.42 These will constitute the basis for many algorithms and applications discussed later on. 3.B. The Radon transforms fulfill data consistency conditions (DCCs)

Without attenuation, the 2-D Radon transform R λ fulfills the Helgason–Ludwig DCCs (see Helgason,43 Theorem 4.1 and Ludwig,44 Theorem 2.1). In their original form, the 2D DCCs make statements about the angular evolution of the moments of the projection profiles Pφ = R λ(·,φ); in particular, in terms of their polynomial expressions in cosφ and sinφ. Specifically, the nth moment of Pφ was found to be a bivariate homogeneous polynomial in cosφ and sinφ (trigonometric polynomial) of degree n. This means that for all k > n, the kth coefficient of the Fourier series of the nth moment is zero,  2π  ∞ dφ eikφ ds s n R λ(s,φ) = 0 ∀k, n ∈ N0, k > n. (8) 0 −∞    nth moment of Pφ

Symmetry arguments show that the same condition further holds for any odd n + k. This is because opposing projection profiles are reversed versions of each other, Pφ (s) = P(φ+π)(−s), such that their nth moments have same (different) signs when n is even (odd). In case of same (different) signs, the Fourier basis functions eikφ can only interfere constructively when having an even (odd) number of cycles per turn k; thus, if n+ k is odd, the Fourier series coefficient must be zero. An illustrative mechanical interpretation offered in Ref. 46 is the following. Let f be a 2-D mass distribution and R f its projections. For n = 0, the zeroth moment of any projection profile is the total mass in the plane and does not depend on the projection angle φ; it is thus constant and has all zero Fourier coefficients for k > 0. For n = 1, any first moment is the center of mass of the projection profile; at the same time, it is the projection of the center of mass of the 2-D mass distribution. As is known, the projection of a point in the plane traces a sinusoidal in the sinogram, hence, can be described by a weighted sum of only cosφ and sinφ. Thus, Fourier coefficients of first moments are zero for k > 1, as well as k = 0 since n + k = 1 is odd. 3.C. The X-ray transform fulfills consistency equations

The Helgason–Ludwig DCCs are global conditions; by contrast, certain parameterizations of the X-ray transform obey local consistency equations (CEs), which take the form of partial differential equations (PDEs). Here, we just reproduce

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a simple example.47 In 3-D, almost all LORs can be defined by two points (x 1, y1,0)⊤ and (x 2, y2,1)⊤. Then, John’s equation45 for a scaled 3-D X-ray transform X of f ,  ∞ Xf = dl f (x 1 + l(x 2 − x 1), y1 + l( y2 − y1),l), (9) l=−∞

reads (

) ∂2 ∂2 − (X f ) = 0. ∂ x 1∂ y2 ∂ y1∂ x 2

(10)

Equation (10) has a geometric interpretation (Fig. 2); applications of the aforementioned global DCCs, including those variants for the attenuated Radon transform, and local CEs to simultaneous reconstruction are further described in Sec. 5. 3.D. Simultaneous reconstruction has nonunique solutions

The problem of inverting Eq. (2) is complicated by the fact that it exhibits several solutions; hence, typical optimization problems based on Eq. (2) are nonconvex.48–50 This had already been appreciated by Natterer,40,51,52 who discussed the nonuniqueness for a finite number of sources in SPECT (Ref. 40) and PET (Ref. 51) as well as for some degenerate

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continuous cases.40,52 For more background, see Sec. 4; here, we just summarize the main results without exterior sources and knowledge of the support of the µ-map. With a finite set of sources, the fan-beam (SPECT) or Radon (PET) transform of µ can be estimated from the data, but only up to an unknown additive constant and only along LORs with sources. The latter restriction means that attenuation information is truncated when attenuation is outside of the convex hull of the activity support.51 With continuous sources, no information can be gained at all when both λ and µ are rotationally symmetric. For the PET case, µ-maps with different assumed values of the unknown constants, and an extreme case with minimum support of the activity sinogram are visualized in Fig. 3. 3.E. Time of flight (TOF) allows simultaneous reconstruction

The situation described in Sec. 3.D is much improved by considering additional information. In fact, the importance of TOF measurements for PET in terms of signal-to-noise ratio (SNR; e.g., Ref. 54) and data redundancy (e.g., Refs. 55–57) had been appreciated by many authors in the past and has only recently been reconfirmed for AC in PET/MRI.58,59 A recent result with direct implications for simultaneous reconstruction has been found by Defrise et al.,60 stating that TOF PET data determine all values of the attenuation sinogram R µ where R λ > 0, up to an additive constant. In contrast to previous works, this includes the cases of continuous source distributions and rotational symmetry of both λ and µ. Hence, all values required for attenuation-corrected reconstruction are known, and λ can be found up to a multiplicative constant— compare Eq. (7). Unfortunately, the remaining nonuniqueness of the solution is inherent to the measurement of true coincidences in PET even with TOF information. Additional information, such as exact knowledge of the µ-map support or external sources, can in principle be used to determine the constant (Ref. 61, Sec. II.C). The proof given in Ref. 60 considers noiseless 2-D and 3-D cases; the results hold for any finite TOF resolution with a Gaussian distribution. A more general proof has subsequently been given by Ahn et al.,49 relaxing the conditions on the shape of the TOF uncertainty distribution. 4. COMMON HISTORY IN PET AND SPECT

F. 2. Geometric interpretation of John’s equation (10). Top. According to Asgeirrson’s mean value theorem, the mean value of the line integrals X f along the families of blue (a) and red (b) lines (generating the same hyperboloid surface, respectively) are equal (Ref. 45, top of p. 305). Bottom. John’s equation can be understood as a localization of this global consistency condition. Each of the two mixed differential operators in Eq. (10) describes a second-order change of the line integral X f from a reference line (green; vertical for ease of presentation) to a line (blue and red, respectively) through points shifted with respect to the reference line by infinitesimal distances δ 1 and δ 2 along different axes. Following Eq. (10), these changes are equal. Note the conceptual similarity of crossing oblique lines. Medical Physics, Vol. 43, No. 2, February 2016

In this section, we follow the roots of early attenuation correction in PET and SPECT using emission data. By “early,” we understand contributions before a (fuzzy) separation line around the year 2000. This choice is less arbitrary than it may seem: before this time, research efforts in SPECT had to focus not only on generation of µ-maps, but also on evaluation and optimization of AC algorithms such as iterative FBP and other postcorrection approaches.62,63 In 2000, the problem of analytic SPECT AC was finally solved after breakthrough work by Novikov,38,39 rendering the aforementioned class of approaches obsolete. More importantly, by this time, iterative image reconstruction, easily incorporating attenuation information, had become a commercial standard option64 with a

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F. 3. Nonuniqueness of simultaneous reconstruction in PET. Left. Inverse 2-D Radon transforms µ C (in cm−1) of three different attenuation sinograms given by (R µ 0) +C · 1suppR µ 0: (a) µ 0, (b) µ 5, (c) µ 10, (d) µ 10 − µ 0. Right. Extreme case of point sources of activity in the center of cold attenuating spherical shells of different radii, thicknesses, and densities. All configurations are compatible with the same PET measurement and cannot be distinguished based on true coincidences: no µ-map can be reconstructed. Reprinted with permission from Y. Berker and V. Schulz, “Scattered PET data for attenuation-map reconstruction in PET/MRI: Fundamentals,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2014), 6pp. Copyright C 2014 IEEE.

similar effect. Finally, the advent of combined PET/CT (and later SPECT/CT) devices1 added to significantly decreasing the interest in generation of attenuation maps based on anything else but CT data for about a decade.65 Before, SPECT and PET had been treated side by side in many papers due to their inherent similarities; however, when research interests in AC reemerged, the clinical availability of TOF detectors with high detection efficiency had overturned the situation for PET, while not so for SPECT. Nonetheless, many current developments share ideas with the works carried out decades earlier. We hence summarize the history of extracting patient outlines from emission data for AC purposes, as well as that of quantitative simultaneous reconstruction in Secs. 4.A and 4.B. 4.A. Segmenting outlines from emission data

The idea of using segmented attenuation maps (instead of transmission scans or model-based µ-maps) was introduced into practical PET imaging as early as 1981.66 ACFs calculated from a segmented attenuation map are less noisy, which is an advantage compared to noisy transmission scans or even to ACF calculated directly from an excessively noisy transmission-based µ-map. More importantly, however, segmented µ-maps have the added advantage that the transmission scan, often contaminated by transmission scatter and simultaneous emission, does not have to be quantitative as long as it allows distinction of different tissue classes. Soon thereafter, it was appreciated that tissue classification could be based on more information than just a transmission scan, and that emission data could improve the generation of attenuation maps. The earliest works focused on cranial imaging by reconstructing a head contour from emission projection data and filling this contour with a model-based attenuation map.67–69 From today’s point of view, it is remarkable that these approaches were validated with a variety of tracers, including 68 Ga-ethylenediaminetetraacetic acid (EDTA) and 15O-water. Medical Physics, Vol. 43, No. 2, February 2016

Subsequent approaches considered the more complex case of whole-body imaging. Following previous ideas, body contours were calculated from projection data; homogeneous filling of these contours70 was later improved upon by considering a fuzzy segmentation of lung tissue.71 More elaborate approaches made use of non-attenuation corrected (NAC) reconstructions to determine body outlines, and augmented this information with lung contours from short transmission scans72 or transmission scans from other subjects.73 NAC PET images are again being used today in truncation compensation for PET/MRI, as discussed in Sec. 7.B.2. In SPECT, developments proceeded in the opposite direction. Approximately attenuation-corrected and NAC SPECT images were thresholded and homogeneously filled74,75 very early on. Interestingly, these early developments comprised the use of scattered coincidences for attenuation correction,76 a field of recent development onto which we focus in Sec. 9. The use of (unscattered) SPECT projection data for generation of patient outlines was only described a few years later,77–79 followed by reports of using radioactive body wraps for body outline detection80 and use of combined transmission and scatter SPECT imaging from external sources.81 The first work employing smoothness and completeness properties of 3-D contours to guide the choice of segmentation thresholds was seen in Ref. 79. Despite their common aim of quantitative nuclear imaging, all approaches described so far are nonquantitative in the sense that, from the emission data, only the presence or nonpresence of attenuating objects is inferred. Implicit assumptions in many of these algorithms are that the attenuating object is solid, convex, and mostly composed of water-equivalent tissue. These are rather broad assumptions; hence, approaches focusing on extracting quantitative attenuation information from emission data attracted significant interest as well. However, as we will see, drawing the line between nonquantitative and quantitative methods in these early days can be challenging.

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4.B. Simultaneous reconstruction

We now summarize the results for early algorithms simultaneously reconstructing attenuation and activity. In contrast to the approaches described in Sec. 4.A, these are mostly optimization approaches. Their formulations often allow a certain degree of versatility with respect to prior information and the quantity to be reconstructed; however, as noted just before, this versatility was not exhausted from the beginning. The first application of simultaneous reconstruction for SPECT was described, as early as the first approach estimating outlines from emission data, by Censor et al.:48 the task was formulated as a discrete feasibility problem and solved with a subgradient projection method. These initial steps sparked a sequence of studies characterizing the underlying continuous problem. The following results were obtained using the analytic data consistency conditions, allowing to separate the estimation of the µ-map from the actual activity reconstruction. If the number of point-like sources is finite in SPECT, the fan-beam transform of µ can be identified up to an additive constant for all sources using data consistency conditions;40 a nonattenuated, unknown calibration source outside the support of µ can be used to fix the unknown constant. Interestingly, the first result does not hold for an infinite number of sources, as was proven by considering circularly symmetric attenuation and activity distributions. These results were extended to 2-D PET (where the Radon transform of µ can be identified) and complemented with a numerical optimization procedure based on Newton’s method.51 These results were again extended toward more realistic, continuous source distributions. When the outline of a region of constant attenuation is known, the constant value of µ in that region can be directly computed in SPECT.82 Focusing again on PET, the problem with a rotationally invariant µ-map and continuous activity within can be solved using a single additional, exterior point source.52 Finally, if the µ-map is constant within an unknown outline, this outline can be recovered from the data and a Newton-based method without the need for an exterior source;52 note the inherent similarity with the methods described in Sec. 4.A. This approach was applied in practical PET imaging83 for constant nonconvex objects and objects with an interior boundary between two tissues (such as bone and brain tissue); these more challenging objects again required the use of external sources. Simplified applications of this idea appeared in the literature years later, estimating the optimal elliptical outline of a uniform attenuator best representing the nonuniform attenuation.84 With a maximumlikelihood procedure, the boundary description of the uniform attenuator was extended from five ellipse parameters to eleven Fourier coefficients.85 Simultaneous reconstruction, finding both µ-map and attenuation-corrected activity λ at the same time, was pursued further using several approximative techniques: a variation of ART was proposed that applied multiplicative updates to both µ and λ based on the ratio of expected and measured SPECT counts.86 Since a mismatch cannot uniquely be attributed to either activity of attenuation, both were modified in opposite Medical Physics, Vol. 43, No. 2, February 2016

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directions using similar updates, weighted according to the sensitivity of the expected counts to the respective quantity. In another work, a linearization of the attenuated Radon transform allowed to combine the data from opposing views and extract some visual features of the µ-map;87 however, this method assumes constant activity and is subject to the theoretical constraints mentioned before, and despite its general approach was found to be limited to finding certain object contours. The practical use of additional transmission information (discussed in more detail in Sec. 7.C) has also been considered early on. In Ref. 88, simultaneous maximum-likelihood reconstruction of activity and attenuation from positron emission and transmission data were proposed, and the idea of alternating updates of activity and attenuation premiered. In various PET or SPECT studies, deformation of a-priori µ-maps based on DCCs was evaluated46,84,89—see also Sec. 7.C.3. Many of the results in this section apply to both PET and SPECT, but as announced, this will not hold true for the remainder of this review. We will thus summarize more recent methods in PET and SPECT separately, with a detailed account of PET methods followed by a briefer one for SPECT. In that respect, there could be no better conclusion to this section than with a method of estimating a µ-map from PET and SPECT data combined.90 The main idea is to measure the rates of coincidences between two detectors, as well as of collimated singles on both ends, e.g., in a SPECT scanner with coincidence counting capabilities. The combined information can be used to solve for the unknown activity and attenuation; however, this algorithmically elegant solution to the problem of attenuation correction opens a vast field of instrumentationrelated challenges in practice. Next we will turn our attention mostly toward PET methods, divided into analytic (Sec. 5) and iterative (Sec. 6) ones. This will be followed by applications and combinations of these methods (Sec. 7), before we summarize methods for SPECT in Sec. 8. 5. ANALYTIC ALGORITHMS BASED ON CONSISTENCY The global DCCs of the 2-D Radon transform without attenuation, given in Eq. (8), can also be applied to the case with attenuation. For PET, only straightforward modifications are necessary, since m P can easily be attenuation-corrected knowing µ: hence, exp(R µ) · m P = a−1 P · m P replaces R λ in Eq. (8); in SPECT, by contrast, exp((I +iH )R µ/2)· m S has to be used, where I is the identity transform and H is the Hilbert transform operating on s (Ref. 91, Theorem 6.2). Other, more complex consistency conditions for the attenuated 2-D Radon transform92 are discussed in Ref. 93, Sec. 4. The general approach is then to guess an initial µ-map and evaluate the left-hand side of Eq. (8). The degree of violation of the DCCs is subsequently quantified by the absolute values of several of these Fourier coefficients. An important feature of this approach is that the formulation only depends on the measured data m and µ, independent of λ (Ref. 52): thus, by varying only the candidate µ-map, the degree of violation can

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be minimized. However, due to the global nature of the condition, the µ-map may need to be parameterized by only a few parameters which can effectively be estimated. These parameters usually describe the contour, tissue attenuation coefficients, registration parameters, or regularization parameters— however, no µ-map is usually reconstructed from scratch using these DCCs. Since the choice of parameterization is highly dependent on the application and allows little insight into the nature of the DCCs, we discuss most of the studies based on global DCCs in Secs. 4, 7, and 8. A much more versatile set of local CEs exists when the data have some inherent redundancy. These CEs can—intuitively— be understood to exist if an m-dimensional object is represented by a function of n parameters, n > m.47 In that case, there are n−m independent consistency equations which characterize the range of the respective transform. This is true, for example, for 3-D non-TOF PET data (m = 3, n = 4);94 in fact, the redundancy of the 3-D X-ray transform and its relation to differential equations, nowadays known as John’s equation, has been appreciated as early as 1938.45 Redundancy is further a feature of 2-D TOF (m = 2, n = 3) and, in particular, 3-D TOF PET data (m = 3, n = 5). The respective CEs are explored in Secs. 5.A and 5.B. 5.A. Consistency equations for 3-D or TOF PET data

As an extension of Eq. (1), nonattenuated 3-D TOF PET data can be parameterized as (compare Fig. 4; Ref. 96),  ∞ p(t,s,φ,z,θ) = dlh(t − l)λ(suˆ + l nˆ + zˆez ) (11) −∞

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with three unit vectors uˆ = (cosφ,sinφ,0)⊤, nˆ = (−cosθ sinφ,cosθ cosφ,sinθ)⊤, eˆ z = (0,0,1)⊤,

(12a) (12b) (12c)

where λ is a 3-D tracer distribution, h is a TOF profile, t is the TOF parameter, s and φ are the 2-D sinogram coordinates as in Eq. (1), z is the axial coordinate of the midpoint of the LOR, and θ is the co-polar angle between the LOR and a transaxial plane. The TOF parameter t (measured in units of distance) is related to the measured TOF time difference ∆T between the two arrival times of the two gammas by t = c∆T/2, where c denotes the speed of light. Note that when h ≡ 1 and z = θ = 0, Eq. (11) reduces to the non-TOF, 2-D case of Eq. (1). In contrast to Refs. 97–99, the co-polar angle θ is used instead of δ = tanθ to parameterize the oblique LORs. Note that we omit the P subscript for PET in this section and Sec. 5.B. A critical assumption for the following calculations is that the TOF profile be described by a Gaussian distribution with standard deviation σ, ( ) t2 1 exp − 2 . (13) h(t) = √ 2σ 2πσ In that case, the standard deviation of h, given by σ = cTFWHM/ (4 2log2), can be given in terms of TFWHM, the full width at half maximum (FWHM) of the measured time difference; this value is on the order of 500–300 ps in current clinical scanners and prototypes.100 The 3-D TOF PET sinogram p has five degrees of freedom, while the object λ has only three—the two degrees of redundancy can be expressed as two independent PDEs and referred to as consistency equations. Different formulations of these consistency equations have been used to exploit the rich structure of TOF PET data, for example, in Fourier rebinning of non-TOF (Ref. 101) or TOF data97,98 from 3-D to 2-D, restoration of missing sinogram data,102,103 rebinning of TOF data to non-TOF data,99 and attenuation correction.60 In fact, consistent TOF PET data satisfy the following two independent consistency equations:96,99 ∂p ∂p ∂p ∂2p ∂p +t cosθ − ssecθ + stanθ + σ 2 cosθ = 0, ∂φ ∂s ∂t ∂z ∂s∂t (14) ∂p ∂p ∂p ∂ p ∂ p +t tanθ −t secθ + σ 2 tanθ 2 − σ 2 secθ = 0, ∂θ ∂t ∂z ∂z∂t ∂t (15) 2

2

where the arguments of p and its derivatives are (t,s,φ,z,θ). ∂ ∂ ∂ Applying secθ ∂z − tanθ ∂t to Eq. (14) and cosθ ∂s to Eq. (15) and adding the results, we obtain a generalization of John’s Eq. (10) for 3-D TOF PET data,96,98 F. 4. Data parameterization for a multiring TOF PET scanner. The LOR between detectors A and B is parameterized in sinogram format by the variables t, s, φ, z, and θ. The TOF profile is centered at the most likely position ˆ and x0 is the center of the LOR. Modified with permission x = x0 + t nˆ along n, from Y. Li, M. Defrise, S. D. Metzler, and S. Matej, “Transmission-less attenuation estimation from time-of-flight PET histo-images using consistency equations,” Phys. Med. Biol. 60, 6563–6583 (2015). Copyright C 2015 IOP Publishing. Medical Physics, Vol. 43, No. 2, February 2016

∂2p ∂2p ∂2p − tanθ − s sec2θ + tan2θ ∂z∂φ ∂φ∂t ∂z∂t 2 2 ∂ p ∂ p + stanθ secθ 2 + stanθ secθ 2 ∂z ∂t 2 ∂p ∂ p − tanθ cosθ + cosθ = 0. ∂s ∂s∂θ secθ

(16)

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Starting from these rather long, general expressions, several special cases can be derived: John’s equation for non-TOF PET data45,101,104 can be isolated from Eq. (16) after removing the three terms including partial derivatives with respect to t,96,99 secθ

∂2p ∂p ∂2p + stanθ secθ 2 − tanθ cosθ ∂z∂φ ∂s ∂z

∂2p = 0. (17) ∂s∂θ By contrast, inserting z = 0 and θ = 0 into Eq. (14) and removing the respective partial derivative terms, we find the consistency equation for 2-D TOF PET data,105 + cosθ

∂p ∂2p ∂p ∂p +t −s + σ2 = 0. (18) ∂φ ∂s ∂t ∂s∂t A unified Fourier theory and a detailed derivation of the consistency equations for TOF-PET data are given in Ref. 96. In Sec. 5.B, we will exemplarily show the application of consistency Eq. (18) to attenuation estimation. Dp =

5.B. Analytic attenuation estimation

TOF PET data contain substantially more information about the unknown attenuation than non-TOF PET data. Defrise et al.60 showed that 2-D TOF PET data can determine the attenuation sinogram up to a constant using consistency Eq. (18). After correction for scattered and random coincidences (but not attenuation), and recalling Eq. (5), we can model the expectation sinogram as m(t,s,φ) = a(s,φ) · p(t,s,φ).106 Inserting p = m/a into Eq. (18) and multiplying by a, one obtains ( ) ∂m ∂ loga ∂ loga mt + σ 2 +m = Dm. (19) ∂t ∂s ∂φ All the terms related to m can be obtained from PET data, and then, we can estimate the two partial derivatives of −loga = R µ by fitting in t in a least-squares sense. This fit can be computed using the data integrated with respect to the TOF parameter; in practice, this involves summation of all (discrete) TOF bins. However, the boundary conditions are not considered in these methods and the attenuation can only be determined up to an additive constant for loga(s,φ), which amounts to a multiplicative constant for the attenuation factors a and activity image λ.60 A preliminary extension of this approach toward 3-D TOF PET has been reported in Ref. 107. In an effort to integrate the least-squares computation of the partial derivatives and their integration, discretized variants of the consistency equations were combined with smoothness priors in a MAP approach,108 increasing robustness toward noise. This has further been extended toward simultaneous estimation of attenuation and activity.109 Alternatively, TOF PET data can be naturally stored in the histo-image format without information loss, and the DIRECT approach can be used for efficient 3-D TOF PET reconstruction.110–112 Thanks to the histo-image parameterization, the consistency equations in histo-image format are more concise Medical Physics, Vol. 43, No. 2, February 2016

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than in the sinogram format. Analytic attenuation estimation from TOF PET histo-images was also studied;95 in addition, a fast solver was developed to estimate the attenuation factors from their derivatives using the discrete sine transform and fast Fourier transform with the consideration of boundary conditions.95 This work showed that the attenuation can be uniquely determined from the TOF PET data, and the scaling constant is naturally solved by considering the boundary conditions when the TOF information is available. However, since the estimate of the directional derivatives of the attenuation is inaccurate for LORs tangent to object boundary, an external source might be needed to give an accurate estimate for such LORs.113

6. ITERATIVE ALGORITHMS In this section, we explore iterative algorithms which, in contrast to those described in Sec. 5, proceed by iteratively refining an initial µ-map estimate until some cost function is sufficiently small. As in activity reconstruction, this general approach can be applied in physically more complex situations than analytic methods, for example, to model the influence of positron range, detector spatial resolution, or depth of interaction, in the system matrix. It further allows to take into account that the measured data have a Poisson distribution— a fact that has been completely neglected in Sec. 5. This, as well as the fact that an efficient, monotonous update rule has been found for the most fundamental case, explains the popularity of maximum-likelihood-based methods34,35 in lowcount experiments such as clinical PET. At the heart of these methods is the notion of the likelihood of a nonattenuated activity sinogram pit and an attenuation sinogram ai , given a measurement mit and assuming its expected value is mit = ai pit + sit. This notation relies on discretized quantities defined for sinogram bins i and TOF bin t (if available114). The nonattenuated activity sinogram pit relates the activity λ j in voxel j through the system matrix cit, j ,  pit = cit, j λ j ; (20) j

similarly, the attenuation factors ai relate to µ j through a (potentially different—in any case, independent of the TOF parameter t) system matrix l i, j and  ai = exp *.− l i, j µ j +/ . (21) , j Finally, sit denotes scattered and random contributions. In this setting, the log-likelihood, after omission the constants with respect to the estimated parameters, reads  (mit logmit − mit). L(λ,µ | m) = (22) i

t

While it has been shown that this optimization problem is convex when the ai are known, the joint problem is nonconvex when optimizing for both λ and a simultaneously—not least due to the scaling ambiguity discussed in Sec. 3.D. In Secs. 6.A and 6.B, we will discuss different iterative approaches to this

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problem, estimating either the µ-map (that is, µ j ) or the attenuation factors ai .

The idea of exploiting the structure of the joint optimization problem had already premiered in 1991 in the joint alternatemaximize (JAM) algorithm.88 This and many other approaches use alternating updates of µ and λ, assuming the respective other quantity fixed. The main motivation for this approach is that a number of algorithms are available to λ reconstruction knowing µ; similarly, µ reconstruction knowing λ can be considered a generalization of transmission tomography, interpreting λ as known sources. Hence, a wide range of well-established algorithms from emission and transmission computed tomography can be used for one quantity (λ or µ) as long as the other is considered fixed. As in activity-only reconstruction, the λ-reconstruction step can be based on MLEM or other algorithms. Similarly, several alternatives have been evaluated for µ reconstruction in the context of PET (Ref. 115) or both PET and SPECT.116,117 We focus on the PET implementations here. Since the attenuation is constant along each LOR, TOF information does not yield additional information to this quantity. Hence, non-TOF    quantities, for example, ci, j = t cit, j , mi = t mit, pi = t pit,  and s i = t sit, are used here to simplify notation in attenuation reconstruction from (TOF-)PET data;61 alternatively, for implementation purposes, algebraic combinations of the above quantities such as ai pit + sit − mit could be summed over the TOF index t. An approach described in Ref. 115 used paraboloidal surrogates coordinate descent (PSCD), which had been proposed for individual emission118 as well as transmission119 tomography, for both λ and µ reconstruction. In Refs. 117, 272, and 273, MLEM was used for λ reconstruction. The MLEM update, using element-wise multiplication and division, reads ) ( ( ) P ⊤ mk λk ⊤ m Pλ +s = P (23) λk+1 = λk · P ⊤ (1) P ⊤1 Pλk + s with P and P ⊤ projection and back-projection operators, respectively, incorporating effects of attenuation. For ease of notation, we will use a “new” superscript instead of k +1, and further omit all k superscripts. Applying the matrix notation introduced before to both  projection operators, (Pλ)it = ai j cit, j λ j and (P ⊤m) j  = it ai cit, j mit, in Eq. (23), we find the well known MLEM update equations for the situations with or without scatter and randoms,  mit ai cit, j ai pit + sit it  λ new = λj (24a) j ai cit, j it

= 

λj ai cit, j

 cit, j mit it

pit

.

it

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The simple and intuitive update for µ reads120  l i, j ai pi i µnew j = µj 

6.A. Estimation of attenuation maps

s it=0

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(24b)

(25) l i, j mi

i

 in our notation: backprojections ( i l i, j ) of expected (ai pi ) and measured (mi ) counts are compared and their quotient is used to update µ. One feature of these multiplicative updates of both λ and µ is that is preserves non-negativity for both quantities. Use of transmission or partial emission (that is, known sources) a-priori information has been envisaged for this algorithm and is discussed in Sec. 7.C. Nuyts et al.116 proposed maximum likelihood reconstruction of attenuation and activity (MLAA) as a combination of MLEM and maximum likelihood gradient ascent for transmission tomography [ML-TRANS, MLTR (Ref. 121)]. Salomon et al.122 were the first to apply this algorithm to TOF data (using MRI segmentation, compare Sec. 7.B.1) and reignited interest in the problem in a TOF setting; subsequently, Rezaei et al.61 extended the characterization of this algorithm while paying particular attention to TOF information, rendering tissuepreference priors used in Ref. 116 mostly obsolete. The MLTR update reads as follows,61,121 with and without scatter and randoms, respectively:  ai pi (ai pi + s i − mi ) l i, j ai pi + s i i new (26a) µj = µj +  (ai pi )2  l i, j l i,ξ ai pi + s i ξ i  l i, j (ai pi − mi ) s i =0

i

= µj +  i

l i, j ai pi



.

(26b)

l i,ξ

ξ

Hence, both MLAA and TOF-MLAA can quite concisely be described as repeated applications of Eqs. (24a) and (26a) to the data, starting from simple initial λ and µ. [If m has been precorrected for scattered and random coincidences, Eqs. (24b) and (26b) may be used, noting that the assumption of Poisson distribution of counts is no longer valid.] To remain within the scheme of alternating updates, it should be made sure that each formula uses the most recent result of the respective other one. Furthermore, it may be beneficial to apply transmission updates more often than emission updates;61,123 however, this finding may not generalize to other implementations122 or other algorithms, especially those not using TOF information.88 For TOF-MLAA (Ref. 61), the influence of the choice of the initial µ-map has been studied,124 as has the influence of different TOF resolutions.125 Some results for non-TOF and TOF-MLAA are shown in Figs. 5 and 6, respectively; note in particular the crosstalk between µ and λ in Fig. 5, manifesting in features of the reference µ-map showing in λ estimated with MLAA (and vice versa). This constitutes a major limitation of non-TOF MLAA and is a feature of many approaches to simultaneous reconstruction. However, as has been shown for analytic approaches in Sec. 5

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F. 5. (a) MLAA reconstruction for the SPECT thorax simulation. From left to right and top to bottom: The true attenuation map, the MLAA attenuation map, the true activity distribution, the MLAA reconstructed activity image, and the MLEM reconstruction without attenuation correction. Reference and MLAA images are displayed with common gray scale. The nonattenuation corrected image is scaled to its own maximum to make it clearly visible. (b) Same for the PET thorax simulation. Reprinted with permission from J. Nuyts, P. Dupont, S. Stroobants, R. Benninck, L. Mortelmans, and P. Suetens, “Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms,” IEEE Trans. Med. Imaging 18, 393–403 (1999). Copyright C 1999 IEEE.

and is confirmed in Fig. 6, TOF information can effectively resolve the µ–λ crosstalk. More abstract approaches focus more directly on the structure of the likelihood function than on using established imagereconstruction methods. In Ref. 126, updates of µ and λ were calculated sequentially (based on the emission data only, in contrast to Ref. 127—see Sec. 7.C) by employing a Newtonlike algorithm applied to the log-likelihood functions. Similarly, Refs. 128 and 129 combined MLEM (for λ) with a Newton–Raphson method with positivity constraint for µ; iterative data refinement was used to enforce consistency of the result with respect to the Radon transform. Although the likelihood function is nonlinear in µ, with the help of an approximate attenuation map µ0, the expression for the AF can be linearized in ∆µ = µ − µ0 and the problem of simultaneous reconstruction can be considered as a convex optimization problem;130 corrections for nonlinearity have also been devised by these authors. In Ref. 49, a framework was developed comprising both the most general case of mostly unconstrained, as well as linear

parameterizations of attenuation maps using prior knowledge.122,131,132 Box constraints and smoothness priors were used to define a penalized-likelihood approach, which was solved using alternating updates: therefore, block-alternating minorization–maximization was used with different surrogate functions for λ and µ, and global convergence to a stationary point was proven using the theory presented in Ref. 133. The alternating direction method of multipliers [ADMM (Ref. 135)], which uses an objective function and partially updates its input variables and thus implicitly decomposes the optimization problem, has also been proposed for this application recently.136 Finally, efforts to separate the likelihood expression in Eq. (22) into additive components have resulted in the method put forward in Refs. 137 and 138, which is based on an extension of the set of latent random variables used in CT reconstruction139 and in a SPECT approach to simultaneous reconstruction.140 This way, the likelihood function can be separated into many one-dimensional components, which can be efficiently optimized for both µ and λ simultaneously, or either of the two alone. 6.B. Estimation of attenuation factors

As suggested in Sec. 1, µ-maps are not strictly required for AC in PET. A recently proposed class of algorithms is conceptually different from the aforementioned ones due to the fact they focus on attenuation factors a instead. As proposed by Nuyts et al.,141 Rezaei et al.,134 and Defrise et al.,50 the update rules of the maximum likelihood attenuation correction factors (MLACF) algorithm comprise TOF-MLEM [Eq. (24a)] for λ and  pit mit s =0 mi it ainew = ai = . (27) p a p + s pi i i it it t F. 6. Transaxial (top), coronal (center), and sagittal (bottom) view of the CT-based attenuation image (left) and attenuation image reconstructed with MLAA (right) from the emission data and CT-based scatter and bed correction. The MLAA image was smoothed with a 3-D Gaussian with 6 mm. The vertical bar represents the gray level lookup table. Reprinted with permission from A. Rezaei, M. Defrise, G. Bal, C. Michel, M. Conti, C.Watson, and J. Nuyts, “Simultaneous reconstruction of activity and attenuation in time-offlight PET,” IEEE Trans. Med. Imaging 31, 2224–2233 (2012). Copyright C 2012 IEEE. Medical Physics, Vol. 43, No. 2, February 2016

An exemplary result is shown in Fig. 7. The algorithm was further analyzed, in particular, in terms of convergence properties. Defrise et al.50 found, assuming absence of randoms and scatter, that the MLACF algorithm monotonically increases the likelihood, that the sequence of iterates is asymptotically regular, and that the limit points of the iteration are stationary points of the likelihood. An alternative way of considering scatter and randoms in this class of algorithms has been proposed in Refs. 142 and 143.

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F. 8. MLACF reconstructions of the activity image (left) and attenuation factors (center) when the detector pair sensitivities are taken into account (top) and when they are ignored (bottom). True attenuation factors (top-right) and their multiplication with detector pair sensitivities (bottom-right) are shown for comparison. Reprinted with permission from A. Rezaei, M. Defrise, and J. Nuyts, “ML-reconstruction for TOF-PET with simultaneous estimation of the attenuation factors,” IEEE Trans. Med. Imaging 33, 1563–1572 (2014). Copyright C 2014 IEEE.

F. 7. Transaxial, coronal, and sagittal views of MLACF (top-left), MLAA (top-right), and MLEM (bottom-left) activity reconstructions of the 4 min 18F-FDG patient data postsmoothed with a Gaussian of 0.6 cm FWHM. MLEM activity reconstruction was obtained taking into account the CTbased attenuation image (bottom-right). The arrows indicate artifacts due to respiratory motion, which are more severe in the MLEM reconstructions. Reprinted with permission from A. Rezaei, M. Defrise, and J. Nuyts, “MLreconstruction for TOF-PET with simultaneous estimation of the attenuation factors,” IEEE Trans. Med. Imaging 33, 1563–1572 (2014). Copyright C 2014 IEEE.

It is an interesting aspect of this algorithm that Eq. (27) involves no forward- or backprojection, such that MLACF has complexity comparable to MLEM. In the ideal case of sit = 0, no more alternating updates are even necessary, and the attenuation factors need not be stored if ai = mi /pi is inserted into Eq. (24b). This separation of estimation of λ and µ in PET is similarly recognized in the global DCCs as a function of µ (independent of p when m is known; compare the first paragraph in Sec. 5), or the differential Eq. (19) as a function of a (independent of p). In all cases, instead of solving a joint optimization problem in two unknown vectors (λ or p and µ or a), a less complex problem in just one vector of variables (µ or a) can be solved. In contrast to algorithms estimating attenuation maps (such as MLAA), anatomical prior information, often defined in the image rather than the sinogram domain, is more difficult to incorporate in the class of MLACF-like algorithms described in this section.143 Furthermore, MLACF does not enforce consistency of the attenuation factors:134 hence, MLACF results may be noisier as can be seen in Fig. 7. However, following the reasoning in Sec. 2.A, this does not necessarily mean that results are less suited for AC. On the upside, lack of consistency enforcement in ACF opens up possibilities to account for additional, inconsistent influences. For example, Medical Physics, Vol. 43, No. 2, February 2016

the system normalization (that is, geometric sensitivity and detection efficiencies) has been estimated as part of the attenuation factors134—compare Fig. 8. Finally, attenuation factors along LORs not crossing the radioactive object cannot be estimated. In algorithms estimating attenuation maps, this missing information is implicitly complemented by enforced consistency; in MLACF, however, this may complicate reconstruction of a µ-map from the attenuation factors. In extreme cases, using highly specific PET tracers in dopamine144 or estrogen receptor PET (Ref. 145), the support of the activity distribution can be much smaller than that of attenuation. This can make additional information indispensable, to which we turn our attention in Sec. 7.

7. APPLICATIONS AND HYBRID METHODS Going beyond the general-purpose approaches discussed before, we now focus on more application-specific implementations. These are mostly defined by requirements specific to the examination of regions of the body, or availability of additional or a-priori information. We divide Sec. 7 into organ-specific applications without other additional information (Sec. 7.A), applications specific to PET/MRI (Sec. 7.B), applications making use of supplemental transmission information (Sec. 7.C), and motion-aware applications (Sec. 7.D). Methods are furthermore summarized for easy comparison in Table I of the supplementary material.146 7.A. Organ-specific applications

Inspired by the optimization procedure in Ref. 46 using 24 Fourier coefficients, an early work focused on refining lung contours, with the body contour estimated from the projection data and an initial lung contour found by warping an atlas to the NAC PET image.147 However, using the sum of only three Fourier coefficients of the global DCCs, the authors concluded

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that the interpatient variability of the lung attenuation coefficients prevents successful application of this method, the main challenge being errors in the used lung attenuation coefficient translating into the estimated lung contour. Building upon algorithms discussed before126,127—using emission data only— an extension toward thorax imaging included an intermediate clustering step of the attenuation map in three clusters (air, lung, soft tissue).148 More recent applications have focused on less complex geometries, such as knee, breast, and brain PET. In Ref. 149, FDG PET was compared to SPECT imaging and bone scintigraphy. For the PET data, the study compared transmissionbased with MLAA-based attenuation correction. In Ref. 150, DCCs were applied to refine breast contours obtained from thresholded projection data, by adapting the threshold based on the squared sum of seven global DCC Fourier coefficients. It should again be stressed, similar to the provisions near the end of Sec. 4.A, that the value of a segmented attenuation map depends not only on correct identification of tissue contours, but equally on the estimation of appropriate µ-values, in particular, in bone and lung tissue. In some brain applications, however, a uniform µ-map estimate from projections as described before in Ref. 68 may be sufficiently accurate.151 Going several steps further, Panin et al.143 used a uniform µ-map obtained from the slopes of brain projection data, but only to calculate an initial estimate of the ACF in MLACF. The ACF were then updated using two TOF bins. This study included both FDG as well as AV-45 patients, and was extended to 57 FDG patients.152 7.B. Applications specific to PET/MRI

As noted in Sec. 1, PET/MRI is a major driver for recent developments. Particularities of MRI-based AC involve the availability of high-resolution contour information for many tissue classes, the use of which is discussed in Sec. 7.B.1; but also the spatial limitation of the transaxial FOV and additional imaging equipment, for which correction methods are summarized in Sec. 7.B.2. It is noteworthy that even without explicit use of any MRI information, MLAA-based µ-maps were found to concord well with transmission-based µ-maps in phantoms and outperform those based on vendor-provided MRI segmentation in patients.153 7.B.1. Using MRI segmentation and classification

Image contrast between different tissues can be obtained using the vast accessory of MRI pulse sequences. This often allows for clustering of voxels into connected image segments (segmentation). Sometimes, but not always, it is also possible to uniquely assign a tissue class to each segment (classification)—for example, distinction between air in paranasal sinuses and cortical bone of the skull is particularly difficult.154 A whole-body segmentation was used in combination with TOF-MLAA (Ref. 122), where the LAC in each of the 3000 segments per patient was initialized with that of water. In order to decrease the number of variables to be estimated and hence the sensitivity to noise, the LAC in each region Medical Physics, Vol. 43, No. 2, February 2016

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was updated after averaging the voxel-wise MLTR updates in that region. Scatter and randoms were iteratively estimated from the current µ-map estimate, which made repetitions of the overall MLAA iteration scheme necessary for improved scatter and randoms correction. When expanding the available MRI information to comprise a tissue classification, several organ-specific applications can be considered, for example, for obtaining patientindividual attenuation coefficients in lungs or cortical bone. For whole-body imaging including the lung area, remember that the cross-talk between lung attenuation coefficient and lung outline has been described as a problem before.147 Benefiting from a MRI segmentation and classification of the subject volume in PET/MRI, stable MLAA estimation of the lung attenuation coefficient was proposed.132 In simulations, it was shown that when empirically correcting for bias from OFOV scatter and randoms, 30 s acquisitions may be sufficient to estimate the mean lung attenuation coefficient with less than 5% uncertainty. Using clinical, standard-protocol PET/CT data, this range of errors of the mean value was confirmed in voxel-wise estimation;155 the method described in this study used lungtissue preference and smoothing priors to estimate continuous lung attenuation coefficients and was able to compensate for respiratory-phase mismatch between PET and CT. A similar idea was followed for brain PET/MRI, where the scale of MLAA updates was driven using MRI information, namely, UTE-based µ-maps and T 1-weighted MR images to benefit from the MRI segmentation and classification.156,157 While the previous work assumed the MRI information is perfectly reliable in regions where attenuation is not updated, this is not necessarily the case. To take into account potential segmentation and classification errors as well as uncertainties in tissue attenuation coefficients, even in regions where reliable information was assumed available by other authors, uncertain regions such as bone, implants, and lungs can be identified and selectively updated through MLAA (Ref. 158). Alternatively, the posterior probabilities for each tissue class have been considered in Ref. 159; in this way, a MR-derived prior can be added to smoothing priors in a MAP approach. Similarly, MLAA has been combined with a Gaussian mixture model (GMM), smoothness priors, and a-priori bone probability maps for improved accuracy of attenuation reconstruction.160 Compared to using only the MRI information, PET reconstruction errors in patients (with respect to CT-based AC) could be reduced.161 A similar approach for explicit use in nonTOF PET/MRI, MR-MLAA (Refs. 162 and 163), incorporated smoothing and intensity priors into the MLAA update formula for µ, leading to essentially the same conclusions using 2-D simulated data. MRI priors can also be used in a more abstract way in the sinogram domain. For instance, in Ref. 164, extending the direct MAP estimation approach based on 2-D TOF consistency equations,108 forward-projected MRI data have been used as an anatomical prior with an information-theoretic similarity metric. While many of the aforementioned methods were evaluated on clinically realistic data, with timing resolutions of

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525–580 ps,122,153 the impact of other TOF resolutions was also studied. In simulation work, MRI-based-like µ-maps were used to initialize TOF-MLAA with timing resolutions of 667, 500, 333, and 167 ps,125 finding more accurate results for better TOF resolution. 7.B.2. Compensating for truncated MRI information

A drawback of MRI information is the transaxial limitation of the effective imaging FOV, leading to truncated anatomical information16 and, consequentially, incomplete MRI-based attenuation maps. Delso et al.165 have found 13.5% pulmonal and mediastinal lesion uptake underestimation in simulations due to truncated arms. A similar issue exists with MRIinvisible imaging equipment that cannot easily be represented in MRI-based µ-maps. While this is not an issue specific to PET/MRI, existing approaches [such as nonlinear extrapolation of the truncated transmission projections in PET/CT (Ref. 166)] are not readily translatable. Three main lines of thought of supplementing incomplete MRI information from emission data are described in this section. If the missing attenuation is soft tissue containing the tracer, the NAC PET image (or a PET image attenuation-correction with a truncated µ-map) may contain enough contour information to detected truncated tissue and complement the µ-map by setting it to the LAC of soft tissue in these regions. This approach was followed in Refs. 167–170, reducing errors from up to 50% to below 10% except near the edges of the FOV (Refs. 169 and 170); the method was further applied to TOF PET data.171 Using the same emission data, more quantitative results can be obtained by application of MLAA in order to take into account varying LACs in the truncated regions. This was

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studied in detail in Ref. 131, again only updating the µ-map in regions of unreliable MRI information (Fig. 9). This approach is also able to identify cold (nonradioactive) objects which the NAC-PET-based methods fail to recover; however, this becomes more challenging the further away these objects are located from the sources. An application of this technique to dynamic and gated PET data has also been described.172 An open question with this type of algorithm is whether exact scatter-correction is necessary—which may require the complete µ-map to begin with—or whether approximate scatter correction may be sufficient. It can be assumed that the answer to this depends highly on the requirements of the application in question, and still requires dedicated quantitative evaluation. As a third approach, scattered PET data may contribute to solving this problem, in particular, for locating cold objects; this recently described idea will be considered in Sec. 9. 7.C. Applications using additional transmission information

Supplemental transmission information can come in three flavors: in the form of a transmission (or combined emission– transmission) sinogram, as partial or only partially reliable µ-maps, or as deformed µ-maps. The former two cases were studied in particular to initialize or complement simultaneous reconstruction of attenuation and activity, and advances will be summarized in Secs. 7.C.1 and 7.C.2. In addition, attenuation properties may be considered sufficiently constant over time and available transmission information reliable enough that an accurate µ-map for PET image reconstruction can be obtained by spatial deformation of available µ-maps (Sec. 7.C.3).

F. 9. Transaxial and coronal slices of the original attenuation image, the truncated image, and the MLAA images obtained with and without scatter correction. Reprinted with permission from J. Nuyts, G. Bal, F. Kehren, M. Fenchel, C. Michel, and C. Watson, “Completion of a truncated attenuation image from the attenuated PET emission data,” IEEE Trans. Med. Imaging 32, 237–246 (2013). Copyright C 2013 IEEE. Medical Physics, Vol. 43, No. 2, February 2016

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7.C.1. Using transmission sinograms

First research in this area was described in Ref. 88, which has been covered in Sec. 6.A. In Ref. 117, source-assisted methods were proposed through a Bayesian adaptation of the µ-map update equation in the aforementioned MLAA-like formalism; the aforementioned method in Ref. 115, minimizing a joint objective function with alternating updates based on paraboloidal surrogates, also considered the transmission scan given knowledge about the blank scan. The same is true for Ref. 127, applying Newton-like algorithms to the loglikelihood functions. Some work considered the emission data only for a minimal amount of information: in Ref. 173, DCCs were used in order to control threshold segmentation of a separate, twominute 68Ge transmission scan in whole-body PET; the authors also envisaged obtaining the attenuation coefficients through this technique, given another reliable segmentation technique would be available. In the same spirit, regularization parameters for reconstruction of low-statistics transmission measurements have been selected based on the emission data and DCCs (Ref. 174). The foundation for modern, TOF-assisted reconstruction of attenuation from simultaneous emission and transmission measurements were laid in Ref. 175. Using a stationary annulus transmission source filled with 18 MBq of FDG, simultaneous emission (EX) and transmission (TX) simulations were separated based on TOF information. However, only TX information was used for reconstruction of attenuation. This approach was subsequently validated in patients using 50 MBq transmission source activities.176 For simultaneous EX and TX measurements, an array of source configurations were proposed. A number of steps toward quantitative, fully simultaneous EX+TX measurement and reconstruction were made in Ref. 177, where a nonstationary 68Ge source rotated around the patient. Separation of EX and TX data using TOF was used only to initialize reconstruction, which was a regularized MAP EX-TX reconstruction of attenuation and activity maps based on MLAA. Integrated EX-TX reconstruction has the advantage of not requiring an explicit, error-prone separation of EX and TX data; not even needing the source to be traced during acquisition. However, TX and EX scatter require elaborate corrections which shall be briefly presented. A TOF TX scatter simulation was required for TX-based µ-map reconstruction. Therefore, scatter was ignored in a preliminary TX-based µ-map reconstruction and the result rescaled. The scatter-contaminated µ-map was then used to simulate and subsequently smooth non-TOF TX scatter, and this TX scatter estimate could be used to reconstruct an improved, scatter-corrected µ-map to initialize EX-TX reconstruction. TOF TX scatter was estimated from non-TOF TX scatter following geometric arguments. TOF EX scatter was subsequently estimated using standard tools after subtraction of the attenuated blank scan and TOF TX scatter from the measurement. In contrast to aforementioned works with rather complete TX information, a sparse transmission (sTX) source consisting Medical Physics, Vol. 43, No. 2, February 2016

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of only few (7–20) fixed line sources was used in Refs. 178 and 179. In measurements, line sources were implemented by 68Ge positron beams;178 the incomplete sTX data were acquired simultaneously with the non-TOF emission data and input to a coupled MLAA-like algorithm (Joint Emission and Transmission Tomography, JETT). As in other works,131,132 the µ-map may be partially known and µ-map updates restricted to regions of unknown attenuation coefficients.178 Interestingly, 20 discrete sources were found to outperform a continuous ring source.179 Results were further compared to EX-only as well as TX-only µ-maps.178,179 Work in Refs. 175 and 176 was extended, again using a FDG annulus source, in that same direction via a variant TOF-MLAA (MLAA+). Similar to Ref. 178, MLAA+ was compared to regular MLTR-MLEM (using only TX data for attenuation) and regular MLAA (using only EX data) in simulation113 and patient180 studies. Finally, an interesting alternative to using artificial transmission sources was investigated in Refs. 181 and 182 by using LSO background radiation emitted during the 176Lu β − decay cascade; the reconstructed µ-maps were used to initialize MLACF in this work, and it was proposed to also use the information for scatter correction. 7.C.2. Using partial or unrealiable µ-maps

In some applications, partial µ-maps may be available and need to be complemented. In the most simple case, a complete attenuation map only shows few localizable artifacts as from a CT contrast agent. Ideally, these artifacts can be roughly localized in a ROI and a binary constraint can be applied, forcing the µ-map value in each pixel of the ROI to either the original value or the LAC of blood. In Refs. 183–185, approaches based on alternative updates of activity and attenuation were proposed; Ref. 186 investigated using several DCCbased methods for choosing between an artifactual and the correct µ-map. A situation similar to that described in Sec. 7.B.2 arises when data are axially (instead of transaxially) truncated. This is the case when overscanning PET in bed positions extending beyond the CT-based µ-map,142 for example, due to radiationdose concerns. For these cases, a variant MLACF including nonzero background events was used to estimate attenuation factors along oblique LORs. This approach was subsequently applied to continuous bed motion (CBM) PET (Ref. 187). For various reasons, even complete CT scans in PET/CT, usually considered the gold standard for AC, may be considered to be of low reliability everywhere. Reasons for this include the sequential nature of PET and CT scanning, or the difficulty of rescaling Hounsfield units for PET attenuation coefficients. Nonetheless, the available information is valuable and shall be used as a prior, if possible. Therefore, Ref. 177 also investigated the case of TX-less reconstruction using a CT-based µ-map (and a CT-based scatter estimate) in the aforementioned MLAA-like method by setting the blank scan measurement to zero. Furthermore, Ref. 188 investigated the use of CT-based µ-maps in quadratic priors in regularized variants of MLACF.

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Note the conceptual similarity to the use of a soft MRI classification in MLAA-GMM (Ref. 160); however, being based on MLACF, algorithms in Ref. 188 are formulated in terms of attenuation factors. As a result, attenuation factors outside of the emission sinogram support could not be estimated and were instead derived from CT-based µ-maps without further modification based on emission data. 7.C.3. Registration of available µ-maps

When available µ-map information is sufficiently reliable, it may only require additional spatial deformation before use in AC. Using DCCs, this application has a long history. In Refs. 84 and 89 (for SPECT) and Ref. 46 (for PET), the attenuation distribution was assumed to be an affine distortion of a known prototype attenuation distribution, and the system of data consistency conditions was solved for this affine distortion using the Levenberg–Marquardt algorithm. These 2-D studies were followed by research on 3-D templates, this time using simplex methods evaluated in simulations and phantoms189 and, later, in cardiac PET/CT patients using rigidbody registration.190 More recent proposals involve TOF PET data and iterative methods. Maximum likelihood reconstruction of the transformation between PET and CT images and activity (MLTA), estimating parameters of a rigid-body transformation, was subject of a 2-D simulation study.191 A 3-D simulation study investigated nonrigid deformation of the attenuation template, where the registration is embedded in an MLAA-like reconstruction scheme;192 parameter updates for the Demons registration algorithm193 were driven by the MLTR updates. In a work already closer to clinical practice, mismatched (shifted or end-inspiration) CT images were used to initialize MLAA for cardiac PET applications;123,194 this approach was able to remove artifacts which were apparent when using the mismatched CT for AC without MLAA. 7.D. Motion-aware applications

Few of the works so far have considered motion as a potential source of artifacts; in most studies, motion is either not included in simulations or its influence boldly ignored. However, its impact especially on whole-body and lung imaging, can be significant. Having, as in Sec. 7.C.3, a static µ-map available, one may assume simultaneous motion of attenuation and activity, estimate a motion-field from NAC PET images, and deform the static µ-map accordingly.195 More in line with the topic of this review, in an integrated approach, others have formulated the likelihood as a function of warped activity and attenuation and deformation (maximum likelihood joint image reconstruction/motion estimation, ML-JRM); one then notes that only an arbitrarily warped µ-map is required to solve for λ and µ in all gates.196 Using a similar algorithm (minimal MRI prior joint image reconstruction/motion estimation, MPJRM), incomplete dynamic MRI information can also be integrated.197 Reliance on prior information makes this approach resemble those in Sec. 7.C.3; one key difference, however, is the amount Medical Physics, Vol. 43, No. 2, February 2016

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of deformation that may be required to match this information to the emission data. In applications such as those in Sec. 7.C.3, this usually involves relatively large deformation to achieve coregistration despite patient repositioning; in this section, by contrast, the focus is more on repetitive motion with smaller amplitudes and exploitation of the consistency of the attenuation and activity distribution across PET gates. Alternative approaches, such as several of the following ones, may not even rely on any prior information. In Ref. 198, it was shown that TOF-MLAA as implemented in Ref. 177, despite ignoring the effect of motion, yields more accurate µ-maps than CT (which is blind to effects of slower motion) in thorax PET/CT. A similar conclusion has been drawn in the context of parametric imaging: MLACF results after initialization with a mismatched (inspiration) µ-map converge to those for the matched (averaged) µ-map.199 In more detail, the effect of three types of motion (that of activity, of attenuation, and simultaneous motion of both activity and attenuation) on TOF-MLAA in nongated PET was studied in Ref. 200, and it was found that the method is nonetheless prone to inconsistencies from motion of attenuation and simultaneous motion. A method for correction of simultaneous, nonrigid motion in gated TOF PET was proposed in Ref. 201: in this work, the transformation parameters for each frame were estimated by joint registration of that frame’s postreconstruction MLAA output (using a common transformation for attenuation and activity) to that of a reference frame. This motion-correction method was subsequently refined by the same group,202 this time integrating it with MLACF. From only gated TOF PET data, this integrated approach reconstructs a single motion-corrected activity image as well as deformation parameters to deform this activity image to the respective frames. Since MLACF does not guarantee availability of µ-maps, the registration is based on the activity maps only. Using an implicit, virtual reference frame slightly improved the performance of this method and eliminated the need to chose an explicit reference frame. An alternative approach, also using intrareconstruction registration of multiple gates, has been described in Ref. 203, using the ADMM to estimate activity distribution and attenuation sinogram. Both approaches used a Demons registration algorithm.193 8. SPECT-SPECIFIC STUDIES Remarkably, the adoption of SPECT/CT, and thus CTbased AC in SPECT, was (and still is) slower than that of PET/CT, which has virtually eliminated standalone-PET (Refs. 204 and 205). Reasons for slow adoption of SPECT/CT include the low number of indications where SPECT/CT is clinically needed,205,206 and hence a lower relative benefit of CT in SPECT, also in terms of acquisition times;205 higher relative cost of CT equipment in SPECT/CT than in PET/CT (Ref. 205); and finally, the trend to transition from SPECT to PET in at least some applications,207 affecting both manufacturers’ and hospitals’ incentives to develop and implement new SPECT technology. For all these reasons, in the prePET/MRI era, intense research on simultaneous reconstruction for SPECT continued for a couple of years longer than for PET,

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with a remarkable number of papers that appeared between 1999 and the early 2000s. For this following section on SPECT-specific literature, it is useful to distinguish the attenuated from the exponential Radon transform (ERT), which is characterized by the common base equation Eq. (2) in combination with a E (s,φ,l) = exp(+µl),

(28)

where µ ∈ R is a constant. While this transform does not offer an immediate physical interpretation, it readily relates to the attenuated Radon transform in the case of (known) constant attenuation in a bounded, convex set.208,209 It appears that the exponential Radon transform (knowing µ) was inverted some 20 years before the attenuated Radon transform;208,210,211 hence, some of the following studies focused on simultaneous SPECT reconstruction assuming constant attenuation in a bounded, convex set—as had before Ref. 82. In that case, the 2-D identification problem can be uniquely solved if and only if λ is not a radial function.212 The case of unknown elliptical, homogeneous attenuation was considered in Ref. 213. Consistency conditions related to the similarities between ERT and the Laplace transform were used to derive a downhill simplex algorithm converting SPECT into ERT data. These ERT data, which can be considered attenuation corrected, were then reconstructed in a separate step—for both 2-D parallel-beam geometry and 3-D rotating slant hole geometry. Similar DCCs for the attenuated Radon transform were used to recover more complicated (including nonconvex) µmap outlines, for example, with the ConTraSPECT algorithm.84 Without a transmission scan, elliptical as well as nonconvex spline curves were sought, as well as sinogram regions for truncation compensation of transmission data.214 The influence of count rates, attenuation coefficients, and photopeak scatter was studied in Ref. 215. To reconstruct nonconstant µ-maps, Tikhonov regularization was often used for smoothness of the result, usually in combination with a Newton algorithm.216 The nonlinear leastsquares problem can also be simplified with the pseudoinverse of R µ to separate out a problem of smaller dimension (with unknown µ only): this was done in two studies for discrete data consistency conditions by Bronnikov.217,218 Again, smoothing priors and a Newton method were applied. Note that the later approach allows a more efficient implementation thanks to a formulation using QR and Cholesky decompositions. Furthermore, while separating out a problem in µ is similar to the application of global DCCs (compare Secs. 3.B and 5), this approach applies to a much wider range of geometries, including 3-D imaging; it is not restricted to the ideal Radon transform and the related CC. In that aspect, it is similar to MLAA and MLACF. Transmission data, even possibly truncated, can serve to define additional linear equality constraints, as was shown in Ref. 219. The studies on general-purpose optimization approaches for the attenuated Radon transform further comprise,220 which compared Gaussian vs Poisson objective functions and several regularizers. Finally, linear optimization was the objective of Ref. 221: after linearization of the attenuated Radon transMedical Physics, Vol. 43, No. 2, February 2016

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form (decomposition into a sum of components linear in λ and bilinear in λ and µ, respectively), Landweber and conjugategradient methods were used for optimization. During that same time, alternating updates of µ and λ have been proposed,116,117,222 the application of which in PET is discussed in Sec. 6. Simultaneous updates of µ and λ have also been pursued: based on ART-intrinsic transmission SPECT (ART-IntraSPECT),223 a simultaneous variant of multiplicative ART, expectation–maximization methods with simultaneous updates were put forward. Examples include EM-IntraSPECT (Ref. 140), which was among the first algorithms to consider both emission and transmission data, as well as OSEMIS, an EM-IntraSPECT extension using ordered subsets.224 An idea that was originally proposed for correction of truncated SPECT transmission measurements is the application of a knowledge set of complete attenuation sinograms and maps.225 Such a knowledge set can be constructed from an a-priori database of µ-map cross-sections by applying principal component analysis, and can be used to approximate µmaps outside the database using a linear combination with few most important coefficients. In Ref. 226, this approach was applied to the case without transmission information: an optimization problem based on the squared norm of the DCCs was constrained to search for linear combinations of most relevant members of the knowledge set. Regularization was performed as a function of the eigenvalues of the knowledge set members and regularization parameters chosen according to the L-curve method.227 A non-DCC-based variant of this method was proposed in Ref. 228: in this work, the attenuated Radon transform was linearized, the µ search space again constrained by the knowledge set, and the optimization problem solved by a conjugate-gradient method. More abstract prior information was considered in Ref. 229, namely, a 3-D organ model and predefined attenuation coefficients to parameterize a µ-map with air, lungs, soft tissue, and bone compartments: a modified MLEM algorithm was used to drive updates of the region boundaries for 99m Tc cardiac SPECT. For dual-isotope SPECT, an idea presented earlier230 was applied to simultaneous reconstruction. The differential attenuation method makes use of the observation that a point source can be identified using two photon energies (such as 75 and 167 keV for 201Tl), even if attenuation is unknown. This is explained by a common activity (before considering branching ratios) and attenuation lengths, but different attenuation coefficients for the two energies. The least squares problem was equipped with several penalty terms for smoothness and tissue preference.231 This study of scatter-free numerical studies was followed by an anthropomorphic phantom study including the effect of scattering.232 A rather empirical approach of finding ACF for SPECT has been described in Ref. 233, using Fourier decomposition of the sinogram of ACF and optimization driven by DCCs. An interesting kind of prior was proposed in Ref. 234: formulating an optimization problem for µ, the authors derived a simple DCC from a reprojected reconstruction and the measured data, and constrained the search space to segmented µ-maps with a

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given topology; therefore, the simulated-annealing algorithm was equipped with a topology-preserving constraint. Finally, a first combination approach using alternating updates (MLEM for λ, and MLEM-like for µ as in Ref. 235) and DCCs was proposed in Ref. 236. In their work, the authors exploited the µ–λ crosstalk for segmentation and used a DCC-based algorithm to find each segment’s optimal µ value. 9. BEYOND TRUE COINCIDENCES As presented in Sec. 3, a theoretical obstacle with some practical implications in PET is posed by incompleteness and nonuniqueness of the estimated attenuation sinogram, and hence the unavailability of a complete µ-map, when estimated from unscattered coincidences. Possibly, imagereconstruction methods established for CT reconstruction from truncated projection data [interior CT (Ref. 237)] may prove valuable in this new setting and recover some of the missing information. Yet, this remains to become an object of future studies. The situation is very similar in SPECT, except that attenuation and emission information are more difficult to separate and algorithms operating on attenuation sinograms are even more difficult to employ. As an alternative solution to the above problem, several groups have considered using scattered emission data as an additional source of information. This extends earlier suggestions of using individual photon energies in PET (Refs. 238 and 239) and, more specifically, applies the idea of Compton scatter (emission) imaging240,241 to SPECT and PET. In the remainder of this section, unless noted otherwise, we use the term scatter for scattered photons (or coincidences involving the same) in the energy range below the photopeak, acquired in one or more separate scatter acceptance windows. Methods are summarized in Table II of the supplementary material.146 As many developments, this idea has first been followed in SPECT, where the data acquisition using additional energy windows is a common approach to photopeak scatter correction.242 In Refs. 76 and 243–246, single photons emitted from 99m Tc were acquired in two windows: a photopeak window around 140.5 keV, and a scatter window covering approximately 90–120 keV, respectively. References 76, 243, and 244 used automatic operations on the lower-energy projection data to derive an outer patient contour; among these, Ref. 243 operated a filter that removed the need for empirical choices of thresholds. In Refs. 245 and 246, photon counts acquired in both windows were independently reconstructed with standard algorithms for activity reconstruction and used to guide interactive segmentation of lungs and nonpulmonary tissues of the chest. This method has also been considered for 201Tl SPECT imaging,247 as have setups detecting Compton-scattered photons from dedicated 99m Tc-MAA injections248 or external sources.249,250 The idea of reconstructing quantitative information from scatter relies on the availability of photon energy measurements, through which a scattering angle can be estimated. It has probably premiered in SPECT (Ref. 251), with an MLEM reconstruction of scattered coincidences from multiple energy windows using a scatter system matrix, which depends on Medical Physics, Vol. 43, No. 2, February 2016

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the current activity estimate. Another MLEM approach using a path-based formalism for scattered coincidences in listmode SPECT was the object of theoretical studies.252,253 An iterative µ-map reconstruction algorithm from scatter named scatter based maximum likelihood gradient ascent [SMLGA (Ref. 254)] has recently been reported and also combined with MLAA for simultaneous reconstruction of activity (from unscattered photons) and attenuation (from unscattered as well as scattered photons). While SMLGA-MLAA was shown to reduce the activity-attenuation crosstalk apparent in MLAA alone, it was unable to completely prevent it. In PET, using scatter for attenuation-map reconstruction is still at an even earlier stage of development. A report on the iterative reconstruction of activity maps from both unscattered and scattered coincidences in TOF PET (Ref. 255) sparked additional work in both this area256–259 and the reconstruction of attenuation maps.260–263 Results suggest that scatter has the potential of resolving both incompleteness261–263 and nonuniqueness261 of attenuation information. However, so far the proposed algorithms show good performance only with relatively point-like activity sources. Also, a high computational demand is indicated by the presentation of low-resolution examples and use of relatively few coincidences. Further, the techniques rely on accurate photon-energy measurements and hence, good to perfect overall energy resolution; consequently, the aforementioned works were carried out using simulated data. Some results261 indicate that focused improvements of PET energy resolution can be beneficial, for example, employing lanthanum() bromide (LaBr3)-based scintillators265 or cadmium zinc telluride (CdZnTe, CZT)based detectors.266 Since PET is based on the principle of electronic collimation (rather than mechanical, as SPECT), the sensitivity of a PET scanner for scattered radiation is more complex. It is a well-known fact that the set of possible annihilation locations for an unscattered coincidence is a LOR. By contrast, the set of possible scattering locations for a single-scattered coincidence, knowing the scattering angle, is represented by the surface of an American-football-like volume (Fig. 10). The geometry of this problem has become the object of several studies, offering dedicated coordinate systems for image reconstruction53,264 compatible with an earlier model of photopeak scatter in a two-detector PET scanner.267 For both SPECT and PET, the influence of photopeak scatter and, in particular, detector scatter and multiply objectscattered lower-energy photons remains to be studied,254,261 as well as potential correction approaches. This may restrict the usable energy window to a region between the Compton edge and the photopeak. Also, scatter simulations need to be evaluated at lower energies. In that aspect, it is interesting to note that the scatter scaling in PET can be estimated using discrete data consistency conditions from all (photopeak) emission counts268 as well as using extensions of MLEM, MLAA, or MLACF by treating the scatter scaling as an additional voxel269,270 in these algorithms. While these methods still require a scatter profile in sinogram space, use of data from secondary energy windows has also been envisaged.269 As a further source of information offering potential

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F. 10. Illustrations of the attenuation information gained from h scattered PET coincidences. Top. 2-D geometry of a single-scattered PET coincidence, knowing the scattered-photon energy and thus the scattering angle. Yellow: annihilation location, red: broken LOR, blue: set of possible scattering locations used for reconstruction of a scatter-based µ-map. Reprinted with permission from H. Sun and S. Pistorius, “Characterization the annihilation position distribution within a geometrical model associated with scattered coincidences in PET,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2014), 3pp. Copyright C 2014 IEEE. Bottom. 3-D surfaces of response, again representing possible scattering locations, for different scattered-photon energies. Reprinted with permission from Y. Berker, F. Kiessling, and V. Schulz, “Scattered PET data for attenuation-map reconstruction in PET/MRI,” Med. Phys. 41, 102502 (13pp.) (2014). Copyright C 2014 AAPM.

for improvement in PET, TOF information of scattered coincidences has yet to be taken into account.261 10. DISCUSSION AND CONCLUSION Simultaneous reconstruction of attenuation and activity has seen a tremendous development in the past nearly 40 years. Without a doubt, the explosion of computational resources has contributed significantly to this development. Meanwhile, the need for quantitative accuracy has grown as well, with clinicians demanding an image quality that would have hardly been dreamable in the beginning. In PET imaging, we have identified TOF information as a crucial enabler to keep up with this growing demand, and we expect that with improving timing performance of future scanners, inherent PET redundancy can be exploited to an even higher degree. This development will comprise the current major driver of simultaneous reconstruction, namely, hybrid and especially simultaneous PET/MRI, as the first TOF PET/MRI systems are commercially available;4 in fact, some PET/MRI applications with highly specific PET tracers may strictly require TOF information. Outside of PET/MRI, timing resolutions down to 300 ps are already commercially available, and there may be potential both for translating this to PET/MRI as well as for even higher performance.100 It will be interesting to see how SPECT will be able to keep up with PET in this respect, and whether the missing TOF information in SPECT will influence ongoing efforts toward simultaneous, clinical SPECT/MRI (Ref. 271). On the one hand, already today, TOF allows for more sophisticated Medical Physics, Vol. 43, No. 2, February 2016

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applications in PET than SPECT. In particular, the MLACF formulation in TOF PET allows to fully integrate recovery of attenuation information and activity reconstruction in the case without randoms and scatter. On the other hand, the µ–λ crosstalk is generally less significant in SPECT than PET (Fig. 5). Two challenges that certainly need to be solved before any large-scale clinical application of simultaneous PET reconstruction are the unknown offset in the attenuation sinogram (Sec. 3.D) and the need to stabilize the attenuation sinogram near the patient boundaries (Sec. 5.B); the latter, while not an issue in theory, is a practical problem related to low statistics on LORs just barely crossing the patient. Both issues are closely linked (Sec. 3.E) and find a possible, common solution in the use of additional transmission information. In PET/MRI, this may make a case for multimodality PET/MRI/low-dose CT scanners, but several other solutions are possible. In terms of algorithms, a downside of current MLACF implementations is that data consistency of µ information is not considered: while this allows normalization to be carried out in passing (Fig. 8), it is still an inherent advantage of MLAAbased implementations that they ensure, at each iteration, that the ACFs correspond to a physically meaningful image-space µ-map through a back-projection into image space. This becomes more of an issue the longer the axial FOV of a scanner is, since the number of LORs grows quadratically, while the number of voxels only linearly. MLAA also simplifies incorporation of prior information (as in truncation compensation), which can, for example, easily be integrated by updating µ values in unreliable regions only. µ-maps (in contrast to just ACFs) are also required for many approaches of scatter correction in activity reconstruction. In addition to that, scatter-correction of photopeak data is required for recovery of quantitative attenuation information, but photopeak scatter is corrected using CT data (or simply ignored) in many studies. In some of the few studies that do consider photopeak scatter, this correction constitutes a major part of the respective work.122,131,177 The hypothesis that photopeak scatter can be corrected using an approximate µmap to yield an improved µ-map131 still needs verification on a larger scale. Another aspect requiring further research is how the performance of various algorithms varies with the radioisotope (due to effects such as photon energy or positron range) as well as the targeting molecule (due to biodistribution). So far, this was mostly considered in the very early days before the predominance of FDG in PET. With tracers more specific than FDG, nonuniqueness and other ambiguities discussed before may be much more relevant. In these cases, there may be no way around using additional information such as transmission or scatter measurements; nonetheless, unscattered emission data may remain to be useful to correct localized or other artifacts, for example, from CT contrast agents or metal implants. For any tracer, large-scale clinical evaluation is needed. Finally, despite their limited direct practical impact, we expect studies on analytic methods to continue to provide insight and guidance for development of advanced iterative methods. Two interesting problems shall be mentioned here.

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First, an energy-resolved single-scatter sinogram has an additional dimension compared to a sinogram of true coincidences: it is thus possible that single-scattered coincidences fulfill consistency conditions similar to those for true coincidences. Such DCCs may be useful both for correcting and using scatter. Second, uses of consistency equations other than attenuation estimation include Fourier-based rebinning: however, to date, rebinning requires attenuation-corrected data to begin with. A combined rebinning–attenuation-estimation would be highly interesting. In summary, future research will certainly be directing along improving PET detector timing, finding the unknown attenuation-sinogram offset and patient boundary, enforcing ACF consistency, correcting and using scatter information, investigating nonFDG tracers, and clinical evaluation. Some of these may require combination of approaches and data from different origins to generate satisfactory results. In the larger picture, any advance may bring closer the not too far aim of 4-D coregistered attenuation correction.

ACKNOWLEDGMENTS The authors thank Dr. Joel Karp for helpful comments on the manuscript, and Dr. Michel Defrise for pointing out Asgeirsson’s mean value theorem. Research reported in this publication was supported by a fellowship within the Postdoc Program of the Deutscher Akademischer Austauschdienst/ German Academic Exchange Service (DAAD), and by the National Cancer Institute of the National Institutes of Health under award number R01CA113941. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Two patent applications relevant to this work have been filed by Koninklijke Philips Electronics N.V. with Y.B. as first inventor.

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Scattering and attenuation correction in emission tomography in the nuclear industry.

MR.

Perfusion quantitation using positron emission tomography.

computed tomography for bone tumors (Review).

Partial-volume effect correction in positron emission tomography brain scan image using super-resolution image reconstruction.

Anatomically derived attenuation coefficients for use in quantitative single photon emission tomography studies of the thorax.

Stimulated emission tomography.

Positron emission tomography.

Positron emission tomography-magnetic resonance imaging: technical review.

Positron emission tomography alone, positron emission tomography-computed tomography and computed tomography in diagnosing recurrent cervical carcinoma: a systematic review and meta-analysis.

Using positron emission tomography to study human ketone body metabolism: a review.

positron emission tomography in neuroscience research.

Positron Emission Tomography (PET) in Oncology.

Fluorodeoxyglucose positron emission tomography in eosinophilic fasciitis.

Positron emission tomography in Shy-Drager syndrome.

Positron Emission Tomography in Breast Cancer.

Positron emission tomography in spinal infections.

Developments in neuroimaging: positron emission tomography.

Positron emission tomography imaging in sarcoidosis.

Positron emission tomography in progressive supranuclear palsy.

Thalamocortical diaschisis: positron emission tomography in humans.

Positron emission tomography in cerebrovascular disorders.

Measurement of radiotracer concentration in brain gray matter using positron emission tomography: MRI-based correction for partial volume effects.

computerized tomography in lung cancer.