The Generalize NEQ an Detectability Inex for Tomosynthesis an Cone-Beam CT: From Cascae Systems Analysis to Human Observers G. J. Gang, a J. Lee, b J. W. Stayman, c D. J. Twar, c W. Zbijewski, c J. L. Prince, b an J. H. Siewersen a,c a Institute of Biomaterials an Biomeical Engineering, University of Toronto, Toronto ON, Canaa M5GM9; b Electrical an Computer Engineering, Johns Hopkins University, Baltimore MD, USA 118 c Dept. of Biomeical Engineering, Johns Hopkins University, Baltimore MD, USA 105 ABSTRACT Purpose: In the early evelopment of new imaging moalities such as tomosynthesis an cone-beam CT (CBCT) an accurate preictive moel for imaging performance is particularly valuable in ientifying the physical factors that govern image quality an guiing system optimization. In this work, a task-base cascae systems moel for etectability inex is propose that escribes not only the signal an noise propagation in the D (projection) an 3D (reconstruction) imaging chain but also the influence of backgroun anatomical noise. The extent to which generalize etectability inex provies a vali metric for imaging performance was assesse through irect comparison to human observer experiments. Methos: Detectability inex ( ) was generalize to inclue anatomical backgroun noise in the same manner as the generalize noise-equivalent quanta (NEQ) propose by Barrett et al. (Proc. SPIE Me. Imaging, Vol. 1090, 1989). Anatomical backgroun noise was measure from a custom phantom esigne to present power-law spectral ensity comparable to various anatomical sites (e.g., breast an lung). Theoretical calculations of as a function of the sourceetector orbital extent (θ tot ) was obtaine from a 3D cascae systems analysis moel for tomosynthesis an cone-beam CT (CBCT). Four moel observers were consiere in the calculation of : prewhitening (PW), non-prewhitening (NPW), prewhitening with eye filter an internal noise (PWE), an non-prewhitening with eye filter an internal noise (NPWE). Human observer performance was measure from 9AFC tests for a variety of iealize imaging tasks presente within a clutter phantom. Theoretical results ( ) were converte to area uner the ROC curve (A z ) an compare irectly to human observer performance as a function of imaging task an orbital extent. Results: Theoretical results emonstrate reasonable corresponence with human observer response for all tasks across the continuum in θ tot ranging from low-angle tomosynthesis (θ tot ~10 o ) to CBCT (θ tot ~180 o ). Both theoretical an experimental A z were foun to increase with acquisition angle, consistent with increase rejection of out-of-plane clutter for larger tomosynthesis angle. Of the four theoretical moel observers consiere, the prewhitening moels tene to overestimate real observer performance, while the non-prewhitening moels emonstrate reasonable agreement. Conclusions: Generalize etectability inex was shown to provie a meaningful metric for imaging performance, helping to brige the gap between real observer performance an prevalent Fourier-base metrics base in first principles of spatial-frequency-epenent NEQ an imaging task. Keywors: cone-beam CT, anatomical noise, anatomical clutter, cascae systems analysis, noise-power spectrum, noise-equivalent quanta, etectability inex 1. INTRODUCTION Recent years have witnesse a ramatic avancement in the scope of avance 3D imaging applications base on flatpanel etectors, such as tomosynthesis an cone-beam CT (CBCT). Each of these approaches aims to improve the conspicuity of subtle lesions through epth localization an the rejection of out-of-plane anatomical clutter, with a broa spectrum of potential applications ranging from breast an chest imaging to image-guie interventions. The knowlegeable evelopment of such moalities stans to benefit from a general quantitative framework for performance Meical Imaging 010: Physics of Meical Imaging, eite by Ehsan Samei, Norbert J. Pelc, Proc. of SPIE Vol. 76, 760Y 010 SPIE CCC coe: 1605-74/10/$18 oi: 10.1117/1.84546 Proc. of SPIE Vol. 76 760Y-1
moeling, assessment, an system optimization. Two common approaches to image quality assessment inclue physical characterization of system performance [e.g., prevalent Fourier-base metrics of spatial-frequency-epenent signal an noise] an human observer-base characterization [e.g., sensitivity an specificity as assesse in a receiver operating characteristic (ROC) test]. A variety of avantages an limitations can be appreciate for each approach for example, the former is simple an practical, yet subject to numerous assumptions of linearity an shift-invariance, while the latter irectly assesses real observer response, but is time consuming, expensive, an epenent on the generation of real (or highly realistic) images. A general quantitative approach that begins with simple Fourier metrics an briges the gap to real observer performance woul be valuable in the esign an optimization of such imaging systems. Previous work has extene cascae systems analysis of the noise-equivalent quanta (NEQ) for projection raiography 1-4 to avance D an 3D imaging moalities 5-8. Recognizing the importance of imaging task in efining meaningful image quality metrics, a task-riven approach is aopte in which NEQ is combine with the imaging task an moel observers to form the etectability inex ( ). The goal of the current stuy is to investigate the extent to which etectability inex is a vali metric for system performance an optimization through comparison to real human observers. The work below represents preliminary stuies spanning initial analysis an experimentation involving a small number of simple imaging tasks, with a more complete evaluation, incluing a greater variety of imaging conitions an imaging tasks, to be pursue in future work.. GENERALIZED DETECTABILITY INDEX.1 Cascae Systems Analysis (CSA) Moel for Tomosynthesis an CBCT Previous work has emonstrate the utility an accuracy of cascae systems analysis of the performance of D x-ray projection imaging systems base on flat-panel etectors. 3,4 Such has more recently been extene to 3D FPD-base CBCT an tomosynthesis systems, 5-8 proviing a general framework for moeling the 3D moulation transfer function (MTF), noise-power spectrum (NPS), NEQ, an etective quantum efficiency (DQE). The basic approach provies quantitative moeling of the signal an noise propagation in the D an 3D imaging chain as illustrate in Fig. 1. The moel consists of the physical processes of D projection image formation followe by the mathematical processes of 3D filtere backprojection reconstruction. The former has been extensively investigate an evelope over the last 15 years. 1-4,9 The reconstruction cascae inclues 7,8 : log-normalization of projection ata (Stage 8); application of the ramp filter (Stage 9); optional application of apoization filters(stage 10); interpolation of the D projection ata (Stage 11); backprojection of D projections, moele as superposition of the D NPS along vanes in the 3D Fourier omain (Stage 1); an sampling of the 3D reconstruction (Stag 13). Such a moel has emonstrate agreement with measurements of the 3D NPS an NEQ for FPD-CBCT over a broa range of experimental conitions an reconstruction techniques 7,8, proviing a moel for imaging performance of tomosynthesis an CBCT as a contimuum represente by varying extent of acquisition angle (θ tot ). Figure. 1. Cascae systems analysis for 3D tomosynthesis an CBCT. Stages 0-7 moel signal an noise transfer through various physical processes in the etector, while Stages 8 to 13 represent mathematical processes of 3D reconstruction by filtere backprojection as etaile in previous work: 0.) incient x-ray quanta; 1.) interaction of x-rays Proc. of SPIE Vol. 76 760Y-
in etector;.) generation of seconary quanta; 3.) sprea of seconary quanta; 4.) coupling of quanta to etector apertures; 5.) integration by etector aperture; 6.) sampling of etector pixels; 7.) reaout with aitive noise; 8.) log normalization; 9.) ramp filter; 10.) apoization filter; 11.) interpolation; 1.) 3D backprojection; an 13.) 3D sampling.. Anatomical Backgroun: The Generalize NEQ an Detectability Inex Such moeling has been shown useful in characterizing the physical performance of systems uner the influence of quantum an electronics noise. However, there is consierable evience that image noise (generally interprete as any ranom or semi-ranom component in an image that impees conspicuity) is often ominate by image fluctuations associate not with the Poisson-istribute incient quanta, but rather those associate with the superposition of anatomical clutter in the D image often referre to as anatomical noise or backgroun noise. 10-13 Similarly in tomosynthesis, epening on the acquisition angle, θ tot, out-of-plane structures are superimpose to varying egrees within a slice of interest, with greater orbital extent reucing the contribution of in-plane anatomical noise. In this way, anatomical noise is expecte to be a ominant source of image egraation at low tomosynthesis angles an to be progressively reuce for a larger arc. Potential traeoffs between quantum noise an anatomical noise are immeiately appreciate: For fixe ose per projection: - a smaller number of projections over a shorter arc will increase quantum noise an anatomical noise - a larger number of projections over a shorter arc will reuce quantum noise, reuce view aliasing noise, an increase anatomical noise - a smaller number of projections over a wier arc will increase quantum noise, increase view aliasing noise, an reuce anatomical noise - a larger number of projections over a wier arc will reuce quantum noise an increase anatomical noise For fixe total ose: - a smaller number of projections over a shorter arc will reuce quantum noise an increase anatomical noise - a larger number of projections over a shorter arc will reuce quantum noise, reuce view aliasing noise, an increase anatomical noise - a smaller number of projections over a wier arc will reuce quantum noise, increase view aliasing noise, an reuce anatomical noise - a larger number of projections over a wier arc will increase quantum noise an reuce anatomical noise Clearly, a moel that quantitatively escribes such traeoffs woul be beneficial in unerstaning the factors that limits imaging performance uner various system configurations. Such traeoffs can be quantitatively appreciate through incorporation of anatomical noise (the so-calle anatomical power spectrum, S B ) in the enominator of the NEQ. Barrett et al. 14 terme this a form of generalize NEQ, written for the 3D case as: MTF GNEQ = S + S + S B Q E (1) where S B can take whatever form is appropriate to the anatomical clutter an is often empirically characterize accoring to a power-law form: SB ( f ) κ = () f β where κ escribe the magnitue an β the frequency content of the anatomical clutter. Moeling such image fluctuations in this manner implies treatment of anatomical clutter as a stochastic noise source, which clearly implies numerous assumptions an limitations. Power-law moeling of anatomical noise has shown reasonable agreement with human observer performance measure with real patient backgroun in breast imaging, 15 proviing a convenient basis for Proc. of SPIE Vol. 76 760Y-3
incorporation of a lumpy backgroun in the generalize NEQ. Such approach has been applie extensively in mammography 1,16-0, chest imaging 1-3, an more recently in tomosynthesis an CBCT 4,5. Just as NEQ may be combine with an iealize task function as escribe in ICRU 54 to yiel the etectability inex: 6 MTF (3) ' = ΔH f f f x y z S + S Q E the generalize NEQ can be similarly extene to yiel a generalize etectability inex: MTF (4) ' = ΔH f f f x y z MTF S + S + S B Q E This iealize form for etectability in the presence of anatomical backgroun is equivalent to the prewhitening (PW) observer moel. It has been shown in previous work 1,7-9 to give a useful metric for imaging performance an system optimization. Eq. (4) escribes a fully 3D etectability inex in which the observer can make full use of the volumetric ata. Such may or may not be a reasonable moel for a real observer scrolling slices; rather, it assumes the observer can instantly appreciate the fully 3D statistics of the volume image. The moel can be reuce to escribe the etectability within a ' single slice of 3D image, enote. Since extraction of a single slice from the 3D spatial omain correspons to a slice convolution over the orthogonal irection in the Fourier omain (i.e., convolution with a sinc function corresponing to the slice thickness), a simple form of slice etectability for a prewhitening observer can be obtaine by integrating the numerator an enominator of the 3D etectability inex as: [ T W f ] ' Task y = f f slice x z T S + S + S f B Q E y (5).3 Moel Observers The simple PW moel of Eq. (4) can be extene to slightly more sophisticate moels that have been shown to better approximate real observers uner various conitions. For example, the non-prewhitening (NPW) observer moel is: ' = [ T W ] f f f Task x y z + + B Q E Task x y z ( T S S S ) [ T W ] f f f Similarly, each moel can be extene to inclue a simple escription of a human eye filter an internal noise, referre to respectively as the prewhitening-eye (PWE) an non-prewhitening-eye (NPWE) moels: (6) ' = E [ T W ] Task + + + B Q E i ( ) E T S S S N f f f x y z (7) ' = E [ T W ] f f f Task x y z E ( T S + S + S ) [ T W ] + N f f f 4 B Q E Task i x y z The eye filter an internal noise aopte in this work are consistent previous work by Burgess 30 : (8) n E( f) = f exp( cf ) (9) Proc. of SPIE Vol. 76 760Y-4
where n = 1.3 an c = 3 gives an eye filter, E(f), that peaks at 4 cycles/eg for a viewing istance of 50 cm. Internal noise is assume to be uncorrelate an is set to a fraction (0.0) of the DC component of the noise-power spectrum at a viewing istance of 100 cm: D N = 0.0 NPS(0,0) i 100 where the viewing istance, D, is set to 50 cm. (10).4 Relating to Human Observer Performance Equations (4) (8) above yiel the etectability inex for a given task an observer moel, with the funamental signal an noise properties escribe by a cascae systems moel for the imaging chain. These forms are almost purely theoretical, with the main semi-empirical components of the moel being: the incient x-ray spectrum; the Poisson excess (Swank noise) of the x-ray converter; an the eye filter parameters (taken from Ref. 30). Relation of erive theoretically from cascae systems analysis calculations can be converte to area-uner the ROC curve (A z ) by the relation: 1 ' A = (1 + erf ( )) (11) Z uner assumption that istribution of cases uner two hypotheses are Gaussian an homosceastic 31. 3. IMAGING TASK 3.1 Task Phantoms To investigate task functions presenting stimuli in a power-law backgroun (anatomical noise), a custom phantom was evelope specifically to present power-law noise. The phantom esign principles an associate backgroun power spectra are etaile in Ref. 3. To summarize, the phantom consists of a ranom assortment of acrylic spheres of iameter ranging 3.18 15.88 mm, with proportions of each sphere selecte to give a power-law backgroun NPS with β ~ 3. For the small-sphere etection task, an assortment of six 3.18 mm iameter Teflon spheres was place within the central coronal (x-z) slice of the clutter phantom as shown in Fig. 3(b). For the large sphere etection task, six 10.3 mm polypropylene spheres were use as stimuli. For the encapsulate-sphere iscrimination task, each stimulus consists of a 10 mm acrylic sphere coate with an 3.18mm outer shell of paraffin wax. Example images of each task phantom are illustrate in Fig. 3(b). Images of each task phantom were acquire over a range in θ tot as escribe above, an 3D images were reconstructe using the Felkamp algorithm for 3D filtere backprojection. A central coronal slice was extracte from the volume image through the central plane of each sphere, giving the slice images use in human observer tests escribe below. The six stimuli give six inepenent trials for each task. 3. Imaging Tasks A brige between simple Fourier metrics of imaging performance (e.g., MTF, NPS, an NEQ) an task-epenent performance metrics can be obtaine through a quantitative escription of the spatial frequencies of interest in performing a given task i.e., through a spatial-frequency-epenent task function or template. A simple form for such task functions is given by the Fourier transform of the ifference in two hypotheses H 1 (x, y, z) an H (x, y, z) where H 1 an H represent the spatial omain representations of, for example, the signal-present an noise-only hypotheses. For example, etection of a given stimulus [H 1 (x,y,z)] against a uniform backgroun [H (x,y,z) = constant] is represente by the task function: WTask =Δμ FT[ H ] (1) 1 where FT[H 1 ] is the Fourier transform of the stimulus, an Δμ are the ifference in attenuation coefficients of the stimulus an backgroun, respectively. Proc. of SPIE Vol. 76 760Y-5
For etection of a given stimulus against the power-law backgroun escribe previously, images presenting H1 contains a stimulus (e.g., a sphere of a ifferent contrast) at the center of the ROI, whereas images presenting H have an acrylic sphere of the same size place at the center. Thus, the two hypotheses can be written as: FT [ H1 ] = μ1 FT [ H1 ] + clutter (13a) FT [ H ] = μ FT [ H1 ] + clutter (13b) Therefore, the task function is given by: WTask = Δμ FT [ H1 ] (13c) Finally, iscrimination of two structures the first ientifie as the stimulus an the secon the normal case is given by: WTask = μ1 FT [ H1 ] μ FT [ H ] (14) where H1 an H represent the two structures to be iscriminate. These three simple task functions represent ifferent regions of the spatial frequency omain an therefore imply ifferent susceptibility to various noise sources. High frequency tasks, such as iscrimination tasks or etection of a small object, are prone to egraation by high frequency quantum an electronics noise, an are therefore more epenent on factors of etector esign an reconstruction technique. Low frequency tasks, such as etection of a large, low-contrast lesion, are more epenent on low-frequency noise characteristics, such as anatomical backgroun. Aitional task functions representing higher-orer tasks an representing a varie richness of spatial frequency content are uner investigation in ongoing work. Example task functions are shown in Fig., along with real images acquire as a function of θtot. Figure. Hypotheses an task functions. Task 1: Detection of a small sphere in clutter. Task : Detection of a large sphere in clutter. Task 3. Discrimination of an encapsulate sphere vs. a uniform sphere. 3.3 Image Acquisition An experimental imaging bench for tomosynthesis an cone-beam CT is illustrate in Fig. 3(a) an escribe in previous work7,3. The main components of the bench inclue: an x-ray tube (Ra 94 in a Sapphire housing; Varian Meical Proc. of SPIE Vol. 76 760Y-6
Systems, Salt Lake City, UT); a flat-panel etector (RID-1640A, Perkin Elmer Optoelectronics, Santa Clara CA), an a motion control system in which a phantom place at isocenter is rotate through an arc corresponing to θtot (6K series translation stages, Parker Daeal, Harrison PA, an Dynaserv rotation motor, Parker Hannifin, Rohnert Park, CA). Images were acquire over a range of θtot spanning a continuum from low-angle tomosynthesis (θtot = 10o) to fully 3D CBCT (180o+fan an 360o). In the experiments reporte below, the angular increment (Δθ) was hel fixe at 0.45o, an the number of projections (Nproj) was varie such that θtot range 10o to 360o. In all cases, the imaging technique was 10 kvp (1.53 mm Al + 1.1 mm Cu ae filtration), an 0.63 mas per projection. The range in Nproj was 3 (at 10o) to 800 (at 360o), corresponing to a range in ose of 0.4 13.8 mgy. Figure 3. (a) Experimental bench showing components an coorinate systems for CBCT an tomosynthesis. (b) Experimental phantom presenting power-law backgroun esigne using principles of fractal self-similarity. Six stimuli were inserte to the central coronal slice. 4. EXPERIMENTAL VALIDATION: HUMAN OBSERVER STUDY Human observer tests were conucte to measure task performance as a function of θtot for each of the imaging tasks escribe above. An efficient test well suite to simple phantom stuies is offere by the multiple-alternative forcechoice (MAFC) methoology. In the stuies escribe below, 9AFC tests were use to measure the proportion correct (Pcorr) which may in turn be relate to as: Pcorr ( ', M ) = 1 π ( x ' ) M 1 [φ ( x) ] x exp (15) where M is the number of alternatives, an Φ is the cumulative Gaussian istribution. Note that for a AFC test, Pcorr = Az uner the assumption of an ieal test in which observer response oes not vary over the course of the test. For the 9AFC stuy, Az corresponing to experimental value of Pcorr is convert to Az via a Pcorr - - Az lookup table given by Eq. (11) an (15). Proc. of SPIE Vol. 76 760Y-7
Figure 4. (a) Observer performing a 9AFC test in a arkene reaing room. (b) Nine images were shown on a 3x3 gri as isplaye by Matlab-base OPTEx software in a ranomize orer. The example shows the encapsulate sphere task, with the stimulus highlighte in the lower-left corner. All tests were performe in a controlle image reaing environment with subue lighting an on a iagnostic quality monochrome isplay (Fig. 4). Reaing orer was ranomize (using the Matlab-base OPTEx utility for human observer stuies), an each test was precee by a training set compose of images rawn from the same task phantom image as a function of θtot with images istinct from those in the actual test. For these simple phantom stuies, physicists / engineers were sufficiently expert to perform each imaging task following the training set. A total of 6 observers were use, giving 5 inepenent trials for each task at each setting (angle, case), an a total time of ~60 min require for each test. Responses from each observer for each setting of a task are assume to be inepenent, i.e. no inter- an intraobserver variability were assume. Mean proportion correct (Pcorr) was therefore calculate from the number of correct responses out of a total of 30 (=5 trials x 6 observers), an converte to Az from the lookup table as mentione previously. A 95% confience interval was calculate for Az an shown in the results below. 5. BRIDGING THE GAP: THEORETICAL AND EXPERIMENTAL OBSERVER PERFORMANCE Theoretical calculations of Az for all four observer moels are plotte in Figure 5 in comparison to Az measure from real observer stuies Pcorr. In each case, an upwar tren with θtot is observe for both theoretical an experimental results, consistent with improve rejection of clutter observe in Figure. Overall, the prewhitening moels (PW an PWE) are seen to overestimate real observer performance in each case, whereas the non-prewhitening (NPW an NPWE) observers show reasonable agreement with experimental results. The large sphere etection (iscrimination of polyethylene an acrylic spheres) task appears to be most ifficult, with observer performance challenge up to θtot ~10o, an with the NPW an NPWE moels bouning the human observer measurements. The small sphere etection an encapsulate sphere iscrimination tasks are a bit more conspicuous, each challenging observers up to θtot ~70-100o an escribe reasonably well by the NPW or NPWE moels. Overall, agreement of theory an measurement is fairly goo, proviing evience that simple Fourier metrics such as NEQ combine with imaging task can offer meaningful escriptions of real observer performance. Proc. of SPIE Vol. 76 760Y-8
Figure 5. Theoretical an experimental imaging performance (A z ) plotte as a function of θ tot for 3 imaging tasks. Reasonable agreement between theory an experiment is observe, with prewhitening observer moels (PW an PWE) appearing to overestimate human response, while non-prewhitening observer moels (NPW an NPWE) emonstrating closer agreement. 6. DISCUSSION AND CONCLUSION Fourier-base physical characterization of imaging systems has been commonly applie in etector esign. In esigning an evaluating meical imaging systems with increasing complexity as in tomosynthesis an CBCT this approach can be extene to provie simple, elegant performance metrics, but correlation of theoretical preictions with human observer response has not been previously emonstrate. On the other han, human observer-base characterization realistically escribes imaging performance, but can be expensive an time-consuming. This work presente generalize etectability inex ( ) as a quantitative metric combining NEQ, anatomical backgroun, imaging task, an moel observers. Theoretical calculations of were valiate with real human observer performance, showing to be a meaningful metric in moeling system performance. It is important to acknowlege a variety of assumptions an limitations associate with in this stuy. Cascae systems analysis of imaging systems assumes linearity, stationarity, an shift invariance. While tomosynthesis an CBCT are recognize to be neither strictly stationary nor shift invariant, we assume the first an secon orer statistics are at least locally stationary an that system response is shift-invariant to the extent that the stuies involve tasks greater than the pixel / voxel size analogous to the D shift-invariance escribe in Ref. 33. In calculating observer performance metrics A z, the istribution of the two hypotheses are assume to be Gaussian an homosceestic. Finally, the evaluation involve a small number of simple, iealize imaging tasks, an the clutter phantom, though measure to present powerlaw backgroun, oes not truly simulate the complexity of structures in human organs. Future work involves the investigation of a broaer range of acquisition techniques, especially in the low-ose limit where etection transitions between anatomical noise-limite an quantum-limite regimes. Other reconstruction techniques will also be stuie, incluing ifferent reconstruction filters an sampling, which have been shown to impart significant effects on system NEQ. More complex an higher-orer tasks will also be investigate as a fuller examination of the extent to which generalize etectability inex emonstrates agreement with real observer performance. 7. ACKNOWLEDGEMENTS The authors acknowlege numerous stimulating conversations an collaboration with Dr. Rebecca Fahrig (Stanfor University), Dr. Sungwon Yoon (Stanfor University), Dr. Angel Pinea (California State University Fullerton), an Dr. Art Burgess (Harvar University, retire). This work was fune by National Institute of Health Grant No. R01-CA- 11163. Proc. of SPIE Vol. 76 760Y-9
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