title: Attributing Uncertainties in the Identification of Hotspots in SPECT Imaging
authors: Costas N. Papanicolas Loizos Koutsantonis Efstathios Stiliaris
text: 
  
  
  
  
  
  []papanicolas@cyi.ac.cy
  The Cyprus Institute, Konstantinou Kavafi 20,  2121 Nicosia, Cyprus
  Physics Department, National and Kapodistrian University of Athens, Ilissia University Campus, 15771 Athens, Greece
  
  
  []l.koutsantonis@cyi.ac.cy
  The Cyprus Institute, Konstantinou Kavafi 20,  2121 Nicosia, Cyprus
  
  
  
  []stiliaris@phys.uoa.gr
  Physics Department, National and Kapodistrian University of Athens, Ilissia University Campus, 15771 Athens, Greece
  The Cyprus Institute, Konstantinou Kavafi 20,  2121 Nicosia, Cyprus
  
  
  
  
  
  
  
  
   In SPECT imaging, the identification and detection of a lesion rely either on visual inspection of the reconstructed
  tomographic images or  post-processing image analysis methods. Both approaches do not provide the
  capability to attribute a quantifiable uncertainty to this identification. We present a framework which allows
  the quantification of this uncertainty and the assignment of a level of confidence to the detection of hotspots. Based on the "Reconstructed Image from Simulations Ensemble"
  (RISE), an image reconstruction method, the presented scheme uses the set of projection measurements to derive the parameters defining the uptake of radioactivity, the position and the size of a hotspot, and as well as their associated uncertainties. The capabilities of the
  proposed method are demonstrated with projection data from GATE phantom simulations.
  
  
  
  Attributing Uncertainties in the Identification of Hotspots in SPECT Imaging
      Efstathios Stiliaris
      October, 2018
  ============================================================================
  
  
  
  
  
  § INTRODUCTION
  
  The task-based image quality assessment is an essential procedure in medical imaging for evaluating the performance of an imaging system used in clinical practice.  The common method for making detection task-based assessments of the image quality    in Single Photon Emission Computed Tomography (SPECT) is by the use of human observers.  In this method, multiple experienced individuals  are asked to rate the detectability of a lesion in a set of reconstructed images based on their confidence that the lesion exists. In this approach, the detection of a lesion is approached as a classification problem, and the Receiver Operating Characteristics (ROC) curve is used to provide an index of the image quality <cit.> concerning the detection task.
  
  The numerical observers, such as the Channelized Hotelling observer, is an alternative strategy for task-based image quality assessments <cit.>. As compared to the human observers, the numerical observers prove to be more practical, especially when a large set of images is to be classified and multiple parameters related to the image quality (e. g. collimator geometry and reconstruction parameters) are to be evaluated. In this approach, a large set of labelled image patterns is used to train an algorithm  to classify the images in two categories, the images containing a lesion and the images presenting only background noise <cit.>. After the training phase, the model observer can be used to assign a test statistic on a new image to quantify the detectability of a lesion.  It is apparent that the performance of the model observer is based on the training phase of the algorithm and especially on the choice of the labeled patterns which are used for this purpose. 
  
  
  
  
  
  In this work, we present the capability of a new image reconstruction method to assign a precise level of confidence  to the detectability of a lesion in its reconstructed images. The method, the Reconstructed Image from Simulations Ensemble (RISE)<cit.>, relies on parametric modelling of the imaged distribution and  Monte-Carlo simulation techniques  to produce tomographic images from the sets of projections. Based on the concepts of the Athens Model Independent Analysis Scheme (AMIAS) <cit.>  the Probability Distribution Functions (PDFs) of the model parameters are derived and used to provide the probability value  for the detection of a lesion. The method is demonstrated using projection data from phantom simulations performed in the GEANT4/GATE for different levels of background activity and varying size of hotspots.
  
  
  
  
  
  § IMAGE RECONSTRUCTION FROM PROBABILITY DISTRIBUTION FUNCTIONS
  
  
  In the implementation of RISE for SPECT, a model described by  a set of parameters {A_ν} is used to represent the image F to be reconstructed.  In this case, a model is chosen to  parametrize the physical characteristics of multiple ellipsoidal "hotspots" and the distribution of background activity <cit.>:
  
      F(x,y) = ∑_i=1^N_sA_i·(exp(r_i-R_i/s_iR_i)+1)^-1+∑_m,nC^m_nZ^m_n(x,y)
  
  The coefficients and parameters of the above model are depicted in Figure <ref> for a single hotspot; their definitions are given in Table <ref>. 
  Given the dataset of projections, the extraction of the model parameters is accomplished employing the AMIAS method <cit.>:
  
  
  
  A. A large ensemble of possible solutions is constructed by random sampling the values of the model parameters {A_ν}. Each solution corresponding to a tomographic image is used in a computerized process to simulate its associated projections {Ỹ_j}. To quantify the linkage of the sampled parameters values to the data, a χ^2 value is calculated for each solution from its  simulated projections and the available set of projection measurements {Y_j}:
  
      χ_j^2 = ∑_k(Y_k-Ỹ^j_k/ϵ_k)^2
  
  where ϵ_k is the error associated to the j^th bin measurement of photon counts. Each set of sampled parameters values ("solution") j in the ensemble is assigned to  a probability  value P(j) calculated by: 
  
      P(j) = exp(-1/2χ_j^2)
   
  
  B. Following the construction of the ensemble of solutions, the probability that the parameter A_ν is equal to a specific value a_ν is calculated  through the equation:
  
      Π (A_ν =a_ν)= ∫_a_ν^a_ν+δ a_ν∑_j dA^j_ν P(j)/∫_-∞^+∞∑_j dA^j_ν P(j)
  
  The above formula is used to  derive the Probability Distribution Function (PDF) of each one of the model parameters representing the data. Mean values and second order moments are derived from the extracted PDFs to determine the "optimum" parameters values and their associated uncertainties. The optimum parameters values are used  in Equation <ref>  to produce the reconstructed tomographic image.
  
  
  
  
  
  
  
  
  
  
  § IDENTIFYING HOTSPOTS IN BACKGROUND ACTIVITY
   
  Besides the determination of the parameters optimum values, the derived PDFs allow the quantification of the uncertainty associated with the reconstructed image. This uncertainty is calculated to provide a level of confidence to specific  tasks such as the localization and the identification of hotspots in the reconstructed image. By the very use of RISE, the first task is achieved: the uncertainty in the position of a hotspot (σ_x_0, σ_y_0) is directly computed from the PDFs of the position parameters (x_0, y_0).
  
  For the second task,  the assignment of a level of confidence to the identification of a hotspot requires a further statistical comparison between the hotspot activity and the background activity. For each hotspot represented by its position solutions (x^j_0, y^j_0), the PDFs of the total activity  F(x,y) and the background activity term B = ∑_n,mC_n^mZ_n^m(x,y) are derived  by selecting the  solutions lying in the Region of Interest (ROI):
  
      (x^j_0, y^j_0) ∈ ([x̅_0 ±3σ_x_0], [y̅_0 ± 3σ_y_0])
  
  where (x̅_̅0̅, y̅_̅0̅) and (σ_x_0,σ_y_0) are the mean values and uncertainties of the position parameters (x_0,y_0) respectively.
  
  
  
   Having derived the PDFs Π(F) and Π(B), the "confidence" C(h_0) of a hotspot  being identified as "true" is defined and quantified by measuring the overlap probability between the two PDFs:
   
      C(h_0) ≡ 1-∫Π(a' ∈ (F∩ B))da'
  
  It is noted that in choosing the ROI to be within a radius of 3σ introduces a model dependence, whose discussion is beyond the scope of this paper.
  
  
  
  
  
  
  
  § PHANTOM SIMULATIONS
  
  Phantom simulations were conducted  to demonstrate the  capability of RISE in assigning a level of confidence to the detection of hotspots. The sets of projection data were generated in GATE <cit.> using the voxelized phantom shown in Figure <ref>. 
  
  
  
  The phantom was defined as a configuration of five circular hotspots and one cold-spot (zero activity) embedded in a uniform  background. The radii of the five hotspots were 1.6 mm, 2.0 mm, 2.8 mm, 3.2 mm and 3.6 mm, where the radius of the cold spot was 2.4 mm. The uniform background was defined on a disk of 25 mm radius. All of the five hotspots had equal activity concentration (20 μCi/ml). In four simulation cases, the Target-to-Background ratio (T:B)  characterizing the 'true' image of the phantom was adjusted to 4:1, 3:2, 2:3 and 1:4.
  
  
  
  The SPECT detector used in GATE for the simulation of data  was described to represent a prototype γ-camera system <cit.> having a Field of View of 50×50 mm^2.
  For each T:B simulation, a set of 24 planar projections, equally distributed over the angular range 0^0-360^0,  was obtained from the phantom. Each set of projections (sinogram) was composed of approximately 278000 counts acquired in the  125-150 keV energy window.
  
  
  
  
  
  
  § RESULTS
  
  
  The tomographic images of the phantom were reconstructed from the projection data using the RISE methodology. Each image was reconstructed as a 128×128 matrix with a 0.4 mm pixel size. For visual comparisons, images were also obtained with the Maximum Likelihood Expectation Maximization (MLEM) method <cit.>. The produced images are shown in Figure <ref> for the different T:B ratios. The quantitative evaluation of the image quality achieved by the two methods  using the appropriate metrics is presented in <cit.>, where it is shown that RISE results compare favourably to those derived using MLEM.
  
  
  
  
  
  
  
  
  
  
  
  
  In Figure <ref>, we provide the correlation-plots of the position parameters (x_0,y_0) of the H1 hotspot. These plots were produced to visualize the ensembles of  "solutions" as they were simulated in RISE for each of the four sets of projection data. Each point on these 2D correlation-plots is color-coded according to its probability value e^-1/2χ^2 value and corresponds to a possible central position of the examined hotspot. The mean values and uncertainties  of the position parameters (x_0,y_0) are depicted for each T:B ratio at the bottom of the corresponding correlation-plot. As expected, the uncertainty in position increases with the increase of the background activity. This increase in uncertainty can be precisely calculated with RISE to evaluate the impact of the background activity on the localization of a hotspot in the reconstructed image.
  
  
  The uncertainty in position provides the physical boundaries (3σ) of a ROI in which a hotspot can be detected. The "solutions" lying in this ROI are selected from the entire ensemble of "solutions" to calculate the PDFs of the total activity (F) and background (B) in the corresponding ROI.
  The PDFs of  F and B derived for the H1 hotspot in the four reconstruction cases are shown in Figure <ref> for the different T:B ratios characterizing the simulated data. The area of the overlapping PDFs was calculated in Equation <ref> to quantify the probability P(h_0) of a  "hotspot" being identified as "true". 
  
  
  
  
  The  values of C(h_0) calculated for the three hotspots H1, H4 and H6 using the proposed methodology are plotted in Figure <ref> over the range of the T:B ratios of the simulated data. The three hotspots having different radii (H1: 1.6 mm, H4: 2.8 mm, H6: 3.6 mm) yielded different C(h_0) curves. The  curve of H1, the hotspot having the smaller radius, shows the steepest descent as the T:B ratio decreases. The identification of the H6 hotspot in the  image (the hotspot having the largest area)  is assigned to the highest level of confidence, which for all T:B ratios is above the value of 0.85.  The strong dependence of the hotspots' detectability on their size and the level of background activity is quantified. 
  
  
  
  
  
  § CONCLUSIONS
  
  In this study, we demonstrate the capability of the RISE method to quantify the uncertainty in the detection of lesions in SPECT imaging. The method was investigated with projection data from GATE simulations using a voxelized phantom containing hotspots of different size. In four distinct simulation cases, the phantom was simulated in different background  
  activity levels. In all cases, we were able to assign a level of confidence to the detection and localization of the hotspots in the reconstructed images. Moreover, the dependence of the hotspot detectability on the size of the hotspots and the background activity was  determined. 
  
  Although the phantom presented here is appropriate for demonstrating the method, further experimentation with real phantoms   is under way for evaluating the potential application of the method  in pre-clinical and clinical studies.
  
  
  This  work was supported by the Cy-Tera Project "NEA IPODOMI/STRATI/0308/31", which is co-funded by the European Regional Development Fund and the Republic of Cyprus through the Research Promotion Foundation. 
  
  
  
  
  
  
  
  
  
  
  
  
  14
  fxundefined [1]
   ifx#1
  
  fnum [1]
   #1firstoftwo
   secondoftwo
   
  fx [1]
   #1firstoftwo
   secondoftwo
   
  
  
  
  
  
  noop [0]secondoftwo
  
  ref[1]@startlink#1@href
  href[1]#1@endlink
  anitize@url [0]`
  12`$12`&12`#12`1̂2`_12`%12
  startlink[1]
  endlink[0]
  
  rl [1]href #1
  
  
  
  
  
  
  
  
  
  
  
  
  @bib@innerbibempty
  
  [Metz(1986)]metz:1986
    author author C. E. Metz, @noop  journal journal
    Investigative radiology volume 21, pages 720 (year 1986)NoStop
  [Myers and Barrett(1987)]myers:1987
    author author K. J. Myers and author H. H. Barrett, @noop  journal journal JOSA
    A volume 4, pages 2447 (year 1987)NoStop
  [Fiete et al.(1987)Fiete,
    Barrett, Smith, and Myers]fiete:1987
    author author R. Fiete, author H. H. Barrett,
    author W. E. Smith,  and  author K. J. Myers, @noop
     journal journal JOSA A volume 4, pages 945 (year 1987)NoStop
  [Barrett et al.(1993)Barrett, Yao, Rolland, and Myers]barrett:1993
    author author H. H. Barrett, author J. Yao,
    author J. P. Rolland,  and  author K. J. Myers, @noop
     journal journal Proceedings of the National
    Academy of Sciences volume 90, pages
    9758 (year 1993)NoStop
  [Gilland et al.(2006)Gilland, Tsui, Qi, and Gullberg]gilland:2006
    author author K. L. Gilland, author B. M. Tsui,
    author Y. Qi,  and author G. T. Gullberg, @noop 
    journal journal IEEE transactions on nuclear
    science volume 53, pages 1200
    (year 2006)NoStop
  [Papanicolas et al.(2018)Papanicolas, Koutsantonis, and Stiliaris]Papanicolas:2018
    author author C. N. Papanicolas, author L. Koutsantonis,  and author E. Stiliaris, @noop  journal journal
    arXiv preprint arXiv:1804.03915  (year 2018)NoStop
  [Markou et al.(2018)Markou,
    Stiliaris, and Papanicolas]Markou:2018
    author author L. Markou, author E. Stiliaris,   and author C. N. Papanicolas,  10.1140/epja/i2018-12549-7 journal journal EPJournal A volume 54, pages
    115 (year 2018)NoStop
  [Papanicolas and Stiliaris(2012)]Papanicolas:2012
    author author C. N. Papanicolas and author E. Stiliaris, @noop  journal journal
    arXiv: 1205.6505  (year 2012)NoStop
  [Stiliaris and Papanicolas(2007)]Stiliaris:2007
    author author E. Stiliaris and author C. N. Papanicolas, in @noop  booktitle AIP
    Conference Proceedings, Vol. volume 904 (organization AIP, year 2007) pp. pages
    257–268NoStop
  [Alexandrou et al.(2015)Alexandrou, Leontiou, Papanicolas, and  Stiliaris]Alexandrou:2015
    author author C. Alexandrou, author T. Leontiou, author C. Papanicolas,  and author E. Stiliaris, @noop  journal journal
    Phys Rev D volume 91, pages 014506
    (year 2015)NoStop
  [Jan et al.(2004)Jan,
    Santin, Strul, Staelens,
    Assie, Autret, Avner,
    Barbier, Bardies, Bloomfield
    et al.]jan:2004
    author author S. Jan, author G. Santin,
    author D. Strul, author S. Staelens, author
    K. Assie, author D. Autret, author S. Avner, author R. Barbier, author M. Bardies, author P. Bloomfield,  et al., @noop  journal journal Phys Med Biol volume
    49, pages 4543 (year 2004)NoStop
  [Spanoudaki et al.(2004)Spanoudaki, Giokaris, Karabarbounis et al.]spanoudaki:2004
    author author V. Spanoudaki, author N. Giokaris, author A. Karabarbounis,  et al., @noop  journal
    journal Nucl Instr Meth Phys Res A volume 527, pages 151 (year
    2004)NoStop
  [Shepp and Vardi(1982)]shepp:1982
    author author L. A. Shepp and author Y. Vardi,  @noop  journal journal IEEE Trans Med
    Imag volume 1, pages 113 (year 1982)NoStop
  [Koutsantonis et al.()Koutsantonis, Stiliaris, and Papanicolas]koutsantonis
    author author L. Koutsantonis, author E. Stiliaris,  and author C. N. Papanicolas, @noop  journal To be published  NoStop
  
  
  