FINSIG 05

Mikko Kervinen † , Marko Vauhkonen, Jari Kaipio and Pasi A. KarjalainenUniversity of KuopioDepartment of Applied PhysicsKuopio, FINLAND† E-mail: Mikko.Kervinen@uku.fihttp://physics.uku.fi/research/biosignalMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 1

FINSIG 2005Kuopio, FinlandOutline of the presentation IntroductionPrinciple of SPECT measurementsNotationsStationary case Time-varying caseState-space equationsTransportation modelsSolution of the state-estimation problem Results ConclusionMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 2

FINSIG 2005Kuopio, FinlandIntroduction In tomographic image reconstruction one basic assumption is thetime–independence of the object to be imaged. In some studiesthe change of the object is, however, the object of interest. In single photon emission tomography the requirement of thetime–independence is often not fulfilled. Several methods havebeen developed to speed up the acquisition process and therebyto reduce the effect of time–variation of the object. Current dynamic SPECT studies are based on the set ofstationary images that are used to determine functionalparameters of metabolism. Truly time–varying reconstruction would increase thetime–resolution of the studies and reduce the artifacts caused bythe assumption of time–independence.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 3

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FINSIG 2005Kuopio, FinlandTomographic measurement Projections are measured around the object by rotating thegamma camera for example 10 ◦ between the measurements. In 2D case each measurement is a vector. Single measurementsare often combined and presented as a sinogram.Projection binsMeasurement number (and time) increasesMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 5

FINSIG 2005Kuopio, FinlandTomographic measurement Projections are measured around the object by rotating thegamma camera for example 10 ◦ between the measurements. In 2D case each measurement is a vector. Single measurementsare often combined and presented as a sinogram. In stationary case all of the projections image identical activitydistribution and differ only in the perspective of the view. In this case the activity distribution can be reconstructed usingclassical projection methods, such as Filtered Back Projection, orusing more sophisticated methods combined with the propermodeling of the measurement system.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 5

FINSIG 2005Kuopio, FinlandNotations Usually the 2D image plane is discretized with N2square pixels and measurement operator, denotedas H, that is the mapping from discretized activitydistribution θ to the projection p, PSfrag is constructed. replacements Construction of observation matrix H may include,for example the effects of geometry, attenuation,scatter and septal penetration in the collimator. Mapping is different for each measurement direction.In the following we will denote the observationmatrix at time t with H t and the correspondingobserved projection vector with p t . Thus t denotesboth time and direction of the camera head.zyxθ 1θ N 2θ Np t1pt2Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 6

FINSIG 2005Kuopio, FinlandStationary case By combining all observations, linear observation model forstationary SPECT can be written in the form⎛⎜⎝p 1.⎞ ⎛⎟⎠ = ⎜⎝H 1.⎞⎟⎠ θ,p TH Twhere p t is the measured projection at time t, θ is the vectorizedactivity distribution (image) and H t is the projection operator thatmaps the activity distribution to the measurements at time t. Even in stationary case the observation model of SPECT is highlyunderdetermined and the reconstruction of the activity distributionrequires novel mathematical methods.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 7

FINSIG 2005Kuopio, Finland In SPECT the error is neither additive nor Gaussian. However, ifthese assumption are made, the observation equation is of theformp = Hθ + v,where v is the measurement error and H the combined projectionoperator and p denotes the stacked observations. Estimator for θcan then be obtained, for example, using constrained leastsquares approach:ˆθ LS = arg minθ>0{‖p − Hθ‖2 }with additional constraints like Tikhonov regularization. Another approach is to take account the Poisson-statistics of theobservations using methods like Maximum Likelihood ExpectationMaximization (MLEM) algorithm.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 8

FINSIG 2005Kuopio, FinlandTime-varying case In many cases the activity distribution is not stationary during thefull tomographic measurement (180 or 360 ◦ rotation) If standard reconstruction methods are used to estimate therapidly changing object, errors are generated due to theinconsistency of the projection data. Thus, the change of the activity distribution should be taken intoaccount in the reconstruction process. Instead of single distribution we must reconstruct multiple imagesusing the same amount of data that is used with the standardmethods.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 9

FINSIG 2005Kuopio, FinlandState-space equations In time-varying case we approximate that the activity distributionis constant during the measurement of a single projection.p t = H t θ t + v t Observations cannot be combined as in stationary case. In order to reconstruct the time-varying activity distribution θ1,...,T ,a model of the time-variation is required. The simplest modeled for the time-variation is the first orderMarkov modelθ t+1 = F t θ t + w twhere F t is the state transition matrix and w t the state noise.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 10

FINSIG 2005Kuopio, FinlandTransportation models Both compartmental transport and diffuse transport can berepresented with the first order Markov model. 1-compartment model can be used to model the activity uptake µof tissues when blood activity B(t) is known:µ m (t) = k m u∫ t0B(τ)e km w (t−τ) dτwhere kum and kw m are the uptake and washout coefficients of thetissue type m. The activity in tissue m is obtained by taking into account thevascular fraction fvm of the corresponding tissue typeθ m (t) = µ m (t) + f m v B(t)Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 11

FINSIG 2005Kuopio, Finland In case of diffuse transport the time–evolution can be written as anon–homogeneous partial differential equationdθ(r, t)dt= ∇ · (ρ(r)∇θ(r, t)) + J(r, t)In case of isotropic homogeneous diffusion ρ(r) is a scalarconstant and ∇ · (ρ(r)∇θ(r, t)) = ρ∇ 2 θ(r, t). The operator ρ∇ 2equals to linear spatial filtering of the activity distribution. By approximating the derivatives with the first difference, the stateequations for both compartmental and diffusion models will be offormθ t+1 = F θ t + GJ twhere the state transition matrix F and the known input GJ t areconstructed according to the desired model.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 12

FINSIG 2005Kuopio, FinlandSolution of the state-estimation problem The time-varying distribution can be solved by applying theKalman Filter to the state–space equations The problem is to find the minimum mean square estimator ˆθt forstate θ t given the observations up to time t, p 1 , p 2 , . . . , p t . Kalman Filter equations can be written in the formˆθ t|t−1 = F t−1 ˆθt−1|t−1C˜θt|t−1= F t−1 C˜θt−1|t−1F T t−1 + C wt−1K t = C˜θt|t−1H T t (H t C˜θt|t−1H T t + C vt ) −1)C˜θt|t−1= ( I − K C˜θt t HtTˆθ t|t = ˆθ(t|t−1 + K t p t − Ht T ˆθ)t|t−1Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 13

FINSIG 2005Kuopio, Finland In SPECT the reconstruction is done usually off–line and all theobservation p 1 , p 2 , . . . , p T are available. The algorithm thatcomputes the estimate based on all observations, i.e. ˆθ t|T iscalled fixed interval Kalman Smoother. In addition to the KalmanFilter equations so called backward run gain matricesA t−1 = C˜θt−1F T t−1C −1˜θ t|t−1must be stored during filtering and the Smoother estimate can becomputed applying equationˆθ t−1|T = ˆθ t−1|t−1 + A t−1 (ˆθ t|T − ˆθ t|t−1 )in time–reversed (backward) order.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 14

FINSIG 2005Kuopio, FinlandAdditional constraints The activity distribution is non-negative by definition. Thisconstraint cannot be included in the Kalman filter directly. To find the optimal non-negative estimates of the activitydistribution, we use the Kalman filter estimates and the errorcovariances, provided by the filter, to find an estimateˆθ ∗ t|t= arg maxθ>0{exp(− 1 }2 (θ − ˆθ t|t )C −1 (θ − ˆθ ˜θ t|t ))t|tthat is, we select the most probable non-negative estimates ateach time-step from the distribution N (ˆθ t|t , C˜θt|t) defined by theKalman filter.Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 15

FINSIG 2005Kuopio, FinlandResults Proposed method is evaluated using simulations Projections from a time-varying activity distribution are acquiredusing 2D parallel hole collimator and single head camera.Attenuation and scatter are not modeled. Our simulated measurements include 3 full rotations using 10◦steps. Observation data is Poisson distributed, also additiveGaussian noise is added to the simulated observations. Conventional method (positivity constrained LS) can be used toreconstruct single image using 180 ◦ data, that is, using 18projections. Kalman smoother is used to estimate the activity distributioncorresponding to each projection measurement.Static imaging sequence might include full 360 ◦ rotation with 3 ◦ steps giving 120projections to be used for reconstruction of a single imageMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 16

FINSIG 2005Kuopio, Finland1020304050603512 410 20 30 40 50 60Phantom consists of 5 tissue types with spatiallyconstant activity in each.Activity/pixel within each tissue type iscomputed using given blood input function(time-activity curve of ROI #2) and the 1-compartment modelFor the Kalman smoother the state equationis a random walk model, that is F t ≡ I.This means that no information about theblood input or transportation is used in thereconstruction.Simulated activitiesNormalized activity concentration10.80.60.40.200 50 100 150t (s)ROI #1ROI #2ROI #3ROI #4ROI #5Projection bin10203040506070809020 40 60 80 100Projection number350300250200150100500Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 17

FINSIG 2005Kuopio, FinlandNormalized activity concentration10.80.60.40.2Least SquaresROI #1ROI #2ROI #3ROI #4ROI #5True Kalman Smoother Least Squares00 50 100 150t (s)Kalman SmootherNormalized activity concentration10.80.60.40.2ROI #1ROI #2ROI #3ROI #4ROI #500 50 100 150t (s)Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 18

FINSIG 2005Kuopio, FinlandReduction of parameters In some cases we are interested in the total activity of someRegions of Interest (usually certain organs) rather than theactivities of each pixel separately. By constructing a matrix Rwhose i’th column is the index set of the pixels that belong to i’thROI, we can compute the total activity of each ROI by equationφ = R T θ If we assume that the activity distribution is approximatelyconstant within each ROI, we can estimate the mean activity ofthe ROI’s directly by modifying the observation model. In this case the number of unknown parameters is reduced to thenumber of ROI’s (compare to the N 2 unknowns in pixel-basedcase.)Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 19

FINSIG 2005Kuopio, Finland When prior information about the ROI’s is used, the model isoverdetermined In this case also the LS-estimates can be computed sequentiallycorresponding to each measurementKalman SmootherLeast SquaresNormalized activity concentration10.80.60.40.2ROI #1ROI #2ROI #3ROI #4ROI #5Normalized activity concentration10.80.60.40.2ROI #1ROI #2ROI #3ROI #4ROI #500 50 100 150t (s)00 50 100 150t (s)Mikko Kervinen, Department of Applied Physics, University of Kuopio Slide 20

FINSIG 2005Kuopio, Finland In case of diffuse transport wehave used both random walkand diffusion models as theevolution model in the statespace equations Use of diffusion model in theestimation clearly removes theeffect of noise, compared to sequentialLS and random walkestimates.TrueKalman Smoother,Kalman Smoother,Random Walk Least Squares LaplaceNormalized activity0.80.70.60.50.40.30.2Simulated ActivityKalman Smoother, Random WalkKalman Smoother, Diffusion0.100 10 20 30 40 50 60PixelMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 21

FINSIG 2005Kuopio, FinlandConclusion A new method for reconstruction of dynamic SPECT images ispresented Method is flexible, both spatial and temporal prior information canbe easily formulated with the state-space equations. State estimation problem was solved with Kalman Smoother,which is (today) computationally too expensive when practicalresolutions are considered. Particle filters could be used to take into account the Poissonstatistics of the observation noise and include the non-negativityconstraint directly in the filtering.→ Computational cost would be still increasedMikko Kervinen, Department of Applied Physics, University of Kuopio Slide 22