A survey of approaches for direct parametric image reconstructionin emission tomography

Charalampos Tsoumpasa�

Hammersmith Imanet, part of GE Healthcare, London, W12 0NN, United Kingdom and PET Methodology,MRC Clinical Sciences Centre, Hammersmith Hospital Campus, Imperial College, London,W12 0NN, United Kingdom

Federico E. TurkheimerPET Methodology, MRC Clinical Sciences Centre, Hammersmith Hospital Campus, Imperial College,London, W12 0NN, United Kingdom

Kris ThielemansHammersmith Imanet, part of GE Healthcare, London, W12 0NN, United Kingdom

�Received 9 March 2008; revised 25 June 2008; accepted for publication 10 July 2008;published 7 August 2008�

The quantitative data obtained by emission tomography are decoded using a number of techniquesand methods in sequence to provide physiological information. Conventionally, the data are recon-structed to produce a series of static images. Then, pharmacokinetic modeling techniques areapplied, and kinetic parameters that have physiological or functional significance are derived.Although it is possible to optimize each estimation step in this process, many simplifying assump-tions have to be introduced to make the methods that are used practicable. Published research hasshown that if the kinetic parameters are estimated directly from the measured data, the parametricimages will have higher quality and lower mean-squared error than if this was done indirectly. Thisreview highlights some aspects of the methods that have been proposed for such direct estimationof pharmacokinetic information from raw emission data. © 2008 American Association of Physi-cists in Medicine. �DOI: 10.1118/1.2966349�

Key words: emission tomography, reconstruction, kinetic modeling

I. INTRODUCTION

The key characteristic of emission tomography �ET�,whether PET or SPECT, is its quantitative capability, whichallows the in vivo estimation of biochemical interactions us-ing radiotracers. In particular, ET can be used to monitor thekinetic behavior of biomolecules, from which physiologicalparameters of interest �e.g., tissue perfusion, metabolic rates,and receptor densities� can then be inferred with the use ofpharmacokinetic models.1

In dynamic ET studies, the changing activity of the in-jected radiotracer is conventionally measured through mul-tiple consecutive time frames. The image of the radioactivitydistribution in each frame is reconstructed independently andthe whole set of frames is then used to estimate the distribu-tion of the physiological parameter of interest by the appli-cation of an appropriate pharmacokinetic model to the time-radioactivity curve either of appropriately selected functionalregions or of each image element. Thus, the kinetic param-eters of the model are obtained indirectly through a two-stepprocess: For example, as described by Wallius et al.2 for PETand by Iida et al. for SPECT.3 When applied to the time-activity curve of each voxel, the whole process generatesimages of the distribution of the parameter of interest: Para-metric images.

However, while it may be relatively simple to optimizeeach step separately, this is not true of the entire indirect

process �Fig. 1, Route I�. For example, it is difficult to accu-

3963 Med. Phys. 35 „9…, September 2008 0094-2405/2008/35

rately estimate the statistical distribution of the voxel valuesin static radioactivity images especially for iterative recon-struction methods. Formulas for first and second order statis-tics are well known,4–7 but they are approximative only wheniterative reconstruction methods are used, and the computa-tions are time-consuming if applied to each voxel, as an es-timation of the Fisher information matrix is necessary.5 Simi-lar formulas for higher order statistics could be derived, butthis would be impractical, so it is difficult to take the fullstatistical nature of the image data into account in the esti-mation of the kinetic parameters. As a result, most kineticmodeling techniques assume Gaussian statistics, which maynot be accurate and can result in an overall increase in themean squared error in the estimated values.

In order to reduce noise in the estimated kinetic param-eters, techniques have been developed that apply spatialregularization to the parametric images during the kineticmodel step to produce images with appreciably less variance:For example, using regions of interest �i.e., assuming thesame mean value for all the voxels of the region�, simplespatial regularization,8 the wavelet transform,9 or Markovrandom fields �i.e., PWLSR�.10 Alternatively, temporal regu-larization can be included into the iterative reconstructionalgorithms: For instance, temporal priors based on compart-mental modeling.11

Another approach to increasing the signal to noise ratio�SNR� in dynamic images is the use of temporal basis func-

tions to accommodate temporal information in the recon-

3963„9…/3963/9/$23.00 © 2008 Am. Assoc. Phys. Med.

3964 Tsoumpas, Turkheimer, and Thielemans: Approaches for direct parametric reconstruction in emission tomography 3964

struction step12 �Fig. 1, Route II�. The time-activity curvesare represented as a sum of temporal functions: For instance,B-splines13–16 or other sophisticated functions.17 These meth-ods estimate the weight of each temporal basis componentfor every voxel and produce activity maps with higher spatialresolution for comparable variance. However, the basis func-tions, and hence the weight images, have no immediatephysiological interpretation. Therefore, the kinetic param-eters must still be estimated from the dynamic images, re-sulting again in a two-step process with loss of information.Moreover, the selection of the optimal type of basis functionand their proper number remains a significant challenge.

Yet another approach to improving SNR that has beenproposed are “projection-based” methods, where the kineticmodeling step is applied directly to the projection data priorto the reconstruction, resulting in “parametric projections”which are then used to reconstruct the parametric images�Fig. 1, Route IV�.18,19 However, this forfeits the Poissondistribution properties of the projection data and conse-quently conventional Poisson-based statistical reconstructionalgorithms are not appropriate. To the best of our knowledge,there have been no attempts to take into account the correctstatistics of the parametric projection data obtained by thismethod during reconstruction.

In the 1980s, Snyder20 and Carson et al.21 independentlymade the original suggestion that the combination of the twosteps in one, i.e., the incorporation of the pharmacokineticsof the measured counts into the reconstruction process �“di-rect reconstruction”�, could result in parametric images ofhigher quality and accuracy.22 In the following sections such“direct” parametric estimation approaches in PET andSPECT are presented in detail. Our main purpose is to clarifythe differences in these methods, identifying their advan-tages, limitations, and key characteristics. Among the dis-cussed methods there are a few that do not clearly belong to

DynamicImages

KineticModelling

DynamicAcquisition

DynamicVector Images

ParametricImages

KineticModelling

ParametricReconstruction

Reconstruction

ParametricProjections

KineticModelling

Denoising

Denoising

DynamicReconstruction

(I) Indirect (II) Vectors (III) Direct (IV) Projection-indirect

FIG. 1. Diagrammatic representation of the methods used for the estimationof parameters.

the “direct” kinetic parameter estimation approaches but

Medical Physics, Vol. 35, No. 9, September 2008

which have, nevertheless, been included as they have thepotential to become practical direct estimation methods inthe future.23–25

II. DIRECT PARAMETRIC RECONSTRUCTIONMETHODS

II.A. Overview

Direct parametric reconstruction approaches estimate thekinetic parameters from the measured emission data �Fig. 1,Route III�. Conceptually, the process is similar to fitting mea-sured data to a model which is parameterized using kineticparameters for each voxel �or region�. The model consists oftwo components: A kinetic model, which translates the time-varying radioactivity for every voxel �or region�, and amodel for the measurement by the tomograph. The fit is de-termined by minimizing an appropriate objective functionrelated to the discrepancy between the measured data and theoutput of the model.

Direct approaches can be categorized into two maingroups according to the statistical model used for the data:Poisson and Gaussian. The former adopt the correspondinglog likelihood while the latter utilize either weighted or non-weighted least-squares as the objective function. It is worthnoting that, while Poisson statistics are a good approximationfor both PET and SPECT raw data, if they are precorrectedfor background counts �e.g., scatter and random events�, or ifthe dead-time effect is severe,26 the Poisson assumption is nolonger accurate and thus the Gaussian approximation is usu-ally employed. Additionally, dynamic SPECT data are incon-sistent for static image reconstruction due to the rotation ofthe scanner. This makes the use of direct methods very ap-pealing for SPECT. In addition to the objective function,there are a few other issues that need to be considered inorder to design a direct reconstruction algorithm appropri-ately. First of all, the kinetic model that describes the tem-poral behavior of the radiotracer affects the choice of thealgorithm. Different implementations will be needed depend-ing on whether this is linear or nonlinear in the kinetic pa-rameters. The tomographic model is normally linear.27

Therefore, most of the algorithms that have already beenused in static image reconstruction can be applied to thedirect parametric case when the pharmacokinetic model islinear in the parameters that need to be estimated. In thiscase, both the Poisson and the Gaussian objective functionsare concave in the kinetic parameters, and hence have no�isolated� local turning-points. On the other hand, nonlinearmodels may have objective functions with local turning-points, so special treatment is required to ensure that thealgorithm converges to the global maximum/minimum value.

Generally, it is expected that the incorporation of the ki-netic model will offer some form of temporal regularization.However, this does not necessarily guarantee the quality ofthe parametric images produced and additional spatial regu-larization may still be required. This is likely to be the casewhen many kinetic parameters are included in the model asthe problem becomes more ill-conditioned. The two most

common approaches for regularization are simply postfilter-

3965 Tsoumpas, Turkheimer, and Thielemans: Approaches for direct parametric reconstruction in emission tomography 3965

ing the parametric images or adding a regularization term tothe objective function, thus penalizing undesired features inthe parametric images.

Finally, there are methods that attempt to estimate notonly the kinetic parameters but also the input function duringthe direct reconstruction. This is usually achieved with alter-nating optimization �i.e., alternating between the “normal”direct reconstruction step and the estimation of the inputfunction�. Nevertheless, even if the objective function is con-cave in each alternating step, this does not preclude the ex-istence of local turning-points. Global convergence usingthese methods does not yet seem to have been demonstrated.

The next sections address the direct reconstruction meth-ods which have been already proposed in the literature indetail. Their features are also summarized in Table I.

II.B. Direct reconstruction methods with Poissonstatistics

II.B.1. Compartmental pharmacokinetic models

Direct reconstruction was first suggested by Snyder.20 Hisalgorithm was designed for multiple kinetic compartments,and the input function was approximated by a Dirac func-tion. The reconstruction algorithm attempted to find themaximum likelihood solution using the general expectationmaximization �MLEM� technique28 assuming that the data,acquired in list-mode, followed inhomogeneous spatiotem-poral Poisson statistics. One characteristic of this methodwas the assumption that each kinetic component is indepen-dent in the derivation of the EM step. The algorithm wasevaluated only by one simple simulation of a two-compartment model.

Independent of Snyder’s approach, Carson and Langeproposed the idea of directly reconstructing the parametersfrom projection data with an EM approach �i.e., EMPIRA�.21

They suggested Fisher scoring or Newton-Raphson as opti-mization algorithms for the maximization step. Their pre-liminary investigation adopted a single-tissue compartmentalmodel with one linear and one nonlinear parameter. Themaximization step was thus reduced to one nonlinear equa-tion and the update was performed by Newton’s method. Theauthors reported that their preliminary implementation wasvery slow in terms of convergence but had the potential toproduce accurate results.

Chiao et al.29,30 presented novel approaches to segmentfunctional regions and concurrently estimate parameters for asingle tissue compartment model. The objective functionused the Poisson log likelihood, to which a regularizationterm was added to reduce the ill-conditioning of the problem.A special characteristic of this method was the possibility ofeither estimating boundary side information by segmentingfunctional regions or incorporating it before processing intothe regularization term. The optimization algorithm wasbased on an iterative scheme using either the Marquardtmethod or the Newton algorithm. The authors claimed thatthis approach was fast and convergent, at least for the simu-lations that they performed. Unfortunately, the boundary in-

formation can only be applied when an anatomical region

Medical Physics, Vol. 35, No. 9, September 2008

coincides with the functional region. The presented resultson simulated cardiac PET data showed an improvement interms of bias and mean squared error.

Recently, another direct method for the reconstruction ofkinetic parameters from time-frame projection data has beenpublished by Kamasak et al.10 This work was the first toimplement a two-tissue compartmental model with four ki-netic parameters. This was used to derive the binding poten-tial �BP�, which is the ratio of bound to free radio-ligand inbrain, and the total volume of distribution �VD�, which is theratio of the volume occupied by the radiotracer at equilib-rium to the total tissue volume. The kinetic parameters wereestimated using the sum of the Poisson log-likelihood and aMarkov random field �MRF� regularization term as the ob-jective function for the kinetic parameters �i.e., using a maxi-mum a posteriori �MAP� technique�. The optimization algo-rithm was based on the iterative coordinate descentmethod,31 so it was designated by the acronym PICD. Thekinetic parameters of the compartmental model were decou-pled as proposed by Phelps et al.,32 so the kinetic model wastransformed to a weighted sum of two exponentials, one foreach compartment, thus making the parameters easier to es-timate. This transformation and the use of a multi-resolutiongrid technique made the reconstruction computationally effi-cient. The algorithm updated the parameters in a two-stagenested optimization. The authors commented that this algo-rithm converges to a local maximum, but they did not inves-tigate the existence of multiple local maxima. The study wasperformed on a single slice grid of a rat’s head phantom. Thepresented results showed improvement in the parametric im-ages for the direct reconstruction method with a particularlylow root-mean-squared error over regions of interest for asingle simulation for all the four kinetic parameters. How-ever, more work seems to be necessary to establish its prac-tical usefulness. A parallel application of the same method toreal data has investigated different approaches to the evalu-ation of the exactness of the fit and the estimation of theerror in the pharmacokinetic parameters.33 The same directreconstruction approach has also been extended to estimatethe pharmacokinetic parameters and the input function evenwhen the latter is not measured.34 Recently, the PICD algo-rithm has been applied to fluorescence optical diffusion to-mography by Milstein et al.35

Another similar approach to the direct reconstruction ofthe two tissue compartmental model has been used to esti-mate also the activity in plasma by Yetik et al.36 The keyfeature of this preliminary study was the direct estimation ofthe input function and the regional compartmental param-eters in two alternating steps using a preconditioned conju-gate gradient reconstruction algorithm for each of them. Theauthors presented results on simulated data where the exactboundaries of three regions were used for the estimation ofthe corresponding values. For a sample of initial values itwas shown that the algorithm converges to the same optimalvalue. A hundred Monte Carlo data sets were used to esti-mate the mean, and the recovery of the regional values was

accurate �up to 5%�. It would be useful to compare these

Key advantagesUnclear issues to

investigate

List-mode More investigation

Concept Convergence rate

t estimation of regions andparameters

Real data

Parametric regularization�ii� Multi-resolution

Real data

ple & practical for real data �i� Convergence rate�ii� Bias

�i� Preconditioner�ii� Regularization

mple & practical for real data

Convergence rate

Simple for real data Convergence rate

estimation of parameters &temporal basis

Ill-conditioned

Estimation of Bv Real data

Estimation of Bv Real data

Concept Convergence rate

nearization of exponentialponent, Estimation of Bv

Global convergence &Real data

r reconstruction of spectrum Highly ill-conditioned

imation of regional shapes Global convergence,constrains & real data

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TABLE I. Reviewed methods for direct reconstruction in emission tomography and their characteristics.

Study ModalityGen. objective

functionOptimization

algorithm Regularization Kinetic model Extension of investigation

Snyder 1984 ToF PET�L-M�

CompletePoisson log-likelihood

MLEM No 2-Tissue Preliminary

Carson andLange 1985

PET Poisson log-likelihood

EM withNewton

No 1-Tissue Not shown, small gridsimulations

Chiao et al. 1994 CardiacECT

Poisson log-likelihood

Marquardt andNewton

No/Yes�Markov�

1-Tissue Low resolution 2Dsimulations

Join

Kamasak et al.2005

PET Poisson log-likelihood

Coordinate Descent MRF 2-Tissue 2D phantom simulations �i�

Matthews et al.1997

PET Poisson log-likelihood

PIR �MLEM� No Linear Phantom & Humanacquisitions

Sim

Wang et al. 2008 PET �Shifted�Poisson log-likelihood

Preconditionedconjugategradient

Yes �MAP withGibbs priors�

Linear �Patlak plot� Phantom & RealmicroPET fully 3D data

�iii� SiTsoumpas et al.2008

PET Poisson log-likelihood

OS-PIR�POSEM�

No Linear�Patlak plot�

Realistic phantomsimulations & clinical data

Reader et al.2006

PET �L-M� Poisson log-likelihood

4D MLEM TemporalConvolution

Linear L-M simulated & clinicaldata

Joint

Huesman et al.1998

SPECT Least squares Unclear No 1-Tissue 3D simulations

Reutter et al.1998

SPECT Least squares Newton-Raphson

No 1-Tissue 2D simulations

Limber et al.1995

SPECT Least squares Levenberg-Marquardt

No 1-Tissue Simulations of small 2Dgrids

Zeng et al.1995

SPECT Least squares Levenberg-Marquardt

No Linearized 1-tissue

Noiseless simulations Licom

Hebber et al.1997

SPECT Least squares �i� LSQR�ii� NNCGLS

Tichonov Spectralanalysis �SA�

2D simulations with 2-tissue model

Linea

Maltz 2000 SPECT Least squares Newton & linesearch

No SVD on SA Phantom simulations Est

3967 Tsoumpas, Turkheimer, and Thielemans: Approaches for direct parametric reconstruction in emission tomography 3967

results with those obtained using an appropriate indirectmethod. Although this study was preliminary and focusedonly on simulated data, it is a good illustration of the poten-tial capabilities of direct reconstruction methods without theneed of a predetermined input function.

II.B.2. Linear approximations of thepharmacokinetic models

For cases in which the model parameters are linear, aninnovative method for direct calculation of parametric im-ages from frame-by-frame projection data was developed byMatthews et al.37 This parametric iterative reconstructionmethod, known also as PIR, was based on the MLEM algo-rithm for image reconstruction,38 but transformed it appro-priately to incorporate the dynamic data. Matthews et al.applied this algorithm to several kinetic modeling ap-proaches, such as population modeling37 and Patlak plot.39

No scatter correction was employed although the scatter frac-tion was around 15%. This may explain some of the ob-served bias in the experimental results. Another possible ex-planation for the bias, especially in the case of Patlak plot,was slow convergence. The authors argued that this might bedue to the similarities of the two basis functions in the linearmodel.

The PIR algorithm was the basis for the direct methodused in a recent investigation where both indirect and directparametric estimation methods were implemented, evaluated,and compared for Patlak plot.40 The main aim of this studywas the investigation of the statistical properties of differentestimation approaches for real and simulated data for dy-namic FDG brain studies. For the indirect estimation, bothfiltered back projection �FBP� and ordered subsets expecta-tion maximization �OSEM� were used and then the pharma-cokinetic parameters were estimated by applying the Patlakplot. These indirect procedures were compared with the sub-set version of PIR called parametric OSEM �POSEM�. Themethods were quantitatively compared for both parametricimages and regions of interest using realistic synthetic data.Results were shown in terms of bias, variance, and root meansquared error. The indirect methods, both analytical and it-erative, showed appreciably higher standard deviation andmean squared error than the direct method. POSEM ap-peared biased only because of slow convergence due to ill-conditioning of the problem and the limited subsets cycle.After a high number of iterations and the use of no subsetsthe result was accurate. It was suggested that algorithms withbetter convergence rate properties may estimate the Patlakplot parameters much faster with a similar level ofaccuracy.41

Another method that used the Patlak model was proposedrecently by Wang et al.,42 who utilized a preconditioned con-jugate gradient algorithm to find the maximum a posteriori�MAP� solution using a regularization term based on Gibbspriors, which were different for the two parameters. Thisalgorithm was evaluated with the use of multiple simulationdata sets and then applied to real fully 3D microPET data.

The results were compared with two indirect approaches.

Medical Physics, Vol. 35, No. 9, September 2008

The first of these implemented the regularization step in theMAP reconstruction, while the second imposed spatial regu-larization during pixel-wise Patlak analysis. The results ofthe simulations showed that, in most instances, the Patlakslope had less bias for the same level of standard deviationwhen using the direct method. However, one of the threeregions that was displayed �white matter� appeared to haveless bias for the indirect case when regularization was im-posed during the Patlak analysis. The Patlak intercept ap-peared much more sensitive than the slope to noise. Theresults were very biased or noisy, depending on the regular-ization parameters. For the real data, a shifted Poissonmodel43 was used as data were precorrected for randomcounts. The corresponding result appeared to be qualitativelysimilar for both the direct method and the best of the indirectmethods; however, no further quantitative analysis was pre-sented. It would have been very interesting to investigate theconvergence rate of the direct algorithm, especially in thecase of real data.

Discrete sets of temporal basis functions may be used torepresent the characteristic exponential response of the tissueand their use can also be extended to direct reconstructionmethods �see Route II of Fig. 1�.25 This approach was inves-tigated for the case of unknown, but non-negative, temporalbasis functions, in which parameters were jointly estimatedfrom list-mode �L-M� data with alternating optimizationsteps.25 Although this is a very promising approach for directestimation, the results obtained by Reader et al. appeared tobe unstable in some cases. In particular, when the methodwas applied to noiseless data with 15 basis functions, or tonoisy data with just three basis functions, the outcome de-pended on the initialization. This non-uniqueness may beexplained by the high level of ill conditioning or by theexistence of multiple local maxima. A further investigationwas recently presented,44 which incorporated integrals con-voluted with the input function as basis functions. In thisapproach better noise properties were found for the dynamicimages compared to conventional reconstruction methods,which resulted in improved estimation of kinetic parameters.However, the final parameters still needed to be estimated byconventional kinetic analysis.

II.C. Direct reconstruction methods with Gaussianstatistics

II.C.1. Compartmental pharmacokinetic models

Two conceptually similar direct estimation methods basedon the minimization of the least-squares objective functionhave been developed by Huesman et al. and Reutter et al.from SPECT projections.45,46 A single tissue compartmentalkinetic model was simulated where the uptake constant andthe portion of the blood �Bv� are linear components. Hues-man et al. removed these conditionally linear parameters us-ing standard techniques47,48 and thus the objective functionwas assumed to be a function of only the nonlinear term.This method succeeded in producing accurate parametric re-constructions of noiseless data but there was still bias in the

case of noisy data even if multiple local minima were not

3968 Tsoumpas, Turkheimer, and Thielemans: Approaches for direct parametric reconstruction in emission tomography 3968

encountered. Reutter et al. used Levenberg-Marquardt forthe minimization of the least-squares. Their method esti-mated all the parameters accurately but they tested it only fornoiseless data.

The Levenberg-Marquardt algorithm has also beenadapted by Limber et al. for the least-squares minimizationof a two-tissue compartmental model which included a back-ground constant term �e.g., blood radioactivity�.49 The modelwas separated into single tissue compartments according totheir kinetics �i.e., fast compartment, single compartment,and a constant term�, so only a single tissue compartmentalmodel was integrated into the direct reconstructions. The re-sults were accurate for normal count cases; however, in verylow count cases some bias was found. Although this ap-proach has some challenging aspects, overall it appearspromising. More investigation of the convergence rate andstopping criteria is desirable.

II.C.2. Linear approximations of thepharmacokinetic models

The concept of direct reconstruction has been extended byHebber et al. to estimate multiple exponential componentsdirectly from SPECT projection data without prior assump-tions about the number of exponentials.23 The approach wasbased on spectral analysis,50 which does not limit the numberof exponentials necessary to represent the data. The spectrumwas discretized into a large number of equally spaced ele-ments in logarithmic space. Two minimization algorithmswere tested for the solution of this large sparse least squaresproblem: The LSQR �most probably corresponding to least-squares QR decomposition� and the non-negative conjugategradient least squares �NNCGLS� iterative techniques.51

These methods appeared to be reasonably stable and robustin the temporal domain both for simulated and real data, butthey were unstable in the spectral domain. The reconstructedspectrum was very different from the original, especially fornoisy data. From the comparison of the two algorithms,LSQR appeared more accurate than NNCGLS. However, thisapproach is ill-conditioned as it attempted to fit a high num-ber of variables �i.e., 64 spectral frequencies for each pixel�to the available data. This might explain why it was notinvestigated further, even though it is a conceptually attrac-tive idea.

Zeng et al. used linear time-invariant system theory forthe estimation of the exponential parameter in SPECT.52 Theestimation of the kinetic parameters was achieved in twoalternating steps. In the first of these, the exponential com-ponent was linearized by the first order Taylor expansion andthen it was estimated by solving the differential equationsystem with a least-squares approach �e.g., with QR factor-ization�. During the second step the coefficients and theblood volume �Bv� term were estimated using the Levenberg-Marquardt algorithm but keeping the exponential factorsconstant. Results were offered only for noiseless simulateddata and appeared more accurate when obtained directly

from projection data than from dynamic reconstructions. One

Medical Physics, Vol. 35, No. 9, September 2008

drawback of this method seemed to be that only one expo-nential parameter could be estimated for each region �i.e.,single-compartment tissue�.

The most recent contribution to direct reconstruction ofregional kinetic parameters in SPECT has been presented intwo pieces of published work by Maltz.24,53 His proposedkinetic model is based on the use of six orthogonal basisfunctions derived with singular value decomposition fromconvolution integrals of spectral analysis, an approach whichhas also been independently suggested by Turkheimer et al.for the indirect case.54 The coefficients of the new basis func-tions for each region were estimated directly from dynamicSPECT data. In his first paper, Maltz53 modeled the shape ofthe regions by ellipses and their geometrical characteristicswere jointly estimated from the data. The objective functionwas the unweighted least-squares error function. The optimi-zation was made feasible by a complex hybrid �global-local�iterative approach.55 This incorporated an adaptive simulatedannealing algorithm for a global optimization search fol-lowed by a local optimization using either a projected �un-preconditioned� gradient descent algorithm or, when the Hes-sian was well-conditioned and positive definite, a projectedstabilized Newton algorithm. The appropriate step-size wascomputed by Armijo’s inexact line search rule.55 This jointestimation of the regional kinetic parameters and their shapewas not very stable. In particular, even in the noiseless case,the multiple local minima did not permit the algorithm tofind the global minimum and the final estimate showed smallbias.53 In his second paper,24 Maltz estimated the coefficientsby an analytic matrix pseudo-inverse operation.56 This esti-mated accurately the time activity curves but not the actualkinetic parameters. The validation of the method for directrecovery of regional tracer kinetics was performed for aphantom and patient study. The results could be accurate, buttheir consistency was unreliable. Some problems appeareddue to low count areas, possibly because the use of least-squares assumes a Gaussian and not Poisson distribution ofthe data. One drawback of both of the methods employed byMaltz,24,53 as well as of other approaches that use basis func-tions for the time-activity curves, is that there is no directcorrespondence with a compartmental model. Finally, an-other issue that should be considered in future investigationsis that each region is likely to be described better with dif-ferent sets of temporal basis functions, which would increasethe level of complexity.

III. CONCLUSION

The current approaches to direct parametric image estima-tion from measured data have been discussed in detail. TableI summarizes some of the features of 14 representative tech-niques that have been reported in the literature. Methods arecategorized according to �i� modality application, �ii� gener-alized objective function �i.e., includes kinetic model�, �iii�optimization algorithm, �iv� regularization approach, �v� ki-netic model, �vi� investigation extension �e.g., only 3D simu-lations, or fully 4D real data�, �vii� key advantages for each

of the methods, and �viii� unclear points to be investigated.

3969 Tsoumpas, Turkheimer, and Thielemans: Approaches for direct parametric reconstruction in emission tomography 3969

The table also highlights some important questions thatfuture investigations should attempt to answer:

• To what extent can direct estimation methods improvethe accuracy, the variance, and mean squared error ofparametric images compared to well-established meth-ods?

• How reproducible are the simulation results in realdatasets?

• Should we expect different convergence properties fordifferent studies or even for different regions of onestudy?

• Does direct estimation depend on the initial parametricimage guess and if so how?

• What are the main factors that may generate bias?• What are the factors or unresolved issues that currently

limit clinical application of direct methods?In regard to the last of these questions, there are a number

of significant obstacles to the implementation and adoptionof direct reconstruction in practical terms:

�1� Image reconstruction �whether by indirect or directmethods� is a statistical parameter estimation problemand therefore entails three requirements:

�a� The model parameters must be identifiable.�b� An estimator of the covariance of the estimated pa-

rameters must exist.�c� A way of measuring how accurately the model fits

to the data must be available.In many respects, direct methods have advantages

over indirect methods in regard to all these require-ments. They provide more accurate estimates of the co-variance and identifiability of the parameters as they arebased on a complete noise model of the data, and model-mismatch can be investigated using residual errors, ei-ther on the image or projection domain.33

�2� A specific kinetic model with a fixed number ofcompartments/parameters is not expected to yield properresults everywhere in the image. Therefore direct recon-struction might need different kinetic models in differentregions. This means that some flexible mechanism ofmodel selection should be incorporated into the process,for example along the lines developed for the indirectcase.57 Special care should be taken regarding identifi-ability and over-parametrization of the model.

�3� In reference region approaches, which are popular toavoid blood sampling, the input function would need tobe estimated from some form of dynamic images beforeinitiating the direct reconstruction process. Alternatively,a joint estimation of the input function and parametricimages could be developed, but this would make thealgorithm more complicated.

�4� Special care must be taken to incorporate patient motioninto the reconstruction procedure. In indirect methods,the images for the time frames are often realigned toeach other using image registration on, for instance,non-attenuation-corrected dynamic images.58 Other

methods track patient movement and obtain motion-

Medical Physics, Vol. 35, No. 9, September 2008

corrected images.59 For direct reconstruction, a motionestimate obtained as in the indirect methods could beintegrated into the projection operators.60

�5� Clinical application will entail the successful resolutionof current issues of computational complexity and con-vergence. One researcher has noted that convergencerate can be significantly improved by the use of off-diagonal preconditioners, which is an avenue of possiblefuture work.61 In practice, a block-diagonal precondi-tioner �i.e., diagonal in spatial domain, but nondiagonalin parametric domain� could provide a good compromisebetween computational complexity and convergencerate.

Further research is also needed to investigate the impactof direct reconstruction algorithms on statistical power �sen-sitivity and specificity� in real scenarios, for example, when adisease cohort is compared to a control group.62,63

Direct iterative reconstruction of list-mode data remains apromising challenge.20,64 This approach is likely to be usefulin cases where radioactivity concentration changes rapidlywith time.

As mentioned earlier, local turning points are likely toappear during the optimization procedure for nonlinear ki-netic models. One technique to avoid them is to initializeclose to the global turning point and also set lower andhigher bounds on the kinetic parameters. The latter approachhas been already used successfully in �indirect� kineticmodeling65,66 but has not been incorporated in direct ap-proaches yet.

As a final note, the methods discussed in this review canbe used to estimate kinetic parameters region-wise, wherethe regions may be defined from structural data �e.g., ana-tomical atlases67�. The advantage of this method is not onlythe significantly fewer variables but also the practical com-putation of the Fisher information matrix and, therefore, thecovariance matrix of the parameters.5,6

ACKNOWLEDGMENT

This work was supported by the Medical Research Coun-cil under Grant No. G78/8306. CT is grateful to Miss AndriTziortzi and Dr. Roy Clements for their thorough comments.

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