IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 1, FEBRUARY 1997 11

Linear Least Squares Compartmental-Model-Independent Parameter Identification in PET

Joseph A. Thie,* Gary T. Smith, and Karl F. Hubner

Abstract—A simplified approach involving linear-regressionstraight-line parameter fitting of dynamic scan data is developedfor both specific and nonspecific models. Where compartmental-model topologies apply, the measured activity may be expressed interms of: its integrals, plasma activity and plasma integrals—allin a linear expression with macroparameters as coefficients.Multiple linear regression, as in spreadsheet software, determinesparameters for best data fits. Positron emission tomography(PET)-acquired gray-matter images in a dynamic scan are ana-lyzed: both by this method and by traditional iterative nonlinearleast squares. Both patient and simulated data were used. Regres-sion and traditional methods are in expected agreement. Monte-Carlo simulations evaluate parameter standard deviations, dueto data noise, and much smaller noise-induced biases. Uniquestraight-line graphical displays permit visualizing data influenceson various macroparameters as changes in slopes. Advantages ofregression fitting are: simplicity, speed, ease of implementationin spreadsheet software, avoiding risks of convergence failures orfalse solutions in iterative least squares, and providing variousvisualizations of the uptake process by straight line graphicaldisplays. Multiparameter model-independent analyses on lesserunderstood systems is also made possible.

Index Terms—Dynamic scans, empirical modeling, parameteridentification, regression.

I. INTRODUCTION

I N both positron emission tomography (PET) and singlephoton emission computed tomography (SPECT) dynamic

scan time-activity data may be fit to models. Fitted parameters,especially when correlated with clinical studies, offer addi-tional information beyond that obtained from simply activityimages. It is the fitting algorithm that is the subject of thework here.

Existing algorithms typically involve curve fitting or spe-cial straight line plots from which model parameters areestimated. Popular in the former class are nonlinear leastsquares (NLS) curve fittings of time activity curves (TAC’s)involving iterative trials of rate constants, and [1].Though quite effective, these can be cumbersome and involvetime-consuming computer programs. They can require goodstarting guesses for success in order to avoid local minima

Manuscript received October 20, 1995; revised October 18, 1996. TheAssociate Editor responsible for coordinating the review of this paper andrecommending its publication was C. J. Thompson.Asterisk indicates corre-sponding author.

*J. A. Thie is with the Department of Nuclear Engineering, University ofTennessee, Knoxville, TN 37996 USA.

G. T. Smith and K. F. Hubner are with the Biomedical Imaging Center,University of Tennessee Medical Center, Knoxville, TN 37920 USA.

Publisher Item Identifier S 0278-0062(97)00976-2.

problems in the least squares fitting. Among the straight-linegraphical methods the most widely used is the Gjedde–Patlakapproach [2]–[4]. Unfortunately graphical methods as popu-larly implemented do not determine a complete set of modelparameters, though this need not be the case [5].

A simple, yet complete, approach to model parameter identi-fication from TAC’s is therefore needed. An attractive methodis one in which the measured tissue activity is directlyproportional to a sum of other measured quantities—andwith combinations of parameters entering as proportionalityconstants. The reason for this is that least squares fitting to datais then possible by solving linear algebraic equations, with noiteration involved. Blomqvist [6] and Evans [7] suggested up totwice integrated plasma and activity as independent variablesto be used in fitting data. This approach was implemented onsimple models by Feng [8], Hubner [9] and Chen [10], withthe latter work also using twice integrated plasma and tissue.Ono [11] used parameter combinations as proportionalityconstants; he determined these by weightings of integrationsperformed once and twice. In other approaches for securingproportionality to parameters Patlak [12], Sawada [13], andThie [5] suggested convolutions, as well as integrals, requiring,however, some limited iterating on a parameter.

The work here extends these prior related efforts to both anyspecific or even unspecified compartmental model. Moreoversimplicity is demonstrated by adapting to the widely usedspreadsheet software capabilities of multiple linear regression.There can be clinical appeal if minimal efforts akin to familiarGjedde–Patlak plotting can be retained, while additionallyfinding applicability to many model types with identificationof all parameters.

II. M ETHODS

A. Regression Analysis

A general compartmental model with an unspecified numberof compartments is shown in Fig. 1. Where the bidirectionalflows are proportional to compartment activities the modelsmay be represented by linear differential equations (see Ap-pendix I). In a usual solution involving plasma concentration

and its convolutions, parameters unfortunately appearnonlinearly and are therefore cumbersome to evaluate in fittingdata.

A preferable solution form would have unknowns appearinglinearly. One used here, coming from integrations of the model

0278–0062/97$10.00 1997 IEEE

12 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 1, FEBRUARY 1997

Fig. 1. The general compartmental model considered here. Intercompart-mental transports proportional to rate constants can possibly exist betweenall pairs of compartments.

equations, is

possibly thrice integrated terms etc. (1)

The are macroparameters which represent combinations ofrate constants and vascular volume, the latter allowing forobserving part of in . The Appendixes obtain explicitexpressions for the for popular models.

In general, the terms having the more multiples of inte-gration have association with the more distant (from plasma)compartments. The more compartments and complexity, themore terms needed. Demonstrations here choose physiolog-ically understood systems. However, the applicability of themethod is more general; (1) can be regarded as an empiricalmodel whose value is in data compression.

Processing dynamic scan data can be convenient usingpopular spreadsheet software. The first step in the method hereis calculating various integrals in adjacent columns for all therows’ data times. Numerical accuracy, especially for noisy dataand/or large fractional changes in and between frames, can bepreserved by the following.

1) Using analytical integrations, such as involving cubicspline or exponential fits to sequential data values.

2) Recognizing that scanners provide exactly inframe increments. Otherwise, exact analytical convolu-tion solutions and computations of integrals may be usedto obtain correction factors on any integrals obtained bynumerical integration approximations within frame timeintervals. A model-specific correction factor is an exact

analytic integral divided by the numerical algorithm’sintegral. Often quite near unity, such corrections basedon representative rate constants and plasma functionscan be suitable.

3) Evaluating effects of noise-induced bias. Monte Carloparameter determinations from similar simulated datawhose noise content is varied from run to run can dothis. In very noisy data where bias might be significant,possibilities of one set of parameter bias corrections forall similar TAC’s in a class of scans could exist.

Except for fitting algorithms developed by Feng [8] and Chen[14], most all others can have parameter bias.

As a next step, terms having fixed known parameters,if any, in (1) are transposed to the left-hand side as anadjusting subtraction to . If there is no weighting the leftside is now multiple linearly regressed against the remainingvariables on the right side. This generalization of regressionagainst one variable (i.e., least squares straight line fitting)is a command available in all popular commercial spread-sheet software. Least squares methods minimize a weightedsum of residuals squared, less terms having fixedparameters remaining right-hand side of (1) at . Theoptimal parameter answers in a regression operation comefrom hidden software solving simultaneous linear equations.Weighting factors are unity if weighting is not beingincorporated.

However, if weighting is used, each term in the squarederror summation will have a known factor. One approachfor accomplishing this can be to transform data and allregressed variables by a factor; then proceed as ifregressing without weights.

If there is a known model, identified rate constantscan generally all be algebraically obtained from there-gression coefficients (see Appendixes). This mapping betweenthese two parameter spaces reflects of course identifiabilityand uniqueness considerations. Otherwise, the’s themselvesrepresent a characterization of the system.

Spreadsheets with ease of repeating similar analyses areconvenient here. But regression with other software solvingsimultaneous linear equations is also possible. Little program-ming would be required using modern mathematical commandlanguage software.

B. Patient Data

It is appropriate that this method be checked against thewidely used Marquardt version [15] of traditional NLS analy-ses using actual data. A female patient with an unspecifiedpsychosis was dynamically PET scanned during conscious se-dation following a 233-MBq [F-18] fluorodeoxyglucose (FDG)injection: six 20-s frames, three 1-min frames, seven 5-minframes, and one 15-min frame. Concurrent arterial samplingat intervals ranging from 15 s to 20 min was performed. A1893-mm brain gray-matter region was selected for. As ameasure of the amount of data noise, root-mean-square (rms)residual errors in fits to models were typically about 8% ofdata values. A vascular volume of 0.04 ml/gm was a fixedanalysis parameter for blood in tissue.

THIE et al.: COMPARTMENTAL-MODEL-INDEPENDENT PARAMETER IDENTIFICATION IN PET 13

Fig. 2. Regression fit for an FDG study.

C. Simulations

As other checks of the method, computer simulated datawas employed. Here, parameters determined in the FDG studywere used in model equations to generate noise-free andnoisy “data.” For the latter, relative Gaussian noise amplitudesadded to were such that the fractional noise in wasproportional to [frame counts] . For the absolute noisemagnitude to be representative it was chosen so as to give thesame sum of fitting error squares as for patient of Fig. 2 (i.e.,corresponds to about 8% fractional rms data noise). In anothersimulation a larger number of parameters was used: four rateconstants and a vascular volume. For these, a set of valuesbased on a representative amino acid (1-aminocyclopentaneC-11-carboxylic acid) PET scan of a dog gracilis muscle wasused.

D. Weighting

All analyses used reciprocal variances of values asweighting factors in the sum of residuals squared, thus pro-viding some consistency in comparisons of examples here.However, it must be noted that this will be an approximationin attempting to achieve a minimum theoretical parametervariance in any method, regression or NLS, if independentvariables ( , its integrals, and ’s integrals) also contributesignificant measurement errors to residuals. In the regressionmethod the total measurement error for a residual is [16]:error in less the sum of products of regression slope andindependent variable errors at point.

The reciprocal variances of this quantity are appropriate foruse in the spreadsheet algorithm as weighting factors. Influ-ences of are common to both NLS and regression methods.But for the latter this expression and (1) show thaterrorsas well enter into independent variables. For regression orany other parameter identification algorithm (especially casesinvolving noisy data leading to large unavoidable parameteruncertainties), where estimations suggest that proper weightingis important, one might consider the method of Feng [8] andChen [10].

Fig. 3. Display of data influence on the regression parameterK1 for FDG.Data points from Fig. 2 have a twice-integrated plasma subtraction and asingle-integral tissue activity addition before plotting here—for purposes ofremoving effects of other parameters. At earlier times, where these removalsbecome small, the line essentially represents proportionality of uptake toCadt.

TABLE ICOMPARISONS OFREGRESSION,NLS, AND SIMULATION *

Regressionresult

NLSresult

Simulationconstants

FDG patientK1 0.0652 0.0662k2 0.1044 0.1056k3 0.0492 0.0479

% standard deviation of fit** 8.6 8.4

FDG simulation(noise free)

K1 0.0662 0.0662k2 0.1056 0.1056k3 0.0479 0.0479

Amino acid simulation(noise free)

V 0.100 0.100K1 0.316 0.316k2 0.1341 0.1345k3 0.0756 0.0759k4 0.0198 0.0200

* Rate constants have units 1/min, except , having ml/gm/min; units of areml/gm.** This is 100 [ (data calculated value)2/ (data)2]1/2.

III. RESULTS

A. Patient and Simulated Data

Fig. 2 shows TAC data of the patient FDG study, correctedfor vascular volume, along with the calculated regression andNLS fits. Table I shows quite close parameter values obtainedby the two methods. This validation is encouraging for theregression method: for independent variables it calculates fromboth noisy ’s and ’s, whereas the NLS requires only thenoisy ’s.

Comparison of methods using noisy data is expectedly notquite perfect. Hence noise-free simulated “data” using thispatient’s parameters is appropriate to analyze. Table I shows

14 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 1, FEBRUARY 1997

TABLE IIREGRESSIONRESULTS OF 500 MONTE CARLO

RUNS OF NOISY SIMULATED FDG DATA*

Simulationvalue

analysesaveraged

value% bias

% standarddeviation

K1 0.0662 0.0648 �2.0 8k2 0.1056 0.1007 �4.6 22k3 0.0479 0.0461 �3.8 17

K1k3=(k2 + k3) 0.02065 0.02040 �1.2 5

* Rate constants have units 1/min, except , having ml/gm/min.

regression analysis giving the same FDG parameters usedin the simulation. It also shows regression results agreeingwith simulation for an amino acid parameter set as well. Thelatter also proved the determination of a model’sandparameters.

With weighting implemented in these numerical compar-isons not necessarily being optimal, it is worth evaluatingits effect. Results for the FDG noisy data without weightingshow the largest changes, occurring in, are only 2% (inregression) to 4% (in NLS), and show considerably smallerchanges in other parameters. Here, for example, weightinginfluences are well below the noise-induced unavoidable pa-rameter standard deviations shown in Table II.

In analyzing results, examination of residuals is traditional.A usual empirical approach is inspecting TAC data pointsalong with the best-fit calculated curve, as Fig. 2. An intenthere, however, is to supplement this with the use of moremeaningful model-oriented displays. In constructing the latter,all but one term on the right side of (1) is transposed to theleft as adjustments to data. The remaining variable is usedas the abscissa. How well data points define a straight line ofslope or though the origin is meaningful to examine.

Fig. 3 shows one such display having a slope(which is, but simply for vascular-volume-corrected

data). Its purpose is to show, noting essentially small randomresiduals here, that this particular parameteris reasonablymodeled and determined.

B. Monte Carlo Calculations

The speed with which spreadsheet software can do regres-sions is a distinct advantage when performing Monte Carloerror calculations. Regressions were performed on 500 sets ofnoisy simulated data. Statistics of the 500 rows of parametersets could then be evaluated: parameter averages, standarddeviations, and correlation coefficients.

Table II shows good agreement of analyses averaged resultswith simulation parameters. Slight bias errors are well belowthe inescapable standard deviations of parameters due toinherent data noise. In other findings for parameter paircorrelations (ranging from 0.69 to 0.95) there is a suggestionthat for the available data here, three parameters are tosome extent more than adequate. One consequence of thesecorrelations is that a defined accumulation ratehas less error.

Though not often seen in published parameter identificationresults, error and correlation determinations as performed

here are recommended. Colinearities (i.e., high correlations),indicative of perhaps over-parameterization for the amount ofnoise in the data, can then be identified for any method used.Thus, small (or even negative) determined parameter valuesrelative to their standard deviations would not be reported dueto a lack of their identifiability.

IV. DISCUSSION

One deterrent to a more widespread use of dynamic analysiscombined with clinical correlations with parameter results isan envisioned complexity of NLS methods. Recent advancesin NLS software [17] have simplified it for users. Nevertheless,it is felt that a method, such as that introduced here, ifconceptually simple and easy to use, will help to encouragemore dynamic analyses.

Already, because of simplicity and ease of use,Gjedde–Patlak analyses have become quite popular. However,it has been shown theoretically that its primary result

is a diagnostic indicator closely related toa final frame standardized uptake value (which is a local tobody-average specific activities ratio), more easily obtainedclinically [18]. On the other hand, more information from acomplete set of parameters obtainable from the same dynamicdata might have more diagnostic potential.

Recognizing the acceptance of Gjedde–Patlak analyses, aprime focus of our work has been to try to mimic the simplicityof that approach while obtaining a complete set of modelparameters. It is easily seen that (1) resembles an extensionof the Gjedde–Patlak equation. If both sides are multipliedby (a time-dependent weighting factor intrinsic to thisgraphical analyses), the result is

additional terms (2)

Thus, regressing -weighted data is operationally akinto performing Gjedde–Patlak analyses. However, with moreand differently defined macroparameters (i.e.,here is notthe Gjedde–Patlak slope), the regression approach gives acomplete set of parameters as a result. Thus, if one has alreadyaccepted a least squares fitting method for Gjedde–Patlak plots,the extension of that computer fitting to a multivariable leastsquares method should seem attractive.

In comparing fitting methods, one should address variousfactors. The standard deviations of individual parameters andmacroparameters may vary among methods, as noted forexample by Feng [8] and Chen [14]. A suggested comparativeratio here might be that of the standard deviation to bias. If it isfeasible to make fixed small bias corrections on generic classesof data, this bias error in many methods including regressionmay be reduced.

When regression and NLS are compared, disadvantages ofthe latter not shared by regression are as follows:

1) a need for reasonable initial parameter guesses to assureconvergence to correct answers;

2) a risk of having multiple minima in the sum of squaredresiduals and not converging to the lowest of these;

3) longer computing times, especially if many iterations onpoor guesses are involved;

THIE et al.: COMPARTMENTAL-MODEL-INDEPENDENT PARAMETER IDENTIFICATION IN PET 15

4) use of a “black box” tool without fully seeing orunderstanding the specifics of data manipulation—incontrast to the high visibility in spreadsheets.

An apparent advantage of NLS could be said to be itsdirect output of rate constants. However, the macroparameteranswers of regression algebraically readily translate into rateconstants. A more fundamental issue in any event is whatset of rate constants and/or macroparameters will give thediagnostician the best discrimination for a given disease forits particular population prevalence. A method’s results shouldtherefore be appropriately transformed for best diagnostic use.

Finally, the model-independent philosophy put forward byGjedde [3] and Patlak [4] should be revisited, but now in anextended form. Equations (1) or (2) in a somewhat empiricalapproach need not have model-based understandings of their

’s. Some prior basis might exist for specifying the numberof parameters. Lacking this, successive fits to data postulatingone, two, three, etc. ’s could be made; statistical analyses,such as the -test and the Akaike information criterion [19],would then help in selecting the number of parameters. Thisempirical approach of data compression can be useful incomplex systems or in cases of unmodeled (or incorrectlymodeled) tracers’ data that might otherwise go under- (orincorrectly) utilized.

Moreover, ’s, as an extension of the Gjedde–Patlak slopeapproach, might, in the same sense, be diagnostic aids. Then atleast some physiological interpretive features might be usefulin the absence of an acceptable model:

1) coefficients associated with or relate, respec-tively, somewhat to tracer entry or exit phenomena incompartments;

2) successive terms represent historical (i.e., integral) con-tributions to . Hence the coefficients of the higherorder integrals relate to compartments more distant intime (i.e., sequential events) from plasma.

V. CONCLUSION

A new approach to characterizing dynamic scans is pre-sented. It can be used with or without a specifically formulatedmodel. Especially attractive is a simplicity and speed compara-ble to least squares fitting of a Gjedde–Patlak plot, yet yieldingall model parameters. Results are as would be obtained fromthe more cumbersome NLS methods, but without risks ofiterating to an incorrect or unconverged parameter set. Addedappeals include: having interpretative straight line graphicalvisualizations of data relating to individual macroparameters;and being able to perform multiparameter model-independentanalyses on lesser understood systems.

APPENDIX ITIME AND LAPLACE DOMAIN

EQUATIONS: THE GENERAL MODEL

In compartmental models, material balances oftissuecompartmental activities lead to a series of linear differ-ential equations of the form

(A1)

where the summation is over additions to from for. The scanner’s measured sums these and a vascular

contribution

(A2)

In a time domain solution approach, each is expressed as alinear combination of integrals (single and multiple) of and

. This is accomplished by integrating these two equationsone or more times and algebraically combining results.

However, an equivalent, but more convenient, Laplacedomain approach transforms these, with appearing onthe left of (A1). Then a set of simultaneous linear equationsis solved by Cramer’s Rule. Substitution into (A2) gives thesystem transfer function relating and in fractional form

(A3)

Here, and are both polynomials up towith parameter combinations as coefficients—coming from theevaluation of determinants in Cramer’s Rule. After multiplying(A3) through by the denominator and also the desiredrelationship linear in parameter combinations results

(A4)

Inverse transformation yields the form

possibly thrice integrated terms etc. (A5)

Appearing are a total number of ’s, made up of rateconstants and . Thus, in a known model the rate constantsmay be determined after data fitting evaluates the’s. In theabsence of a known model this equation then offers at least ageneral representation suitable for data fitting by regression.

In (A5) plasma function additions and activity functionsubtractions may be interpreted in terms of model structure.The accumulated activity is seen expectedly to: increase withexposure (integrations) to plasma; decrease due to exposedpresence (integrations) of activity in compartments due to exitrate constants.

APPENDIX IITWO-COMPARTMENT MODEL EXAMPLE

As an example of the time domain approach, one mayintegrate (A1), designating and . Then

and its integral are replaced by expressions from (A2) andits integral, with the desired result

(A6)

Comparison with (A5) defines its and explicitly.

16 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 1, FEBRUARY 1997

APPENDIX IIITHREE-COMPARTMENT MODEL EXAMPLE

To illustrate using the Laplace method, the case of plasmaand tissue compartments has for (A1)

(A7)

(A8)

using traditional rate-constant notation with . Whensolved in the form of (A4), the result is

(A9)

With inverse transformation, the and become inte-grations, once and twice, respectively. Themacroparametersin (A5) are now identified in brackets here.

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