Generalized Linear Least Squares Research Articles

Applying approximate Bayesian computation to reduce uncertainty in multigroup 235-U cross-sections using ICSBEP experimental data

The multiplication factor (keff) and its uncertainty are critical design parameters in nuclear reactors. The keff uncertainty must be considered for operation, safety, and economic reasons. Consequently, reducing uncertainty in keff has been of interest to the nuclear industry for as long as nuclear reactors were designed. Two methods of reducing this uncertainty are in use – Generalized Linear Least Squares (GLLS) and A General Monte Carlo-Bayes Procedure for Improved Predictions of Integral Functions of Nuclear Data (MOCABA). Both methods have the ability to reduce the keff uncertainty by reducing uncertainty in cross-sections through the assimilation of measured critical systems’ or nuclear reactors’ operational data. The algorithms require means of simulating the experimental data and that the calibrated parameters are described by specific statistical distributions. GLLS is limited to linear models and multivariate normal prior and posterior, while MOCABA can use any (non-linear) model but is also limited to multivariate normal prior and posterior. This work implements a universal and rigorous algorithm called Sequential Monte Carlo – Approximate Bayesian Computation (SMC-ABC) for the same application. It is found that SMC-ABC, GLLS, and likely MOCABA give essentially the same results for the same problems. The paper also presents a reliable method for validating the results based on so-called “synthetic experiments”. Synthetic experiments are computationally generated data used in place of experiments. The method allows for verifying whether there is a risk of overfitting the calibrated parameters (cross-sections in case of this work) during the data assimilation. This novel method shows the potential of using SMC-ABC and other Bayesian calibration methods to generate improved general-purpose neutron cross-section libraries.

Non-intrusive stochastic approach for nuclear cross-sections adjustment

Bayesian adjustment of nuclear cross-section data within their prior uncertainties has been well-established in the neutronic community as the most mathematically-disciplined approach for improving the quality of neutronic calculations. The premise is that nuclear data are believed to contribute the most to the observed discrepancies between code calculations and measurements for key neutronic performance metrics such as critical eigenvalue, flux, power distribution, isotopic concentrations, etc. In its standard rendition, the adjustment procedure requires the solution of an optimization problem to calculate the optimum adjustments, with the optimality denoting the best possible agreement between measurements and predictions that constrains the adjustments to be statistically consistent with the prior uncertainties. To achieve that the optimization problem requires the derivatives of the responses of interest, e.g., keff, spectral indices, etc., with respect to all nuclear data. In the absence of an adjoint sensitivity capability, the calculation of these derivatives is computationally infeasible, thereby limiting applicability to adjoint-enabled codes only. This manuscript proposes a stochastic non-intrusive approach to preclude the need for derivatives, thereby allowing general neutronic codes with no adjoint capability to benefit from the nuclear data adjustment applications. Furthermore, the proposed approach serves as an independent tool for verifying the implementation of other adjustment capabilities, as it only requires non-intrusive access to the neutronic codes. This manuscript develops the theoretical basis for the new approach and compares its performance to an existing rendition, denoted by the generalized linear least-squares (GLLS) methodology, under the SCALE’s TSURFER module, for a criticality safety application.

On data assimilation with Monte-Carlo-calculated and statistically uncertain sensitivity coefficients

Sensitivity coefficients from Monte Carlo neutron transport codes have uncertainties that can affect nuclear data adjustments with integral experiments. This paper presents an extended version of Generalized Linear Least Squares (GLLS), called xGLLS, that accounts for these uncertainties. With very large sensitivity uncertainties, xGLLS constrains the nuclear data adjustments so that the posterior biases and uncertainties are larger than with GLLS. However, for the range of sensitivity uncertainties realistically encountered, xGLLS does not produce adjustments different from GLLS. This indicates that sensitivity uncertainties are not important compared to experimental, modeling, methodological, and nuclear data uncertainties. To balance a simulation’s accuracy with its computational cost, we recommend stopping a simulation once the uncertainty of a calculated integral parameter, caused by modeling and methodologies and by the sensitivities, is an order of magnitude smaller than that caused by nuclear data.

Applications of Multivariate Normal Bayesian Models in Nuclear Engineering

A general Bayesian framework for best-estimate plus uncertainty predictions of multidimensional continuous observables is presented. Parameterizing uncertainties in terms of multivariate normal distribution models, this Multivariate normal Bayesian model (MNBM) framework allows one to include both measured data and linear constraints in a mathematically consistent way. The resulting updating formulas are generalizations of the updating formulas of the Generalized Linear Least Squares (GLLS) framework, which is widely used for the generation of adjusted nuclear data libraries. While the GLLS methodology is restricted to first-order perturbation theory, there is no such restriction for the considered MNBM framework. This makes it possible to use Monte Carlo uncertainty propagation and to apply the updating formulas directly to the observables of interest without having to first update the input parameter distributions. After a general presentation of the MNBM framework and a brief discussion of its possible applications, the generation of bounding burnup-dependent axial burnup profiles of light water reactor fuel assemblies for the purpose of criticality safety analysis is discussed as an example application.

Performance assessment of adjusted nuclear data along with their covariances on the basis of fast reactor experiments

In view of fast reactor analyses, it is shown that efficient nuclear data adjustments can be obtained on a limited assimilation database consisting of just six well documented integral parameters, i.e. the central spectral indices measured in Godiva and ZPPR-9. This study uses a Generalized Linear Least-Squares (GLLS) based data assimilation method by means of Asymptotic Progressing Incremental nuclear data Adjustment (APIA) simulations with two incremental steps, one involving Godiva; the other one ZPPR-9. Consistent JEFF-3.3 and TENDL based prior data including their covariances are used; correspondingly, the assimilation leads to posterior JEFF-3.3 and TENDL data. 34 target experiments are then investigated by means of both prior and posterior data. These experiments consist of spectral indices as well as multiplication factors which pertain to 11 fast spectrum configurations including the six integral parameters which are part of the assimilation.It is found that (1) after adjustment the mean χ2 is strongly reduced to values smaller than 2, in each case. (2) The performance of the adjustment is comparable between JEFF-3.3 and TENDL also in terms of the Gaussian Coverage Factor (GCF), which is the common surface spanned below two normal probability density functions associated with data means and variances.Correspondingly it is found by comparing JEFF-3.3 and TENDL data among each other in a similar way by computing GCFs of cross-sections, that (3) posterior data overall appears less deviating than prior data.It seems worthwhile investigating whether similar promising results and trends assessed based upon a deterministic code, namely ERANOS, are reproducible with a stochastic method which is deemed to be a reference tool.

Adaptive Weighted Nonlinear Least Squares Method for Fluorodeoxyglucose Positron Emission Tomography Quantification

This study developed an adaptive weighted nonlinear least squares (AWNLS) method for solving the problem of high variability in the estimates of the microrate constants of fluorodeoxyglucose (FDG) kinetics caused by measurement noise. In the AWNLS method for adaptive quantitative analysis, the cost function is adjusted according to the characteristics of the tissue time-activity curve (TTAC). Specifically, the average of the early part of the TTAC was used to modify the cost function when fitting the FDG model to the TTAC. A computer simulation study applying different sets of parameter values and noise conditions was conducted. The accuracy and reliability of the parameter estimates obtained using AWNLS were compared with those of nonlinear least squares (NLS), weighted nonlinear least squares (WNLS), linear least squares (LLS), and generalized linear least squares (GLLS). The errors in k1–k3 obtained using NLS indicate this method’s poor precision in the presence of high noise levels. NLS and WNLS were sensitive to the initial values. Moreover, the results of k4 estimated using LLS and GLLS were inaccurate because of large bias. By contrast, the microrate constants (k1–k4), the FDG metabolic rate (K), and the volume of distribution (k1/k2) obtained using AWNLS were stable and accurate regardless of the noise level and initial values. The AWNLS method could estimate the FDG metabolic rate (K) and the microrate constants (k 1–k 4) of the FDG model accurately at various noise levels, irrespective of the initial values.

Comparison of progressive incremental adjustment sequences for cross-section and variance/covariance data adjustment by analyzing fast-spectrum systems

An attempt is made to improve our confidence in current adjustment techniques based on the Generalized Linear Least-Squares (GLLS) method. Instead of involving all considered experiments simultaneously, the adjustment is made progressively by considering at a time small groups of correlated and well documented experiments which have been performed preferably in the same configuration. Nonlinearity is accounted for implying that the required sensitivity coefficients are recalculated within an iterative procedure. The individual experiments may refer to spectral indices, reaction rate traverses, as well as local reactivity effects in fast-spectrum systems. It is shown that specific Progressing Incremental nuclear data Adjustment (PIA) techniques developed in compliance with ideas of Palmiotti and Salvatores, which specifically make use of preprocessed a priori cross-sections and variance/covariance data in 33 neutron groups as a basis for analyzes in the framework of the deterministic code system ERANOS (Edition 2.2-N), allow to a large extent separating effects and also providing automatic criteria for the inclusion or exclusion of experiments from the adjustment. Other important results are that (1) global parameters such as effective multiplication factors may be considered but only marginally in the adjustment and (2) only the adjustment of those cross-sections resulting independent of the PIA sequence may be considered as a basis for reliable adjustments, provided that nonlinearity is accounted for.

Resonance Parameter Adjustment Based on Integral Experiments

This project seeks to allow coupling of differential and integral data evaluation in a continuous-energy framework and to use the generalized linear least-squares (GLLS) methodology in the TSURFER module of the SCALE code package to update the parameters of a resolved resonance region evaluation. Recognizing that the GLLS methodology in TSURFER is identical to the mathematical description of a Bayesian update in SAMMY, the SAMINT code was created to use the mathematical machinery of SAMMY to update resolved resonance parameters based on integral data. Traditionally, SAMMY used differential experimental data to adjust nuclear data parameters. Integral experimental data, such as in the International Criticality Safety Benchmark Experiments Project, remain a tool for validation of completed nuclear data evaluations. SAMINT extracts information from integral benchmarks to aid the nuclear data evaluation process. Later, integral data can be used to resolve any remaining ambiguity between differential data sets, highlight troublesome energy regions, determine key nuclear data parameters for integral benchmark calculations, and improve the nuclear data covariance matrix evaluation. SAMINT is not intended to bias nuclear data toward specific integral experiments but should be used to supplement the evaluation of differential experimental data. Using GLLS ensures proper weight is given to the differential data.

MOCABA: A general Monte Carlo–Bayes procedure for improved predictions of integral functions of nuclear data

MOCABA is a combination of Monte Carlo sampling and Bayesian updating algorithms for the prediction of integral functions of nuclear data, such as reactor power distributions or neutron multiplication factors. Similarly to the established Generalized Linear Least Squares (GLLS) methodology, MOCABA offers the capability to utilize integral experimental data to reduce the prior uncertainty of integral observables. The MOCABA approach, however, does not involve any series expansions and, therefore, does not suffer from the breakdown of first-order perturbation theory for large nuclear data uncertainties. This is related to the fact that, in contrast to the GLLS method, the updating mechanism within MOCABA is applied directly to the integral observables without having to “adjust” any nuclear data. A central part of MOCABA is the nuclear data Monte Carlo program NUDUNA, which performs random sampling of nuclear data evaluations according to their covariance information and converts them into libraries for transport code systems like MCNP or SCALE. What is special about MOCABA is that it can be applied to any integral function of nuclear data, and any integral measurement can be taken into account to improve the prediction of an integral observable of interest. In this paper we present two example applications of the MOCABA framework: the prediction of the neutron multiplication factor of a water-moderated PWR fuel assembly based on 21 criticality safety benchmark experiments and the prediction of the power distribution within a toy model reactor containing 100 fuel assemblies.

Optimization of methods for quantification of rCBF using high-resolution [15O]H2O PET images

This study aimed to derive accurate estimates of regional cerebral blood flow (rCBF) from noisy dynamic [15O]H2O PET images acquired on the high-resolution research tomograph, while retaining as much as possible the high spatial resolution of this brain scanner (2–3 mm) in parametric maps of rCBF. The PET autoradiographic method and generalized linear least-squares (GLLS), with fixed or extended to include spatially variable estimates of the dispersion of the measured input function, were compared to nonlinear least-squares (NLLS) for rCBF estimation. Six healthy volunteers underwent two [15O]H2O PET scans with continuous arterial blood sampling. rCBF estimates were obtained from three image reconstruction methods (one analytic and two iterative, of which one includes a resolution model) to which a range of post-reconstruction filters (3D Gaussian: 2, 4 and 6 mm FWHM) were applied. The optimal injected activity was estimated to be around 11 MBq kg−1 (800 MBq) by extrapolation of patient-specific noise equivalent count rates. Whole-brain rCBF values were found to be relatively insensitive to the method of reconstruction and rCBF quantification. The grey and white matter rCBF for analytic reconstruction and NLLS were 0.44 ± 0.03 and 0.15 ± 0.03 mL min−1 cm−3, respectively, in agreement with literature values. Similar values were obtained from the other methods. For generation of parametric images using GLLS or the autoradiographic method, a filter of ⩾4 mm was required in order to suppress noise in the PET images which otherwise produced large biases in the rCBF estimates.

Performance evaluation of kinetic parameter estimation methods in dynamic FDG-PET studies

To evaluate several popular parameter estimation methods for determining the cerebral metabolic rate for glucose and individual kinetic rate constant parameters in 2-deoxy-2-[18F]fluoro-D-glucose positron emission tomography studies. These methods can be divided into two categories: nonlinear estimation methods and linear estimation methods. The nonlinear estimation methods include nonlinear least squares (NLLS), weighted NLLS using noisy tissue time-activity data (WNLLS-N), weighted NLLS using noise-free tissue time-activity data (WNLLS-NF), iteratively reweighted NNLS (IRWNLLS) and nonlinear ridge regression (NLRR) method, whereas the linear estimation methods include Patlak-Gjedde graphical analysis (PGA), linear least squares (LLS), generalized LLS (GLLS), total least squares (TLS) and the basis functions (BF) method. Simulation studies are presented. There are several findings: (i) when the noise level is low, GLLS performs well. However, it exhibits large bias and poor precision especially in k*3 and k*4 when the noise level is high. (ii) BF is a promising method with superior bias and precision properties, and is less affected by the scan duration used. (iii) The weighting factors in the nonlinear estimation methods are important: a good choice of weights can help to make the estimates more accurate and reliable. Weighting based on noisy data should be avoided. (iv) It confirms that PGA is little affected by noise, but the assumptions of PGA could induce bias. It also confirms that 60 min is not long enough to give reliable estimates of k*4 especially for the linear estimation methods LLS, TLS, and GLLS.

Generalized Linear Least-Squares Adjustment, Revisited

Abstract Tsurfer, a generalized linear least-squares (Glls) code, is a new module of the SCALE system. After a short introduction outlining the history and applications of the Glls methodology in reactor physics, a new application of the Glls methodology in criticality safety is discussed. Some characteristic Tsurfer input data are discussed in detail.

SU‐FF‐J‐91: Linear Methods for Hepatic Perfusion Imaging

Purpose: Hepatic perfusion imaging could be valuable for assessment of therapeutic response and normal tissue toxicity in patients undergoing treatment for hepatic lesions. However, visual‐by‐visual estimation of liver perfusion using non‐linear least squares (NLLS) fits of dynamic contrast‐enhanced (DCE) CT or MRI data to a compartmental model with dual input functions is a computational expensive process. The purpose of this study is to develop and evaluate efficiency, stability, and bias of three linear methods for computation of hepatic perfusion parameters. Method and Materials: Through simple mathematical manipulation, the single compartmental model is converted into a linear equation (linear in three unknown parameters: arterial perfusion ka, portal vein perfusion kp, and the outflow rate k2). The ordinary least squares (OLS), total least squares (TLS), generalized linear least squares (GLLS), and NLLS methods were used for estimates of the perfusion parameters from simulated data that mimic normal liver perfusion. Gaussian‐distributed random noise was added into the theoretic time‐contrast concentration curves. The stability and bias of the three parameters were calculated from 5000 simulations as a function of contrast‐to‐noise ratio of the liver parenchyma, CNRL, and the temporal sampling interval, Δt, for each of the four methods. Results: Dependencies of the stability on CNRL (varied from 10 to 20) and Δt (1 to 8 seconds) were similar among all four methods, with NLLS and GLLS having slightly better stability. The increase in bias with Δt was greater for NLLS and GLLS than for TLS and OLS, particularly for estimates of ka. However, the computation time for NLLS was 10–100 times greater than for the linear methods. Conclusion: This study suggests that the linear methods can achieve the stability and bias similar to or better than NLLS, but substantially reduce the computation time, making liver perfusion imaging practical for use in clinical trials.

Evaluation of objective functions for estimation of kinetic parameters

There is growing interest in quantitatively analyzing in vivo image data, as this facilitates objective comparisons and measurement of effect. In this regard, people increasingly turn to pharmacokinetic models and estimation of parameters of such models. In this work several parameter estimation methodologies were compared within the context of the most common pharmacokinetic model used in positron emission tomography imaging to describe glucose metabolism and receptor-ligand interactions at tracer concentrations. Simulated data were generated with 1000 realizations at each of 5 different noise levels. Estimates of the kinetic parameters were made for each realization using seven iterative, nonlinear estimation methodologies: ordinary least squares (OLS), weighted least squares (WLS), penalized weighted least squares (PWLS), iteratively reweighted least squares (IRLS), and variations of extended least squares (ELS0, ELS1, ELS3). Additionally, generalized linear least squares (GLLS) was also used. With relatively noise-free data, the iterative nonlinear estimation methods generally produced low-bias, high-precision parameter estimates, whereas with GLLS the bias was more prominent. Greater distinction between the estimation methods was seen at the higher, more realistic noise levels, with ELS and IRLS methods generally achieving better precision than the other methods. At the high noise levels WLS, GLLS, and PWLS yielded parameter estimates with large bias (>200%) for some kinetic parameters. In general, there are more favorable estimator methodologies than the frequently employed WLS. Methods that determine values of weights based on model output--IRLS, ELS0, ELS1 and ELS3--generally perform better than methods that determine values of weights based directly on the experimental data.

A STUDY OF PARTIAL VOLUME EFFECTS ON CLUSTERING-AIDED PARAMETRIC IMAGES

Cluster analysis has been applied in functional imaging to derive parametric images with improved signal to noise ratio. However, the impact of partial volume effects (PVE) on the parametric images aided by clustering is still uncertain. Computer simulations were performed to generate simulated dynamic data at various noise levels. Reconstructed data were processed by cluster analysis. The generalized linear least squares (GLLS) method was used to estimate parametric images. The results were compared with the parametric images generated without clustering. The results demonstrate that clustering does not enhance PVE, but reduces PVE. Furthermore, clustering-aided parametric images are shown to be insensitive to noise.

Evaluation of Basis Function and Linear Least Squares Methods for Generating Parametric Blood Flow Images Using 15O-Water and Positron Emission Tomography

Parametric analysis of (15)O-water positron emission tomography (PET) studies allows determination of blood flow (BF), perfusable tissue fraction (PTF), and volume of distribution (V (d)) with high spatial resolution. In this paper the performance of basis function and linear least squares methods for generating parametric flow data were evaluated. Monte Carlo simulations were performed using typical perfusion values for brain, tumor, and heart. Clinical evaluation was performed using seven cerebral and 10 myocardial (15)O-water PET studies. Basis function (BFM), linear least squares (LLS), and generalized linear least squares (GLLS) methods were used to calculate BF, PTF, or V(d). Monte Carlo simulations and human studies showed that, for low BF values (<1 ml/min(-1)ml(-1), BF, PTF, and V(d) were calculated with accuracies better than 5% for all methods tested. For high BF (>2 ml/min(-1)ml(-1)), use of BFM provided more accurate V(d) compared with (G)LLS. In general, BFM provided the most accurate estimates of BF, PTF, and V(d).

On the Convergence of the Generalized Linear Least Squares Algorithm

This paper considers the issue of parameter estimation for biomedical applications using nonuniformly sampled data. The generalized linear least squares (GLLS) algorithm, first introduced by Feng and Ho (1993), is used in the medical imaging community for removal of bias when the data defining the model are correlated. GLLS provides an efficient iterative linear algorithm for the solution of the non linear parameter estimation problem. This paper presents a theoretical discussion of GLLS and introduces use of both Gauss Newton and an alternating Gauss Newton for solution of the parameter estimation problem in nonlinear form. Numerical examples are presented to contrast the algorithms and emphasize aspects of the theoretical discussion.

Strategy for the formation of parametric images under conditions of low injected radioactivity applied to PET studies with the irreversible monoamine oxidase A tracers [11C]clorgyline and deuterium-substituted [11C]clorgyline.

The construction of parametric positron emission tomography images of enzyme or receptor concentration obtained using irreversibly binding radiotracers presents problems not usually encountered with reversibly binding radiotracers. Difficulties are most apparent in brain regions having low blood flow and/or high enzyme or receptor concentration and are exacerbated with noisy data. This is especially true when minimal doses of radiotracer are administered. A comparison was recently reported of the irreversible monoamine oxidase A (MAO A) radiotracers [11C]clorgyline (CLG) and deuterium-substituted [11C]clorgyline (CLG-D) in the human brain using region of interest (ROI) analysis in which the authors observed an unexpected loss of image contrast with CLG-D compared with CLG. In order to more fully investigate patterns of binding of these irreversibly binding radiotracers, a strategy was devised to reduce noise in the generation of parametric images of the model term related to enzyme or receptor concentration. The generalized linear least squares (GLLS) method of Feng et al. (1995), a rapid linear method that is unbiased, was used for image-wide parameter estimation. Since GLLS can fail in the presence of large amounts of noise, local voxels were grouped within the image to increase the signal, and the GLLS method was combined with the standard nonlinear estimation methods when necessary. Voxels were grouped together depending on their proximity and whether they fell within a specified range of the time-integrated image. It was assumed that voxels meeting both criteria are functionally related. Simulations reflecting varying enzyme concentrations were performed to assess precision and accuracy of parameter estimates in the presence of varying amounts of noise. Using this approach, images were generated of the combination parameter lambdak3 (lambda = K1/k2, where K1 and k2 are plasma-to-tissue and tissue-to-plasma transport constants, respectively) that is related to enzyme concentration as well as images of the transport constant K1 for individual subjects. Reasonably high-quality images of both K1 and lambdak3 were obtained for CLG and CLG-D for individual subjects even with low injected doses averaging 6 mCi. While there were no differences in the K1 images, the lambdak3 images revealed the loss of contrast previously reported for CLG-D using the ROI analysis. This method should be generalizable to other tracers and should facilitate the analysis of group differences.

Generalized linear least squares algorithms for modeling glucose metabolism in the human brain with corrections for vascular effects

The generalized linear least squares (GLLS) algorithm has been found useful in image-wide parameter estimation for the generation of parametric images with positron emission tomography (PET) as it is computationally efficient and statistically reliable. However, the original algorithm was designed for parameter estimation with non-uniformly sampled instantaneous measurements. When dynamic PET data are sampled with the optimal image sampling schedule (OISS) to reduce memory and storage space, only a few temporal image frames are recorded. As a result, the direct application of GLLS is no longer appropriate. In this paper, we extend the GLLS algorithm to a five parameter model for the study of human brain metabolism, which accounts for the effect of cerebral blood volume (CBV), using OISS sampled data, with as few as five temporal samples. The formulation for this new GLLS algorithm is developed, and its computational efficiency and statistical reliability are investigated and validated using computer simulations and clinical PET [18F]-2-fluoro-2-deoxy- d-glucose (FDG) data.

A strategy for removing the bias in the graphical analysis method.

The graphical analysis method, which transforms multiple time measurements of plasma and tissue uptake data into a linear plot, is a useful tool for rapidly obtaining information about the binding of radioligands used in PET studies. The strength of the method is that it does not require a particular model structure. However, a bias is introduced in the case of noisy data resulting in the underestimation of the distribution volume (DV), the slope obtained from the graphical method. To remove the bias, a modification of the method developed by Feng et al. (1993), the generalized linear least squares (GLLS) method, which provides unbiased estimates for compartment models was used. The one compartment GLLS method has a relatively simple form, which was used to estimate the DV directly and as a smoothing technique for more general classes of model structures. In the latter case, the GLLS method was applied to the data in two parts, that is, one set of parameters was determined for times 0 to T1 and a second set from T1 to the end time. The curve generated from these two sets of parameters then was used as input to the graphical method. This has been tested using simulations of data similar to that of the PET ligand [11C]-d-threo-methylphenidate (MP, DV = 35 mL/mL) and 11C raclopride (RAC, DV = 1.92 mL/mL) and compared with two examples from image data with the same tracers. The noise model was based on counting statistics through the half-life of the isotope and the scanning time. Five hundred data sets at each noise level were analyzed. Results (DV) for the graphical analysis (DV(G)), the nonlinear least squares (NLS) method (DV(NLS)), the one-tissue compartment GLLS method (DV(F)), and the two part GLLS followed by graphical analysis (DV(FG)) were compared. DV(FG) was found to increase somewhat with increasing noise and in some data sets at high noise levels no estimate could be obtained. However, at intermediate levels it provided a good estimation of the true DV. This method was extended to use a reference tissue in place of the input function to generate the distribution volume ratio (DVR) to the reference region. A linearized form of the simplified reference tissue method of Lammertsma and Hume (1996) was used. The DVR generated directly from the model (DVR(FL)) was compared with DVR(FG) (determined from a "smoothed" uptake curve as for DV(FG)) using the graphical method.