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Enhanced parameter estimation from noisy PET data: Part II--evaluation.

RATIONALE AND OBJECTIVES: Positron emission tomography (PET) is a minimally invasive imaging modality that provides three-dimensional distribution data for a radioactive tracer concentration within the body. Local functional parameters are estimated from these images by fitting tracer kinetic data with mathematical models. However, in some applications, the reliability of parameter estimates may be hindered by the presence of noise. In the accompanying report in this issue of Academic Radiology, a novel method using principal component analysis (PCA) was presented and used for deriving parametric images of lung function from imaged tracer kinetics of intravenously injected nitrogen 13 (13NN) in saline solution. The PCA method averages 13NN concentrations from groups of voxels (volume elements) selected for their similarity in kinetics, rather than their spatial proximity. The goal of this study is to conduct a Monte Carlo simulation to evaluate the robustness to noise of parameters derived by means of the PCA method. MATERIALS AND METHODS: This evaluation involved: (1) generating "noise-free" synthetic PET images from experimental PET data, (2) adding noise to these images, (3) applying the PCA method to yield parametric images, and (4) comparing these parametric images with original noise-free images. RESULTS: Local parameters recovered by using the PCA method deviated from noise-free parameters on average by less than 1% for up to 32-fold of expected noise levels. These deviations were much less than those (>10%) recovered by using a direct curve-fitting method. CONCLUSION: The novel PCA approach provides robust parametric lung functional images while preserving the spatial resolution of the original images.

Humans↗

Sensitivity analysis of respiratory parameter estimates in the constant-phase model.

The constant-phase model is increasingly used to fit low-frequency respiratory input impedance (Zrs), highlighting the need for a better understanding of the use of the model. Of particular interest is the extent to which Zrs would be affected by changes in parameters of the model, and conversely, how reliable are parameters estimated from model fits to the measured Zrs. We performed sensitivity analysis on respiratory data from 6 adult mice, at functional residual capacity (FRC), total lung capacity (TLC), and during bronchoconstriction, obtained using a 1-25 Hz oscillatory signal. The partial derivatives of Zrs with respect to each parameter were first examined. The limits of the 95% confidence intervals, 2-dimensional pairwise and p-dimensional joint confidence regions were then calculated. It was found that airway resistance was better estimated at FRC, as determined by the confidence region limits, whereas tissue damping and elastance were better estimated at TLC. Airway inertance was poorly estimated at this frequency range, as expected. During methacholine-evoked pulmonary constriction, there was an increase in the uncertainty of airway resistance and tissue damping, but this can be compensated for by using the relative (weighted residuals) in preference over the absolute (unweighted residuals) fitting criterion. These results are consistent with experimental observation and physiological understanding.

Airway Resistance↗

Source parameter estimation in inhomogeneous volume conductors of arbitrary shape.

In this paper it is demonstrated that the use of a direct matrix inverse in the solution of the forward problem in volume conduction problems greatly facilitates the application of standard, nonlinear parameter estimation procedures for finding the strength as well as the location of current sources inside an inhomogeneous volume conductor of arbitrary shape from potential measurements at the outer surface (inverse procedure). This, in turn, facilitates the inclusion of a priori constraints. Where possible, the performance of the method is compared to that of the Gabor-Nelson method. Applications are in the fields of bioelectricity (e.g., electrocardiography and electroencephalography).

Electric Conductivity↗

Influence of flow pattern on the parameter estimates of a simple breathing mechanics model.

The first-order model of breathing mechanics is widely used in clinical practice to assess the viscoelastic properties of the respiratory system. Although simple, this model takes the predominant features of the pressure-flow relationship into account but gives highly systematic residuals between measured and model-predicted variables. To achieve a better fit of the entire data set, an approach hypothesizing deterministic time-variations of model parameters, summarized by information-weighted histograms was recently proposed by Bates and Lauzon. The present study uses flow and pressure data measured in intensive care patients to evaluate the real potential of this approach in clinical practice. Information-weighted histograms of the model parameters, estimated by an on-line identification algorithm, were first constructed by taking into account the parameter percentage standard deviations. Then, the influence of the respiratory flow pattern on the calculated histograms was evaluated by the Kolmogorov-Smirnov statistical test. The results show that the method gives good reproducibility under stable experimental conditions. In addition, for a given airflow waveform, an increase in respiratory frequency shifts the histograms representing time-varying viscous properties strongly versus lower values, whereas it shifts the histograms representing time-varying elastic properties slightly versus higher values. On the other hand, the same histograms were highly dependent on the airflow waveform, especially for the viscous properties. Even in a limited experimental work, in all the conditions considered, the method provides results which agree well with the physiological knowledge of nonlinear and multicompartment behavior of respiratory mechanics.

Algorithms↗

On the parameter estimation for diffusion models of single neuron's activities. I. Application to spontaneous activities of mesencephalic reticular formation cells in sleep and waking states.

For the Ornstein-Uhlenbeck neuronal model a quantitative method is proposed for the estimation of the two parameters characterizing the unknown input process, namely the neuron's mean input per unit time mu and the infinitesimal standard deviation per unit time sigma. This method is based on the experimentally observed first- and second-order moments of interspike intervals. The dependence of the estimates mu and ŝigma on the moments of the observed interspike intervals and on the neuronal parameters is clarified, and a comparison is made between the estimates based on the classical Wiener model and those yielded by the Ornstein-Uhlenbeck model. Comprehensive tables are included in which the displayed values of mu and ŝigma have been calculated in terms of physiologically realistic pairs of first- and second-order moments. Our method is finally applied to interspike interval data recorded from neurons in the mesencephalic reticular formation of the cat during hypothetical sleep, slow-wave sleep stage, and wake stage.

Animals↗

Identification and parameter estimation of the mechanical ventilatory system.

Oesophageal pressure and mouth volume were measured by oesophageal balloon and pneumotachography, respectively, in 10 men (both normal and diseases) under different experimental conditions, i.e. spontaneous ventilation, quasi-static manoeuvres and respiration at high frequencies. The recorded data were then analyzed by an identification and parameter estimation computer programme, to determine a mathematical model for the process. Contrary to other recently published papers dealing with frequency analysis, we studied a time-dependent model, represented by a linear time-invariant differential equation. The results were quite satisfactory since a relatively second-order model seems to be adequate for describing the process. Differences in the parameter values between normal and diseased subjects were characterized by increases in the patients' lung viscous component. In conclusion, our identification test results in the mechanical ventilatory system seem to indicate that a simple second-order mathematical model is consistent with the experimental data.

Adult↗

The effect of random measurement errors on kinetic transport parameter estimation.

Saturation kinetics experiments, in which uptake U of a substance across a transport barrier is measured as a function of initial concentration difference C, are used to describe transport of nutrients. Many such processes are characterized by low- and high-affinity systems in which kinetic parameters Vmax and Km differ by orders of magnitude. Transformations of equations to straight-line relationships between U and C are popular methods of parameter estimation. The aims of this study are (1) to show effects of random errors in U measurement on Vmax and Km estimation in a two-affinity process under several transformations: Lineweaver-Burk (1/U vs. 1/C), Hanes (C/U vs. C), Eadie-Hofstee (U/C vs. U), and Wolff (U vs. U/C), and (2) to indicate strategies for minimizing effects of errors. Two transport properties will illustrate: an ideal process of low- (Vmax = 100, Km = 10) and high-affinity (Vmax = 1, Km = .1) systems to which random error is added, and experimental uptake of 5-methyltetrahydrofolic acid by isolated hepatocytes.

Age Factors↗

Novel parameter estimation methods for 11C-acetate dual-input liver model with dynamic PET.

The successful investigation of 11C-acetate in positron emission tomography (PET) imaging for marking hepatocellular carcinoma (HCC) has been validated by both clinical and quantitative modeling studies. In the previous quantitative studies, all the individual model parameters were estimated by the weighted nonlinear least squares (NLS) algorithm. However, five parameters need to be estimated simultaneously, therefore, the computational time-complexity is high and some estimates are not quite reliable, which limits its application in clinical environment. In addition, liver system modeling with dual-input function is very different from the widespread single-input system modeling. Therefore, most of the currently developed estimation techniques are not applicable. In this paper, two parameter estimation techniques: graphed NLS (GNLS) and graphed dual-input generalized linear least squares (GDGLLS) algorithms were presented for 11C-acetate dual-input liver model. Clinical and simulated data were utilized to test the proposed algorithms by a systematic statistical analysis. Compared to NLS fitting, these two novel methods achieve better estimation reliability and are computationally efficient, and they are extremely powerful for the estimation of the two potential HCC indicators: local hepatic metabolic rate-constant of acetate and relative portal venous contribution to the hepatic blood flow.

Acetates↗

Genetic parameter estimates for serum insulin-like growth factor I concentration and performance traits in Angus beef cattle.

Data for this study were obtained from an experiment involving divergent selection for blood serum IGF-I concentration in beef cattle. Multiple trait derivative-free REML procedures were used to obtain genetic parameter estimates for IGF-I concentration at d 28, 42, and 56 of the postweaning period and for mean IGF-I concentration, as well as for weights and gains. Included in the analysis were 1,563 animals in the A-1 matrix, 731 of which had valid records for mean IGF-I concentration. Direct heritabilities (hd2) were .42 +/- .13, .53 +/- .15, .71 +/- .16, and .48 +/- .13 for IGF-I at d 28, 42, and 56 of the postweaning period and for mean IGF-I, respectively. Heritability of maternal genetic effects (hm2) ranged from .02 to .12, whereas the proportion of the total variance due to the maternal permanent environmental effect (c2) was essentially zero for all measures of IGF-I. Genetic correlations of IGF-I with weaning and postweaning weights and with postweaning weight gain ranged from -.21 to -.54 and averaged -.38. The environmental correlation between IGF-I and performance traits varied from .10 to .35 and averaged .22. Phenotypic correlations of IGF-I concentrations with weaning weight and postweaning weights and gains ranged from -.01 to .12 and averaged .04. Estimates of hd2 indicate that it should be possible to change IGF-I concentration in beef cattle via selection. Negative genetic correlations imply that, if the goal is to make genetic improvement in weaning weights, postweaning weights and (or) postweaning gain in beef cattle, selection should be for decreased postweaning serum IGF-I concentration.

Animals↗

Parameter estimation of feedback gain in a stochastic model of renal hemodynamics: differences between spontaneously hypertensive and Sprague-Dawley rats.

Proximal tubular pressure shows periodic self-sustained oscillations in normotensive rats but highly irregular fluctuations in spontaneously hypertensive rats (SHR). Although we have suggested that the irregular fluctuations in SHR represent low-dimensional deterministic chaos in tubuloglomerular feedback (TGF), they could also arise from other mechanisms, such as intrinsic instabilities in preglomerular vessels or inputs from neighboring, coupled nephrons. To test this possibility, we applied a parameter estimation procedure to a model of TGF, where a stochastic process was added to represent mechanisms not included explicitly in the model. In its deterministic version, the model can have chaotic dynamics arising from TGF. The model introduces random fluctuations into a parameter that determines the gain of TGF. The model shows a rich variety of dynamics ranging from low-dimensional deterministic oscillations and chaos to high-dimensional random fluctuations. To fit the data from normotensive rats, the model must introduce only a small variation in the feedback gain, and its estimates of that gain agree well with experimental values. These results support the use of the deterministic model of nephron dynamics in normotensive rats. In contrast, the irregular tubular pressure fluctuations in SHR were best described by a model dominated by random parameter fluctuations. The results point to the failure of simple mathematical models of nephron dynamics adequately to describe processes that are important for the irregular tubular pressure fluctuations and the need to consider other factors, such as differences in vascular function or nephron-nephron interactions, in further work on this problem.

Animals↗

Parameter estimation in six numeric models of transperitoneal transport of glucose.

Six competing kinetic models of transperitoneal glucose transport were formulated and validated. The models were designed to elucidate the presence or absence of diffusive, nonlymphatic convective and lymphatic convective solute transport. The validation procedure included an assessment of theoretical and practical identifiability, goodness of fit, residual error analysis, and plausibility of parameter estimates. Experimental results were obtained from 21 patients without diabetes. The validation procedure demonstrated that the model that only included diffusion was superior to the other models. Theoretically, both nonlymphatic convective and lymphatic convective transports might exist. However, neither the ultrafiltration sieving coefficient nor the lymphatic flow rate were practically identifiable, probably because any amount of glucose transported by nonlymphatic convective and lymphatic convective transport mechanisms was negligible compared with the amount transported by diffusion. Based on these results, there appear to be problems measuring convective solute transport parameters when the solute transport is in the dialysate-to-blood direction while the fluid transport is in the blood-to-dialysate direction.

Adult↗

Quantitative analyses of anaerobic wastewater treatment processes: identifiability and parameter estimation.

We investigated the problem of identifying the parameters of a nonlinear fifth order model describing the population dynamics of two main bacterial groups in an anaerobic wastewater treatment process. In addition to addressing problems concerning structural and practical identifiability, we also analyzed how mathematical descriptions of bacterial population dynamics can model real data. Using three data sets recorded under different experimental conditions, we estimated important biochemical parameters and demonstrated that our model could describe the data successfully. Parameters, which are simultaneously determined using information from all three experiments, have more reliable estimates. We conclude that, after appropriate estimation, this model can be used for optimization and the control of continuous processes.

Anaerobiosis↗

A model of fluid resuscitation following burn injury: formulation and parameter estimation.

A dynamic compartmental model is developed to describe the redistribution of fluid and albumin between the circulation and the intact and injured interstitia following burn injury in humans. Transcapillary fluid and albumin exchange is described by a coupled Starling mechanism, while the effect of the burn is represented by time-dependent perturbations to all three compartments. The unknown model parameters are determined for two groups of patients, having less than and greater than 25% total body surface area burns, by statistical fitting of model predictions to patient data from two sources. The parameters include the perturbations to the fluid filtration coefficients in uninjured and injured tissue, GkF,Tl and GkF,BT, respectively, the relaxation coefficient, r, which describes the exponential decay of the perturbations, and the exudation factor, EXFAC, which relates the protein concentration in the exudate to that in the injured tissue. Perturbations to other parameters, including the membrane permeability-surface area product and the albumin reflection coefficient in the injured and uninjured tissues, are determined based on interrelationships with GkF,Tl and GkF,BT. The values of GkF,BT, when corrected for tissue destruction and decreased post-injury perfusion, are in reasonable agreement with the limited experimental data available from the literature. The model and its parameters are further validated by comparing the simulated patient responses to the clinical data used in the parameter estimation as well as to data available from two additional sources.

Body Fluid Compartments↗

Bayesian parameter estimates of nelfinavir and its active metabolite, hydroxy-tert-butylamide, in infants perinatally infected with human immunodeficiency virus type 1.

The objective of the present study was to develop a population pharmacokinetic model for nelfinavir mesylate (NFV) and nelfinavir hydroxy-tert-butylamide (M8), the most abundant metabolite of NFV, in infants vertically infected with human immunodeficiency virus type 1 and participating in the Paediatric European Network for Treatment of AIDS 7 study. Plasma NFV concentrations were determined during repeated NFV administrations (two to three times a day). Eighteen infants younger that age 2 years participated in this study. The doses administered ranged from 71 to 203 mg/kg of body weight/day. Pharmacokinetic parameter estimates were obtained by a compartmental approach by using a kinetic model to simultaneously fit NFV and M8 (active metabolite) concentrations. M8 was shown to be formation rate limited and was characterized by first-order rate constants of formation and elimination. Body weight was found to be a more appropriate predictor than age of the changes in (i) the rate of metabolism, (ii) the elimination rate constant of NFV, and (iii) NFV clearance. Population parameters were computed to account for the relationship between the rate of metabolism and body weight. The estimated NFV and M8 elimination half-lives were 4.3 and 2.04 h, respectively. The estimated NFV clearance was 2.13 liters/h/kg. The M8 concentration-to-NFV concentration ratio was 0.64 +/- 0.44. In conclusion, the population pharmacokinetic model describing the dispositions of NFV and M8 should facilitate the design of future studies to elucidate the relative contributions of the parent compound and M8 to the pharmacological and toxic effects of NFV therapy.

Aging↗

Aspects of parameter estimation in ascertainment sampling schemes.

It has recently been suggested that ascertainment sampling estimation procedures commonly used are not fully efficient in that the number of unobserved families is an unknown parameter that should be estimated (contrary to common practice) along with the genetic parameters for fully efficient estimation. It has also been suggested that the frequency distribution of family size contains unknown parameters that should similarly be estimated with the genetic parameters. These two suggestions are considered in this paper. It is shown by means of an equivalence theorem that in both cases the estimates and their variances obtained by adopting the suggested procedure are identical with those found by ignoring the unobserved families and by ignoring the family-size distribution. This demonstration leads to a formal justification of further procedures, in particular: (1) use of "method-of-moments" estimators, (2) ignoring the ascertainment scheme in some cases when estimating parameters, and (3) forming estimates of parameters when various parts of the data are obtained by different ascertainment schemes.

Family Characteristics↗

Information gain for genetic parameter estimation with incorporation of marker data.

Genetic marker data has been increasingly incorporated into segregation analysis, as combined segregation and linkage analysis has been performed more frequently. In this article, we study the extent of information gains with incorporation of marker data in segregation analysis, a topic that has not been investigated rigorously. Specifically, the current study is to investigate the influence of marker data on genetic model parameter estimation. A variance matrix criterion (as the inverse of the Fisher information matrix) and a relative entropy criterion (a measure of flatness of expected log-likelihood surface) are used to quantify the information gains. Our results indicate that substantial information gain can be achieved with the incorporation of marker data. The amount of variance reduction increases as the heterozygosity of the linked marker increases and as the trait gets closer to the linked marker(s). Incorporation of marker data in larger pedigrees also yields greater information gains based on both criteria. The effect of pedigree structure is also studied.

Data Interpretation, Statistical↗

[Ion single channel signal restoration and parameters' estimation based on the hidden Markov models].

The single ion channel signal is stochastic ionic current on the order of 1 pA recorded by patch clamp. Because the weakness of the signal, the background noise always dominates in the recordings, the threshold detector traditionally used in patch clamp to denoise and restore the channel signal can't work satisfactorily. This problem was analyzed mathematically, and a signal restoring and parameters estimating scheme called HMM algorithm was studied. The algorithm has been validated by simulation and the results suggest it performs effectively in the situation of low signal to noise ratio where the threshold detector fails completely.

Algorithms↗

A calculator program for least-squares parameter estimation according to the one-compartment kinetic model with zero-order input.

A calculator program that performs a nonlinear least-squares fit to data conforming to the one-compartment model with zero-order input is described. The program, which is designed for the Hewlett-Packard HP-41 CV calculator, is based on the Gauss-Newton iterative algorithm as modified by Hartley. A subroutine for calculation of initial parameter estimates is incorporated into the program. Plasma concentration data relative to a single oral dose of a sustained-release theophylline formulation are used to demonstrate the practical application of the program.

Computers↗