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Results for “parameter estimation”

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PubMed1998

A realistic closed-form radiobiological model of clinical tumor-control data incorporating intertumor heterogeneity.

PURPOSE: To investigate the role of intertumor heterogeneity in clinical tumor control datasets and the relationship to in vitro measurements of tumor biopsy samples. Specifically, to develop a modified linear-quadratic (LQ) model incorporating such heterogeneity that it is practical to fit to clinical tumor-control datasets. METHODS AND MATERIALS: We developed a modified version of the linear-quadratic (LQ) model for tumor control, incorporating a (lagged) time factor to allow for tumor cell repopulation. We explicitly took into account the interpatient heterogeneity in clonogen number, radiosensitivity, and repopulation rate. Using this model, we could generate realistic TCP curves using parameter estimates consistent with those reported from in vitro studies, subject to the inclusion of a radiosensitivity (or dose)-modifying factor. We then demonstrated that the model was dominated by the heterogeneity in alpha (tumor radiosensitivity) and derived an approximate simplified model incorporating this heterogeneity. This simplified model is expressible in a compact closed form, which it is practical to fit to clinical datasets. Using two previously analysed datasets, we fit the model using direct maximum-likelihood techniques and obtained parameter estimates that were, again, consistent with the experimental data on the radiosensitivity of primary human tumor cells. This heterogeneity model includes the same number of adjustable parameters as the standard LQ model. RESULTS: The modified model provides parameter estimates that can easily be reconciled with the in vitro measurements. The simplified (approximate) form of the heterogeneity model is a compact, closed-form probit function that can readily be fitted to clinical series by conventional maximum-likelihood methodology. This heterogeneity model provides a slightly better fit to the datasets than the conventional LQ model, with the same numbers of fitted parameters. The parameter estimates of the clinically important time factors and lag periods are very similar to those obtained from the conventional LQ model, but with slightly narrower confidence intervals, reflecting the better fit to the clinical data. DISCUSSION: We have demonstrated, as have others, the importance of intertumor heterogeneity in the response of patient populations to radiotherapy. With the possible inclusion of a radiosensitivity-modifying factor (in vitro/in vivo) of around 1.7, the in vivo data can be made consistent with the in vitro SF2 and Tpot data. Fitting two previously analyzed multicenter datasets indicated that previous analyses based on conventional LQ models gave results for clinically important time factors and lags periods that were not significantly biased by the failure to include intertumor heterogeneity, with slightly narrower confidence intervals, reflecting the better fit to the clinical data. The simple closed-form model we have developed allows direct estimation of the heterogeneity in radiosensitivity within clinical series, and should prove useful in the analysis of other clinical series.

Dose-Response Relationship, Radiation↗

PubMed2006

Estimation of dynamical model parameters taking into account undetectable marker values.

BACKGROUND: Mathematical models are widely used for studying the dynamic of infectious agents such as hepatitis C virus (HCV). Most often, model parameters are estimated using standard least-square procedures for each individual. Hierarchical models have been proposed in such applications. However, another issue is the left-censoring (undetectable values) of plasma viral load due to the lack of sensitivity of assays used for quantification. A method is proposed to take into account left-censored values for estimating parameters of non linear mixed models and its impact is demonstrated through a simulation study and an actual clinical trial of anti-HCV drugs. METHODS: The method consists in a full likelihood approach distinguishing the contribution of observed and left-censored measurements assuming a lognormal distribution of the outcome. Parameters of analytical solution of system of differential equations taking into account left-censoring are estimated using standard software. RESULTS: A simulation study with only 14% of measurements being left-censored showed that model parameters were largely biased (from -55% to +133% according to the parameter) with the exception of the estimate of initial outcome value when left-censored viral load values are replaced by the value of the threshold. When left-censoring was taken into account, the relative bias on fixed effects was equal or less than 2%. Then, parameters were estimated using the 100 measurements of HCV RNA available (with 12% of left-censored values) during the first 4 weeks following treatment initiation in the 17 patients included in the trial. Differences between estimates according to the method used were clinically significant, particularly on the death rate of infected cells. With the crude approach the estimate was 0.13 day-1 (95% confidence interval [CI]: 0.11; 0.17) compared to 0.19 day-1 (CI: 0.14; 0.26) when taking into account left-censoring. The relative differences between estimates of individual treatment efficacy according to the method used varied from 0.001% to 37%. CONCLUSION: We proposed a method that gives unbiased estimates if the assumed distribution is correct (e.g. lognormal) and that is easy to use with standard software.

AIDS-Related Opportunistic Infections↗

PubMed2005

Multiple imputation of the Glasgow Coma Score.

BACKGROUND: To investigate whether multiple imputation (MI) of missing Glasgow Coma Scale (GCS) values generates more accurate GCS frequency distributions and less biased parameter estimates in logistic regression models predicting mortality than the standard procedure of excluding observations with missing GCS values. METHODS: The study population consisted of 5,065 patients with complete GCS information from the trauma registry of a Level 1 trauma center. Missing GCS values were imposed on the data set, and the performance of MI (extrapolating missing GCS from a data prediction model) and of deleting all data observations with missing GCS (list-wise deletion) were evaluated. GCS and Trauma and Injury Severity Score (TRISS) frequency distributions and parameter estimates were compared with true values from the original data set. RESULTS: GCS and TRISS frequency values generated by MI were much more accurate than those generated by list-wise deletion. GCS and TRISS parameter estimates generated by MI all had acceptable bias and coverage rates when compared with true values. List-wise deletion provided biased parameter estimates for the GCS, the Revised Trauma Score, and the Injury Severity Score. CONCLUSION: MI is a valid solution to the problem of missing GCS data in trauma research. It allows the conservation of precious data observations and leads to unbiased estimates in consequent analyses. Analyses, which exclude observations with missing GCS data, provide biased results.

Biomedical Research↗

PubMed2003

Comparison of alpha/beta estimates from homogeneous (individual) and heterogeneous (population) tumor control models for early stage prostate cancer.

Radiobiological parameter estimates for prostate cancer are obtained from both a homogeneous (individual) and heterogeneous (population) tumor control model based on Poisson statistics and the linear quadratic model of cell survival. Parameter estimates for both models are highly correlated: statistically equivalent fits are achievable using either (1) linear quadratic (LQ) parameters with low numbers of radioresistant tumor stem cells, or (2) LQ parameters with corresponding larger number of radiosensitive tumor stem cells. A theoretical framework is developed to explain this correlation. A Monte Carlo error analysis based on binomial statistics is used to estimate confidence intervals for all parameter estimates. It was found that both the homogeneous and heterogeneous models produce approximately equivalent estimates of radiobiological parameters, including the alpha/beta ratio. However, the 95% confidence interval for the alpha/beta ratio derived from the heterogeneous model are considerably larger than those derived from the homogeneous model, which indicate the homogeneous model overestimates the statistical significance of the alpha/beta estimate.

Cell Survival↗

PubMed1969

Observations on programs to estimate the parameters of enzyme kinetics.

This paper gives a critical account of the two major programs so far published specifically to estimate the parameters of enzyme kinetics. The dangers of submitting data to the programs without proper checks are discussed, and a screening test is described to identifty sets of data which may not be best fitted by a rectangular hyperbola.

Journal Article↗

PubMed2005

Applicability of an extant batch respirometric assay in describing dynamics of ammonia and nitrite oxidation in a nitrifying bioreactor.

Several techniques have been proposed for biokinetic estimation of nitrification. Recently, an extant respirometric assay has been presented that yields kinetic parameters for both nitrification steps with minimal physiological change to the microorganisms during the assay. Herein, the ability of biokinetic parameter estimates from the extant respirometric assay to adequately describe concurrently obtained NH4+-N and NO(2-)-N substrate depletion profiles is evaluated. Based on our results, in general, the substrate depletion profiles resulted in a higher estimate of the maximum specific growth rate coefficient, micro(max) for both NH4+-N to NO(2-)-N oxidation and NO(2-)-N to NO(3-)-N oxidation compared to estimates from the extant respirograms. The trends in the kinetic parameter estimates from the different biokinetic estimation techniques are paralleled in the nature of substrate depletion profiles obtained from best-fit parameters. Based on a visual inspection, in general, best-fit parameters from optimally designed complete respirograms provided a better description of the substrate depletion profiles than estimates from isolated respirograms. Nevertheless, the sum of the squared errors for the best-fit respirometry based parameters was outside the 95% joint confidence interval computed for the best-fit substrate depletion based parameters. Notwithstanding the difference in kinetic parameter estimates determined in this study, the different biokinetic estimation techniques still are close to estimates reported in literature. Additional parameter identifiability and sensitivity analysis of parameters from substrate depletion assays revealed high precision of parameters and high parameter correlation. Although biokinetic estimation via automated extant respirometry is far more facile than via manual substrate depletion measurements, additional sensitivity analyses are needed to test the impact of differences in the resulting parameter values on continuous reactor performance.

Ammonia↗

PubMed2005

A method for estimating the parameters of electrodynamic drivers in thermoacoustic coolers.

The electroacoustic efficiency of high-power actuators used in thermoacoustic coolers may be estimated using a linear model involving a combination of six parameters. A method to identify these equivalent driver parameters from measured total electrical impedance and velocity-voltage transfer function data was developed. A commercially available, moving-magnet driver coupled to a functional thermoacoustic cooler was used to demonstrate the procedure experimentally. The method, based on linear electrical circuit theory, allowed for the possible frequency and amplitude dependence of the driver parameters to be estimated. The results demonstrated that driver parameters measured in vacuo using this method can be used to predict the driver efficiency and performance for operating conditions which may be encountered under load.

Journal Article↗

PubMed2006

Tracking transplanted cells using dual-radionuclide SPECT.

The purpose of this study was to characterize the performance of single photon emission computed tomography (SPECT) in tasks associated with tracking transplanted cells. Previous studies identified matters of hardware design, whereas we focus on biological variables impacting system performance, such as cell colony growth and non-specific radiolabelling. Using experimental data, a digital phantom was developed of in vitro 111In-radiolabelled stem cells, transfected with a reporter gene, transplanted into canine infarcted myocardium and interrogated using a peripherally injected 131I-radiolabelled reporter probe. Single- and dual-head SPECT acquisition was simulated. Performance was characterized using an estimation task, where the precision of parameter estimates (111In and 131I radiolabel quantity, cell colony size and location, and background) was tracked as the phantom evolved to simulate 111In-label efflux, cell colony growth and improved reporter probe specificity. In vitro pre-labelling of transplanted cells improved precision of parameter estimates via a priori size and location information. Precision of radiolabel quantity estimates improved with cell colony growth, despite 111In radiolabel dilution; size and location parameters were influenced little. Precision of radiolabel quantity estimates improved with reduced reporter probe non-specific uptake. The performance of SPECT in cell tracking is influenced strongly by biological variables. These should be considered when planning experiments or developing SPECT technology for cell tracking.

Animals↗

PubMed1997

A comparison of the standard approach and the NONMEM approach in the estimation of bioavailability in man.

There has recently been concern about confidence intervals calculated using the standard error of parameter estimates from NONMEM, a computer program that uses a non-linear mixed-effects model to calculate relative bioavailability (F), because of possible downward bias of these estimates. In this study an alternate approach, the log-likelihood procedure, was used to calculate the confidence intervals for F from NONMEM. These were then compared with those calculated using the standard error of the parameter estimates, the traditional NONMEM approach, and the standard model-independent method, to determine whether bias exists. By use of data from a single dose, open cross-over study of ibuprofen using 14 healthy male volunteers, NONMEM was shown to give results consistent with those obtained using the standard model-independent method of analysis and could be a useful tool in the determination of F where conditions for using the standard method of analysis are not optimum. The width of the confidence interval for F using the log-likelihood procedure was narrower and non-symmetrical when compared with that obtained using the traditional NONMEM approach. The width of the confidence interval obtained using the traditional NONMEM method was similar to that from the standard approach, however the parameter estimate for F was higher than that obtained from the standard method. This could have been because of an outlier in the data set to which the standard approach is more sensitive. No downward bias was found in the confidence intervals from NONMEM. The bioavailability data set was of relatively low variability and more research with highly variable data is necessary before it can be concluded that the confidence intervals calculated from NONMEM can be used for hypothesis testing.

Anti-Inflammatory Agents, Non-Steroidal↗

PubMed1994

Prediction of acetaminophen concentrations in overdose patients using a Bayesian pharmacokinetic model.

A pharmacokinetic program using population-based parameter estimates and a Bayesian forecasting model was retrospectively evaluated for predicting acetaminophen serum concentrations in overdose patients. Dynamic disposition factors known to affect acetaminophen disposition (emesis, activated charcoal, N-acetylcysteine, etc.) were included in the program. Twenty six patients who reported an acetaminophen ingestion of at least 70 mg/kg within 24 h of presentation to the hospital and had at least one measured acetaminophen concentration were included. Prediction of initial acetaminophen concentrations using only population-based parameter estimates resulted in a percent mean error (%ME) and percent mean absolute error (%MAE) of 9.3 and 42.2, respectively. Using only the initial concentration as feedback, the Bayesian forecasting model accurately predicted the second acetaminophen concentration (%ME = 4.0, %MAE = 23.6). The Bayesian model also accurately predicted all concentrations within 8 h of the ingestion (%ME = 10.6, %MAE = 24.0). The prediction of concentrations between 2 to 4 h and 4 to 4.5 h after ingestion with only population-based parameter estimates resulted in %ME of 17.0 and 13.2, respectively, and %MAE of 36.5 and 35.1, respectively. Our data suggests that acetaminophen serum concentrations occurring within the first 4.5 h after ingestion can be reliably predicted by the set of population-based parameter estimates evaluated. Once a single acetaminophen concentration is available, the Bayesian forecasting model can accurately predict subsequent concentrations within the first 8 h after an acetaminophen ingestion.

Acetaminophen↗

PubMed1988

Estimation of the rate constants in a data-sparse environment: comparison of a mathematical method and least squares analysis.

A new method is presented for estimating the rate constants for the one-compartment open model with first-order absorption in a data-sparse environment. It is based on the principles of matrix algebra and system theory and requires only three to four plasma samples drawn at equal time intervals (a minimum of at least one sample in the absorption phase and one sample in the elimination phase). The utility of the technique is illustrated by comparing the parameter estimates from the matrix method with the estimates from a nonlinear computer parameter estimation program. In the preliminary evaluation of the method with both perfect data and data with randomly distributed error, the parameter estimates from the matrix method proved to be as good as the computer estimation for data-sparse systems. In a more realistic comparison with published patient data, the matrix method resulted in 0 to 456% better estimates of kel and ka than the computer estimation. Since the matrix method is mathematically simple and requires only three to four blood samples, it should prove very useful in data-sparse systems (e.g., clinical or small animal situations) where a minimum amount of blood samples can be drawn.

Absorption↗

PubMed1993

Exploring the HDL likelihood surface.

Using random initial parameter estimates, three segregation analysis models of the inheritance of HDL2 in the Berkeley GAW8 data set were maximized 5000 times each. Initial parameter estimates were assumed to be uniformly distributed on intervals formed by parameter boundaries. The three models were unrestricted, environmental, and Mendelian regressive type A models. Likelihood ratio tests of the global maxima rejected the Mendelian model and accepted the environmental model. However, tests using local maxima accepted the Mendelian model and both rejected and accepted the environmental model. Patterns among the initial parameter estimates of convergent runs were examined to develop empirical rules to increase the frequency of convergence. These rules were tested using data on apoAI in the Berkeley GAW8 data set.

Genetic Predisposition to Disease↗

PubMed2005

Estimating marginal and incremental effects on health outcomes using flexible link and variance function models.

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.

Adult↗

PubMed2004

Estimating genetic parameters in natural populations using the "animal model".

Estimating the genetic basis of quantitative traits can be tricky for wild populations in natural environments, as environmental variation frequently obscures the underlying evolutionary patterns. I review the recent application of restricted maximum-likelihood "animal models" to multigenerational data from natural populations, and show how the estimation of variance components and prediction of breeding values using these methods offer a powerful means of tackling the potentially confounding effects of environmental variation, as well as generating a wealth of new areas of investigation.

Analysis of Variance↗

PubMed2002

Factorial hidden Markov models and the generalized backfitting algorithm.

Previous researchers developed new learning architectures for sequential data by extending conventional hidden Markov models through the use of distributed state representations. Although exact inference and parameter estimation in these architectures is computationally intractable, Ghahramani and Jordan (1997) showed that approximate inference and parameter estimation in one such architecture, factorial hidden Markov models (FHMMs), is feasible in certain circumstances. However, the learning algorithm proposed by these investigators, based on variational techniques, is difficult to understand and implement and is limited to the study of real-valued data sets. This chapter proposes an alternative method for approximate inference and parameter estimation in FHMMs based on the perspective that FHMMs are a generalization of a well-known class of statistical models known as generalized additive models (GAMs; Hastie & Tibshirani, 1990). Using existing statistical techniques for GAMs as a guide, we have developed the generalized backfitting algorithm. This algorithm computes customized error signals for each hidden Markov chain of an FHMM and then trains each chain one at a time using conventional techniques from the hidden Markov models literature. Relative to previous perspectives on FHMMs, we believe that the viewpoint taken here has a number of advantages. First, it places FHMMs on firm statistical foundations by relating them to a class of models that are well studied in the statistics community, yet it generalizes this class of models in an interesting way. Second, it leads to an understanding of how FHMMs can be applied to many different types of time-series data, including Bernoulli and multinomial data, not just data that are real valued. Finally, it leads to an effective learning procedure for FHMMs that is easier to understand and easier to implement than existing learning procedures. Simulation results suggest that FHMMs trained with the generalized backfitting algorithm are a practical and powerful tool for analyzing sequential data.

Algorithms↗

PubMed1993

A comparison of methods for estimating mortality parameters from survival data.

The Gompertz mortality function, Rm = R0e alpha t, is frequently used to describe changes in mortality rate (Rm) with time (t). In this paper, four methods for determining the best fit values of the two parameters, R0 and alpha, are compared. Three of the four methods use the Gompertz mortality function with mortality rate estimates derived from survival data to determine the best fit values for the two parameters. All three confront problems. The fourth method uses the Gompertz survival function, which can be derived from the Gompertz mortality function and which allows one to use survival data directly. It thereby avoids the problems and generally gives the best estimates for the two parameters. The use of the mortality function, with mortality rate estimates, confronts four distinct problems. One of these is caused by time intervals when zero organisms die. A second is caused by errors produced in estimating mortality rates from survival data. If too high a proportion of a population die in a given time interval, the mortality rate estimates are too low. A third problem is the sensitivity of the mortality-equation-based analyses to values at the end of the survival curve, where scatter in mortality values tends to be greater. A final problem occurs when time intervals greater than one time unit (day, week, year, etc.) are used in the analysis. Such problems with the use of mortality rates to estimate parameter values are revealed when the calculated parameters are used to produce a survival curve, or when known values of R0 and alpha are used to generate survival data. This paper introduces a non-linear regression analysis, using a Simplex algorithm to fit parameters R0 and alpha in the Gompertz Survival function and concludes that it gives more reliable and consistent results with a variety of data than do three methods that use the mortality function.

Animals↗

PubMed2001

Estimating synaptic parameters from mean, variance, and covariance in trains of synaptic responses.

Fluctuation analysis of synaptic transmission using the variance-mean approach has been restricted in the past to steady-state responses. Here we extend this method to short repetitive trains of synaptic responses, during which the response amplitudes are not stationary. We consider intervals between trains, long enough so that the system is in the same average state at the beginning of each train. This allows analysis of ensemble means and variances for each response in a train separately. Thus, modifications in synaptic efficacy during short-term plasticity can be attributed to changes in synaptic parameters. In addition, we provide practical guidelines for the analysis of the covariance between successive responses in trains. Explicit algorithms to estimate synaptic parameters are derived and tested by Monte Carlo simulations on the basis of a binomial model of synaptic transmission, allowing for quantal variability, heterogeneity in the release probability, and postsynaptic receptor saturation and desensitization. We find that the combined analysis of variance and covariance is advantageous in yielding an estimate for the number of release sites, which is independent of heterogeneity in the release probability under certain conditions. Furthermore, it allows one to calculate the apparent quantal size for each response in a sequence of stimuli.

Analysis of Variance↗

PubMed1969

Some experiences in the estimation of parameters in non-linear differential equations.

The author describes a procedure developed by himself and his colleagues for obtaining estimates of the parameters of rate equations, together with information about confidence regions for the estimates. The program has been used successfully for processing results from the chemical engineering industry, with highly non-linear model systems, particularly since temperature was a variable, and the "rate constants" were non-linear combinations of other constants. In biochemical situations, in which investigations are almost always at constant temperature, the non-linearity should not be so extreme, and the procedure may well be capable of dealing with more than 5 to 7 parameters for which it is recommended.

Journal Article↗