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PubMed1999

Sensitivity of biological models to errors in parameter estimates.

Since A. M. Turing's paper proposing a mathematical basis for pattern formation in developing organisms many mathematical approaches have been proposed to model biological phenomenon. Continued laboratory study and recent improvements in measurement capabilities have provided an immense quantity of raw gene expression data. The level of data now available demands the development of well-characterized and tested computational tools. Thus, we have examined one mathematical model's sensitivity to errors in estimating its' parameters. Errors in parameter estimation can arise from noise in the laboratory measurements and recasting of laboratory data. We elected to examine the rule-based mathematical model of Mjolsness et al for its' sensitivity to errors in estimated parameters. We have used the technique of sensitivity equations as generally applied in nonlinear systems analysis.

Computational Biology↗
PubMed2000

Dynamic algorithm for parameter estimation and its applications

We consider a dynamic method, based on synchronization and adaptive control, to estimate unknown parameters of a nonlinear dynamical system from a given scalar chaotic time series. We present an important extension of the method when the time series of a scalar function of the variables of the underlying dynamical system is given. We find that it is possible to obtain synchronization as well as parameter estimation using such a time series. We then consider a general quadratic flow in three dimensions and discuss the applicability of our method of parameter estimation in this case. In practical situations one expects only a finite time series of a system variable to be known. We show that the finite time series can be repeatedly used to estimate unknown parameters with an accuracy that improves and then saturates to a constant value with repeated use of the time series. Finally, we suggest an important application of the parameter estimation method. We propose that the method can be used to confirm the correctness of a trial function modeling an external unknown perturbation to a known system. We show that our method produces exact synchronization with the given time series only when the trial function has a form identical to that of the perturbation.

Journal Article↗
PubMed2005

Estimating the extent of parameter bias in the classical twin design: a comparison of parameter estimates from extended twin-family and classical twin designs.

The classical twin design (CTD) circumvents parameter indeterminacy by assuming (1) negligible higher-order epistasis; and (2) either nonadditive genetic or common environmental effects are nonexistent, creating two potential sources of bias (Eaves et al., 1978; Grayson, 1989). Because the extended twin-family design (ETFD) uses many more unique covariance observations to estimate parameters, common environmental and nonadditive genetic parameters can be simultaneously estimated. The ETFD thereby corrects for what is likely to be the largest of the two sources of bias in CTD parameter estimates (Keller & Coventry, 2005). In the current paper, we assess the extent of this and other potential sources of bias in the CTD by comparing all published ETFD parameter estimates to CTD parameter estimates derived from the same data. CTD estimates of the common environment were lower than ETFD estimates of the common environment for some phenotypes, but for other phenotypes (e.g., stature in females and certain social attitudes), what appeared as the common environment was resolved to be assortative mating in the ETFD. On average, CTD estimates of nonadditive genetic factors were 43% lower, and additive genetic factors 63% higher, than ETFD estimates. However, broad-sense heritability estimates from the CTD were only 18% higher than ETFD estimates, highlighting that the CTD is useful for estimating broad-sense but not narrow-sense heritability. These results suggest that CTD estimates can be misleading when interpreted literally, but useful, albeit coarse, when interpreted properly.

Genetic Variation↗
PubMed2002

Achievable accuracy of parameter estimation for multidimensional NMR experiments.

A fundamental issue in NMR spectroscopy is the estimation of parameters such as the Larmor frequencies of nuclei, J coupling constants, and relaxation rates. The Cramer-Rao lower bound provides a method to assess the best achievable accuracy of parameter estimates resulting from an unbiased estimation procedure. We show how the Cramer-Rao lower bound can be calculated for data obtained from multidimensional NMR experiments. The Cramer-Rao lower bound is compared to the variance of parameter estimates for simulated data using a least-squares estimation procedure. It is also shown how our results on the Cramer-Rao lower bound can be used to analyze whether an experimental design can be improved to provide experimental data which can result in parameter estimates with higher accuracy. The concept of nonuniform averaging in the indirect dimension is introduced and studied in connection with nonuniform sampling of the data.

Least-Squares Analysis↗
PubMed2006

Parameter estimation in biochemical systems models with alternating regression.

BACKGROUND: The estimation of parameter values continues to be the bottleneck of the computational analysis of biological systems. It is therefore necessary to develop improved methods that are effective, fast, and scalable. RESULTS: We show here that alternating regression (AR), applied to S-system models and combined with methods for decoupling systems of differential equations, provides a fast new tool for identifying parameter values from time series data. The key feature of AR is that it dissects the nonlinear inverse problem of estimating parameter values into iterative steps of linear regression. We show with several artificial examples that the method works well in many cases. In cases of no convergence, it is feasible to dedicate some computational effort to identifying suitable start values and search settings, because the method is fast in comparison to conventional methods that the search for suitable initial values is easily recouped. Because parameter estimation and the identification of system structure are closely related in S-system modeling, the AR method is beneficial for the latter as well. Specifically, we show with an example from the literature that AR is three to five orders of magnitudes faster than direct structure identifications in systems of nonlinear differential equations. CONCLUSION: Alternating regression provides a strategy for the estimation of parameter values and the identification of structure and regulation in S-systems that is genuinely different from all existing methods. Alternating regression is usually very fast, but its convergence patterns are complex and will require further investigation. In cases where convergence is an issue, the enormous speed of the method renders it feasible to select several initial guesses and search settings as an effective countermeasure.

Computational Biology↗
PubMed1997

The impact of arteriovenous concentration differences on pharmacodynamic parameter estimates.

In many pharmacodynamic investigations venous drug concentrations are measured and linked to effect-site concentrations by means of a traditional first-order effect-compartment model to estimate pharmacodynamic (PD) parameters. This analysis ignores the underlying physiology that arterial blood supplies both the venous sampling site and effect site. Recently, an extended effect-compartment model has been proposed that reflects physiology by postulating a first-order rate constant of equilibrium between arterial and effect-site concentrations (ke0) as well as first-order rate constant between arterial and venous concentrations (kv0). In the current paper, we evaluate the bias in PD parameter estimates if venous drug concentrations are measured and linked to effect-site concentrations by a traditional effect compartment as a function ke0, kv0, and the drug's elimination half-life (T1/2); we present an analytical solution to the differential equations characterizing the extended effect-compartment model; and we evaluate the performance of the extended effect-compartment model to estimate pharmacodynamic parameters on the basis of venous drug concentrations. Time profiles of venous drug concentrations and drug effect were simulated for a wide range of different values of the half-life of ke0 (T1/2,e0), the half-life of kv0 (T1/2,v0), and T1/2. The simulations showed that a significant bias (up to 90%) in PD parameter estimates occurred for certain values of T1/2,e0, T1/2,v0, and T1/2 if venous drug concentrations are linked to effect-site concentrations by a traditional effect-compartment model. This model misspecification is not apparent from the results of the fitting procedure. The extended effect-compartment model provided unbiased but imprecise PD parameter estimates. The extended effect-compartment model was also able to analyze instances in which the venous concentrations equilibrate slower with the arterial concentrations than the effect-site concentrations, and proteresis is observed in the concentration--effect relationship. It is concluded that if the apparent T1/2 of the drug in the time period in which the decline in pharmacological effect is most pronounced is greater than 5 times T1/2,e0 and T1/2,e0 is greater than T1/2,v0 there is no need to model the underlying arteriovenous equilibrium delay. Under these conditions a traditional first-order link between venous and effect-site concentrations will yield accurate and reliable (less than 10% bias) estimates of the PD parameters such as Emax, EC50 and N. If T1/2 is less than 5 times T1/2,e0 or if T1/2,v0 is greater than T1/2,e0, the underlying arteriovenous equilibration delay needs to be taken into account in the model to obtain unbiased estimates of the PD parameters. This applies for almost all values of T1/2.v0. Arteriovenous equilibration delay can be best taken into account by measuring arterial blood concentrations. If this is not possible, the extended effect-compartment link model can be used. However, a large number of effect measurements needs to be obtained to estimate the model parameters accurately.

Arteries↗
PubMed2007

Evaluation of uncertainty parameters estimated by different population PK software and methods.

The uncertainty associated with parameter estimations is essential for population model building, evaluation, and simulation. Summarized by the standard error (SE), its estimation is sometimes questionable. Herein, we evaluate SEs provided by different non linear mixed-effect estimation methods associated with their estimation performances. Methods based on maximum likelihood (FO and FOCE in NONMEM, nlme in Splus, and SAEM in MONOLIX) and Bayesian theory (WinBUGS) were evaluated on datasets obtained by simulations of a one-compartment PK model using 9 different designs. Bootstrap techniques were applied to FO, FOCE, and nlme. We compared SE estimations, parameter estimations, convergence, and computation time. Regarding SE estimations, methods provided concordant results for fixed effects. On random effects, SAEM and WinBUGS, tended respectively to under or over-estimate them. With sparse data, FO provided biased estimations of SE and discordant results between bootstrapped and original datasets. Regarding parameter estimations, FO showed a systematic bias on fixed and random effects. WinBUGS provided biased estimations, but only with sparse data. SAEM and WinBUGS converged systematically while FOCE failed in half of the cases. Applying bootstrap with FOCE yielded CPU times too large for routine application and bootstrap with nlme resulted in frequent crashes. In conclusion, FO provided bias on parameter estimations and on SE estimations of random effects. Methods like FOCE provided unbiased results but convergence was the biggest issue. Bootstrap did not improve SEs for FOCE methods, except when confidence interval of random effects is needed. WinBUGS gave consistent results but required long computation times. SAEM was in-between, showing few under-estimated SE but unbiased parameter estimations.

Bayes Theorem↗
PubMed1992

Methods of segregation analysis for animal breeding data: parameter estimates.

Three approximations to the combined model likelihood (including variation due to both polygenes and a major gene) used in segregation analysis were compared in their ability to estimate the parameters describing the model. Paternal half-sib data simulated under four different combined models were used. In general, when a combined model provides the best explanation of the data the parameter estimates from all three approximations investigated are, on average, in good agreement with the values simulated. The use of Hermite integration to replace the integration in the combined model likelihood provided the parameter estimates closest to those stimulated. The other two approximations, based on extensions of linear mixed-model theory, gave estimates further from the expected values, however, these latter two methods can estimate transmitting abilities for the sires directly from the analyses. The three approximations are similar in their ability to genotype sires at the major locus. On average, with the models studied here, about 65 per cent of sires can be successfully genotyped.

Alleles↗
PubMed2004

Optimal sampling time selection for parameter estimation in dynamic pathway modeling.

Systems Biology is an emerging research area, which considers mathematical representations of inter- and intra-cellular dynamics. Among the many research problems that have been addressed, dynamic modeling of signal transduction pathways has received increasing attention. The usual approach to represent intra-cellular dynamics are nonlinear, usually ordinary, differential equations. The purpose of the models is to test and generate hypothesis of specific pathways and it is therefore required to estimate model parameters from experimental data. The experiments to generate data are complex and expensive, as a consequence of which the time series available are usually rather short, with few if any replicates. Almost certainly, not all variables one would like to include in a model can be measured. Parameter estimation is therefore an important research problem in Systems Biology and the focus of this paper. In particular, we are interested in optimizing the sampling time selection in order to minimize the variance of the parameter estimation error. With few sampling time points feasible, their selection is of practical importance in experimental design. Finally, the theoretical results are supported with an application.

Computer Simulation↗
PubMed2006

Parameter estimation for stiff equations of biosystems using radial basis function networks.

BACKGROUND: The modeling of dynamic systems requires estimating kinetic parameters from experimentally measured time-courses. Conventional global optimization methods used for parameter estimation, e.g. genetic algorithms (GA), consume enormous computational time because they require iterative numerical integrations for differential equations. When the target model is stiff, the computational time for reaching a solution increases further. RESULTS: In an attempt to solve this problem, we explored a learning technique that uses radial basis function networks (RBFN) to achieve a parameter estimation for biochemical models. RBFN reduce the number of numerical integrations by replacing derivatives with slopes derived from the distribution of searching points. To introduce a slight search bias, we implemented additional data selection using a GA that searches data-sparse areas at low computational cost. In addition, we adopted logarithmic transformation that smoothes the fitness surface to obtain a solution simply. We conducted numerical experiments to validate our methods and compared the results with those obtained by GA. We found that the calculation time decreased by more than 50% and the convergence rate increased from 60% to 90%. CONCLUSION: In this work, our RBFN technique was effective for parameter optimization of stiff biochemical models.

Algorithms↗
PubMed1994

Optimal sampling theory: effect of error in a nominal parameter value on bias and precision of parameter estimation.

The authors examined the robustness of optimal sampling theory in estimating the parameter values of two different populations of patients receiving a constant rate, half-hour intravenous infusion of theophylline. One population consisted of smokers; the other included nonsmokers. The smoking population was predicted to have a serum clearance approximately 50% greater than the nonsmokers because of an induction of the cytochrome P450 system. After an initial study to provide both patient-specific and population mean parameter values, optimal sampling strategies that were derived from each population (seven sample split designs) and the patient's seven sample and four sample design were determined. A second study was performed with an overall sampling strategy that was superset of all the above strategies. The analysis of all samples served as the reference for the parameter values. Bias and precision of the values determined with each of the optimal sampling sets (seven sample sets based on the "correct" and "wrong" populations, the patient's seven and four sample sets) were determined relative to these reference values. Irrespective of the sample set used for analysis, unbiased and precise parameter estimates, particularly of hybrid parameters were provided. With the patient's four sample set, Vss was significantly biased, but the value of (2.2%) was clinically insignificant. The authors conclude that optimal sampling theory, as implemented in this study, provides robust estimates of important pharmacokinetic parameter values, even when errors of 50% are present in the clearance of the population used to calculate the optimal sampling design.

Adolescent↗
PubMed2006

SBML-PET: a Systems Biology Markup Language-based parameter estimation tool.

UNLABELLED: The estimation of model parameters from experimental data remains a bottleneck for a major breakthrough in systems biology. We present a Systems Biology Markup Language (SBML) based Parameter Estimation Tool (SBML-PET). The tool is designed to enable parameter estimation for biological models including signaling pathways, gene regulation networks and metabolic pathways. SBML-PET supports import and export of the models in the SBML format. It can estimate the parameters by fitting a variety of experimental data from different experimental conditions. SBML-PET has a unique feature of supporting event definition in the SMBL model. SBML models can also be simulated in SBML-PET. Stochastic Ranking Evolution Strategy (SRES) is incorporated in SBML-PET for parameter estimation jobs. A classic ODE Solver called ODEPACK is used to solve the Ordinary Differential Equation (ODE) system. AVAILABILITY: http://sysbio.molgen.mpg.de/SBML-PET/. The website also contains detailed documentation for SBML-PET.

Algorithms↗
PubMed2002

Ascertainment-adjusted parameter estimates revisited.

Ascertainment-adjusted parameter estimates from a genetic analysis are typically assumed to reflect the parameter values in the original population from which the ascertained data were collected. Burton et al. (2000) recently showed that, given unmodeled parameter heterogeneity, the standard ascertainment adjustment leads to biased parameter estimates of the population-based values. This finding has important implications in complex genetic studies, because of the potential existence of unmodeled genetic parameter heterogeneity. The authors further stated the important point that, given unmodeled heterogeneity, the ascertainment-adjusted parameter estimates reflect the true parameter values in the ascertained subpopulation. They illustrated these statements with two examples. By revisiting these examples, we demonstrate that if the ascertainment scheme and the nature of the data can be correctly modeled, then an ascertainment-adjusted analysis returns population-based parameter estimates. We further demonstrate that if the ascertainment scheme and data cannot be modeled properly, then the resulting ascertainment-adjusted analysis produces parameter estimates that generally do not reflect the true values in either the original population or the ascertained subpopulation.

Genetic Diseases, Inborn↗
PubMed1994

Experimental design for parameter estimation through sensitivity analysis.

Parameter estimates can be obtained by fitting a numerical simulation model to experimental data, but these estimates may be biased and/or imprecise because of noise in the experimental data. Appropriate choice of experimental conditions, such as exposure or substrate concentrations and sampling times, can minimize the effect of experimental noise on parameter estimates, thus reducing bias and improving precision. This article describes a technique for selecting experimental (initial) conditions and measurement times for optimal parameter estimation. The technique makes use of a user-supplied mathematical simulation model for the process under study with a set of "current" parameter values specified. These "current" parameter values are the best that can be obtained using all available experimental data and/or literature information at the time when design calculations are performed. Early in a modeling study, the "current" parameter values will be tentative--based on a relatively small amount of information. Later in a study, the "current" parameter values may be known to reasonable accuracy, but final confirmation is desired. The technique uses the simulation model to calculate a numerical index for each possible experimental design. The numerical index, or Information Index, is a measure of the response of a simulation model to changes in parameter values, described by Kalogerakis and Luus (1983, 1984). The experimental design with the greatest value of Information Index is the one under which parameters can be most precisely estimated. Computation of the Information Index, described in detail, can be somewhat complicated, depending on the software available. The results, however, are simple to interpret and provide valuable information on the quality of alternate proposed experiments. The technique is applicable to a broad range of dynamical systems. Its use is demonstrated by application to a simulation model being developed to describe the in vitro metabolism of benzene by mouse liver microsomes.

Animals↗
PubMed1999

Biorthogonal wavelet transforms for ECG parameters estimation.

The parameters of various morphologies of ECG waveform are basic in characterizing them as normal or otherwise. The use of multiscale analysis, through biorthogonal wavelets presented in this paper, appears very promising for such a characterization. This is on account of the fact that various morphologies are excited better at different scales. From these different scales, amplitudes, durations and various segments, widths can be determined more accurately. Simulation studies, with real ECG data, have shown that even when the signal-to-noise ratios are poor, the proposed technique can be used to accurately estimate the said parameters.

Biomedical Engineering↗
PubMed1999

Parameter estimation using a direct solution of the integrated Michaelis-Menten equation.

A novel method of estimating enzyme kinetic parameters is presented using the Lambert omega function coupled with nonlinear regression. Explicit expressions for the substrate and product concentrations in the integrated Michaelis-Menten equation were obtained using the omega function which simplified kinetic parameter estimation as root-solving and numerical integration of the Michaelis-Menten equation were avoided. The omega function was highly accurate in describing the substrate and product concentrations in the integrated Michaelis-Menten equation with an accuracy of the order of 10(-16) when double precision arithmetic was used. Progress curve data from five different experimental systems were used to demonstrate the suitability of the omega function for kinetic parameter estimation. In all cases, the kinetic parameters obtained using the omega function were almost identical to those obtained using the conventional root-solving technique. The availability of highly efficient algorithms makes the computation of omega simpler than root-solving or numerical integration. The accuracy and simplicity of the omega function approach make it an attractive alternative for parameter estimation in enzyme kinetics.

Algorithms↗
PubMed2000

Problems in Extinction Model Selection and Parameter Estimation.

It is a vexing problem to achieve a consensus about the proper scientific way to assess population viability for habitat conservation plans. Rather than a hypothesis-testing approach, here it is proposed to select population models, estimate extinction parameters, and assess prediction uncertainty using a pragmatic, empirical Bayesian approach. The simplest usable models include the effects of population growth, r; carrying capacity, K; Allee threshold, N(A); and environmental stochasticity, v(r). Analytic predictions of expected extinction times are available for such models. Models that are more complex can be elaborated from this basis. Selection from a hierarchy of nesting population models can often be done through the evaluation of parameters. The estimation of the most important extinction parameters can be undertaken in a variety of ways. Time series can be analyzed to estimate r(d), v(r), rho, and K. Habitat models and individualistic population models may help estimate N(A) and K and demographic stochasticity. Fine-scale biogeography and climatological data may be useful in the estimation of a variety of parameters. Because it takes many years to estimate extinction parameters accurately for a given population of interest, the most efficient estimation procedures are desirable. I propose the use of prior information from an (as yet nonexistent) population biology database. The accumulation of local information through monitoring will improve our estimates allowing adaptive management. Uncertainty in the estimates will always remain, but it may be quantified by the posterior distributions. A crude example is discussed using treefrog population data. Although the motivations, beliefs, and biases of competing stakeholders will differ, a habitat conservation plan could accommodate this variation in the prior distributions. Field experience from monitoring will increasingly clear up any discrepancies between the opposing beliefs and the real ecosystem. As the world is an uncertain place and because there is no universal scientific method, there will always be controversy and surprises. The best we can do is (1) agree about our prior information, (2) agree about the strategy of model selection and parameter estimation, and (3) agree about our strategy for adaptive management. Perhaps the greatest impediment to such prior agreements for HCPs is the likely paranoia inspired by the use of unfamiliar statistical methodology. We need to train students of ecology in a more flexible and deeper understanding of statistics and philosophy of science.

Journal Article↗
PubMed1994

Nonlinear parameter estimation by linear association: application to a five-parameter passive neuron model.

Linear associative memories (LAM) have been intensely used in the areas of pattern recognition and parallel processing for the past two decades. Application of LAM to nonlinear parameter estimation, however, has only been recently attempted. The process consists in converting the nonlinear function in the parameters into a set of linear algebraic equations. The nature of the linearized system and the factors influencing the accuracy of the parameter estimates have not yet been fully investigated. In this paper, LAM is applied to a nonlinear five-parameter model of the neuron. Ill-conditioning, which is often exhibited in LAM, is treated with the method of regularization as well as by the singular value decomposition (SVD). Simulation results indicate that the parameters estimated by LAM exhibit a remarkable robustness against additive white noise in comparison with the classical gradient optimization technique. Moreover, it is shown that regularization can be superior to SVD under certain conditions. Our results suggest that LAM can be used both as a noise reduction technique and as a stand-alone nonlinear parameter estimation algorithm. The comparison between LAM and a gradient technique show that, for this estimation problem, the LAM method can give more reliable estimates. Further improvements in estimation quality may still be achieved by the use of other forms of regularizing functions.

Linear Models↗