Optimal design of experiments for the estimation of precise hyperbolic kinetic and binding parameters

Optimal designs were evaluated for the estimation of precise parameters in kinetic and binding experiments in which the concentration dependence of the response (reaction velocity or bound ligand concentration) is characterized by a hyperbola. The designs were evaluated by maximizing the determinant of the appropriate information matrix. It is demonstrated that highest precision is obtained by replicating basic two-point designs. However, the calculated optimal designs should serve only as guidelines and not rules for experimentation.In the presence of constant variance, half of the observations should be obtained at the highest practically attainable concentration. In kinetic studies, the remaining measurements should yield half of the maximally attainable velocity. In binding experiments, the second half of observations for either bound (b) or free (f) ligand concentrations should be made at a total concentration of c = P + K (K is the dissociation constant, and P the binding capacity).With constant coefficient of variation (constant relative error), kinetic experiments should be performed, at equal frequency, at the highest and lowest attainable concentration. Half of the binding experiments should be made at their lowest precision: at the highest possible concentration when b is measured, and at the smallest feasible concentration when f is obtained. The other half of the readings should be taken at c = P ? K (if P > K) for the measurement of b, and at c = P + K for observations of f.Deviations from the calculated optimal designs result in diminished efficiency (increased variance) of the estimated parameters. The reduction can be substantial with some frequently used experimental designs, especially when the number of observations is not very small. Therefore, the first few readings could consider the validation of the model, but additional measurements should follow the guidelines presented here.     

Determination of enzyme kinetic parameters by continuous addition of substrate to a single reaction mixture and analysis by a tangent-slope procedure: I. Analysis of the method using computed progress curves

A new method has been developed which provides reliable estimates of enzyme kinetic constants from single reaction progress curves recorded under conditions of continuously increasing substrate concentration. Equally spaced data points simulating such progress curves and containing known amounts of superimposed random noise were fit to the Hill equation by (i) direct nonlinear curve-fitting of raw data, and (ii) a tangent-slope technique in which the raw data are numerically differentiated, transformed into substrate versus velocity data, and then analyzed as linear plots. Both integral and differential procedures provided accurate and precise estimates of the Hill parameters (S_0.5, V, and n) from single reaction mixtures. However, the tangent-slope method was at least 10-fold faster to compute and was not dependent on accurate initial guesses of the Hill parameters or integration of the rate equation. With the tangent-slope method, the optimal number of data points used in calculating tangent slopes was found to be 9 or 11. The reliability of the Hill parameters determined with the tangent-slope method was relatively insensitive to the maximum substrate concentration over a range of $\text{S}{}_{\mathrm{<mtext>max</mtext>}}\text{S}{}_{0.5}$ of 1.5 to 10; the optimal value was 3. Through further analysis of simulated data, it was found that slow enzyme inactivation (<4% loss during the assay), or product competitive inhibition (maximum product concentration < 30% of the inhibitor dissociation constant) does not produce serious errors in the Hill parameters. Methods are presented to detect and distinguish enzyme inactivation and product competitive inhibition. It is suggested that continuous addition methodology combined with tangent-slope analysis provides the basis for a flexible system for kinetic characterization of enzymes which has wider applicability and other advantages over multicuvette or conventional progress curve methodology. A major advantage in contrast to the progress curve approach is that product accumulation and associated product effects are lowest at lower substrate concentrations.     

Internal pilots for observational studies

Study planning often involves selecting an appropriate sample size. Power calculations require specifying an effect size and estimating “nuisance” parameters, e.g. the overall incidence of the outcome. For observational studies, an additional source of randomness must be estimated: the rate of the exposure. A poor estimate of any of these parameters will produce an erroneous sample size. Internal pilot (IP) designs reduce the risk of this error ‐ leading to better resource utilization ‐ by using revised estimates of the nuisance parameters at an interim stage to adjust the final sample size. In the clinical trials setting, where allocation to treatment groups is pre‐determined, IP designs have been shown to achieve the targeted power without introducing substantial inflation of the type I error rate. It has not been demonstrated whether the same general conclusions hold in observational studies, where exposure‐group membership cannot be controlled by the investigator. We extend the IP to observational settings. We demonstrate through simulations that implementing an IP, in which prevalence of the exposure can be re‐estimated at an interim stage, helps ensure optimal power for observational research with little inflation of the type I error associated with the final data analysis.     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Design issues for the Michaelis-Menten model

We discuss design issues for the Michaelis-Menten model and use geometrical arguments to find optimal designs for estimating a subset of the model parameters, or a linear combination of the parameters. We propose multiple-objective optimal designs when the parameters have different levels of interest to the researcher. In addition, we compare six commonly used sequence designs in the biological sciences for estimating parameters and, propose optimal choices for the parameters for geometric designs using closed-form efficiency formulas.     

Discrete design of enzyme kinetic experiments

The usefulness of discrete designs in enzyme kinetics as an alternative to continuous designs is discussed in this paper, focusing on designs satisfying the D-optimality criterion. This study has been carried out using a program called DODID, specifically devised for this purpose, which is available by request to the authors. The results presented in this paper show that the relative efficiency of the D-optimal discrete designs with respect to the continuous ones increases rapidly when increasing the number of possible values for the control variables. Relative efficiencies higher than 0.98 are achieved when using 20 possible values for each variable. The power of the tools provided by the computational approach of this work is proved by the analysis made on the robustness of different designs for estimating the kinetic parameters when a wrong assumption on the error structure has been made. The robustness of the designs made assuming medium constant error (error variance proportional to the true response) is thus confirmed. A comparative study of several discriminating designs is also presented. The results obtained show that the designs produced by adding the optimal discrete designs corresponding to both candidate models plus the point where the weighted difference between the predicted values is maximum, is a good choice when designing an experiment for discrimination.     

Quantifying the relative importance of experimental data points in parameter estimation

Background

Ordinary differential equations (ODEs) are often used to understand biological processes. Since ODE-based models usually contain many unknown parameters, parameter estimation is an important step toward deeper understanding of the process. Parameter estimation is often formulated as a least squares optimization problem, where all experimental data points are considered as equally important. However, this equal-weight formulation ignores the possibility of existence of relative importance among different data points, and may lead to misleading parameter estimation results. Therefore, we propose to introduce weights to account for the relative importance of different data points when formulating the least squares optimization problem. Each weight is defined by the uncertainty of one data point given the other data points. If one data point can be accurately inferred given the other data, the uncertainty of this data point is low and the importance of this data point is low. Whereas, if inferring one data point from the other data is almost impossible, it contains a huge uncertainty and carries more information for estimating parameters.

Results

G1/S transition model with 6 parameters and 12 parameters, and MAPK module with 14 parameters were used to test the weighted formulation. In each case, evenly spaced experimental data points were used. Weights calculated in these models showed similar patterns: high weights for data points in dynamic regions and low weights for data points in flat regions. We developed a sampling algorithm to evaluate the weighted formulation, and demonstrated that the weighted formulation reduced the redundancy in the data. For G1/S transition model with 12 parameters, we examined unevenly spaced experimental data points, strategically sampled to have more measurement points where the weights were relatively high, and fewer measurement points where the weights were relatively low. This analysis showed that the proposed weights can be used for designing measurement time points.

Conclusions

Giving a different weight to each data point according to its relative importance compared to other data points is an effective method for improving robustness of parameter estimation by reducing the redundancy in the experimental data.

     

Optimal design of longitudinal data analysis using generalized estimating equation models

Longitudinal studies are often applied in biomedical research and clinical trials to evaluate the treatment effect. The association pattern within the subject must be considered in both sample size calculation and the analysis. One of the most important approaches to analyze such a study is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which “working correlation structure” is introduced and the association pattern within the subject depends on a vector of association parameters denoted by ρ. The explicit sample size formulas for two‐group comparison in linear and logistic regression models are obtained based on the GEE method by Liu and Liang. For cluster randomized trials (CRTs), researchers proposed the optimal sample sizes at both the cluster and individual level as a function of sampling costs and the intracluster correlation coefficient (ICC). In these approaches, the optimal sample sizes depend strongly on the ICC. However, the ICC is usually unknown for CRTs and multicenter trials. To overcome this shortcoming, Van Breukelen et al. consider a range of possible ICC values identified from literature reviews and present Maximin designs (MMDs) based on relative efficiency (RE) and efficiency under budget and cost constraints. In this paper, the optimal sample size and number of repeated measurements using GEE models with an exchangeable working correlation matrix is proposed under the considerations of fixed budget, where “optimal” refers to maximum power for a given sampling budget. The equations of sample size and number of repeated measurements for a known parameter value ρ are derived and a straightforward algorithm for unknown ρ is developed. Applications in practice are discussed. We also discuss the existence of the optimal design when an AR(1) working correlation matrix is assumed. Our proposed method can be extended under the scenarios when the true and working correlation matrix are different.     

Parameter estimation in stochastic biochemical reactions

Gene regulatory, signal transduction and metabolic networks are major areas of interest in the newly emerging field of systems biology. In living cells, stochastic dynamics play an important role; however, the kinetic parameters of biochemical reactions necessary for modelling these processes are often not accessible directly through experiments. The problem of estimating stochastic reaction constants from molecule count data measured, with error, at discrete time points is considered. For modelling the system, a hidden Markov process is used, where the hidden states are the true molecule counts, and the transitions between those states correspond to reaction events following collisions of molecules. Two different algorithms are proposed for estimating the unknown model parameters. The first is an approximate maximum likelihood method that gives good estimates of the reaction parameters in systems with few possible reactions in each sampling interval. The second algorithm, treating the data as exact measurements, approximates the number of reactions in each sampling interval by solving a simple linear equation. Maximising the likelihood based on these approximations can provide good results, even in complex reaction systems.     

Sampling designs of insect time series data: are they all irregularly spaced?

Time series data are commonly obtained by trapping over a standardized period of time, for example daily or weekly. In this paper we present evidence that such sampling designs are inherently irregularly spaced due to the varying developmental rates and population parameters caused by changing temperatures during a sampling season. We modeled an exponentially growing population based on stable fly population growth rates, and then compare different sampling regimes to determine which produces the best estimate of population growth rate. These results are then compared to field data based on weekly sampling at three dairy farms in Ontario over two summers. Transforming catch numbers (N) to ln(N)/(number of degree days within the sampling period) corrects for the irregular spaced sampling in these data. These results support the use of measuring population parameters such as population growth rates in terms of degree days.     

Optimal designs for estimating and testing interaction among multiple loci in complex traits by a Gibbs sampler

A simulation study was conducted to provide a practical guideline for experimental designs with the Bayesian approach using Gibbs sampling (BAGS), a recently developed method for estimating interaction among multiple loci. Various data sets were simulated from combinations of number of loci, within-genotype variance, sample size, and balance of design. Mean square prediction error (MSPE) and empirical statistical power were obtained from estimating and testing the posterior mean estimate of combination genotypic effect. Simultaneous use of both MSPE and power was suggested to find an optimal design because their correlation estimate (− 0.8) would not be large enough to ignore either of them. The optimal sample sizes with MSPE > 2.0 and power > 0.8 with the within-genotype variance of 30 were 135, 675, and > 8100 for 2-, 3-, and 4-locus unbalanced data. The BAGS was suggested for interaction effects among limited number (4 or less) of loci in practice. A practical guideline for determining an optimal sample size with a given power or vise versa is provided for BAGS.     

不同采样设计评估鱼类群落效果比较

鱼类群落生态学研究结果的准确性很大程度上依赖于采样设计的合理性和准确性,正确的采样调查设计不仅可以降低调查成本,其结果也对渔业资源的评估或者管理起到相当重要的作用.本文利用计算机模拟定点采样、简单随机采样和分层采样,比较了3种采样设计的采样效果、相对误差及相对偏差.结果表明： 定点采样设计的采样效果 (采样效果平均值为3.37)要弱于简单随机采样和分层随机采样 (采样效果平均值为0.961).3种采样设计中,分层采样设计在鱼类群落丰富度评估时表现最好,其采样效果、相对误差和相对偏差表现最佳.随着采样数的增加,分层采样设计的采样效果有所下降,但其采样精度提高.     

Enzymatic assays of L-ornithine and L-delta 1-pyrroline-5-carboxylate in tissues, and orniathine-load test in human subjects

The efficiencies of estimates obtained from the direct linear plot (A. Cornish-Bowden and R. Eisenthal, 1978, Biochem. Biophys, Acta, 523, 268) are shown to be dependent on the spacing of substrate concentrations. When substrate values are harmonically spaced, the direct linear plot should not be used. The nonparametric confidence limits based on the direct linear plot are accurate in their confidence coefficient, but their efficiencies are shown to be dependent on substrate spacing. Harmonic spacing is, in general, a more efficient experimental design for estimating K_m than arithmetic spacing when the appropriate estimation methods are used. If assumptions about the error structure cannot be made, the best procedure for estimating K_m is to have harmonic spacing of substrate values and use weighted least squares for estimation. The most accurate and precise estimation of enzyme kinetic parameters requires knowledge of the error structure and utilization of the appropriate nonlinear regression.     

Optimal designs for a multiresponse Emax model and efficient parameter estimation

The aim of dose finding studies is sometimes to estimate parameters in a fitted model. The precision of the parameter estimates should be as high as possible. This can be obtained by increasing the number of subjects in the study, N, choosing a good and efficient estimation approach, and by designing the dose finding study in an optimal way. Increasing the number of subjects is not always feasible because of increasing cost, time limitations, etc. In this paper, we assume fixed N and consider estimation approaches and study designs for multiresponse dose finding studies. We work with diabetes dose–response data and compare a system estimation approach that fits a multiresponse Emax model to the data to equation‐by‐equation estimation that fits uniresponse Emax models to the data. We then derive some optimal designs for estimating the parameters in the multi‐ and uniresponse Emax model and study the efficiency of these designs.     

基于湿度分布特征的小尺度土壤碳通量空间采样策略

由于土壤碳通量的空间异质性很强,传统的随机抽样方法无法对区域土壤碳通量进行准确估算,而多点采样需耗费大量的人力及设备成本,因此确定适当的采样点数量及分布策略对于区域土壤碳通量的测算非常重要。提出一种基于湿度空间分布特征的小尺度土壤碳通量空间采样策略:首先利用无线传感网密集测量区域的土壤湿度,根据湿度数据的空间分布特征划分监测区域,通过Hammond Mc Cullagh方程计算各子区域内的最优采样点数量,最终确定整个监测区域的空间采样点部署策略。提出的方法考虑了各子区域间土壤碳通量空间分布的差异,使得采样点的部署位置与土壤碳通量的分布具有较好的相关性。研究结果证明:土壤碳通量部署策略能够获得比均匀部署策略、随机部署策略更高的区域土壤碳通量估算准确度。     

Design and analysis of protein binding experiments

The design and analysis of protein binding experiments for obtaining precise parameter estimates for a one-site and a two-site model treating fu, the fraction unbound as the experimentally determined quantity was investigated. Total drug concentrations were chosen at which the binding isotherm is determined to yield the most information about the parameters under study. The D-optimization information criterion was used to achieve this although other criteria are also discussed. For both the one-site and the two-site models the number of design points was always equal to the number of parameters being estimated. The results arrived at when dealing with constant variance and unconstrained total drug concentration were rather unique in that in most of the cases studied, all the optimal design points were away from the boundary conditions. For constant relative variance and unconstrained total drug concentrations, one of the design points was always placed at the smallest possible value of fu, the fraction unbound. For the one-site model the second point was always given by K(-1) + nP. The optimal designs arrived at lead to lower theoretical coefficients of variation in the parameters than the corresponding conventional ones. Simulated experiments supported these theoretical findings for both the one-site and the two-site models. For the one-site model, results from nonlinear regression were compared with Scatchard analysis and the optimal designs were also optimal in Scatchard space. We also show that using Scatchard analysis with the conventional strategy leads to poorly determined estimates particularly when the number of observations is low.     

A model-based meta-analysis of monoclonal antibody pharmacokinetics to guide optimal first-in-human study design

The objectives of this retrospective analysis were (1) to characterize the population pharmacokinetics (popPK) of four different monoclonal antibodies (mAbs) in a combined analysis of individual data collected during first-in-human (FIH) studies and (2) to provide a scientific rationale for prospective design of FIH studies with mAbs. The data set was composed of 171 subjects contributing a total of 2716 mAb serum concentrations, following intravenous (IV) and subcutaneous (SC) doses. mAb PK was described by an open 2-compartment model with first-order elimination from the central compartment and a depot compartment with first-order absorption. Parameter values obtained from the popPK model were further used to generate optimal sampling times for a single dose study. A robust fit to the combined data from four mAbs was obtained using the 2-compartment model. Population parameter estimates for systemic clearance and central volume of distribution were 0.20 L/day and 3.6 L with intersubject variability of 31% and 34%, respectively. The random residual error was 14%. Differences (> 2-fold) in PK parameters were not apparent across mAbs. Rich designs (22 samples/subject), minimal designs for popPK (5 samples/subject), and optimal designs for non-compartmental analysis (NCA) and popPK (10 samples/subject) were examined by stochastic simulation and estimation. Single-dose PK studies for linear mAbs executed using the optimal designs are expected to yield high-quality model estimates, and accurate capture of NCA estimations. This model-based meta-analysis has determined typical popPK values for four mAbs with linear elimination and enabled prospective optimization of FIH study designs, potentially improving the efficiency of FIH studies for this class of therapeutics.     

Optimization of cDNA microarray experimental designs using an evolutionary algorithm

The cDNA microarray is an important tool for generating large datasets of gene expression measurements.An efficient design is critical to ensure that the experiment will be able to address relevant biologicalquestions. Microarray experimental design can be treated as a multicriterion optimization problem. For thisclass of problems evolutionary algorithms (EAs) are well suited, as they can search the solution space andevolve a design that optimizes the parameters of interest based on their relative value to the researcher undera given set of constraints. This paper introduces the use of EAs for optimization of experimental designs ofspotted microarrays using a weighted objective function. The EA and the various criteria relevant to designoptimization are discussed. Evolved designs are compared with designs obtained through exhaustive searchwith results suggesting that the EA can find just as efficient optimal or near-optimal designs within atractable timeframe.     

A model-based meta-analysis of monoclonal antibody pharmacokinetics to guide optimal first-in-human study design

The objectives of this retrospective analysis were (1) to characterize the population pharmacokinetics (popPK) of four different monoclonal antibodies (mAbs) in a combined analysis of individual data collected during first-in-human (FIH) studies and (2) to provide a scientific rationale for prospective design of FIH studies with mAbs. The data set was composed of 171 subjects contributing a total of 2716 mAb serum concentrations, following intravenous (IV) and subcutaneous (SC) doses. mAb PK was described by an open 2-compartment model with first-order elimination from the central compartment and a depot compartment with first-order absorption. Parameter values obtained from the popPK model were further used to generate optimal sampling times for a single dose study. A robust fit to the combined data from four mAbs was obtained using the 2-compartment model. Population parameter estimates for systemic clearance and central volume of distribution were 0.20 L/day and 3.6 L with intersubject variability of 31% and 34%, respectively. The random residual error was 14%. Differences (> 2-fold) in PK parameters were not apparent across mAbs. Rich designs (22 samples/subject), minimal designs for popPK (5 samples/subject), and optimal designs for non-compartmental analysis (NCA) and popPK (10 samples/subject) were examined by stochastic simulation and estimation. Single-dose PK studies for linear mAbs executed using the optimal designs are expected to yield high-quality model estimates, and accurate capture of NCA estimations. This model-based meta-analysis has determined typical popPK values for four mAbs with linear elimination and enabled prospective optimization of FIH study designs, potentially improving the efficiency of FIH studies for this class of therapeutics.     

Design of validation studies for estimating the odds ratio of exposure-disease relationships when exposure is misclassified

We compared several validation study designs for estimating the odds ratio of disease with misclassified exposure. We assumed that the outcome and misclassified binary covariate are available and that the error-free binary covariate is measured in a subsample, the validation sample. We considered designs in which the total size of the validation sample is fixed and the probability of selection into the validation sample may depend on outcome and misclassified covariate values. Design comparisons were conducted for rare and common disease scenarios, where the optimal design is the one that minimizes the variance of the maximum likelihood estimator of the true log odds ratio relating the outcome to the exposure of interest. Misclassification rates were assumed to be independent of the outcome. We used a sensitivity analysis to assess the effect of misspecifying the misclassification rates. Under the scenarios considered, our results suggested that a balanced design, which allocates equal numbers of validation subjects into each of the four outcome/mismeasured covariate categories, is preferable for its simplicity and good performance. A user-friendly Fortran program is available from the second author, which calculates the optimal sampling fractions for all designs considered and the efficiencies of these designs relative to the optimal hybrid design for any scenario of interest.