Robust and Efficient Designs for the Michaelis–Menten Model

An equivalence theorem for design optimality with respect to a multi-objective criterion

Maxi-min efficiency criteria are a kind of multi-objective criteria, since they enable us to take into consideration several tasks expressed by different component-wise criteria. However, they are difficult to manage because of their lack of differentiability. As a consequence, maxi-min efficiency designs are frequently built through heuristic and ad hoc algorithms, without the possibility of checking for their optimality. The main contribution of this study is to prove that the maxi-min efficiency optimality is equivalent to a Bayesian criterion, which is differentiable. In addition, we provide an analytic method to find the prior probability associated with a maxi-min efficient design, making feasible the application of the equivalence theorem. Two illustrative examples show how the proposed theory works.

Optimal Design of Experiments for Hybrid Nonlinear Models, with Applications to Extended Michaelis–Menten Kinetics

Biochemical mechanism studies often assume statistical models derived from Michaelis–Menten kinetics, which are used to approximate initial reaction rate data given the concentration level of a single substrate. In experiments dealing with industrial applications, however, there are typically a wide range of kinetic profiles where more than one factor is controlled. We focus on optimal design of such experiments requiring the use of multifactor hybrid nonlinear models, which presents a considerable computational challenge. We examine three different candidate models and search for tailor-made D- or weighted-A-optimal designs that can ensure the efficiency of nonlinear least squares estimation. We also study a compound design criterion for discriminating between two candidate models, which we recommend for design of advanced kinetic studies.

Supplementary materials accompanying this paper appear on-line

Optimal two-stage sampling for mean estimation in multilevel populations when cluster size is informative

To estimate the mean of a quantitative variable in a hierarchical population, it is logistically convenient to sample in two stages (two-stage sampling), i.e. selecting first clusters, and then individuals from the sampled clusters. Allowing cluster size to vary in the population and to be related to the mean of the outcome variable of interest (informative cluster size), the following competing sampling designs are considered: sampling clusters with probability proportional to cluster size, and then the same number of individuals per cluster; drawing clusters with equal probability, and then the same percentage of individuals per cluster; and selecting clusters with equal probability, and then the same number of individuals per cluster. For each design, optimal sample sizes are derived under a budget constraint. The three optimal two-stage sampling designs are compared, in terms of efficiency, with each other and with simple random sampling of individuals. Sampling clusters with probability proportional to size is recommended. To overcome the dependency of the optimal design on unknown nuisance parameters, maximin designs are derived. The results are illustrated, assuming probability proportional to size sampling of clusters, with the planning of a hypothetical survey to compare adolescent alcohol consumption between France and Italy.

Data science for the scientific life cycle

Data science can be incorporated into every stage of a scientific study. Here we describe how data science can be used to generate hypotheses, to design experiments, to perform experiments, and to analyse data. We also present our vision for how data science techniques will be an integral part of the laboratory of the future.

Standardized maximim D-optimal designs for enzyme kinetic inhibition models

Locally optimal designs for nonlinear models require a single set of nominal values for the unknown parameters. An alternative is the maximin approach that allows the user to specify a range of values for each parameter of interest. However, the maximin approach is difficult because we first have to determine the locally optimal design for each set of nominal values before maximin types of optimal designs can be found via a nested optimization process. We show that particle swarm optimization (PSO) techniques can solve such complex optimization problems effectively. We demonstrate numerical results from PSO can help find, for the first time, formulae for standardized maximin D-optimal designs for nonlinear model with 3 or 4 parameters on the compact and nonnegative design space. Additionally, we show locally and standardized maximin D-optimal designs for inhibition models are not necessarily supported at a minimum number of points. To facilitate use of such designs, we create a web-based tool for practitioners to find tailor-made locally and standardized maximin optimal designs.

Maximin optimal designs for cluster randomized trials

We consider design issues for cluster randomized trials (CRTs) with a binary outcome where both unit costs and intraclass correlation coefficients (ICCs) in the two arms may be unequal. We first propose a design that maximizes cost efficiency (CE), defined as the ratio of the precision of the efficacy measure to the study cost. Because such designs can be highly sensitive to the unknown ICCs and the anticipated success rates in the two arms, a local strategy based on a single set of best guesses for the ICCs and success rates can be risky. To mitigate this issue, we propose a maximin optimal design that permits ranges of values to be specified for the success rate and the ICC in each arm. We derive maximin optimal designs for three common measures of the efficacy of the intervention, risk difference, relative risk and odds ratio, and study their properties. Using a real cancer control and prevention trial example, we ascertain the efficiency of the widely used balanced design relative to the maximin optimal design and show that the former can be quite inefficient and less robust to mis-specifications of the ICCs and the success rates in the two arms.

A MINE alternative to D-optimal designs for the linear model

Doing large-scale genomics experiments can be expensive, and so experimenters want to get the most information out of each experiment. To this end the Maximally Informative Next Experiment (MINE) criterion for experimental design was developed. Here we explore this idea in a simplified context, the linear model. Four variations of the MINE method for the linear model were created: MINE-like, MINE, MINE with random orthonormal basis, and MINE with random rotation. Each method varies in how it maximizes the MINE criterion. Theorem 1 establishes sufficient conditions for the maximization of the MINE criterion under the linear model. Theorem 2 establishes when the MINE criterion is equivalent to the classic design criterion of D-optimality. By simulation under the linear model, we establish that the MINE with random orthonormal basis and MINE with random rotation are faster to discover the true linear relation with regression coefficients and observations when . We also establish in simulations with , , and 1000 replicates that these two variations of MINE also display a lower false positive rate than the MINE-like method and additionally, for a majority of the experiments, for the MINE method.

Using Animal Instincts to Design Efficient Biomedical Studies via Particle Swarm Optimization

Particle swarm optimization (PSO) is an increasingly popular metaheuristic algorithm for solving complex optimization problems. Its popularity is due to its repeated successes in finding an optimum or a near optimal solution for problems in many applied disciplines. The algorithm makes no assumption of the function to be optimized and for biomedical experiments like those presented here, PSO typically finds the optimal solutions in a few seconds of CPU time on a garden-variety laptop. We apply PSO to find various types of optimal designs for several problems in the biological sciences and compare PSO performance relative to the differential evolution algorithm, another popular metaheuristic algorithm in the engineering literature.

Design optimality for models defined by a system of ordinary differential equations

Many scientific processes, specially in pharmacokinetics (PK) and pharmacodynamics (PD) studies, are defined by a system of ordinary differential equations (ODE). If there are unknown parameters that need to be estimated, the optimal experimental design approach offers quality estimators for the different objectives of the practitioners. When computing optimal designs the standard procedure uses the linearization of the analytical expression of the ODE solution, which is not feasible when this analytical form does not exist. In this work some methods to solve this problem are described and discussed. Optimal designs for two well-known example models, Iodine and Michaelis-Menten, have been computed using the proposed methods. A thorough study has been done for a specific two-parameter PK model, the biokinetic model of ciprofloxacin and ofloxacin, computing the best designs for different optimality criteria and numbers of points. The designs have been compared according to their efficiency, and the goodness of the designs for the estimation of each parameter has been checked. Although the objectives of the paper are focused on the optimal design field, the methodology can be used as well for a sensitivity analysis of ordinary differential equation systems.

Quantitation of glucose uptake in tumors by dynamic FDG-PET has less glucose bias and lower variability when adjusted for partial saturation of glucose transport

Background

A retrospective analysis of estimates of tumor glucose uptake from 1,192 dynamic 2-deoxy-2-(¹⁸F)fluoro-D-glucose-positron-emission tomography [FDG-PET] scans showed strong correlations between blood glucose and both the uptake rate constant [K_i] and the metabolic rate of glucose [MRGluc], hindering the interpretation of PET scans acquired under conditions of altered blood glucose. We sought a method to reduce this glucose bias without increasing the between-subject or test-retest variability and did this by considering that tissue glucose transport is a saturable yet unsaturated process best described as a nonlinear function of glucose levels.

Methods

Patlak-Gjedde analysis was used to compute K_i from 30-min dynamic PET scans in tumor-bearing mice. MRGluc was calculated by factoring in the blood glucose level and a lumped constant equal to unity. Alternatively, we assumed that glucose consumption is saturable according to Michaelis-Menten kinetics and estimated a hypothetical maximum rate of glucose consumption [MRGluc^MAX] by multiplying K_i and (K_M + [glucose]), where K_M is a half-saturation Michaelis constant for glucose uptake. Results were computed for 112 separate studies of 8 to 12 scans each; test-retest statistics were measured in a suitable subset of 201 mice.

Results

A K_M value of 130 mg/dL was determined from the data based on minimizing the average correlation between blood glucose and the uptake metric. Using MRGluc^MAX resulted in the following benefits compared to using MRGluc: (1) the median correlation with blood glucose was practically zero, and yet (2) the test-retest coefficient of variation [COV] was reduced by 13.4%, and (3) the between-animal COVs were reduced by15.5%. In statistically equivalent terms, achieving the same reduction in between-animal COV while using the traditional MRGluc would require a 40% increase in sample size.

Conclusions

MRGluc appeared to overcorrect tumor FDG data for changing glucose levels. Applying partial saturation correction using MRGluc^MAX offered reduced bias, reduced variability, and potentially increased statistical power. We recommend further investigation of MRGluc^MAX in quantitative studies of tumor FDG uptake.

Curvature-adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models

In pharmacokinetics, compartment models are often used to describe the time course of blood concentration after the administration of a drug. In this article, we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non-sequential design of sampling times of blood drug concentration data in individual pharmacokinetics. The proposed optimal design criterion minimizes the determinant of the mean-squared error matrix of the parameter estimator that is quadratically approximated by the curvature array. Therefore, the proposed criterion considers the intrinsic and parameter-effects nonlinearity underlying the compartment model, and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small.

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.

Optimal designs for Michaelis-Menten kinetic studies

Many reactions in enzymology are governed by the Michaelis-Menten equation. Characterising these reactions requires the estimation of the parameters K(M) and V(max) which determine the Michaelis-Menten equation and this is done by observing rates of reactions at a set of substrate concentrations. The choice of substrate concentrations is investigated by determining Bayesian D-optimal designs for a model in which residuals have a normal distribution with constant variance. Designs which focus on alternative quantities, such as K(M) or the ratio V(max)/K(M) are also considered. The effect on the optimal designs of alternative error distributions is also considered.