Nonlinear mixed approach

Branch and bound (BB) is a class of methods for linear and nonlinear mixed-integer programming. If carried to completion, it is guaranteed to find an optimal solution to linear and convex nonlinear problems. It is the most popular approach and is currently used in virtually all commercial MILP software (see Chapter 7). 

There are two common methods for obtaining estimates of the fixed effects (the mean) and the variability the two-stage approach and the nonlinear, mixed-effects modeling approach. The two-stage approach involves multiple measurements on each subject. The nonlinear, mixed-effects model can be used in situations where extensive measurements cannot or will not be made on all or any of the subjects. 

Pharmacokinetic/pharmacodynamic model using nonlinear, mixed-effects model in two compartment, best described time course of concentration strong correlation with creatinine clearance predicted concentration at the efi ect site and in reduction of heart rate during atrial fibrillation using population kinetic approach... 

Section 10.2 describes the MINLP approach of Kokossis and Floudas (1990) for the synthesis of isothermal reactor networks that may exhibit complex reaction mechanisms. Section 10.3 discusses the synthesis of reactor-separator-recycle systems through a mixed-integer nonlinear optimization approach proposed by Kokossis and Floudas (1991). The problem representations are presented and shown to include a very rich set of alternatives, and the mathematical models are presented for two illustrative examples. Further reading material in these topics can be found in the suggested references, while the work of Kokossis and Floudas (1994) presents a mixed-integer optimization approach for nonisothermal reactor networks. 

Various methods are available to estimate population parameters, but today the nonlinear mixed effects modeling approach is the most common one employed. Population analyses have been performed for mAbs such as basiliximab, daclizu-mab and trastuzumab, as well as several others in development, including clenolixi-mab and sibrotuzumab. Population pharmacokinetic models comprise three submodels the structural the statistical and covariate submodels (Fig. 3.13). Their development and impact for mAbs will be discussed in the following section. 

The models are built similar to the descriptive mechanism-based PD models. Most of them are also estimated by the nonlinear mixed effects modeling approach considering interindividual and residual variability. In addition, covariates influencing the disease progression can also be investigated. 

It is important to be able to test the FM laser spectrum to see how closely it approaches that of an ideal FM oscillator. Mode analysis of the FM spectrum is too tedious. We have developed methods of testing the quality of the FM spectrum which involve nonlinear mixing of the FM beams. If we phase shift a part of the FM beam by 0 we obtain a modified field,... 

Kocis, G. R., A Mixed-Integer Nonlinear Programming Approach to Structural Flowsheet Optimization. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 1988. 

The population pharmacokinetic aproach assesses the impact of various covariates on the pharmacokinetic of a drug. Nonlinear mixed effects modeling may be used to model the relationship between various covariates and pharmacokinetic parameters. Age or age group may be one of the covariates. This type of approach has its advantages as it involves assessment of the effect of age on the pharmacokinetics in the target population. 

Steimer, J.L. Mallet, A. Golmard, J.L. Boisvieux, J.F. Alternative approaches to estimation of population pharmacokinetic parameters comparison with nonlinear mixed-effect model. Drug Metab. Rev. 1984,15 (1-2), 265-292. 

The number of samples per subject used for this approach is typically small, ranging from one to six. As does the pooled analysis technique, nonlinear mixed-effects modeling approaches analyze the data of all individuals at once, but take the interindividual random effects structure into account. This ensures that confounding correlations and imbalance that may occur in observational data are properly accounted for. 

The first attempt at estimating interindividual PK variability without neglecting the difficulties (data imbalance, sparse data, subject-specific dosing history, etc.) associated with data from patients undergoing drug therapy was made by Sheiner and co-workers (44) using the nonlinear mixed-effects model approach. The vector 6 of population characteristics is composed of aU quantities of the first two moments of the distribution of the parameters the mean values (fixed effects) and the elements of the variance-covariance matrix that characterize random effects (19, 20, 45-47). 

Most of the nonlinear mixed effects modeling methods estimate the parameters by the maximum likelihood approach. The probability of the data under the model is written as a function of the model parameters, and parameter estimates are chosen to maximize this probability. This amounts to asserting that the best parameter estimates are those that render the observed data more probable than they would be under any other set of parameters. 

K. M. Higgins, M. Davidian, and D. M. Giltinan, A two-step approach to measurement error in time-varying covariates in nonlinear mixed-effects models, with apphcation to IGF-1 pharmacokinetics. J Am Stat Assoc 92 436 48 (1997). 

Until recently, no method of comparing nonhierarchical regression models has been available. The bootstrap has been proposed because it may estimate the distribution of a statistic under weaker conditions than do the traditional approaches. In general, for nonlinear mixed effects models that are not hierarchical, the preferred model has simply been selected as that with the lower objective function (2). A more rational approach has been proposed for comparing nonhierarchical models, which is an extension of Efron s method (2, 30). The test statistic is the difference between the objective functions (log-likelihood difference—LED) of the two nonhierarchical models. The method consists of constructing the confidence interval for the LLDs. 

NONLINEAR MIXED EFFECTS MODELING APPROACH TO THE ANALYSIS OF NONRANDOMLY CENSORED ORDERED CATEGORICAL LONGITUDINAL DATA FROM ANALGESIC TRIALS... 

In the subsequent sections we present the nonlinear mixed effects model approach for analyzing analgesic data and apply it to simulated analgesic study data. 

The approach involves a semimechanistic or mechanistic model that describes the joint probability of the time of remedication and the pain relief score (which is related to plasma drug concentrations). This joint probability can be written as the product of the conditional probability of the time of remedication, given the level of pain relief and the probability of the pain relief score. First, a population pharmacokinetic (PK) model is developed using the nonlinear mixed effects modeling approach (16-19) (see also Chapters 10 and 14 of this book). With this approach both population (average) and random (inter- and intraindividual) effects parameters are estimated. When the PK model is linked to an effect (pharmacodynamic (PD) model), the effect site concentration (C ) as defined by Sheiner et al. (20) can be obtained. The effect site concentration is useful in linking dose to pain relief and subsequently to the decision to remedicate. 

In characterizing hypnograms using the nonlinear mixed effects modeling approach, it is important to test for correlations between r values of one transition model and those from another model using individual estimates of r values. Correlations detected should be accounted for in the model. Correlations with correlation coefficient (r) > 0.75 are termed high correlations and correlations with r values between 0.5 and 0.75 are moderate correlations (25). Not accounting for such correlations may yield parameter estimates with poor precision. 

The aim of this chapter is to equip the pharmacometrician with sufficient theory and application to confidently approach the PK/PD-based analysis of count data and thus derive the maximum return on investment from clinical study data. Section 27.2 provides a motivating example and Section 27.3 presents relevant definitions and theory. Section 27.4 applies the theory to the example and introduces diagnostics methods. Throughout the chapter, the focus is on population approaches using nonlinear mixed effects models. Code segments of NONMEM control files are presented in the appendix. Mixed effects analysis methodology is described in detail in Chapter 4 of this text. 

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