Structured population model

A model that accounts for individual variation in one or more characteristics - such as age, size, or class - is often called a structured population model, and the particular characteristics allowed to vary are called the structure variables. In this chapter, a size-structured population model is presented. There is a large and rapidly developing literature on structured... 

Vol. 1936 P. Magal, S. Ruan (Eds.), Structured Population Models in Biology and Epidemiology (2008)... 

With time, the relevance of in vivo mutations with respect to structural in vitro data will provide a detailed population model of the receptors of family A, which share structnral similarity to rhodopsin. Comparison of these data with the family B GPCRs, the glncagon-like receptors (255,256), and the family C receptors, such as the CASR, may provide the detail necessary to model how GPCR structure and function are altered by common genetic variants. 

Using the NLME, the population model contains three components the structural model, the statistical model and - if necessary - the (integrated) patient covariate model (Fig. 17.2). 

Statistical thermodynamic treatments of defect populations have lead to an explanation of existence of grossly nonstoichio-metric crystals in terms of microdomains of ordered structure. The model considers that the nonstoichiometric matrix is made up of a mosaic of small regions of ordered defect-free structures, the microdomains. To account for stoichiometric variation, one can postulate that at least two microdomains with different compositions occur. However, compositional change might simply arise at the surface of the domain. For example, if there are compositionally identical microdomains, one of which is bounded by an anion surface and one by a cation surface, variation in the two populations can give rise to compositional variation. In a strict sense, as each microdomain is ordered, the concept of a defect is redundant, except for... 

Assume an experiment in which a group of subjects selected to represent a spectrum of severity of some condition (e.g., renal insufficiency) is given a dose of drug, and drug concentrations are measured in blood samples collected at intervals after dosing. The structural kinetic models used when performing a population analysis do not differ at all from those used for analysis of data from an individual patient. One still needs a model for the relationship of concentration to dose and time, and this relationship does not depend on whether the fixed-effect parameter changes... 

Any effects on populations may ultimately be manifested as effects on communities because, by definition, communities are collections of interacting populations of several species (e.g., an aquatic community may consist of populations of fish, worms, plants, insects). Individual populations within a community may interact by competing for resources (food, habitat, etc.) or by predator/prey relationships. Environmental contaminants can affect the structure of communities as well as the interactions of species within them. For example, it is well known that exposure to chemicals may cause a reduction in community diversity (e.g., relative number of species), and changes in community composition. In addition, the trophic structure of fish and invertebrate communities may also be affected by exposure to anthropogenic chemicals. Changes in community structure and diversity may be determined by field sampling or manipulative studies. Alternatively, computer simulations using food web or linked population models may be used to assess community-level effects. 

Barnthouse and colleagues (Barnthouse 1993 Bamthouse et al. 1990, 1989) have explored the use of conventional population models to explore the interactions among toxicity, predation, and harvesting pressure for fish populations. These studies are excellent illustrations of the use of population models in the estimation of toxicant impacts. These models employed the use of information concerning the life history and age structure of the organism being modeled. 

Life History diagram for an age-structured population. The numbers of organisms in the population at time t, is dependent on the numbers of the one-year younger-age class of the year before and the survivorship percentage from t0 to tj. The numbers are also dependent upon the number of offspring from the previous year surviving up to age 1. This is a general model for many plant and animal populations. 

Goodness-of-Fit. It is implied in steps 2 to 6 above that diagnostic plots (e.g., weighted residual versus time, weighted residual versus predicted observations, population observed versus predicted concentrations, individual observed versus predicted concentrations) and a test statistic such as the likelihood ratio test would be used in arriving at the base model (see Section 8.6.1.1 for goodness of fit). Once the base model (with optimized structural and variance models) has been obtained, the next step in the PM model identification process is the development of the population model. 

For example, a two parallel first-order absorption type can be one where the first process starts without a lag time and the second process starts with a lag time (ALAG2). Without lag time this absorption is called simultaneous first-order absorption. In population modeling, the presence of more than one lag time often makes the model difficult to identify and in case the between-subject variability in lag time is important, there might be an identifiability problem for the population absorption model. Prior information, from preclinic or previous PK studies, may help in defining the structure of the model. 

Simulations were conducted under the following population model motivated by the broncodilation data from application Example 1 in Section 16.7.1. Equation (16.2) was used as the structural model, lognormal interindividual variability terms were placed on (eo, Emax, ed50), and the residual variability was assumed log-normally distributed. Estimation was carried out in NONMEM. Fixed effects were estimated as (f, eo, Emax, ed5o) = (o.V9, 0.92, 60.3, 2.99). The variance-covariance matrix of interindividual variability terms was estimated as... 

Acceptable population models resulted in successful minimization, with at least three significant digits for any parameter, a successful estimation of the covariance, and the absolute value of last iteration gradients greater than 0.001 but smaller than 100. Confidence intervals of structural parameters should not include value zero correlation between any two structural parameters should never be greater than 0.95. Acceptable models should not lead to trends in the distribution of weighted residuals versus model predictions and versus independent variable. They should not be oversensitive to initial estimates nor lead to differences between the population parameters and the corresponding medians of individual POSTHOC parameters. The predictions versus observations data should be evenly distributed around the unit line. If constraints were applied on parameters, no final estimate should be equal to one of the boundaries. 

In an age-structured matrix model, the state of a population at a certain time t is described by a vector consisting of m elements, each of the elements describing the number of individuals in one of m age classes that is, nl is the number of individuals in age class 1, and so forth ... 

Long-term effects of sublethal effects Often the sublethal effects of a PPP are described well on the level of individuals, whereas the likely impact on population level is less well known. In some cases, the effects of density dependence, for example, will make the effects much less pronounced on the population level than on the individual level. In other cases, the sublethal effects may have pronounced effects on population density and structure. Ecological models can be used to put the effects observed on the individual level into the more ecologically relevant population level. 

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