Nonlinear Population Model Research Articles

Remifentanil Pharmacokinetics in Obese Versus Lean Patients

Background Remifentanil is a short-acting opioid whose pharmacokinetics have been characterized in detail. However, the impact of obesity on remifentanil pharmacokinetics has not been specifically examined. The goal of this study was to investigate the influence of body weight on remifentanil pharmacokinetics. Methods Twelve obese and 12 matched lean subjects undergoing elective surgery received a 1-min remifentanil infusion after induction of anesthesia. Arterial blood samples were collected for determination of remifentanil blood concentrations. Each subject's pharmacokinetic parameters were estimated by fitting a two-compartment model to the concentration versus time curves. Nonlinear mixed-effects population models examining the influence of lean body mass (LBM) and total body weight (TBW) were also constructed. Clinical simulations using the final population model were performed. Results The obese patient cohort reached substantially higher remifentanil concentrations. The individual pharmacokinetic parameters of a two-compartment model were not significantly different between the obese versus lean cohorts (unless normalized to TBW). The final population model scaled central clearance and the central and peripheral distribution volumes to LBM. The simulations illustrated that remifentanil pharmacokinetics are not grossly different in obese versus lean subjects and that TBW based dosing in obese patients can result in excessively high remifentanil concentrations. Conclusions The essential findings of the study are that remifentanil's pharmacokinetics are not appreciably different in obese versus lean subjects and that remifentanil pharmacokinetic parameters are therefore more closely related to LBM than to TBW. Clinically this means that remifentanil dosing regimens should be based on ideal body weight (or LBM) and not TBW.

A procedure for generating bootstrap samples for the validation of nonlinear mixed-effects population models

A method is presented for automated preparation of bootstrap data samples and their presentation to NONMEM for use in the validation of population pharmacokinetic or pharmacodynamic models, which have been developed with relatively small numbers of subjects. The bootstrap sampling procedure involves the use of an MS-DOS batch file and a script file for use with the public-domain text processing program, AWK, which is a single EXE file (48k in size for the version used in this report). A UNIX version of AWK is also available and UNIX users can adapt the process to their needs. Its use obviates the need for expensive, high-end statistical packages and associated script files written using in-built programming languages. The method can easily be adapted to bootstrap sampling requirements in other biomedical modelling applications.

On the numerical integration of non-local terms for age-structured population models

We formulate explicit second-order finite difference schemes for the numerical integration of non-linear age-dependent population models. These methods have been designed by means of a representation formula for the theoretical solution of the integro-differential equation joint with open quadrature formulae for the numerical approximation of non-local terms. The schemes are analyzed and some numerical experiments are also reported in order to show numerically their accuracy.

Stability conditions for equilibra of nonlinear matrix population models

In this paper we develop stability and instability criteria for equilibria of nonlinear matrix population models in which density dependence (i.e. nonlinearity) arises through a dependence of vital rates on a weighted total population size w. These criteria are based upon the net reproductive number n = n(w) as a function of w and partially address an earlier conjecture concerning the realtionship between stability and the derivative n = n′(w) (Cushing 1988a). Under rather general conditions it is shown that n′(w) > 0 at equilibrium implies instability and hence that n′(w)≤ 0 is necessary for stability. In generaln′(w)< 0 is not sufficient for stability. Conditions under which n′(w) < 0 does imply stability and also given. A variety of applications is given.

A Bayesian Population Model with Hierarchical Mixture Priors Applied to Blood Count Data

Population pharmacokinetic and pharmacodynamic studies require analyzing nonlinear growth curves fit to multiple measurements from study subjects. We propose a class of nonlinear population models with nonparametric second-stage priors for analyzing such data. The proposed models apply a flexible class of mixtures to implement the nonparametric second stage. The discussion is based on a pharmacodynamic study involving longitudinal data consisting of hematologic profiles (i.e., blood counts measured over time) of cancer patients undergoing chemotherapy. We describe a full posterior analysis in a Bayesian framework. This includes prediction of future observations (profiles and end points for new patients), estimation of the mean response function for observed individuals, and inference on population characteristics. The mixture model is specified and given a hyperprior distribution by means of a Dirichlet processes prior on the mixing measure. Estimation is implemented by a combination of various Markov chain Monte Carlo schemes, including a novel independence chain scheme for a logistic regression. The discussion is motivated by a pharmacodynamic case study; however, the concepts are more generally applicable to the wider class of population models.

The effect of periodic habitat fluctuations on a nonlinear insect population model

Laboratory data show that populations of flour beetles (Tribolium), when grown in a periodically fluctuating volume of flour, can exhibit significant increases in numbers above those attained when grown in a constant volume (of the same average). To analyze and explain this phenomenon a discrete stage-structured model of Tribolium dynamics with periodic environmental forcing is introduced and studied. This model is an appropriately modified version of an experimentally validated model for flour beetle populations growing in a constant volume of flour, in which cannibalism rates are assumed inversely proportional to flour volume. This modeling assumption has been confirmed by laboratory experiments. Theorems implying the existence and stability of periodic solutions of the periodically forced model are proved. The time averages of periodic solutions of the forced model are compared with the equilibrium levels of the unforced model (with the same average flour volume). Parameter constraints are determined for which the average population numbers in the periodic environment are greater than (or less than) the equilibrium population numbers in the associated constant environment. Sample parameter estimates taken from the literature show that these constraints are fulfilled. These theoretical results provide an explanation for the experimentally observed increase in flour beetle numbers as a result of periodically fluctuating flour volumes. More generally, these integrated theoretical and experimental results provide the first convincing example illustrating the possibility of increased population numbers in a periodically fluctuating environment.

Dynamics of constant and variable stepsize methods for a nonlinear population model with delay

Hutchinson's equation is a reaction-diffusion model where the quadratic reaction term involves a delay. It is a natural extension of the logistic equation (no diffusion, no delay) and Fisher's equation (no delay), both of which have been used to illustrate the potential for spurious long-term dynamics in numerical methods. For the case where initial conditions and periodic boundary conditions are supplied, we look at the use of central differences in space and either Euler's method or the Improved Euler method in time. Our aim is to investigate the impact of the delay on the long-term behaviour of the scheme. After studying the fixed points of the methods in constant stepsize mode, we consider an adaptive time-stepping approach, using a standard local error control strategy. Applying ideas of Hall (1985) we are able to explain the fine detail of the time-step selection process.

Second Order Schemes for Age-Structured Population Equations

We formulate second order finite difference schemes based on Padé rational approximations for the numerical integration of nonlinear age-dependent population models. The schemes are completely analysed and some numerical experiments are also reported in order to show numerically their accuracy.

An implicit finite difference scheme for the nonlinear size-structured population model

In this paper we consider the "fully nonlinear" size structured population model. We develop an implicit finite difference scheme to approximate the solution of this nonlinear partial differential equation. The convergence of this approximation to a unique bounded variation solution of this model is obtained. Numerical results to an example problem are presented.

Overcompensatory recruitment and generation delay in discrete age-structured population models

The effect of overcompensatory recruitment and the combined effect of overcompensatory recruitment and generation delay in discrete nonlinear age-structured population models is studied. Considering overcompensatory recruitment alone, we present formal proofs of the supercritical nature of bifurcations (both flip and Hopf) as well as an extensive analysis of dynamics in unstable parameter regions. One important finding here is that in case of small and moderate year to year survival probabilities there are large regions in parameter space where the qualitative behaviour found in a general n+1 dimensional model is retained already in a one-dimensional model. Another result is that the dynamics at or near the boundary of parameter space may be very complicated. Generally, generation delay is found to act as a destabilizing effect but its effect on dynamics is by no means unique. The most profound effect occurs in the n-generation delay cases. In these cases there is no stable equilibrium X* at all, but whenever X* small, a stable cycle of period n+1 where the periodic points in the cycle are on a very special form. In other cases generation delay does not alter the dynamics in any substantial way.

Discussion of the Paper by Lee and Nelder

Discussion of the Paper by Lee and Nelder

Unexpected dominance of high frequencies in chaotic nonlinear population models.

Because water has a higher heat capacity than air, large bodies of water fluctuate in temperature more slowly than does the atmosphere. Marine temperature time series are 'redder' than atmospheric temperature time series by analogy to light: in red light, low-frequency variability has greater amplitude than high-frequency variability, whereas in white light all frequencies have the same amplitude. Differences in the relative importance of high-and low-frequency variability in different habitats affect the population dynamics of individual species and the structure of ecological communities. Population dynamics of individual species are thought to be dominated by low-frequency fluctuations, that is, to display reddened fluctuations. Here I report, however, that in eight nonlinear, iterative, deterministic, autonomous, discrete-time population models, some of which have been used to model real biological populations, the power spectral densities of chaotic trajectories are neither white nor reddened but are notably blue, with increasing power at higher frequencies.

Bayesian analysis of linear and non-linear population models by using the Gibbs sampler. : Wakefield J.C., Smith A.F.M., Racine-Poon A., Gelfand A.E., Imperial College of Science, Technology and Medicine, London, UK, Ciba-Geigy AG, Basle, Switzerland, University of Connecticut, Storrs, USA, fJournal of the Royal Statistical Society: Applied Statistics, Vol. 43, nr. 1, 1994, pp. 201–221.

Bayesian analysis of linear and non-linear population models by using the Gibbs sampler. : Wakefield J.C., Smith A.F.M., Racine-Poon A., Gelfand A.E., Imperial College of Science, Technology and Medicine, London, UK, Ciba-Geigy AG, Basle, Switzerland, University of Connecticut, Storrs, USA, fJournal of the Royal Statistical Society: Applied Statistics, Vol. 43, nr. 1, 1994, pp. 201–221.

Bayesian Analysis of Linear and Non-Linear Population Models by Using the Gibbs Sampler

Abstract : A fully Bayesian analysis of linear and nonlinear population models has previously been unavailable, as a consequence of the seeming impossibility of performing the necessary numerical Integrations in the complex multi- parameter structures typically arising in such models. It is demonstrated that, for a variety of linear and nonlinear population models, a fully Bayesian analysis can be implemented in a straightforward manner using the Gibbs sampler. The approach is illustrated with examples involving challenging problems of outliers and mean-variance relationships in population modelling.

Maximum sustainable yield of a nonlinear population model with continuous age structure

Here we investigate the maximum sustainable yield problem for an age-structured population whose dynamics are density dependent. We use the nonlinear version of the McKendrick model of population dynamics that was introduced by Gurtin and MacCamy and do not constrain the magnitude of the harvesting term. We show that this problem has an optimal solution and that the optimum is attainable by a bimodal harvesting policy. This result is consistent with the results obtained by Grey for the nonlinear Leslie model.

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators.

Nonlinear age-dependent models for prediction of population growth

In this paper, some new algorithms are proposed to estimate parameter functions in nonlinear age-dependent population models by practical data. These algorithms together with a numerical method are applied to compute the human population using the data provided by the United Nations Demographic Yearbook.

Global behaviour of age-dependent logistic population models

We study the large time behaviour of a nonlinear population model with a general logistic term. It is proved that every solution must have a limit when time becomes infinite. We present conditions that guarantee the boundedness of the solution. Furthermore, we prove that in general no oscillation is possible for the total number of population. This is in sharp contrast to the linear case.

A numerical method for nonlinear age dependent population models

A numerical method for nonlinear age dependent population models

Hopf bifurcation and the stability of non-linear age-dependent population models

The possibility of Hopf bifurcation into stable orbits is considered for the Gurtin-MacCamy model of age-dependent population dynamics, in which the mortality function depends only on the population while fertility is a fairly general function of age as well as population. In addition an algorithm is produced which provides a necessary condition for Hopf bifurcation to occur for arbitrary systems of ordinary differential equations.