Mathematical pharmacokinetic data

DWA Bourne. Mathematical Modeling of Pharmacokinetic Data. Lancaster, PA Technomic Publishing Company, 1995. 

A posteriori identifiability is linked to the theory of optimization in mathematics because one normally uses a software package that has an optimization (data-fitting) capability in order to estimate parameter values for a multicompartmental model from a set of pharmacokinetic data. One obtains an estimate for the parameter values, an estimate for their errors, and a value for the correlation (or covariance) matrix. The details of optimization and how to deal with the output from an optimization routine are beyond the scope of this chapter, and the interested reader is referred to Cobelli et al. (12). The point to be made here is that the output from these routines is crucial in assessing the goodness-of-fit — that is, how well the model performs when compared to the data — since inferences about a drug s pharmacokinetics will be made from these parameter values. 

Bourne, D.W.A. Mathematical Modeling of Pharmacokinetic Data Technomic Publishing Lancaster, PA, 1995. 

Much more complex mathematical models have been derived for the quantitative description of pharmacokinetic data of acids and bases at different pH values. Applications to practical examples are discussed in chapter 7.3 and in refs. [41, 156, 175, 459, 477-479]. 

Compartmental modeling involves the specification of a structural mathematical model (commonly using either explicit or ordinary differential equations) and system parameters are estimated from fitting the model to pharmacokinetic data via non linear regression analysis or population mixed effects modeling. One popular structural model is the open two-compartment model shown in Figure 6.10. 

In order to obtain an in vitro-in vivo relationship two sets of data are needed. The first set is the in vivo data, usually entire blood/plasma concentration profiles or a pharmacokinetic metric derived from plasma concentration profile (e.g., cmax, tmax, AUC, % absorbed). The second data set is the in vitro data (e.g., drug release using an appropriate dissolution test). A mathematical model describing the relationship between these data sets is then developed. Fairly obvious, the in vivo data are fixed. However, the in vitro drug-release profile is often adjusted by changing the dissolution testing conditions to determine which match the computed in vivo-release profiles the best, i.e., results in the highest correlation coefficient. 

Combined pharmacokinetic-dynamic studies seek to characterize the time course of drug effects through the application of mathematical modeling to dose-effecttime data. This definition places particular emphasis on the time course of drug... 

The process of applying mathematical constructs to describe experimental results often reveals patterns in the agent s pharmacokinetics or dynamics that might not otherwise be discernible. Failure of a model s simulations to predict experimental measurements sometimes prompts questioning of the data, such as the reliability of the quantitative methods, or sample collection or exposure techniques. More often, it may indicate that greater complexity in the model s structure is required to capture the data s behavior. This is another primary reason for developing models to create hypotheses (model structures) that are falsifiable, leading to improved models and improved predictions in an iterative process. 

Once the health-effect endpoint and data points describing the exposure concentration-duration relationship have been selected, the values are plotted and fit to a mathematical equation from which the AEGL values are developed. There may be issues regarding the placement of the exponential function in the equation describing the concentration-duration relationship (e.g., C x t = k vs C X t = k2 vs X E = k3>. It is clear that the exposure concentration-duration relationship for a given chemical is directly related to its pharmacokinetic and pharmacodynamic properties. Hence, the use and proper placement of an exponent or exponents to describe these properties quantitatively is highly complex and not completely understood for all materials of concern. 

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