Two exceptional sets of physiological clearance curves and their mathematical form: test cases?

Advanced pharmacokinetic models based on organ clearance, circulatory, and fractal concepts

Three advanced models of pharmacokinetics are described. In the first class are physiologically based pharmacokinetic models based on in vitro data on transport and metabolism. The information is translated as transporter and enzyme activities and their attendant heterogeneities into liver and intestine models. Second are circulatory models based on transit time distribution and plasma concentration time curves. The third are fractal models for nonhomogeneous systems and non-Fickian processes are presented. The usefulness of these pharmacokinetic models, with examples, is compared.

Predicting Plutonium Decorporation Efficacy after Intravenous Administration of DTPA Formulations: Study of Pharmacokinetic–Pharmacodynamic Relationships in Rats

PURPOSE

The objectives of this study were: 1) to assess the relationship between plutonium decorporation (increased excretion and reduced retention in main organs of deposition) induced by intravenous liposome formulations of the chelating agent diethylene triamine pentaacetic acid (DTPA) and its pharmacokinetics, and 2) to model the renal excretion of plutonium after treatment with liposome-encapsulated DTPA in order to predict its efficacy and to optimise treatment schedules.

MATERIALS AND METHODS

Pharmacokinetic parameters from plasma or urinary data (days 0-16 sample collections) were modelled versus decorporation efficacy, and best correlations were selected for their goodness of fit.

RESULTS

The plutonium decorporation enhancement by DTPA liposomal formulations was well described by logistic models and the best correlation was observed with the area under the DTPA concentration curve of each formulation. The plutonium urinary excretion rates decreased mono-exponentially as a function of time after a single dose and the proposed model allowed a simple determination of the elimination half-life of the Pu-DTPA complex, a reasonably good approximation of the long-term efficacy of the treatments from truncated urinary data.

CONCLUSIONS

Both liposomal formulations of chelating agents and pharmacokinetic approaches to plutonium decorporation should be helpful in optimising treatment protocols.

Noncompartmental models of whole-body clearance of tracers: a review

Noncompartmental models are defined as models that allow for transport of material through regions of the body that are not necessarily well-mixed or of uniform concentration. The clearance of a substance of interest (metabolite or drug) from a noncompartmental system will not necessarily be governed by a sum of exponentials or even be describable by a set of ordinary differential equations. The model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. It may use the theory of linear systems to obviate the need for defining any precise anatomical structure. A number of the models reviewed deal with plasma clearance curves that are best described by power functions of time. Circulatory models are reviewed from their inception to the present. Recent studies on clearance as a fractal process are introduced.

Cycle-time and residence-time density approximations in a stochastic model for circulatory transport

The concentration of a drug in the circulatory system is studied under two different elimination strategies. The first strategy--geometric elimination--is the classical one which assumes a constant elimination rate per cycle. The second strategy--Poisson elimination--assumes that the elimination rate changes during the process of elimination. The problem studied here is to find a relationship between the residence-time distribution and the cycle-time distribution for a given rule of elimination. While the presented model gives this relationship in terms of Laplace-Stieltjes transform., the aim here is to determine the shapes of the corresponding probability density functions. From experimental data, we expect positively skewed, gamma-like distributions for the residence time of the drug in the body. Also, as some elimination parameter in the model approaches a limit, the exponential distribution often arises. Therefore, we use Laguerre series expansions, which yield a parsimonious approximation of positively skewed probability densities that are close to a gamma distribution. The coefficients in the expansion are determined by the central moments, which can be obtained from experimental data or as a consequence of theoretical assumptions. The examples presented show that gamma-like densities arise for a diverse set of cycle-time distribution and under both elimination rules.

A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics

PURPOSE

To point out the importance of heterogeneity in drug distribution processes and develop a noncompartmental approach for the description of the distribution of drug in the body.

METHODS

A dichotomous branching network of vessels for the arterial tree connected to a similar venous network was used to describe the heterogeneity of blood flow in the successive generations of the networks. The relevant kinetics of drug distribution in the well perfused and the deep tissues was considered to take place under well stirred (homogeneous) and understirred (heterogeneous) conditions, respectively.

RESULTS

A "homogeneous model" with classical kinetics (which is mathematically equivalent with the one-compartment model) was developed for these drugs which are confined to well perfused ("well stirred") spaces. A "heterogeneous model" was proposed for the drugs reaching understirred spaces using a decreasing with time rate coefficient (fractal kinetics) to model the diffusion of drug under heterogeneous conditions. The analysis of the model equations revealed that the homogeneous model can be considered as a special case of the heterogeneous model. Concentration-time plots of multiexponential type were generated using the heterogeneous model equation. The empirically used power functions of time for the analysis of calcium clearance curves, were found to be similar to the equation adhering to the heterogeneous model. Fittings comparable to multiexponential models were obtained when the heterogeneous model equation with only one adjustable parameter was applied to six sets of long period calcium data.

CONCLUSIONS

The heterogeneous processes of drug distribution in the body can obey the principles of fractal kinetics. Calcium clearance curves were analysed with the heterogeneous model. The validity of multicompartmental models which are based on the concept of homogeneity to describe drug distribution should be reconsidered.

A stochastic model for circulatory transport in pharmacokinetics

A new stochastic model for the residence time distribution of a drug injected instantaneously into the circulatory system is proposed and analyzed. The properties of the residence time are derived from the assumptions made about the cycle time distribution and the rule for elimination. This rule is given by the probability distribution of the number of cycles needed for elimination of a randomly selected molecule of the drug. Only the geometric distribution has been previously used for this purpose. Its transformation is applied here to get a boundary for the residence time. Other discrete distributions are applied with a view to describing different experimental situations. Suitable continuous probability distributions for the cycle time description are discussed.