Subcellular pharmacokinetics: One-compartment model
Innovations in Teaching
Subcellular pharmacokinetics aims at descriptions of the time courses of drug concentrations in individual physically distinct subcellular compartments, as are the extra- and intracellular aqueous phases and membranes. Full descriptions work well for cell suspensions and other small biological systems with a few subcellular compartments. For tissues, organs, and organisms, containing a huge number of subcellular compartments, the modeling principles for large systems (time hierarchy, lumping) are applied to arrive at simplified descriptions that have complexity commensurate with available experimental information. Here, the derivation of a one-compartment model at the subcellular level is described in detail. The resulting description provides the expressions for pharmacokinetic parameters (the distribution volume, clearance, and the elimination rate constant) as related to body composition and drug properties. The principles of subcellular pharmacokinetics have been used in teaching pharmacokinetics in both the PharmD and graduate programs and contributed to a better understanding of drug disposition.
Classical pharmacokinetics(1) deals with phenomenological descriptions of the time courses of drug concentrations in tissues, organs and organisms. The biological system is represented by a variably structured set of kinetically distinct macroscopic compartments, which may correspond to blood plasma or organs, but in general have very loosely defined morphological basis. Drug disposition is expressed in terms of the space-averaged drug concentrations in the non-homogeneous compartments. Though these concentrations are of great value for the practical purposes of chemotherapy, the analysis of the drug effects at the molecular level requires the use of actual drug concentrations in immediate surroundings of the receptors. Classical pharmacokinetics is mainly interested in the fates of a single drug in the body, therefore the aspect of time has been stressed in its development. The phenomenological parameters in the classical pharmacokinetic models may not bear any obvious relation to body composition and drug properties. These relations are sought additionally from partial models for individual pharmacokinetic quantities.
Subcellular pharmacokinetics tackles the problem of drug disposition from the opposite direction. The biological system is represented by a large set of subcellular compartments, as are extra- and intracellular aqueous phases and the bilayer parts of membranes. This representation is physically correct because: (i) drug concentrations usually differ dramatically in flanking subcellular compartments due to different solvation properties of water and nonpolar phases, and (ii) the compartments are small enough for the drug molecules to achieve practically homogeneous distribution in their whole volume within a fraction of a second(2). The detailed models, however, can be treated in full detail for small biological systems (subcellular particles and cell suspensions) only. For larger systems (tissues, organs, organisms), the observed time courses of drug concentrations are much simpler than those resulting from complex models. Apparently, the time hierarchy of the processes determining drug disposition causes many contributing drug processes to collapse into several main streams. Application of the time hierarchy principles to the description of detailed models of subcellular pharmacokinetics results in expressions that provide similar concentration/time dependencies as the models of classical pharmacokinetics. However, pharmacokinetic parameters in the expressions are given in terms of body composition and drug properties due to the detailed structure of subcellular models.
Application of the time hierarchy for simplification of differential equations describing realistic subcellular models can be mathematically challenging for some scenarios. However, for a mono-exponential concentration/time dependence, corresponding to the one-compartment model, the derivation is sufficiently simple and, I believe, instructive, to be taught in basic pharmacokinetic courses. The derivation of the one-compartment model at subcellular level is given below. The resulting descriptions of pharmacokinetic parameters are discussed with regard to their dependencies on body composition and drug properties.
If the drug concentrations in plasma or other body fluids decrease mono-exponentially in time, absorption is apparently absent or very fast. Since distribution is usually faster than elimination, this situation can be envisaged as follows (Figure 1). Drug molecules are quickly transported into all accessible parts of the biological system and subsequently eliminated by excretion and metabolism. The equilibria for ionization, ionpairing, protein binding, and membrane/water partitioning of the drug molecules (Figure 2) are achieved practically immediately. No kinetics need to be considered for these processes; they are completely characterized by the respective equilibrium constants. As drug concentrations in individual subcellular compartments decrease due to elimination, the equilibria are disturbed and the drug molecules are quickly released from the membranes or from complexes with proteins or other body constituents. These releases cause the decrease in drug concentrations to be slower than that caused solely by elimination. The extent of the releases depends on the composition of the biological system (content of proteins, phospholipids, lipids, and ions, pH of aqueous phases) and also on the affinities of drug molecules to body constituents.
As Eq. (18) shows, metabolic clearance can be regarded as a weighed sum of the first order rate constants of metabolism. The weighing factors correspond to the volumes of individual phases where the metabolic reactions take place that are multiplied by the dissociation factors di. if the reaction affects ionized molecules. It is also obvious that the overall clearance is the sum of clearances of individual processes. Eq. (18) can also be used as a basis for interspecies scaling and population pharmacokinetics.
Elimination Rate Constant
The elimination rate constant in the one-compartment model is the ratio of clearance and the volume of distribution. Inspection of Eq. (22), with Eqs. (18) and (19) defining clearance, shows that the overall elimination rate constant is a weighed sum of individual rate constants. The weights are the products of the individual volumes and the dissociation factors di., divided by the distribution volume. The rate constants can directly be summed up only if the metabolic processes take place in the same phase. Otherwise, the rate constants of metabolic processes need to be scaled before the summation is performed.
The one-compartment model treated at subcellular level has been taught in pharmacokinetics courses in both the PharmD and graduate programs. Its use seemed to contribute to the students’ understanding of drug disposition, as reflected in 3-5 percent improvement in the grades of the tests of similar quality. The more difficult pharmacokinetic situations were explained with the help of a web-based simulator. The students liked the teaching innovation as evidenced by a significant increase in the students’ rating of the course. The rating increased in most categories by 0.8-0.9 points on the five-point scale what is more than would be expected on the basis of improved grades.
Acknowledgements. Although the presented models resulted from the research that has been funded in parts by US EPA (R82-6652-011) and NIH NCRR (1 P20 RR15566-01), they have not been subjected to the agencies’ peer and policy reviews and therefore do not necessarily reflect the views of the agencies and no official endorsement should be inferred.
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College of Pharmacy, North Dakota State University, Sudro 222B, Fargo ND 58105-5055
Am. J. Pharm. Educ., 66, 66-71(2002); received 12/31/01, accepted 1/14/02.
1Manuscript based on portfolio submitted to the 2001 Council of Faculties Innovations in Teaching competition.
Copyright American Association of Colleges of Pharmacy Spring 2002
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