ALBERT

Notes: Abstract Lumping is a common pragmatic approach aimed at the reduction of whole-body physiologically based pharmacokinetic (PBPK) model dimensionality and complexity. Incorrect lumping is equivalent to model misspecification with all the negative consequences to the subsequent model implementation. Proper lumping should guarantee that no useful information about the kinetics of the underlying processes is lost. To enforce this guarantee, formal standard lumping procedures and techniques need to be defined and implemented. This study examines the lumping process from a system theory point of view, which provides a formal basis for the derivation of principles and standard procedures of lumping. The lumping principle in PBPK modeling is defined as follows: Only tissues with identical model specification, and occupying identical positions in the system structure should be lumped together at each lumping iteration. In order to lump together parallel tissues, they should have similar or close time constants. In order to lump together serial tissues, they should equilibrate very rapidly with one another. The lumping procedure should include the following stages: (i) tissue specification conversion (when tissues with different model specifications are to be lumped together); (ii) classification of the tissues into classes with significantly different kinetics, according to the basic principle of lumping above; (iii) calculation of the parameters of the lumped compartments; (iv) simulation of the lumped system; (v) lumping of the experimental data; and (vi) verification of the lumped model. The use of the lumping principles and procedures to be adopted is illustrated with an example of a commonly implemented whole-body physiologically based pharmacokinetic model structure to characterize the pharmacokinetics of a homologous series of barbiturates in the rat.

Notes: Abstract Sensitivity analysis studies the effects of the inherent variability and uncertainty in model parameters on the model outputs and may be a useful tool at all stages of the pharmacokinetic modeling process. The present study examined the sensitivity of a whole-body physiologically based pharmacokinetic (PBPK) model for the distribution kinetics of nine 5-n-alkyl-5-ethyl barbituric acids in arterial blood and 14 tissues (lung, liver, kidney, stomach, pancreas, spleen, gut, muscle, adipose, skin, bone, heart, brain, testes) after iv bolus administration to rats. The aims were to obtain new insights into the model used, to rank the model parameters involved according to their impact on the model outputs and to study the changes in the sensitivity induced by the increase in the lipophilicity of the homologues on ascending the series. Two approaches for sensitivity analysis have been implemented. The first, based on the Matrix Perturbation Theory, uses a sensitivity index defined as the normalized sensitivity of the 2-norm of the model compartmental matrix to perturbations in its entries. The second approach uses the traditional definition of the normalized sensitivity function as the relative change in a model state (a tissue concentration) corresponding to a relative change in a model parameter. Autosensitivity has been defined as sensitivity of a state to any of its parameters; cross-sensitivity as the sensitivity of a state to any other states' parameters. Using the two approaches, the sensitivity of representative tissue concentrations (lung, liver, kidney, stomach, gut, adipose, heart, and brain) to the following model parameters: tissue-to-unbound plasma partition coefficients, tissue blood flows, unbound renal and intrinsic hepatic clearance, permeability surface area product of the brain, have been analyzed. Both the tissues and the parameters were ranked according to their sensitivity and impact. The following general conclusions were drawn: (i) the overall sensitivity of the system to all parameters involved is small due to the weak connectivity of the system structure; (ii) the time course of both the auto- and cross-sensitivity functions for all tissues depends on the dynamics of the tissues themselves, e.g., the higher the perfusion of a tissue, the higher are both its cross-sensitivity to other tissues' parameters and the cross-sensitivities of other tissues to its parameters; and (iii) with a few exceptions, there is not a marked influence of the lipophilicity of the homologues on either the pattern or the values of the sensitivity functions. The estimates of the sensitivity and the subsequent tissue and parameter rankings may be extended to other drugs, sharing the same common structure of the whole body PBPK model, and having similar model parameters. Results show also that the computationally simple Matrix Perturbation Analysis should be used only when an initial idea about the sensitivity of a system is required. If comprehensive information regarding the sensitivity is needed, the numerically expensive Direct Sensitivity Analysis should be used.

Notes: Abstract An extension of the plasma area method of Chiou to the evaluation of accumulation profiles during quasiuni-form multiple dosing regimens (i.e., regimens with varying doses and/or dosing intervals within a certain time period which are repeated in the successive periods throughout the administration course) is presented. The derivations show, that the equations for the accumulation and steady state curves during quasi-uniform multiple dosing are similar or identical to these for the uniform multiple dosing case, if the constant time interval in the latter is replaced by the constant time period in the former. The equations derived can be used for rapid estimation of the time, required to reach a prespecified fraction of quasi steady state plasma level during quasi-uniform multiple dosing.

Notes: Abstract As pan of an overall program to develop a framework for evaluating the contribution of structural and physicochemical properties to pharmacokinetics, the distribution kinetics of nine 5-n-alkyl-5-ethyl barbituric acids in arterial blood and 14 tissues (lung, liver, kidney, stomach, pancreas, spleen, gut, muscle, adipose, skin, bone, heart, brain, testes) was examined after iv bolus administration in rats. The barbituric acids studied form a true homologous series; therefore any differences in pharmacokinetics, noted between congeners, can be directly linked to the increase in lipophilicity, resulting from the addition of a methylene group. A whole-body physiologically based pharmacokinetic model has been developed, assuming most of the tissues to be well-stirred compartments. Brain and testes, in which distribution for the lower homologues was permeability rate-limited, were represented by two compartments. For each homologue, the model parameters have been optimized, using the tissue concentration–time data. The initial distribution processes in the system were very rapid, making it quite stiff, and essentially over before the first samples were taken. A progressively increasing redistribution from lean tissues into adipose on ascending the homologous series was observed, characterized by a tendency for a progressive decrease in the magnitude of the concentration–time profiles for some of the lean and well-perfused tissues, an increase in the adipose concentration–time profile, and an increase in the time to reach the maximum adipose concentration. A shift from permeability rate limitation to perfusion rate limitation of the distribution processes for brain and testes, as well as an increase in the intrinsic hepatic clearance and decrease in the renal clearance with the increase of lipophilicity of the homologues, were quantified. An increase in the total unbound volume of distribution on ascending the homologous series was also observed. Muscle was found to be the major drug depot at steady state, accounting for approximately 50% of the total unbound volume of distribution, regardless of the lipophilicity of the homologue; the unbound volume of distribution of adipose increases more than 10-fold with the increase of lipophilicity.

Notes: Abstract Based on a frequency response approach to the sensitivity analysis of pharmacokinetic models, the concept of structural sensitivity is introduced. The core of this concept is the factorization of the system sensitivity into two multipliers. The first one, called structural sensitivity index, has an analytical form, which depends solely on the structure and connectivity of the system and does not depend on the drug administered or the factor perturbed. The second multiplier, the parameter sensitivity index, depends on the drug properties, the tissue of interest and the parameter perturbed, but is largely independent of the structure of the system. The structural and parametric sensitivity indices can be evaluated and analyzed separately. The most important feature of the proposed approach is that the conclusions drawn from the analysis of the structural sensitivity index are valid across all mammalian species, as the latter share a common anatomical and physiological structure. The concept of structural sensitivity is illustrated on the commonly used structure of the whole body physiologically based pharmacokinetic models by showing that the factorization of the sensitivity carried out arises naturally from the mechanism of the distribution of perturbations throughout the organism. The concept of structural sensitivity has interesting practical implications. It enables the formal proof of relationships and facts that have been observed previously. Moreover, the conclusions drawn introduce in fact a ranking of the tissues or subsystems with respect to their impact on the model outputs. From this ranking, direct recommendations regarding the design of experiments for whole-body physiologically based pharmacokinetic models are derived.