Accuracy of numerical inversion of Laplace transforms for pharmacokinetic parameter estimation

Implementing PRED Subroutine of NONMEM for Versatile Pharmacokinetic Analysis Using Fast Inversion of Laplace Transform (FILT)

In pharmacokinetic (PK) analysis, conventional models are described by ordinary differential equations (ODE) that are generally solved in their Laplace transformed forms. The solution in the Laplace transformed forms is inverse Laplace transformed to derive an analytical solution. However, inverse Laplace transform is often mathematically difficult. Consequently, numerical inverse Laplace transform methods have been developed. In this study, we focus on extending the modeling functions of Nonlinear Mixed Effect Model (NONMEM), a standard software for PK and population pharmacokinetic (PPK) analyses, by adding the Fast Inversion of Laplace Transform (FILT) method, one of the representative numerical inverse Laplace transform methods. We implemented PREDFILT, a specialized PRED subroutine, which functions as an internal model unit in NONMEM to enable versatile FILT analysis with second-order precision. The calculation results of the compartment models and a dispersion model are in good agreement with the ordinary analytical solutions and theoretical values. Therefore, PREDFILT ensures enhanced flexibility in PK or PPK analyses under NONMEM environments.

Diffusion modeling of percutaneous absorption kinetics: 3. Variable diffusion and partition coefficients, consequences for stratum corneum depth profiles and desorption kinetics

Stratum corneum (SC) desorption experiments have yielded higher calculated steady-state fluxes than those obtained by epidermal penetration studies. A possible explanation of this result is a variable diffusion or partition coefficient across the SC. We therefore developed the diffusion model for percutaneous penetration and desorption to study the effects of either a variable diffusion coefficient or variable partition coefficient in the SC over the diffusion path length. Steady-state flux, lag time, and mean desorption time were obtained from Laplace domain solutions. Numerical inversion of the Laplace domain solutions was used for simulations of solute concentration-distance and amount penetrated (desorbed)-time profiles. Diffusion and partition coefficients heterogeneity were examined using six different models. The effect of heterogeneity on predicted flux from desorption studies was compared with that obtained in permeation studies. Partition coefficient heterogeneity had a more profound effect on predicted fluxes than diffusion coefficient heterogeneity. Concentration-distance profiles show even larger dependence on heterogeneity, which is consistent with experimental tape-stripping data reported for clobetasol propionate and other solutes. The clobetasol propionate tape-stripping data were most consistent with the partition coefficient decreasing exponentially for half the SC and then becoming a constant for the remaining SC.

Hepatic Kupffer cell phagocytotic function in rats with erythrocytic-stage malaria

BACKGROUND

In the erythrocytic phase of malaria, Kupffer cells show marked hypertrophy and hyperplasia and are filled with malarial pigment. However, phagocytic function in this state has not been well characterized. The aim of the present study was to use mouse Plasmodium berghei to infect rats with malaria and study the phagocytic function and morphology of Kupffer cells.

METHODS

We used a recirculating isolated perfused rat liver (IPRL) to quantitate Kupffer cell phagocytic clearance of radiolabeled albumin-latex over 120 min in high parasitemia (53 +/- 6%; n = 7) and low parasitemia (approximately 1%; n = 4) malaria-infected rats and littermate controls (n = 7 and n = 4, respectively). In a further group of high-parasitemic rats, perfusion was ceased after 7 min and liver radioactivity also measured. Electron microscopy was performed after perfusions.

RESULTS

In high-parasitemia malaria rats, clearance of radiolabeled latex from IPRL perfusate over 120 min was significantly (P < 0.01) faster than in controls, with a lower area under the curve (0.19 +/- 0.02 vs 0.43 +/- 0.07 /mL per min, respectively) and shorter half-life (t1/2k; 2.4 +/- 0.6 vs 10.0 +/- 2.3 min, respectively). Low-parasitemia rats were identical to controls. After 7 min perfusion in high-parasitemic rats (n = 4), total radioactivity in liver homogenates was higher than in controls (n = 4; 33.1 +/- 6.2 vs 18.4 +/- 1.9% of injected radiolabel; P < 0.05). Electron microscopy showed latex in Kupffer cells, more abundantly seen in high-parasitemic animals.

CONCLUSIONS

Total Kupffer cell phagocytic activity of the liver is markedly increased in rats with a high parasitemic load of malarial P. berghei infection. This is presumed to reflect an upregulation of scavenger activity phagocytosing erythrocytes and their breakdown products.

Diffusion modeling of percutaneous absorption kinetics. 1. Effects of flow rate, receptor sampling rate, and viable epidermal resistance for a constant donor concentration

A diffusion model for the percutaneous absorption of a solute through the skin is developed for the specific case of a constant donor concentration with a finite removal rate from the receptor due to either perfusion rate or sampling. The model has been developed to include a viable epidermal resistance and a donor-stratum corneum interfacial resistance. Numerical inversion of the Laplace domain solutions were used for simulations of solute flux and cumulative amount absorbed and to model specific examples of percutaneous absorption. Limits of the Laplace domain solutions were used to define the steady-state flux, lag time, and receptor concentration. Steady-state approximations obtained from the solutions were used to relate the steady-state flux and the effective permeability coefficient to the viable epidermis resistance, a donor-stratum corneum interfacial resistance, receptor removal rate, and partitioning between the receptor and donor phases. The lag time was shown to be dependent on these parameters and on the volume of the receptor phase. It is concluded that curvilinear cumulative amount and flux-time profiles are dependent on the processes affecting percutaneous absorption, the shapes of the profiles reflecting the processes most determining transport.

Modeling of hepatic elimination and organ distribution kinetics with the extended convection-dispersion model

The conventional convection-dispersion (also called axial dispersion) model is widely used to interrelate hepatic availability (F) and clearance (Cl) with the morphology and physiology of the liver and to predict effects such as changes in liver blood flow on F and Cl. An extended form of the convection-dispersion model has been developed to adequately describe the outflow concentration-time profiles for vascular markers at both short and long times after bolus injections into perfused livers. The model, based on flux concentration and a convolution of catheters and large vessels, assumes that solute elimination in hepatocytes follows either fast distribution into or radial diffusion in hepatocytes. The model includes a secondary vascular compartment, postulated to be interconnecting sinusoids. Analysis of the mean hepatic transit time (MTT) and normalized variance (CV2) of solutes with extraction showed that the discrepancy between the predictions of MTT and CV2 for the extended and unweighted conventional convection-dispersion models decreases as hepatic extraction increases. A correspondence of more than 95% in F and Cl exists for all solute extractions. In addition, the analysis showed that the outflow concentration-time profiles for both the extended and conventional models are essentially identical irrespective of the magnitude of rate constants representing permeability, volume, and clearance parameters, providing that there is significant hepatic extraction. In conclusion, the application of a newly developed extended convection-dispersion model has shown that the unweighted conventional convection-dispersion model can be used to describe the disposition of extracted solutes and, in particular, to estimate hepatic availability and clearance in both experimental and clinical situations.

Structure-hepatic disposition relationships for phenolic compounds

Phenolic compounds are widely used in therapeutic, environmental, and industrial applications. The present work seeks to define the hepatic disposition of 11 phenolic compounds with varying lipophilicities and molecular weights. The hepatic disposition kinetics were studied in a once-through in situ rat liver perfusion preparation in order to avoid extra-hepatic metabolism and recirculation effects. The phenols were administered using the impulse-response technique and the time course of hepatic venous effluent concentration was examined by moments and a two-compartment dispersion model. While the extraction of the phenolic compounds was relatively independent of lipophilicity, the estimated permeability-surface area (PS) product for influx of solutes into the hepatocytes could be related to the compounds' octanol-buffer partition coefficients (log Papp). This log PS-logPapp relationship was consistent with that reported earlier for another series of solutes with a wide range of lipophilicity. The metabolites produced from each of the phenolic compounds used in this study had mean transit times similar to those of their corresponding parent phenols, suggesting that the metabolites were not trapped in the liver as a consequence of their higher polarity. It is concluded that the strong solute lipophilicity-toxicity and lipophilicity-skin penetration relationships often seen for aqueous solutions of phenols are not evident for the hepatic extraction of these compounds. Such a conclusion is consistent with the hepatic extraction of phenolic compounds being mainly determined by a blood flow limitation in delivery of the phenol to the liver, rather than the intrinsic liver metabolic enzyme activities at the doses injected.

An analysis of solute structure-human epidermal transport relationships in epidermal iontophoresis using the ionic mobility: pore model

This study sought to examine the extent the ionic mobility-pore model, used to describe epidermal iontophoretic structure-permeability relationships, could describe a range of published iontophoretic data. The model incorporates, as determinants of iontophoretic transport, solute size, solute mobility, total current applied, presence of extraneous ions, determined by conductivities of both donor and receptor solutions, permselectivity of the epidermis, as well as a solute pore interaction term which together provided an excellent regression for iontophoretic permeability. The 'pore' radii for solute transport estimated from literature iontophoretic permeabilities using the model ranged from 6.8 to 17 A depending on the degree of hydration and conformation of solute assumed. The pore size range is consistent with transport through the polar intercellular and transappendageal pathway for transport. The pore restriction form of the model better describes the data obtained to date than other models described previously (Yoshida, N.H., Roberts, M.S., Solute molecular size and transdermal iontophoresis across excised human skin. J. Control. Release 25 (1993) 177-195).

Pharmacokinetic curve fitting using numerical inverse Laplace transformation

The combination of the nonlinear regression program ADAPT II with Talbot's method of numerical Laplace transformation, that allows parameter estimation when the model function is given only in the Laplace domain, is described and successfully applied to pharmacokinetic problems. The accuracy and precision of the method has been found satisfactory; its performance is comparable to that achieved in parameter estimation based on functions defined in the time domain.

Lateral iontophoretic solute transport in skin

PURPOSE

The lateral iontophoretic transport of three solutes (sodium, ethanolamine, lidocaine) from an active electrode through skin and other tissues to an indifferent electrodes was investigated.

METHODS

Anodal epidermal iontophoresis was carried out on an in vivo rat model using constant direct current of 0.38 mA/cm2. Cells were fixed on the epidermis of anesthetized rats at distances of adjacent, 3 cm and 7 cm apart. After iontophoresis, tissues were dissected at I cm intervals between the electrodes. Concentrations of the radiolabelled solutes in tissues were determined by liquid scintillation counting or gamma counting.

RESULTS

The concentration of each solutes in the epidermis, dermis and other tissues was found to decrease in an exponential manner with lateral distance from the active electrode to the indifferent electrode. The detectable lateral distance for ethanolamine and lidocaine was less than 2 cm from the donor sites, at which distance the concentrations were not significantly different to those found in the corresponding contralateral site. The lateral drift velocities for all solutes in the epidermis and dermis were consistent with diffusivities of the order of 10(-6) cm2/s. The drift velocity of sodium was greater than either lidocaine or ethanolamine.

CONCLUSIONS

The decline in solute concentration with lateral distance is mainly due to clearance from the site of application by the skin's microcirculation and decreases with distance from the active electrode until a baseline concentration, similar to the contralateral tissue concentration is reached.

Analysis of nonlinear and nonsteady state hepatic extraction with the dispersion model using the finite difference method

A numerical calculation method for dispersion models was developed to analyze nonlinear and nonsteady hepatic elimination of substances. The finite difference method (FDM), a standard numerical calculation technique, was utilized to solve nonlinear partial differential equations of the dispersion model. Using this method, flexible application of the dispersion model becomes possible, because (i) nonlinear kinetics can be incorporated anywhere, (ii) the input function can be altered arbitrarily, and (iii) the number of compartments can be increased as needed. This method was implemented in a multipurpose nonlinear least-squares fitting computer program, Napp (Numeric Analysis Program for Pharmacokinetics). We simulated dilution curves for several nonlinear two-compartment hepatic models in which the saturable process is assumed in transport or metabolism, and investigated whether they could definitely be discriminated from each other. Preliminary analysis of the rat liver perfusion data of a cyclic pentapeptide, BQ-123, was performed by this method to demonstrate its applicability.

Epidermal iontophoresis: I. Development of the ionic mobility-pore model

PURPOSE

An integrated ionic mobility-pore model for epidermal iontophoresis is developed from theoretical considerations using both the free volume and pore restriction forms of the model for a range of solute radii (rj) approaching the pore radii (rp) as well as approximation of the pore restriction form for rj/rp < 0.4. In this model, we defined the determinants for iontophoresis as solute size (defined by MV, MW or radius), solute mobility, solute shape, solute charge, the Debye layer thickness, total current applied solute concentration, fraction ionized, presence of extraneous ions (defined by solvent conductivity), epidermal permselectivity, partitioning rates to account for interaction of unionized and ionized lipophilic solutes with the wall of the pore and electroosmosis.

METHODS

The ionic mobility-pore model was developed from theoretical considerations to include each of the determinants of iontophoretic transport. The model was then used to reexamine iontophoretic flux conductivity and iontophoretic flux-fraction ionized literature data on the determinants of iontophoretic flux.

RESULTS

The ionic mobility-pore model was found to be consistent with existing experimental data and determinants defining iontophoretic transport. However, the predicted effects of solute size on iontophoresis are more consistent with the pore-restriction than free volume form of the model. A reanalysis of iontophoretic flux-conductivity data confirmed the model's prediction that, in the absence of significant electroosmosis, the reciprocal of flux is linearly related to either donor or receptor solution conductivity. Significant interaction with the pore walls, as described by the model, accounted for the reported pH dependence of the iontophoretic transport for a range of ionizable solutes.

CONCLUSIONS

The ionic mobility-pore iontophoretic model developed enables a range of determinants of iontophoresis to be described in a single unifying equation which recognises a range of determinants of iontophoretic flux.

Epidermal iontophoresis: II. Application of the ionic mobility-pore model to the transport of local anesthetics

PURPOSE

An in vitro study was carried out to determine the iontophoretic permeability of local anesthetics through human epidermis. The relationship between physicochemical structure and the permeability of these solutes was then examined using an ionic mobility-pore model developed to define quantitative relationships.

METHODS

The iontophoretic permeability of both ester-type anesthetics (procaine, butacaine, tetracaine) and amide-type anesthetics (prilocaine, mepivacaine, lidocaine, bupivacaine, etidocaine, cinchocaine) were determined through excised human epidermis over 2 hrs using a constant d.c. current and Ag/AgCl electrodes. Individual ion mobilities were determined from conductivity measurements in aqueous solutions. Multiple stepwise regression was applied to interrelate the iontophoretic permeability of the solutes with their physical properties to examine the appropriateness of the ionic mobility-pore model and to determine the best predictor of iontophoretic permeability of the local anesthetics.

RESULTS

The logarithm of the iontophoretic permeability coefficient (log PC(j,iont)) for local anesthetics was directly related to the log ionic mobility and MW for the free volume form of the model when other conditions are held constant. Multiple linear regressions confirmed that log PC(j,iont) was best defined by ionic mobility (and its determinants: conductivity, pKa and MW) and MW.

CONCLUSIONS

Our results suggest that of the properties studied, the best predictors of iontophoretic transport of local anesthetics are ionic mobility (or pKa) and molecular size. These predictions are consistent with the ionic mobility pore model determined by the mobility of ions in the aqueous solution, the total current, epidermal permselectivity and other factors as defined by the model.

Relative dispersions of intra-albumin transit times across rat and elasmobranch perfused livers, and implications for intra- and inter-species scaling of hepatic clearance using microsomal data

It is recognized that vascular dispersion in the liver is a determinant of high first-pass extraction of solutes by that organ. Such dispersion is also required for translation of in-vitro microsomal activity into in-vivo predictions of hepatic extraction for any solute. We therefore investigated the relative dispersion of albumin transit times (CV2) in the livers of adult and weanling rats and in elasmobranch livers. The mean and normalized variance of the hepatic transit time distribution of albumin was estimated using parametric non-linear regression (with a correction for catheter influence) after an impulse (bolus) input of labelled albumin into a single-pass liver perfusion. The mean+/-s.e. of CV2 for albumin determined in each of the liver groups were 0.85+/-0.20 (n = 12), 1.48+/-0.33 (n = 7) and 0.90+/-0.18 (n = 4) for the livers of adult and weanling rats and elasmobranch livers, respectively. These CV2 are comparable with that reported previously for the dog and suggest that the CV2 of the liver is of a similar order of magnitude irrespective of the age and morphological development of the species. It might, therefore, be justified, in the absence of other information, to predict the hepatic clearances and availabilities of highly extracted solutes by scaling within and between species livers using hepatic elimination models such as the dispersion model with a CV2 of approximately unity.

Impulse-response studies on tracer doses of [14C]lignocaine and its multiple metabolites in the perfused rat liver

The outflow-concentration-time profiles for lignocaine (lidocaine) and its metabolites have been measured after bolus impulse administration of [14C]lignocaine into the perfused rat liver. Livers from female Sprague-Dawley rats were perfused in a once-through fashion with red-blood-cell-free Krebs-Henseleit buffer containing 0 or 2% bovine serum albumin. Perfusate flow rates of 20 and 30 mL min-1 were used and both normal and retrograde flow directions were employed. Significant amounts of metabolite were detected in the effluent perfusate soon after lignocaine injection. The early appearance of metabolite contributed to bimodal outflow profiles observed for total 14C radioactivity. The lignocaine outflow profiles were well characterized by the two-compartment dispersion model, with efflux rate << influx rate. The profiles for lignocaine metabolites were also characterized in terms of a simplified two-compartment dispersion model. Lignocaine was found to be extensively metabolized under the experimental conditions with the hepatic availability ranging between 0.09 and 0.18. Generally lignocaine and metabolite availability showed no significant change with alterations in perfusate flow rate from 20 to 30 mL min-1 or protein content from 0 to 2%. A significant increase in lignocaine availability occurred when 1200 microM unlabelled lignocaine was added to the perfusate. Solute mean transit times generally decreased with increasing flow rate and with increasing perfusate protein content. The results confirm that lignocaine pharmacokinetics in the liver closely follow the predictions of the wellstirred model. The increase in lignocaine availability when 1200 microM unlabelled lignocaine was added to the perfusate is consistent with saturation of the hydroxylation metabolic pathways of lignocaine metabolism.

Metabolite mean transit times in the liver as predicted by various models of hepatic elimination

Predicted area under curve (AUC), mean transit time (MTT) and normalized variance (CV2) data have been compared for parent compound and generated metabolite following an impulse input into the liver. Models studied were the well-stirred (tank) model, tube model, a distributed tube model, dispersion model (Danckwerts and mixed boundary conditions) and tanks-in-series model. It is well known that discrimination between models for a parent solute is greatest when the parent solute is highly extracted by the liver. With the metabolite, greatest model differences for MTT and CV2 occur when parent solute is poorly extracted. In all cases the predictions of the distributed tube, dispersion, and tanks-in-series models are between the predictions of the tank and tube models. The dispersion model with mixed boundary conditions yields identical predictions to those for the distributed tube model (assuming an inverse gaussian distribution of tube transit times). The dispersion model with Danckwerts boundary conditions and the tanks-in series models give similar predictions to the dispersion (mixed boundary conditions) and the distributed tube. The normalized variance for parent compound is dependent upon hepatocyte permeability only within a distinct range of permeability values. This range is similar for each model but the order of magnitude predicted for normalized variance is model dependent. Only for a one-compartment system is the MTT for generated metabolite equal to the sum of MTTs for the parent compound and preformed metabolite administered as parent.

A novel extravascular input function for the assessment of drug absorption in bioavailability studies

PURPOSE

Flexible parametric models describing the input process after extravascular drug administration are needed for the assessment of absorption rate and the use of population methods in bioavailability and bioequivalence studies.

METHODS

The oral concentration-time curve modeled as the product of the input and disposition function in the Laplace domain was obtained by numerical inversion methods for parameter estimation. The utility of the inverse Gaussian input density was examined using bioavailability data of an extended-release dosage form. Measures of rate of absorption and the cumulative absorbed amount profile were defined in terms of the estimated model parameters.

RESULTS

Accurate estimation of absorption parameters was achieved by simultaneous fitting of the extravascular and intravascular data (describing the latter by a triexponential function). The new input function allowed a direct estimation of both extent of absorption and mean absorption time.

CONCLUSIONS

The findings suggest that the inverse Gaussian density is a useful input function. Its flexibility may reduce the effect of model misspecification in parameter estimation. All parameters can be readily interpreted in terms of the absorption process.

Modeling solute sorption into plastic tubing during organ perfusion and intravenous infusions

The uptake of solutes into plastic infusion and perfusion tubing has been well documented, but the kinetics of the uptake process is not well-defined. Three mathematical models have been developed to describe the outflow fraction concentration--time profiles for solutes sorbed into the plastic tubing during infusion and perfusions. The models are referred to as model 1, convection--diffusion; model 2, convention-- interfacial resistance--diffusion; ad model 3, convection--interfacial resistance--infinite sink models. In each model, plug flow is assumed and, in order to minimize the number of variables required, solutions are limited to early times when the plastic behaves as an infinite sink. Initial conditions of (i) no solution in the tubing and (ii) a preloading of tubing with drug solution are considered for each of the three models. Two parameters, one being the transit time of solution through tubing (tmin) and the other a measure of the affinity and diffusivity of the solute in the plastic (SN), are sufficient to describe the outflow concentration--time profiles for solutes with sorption into tubing being limited by diffusion in the plastic (model 1). A single parameter, which is the effective interfacial permeability coefficient (H), is sufficient to describe the outflow concentration--time profiles for solutes with sorption into tubing being limited by an aqueous--plastic interfacial barrier (model 3). The three parameters (tmin, SN, and H) are required when uptake into tubing is limited by a combination of diffusion into plastic and an interfacial resistance (model 3). Each model has a characteristic outflow concentration--time profile determined by the relative magnitude of diffusivity of the solute in the plastic to that across the interfacial barrier. The sorption of nitroglycerin and isosorbide dinitrate are adequately described by the convection--diffusion model (model 1 (ii)) whereas the convection--interfacial resistance--diffusion model (model 2 (ii)) is required to describe the sorption of diazepam and chlorpromazine.

Tissue distribution kinetics as determinant of transit time dispersion of drugs in organs: application of a stochastic model to the rat hindlimb

A stochastic theory of drug transport in a random capillary network with permeation across the endothelial barrier is coupled with a model of tissue residence time of drugs assuming radial intratissue diffusion. Axial diffusion is neglected both in tissue as well as in the radially well-mixed vascular phase. The convective transport through the microcirculatory network is characterized by an experimentally determined transit time distribution of a nonpermeating vascular indicator. This information is used to identify three adjustable model parameters characterizing permeation, diffusion, and steady-state distribution into tissue. Predictions are made for the influence of distribution volume, capillary permeability, and tissue diffusion on transit time distributions. The role of convection (through the random capillary network), permeation, and diffusion as determinants of the relative dispersion of organ transit times has been examined. The relationship to previously proposed models of capillary exchange is discussed. Results obtained for lidocaine in the isolated perfused hindleg in rats indicate that although the contribution of intratissue diffusion to the dispersion process is relatively small in quantitative terms, it has a pronounced influence on the shape of the impulse response curve. The theory suggests that the rate of diffusion in muscle tissue is about two orders of magnitude slower than in water.