A model of solute transport through stratum corneum using solute capture and release.

A one-dimensional model of solute transport through the stratum corneum is presented. Solute is assumed to diffuse through lipid bi-layers surrounding impermeable corneocytes. Transverse diffusion (perpendicular to the skin surface) through lipids separating adjacent corneocytes, is modeled in the usual way. Longitudinal diffusion (parallel to the skin surface) through lipids between corneocyte layers, is modeled as temporary trapping of solute, with subsequent release in the transverse direction. This leads to a linear equation for one-dimensional transport in the transverse direction. The model involves an arbitrary function whose precise form is uncertain. For a specific choice of this function, closed form expressions for the Laplace transform of solute out-flux at the inner boundary, and for the time lag are obtained in the case that a constant solute concentration is maintained at the outer skin surface, with the inner boundary of the stratum corneum kept at zero concentration, and with the stratum corneum initially free of solute.

Interconnected-tubes model of hepatic elimination: steady-state considerations.

In the interconnected-tubes model of hepatic transport and elimination, intermixing between sinusoids was modelled by the continuous interchange of solutes between a set of parallel tubes. In the case of strongly interconnected tubes and for bolus input of solute, a zeroth-order approximation led to the governing equation of the dispersion model. The dispersion number was expressed for the first time in terms of its main physiological determinants: heterogeneity of flow and density of interconnections. The interconnected-tubes model is now applied to steady-state hepatic extraction. In the limit of strong interconnections, the expression for output concentrations is predicted to be similar in form to those predicted by the distributed model for a narrow distribution of elimination rates over sinusoids, and by the dispersion model in the limit of a small dispersion number D(N). More generally, the equations for the predicted output concentrations can be expressed in terms of a dimensionless 'heterogeneity number'H(N), which characterizes the combined effects of variations in enzyme distribution and flow rates between different sinusoids, together with the effects of interconnections between sinusoids. A comparative analysis of the equations for the dispersion and heterogeneity numbers shows that the value of H(N)can be less than, greater than or equal to the value of D(N)for a correlation between distributions of velocities and elimination rates over sinusoids, anticorrelation between them, and when all sinusoids have the same elimination rate, respectively. Simple model systems are used to illustrate the determinants of H(N)and D(N).

Interconnected-tubes model of hepatic elimination.

The distributed-tubes model of hepatic elimination is extended to include intermixing between sinusoids, resulting in the formulation of a new, interconnected-tubes model. The new model is analysed for the simple case of two interconnected tubes, where an exact solution is obtained. For the case of many strongly-interconnected tubes, it is shown that a zeroth-order approximation leads to the convection-dispersion model. As a consequence the dispersion number is expressed, for the first time, in terms of its main physiological determinants: heterogeneity of flow and density of interconnections between sinusoids. The analysis of multiple indicator dilution data from a perfused liver preparation using the simplest version of the model yields the estimate 10.3 for the average number of interconnections. The problem of boundary conditions of the dispersion model is considered from the viewpoint that the dispersion-convection equation is a zeroth-order approximation to the equations for the interconnected-tubes model.

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.

Flux ratio theorems for nonstationary membrane transport with temporary capture of tracer.

It has been shown recently that the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane which has transport properties varying arbitrarily with the distance from a boundary, is independent of time from the very first appearance of the two outfluxes from the membrane. This surprising proposition has been proved for boundary conditions defining standard unidirectional fluxes, and then generalized to classes of time-dependent boundary conditions. The operational meaning of all the resulting theorems is that when any of them appear to be refuted experimentally, the presence of more than one parallel transport pathway (that is, of membrane heterogeneity transverse to the direction of transport) can be inferred and analyzed. Recent experimental data have been interpreted accordingly. However, the proofs of the theorems given so far have not taken into account the possibility of temporary capture of tracer at sites fixed in the membrane (including also entrances to microscopic culs-de-sac). The possible presence of such a process, which would not affect fluxes in the steady state, left a fundamental gap in the aforementioned inferences. It is shown here that all the theorems previously proved for the flux ratio under unsteady conditions remain valid when temporary capture of tracer is admitted, no matter how the rate of capture, and the probability distribution of residence times of tracer at capture sites, may depend on the distance from a membrane boundary. The validity of the aforementioned inferences from observed time-dependence of the flux ratio is thereby extended to a much wider class of membrane transport processes.

Asymptotic forms of tracer clearance curves: theory and applications of improved extrapolations.

Tracer clearance curves are conventionally extrapolated beyond times of observation by using monoexponential asymptotic forms. The inadequacy of the resulting predictions, especially as to the mean transit time and quantities derived from it, has been previously demonstrated experimentally. Here improvements in extrapolations and in the resulting predictions are derived theoretically and tested on previously published data, venous as well as externally recorded. First, secure lower bounds on the mean transit times are constructed, and shown to be much higher than conventional outright estimates for venous data (twice as high in some cases). Next, new asymptotic forms of tracer clearance curves from kinetically heterogeneous systems are derived; they are not monoexponential, but they are as robust, contain as few parameters and are as easily connected to data. It is shown theorectically that for real organs these new asymptotic forms should extrapolate and predict better than monoexponentials, and this is demonstrated on previously published venous data from perfused muscle. In particular, the resulting outright predictions of mean transit times are substantially better than the best lower bounds. Furthermore, a correction is derived to the standard estimate of the rate of regional cerebral blood flow. In an application to previously published data recorded externally, that correction reduces the estimated flow rate by 4%.