First-order absorption models model parameter estimation

The principal parameter used to indicate the rate of drug absorption is Cmax, even though it is also influenced by the extent of absorption the observed fmaX is less reliable. Because of the uncertainty associated with Cmax, it has been suggested (Endrenyi Yan, 1993 Tozer, 1994) that Cmax/AUCo-loq/ where AUCo-loq is the area under the curve from time zero to the LOQ of the acceptable analytical method, may more reliably measure the rate of drug absorption, except when multiexponential decline is extensive. Estimation of the terms should be based on the observed (measured) plasma concentrationtime data and the use of non-compartmental methods rather than compart-mental pharmacokinetic models. MRTs, from time zero to the LOQ of the analytical method, for the test and reference products can be compared, assuming that first-order absorption and disposition of the drug apply (Jackson Chen, 1987). 

A one-compartment model with first-order absorption and elimination with a lag time adequately described the nelfinavir PK profile. Although the nelfinavir elimination is mainly metabolic, a model with a Michaelis-Menten elimination was not considered because (a) only one dose of nelfinavir was tested leading to a concentration range between peak and trough not large enough to estimate Michaelis-Menten parameters and (b) there were no signs of saturation of the elimination on the plots. No saturable mechanism has been reported in the literature for nelfinavir. 

A necessary and sufficient condition for identifia-bility is the concept that with p-estimable parameters at least p-solvable relations can be generated. Traditionally, for linear compartment models this involves using Laplace transforms. For example, going back to the 1-compartment model after first-order absorption with complete bioavailability, the model can be written in state-space notation as... 

Estimation of one-compartment first-order absorption parameters from measured plasma samples ends up being a bit more complicated and laborious than the previous models, but the methods are stUl relatively straightforward. Proper parameter evaluation ideally... 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Keys et al. (1999) note that certain model parameter values were estimated by applying a step-wise parameter optimization routine to data on blood or tissue levels following oral or intravenous exposure to DEHP and MEHP. The parameters estimated included the km and Vmax values for metabolism of DEHP and MEHP, and first order rate constants for the following parameters metabolism of DEHP (e.g., liver), absorption of DEHP and MEHP in the small intestine, intracellular-to-extracellular transfer of nonionized MEHP, and biliary transfer of MEHP from liver to small intestine (these values are not provided in the profile because they are derived from optimization procedures and might not be directly useful for other models). Keys et al. (1999) do not explicitly cite or describe the data sets used to optimize model parameter values, or distinguish the data used in optimization from data used in validation exercises. Based on Table 5 of their report, it appears that at least some data from Pollack et al. (1985b) were used to optimize the model. 

The FDS5 pyrolysis model is used here to qualitatively illustrate the complexity associated with material property estimation. Each condensed-phase species (i.e., virgin wood, char, ash, etc.) must be characterized in terms of its bulk density, thermal properties (thermal conductivity and specific heat capacity, both of which are usually temperature-dependent), emissivity, and in-depth radiation absorption coefficient. Similarly, each condensed-phase reaction must be quantified through specification of its kinetic triplet (preexponential factor, activation energy, reaction order), heat of reaction, and the reactant/product species. For a simple charring material with temperature-invariant thermal properties that degrades by a single-step first order reaction, this amounts to -11 parameters that must be specified (two kinetic parameters, one heat of reaction, two thermal conductivities, two specific heat capacities, two emissivities, and two in-depth radiation absorption coefficients). 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

The parameter estimates were fixed from Model 1 and model development proceeded with the inhalational (si) route of administration. The initial model treated the inhaled dose as going directly into an absorption compartment with first-order transfer (ka) into the central compartment (Model 2 in Fig. 5.5). To account for the fact that some of the inhaled cocaine was exhaled and that there may have been first-pass metabolism prior to reaching the sampling site, a bioavailability term was included in the model (Fsj). 

Wade et al. (1993) simulated concentration data for 100 subjects under a one-compartment steady-state model using either first-or zero-order absorption. Simulated data were then fit using FO-approximation with a first-order absorption model having ka fixed to 0.25-, 0.5-, 1-, 2-, 3-, and 4 times the true ka value. Whatever value ka was fixed equal to, clearance was consistently biased, but was relatively robust with underpredictions of the true value by less than 5% on average. In contrast, volume of distribution was very sensitive to absorption misspecification, but only when there were samples collected in the absorption phase. When there were no concentration data in the absorption phase, significant parameter bias was not observed for any parameter. The variance components were far more sensitive to model misspecification than the parameter estimates with some... 

Estimation of multicompartment model parameters from measured plasma samples is very similar to the procedures described previously for the two-compartment first-order absorption model. The first step is to calculate bi(C ) for each of the measured plasma sample concentrations. The values of In(C ) are then plotted versus time (t), and the points on the terminal line are identified. Linear regression analysis of the terminal line provides values for B (B = c ) and In = —m). The first residual (/ i) values are then calculated as the difference between the measured plasma concentrations and the terminal line for points not used on the terminal line. A plot of ln(i i) versus t is then employed to identify points on the next terminal line, with linear regression analysis of this line used to determine and X -. Successive method of residuals analyses are then used to calculate the remaining B and A, values, with linear regression of the n-1 residual (Rn-i) values providing the values of Bi and Aj. If a first-order absorption model is being used, then one more set of residuals (R ) are calculated, and the linear regression analysis of these residuals then provides and kg. This type of analysis is typically performed by specialized PK software when the model contains more than two compartments. 

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