First-order absorption models

C(t) modeled according to two-compartment model with zero-order and first-order absorption Pharmacokinetic/pharmacodynamic relationship modeled using Hill model with first-order absorption. Modeled parameters matched experimental parameters when bicompartmental model with zero-order input was used. Linear PKs, anticlockwise hysteresis loop established for all doses studied. Apomorphine and growth hormone concentration predicted with good accuracy... 

First-order absorption model with or without lag time... 

Optimize the Structural Model It is important to ensure that the structural model describes the underlying patterns in the data. If the data speaks to the existence of two clearly distinct absorption profiles, then a mixture model should be tested to ensure that the absorption phase of the profile is well characterized. If absorption, for example, could be better characterized with sequential first-order absorption models instead of a simple first-order model, the appropriate model should be used to eliminate bias due to model misspecification. 

Example 3 Two Parallel First-Order Absorption Models... 

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... 

Applying these results to this simulation, it can be assumed that bioavailability would be 100% and that a first-order absorption model would apply such that the time to maximal concentrations was 1 h (which corresponds to an absorption rate constant of 1.2 per hour). Under these assumptions, the simulated concentrationtime profile after repeated administration of 1 mg/kg every 8 h is shown in Fig. 9.22. Maximal concentrations were attained 1.1 h. after intramuscular administration and were about half the maximal concentration attained after intravenous administration. Trough concentrations at steady-state after intramuscular administration were very near trough concentrations after intravenous administration. Hence, based on this simulation, the physician could either keep the dose as is or increase the dose to more attain the concentrations seen after intravenous administration. 

Zhi, J. Unique pharmacokinetic characteristics of the 1-compartment first-order absorption model with equal absorption and elimination rate constants. Journal of Pharmaceutical Sciences 1990 79 652-654. 

Organization of Single-Dose Pharmacokinetic First-Order Absorption Model 239... 

Special Cases of the Two-Compartment First-Order Absorption Model 260... 

The standard one-compartment first-order absorption model makes three inherent assumptions about the ADME processes that occur during and after drug delivery. The specific nature and implications of each of these assumptions are described in this section. 

As in all first-order absorption models, the rate at which drug enters the systemic circulation is taken to be proportional to the amount of drug remaining to... 

Figure 10.43 Graphical representation of the absorption phase (rising concentration) and elimination phase (falling concentration) for a one-compartment first-order absorption model.

Elimination parameters are determined by linear regression analysis of the measured plasma concentration data falling on the terminal line. As always, the first step is to calculate the natural logarithm of each of the measured plasma concentration values. The values of In(C ) are then plotted versus time (t). If the plot shows later points falling near a straight terminal line with no early points above the terminal line, then the data can be well represented by the one-compartment first-order absorption model. As with previous one-compartment models, early high points above the terminal line indicate that the one-compartment model is not the best PK model for the data, and erratic late data points could mean the values are unreliable, as illustrated in Figure 10.47. 

First-order absorption processes end up being one of the cases where Af/C values are needed in order to fully evaluate all model parameters. The AUC for the one-compartment first-order absorption model is represented graphically as the shaded area in Figure 10.51. The value of AUC for this model can be calculated directly by the equation... 

Three special cases are considered for the one-compartment first-order absorption model. Eirst is a relatively rare situation known as a flip-flop situation. Second is the use of the one-compartment first-order absorption model to approximate the plasma concentrations of drugs that follow two-compartment kinetics. The last case considered is the identification of conditions when first-order drug delivery with rapid absorption can be modeled as an instantaneous absorption process. 

The two-compartment first-order absorption model will be covered in Section 10.12. First-order absorption of a drug that follows two-compartment kinetics yields plasma concentrations near the peak concentration that are above the terminal line, as illustrated in Figure 10.47. The PK model for two-compartment first-order absorption is more complicated than the one-compartment first-order absorption case. It turns out that if the peak area high points are only slightly above the terminal line, then the one-compartment model provides a reasonable approximation to the actual plasma concentrations. As mentioned several times previously, the simpler one-compartment model is often used even when it does not apply particularly well to a given drug. When this simplification is used for first-order absorption, the one-compartment first-order absorption equations can be employed without modification. 

This assumption is the same for all first-order absorption models. See Section 10.9.1.1 for the details regarding this assumption. 

Ai) multiplied by a micro elimination rate constant ( lo). A schematic representation of this standard two-compartment first-order absorption model is provided in Figure 10.76. The mass balance equations for each compartment are then... 

Figure 10.77 Graphical representation of the amount of drug in compartment 1 (/ i) versus time (t) and ln(/4i) versus t for a two-compartment first-order absorption model.

Unlike the one-compartment first-order absorption model, the time hax of the maximum plasma concentration (Cmax) cannot be written in a solved equation form. Thus determination of hax and C ax for this model requires a trial-and-error approach in which a time (<) is guessed and the plasma concentration (Cp) is calculated repeatedly until the C ax and hax are identified. 

The plasma concentration versus time equation for the two-compartment first-order absorption model contains three exponential decay terms, and three corresponding phases in Figure 10.80. The first exponential decay term contains the larger hybrid rate constant (/li), which dominates just after t ax during the distribution phase. Hence li is called the hybrid distribution rate constant. The second exponential decay term contains the smaller hybrid rate constant I2), which dominates at later times during the elimination phase. The third exponential decay curve contains and dominates the early rising absorption phase. The two-compartment elimination half-lrfe is then written in terms of as... 

The remaining model parameters still to be determined include the overall clearance (CL) and the two-compartment distribution volumes (Fi, V s, Vauc)-As in the one-compartment first-order absorption model, these remaining model parameters cannot be calculated until the bioavailability (F) has been evaluated. This will require AUC calculations and a comparison to IV drug delivery results, as described in the next two sections. 

Evaluation of the bioavailability for a drug following two-compartment kinetics is identical to the methods employed for the one-compartment first-order absorption model. The ratio of bioavailabiUty values F1/F2) for a drug delivered by two different routes is called the relative bioavailability. The relative bioavailability can be determined from the dose D and D2) and AUC values AUCi and AUC ) of each route by the relationship... 

The two-compartment first-order absorption model is significantly harder to work with than the one-compartment first-order absorption model. Thus the one-compartment model often is used when it provides a reasonable approximation to the two-compartment values. In fact, the one-compartment model is often used even when a drug is known to significantly deviate from single compartment kinetics. Just as in the case of the two-compartment bolus IV injection model in Section 10.10.5.3, as a general rule of thumb the one-compartment model can be employed with reasonable accuracy for Li < 2 B. When this simplification is used, the one-compartment first-order absorption model equations can be used without modification. 

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. 

First-order absorption processes require AUC values in order to evaluate the bioavailability (F). The AUC for the multicompartment extravascular first-order absorption model can be calculated by the simple equation... 

A comparison of the two-compartment first-order absorption model fit to measured plasma concentration data from a traditional method of residuals analysis and a nonlinear regression analysis is provided in Figure 10.99. This figure illustrates the fact that both methods offer a very reasonable fit to the measured data. It also demonstrates that there is not a large difference between the fit provided by the two different techniques. Close examination does reveal, however, that the nonlinear regression analysis does provide a more universal fit to all the data points. This is likely due to the fact that nonlinear regression fits all the points simultaneously, whereas the method of residuals analysis fits the data in a piecewise manner with different data points used for different regions of the curve. 

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