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Exploring Potential Models

The first step towards successfully modeling pharmacokinetic data is to consider the route of administration and the data available. Data collected after an intravenous (IV) bolus may be the easiest to analyze as there are no absorptions steps to consider. The bolus dose should be well defined and it should not be necessary to estimates its value. An IV infusion adds an administration step, a rate of infusion and possibly a duration if the infusion is stopped. Again, both of these parameters should be well know. A dose given by mouth or as a intramuscular (IM) injection require the addition of an administration step. This might be a simple one compartment process or it could be more complex. Solubility, stability and site of absorption can add to the complexity of the absorption step and the overall model. Samples other than blood or plasma may be available. Unchanged drug and metabolite in urine samples can add another dimension to the model selection process. The absorption, distribution, metabolism and excretion (ADME) processes may not be all first order. Data collected after different doses can be useful, as seen in the figure below.

Plot of Cp versus time after three different doses — **Semi-log plot of concentration versus time after three different doses**

AUC versus dose — **Linear plot of AUC achieved after various doses**

Figure 6.6.5 Data Suggesting the Need to include Non-Linear Components to the Model

Data collected after multiple dosing also adds the model detail. At this point we could envisage potential models with rapid distribution and a single compartment representing the body.

We might now move to the consideration of data plotted on linear and semi-log graphs. These graphs should confirm our thought regarding the administration and elimination, metabolism and excretion, of the drug and metabolite. A distribution phase may suggest a multi compartment pharmackinetic model. Even after extravascular administration such as oral dosing a distribution phase may be evident in the semi-log plot. Compare the plots in Figures 6.6.6 and 6.6.7. The early data in the second semi-log plot indicate a deviation from the terminal straight line at early time points, leading one to consider a two or three compartment distribution as part of the model.

Linear plot - One compartment — **Linear Plot**

Semi-log plot - One compartment — **Semi-log Plot**

Figure 6.6.6 Data representing a One Compartment Model

Linear plot - Two compartment — **Linear Plot**

Semi-log plot - Two compartment — **Semi-log Plot**

Figure 6.6.7 Data representing a Two Compartment Model

Initial Estimates

Fitting any model to pharmacokinetic data with any computer software is more efficient and more likely to succeed if you can provide good initial estimates. A number of techniques can be useful.

For a simple one compartment model after an IV bolus the equation for concentration versus time can be expressed in logarithmic form as a straight line as illustrated by the right hand plot in Figure 6.6.6. The slope and intercept can provide estimates of V and kel. Estimating the area under the concentration versus time curve (AUC) can provide an estimate of clearance. Initial estimates for the parameters of a two compartment model can be determined by the method of residual (aka: curve striping or feathering the curve). In a similar fashion the absorption and elimination rates constant for oral administration, one compartment model can be estimated using the method of residuals.

Another approach that can be quite useful is to perform a non compartmental analysis (NCA) of the data and derived estimates in the process.

Some computer software can use a range of typical parameter values and preform a multi-dimensional grid search to find a region near the minimum sum of the weighted residuals.

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