Kinetica - Turnover Model

where:

• k – second-order inactivation rate constant
• C_p – plasma drug concentration
• R – measured drug response

For the nonlinear inactivation

where:

• K_max – maximum inactivation
• KC₅₀ – drug concentration producing 50% of maximum inactivation

The regeneration of the response is described by the zero-order rate constant (K_in) and lost by the first-order rate constant K_out). The model is shown in Fig.1.


Figure 1. Basic scheme for Turnover Model with Irreversible Inactivation.


The equation describing Turnover Model with Irreversible Inactivation is as follows:

where:

• R – drug response
• K_in – zero-order rate constant for production of drug response
• K_out – first-order rate constant for loss of drug response
• f(C_p) – first-order inactivation function

In the absence of drug the response stays at the baseline value (R0):

Estimation of model parameters
It is assumed that drug plasma concentrations (C_p) are known prior to pharmacodynamic data analysis. The drug kinetics is incorporated into the model via explicit equation for C_p e.g. for the monoexponential kinetics:

where:

• D – dose
• V – volume of distribution
• K_el – elimination rate constant

or via differential equation:

The entire set of pharmacokinetic parameters (e.g., D, V, K_el) must be known and available for an input. Pharmacodynamic parameters for Turnover Model with Irreversible Inactivation are: R₀, k_out, k (or k_max and kC₅₀), and k_in.

• R₀: If the baseline response is known, then R0 must be fixed at the baseline value, otherwise R₀ can be estimated
• k_out, k, or k_max and kC₅₀: These parameters are recommended to be fitted
• k_in: The rate of production of response is calculated from the baseline equation: k_in=k_out.R₀

The initial values of the pharmacodynamic parameters can be derived from the observed response data.

For Irreversible Linear Inactivation Model a single response data for large dose should suffice to estimate the values of model parameters. For Irreversible Nonlinear Inactivation Model, because of the large number of model parameters, a single response data might yield high standard deviations of some parameters. It is recommended to apply this model to two or more response data set to avoid high inaccuracy in parameter estimation.

Linear inactivation
The Eq. 5 for monoexponential kinetics and Eq. 1and 3 for Turnover Model with Irreversible Inactivation were used for simulations. The values of D=100 mg, V=1l, and K_el=0.3 h^-1 were used to simulate the plasma concentration-time profile, whereas R₀=100 units, k_out=0.1 h^-1, and k=0.01 l/mg.h were used to simulate the response versus time profile. The 10% noise was introduced to the simulated data of drug response. Kinetica was utilized to estimate pharmacodynamic parameters by fitting Eq. 1 and 3 to the noisy data. The model parameters are listed in Table 1 and Fig. 2 shows the predicted response vs. time curve.

Table 1. Pharmacodynamic parameters for Signal Transduction Model estimated with Kinetica (R₀=100 units, D=100 mg, V=1l, and k_el=0.3 h^-1)


Figure 2. Turnover Model with Irreversible Inactivation fitting (solid line) to the simulated data with 10% random noise (solid circles) for a drug with monoexponential pharmacokinetics


Nonlinear inactivation
The Eq. 5 for monoexponential kinetics and Eq. 2 and 3 for Turnover Model with Irreversible Inactivation were used for simulations. The values of pharmacokinetic parameters used to simulate the plasma concentration-time profile were analogous as for linear inactivation model, whereas whereas R₀=100 units, k_out=0.1 h^-1, k_max=1.0 h^-1 and kC₅₀=10.0 l/mg . h were used to simulate the response versus time profile. The 10% noise was introduced to the simulated data of drug response. Kinetica was utilized to estimate pharmacodynamic parameters by fitting Eq. 2 and 3 to the noisy data. The model parameters are listed in Table 2 and Fig. 3 shows the predicted response vs. time curve.

Table 2. Pharmacodynamic parameters for Signal Transduction Model estimated with Kinetica (R₀=100 units, D=100 mg, V=1l, and k_el=0.3 h^-1)

Figure 3. Turnover Model with Irreversible Inactivation fitting (solid line) to the simulated data with 10% random noise (solid circles) for a drug with monoexponential pharmacokinetics