Kinetica - Indirect Response 1 - 2 Doses

Model description
Indirect response (IDR) Model I describes the time pattern of the mediator or response variable (R) that is affected by a drug, which inhibits the production process (K_in), normally controlling endogenous levels of R [1]. The response is produced at the zero-order rate and lost at the first-order rate (K_out) as shown in Fig.1.


Figure 1. Basic scheme for indirect response Model I.


The equation describing IDR Model I 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
• C_p – plasma drug concentration
• I_max – maximum inhibitory factor attributed to drug (0<I_max≤1)
• IC₅₀ – drug concentration producing 50% of maximum inhibition

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

Estimation of model parameters
It is assumed that drug plasma concentrations (Cp) for two doses are known prior to pharmacodynamic data analysis. The drug kinetics is incorporated into the model via explicit equation for Cp 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.,D₁, D₂, V, K_el) must be known and available for an input. Pharmacodynamic parameters for IDR Model I are: R₀, I_max, IC₅₀, K_out, and K_in.

• R₀: If the baseline response is known, then R0 must be fixed at the baseline value, otherwise R0 can be estimated.
• I_max : The maximum inhibition value must lie between 0 and 1. If the drug shows strong inhibition one can fix Imax=1.
• K_out and IC₅₀: 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 [2].

Equation 3 for monoexponential kinetics and Equation 1 for IDR Model I were used for simulations. The values of D₁=100 mg, D₂ =10 000 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, IC₅₀=1.0 mg/l and I_max=0.8 were used to simulate the response versus time profile. The 10% noise was introduced to the simulated data of drug response. The model parameters are listed in Table 2 and Fig. 2 shows the predicted response vs. time curve.

 

Table 1. Pharmacodynamic parameters for IDR Model I estimated with Kinetica (R₀=100 units, D₁=100 mg, D₂=10 000 mg, V=1l, and k_el=0.3 h^-1).




Fgure 2. IDR Model I fitting for two doses: D₁=100 mg ( • ) and D₂ =10 000 mg (0) to the simulated data with 10% random noise for a drug with monoexponential pharmacokinetics.


References

1. Dayneka NL, Garg V, JuskoWJ. Comparison of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 1993;21:457-478.
2. Sharma A, Jusko WJ. Characterization of four basic models of indirect pharmacodyna¬mic responses. J Pharmacokinet Biopharm 1996;24:611-635.