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 (kin), normally controlling endogenous levels of R [1]. The response is produced at the zero-order rate and lost at the first-order rate (kout) 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 plasma drug concentrations (C_p) are known prior to pharmacodynamic data analysis. The drug kinetics is incorporated into the model via an 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 IDR Model 1 are: R₀, I_max, IC_50, 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 I_max=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].

The classic example of IDR Model I is the inhibition of prothrombin complex activity by warfarin, an oral anticoagulant used in thrombophlebitis and pulmonary embolism. It blocks the vitamin K epoxide reductase, enzyme that reduces vitamin K epoxide to vitamin K, which is a cofactor for carboxylation of the clotting factor such as factor II (prothrombin), VII, IX, and X. The blockade of the reductase activity by warfarin leads to the inhibition of coagulation. This effect is usually measured with the use of prothrombin time. It is assumed that the clotting factors are synthesized with a zero-order rate constant (kin) and degraded with a first-order rate constant (kout) as shown in Fig 2.


Figure 2. Schematic representation of IDR Model I for inhibition of prothrombin complex activity by warfarin.


After oral administration of 1.5 mg/kg sodium warfarin to five healthy subjects, the disposition of warfarin was found to be biexponential [3]. Pharmacodynamic response was defined as a decrease of normal prothrombin complex activity. These data were reanalyzed by Jusko and Ko [3].
Plasma warfarin concentrations were described by the first-order absorption and biexponential disposition equation [3]:

where: A, B – intercept coefficients
α,β - slopes
K_a - first-order absorption rate constant


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 pharmacodynamic responses. J Pharmacokinet Biopharm 1996;24:611-635.
3. JuskoWJ, Ko HC. Physiologic indirect response models characterize diverse types of pharmacodynamic effects. Clin Pharmacol Ther 1994;56:406-419.
4. Nagashima R, O’Reilly RA, Levy G. Kinetics of pharmacologic effects in man: the anticoagulant action of warfarin. Clin Pharmacol Ther 1969;10:22-35.