Kinetica - Signal Transduction Model

Model description
Pharmacological effects of endogenous compounds (hormones) and exogenous substances (therapeutic drugs) are often produced via signal transduction processes. These cascade responses are initiated by the interactions between hormone or drug molecules and their specific receptors. For some transduction pathways, second messengers (such as cyclic AMP, calcium ion, etc.) are involved in the processes. These messengers play important roles in regulating the cascade steps in multiple processes leading to their pharmacological end points.

The Signal Transduction Model links the pharmacokinetic profile of the tested compound, receptor occupancy, and cascade steps for the signal transduction process as shown in Fig. 1. In this model drug (D) binds to the receptor (R) forming drug-receptor complex (DR). According to the law of mass action and assuming equilibrium conditions and reversible binding, DR concentrations can be defined as

DR = B_maxD/(K_D+D)

where B_max is the total concentration of receptors and K_D is the equilibrium dissociation constant for this process. The complex initiates the signal transduction process represented here as a single transit compartment at the receptor site. Based on receptor occupancy theory, the biological effector signal directly proportional to the concentration of DR is E*=ε.DR, where ε is the intrinsic activity of the drug. The maximum induced signal (E_max) is assumed to occur when all receptors are occupied, thus E_max = ε.B_max and

where:

• C_p – plasma drug concentration
• E_max – maximum induced signal
• EC₅₀ – drug concentration producing 50% of maximum induced signal (EC₅₀=K_D)

The signal is delayed by the mean transit time (τ), producing an observed effect (E). The production and loss of E is dependent on first-order rate constants which are equivalent to the reciprocal of the transit time. The observable effect can be described as follows:

where:

• E – measurable drug effect
• τ –mean transit time

Baseline values of the PD marker (E₀) can be reflected by allowing the measured effect to equal E₀+E.

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Figure 1. Basic Scheme for Signal Transduction Model.


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 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 Signal Transduction Model are: τ, EC₅₀, and E_max and it is recommended that they are fitted.
For the Signal Transduction Model, because of the large number of model parameters, a single response profile might yield high standard deviations of some parameters. It is recommended to apply this model to two or more response data sets to avoid inaccuracy in parameter estimation. The initial values of the pharmacodynamic parameters can be derived from the observed response data.

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

Figure 2. Signal Transduction Model (Eq. 3) fitting (solid line) to the simulated data with 10% random noise (solid circles) for a drug with monoexponential pharmacokinetics (Eq. 4).


References
1. Y-N. Sun and W. J. Jusko, Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics, J. Pharmaceut. Sci. 87: 732-737 (1998).