Kinetica - Cell Proliferation Models

Model description
This Cell Proliferation Model with Irreversible Inactivation describes irreversible reaction of a drug with receptors of the target cells, which produces inactivation or other loss of these cells. At the same time viable cells increase in number at their natural mitotic rate (ks). This is a basic model for the quantitation of the effects of cell-phase-nonspecific chemotherapeutic agents [1]. The pharmacodynamic response R becomes the cell number. The inactivation rate f(C_p) can be a linear or nonlinear function of the drug plasma concentration Cp. If the inactivation is linear, then

f(C_p) = k.C_p (1)

where: k = second-order inactivation rate constant,
Cp = plasma drug concentration,

For the nonlinear inactivation:

where: K_max = maximum inactivation,
KC₅₀ = drug concentration producing 50% of maximum inactivation.

The equation describing Cell Proliferation Model with Irreversible Inactivation is as follows:

where: R = drug response (cell number)
K_s = first-order cell-proliferation rate constant,
f(C_p) = first-order inactivation function.

This model predicts exponential growth of the response and therefore there is no baseline value. The initial condition R₀ for the above equation should be equal to the number of cells at time t = 0.


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, kel) must be known and available for an input. Pharmacodynamic parameters for Cell Proliferation Model with Irreversible Inactivation are: ks, and k (or K_max and KC₅₀).

ks , k , (or K_max and KC₅₀): These parameters are recommended to be fitted.

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

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


References
1. Jusko WJ. Pharmacodynamics of chemotherapeutic effects: Dose-time-response relationship for phase-nonspecific agents. J Pharm Sci 1971;60; 892-895.