Problem Description
Target-mediated drug disposition (TMDD) is the process in which a drug binds with high affinity to its pharmacological target (for example, a receptor) to such an extent that this affects its pharmacokinetic characteristics.
A key characteristic of TMDD systems is that the pharmacokinetic behavior depends on thd dose. The figure below shows a typical free ligand concentration-time course with several doses of different magnitudes, using a bolus administration and a single compartment for the ligand.
PK compartments include free ligand in tissue and free ligand in plasma. PD compartments include free receptor and ligand-receptor conmplex
Resulting in nonlinear system of four differential equations (\(V\) is a volume of distribution)
\[\begin{array}{rcl} {d A \over d t} & = & k_{21} A \: + \: k_{12} L V \\ {d L \over d t} & = & {K_{21} A \over V} \: - \: \left ( k_{12} \: + \: k_{el} \right ) L \: - \: k_{on} L R \: + \: k_{off} P \\ {d R \over d t} & = & k_{syn} \: - \: k_{deg} R \: - \: k_{on} L R \: + \: k_{off} P \\ {d P \over d t} & = & k_{on} L R \: - \left ( k_{off} + k_{int} \right ) P \end{array}\]
Response Function Model
Results
Ligand doses of 0.1, 1.0, 10.0, 100.0, and 1000.0 are considered for this test.
The “data” points represent the \(L(t)\) points from the nolinear system of differential equations, using each of the 5 ligand input doses
Resulting \(L(t)\) curves are generated from the nonlinear response function model, using each of the 5 ligand input doses
Predictions are made for \(L(t)\) curves using ligand input doses of 40.0 and 400.0
\(L(t)\) curves from the nonlinear response model are compared to the data points from the nonlinear differential equation system. \(L(t)\) curves are labeled L1, L2, L3, L4, and L5. Prediction curves for the 2 prediction ligand doses are labeled P1 and P2. These resulting plots are shown here.
Using logged data, \(L(t)\) curves from the nonlinear response model are compared to the data points from the nonlinear differential equation system. \(L(t)\) curves are labeled L1, L2, L3, L4, and L5. Prediction curves for the 2 prediction ligand doses are labeled P1 and P2. These resulting plots are shown here.
Conclusions
The nonlinear response function provides accurate fits to data derived from the traditional nonlinear system of differential equations approach
The nonlinear response function model requires no additional fitting of parameters beyond those which are contained in the response function
The nonlinear response function model provides expected inter-dose time-course predictions
Given the oder-of-magnitude differences in ligand doses, a logged model provides the most accurate predictions across the entire dosing range