Methods of solving rapid binding target-mediated drug disposition model for two drugs competing for the same receptor

Abstract

The target-mediated drug disposition (TMDD) model has been adopted to describe pharmacokinetics for two drugs competing for the same receptor. A rapid binding assumption introduces total receptor and total drug concentrations while free drug concentrations C (A) and C (B) are calculated from the equilibrium (Gaddum) equations. The Gaddum equations are polynomials in C (A) and C (B) of second degree that have explicit solutions involving complex numbers. The aim of this study was to develop numerical methods to solve the rapid binding TMDD model for two drugs competing for the same receptor that can be implemented in pharmacokinetic software. Algebra, calculus, and computer simulations were used to develop algorithms and investigate properties of solutions to the TMDD model with two drugs competitively binding to the same receptor. A general rapid binding approximation of the TMDD model for two drugs competing for the same receptor has been proposed. The explicit solutions to the equilibrium equations employ complex numbers, which cannot be easily solved by pharmacokinetic software. Numerical bisection algorithm and differential representation were developed to solve the system instead of obtaining an explicit solution. The numerical solutions were validated by MATLAB 7.2 solver for polynomial roots. The applicability of these algorithms was demonstrated by simulating concentration-time profiles resulting from exogenous and endogenous IgG competing for the neonatal Fc receptor (FcRn), and darbepoetin competing with endogenous erythropoietin for the erythropoietin receptor. These models were implemented in ADAPT 5 and Phoenix WinNonlin 6.0, respectively.

Fig. 1

Target-mediated drug disposition for two drugs competing for the same receptor. Symbols are defined in the theoretical.

Fig. 2

Simulated concentration-time profiles for escalating IV bolus doses (100, 500, 1000 units for both A and B) using TMDD model with two ligands competitively binding to the same target. V_c = 10, k_elA = k_elB = 0.01, k_ptA = k_ptB = k_tpA = k_tpB = 0, C_A0 = C_B0 = 0, k_intA = k_intB = 0.1, R_tot0 = 50, k_deg = 0.02. For upper two panels, K_DA = K_DB = 1. For lower two panels, K_DA = 1, K_DB = 0.1. Other parameters used for simulations are listed in Table 1. Simulations were performed in MATLAB using the algebraic solution of equilibrium equations.

Fig. 3

A flow chart illustrating the bisection algorithm for solving equation f(z) = 0. The START and STOP steps denote the beginning and end of the algorithm, respectively. The rectangular boxes represent the assignment steps whereas the diagonal boxes refer to conditional statement with two possible outcomes Yes (if condition is true), and No (if condition is false). The arrows indicate the next steps. The meanings of the symbols a_A, a_B, acc, and N_max are explained in the methods and results.

Fig. 4

Model diagram for IgG pharmacokinetics with exogenous and endogenous IgG competing for FcRn receptor. Symbols are defined in Example 1.

Fig. 5

Endogenous IgG plasma concentration profiles (upper panel) and exogeneous plasma concentration profiles (lower panel) after administration of 10 mg/kg (6.67 nmole/kg) exogenous IgG the equilibrium dissociation constant K_DB = K_DA, 0.1K_DA, and 0.01K_DA. Simulations were performed in ADAPT 5 using the differential solution of equilibrium equations.

Fig. 6

Model diagram for target-mediated drug disposition for darbepoetin competing for EPO receptor with endogenous erythropoietin. Symbols are defined in Example 2.

Fig. 7

Simulated concentration-time profiles for escalating IV bolus doses (0.1, 0.02, 0.002 nmol/kg) of darbepoetin (DA). Upper panel: concentration-time profiles of endogenous EPO. Dash-dot line represents the baseline EPO level. Middle panel: concentration-time profiles of darbepoetin. Lower panel: concentration-time profile of the sum of DA and EPO. Solid lines represent these profiles when k_EPO = 0.00043 nM h^-1. Dash lines represent these profiles when k_EPO = 0. Other parameters for simulation are listed in Table 2. Simulations were performed in Phoenix WinNonlin using the bisection method of solving equilibrium equations.

Fig. 8

Graphical representation of solutions of the equilibrium equations when K_DA ≠ K_DB (upper panel) and K_DA = K_DB (lower panel). The equilibrium equations Eqs. 27 and 28 are equivalent to a system of two hyperbolic equations represented by the solid lines. Their intersection coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃) (upper panel), and (x₁, y₂), (x₂, y₂) (lower panel) are all possible solutions of the system. The dashed lines represent the asymptotes for the hyperbolas. In case K_DA = K_DB (equivalent to k_A = k_B) the diagonal asymptotes collapse to a single one reducing the number of solutions to two. The vertical and horizontal asymptotes intersect the axes at a_A and a_B, respectively. Only the solution (x₁, y₁) is inside the rectangle of vertices defined by 0, a_A, and a_B.