Kinetic‐pharmacodynamic model for drugs with non‐linear elimination: Parameterisation matters

1. INTRODUCTION

Kinetic‐pharmacodynamic (KPD) models are used to predict the time course and magnitude of drug effects in the absence of pharmacokinetic (PK) data.1, 2, 3, 4 They have found utility in situations where it is not possible or feasible to collect plasma concentration data, but where a pharmacodynamic (PD) marker can be measured. In this case, a KPD model can be fitted to the PD data while the ‘kinetic’ behaviour of the drug is inferred. In most cases, the ‘kinetic portion of the KPD model is represented by a standard one‐compartment model with intravenous bolus input and linear elimination.

Two parameterisations have been proposed for linking the ‘kinetic’ and PD portions in a KPD model. Most commonly, the elimination rate of the drug from the ‘kinetic’ compartment is estimated and used to drive the PD effect.3 For an inhibitory function this is given by;

$$I=1-\frac{I_{\mathit{max}}·\mathit{ER}}{\mathit{ED}{K}_{50}+\mathit{ER}},$$ (1)

where ER denotes the drug elimination rate, EDK₅₀ is the elimination rate that gives half‐maximal effect (analogous to the C₅₀ value in a standard E_max model), and I_max is the maximum inhibitory effect. An alternative parameterisation is to use the amount of drug in the body (A) to drive the PD effect1 given by;

$$\begin{array}{c}I=1-\frac{I_{\mathit{max}}·A}{A_{50}+A}.\end{array}$$ (2)

where A₅₀ is the amount of drug that gives half maximal effect. Both parameterisations perform equivalently for a drug with linear pharmacokinetics.

In a recent paper, Gonzales‐Sales and colleagues5, 6 performed a simulation‐based study looking at the predictive performance of KPD models and concluded that they should not be used in situations where the drug exhibits non‐linear elimination. Other authors have also proposed that KPD models should only be used in restricted settings, e.g. in the setting of linear pharmacokinetics and in the absence of any delayed drug effects or system recursiveness (i.e. feedback, tolerance or rebound).4, 5

We hypothesise that the choice of parameterisation (A₅₀ or EDK₅₀) will impact KPD model performance, particularly for drugs with non‐linear PK. In this commentary, we will therefore explore the performance of KPD models with both the EDK₅₀ and A₅₀ parameterisations in the setting of non‐linear elimination.

A stochastic simulation and estimation (SSE) study was conducted. Simulation and estimation were performed using NONMEM (v.7.3, ICON Development Solutions, Ellicott City, MD, USA).

2. SIMULATION OF REPLICATION DATASETS

Simulations of a single dose were conducted at 4 mg, 8 mg, and 16 mg. A total of 90 patients were simulated with equal number of patients (n = 30) in each dose stratum. For each patient, nine PD biomarker observations were made at 0, 6, 12, 24, 48, 72, 96, 120, and 144 hours. 500 datasets were simulated (termed replication datasets).

In the first step, the reference datasets were simulated using a full pharmacokinetic‐pharmacodynamic (PKPD) model for a hypothetical drug with non‐linear elimination and delayed drug effects. The time course of the hypothetical PD response marker, R, was described using a turnover model with a zero‐order input (R_in) and a first‐order output (rate constant k_out). The drug effects were assumed to inhibit R_in. The drug concentrations were given by a one‐compartment model with an intravenous bolus input and with a non‐linear elimination represented by the Michaelis–Menten equation. The drug concentrations, C, were used to drive the I_max model such that:

$$\begin{array}{c}I=1-\frac{I_{\mathit{max}}·C}{C_{50}+C},\end{array}$$ (3)

where C₅₀ denotes the drug concentration that results in half the I_max. The system of ordinary differential equations is as follows:

$$\begin{array}{c}\frac{\mathrm{dC}}{\mathrm{dt}}=-\frac{\frac{{\mathrm{V}}_{\max }}{\mathrm{V}}}{{\mathrm{K}}_{\mathrm{M}}+\mathrm{C}}·\mathrm{C};{\mathrm{C}}_{\mathrm{t}=0}=\frac{\mathrm{D}}{\mathrm{V}}\\ \frac{\mathrm{dR}}{\mathrm{dt}}={\mathrm{R}}_{\text{in}}·\mathrm{I}-{\mathrm{k}}_{\text{out}}·\mathrm{R};{\mathrm{R}}_{\mathrm{t}=0}=\frac{{\mathrm{R}}_{\text{in}}}{{\mathrm{k}}_{\text{out}}}.\end{array}$$ (4)

Here, D is the dose, t is the time, and V is the volume of distribution. The non‐linear elimination is described by the Michaelis–Menten equation where V_max is the maximum drug's elimination rate and K_M is the Michaelis–Menten constant.

Individual parameters were assumed to be lognormally distributed, ${\theta }_i=\theta ·e^{{\eta }_i}$, where η_i~N(0, ω²) and η_i represents the random between‐subject effects for the i ^th individual. The residual error was described using a proportional error model, y_ij = f_ij(θ_i, t_ij) · (1+ε_ij), where f_ij represents the individual prediction for the i ^th individual and j ^th time point, t_ij is the time, y_ij is the observation, and ε_ij is the error terms where ${\epsilon }_{\mathit{ij}}~N\left(0,{\sigma }_{\mathit{\text{prop}}}^2\right)$.

The parameter values for the reference model were as follows: V_max = 0.08 mg/h, K_M = 0.1 mg/L, V = 10 L, R_in = 7 unit/h, k_out = 0.1/h, I_max = 1, C₅₀ = 0.4 mg/L, ω² (V_max) = 0.1, ω² (K_M) = 0.1, ω² (V) = 0.1, ω² (R_in) = 0.1, ω² (k_out) = 0.1, ${\sigma }_{\mathit{\text{prop}}}^2\left(C\right)=0.01$, and ${\sigma }_{\mathit{\text{prop}}}^2\left(R\right)=0.01$.

3. ESTIMATION

Two models were fitted to the replication datasets, a) a KPD model with the EDK₅₀ parameterisation (termed EDK50 model),

$$\begin{array}{c}\begin{array}{c}\frac{\mathit{dA}}{\mathit{dt}}=-\frac{V_{\mathit{max}}}{K_M^{*}+A}·A;A_{t=0}=D\\ \frac{\mathit{dR}}{\mathit{dt}}=R_{\mathit{\text{in}}}·I-k_{\mathit{\text{out}}}·R;R_{t=0}=\frac{R_{\mathit{\text{in}}}}{k_{\mathit{\text{out}}}},I=1-\frac{I_{\mathit{max}}·\mathit{ER}}{{\mathit{EDK}}_{50}+\mathit{ER}},\mathit{ER}=\frac{V_{\mathit{max}}}{K_M^{*}+A}·A,\end{array}\end{array}$$ (5)

and b) a KPD model with the A₅₀ parameterisation (termed A50 model),

$$\begin{array}{c}\begin{array}{c}\frac{\mathit{dA}}{\mathit{dt}}=-\frac{V_{\mathit{max}}}{K_M^{*}+A}·A;A_{t=0}=D\\ \frac{\mathit{dR}}{\mathit{dt}}=R_{\mathit{\text{in}}}·I-k_{\mathit{\text{out}}}·R;R_{t=0}=\frac{R_{\mathit{\text{in}}}}{k_{\mathit{\text{out}}}},I=1-\frac{I_{\mathit{max}}·A}{A_{50}+A}.\end{array}\end{array}$$ (6)

$K_M^{*}$ is the amount‐version of the Michaelis–Menten constant (unit: milligram). The first‐order conditional estimation method with interaction was used with the convergence criterion and the precision of integration solution set to three (SIG = 3) and nine (TOL = 9) significant digits, respectively.

4. RESULTS

The Akaike's Information Criterion (AIC), parameter estimates, and relative standard error (RSE) were summarised across the 500 replication datasets using the 2.5^th, 50^th (median), and 97.5^th percentiles. The 2.5^th and 97.5^th percentiles gave the 95% credible interval (CrI) for the quantity of interest. On the basis of the AIC, the A50 model provided significantly better fit to the data compared to the EDK50 model (ΔAIC = 435 [95 % CrI 233,  638]). Visual inspection of visual predictive checks and the residual plots showed no discernible deviation in model fit for the A50 model while misspecification was evident for the EDK50 model (not shown). The A50 model was also associated with better rate of successful convergence (100% vs 97.4%) and covariance step (97.4% vs 76.8%) compared to the EDK50 model.

The A50 model produced unbiased parameter estimates (see Figure 1) while R_in, k_out, and V_max were biased for the EDK50 model. The parameters of the A50 model were also precisely estimated with %RSE <30% for fixed‐effects parameters and < 50% for random‐effects parameters. By contrast, the between‐subject variance (BSV) parameters for the EDK50 model were poorly estimated with %RSE (95% CrI) values of: a) $\%\mathit{RS}{E}_{\mathit{BSV}\left(V_{\mathit{max}}\right)}=49.6\left(95\%\mathit{CrI}18.9,640\right)\%$; b) $\%\mathit{RS}{E}_{\mathit{BSV}\left(R_{\mathit{\text{in}}}\right)}=30.6\left(95\%\mathit{CrI}16.9,251\right)\%$; c) $\%\mathit{RS}{E}_{\mathit{BSV}\left(k_{\mathit{\text{out}}}\right)}=30.7\left(95\%\mathit{CrI}16.0,195\right)\%$. It is important to note that $K_M^{*}$ and V_max estimates were highly correlated for both the A50 model (ρ = 0.880 [95 % CrI 0.745,0.961]) and the EDK50 model (ρ = 0.820 [95 % CrI 0.675,0.897]), indicating that independent estimation of $K_M^{*}$ and V_max was not possible given the available data. Hence, $K_M^{*}$ was heuristically fixed to one in the final model.

Figure 1.

Relative estimation error (REE) of parameter estimates of the A50 model (blue boxplot) and EDK50 (yellow boxplot). REE was calculated as the percentage difference of the parameter estimate from the reference parameter value. C₅₀ is the drug concentration in the body that gives half‐maximal effect. C₅₀ was derived for the A50 model based on C₅₀ = A₅₀/V and with the true value of V = 10 L assumed. C₅₀ was unable to be derived for the EDK50 model because CL varies with the amount of drug in the body, A, such that $\mathit{CL}=V_{\mathit{max}}·V/\left(K_M^{*}+A\right)$ i.e. CL is not a constant. The A50 model is the KPD model with the A₅₀ parameterisation. The EDK50 model is the KPD model with the EDK₅₀ parameterisation. $K_M^{*}$ Michaelis–Menten constant (amount‐version). R_in is the zero‐order input rate of response. k_out is the first‐order degradation rate of response. V_max is the maximum rate of elimination

5. DISCUSSION AND CONCLUSIONS

In this commentary, we have generated evidence to contest the view that KPD models cannot be used in the setting of non‐linear drug elimination. Our findings suggest that this is an issue of parametrisation, with the A₅₀ model producing unbiased and more precise parameter estimates than the EDK₅₀ model. This is because the EDK₅₀ parameter is a composite of CL and C₅₀ i.e. EDK₅₀ = CL · C₅₀ 3 so in the setting of non‐linear elimination where $\mathit{CL}=\frac{V_{\mathit{max}}}{K_M+C}$, the EDK₅₀ parameter becomes dependent on the drug concentration and hence varies with time. We suggest that it is incorrect in principal to estimate EDK₅₀ as a constant in the setting of non‐linear elimination. By contrast, the A₅₀ parameter is a composite of V and C₅₀ 3 and will be unaffected by non‐linearity in CL.

We note that the A50 parameterisation performs equivalently to the EDK50 approach when the drug has linear elimination (not shown).

We acknowledge some limitations that should be considered when interpreting our work. We have explored only a simple case of non‐linear elimination and have not examined other sources of non‐linearity in PK, e.g. non‐linear absorption. The PD model explored here assumed that the drug will inhibit the rate of biomarker production, but other scenarios (including a nonlinear PD model) for describing the PD effect exist and should be explored in future work.

In summary, a KPD model with a non‐linear ‘kinetic’ structure and A₅₀ parameterisation, provided unbiased and precise parameter estimates for a drug with non‐linear elimination. Our results suggest that KPD models with the A₅₀ parameterisation can be considered in the setting of non‐linear elimination.

COMPETING INTERESTS

There are no competing interests to declare.

Ooi QX, Hasegawa C, Duffull SB, Wright DFB. Kinetic‐pharmacodynamic model for drugs with non‐linear elimination: Parameterisation matters. Br J Clin Pharmacol. 2020;86:196–198. 10.1111/bcp.14154

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.