Population PK/PD analysis of metformin using the signal transduction model

Introduction

Metformin, a biguanide glucose lowering agent, is commonly used to manage type 2 diabetes [1]. Metformin is used as monotherapy, as an adjunct to diet for managing type 2 diabetes mellitus in patients whose hyperglycaemia cannot be controlled by diet alone [2]. Metformin may also be used in combination with other antidiabetic agents in patients with type 2 diabetes who do not achieve adequate glycaemic control with a sulfonylurea agent alone [3]. The glucose lowering effect of metformin is primarily the result of reduced hepatic glucose output through inhibition of gluconeogenesis and glycogenolysis [4]. The glucose lowering effect of metformin vs. the plasma concentration curve shows a counter clock-wise hysteresis loop [5].

Biophase models are most appropriate when delay in drug action occurs from the distribution site to the site of action [6]. However, this modelling approach is often applied inappropriately when the most relevant underlying process that causes the delay is not drug distribution. Other reasons, such as glucose turnover, signalling pathways or translocation of glucose transporters, may be more relevant to the action of antidiabetic drugs [7]. Moreover, the turnover and homeostasis of glucose and insulin is not accounted for by biophase models. Signal transduction models describe a drug mechanism that immediately alters the production or loss of endogenous substances, and, therefore, are the most useful when turnover (glucose concentration) can be measured directly [8].

The pharmacokinetics (PK) and pharmacodynamics (PD) of metformin have been investigated in healthy humans and patients with type 2 diabetes mellitus [5, 9–11], and modelling has been performed using an indirect response model [5]. However, the indirect response model is insufficient to explain the PK/PD relationship and describe the visual inspection of metformin. No reported study clearly describes the glucose lowering effect and the plasma concentrations of metformin in healthy humans using the NONMEM program and the signal transduction model method.

The objectives of this study were to examine the relationship between the plasma concentration of metformin and its antihyperglycaemic effect in healthy humans following administration of a single 500 mg metformin tablet. A Monte Carlo simulation was performed using the ADAPT 5 program (Biomedical Simulation Resource, Los Angeles, CA, USA) to predict plasma glucose concentrations in patients with diabetes. The model was used to predict the antihyperglycaemic effect in patients with type 2 diabetes. The predicted plasma glucose concentration value was similar to that of previous studies [12, 13] in patients with diabetes. Thus, the proposed model was able to predict the antihyperglycaemic effect.

Methods

Study design

All subjects fasted for at least 12 h prior to dosing. At time zero, an intravenous cannula was inserted into a forearm vein and blank blood samples were collected. After baseline blood sampling, the metformin tablet (Diabex 500 mg, Daewoong) was orally administered with 200 ml water. All volunteers consumed 12 g of sugar 20 min after drug administration. Blood samples to determine plasma metformin were taken at 0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 10 and 12 h after drug administration. In addition, plasma glucose concentration was measured at the same time after drug administration. All subjects abstained from food until 4 h after drug administration. The blood samples were collected in heparinized tubes, immediately centrifuged (10 min, 3000 rev min⁻¹), and stored at −20°C until LC-MS/MS analysis. An identical study was performed without metformin administration 1 week later to determine the glucose concentration without metformin (Figure 1).

Population PK/PD analysis

Pooled data from the healthy volunteers were analyzed with NONMEM VI (ICON Development Solutions, Ellicott City, MD, USA) [15] with a G77 FORTRAN compiler. Analysis and post processing were performed with the aid of the PsN toolkit [16] and Xpose (ver. 4) [17], programmed in the statistics package R.

A sequential modelling approach was used to facilitate the data analysis. The population PK analysis was performed in the first step. Using the individual PK parameter estimates as part of the input, population PD analysis was performed. An exploratory two-stage PK analysis was performed to identify the optimal structure model, using a one compartment body model with first order absorption and elimination. The differential equations used to describe the PK model were as follows:

where X is the mass of the compound in each compartment (1: absorption, 2: central), K_a is the absorption rate constant (units of inverse time), V₂ is the apparent volume of the central compartment, and CL is the elimination clearance (units of volume per time).

The between-subject variability for the PK parameters was developed as an exponential error model as follows:

where P_i is the parameter for the ith subject, θ is the typical population value of the parameter, and η_i is the random intersubject effect with a mean of 0 and variance ω². The within-subject variability was developed as an additive error model, as follows:

where Y_ij and Y p_ij present the i^th subject's j^th observed and predicted concentration, respectively, and ε is the random residual error, which is normally distributed with a mean of 0 and variance σ². A first order conditional estimation method applying an additive error model was used to estimate the parameters. Individual subject PK parameters were calculated using the NONMEM posterior conditional estimation technique. In general, the additive error model is applied when the variance is assumed to have a constant absolute magnitude and independent for all measurements. In the case of PK concentrations, where a wide range of concentrations is measured, the error in measurement based on the analytical method is usually a combination of additive and proportional error. However, when we compared the additive, proportional and combination error model, the additive model was the most appropriate for the PK of metformin.

The model that was previously suggested to describe signal transduction processes [18, 19] was used to model the delayed antihyperglycaemic effect of metformin (Figure 2). Drug concentration in the central compartment (C_p= X₂/V₂), derived from the PK model, was linked to mean response vs. time data using the follow set of equations:

where DR represents a hypothetical drug–receptor complex that produces M₁, a secondary messenger, E_max is maximal M₁ production and EC₅₀ is the drug concentration that produces 50% of E_max. This model represents a cascade process in which M₁ is transformed into another secondary messenger M₂ and M₂ is further transformed into M₃. The most parsimonious three-step signal transduction model was applied, assuming that the mean transit time (τ) of the biosignal through all transit compartments, M₁, M₂, and M₃, was identical [20].

M3 represents the change in the antihyperglycaemic effect. The measured response is:

The proportional error was used for pharmacodynamics part of the model.

Results

PK/PD

The individual posterior Bayesian estimates of the PK parameters (CL/F and V/F) generated from the base model were plotted against the characteristics of healthy volunteers (gender, age, height, weight, CL_cr, total bilirubin, and haemoglobin). CL_cr was highly correlated with CL/F and the incorporation of CL_cr into the base model produced a significant decrease in the objective function value (Δ OFV =−9.22, P < 0.01). Comparing base model estimates with final model estimates, the η shrinkage for CL/F was relatively small (26.22%). It was considered appropriate to use the plots of the random effects (η) of PK parameters vs. covariates for screening potential covariates [26]. The structure of the final PK model was:

PK parameter estimates obtained from this model and the 95% CIs from the bootstrap analysis are shown in Table 2[27, 28]. The mean apparent volume of distribution (V/F) was 113 l (RSE, 4.18%). K_a and apparent clearance (CL/F) were estimated as 0.41 h⁻¹ (2.43%) and 52.6 l h⁻¹ (4.18%). Most parameters showed a reasonable amount of inter-individual variability (≤41%). The observed bootstrap means were generally consistent with the population mean estimates. The coefficient of variation for the random residual constant was reasonable (23%). The incorporation of CL_cr into the base model explained part of the inter-individual variability of CL/F, with its value decreasing, from 47% to 41%.

Table 2. Pharmacokinetic and pharmacodynamic parameters and estimates of variability

Parameter, unit	Definition	Population mean (% RSE)	Bootstrap median (95% CI*)	Interindividual variability (IIV) CV%	Bootstrap IIV CV% (95% CI*)
Pharmacokinetic
CL (l h−1)	Apparent clearance	52.6 (4.18)	52.9 (48.5, 56.7)	29.7	27.9 (26.5, 29.1)
V (l)	Apparent volume of distribution	113 (56.6)	113 (100, 126)	22.1	22.2 (20.2, 24.1)
Ka (h−1)	Absorption rate constant	0.41 (2.43)	0.41 (0.39, 0.43)	–†	–†
Residual error (ng ml−1)		23.0 (11.7)	23.1 (21.7, 25.2)
Pharmacodynamic
τ	Mean transit time	0.50 (2.97)	0.48 (0.48, 0.51)	–†	–†
Emax	Maximum simulation	19.8 (3.17)	20.3 (19.2, 20.4)	–†	–†
EC50	Simulation constant	3.68 (3.89)	3.68 (3.49, 3.81)	–†	–†
r	Hill coefficient	0.55 (9.05)	0.59 (0.50, 0.60)	4.05	4.07 (3.86, 4.22)
Residual error, %		40.4 (1.52)	40.4 (38.2, 42.9)

• *CI, confidence interval calculated from 1000 bootstrap resamplings. †Not estimated. % RSE, relative standard error for estimate; CV, coefficient of variation.

Goodness of fit plots for the final PK/PD model are shown in 3, 4. The plots of observed concentration vs. population predicted concentrations (A) and individual predicted concentrations (B) showed the better visual agreement between predicted and observed data. These plots show an improvement in fit with the latter (B) including covariates, observed as tighter and more random scatter about the identity line and good concordance. The diagnostic plots of the final population PK/PD model revealed no systemic bias. The conditional weighted predictions for the final population PK/PD model were generally distributed around zero and were relatively symmetrical.

The signal transduction model was sequentially fitted to metformin concentrations and the antihyperglycaemic effect. A population PD analysis was performed using the individual PK parameter estimates as part of the input. The summary of the population PD parameters obtained from the final PK/PD model is listed in Table 2. The population means for efficacy (E_max) and potency (EC₅₀) were estimated to be 19.8 and 3.68 µg ml⁻¹. The estimated transit time (τ) was about 0.5 h. The Hill coefficient (r) was estimated to be 0.55 and between-subject variability was 4.05%. Bootstrap CIs for PD parameters were obtained (Table 2). The coefficient of variation for the random residual constant was 40.4%.

Monte Carlo simulations were performed with the final covariate model to compare the distribution of simulated metformin concentrations and fasting plasma glucose (FPG) cconcentrations with that of the observed data following administration of metformin 500 mg in healthy humans (Figure 5). With the exception of some metformin and FPG concentrations, most of the observed data fell into the range between the 5^th to 95^th percentiles of the simulated values. Overall, the final model was able to describe the observed metformin and FPG concentrations reasonably well.

Discussion

The molecular mechanisms of metformin have not been fully identified, but turnover of biomarkers such as glucose and signalling pathways or translocation of glucose transporters are closely related to the glucose-lowering effects of metformin. Furthermore, the counterclockwise hysteresis loops observed in the metformin plasma concentration−glucose concentration changes indicate the presence of a time delay between the change in plasma concentration and the effect of the drug. Compared with the indirect response model, the signal transduction model had better goodness of fitness and lower objective function value (Δ OFV =−14.21). Thus, in this study, the population PK/PD analysis using a signal transduction model in healthy humans was developed and used. Moreover, the simulation was performed to predict metformin plasma concentrations and the glucose lowering effects in patients with diabetes using a Monte Carlo simulation.

The PK of metformin were best described by a one compartment model with first order absorption and elimination. CL_cr was an influential covariate explaining part of the variability in the CL/F of metformin. This is because, in accordance with clinical observations, metformin is not metabolized and is primarily eliminated unchanged, through renal excretion [12, 13, 29, 30].

Despite the limited dose range and interindividual variability, the estimated EC₅₀ (3.68 µg ml⁻¹) was comparable with values reported in previous studies (2.26 µg ml⁻¹[5], 4.23 µg ml⁻¹[11]), in which the plasma glucose time course was obtained and the glucose-lowering effect of metformin was investigated after administering metformin to healthy volunteers and patients with type 2 diabetes mellitus.

To predict a patient's FPG values following metformin administration as defined by the final structure signal transduction model, Monte Carlo simulations were performed (n= 1000) for the regimen of 500 mg metformin orally administered twice daily for 2 weeks. Monte Carlo methods were generated using the final model based on the central tendency and dispersion of each PK and PD parameter. The simulated PK and PD of the drug represent the wide spectrum of subjects with better predictive capabilities for the PK/PD of drugs. Due to this benefit, Monte Carlo simulations are being used increasingly to predict the PK/PD variability of diabetic agents in a population [11, 31]. In a Monte Carlo simulation, the distribution of the model parameter values must be known and used as inputs. Diabetes mellitus diagnostic criteria values were used to simulate the glucose concentration in patients with diabetes. The FPG of a patient with diabetes is >126 mg dl⁻¹ and the 2 h postprandial glucose is >200 mg dl⁻¹, so the initial FPG value for patients was set between 126 mg dl⁻¹ and 200 mg dl⁻¹. In a previous study, patients with type 2 diabetes (n= 182) received 500 mg metformin twice daily for 104 weeks and changes in glucose plasma were observed [32]. The FPG value was initially 178.1 mg dl⁻¹. After 104 weeks, the glucose concentration was 141.3 mg dl⁻¹. In another study, patients with type 2 diabetes (n= 435) received 500 mg metformin twice daily for 24 weeks and changes in plasma glucose were observed [33]. The FPG value was initially 142.1 mg dl⁻¹. After 24 weeks, the glucose value was 122.8 mg dl⁻¹. The mean values and SDs of the parameters (n= 42) obtained from our final signal transduction model were used as inputs for a Monte Carlo simulation, and the baseline plasma glucose concentration from a previous study (plasma glucose = 142.1 mg dl⁻¹, 178.1 mg dl⁻¹) was set as the initial plasma glucose value, and reasonable PK/PD predictions (n= 1000) for 500 mg metformin orally administered twice daily for 2 weeks were simulated. The steady-state predicted FPG concentrations were compared with those reported in a previous study. Simulated FPG concentrations (initial FPG baseline, 178.1 mg dl⁻¹, 142.1 mg dl⁻¹) decreased by about Δ 30.2 mg dl⁻¹ (95% CI 22.8, 38.1) and Δ 24.0 mg dl⁻¹ (95% CI 16.7, 31.3), respectively. Both values were similar to the values of a previous study of Δ 36.8 mg dl⁻¹ (95% CI 28.3, 43.5) [32] and Δ 19.3 mg dl⁻¹ (95% CI 12.5, 26.1), respectively [33]. Thus, the model we developed for PK/PD of patients with diabetes seemed reliable.

In conclusion, a population signal transduction model was developed and evaluated using metformin in healthy volunteers. Model evaluation using nonparametric bootstrap analysis suggested that the proposed model was robust, and that the values were estimated with good precision. Although the model was developed initially in healthy volunteers with normal renal function, the simulation results revealed that the model was useful to predict FPG concentrations in patients with type 2 diabetes. The model was appropriate to predict the time course of plasma metformin and of FPG concentrations in patients with type 2 diabetes and may be useful in the design and analysis of future studies evaluating metformin in combination with other drugs for treating type 2 diabetes mellitus.

Competing Interests

There are no competing interests to declare.