The design of the simulation study is shown in Fig. 1. Steps 1 to 5 are explained into detail below.
Simulation of pharmacokinetic and pharmacodynamic data (step 1)
PK and PD data were simulated for nine compounds with various pharmacokinetic profiles. (Figure 2) A three-compartment model was used with or without saturable elimination and with or without auto-induction. (Figure 3) Drug elimination was described by the Michaelis Menten parameters V
max (maximal elimination rate) and K
m (concentration related to half-maximal elimination). K
m was relatively low (1 mg/L) or relatively high (100 mg/L) representing saturable and linear drug elimination, respectively. Auto-induction was described by an enzyme turnover model. Drug concentrations in the central compartment were related to an increased input rate into the enzyme compartment, resulting in an increased concentration of a hypothetical amount of enzyme and a proportional increase of the drug elimination rate. Drug elimination was maximally induced by 100% (I
max), the effective concentration related to half-maximal induction (ECI50) was 1 mg/L and the mean turnover time of the enzyme (MTTenzyme) was 24 h. The distribution volumes of the peripheral compartments were relatively small (1 L) or relatively large (10 L) to vary the magnitude of drug distribution. Inter-individual and residual random effects were moderate or large. (Table 1) The differential equations of the pharmacokinetic models are given below.
$$V_{1} \cdot \frac{{{\text{d}}C_{1} }}{{{\text{d}}t}} = - V_{{\max }} \cdot \frac{{C_{1} }}{{C_{1} + K_{m} }} \cdot A_{4} - Q_{2} \cdot C_{1} + Q_{2} \cdot C_{2} - Q_{3} \cdot C_{1} + Q_{3} \cdot C_{3} $$
(1)
$$V_2 \cdot \frac{{{\text{d}}C_2 }}{{{\text{D}}t}} = Q_2 \cdot C_1 - Q_2 \cdot C_2 $$
(2)
$$V_3 \cdot \frac{{{\text{d}}C_3 }}{{{\text{d}}t}} = Q_3 \cdot C_1 - Q_3 \cdot C_3 $$
(3)
$$\frac{{{\text{d}}A_4 }}{{{\text{d}}t}} = k_{{\text{enz in}}} \cdot \left( {1 + I_{\max } \cdot \frac{{C_1 }}{{C_1 + {\text{ECI}}_{50} }}} \right) - k_{{\text{enz out}}} \cdot A_4 {\text{ }}$$
(4)
Table 1 Pharmacokinetic and pharmacodynamic parameters of the nine virtual compounds
The dose-limiting toxicity of all virtual compounds was neutropenia. The time course of drug-related neutropenia was described by a semi-physiological population pharmacodynamic model that was developed by Friberg et al. [9] (Fig. 3). This model comprised a progenitor compartment of proliferating blood cells, three transit compartments representing the maturation chain in the bone marrow and a compartment corresponding to the central circulation. The model comprised two system related parameters: the mean transit time (MTT) and a feedback parameter gamma (γ). The MTT was the average time between neutrophil proliferation and completion of maturation and was related to the first order transition rate constant k
tr (=4/MTT). The feedback parameter γ represented the induction of stem cell proliferation by endogenous growth factors and/or cytokines [10]. The baseline blood cell count was estimated from the observed baseline value and a residual error. The proliferation rate was reduced by drug exposure in the central pharmacokinetic compartment according to a linear function (Eq. 5).
$$k_{{\text{prol}}} {\text{ = }}k_{{\text{tr}}} \cdot \left( {{\text{1}} - {\text{slope}} \cdot C_{\text{1}} } \right)$$
(5)
Using the PK–PD parameters from Table 1, data for dose escalation studies were simulated. The design of the simulated dose escalation studies was in accordance with the design of classical phase I studies [11]. Patients were treated in silico in three-patient cohorts, that were extended to six-patient cohorts if significant (≥CTC grade 2) toxicity was observed. Neutrophil counts were simulated with alternating 3 and 4 day intervals (twice weekly). A modified Fibonacci-like dose escalation strategy was used [1]. The dose was escalated by 100% if no or minimal (CTC grade 0 or 1) neutropenia was observed. If one or more patients had CTC grade 2 toxicity, the dose was escalated by 50% increments and if serious (CTC grade 3 or 4) neutropenia was observed, the dose was escalated by 25%. Dose-limiting neutropenia was defined as CTC grade 4 neutropenia (absolute neutrophil count ≤0.5 × 109 per liter) during at least two subsequent hematological measurements (∼1 week). If two or more patients in a six-patient cohort experienced dose-limiting neutropenia, this dose level was defined as the non-tolerated dose (NTD). The recommended dose for future studies was the dose level immediately below the NTD.
Dose escalation studies were simulated for five administration regimens:
-
1.
A single 1-h infusion (D × 1) every 3 weeks
-
2.
Daily 1-h infusions on days 1–5 (D × 5) every 3 weeks
-
3.
Weekly 1-h infusions on days 1, 8, 15 and 22 (W × 4) every 6 weeks
-
4.
A 120-h infusion (120 H) every 3 weeks
-
5.
A continuous infusion during 3 weeks (CI)
Regimens 1–4 were evaluated in the phase I program of indisulam, that was used in this study as a real dataset for a retrospective evaluation of the two-stage model-based design [12]. These regimens comprised short and long infusions, single and multiple drug administrations and were therefore selected for the current study. Schedule 5, the continuous infusion, was added to cover the full range of short exposure to continuous exposure regimens.
Pharmacokinetic data were simulated according to rich sampling designs as commonly used in phase I trials. PK samples were taken during infusion at fixed time points. Additional PK samples were taken after infusion and were logarithmically spaced between the end of infusion and the last sampling point. The last sampling point corresponded to five times the mean residence time (MRT) after the end of infusion. To mimic the fact that limited knowledge may be available at the start of first in human studies, the true MRT was perturbed by a geometric standard deviation of 1.70 to account for prediction uncertainty. This standard deviation was derived from previous publications [13, 14]. The limit of quantitation (LOQ) was 50 ng/mL for all compounds. Data below the LOQ were excluded from data analysis.
Dose escalation studies with a modified Fibonacci-like escalation scheme (see above) were performed in silico for five administration regimens for each of the nine compounds. In total, 45 phase I studies were simulated using NONMEM (version VI, GloboMax, Hanover, MD, USA). Parameters with inter-individual variability were log-normally distributed. Data were logarithmically transformed and residual errors were additive on a logarithmic scale.
PK–PD model development (step 2)
Pharmacokinetic and pharmacodynamic models were developed for each of the 45 simulated data sets. The investigator was blinded to the ‘true’ PK–PD models that were defined by the parameters in Table 1.
Pharmacokinetic and pharmacodynamic data were analyzed sequentially. For PK–PD analysis, a data set was used that contained all dosing information and all pharmacokinetic and pharmacodynamic measurements. Pharmacokinetic parameters were fixed to the estimates of the pharmacokinetic analysis. This method has been described into more detail by Zhang et al. [15].
The estimation of the system parameters MTT and γ and the interindividual variability of MTT was supported by prior knowledge of the values of these parameters. The Bayesian priors were the geometric means of the estimates that were reported for various other anticancer agents by Friberg et al. (MTT = 116 h, γ = 0.167, IIV MTT = 22.4%). Prior uncertainty corresponded to the geometric standard deviation of these estimates (MTT 18%, γ 59%, IIV MTT 20%) [9].
All parameters (P) were assumed to be log-normally distributed in the study population. Consequently, interindividual variability was estimated using an additive function on a logarithmic scale (ln(P
i) = ln(TVP) + η
i) to describe the individual deviation (η
i) from the population typical value (TVP). Differences between observed and individual predicted values were modeled as additive residual errors on a logarithmic scale. Data were analyzed with the first-order conditional method using NONMEM (version VI, GloboMax, Hanover, MD, USA). Discrimination between hierarchical models was based on the objective function value, goodness of fit plots and standard errors of parameter estimates.
Standard errors and measures of correlation between parameter estimates were obtained using the COVARIANCE option of NONMEM.
Prediction of safe starting dose (step 3)
The PK–PD models, that were developed in step 2 using data from a single in silico phase I study (simulated in step 1), were used to predict the outcome of subsequent dose escalation studies. For instance, the model that described the PK–PD data of compound 1 after a single 1-h infusion, was used to predict the recommended doses for the other four administration regimens for compound 1. Trial simulations were carried out to make these predictions. For each prediction, a dose escalation trial was simulated in 40,000-fold to account for uncertainty in the PK–PD parameter estimates and for variability between patients. A total number of 200 sets of PK–PD parameters from the final parameter estimates, their geometric standard errors and the correlation matrix were selected. For each set of PK–PD parameters, 200 trials were simulated with different random selections of patients.
One tenth of the predicted non-tolerated dose is conventionally considered a safe starting dose for dose escalation studies. Therefore, in order to determine the starting dose of a dose escalation trial, the non-tolerated dose of the previous trial was multiplied by 10% and perturbed using a geometric standard deviation of 1.54. The perturbation accounted for uncertainty in the predicted tolerability of anticancer agents prior to first in man studies. The geometric standard deviation was calculated from predicted and empirically determined non-tolerated doses of 21 anticancer drugs [16–19]. Patients were treated in three- or six-patient cohort and the modified Fibonacci-like dose escalation strategy was used (see above for details).
The median value of each set of 40,000 recommended doses was the predicted recommended dose. The five and 95 percentiles formed the 90% confidence intervals of the prediction. The lower boundary was considered the safe starting dose for an in vivo clinical dose escalation study.
Evaluation of the two-stage model-based design using simulated data (step 4)
Using the true PK–PD parameters (Table 1) and starting doses that were considered safe for clinical dose escalation studies, 1,000 replicate trials were simulated (two-stage model-based design). In addition, 1,000 replicate trials were simulated with starting doses perturbed around 10% of the recommended dose (conventional method).
The number of patients with dose-limiting neutropenia was assessed for both methods as a measure of safety of the two-stage model-based design. The median number of patients treated with a dose below the recommended dose was compared between the new and the conventional methods, for each virtual compound and for each administration regimen, as a measure of efficiency of the two-stage model-based design. The two-stage model-based design was considered successful if it did not result in an increased number of patients with dose-limiting neutropenia and if the number of patients treated with a dose below the recommended dose was reduced.
Post-hoc determination of recommended dose (step 5)
Based on the phase I program of a novel anticancer agent, a recommended dose must be defined for further testing in phase II clinical studies. In a conventional study design, the selection of the recommended dose is empirically based on clinical outcome and is defined as the highest dose level with more than two out of six patients with dose-limiting toxicity. In the two-stage model-based design, the recommended dose for phase II studies is based on a PK–PD analysis using all PK–PD data from phase I. This PK–PD analysis should be performed after finalization of the clinical phase I program and is therefore termed a post-hoc analysis.
To evaluate this part of the two-stage model-based design, the PK–PD models of each compound were updated using the data from all five phase I studies that were simulated in step 1 (Fig. 1). The updated models were used for the simulation of 40,000 dose escalation studies for each administration regimen. Parameter uncertainty and variability between patients were taken into account. The median value of the 40,000 simulated recommended doses was proposed for further testing in phase II studies. This method was used to select a recommended dose for all compounds (n = 9) and for all administration regimens (n = 5). The 45 selected recommended doses (D
rec.,pred) were used to determine the precision of this strategy. The root mean squared relative prediction error (RMSE%) was calculated as a measure of precision (Eq. 6). The true recommended dose (D
rec.,true) was defined as the median value of 1,000 simulated recommended doses using the true PK–PD model, conventional starting doses (i.e. the recommended dose of the previous trial multiplied by 10% and perturbed using a geometric standard deviation of 1.54) and the modified Fibonacci-like dose escalation strategy.
$${\text{RMSE\% = }}\sqrt {\left( {\sum {\left( {\frac{{\left( {D_{{\text{rec}}{\text{.,pred}}} - D_{{\text{rec}}{\text{.,true}}} } \right)}}{{\left( {D_{{\text{rec}}{\text{.,true}}} } \right)}}} \right)^2 } \times {\text{ }}n^{ - 1} } \right)} {\text{ }}$$
(6)
The precision of the two-stage model-based design was compared to the precision of the conventional design, where the recommended dose was based on the clinical outcome of a dose escalation study. The RMSE% of the conventional design was calculated for the 1,000 simulated recommended doses from the true PK–PD model (D
rec.,pred) and their median value (D
rec.,true).
The precision of the selection of the recommended dose for phase II evaluation was also assessed for the PK–PD models that were developed using data from a single phase I study.
Evaluation of the two-stage model-based design using clinical data of indisulam
The previously conducted phase I program of the investigational anticancer agent consisted of four dose escalation studies of four administration regimens: D × 1, D × 5, W × 4, 120 H [20–23]. Neutropenia was identified as the dose-limiting toxicity [20–23]. It was verified retrospectively if the number of patients treated with a dose below the recommended dose could have been reduced by the proposed two-stage model-based design. For each of the phase I studies, a PK–PD model was developed using indisulam plasma concentrations and absolute neutrophil counts that were measured during the first treatment cycle. The models were used to simulate 40,000 dose escalation studies for the alternative administration regimens, taking into account parameter uncertainty (×200) and variability between patients (×200). The median values were selected as the predicted recommended doses and the five percentiles were considered safe starting doses for the other three administration regimens. The selected starting doses were compared to the clinically determined recommended and non-tolerated doses. The number of patients treated at dose levels below the clinically determined recommended doses was assessed. In this retrospective evaluation, the two-stage model-based design was considered successful if all selected starting doses were below the highest administered dose levels and if the number of patients treated with at a dose level below the recommended dose was reduced by at least 10%.