Chapter 5

Bayesian Dose-Finding Designs in Healthy Volunteers

Yinghui Zhou

5.1 Introduction

Phase I studies have unique characteristics that distinguish them from other phases of clinical research. This is the phase when new experimental drugs are given to human subjects for the first time: A more explicit name for such trials is “first-into-man” (FIM) studies. Although intensive toxicologic studies have been carried out at preclinical trials, the primary concern for FIM studies is always the safety of the participating subjects [1], Safety can be assessed by the incidence of dose-limiting events (DLE), such as moderate or severe adverse events, clinically significant changes in cardiac function, exceeding the maximum exposure limits of pharmacokinetic profiles such as the area under the curve (AUC) or the maximum concentration (C_max), and changes in pharmacodynamic parameters such as blood flow or heart rate. The majority of FIM studies are conducted on healthy volunteers for whom DLEs should generally be avoided. Only a small proportion of FIM studies is conducted on late-stage patients in cancer trials. For these patients, standard therapies have failed, and low, safe doses will not achieve therapeutic effects, while high, unsafe doses will cause toxicity. A small risk of a DLE is therefore permitted to gain some therapeutic effect.

The primary objective of FIM studies is to find an optimal dose, both safe and potentially efficacious, for later phases of clinical research. This involves a dose-escalation scheme: A fixed range of discrete doses, d₁ < · · · < d_k for some integer k, is predefined by the investigators to be administered in turn to subjects to assess the safety and tolerability of the compound. For healthy volunteer studies, the optimal dose is the maximum safe dose leading to an acceptable concentration of drug in plasma or to an adequate description of the biological effects.

The majority of phase I studies are conducted in healthy volunteers. Current dose-escalation designs for healthy volunteer studies are usually crossover designs in which each subject receives a dose in each of a series of consecutive treatment periods separated by washout periods. Such a design is illustrated in Table 1. Groups of subjects, known as cohorts or panels, some on active doses and others on placebo, are treated simultaneously. The lowest dose, d₁, is normally regarded as the only “safe” dose to be used for the first cohort of subjects. Safety data along with pharmacokinetic/pharmacodynamic (PK/PD) data from each treatment period are collected and then summarized in tables, listings, and graphs for a safety review committee to make decisions regarding the doses to be given during the next period. The committee will normally assign doses according to some predefined dose-escalation scheme, but they may alter the scheme by repeating the previous doses or de-escalating one dose level; they may even stop the trial if its safety is in question. This is essentially a PK/PD “clinical judgment”-guided design. Response variables are analyzed either by simple analysis of variance (ANOVA) approaches [2] or by repeated-measures ANOVA. The PK/PD models may also be estimated and presented at the end of a trial.

Table 1: An Example of a Crossover Design

Under conventional dose-escalation rules, it is likely that healthy volunteers will be treated at subtherapeutic doses. Consequently, information gathered from a trial is mostly not relevant for identifying the optimal dose for phase II studies. Bayesian decision-theoretic designs, motivated by statistical model-based designs for cancer trials, have been proposed to enhance the precision of the optimal dose for phase II studies, to increase the overall efficiency of the dose-escalation procedure while maintaining the safety of subjects [3–6].

5.2 A Bayesian Decision-Theoretic Design

In healthy volunteer studies, several PK/PD measurements are normally monitored and recorded. The methodology described here, however, focuses on a single pharmacokinetic variable derived from the curve relating the concentration of the drug in plasma to the time since administration [7]. Commonly used summaries such as area under the curve (AUC) or peak drug concentration (C_max) are often modeled by the normal distribution after a logarithmic transformation [8]. The mean value of y, denoting either log(AUC) or log(C_max), is modeled as linear in log dose: E(y) = θ₁ + θ₂ log(dose), where θ₁ and θ₂ are unknown population parameters. A maximum exposure level, L, is defined before the start of the study based on the toxicity profile for the compound observed in the most sensitive animal species. Any concentration of the drug leading to an AUC or C_max in excess of this level will be unacceptable.

As each subject receives more than one dose, the multiple responses of each subject are correlated. Furthermore, different subjects vary considerably in terms of pharmacokinetics. Therefore, both random subject effects and random errors are included in the model. A log–log mixed effects model [3–6] is fitted to y_ij, the response following the jth dose to the ith subject:

(1)

The logarithm of the jth active dose received by the ith subject is denoted by l_ij. The term s_i is a random effect relating to the ith subject. The s_i and ε_ij are modeled as mutually independent, normally distributed random variables with mean zero and variances τ² and σ², respectively. The correlation, ρ, between two responses on the same subject can be equal to τ²/(σ² + τ²). Placebo administrations are ignored in this model because there will be no drug detected in plasma.

Bayesian decision theory supplies a general framework for making decisions under uncertainty, which can be applied in many scientific and business fields [9–11]. Let θ = (θ₁, θ₂)′ be the vector of the unknown parameters. Some information or expert opinion about θ, from either preclinical data or experience of similar compounds, will be available before the dose-escalation study begins. This information can be formulated as a probability density function for θ, denoted by h₀(θ) and known as the “prior density.” Let x denote the data collected in the dose-escalation study, and f(x|θ) be its likelihood function of the data. Then, the posterior density function of θ, h(θ|x), can be derived using Bayes’ theorem:

This posterior represents the opinion about θ formed by combining the prior opinion with the data.

Potentially, one could treat θ₁, θ, τ², and σ² all as unknown parameters. However, the more parameters that are modeled, the less accurate the resulting estimates, especially as sample sizes are small in phase I trials. One way to solve the problem is to give some parameters fixed values as though they are known. In particular, the within-subject correlation, ρ, might be set to a value such as 0.6. Hence, there will be three unknown parameters in the model [1] instead of four parameters, namely, θ₁, θ₂, and σ². Fixing ρ is convenient, and the effect of doing so has been studied by Whitehead et al. [5] by taking the alternative strategy of specifying discrete priors on ρ. In the resulting analyses, ρ is underestimated by the Bayesian procedure and the estimate of σ² is overestimated. Consequently, it was concluded that fixing ρ was a reasonable policy for practical use.

Conjugate priors have been proposed for θ₁, θ₂, and v [3–6], where v is the within-subject precision v = σ⁻². In particular, the conditional distribution of θ given v, and the marginal distribution of v can be taken to be

(2)

where N denotes a normal distribution and Ga a gamma distribution, and the values of μ₀, Q₀, α₀, and β₀ are chosen to represent prior knowledge. Being a conjugate prior, the posterior distribution shares the same form.

Suppose that n subjects have been treated. The ith subject has received p_i periods of treatment, i = 1, …, n, and so a total of p₁ + · · · + p_n = p observations are available. Let the p-dimensional y denote the vector of responses with elements y_ij ordered by subject and by period within subject. The (p × 2) design matrix X is made up of rows of the form (1, l_ij), ordered in the same way as y. The (p × n) matrix U, the design matrix of the random subject effect, is defined as having a 1 in the ith column of rows p₁ + · · · + p_i1 + 1, …, p₁ + · · · + p_i for i = 1, …, n, and zeros elsewhere. The identity matrix is denoted by I. The (p × p) matrix P is defined as P = [I + ρ/(1 − ρ)UU’]. Posterior distributions for θ and v are

(3)

where α = α₀ + p/2, β = β0 + (y’Py + μ₀Q₀μ₀ − μ’Qμ/2, μ = (Q₀ + X’PX)⁻¹ (Q’₀μ₀ + X’Py), and Q − (Q₀ + X’PX). Priors reflect not only one’s opinions but also how strongly they are held. Here, a small value for α₀ represents a weak opinion. Consequently, dose escalation may be quick at the beginning, as a few safe observations will soon overcome any prior reservations. A bigger value for α₀ represents a strong prior, and the resulting dose escalation will be conservative, as prior concerns will only be removed by a clear demonstration of safety.

A safety constraint can be used to help control overdosing [3–6]. This requires that no dose be given if the predicted response at this dose is likely to exceeds the safety limit L. Mathematically this can be expressed as

(4)

where π₀, the tolerance level, can be set at a low value such as 0.05 or 0.20. The dose at which the above probability is equal to π₀ is called the maximum safe dose for the ith subject following the jth observation. The maximum safe dose is subject related, and posterior estimates may differ among subjects who have already been observed, being lower for a subject who previously had absorbed more drug than average and higher if the absorption was less. After each treatment period of the dose-escalation procedure, the posterior predictive probability that a future response will lie above the safety limit is updated.

The decision of which dose to administer to each subject in each dosing period is made using a predefined criterion. This criterion can be based on safety; for example, one could use the maximum safe dose as the recommended dose. It can also be based on accuracy of estimates of unknown parameters; for example, the optimal choice of doses is that which will maximize the determinant of the posterior variance-covariance matrix of the joint posterior distribution or minimize the posterior variance of some key parameter.

5.3 An Example of Dose Escalation in Healthy Volunteer Studies

An example, described in detail by Whitehead et al. [5], in which the principal pharmacokinetic measure was C_max is outlined briefly here. The safety cutoff for this response was taken to be y_L = log(200). Seven doses were used according to the schedule: 2.5, 5, 10, 25, 50, 100, 150 μg. The actual trial was conducted according to a conventional design, and the dosing structure and the resulting data are listed in Table 2. From a SAS PROC MIXED analysis, maximum likelihood estimates of the parameters in model [1] are

Table 2: Real Data from a Healthy Volunteer Trial in Whitehead et al. (2006)

(5)

Note that it follows that the within-subject correlation is estimated as 0.579.

As an illustration, the Bayesian method described above is applied retrospectively to this situation. Conjugate priors must first be expressed. It is assumed that prior expert opinion suggested that the C_max values would be 5.32 and 319.2 pg/ml at the doses 2.5 and 150 μg, respectively. This prior indicates that the highest dose, 150 μg, is not a safe dose because the predicted C_max exceeds the safety limit of 200 pg/ml. The value for ρ is set as 0.6. This forms the bivariate normal distributions for θ,

The value for α₀ is set as 1, as suggested by Whitehead et al. [5]. The value for β₀ can be found via the safety constraint: P(y₀₁ > log L|d₀₁ = 2.5) = 0.05. This implies that dose 2.5 μg will be the maximum safe dose for all new subjects at the first dosing period in the first cohort. Therefore, v ~ Ga(l, 0.309).

To illustrate Bayesian dose escalation, data are simulated based on the parameters found from the mixed model analysis given by Whitehead et al. [5]. Thus, C_max values are generated from three cohorts of eight healthy volunteers, each treated in four consecutive periods and receiving three active doses and one randomly placed placebo. A simulated dose escalation with doses chosen according to the maximum safe dose criterion is shown in Table 3. The first six subjects received the lowest dose, 2.5 μg. All subjects at the next dosing period received dose 10 μg, in which dose 5 μg was skipped for subjects 1 to 4. Subjects 7 and 8, who were on placebo in the first dosing period, skipped two doses, 2.5 and 5 μg, to receive 10 μg in the second dosing period. If this dosing proposal was presented to a safety committee in a real trial, the committee members might wish to alter this dose recommendation. The Bayesian approach provides a scientific recommendation for dose escalations. However, the final decision on which doses are given should come from a safety committee. The procedure would be able to make use of results from any dose administered. In Table 3, it is shown that the maximum safe dose for a new subject at the beginning of the second cohort is 25 μg. All subjects in the second cohort received 50 μg at least twice. Subjects 17 to 22 in the first dosing period of the final cohort received 50 μg. However, they all had a high value of C_max. The Bayesian approach then recommended a lower dose, 25 μg, for subjects 17 to 20 (subjects 21 and 22 were on placebo). This shows that the Bayesian approach can react to different situations quickly: When escalation looks safe, dose levels can be skipped; when escalation appears to be unsafe, lower doses are recommended. Two high doses that were administered in the real trial, 100 and 150 μg, were never used in this illustrative run. The posterior distributions for θ at the end of the third cohort are

Table 3: A Simulated Dose Escalation

and v ~ Ga(37, 2.625).

Figure 1 shows the doses administered to each subject and the corresponding responses. Table 4 gives the maximum likelihood estimates from the real data in Table 2 that were used as the true values in the simulation, together with the maximum likelihood estimates from the simulated data in Table 3. Results show that σ² and τ² were underestimated from the simulated data, with there being no evidence of between-subject variation. Consequently, the estimated correlation from the simulated data is zero, in contrast to the true value of 0.579 used in the simulation. This is a consequence of the small data set, and it illustrates the value of fixing ρ during the escalation process.

Figure 1: An illustration of a simulated dose escalation (using data from Table 3)

Table 4: Maximum Likelihood Estimates (MLE) and Bayesian Model Estimates of Simulated Data in Table 3 (with Standard Errors or Standard Deviations)

Different prior settings will result in different dose escalations. For example, if the value for α₀ is changed from 1.0 to 0.1, then the dose escalation will be more rapid. Table 5 summarizes the recommended doses and simulated responses from another simulation run where α₀ = 0.1. In the second dosing period of the first cohort, subjects 1, 2, and 4 skipped two doses, 5 and 10 μg. Subjects 3, 7, and 8 skipped three doses in that cohort. The starting dose for all subjects in the first dosing period of the second and third cohorts was 50 μg (25 μg and 50 μg were the doses in Table 3). All subjects, except subject 13 at the fourth period, repeatedly received 50 μg during the second cohort. In the third cohort, the dose of 100 μg was used quite frequently. The highest dose, 150 μg, was never used. This example shows that different prior settings will affect dose-escalation procedures. Multiple simulations should therefore be conducted to gain a better understanding of the properties of a design. Different scenarios should be tried to ensure that the procedure has good properties, whether the drug is safe, unsafe, or only safe for some lower doses.

Table 5: A Simulated Dose Escalation (the value for α₀ is 0.1)

5.4 Discussion

Bayesian methods offer advantages over conventional designs. Unlike conventional designs, more doses within the predefined dose range, or even outside of the predefined dose range, can be explored without necessarily needing extra dosing time as dose level skipping can be permitted. From the simulation runs in Tables 3 and 5, more doses, such as 40, 60, and 80 μg, could perhaps be included in the dose scheme. Simulations have shown that skipping dose levels does not affect either safety or accuracy; on the contrary, it improves safety or accuracy, as a wider range of doses is used [12]. Providing a greater choice of doses, while allowing dose skipping, leads to procedures that are more likely to find the target dose and to learn about the dose-response relationship curve efficiently. Ethical concerns can be expressed through cautious priors or safety constraints. Dose escalation will be dominated by prior opinion at the beginning, but it will soon be influenced more by the accumulating real data.

Before a Bayesian design is implemented in a specific phase I trial, intensive simulations should be used to evaluate different prior settings, safety constraints, and dosing criteria under a range of scenarios. Investigators can then choose a satisfactory design based on the simulation properties that they are interested in. For instance, they may look for a design that gives the smallest number of toxicities or the most accurate estimates. Until recently, such prospective assessment of design options was not possible, but now that the necessary computing tools are available, it would appear inexcusable not to explore any proposed design before it is implemented.

Although the methodology described here is presented only for a single pharmacokinetic outcome, the principles are easily generalized for multiple end points. Optimal doses can be found according to the safety limits for each of the endpoints, and then the lowest of these doses can be recommended. The Bayesian decision-theoretic approach has also been extended for application to an attention deficit disorder study [13], where a pharmacodynamic response (heart rate change from baseline), a pharmacokinetic response (AUG), and a binary response (occurrence of any dose-limiting events) are modeled. A one-off simulation run indicates that the Bayesian approach controls unwanted events, dose-limiting events, and AUC levels exceeding the safety limit, while achieving more heart rate changes within the therapeutic range.

Bayesian methodology only provides scientific dose recommendations. These should be treated as additional information for guidance of the safety committee of a trial rather than dictating final doses to be administered. Statisticians and clinicians need to be familiar with this methodology. Once the trial starts, data need to be available quickly and presented unblinded with dose recommendations to the safety committee. They can then make a final decision, based on the formal recommendations, together with all of the additional safety, laboratory, and other data available.

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