Escalation with Overdose Control (EWOC) design

Like the CRM, the EWOC design is able to provide an estimate of the MTD as a dose other than doses actually studied in the clinical trial (Piantadosi et al., 1998). The EWOC was introduced by Babb et al. (1998) as an alternative to the CRM that sought to control the chance of over-dosing patients during dose escalation. In brief, there are 3 key differences from the Goodman et al. (1995) formulation of the CRM, namely:

• 2-parameter Bayesian logistic model replaces the 1-parameter

• a loss function is incorporated formally to identify the dose for subsequent patients

• dose (or standardised dose e.g. each dose is divided by a reference dose) values can be modelled directly

The 2-parameter Bayesian logistic model had been considered earlier by other authors e.g. Gatsonis & Greenhouse (1992) but Babb et al. described a criteria that could estimate the probability of over-dosing subsequent patients. In more detail, and as with CRM, suppose the target toxicity is represented by θ and this corresponds to a dose level 𝑥^∗ (not necessarily an available dose). Babb et al. (1998) consider the general 2-parameter dose-response model:

P(𝑌_𝑗 =1 | Dose=𝑥_𝑖) = F(𝛽₀+ 𝛽₁𝑥_𝑖) ( 2.6 )

where F(·) is a specified cumulative distribution function, for example, logistic leads to:

- 41 -

P(𝑌_𝑗 =1 | Dose=𝑥_𝑖) =

�

�

This model is represented in Figure 3 for a range of values of the parameters 𝛽₀ and 𝛽₁.

Figure 3 Examples of Two-parameter Logistic Dose-Response Models

Analogous to the Bayesian approach for CRM (see Equation ( 2.1 )), the likelihood can be expressed in terms of the parameters 𝛽0 and 𝛽1 then combined with priors (for 𝛽0

and 𝛽1) to obtain posteriors, however, Babb et al. (1998) chose to re-parameterise the logistic model because specifying priors for regression parameters is not

straightforward (Kadane et al., 1980). The alternative formulation is in terms of 𝜌0 and γ, where 𝜌0 is the probability of DLT at the minimum dose (denoted Xmin) and γ is the MTD. For target response rate, θ, the logistic model can now be expressed as:

logit (𝜌0) = 𝛽0 + 𝛽1Xmin ( 2.8 )

and logit (θ) = 𝛽0 + 𝛽1γ ( 2.9 )

By subtraction, these two equations for 𝛽0 and 𝛽1 become:

- 42 - 𝛽₀ = _{𝛾− 𝑋}¹

𝑚𝑖𝑚 {𝛾 𝑙𝑙𝑙𝑙𝑙(𝜌₀) − 𝑋_𝑚𝑖𝑚𝑙𝑙𝑙𝑙𝑙(𝜃)} ( 2.10 ) and 𝛽₁ =_{𝛾− 𝑋}¹

𝑚𝑖𝑚 {𝑙𝑙𝑙𝑙𝑙(𝜃) − 𝑙𝑙𝑙𝑙𝑙(𝜌₀)} ( 2.11 ) In the original article, Babb et al. (1998) considered placing (bounded) independent Uniform priors on 𝜌0 and γ. This was later extended by Tighiouart et al. (2005) to include a set of four alternative priors of which two incorporated correlation between 𝜌₀ and γ.

As with CRM (see Equation ( 2.1 )), the first patient receives the first dose i.e. 𝑥1=𝑋𝑚𝑖𝑚

and provides (binary) response data 𝑌𝑖 then the Likelihood (in terms of 𝜌0 and γ) from the first j patients can be written as:

Although these expressions appear unwieldy, it does allow us to specify priors on parameters that are intuitively more appealing to clinical researchers since the two parameters reflect observable quantities (Bedrick et al., 1996; Huson & Kinnersley, 2009; O’Hagan et al., 2006, p.145).

After the responses from j patients are available the (marginal) posterior distribution for γ can be written in a similar manner to Equation ( 2.2 ) for which the cumulative distribution function (CDF) for γ will be denoted by ∏ (𝑥)_𝑗 . Assuming the first patient has no toxicity on dose 𝑥1=𝑋𝑚𝑖𝑚 the dose for the next patient is identified by selecting a probability (denoted alpha) for which the chance of over-dosing is bounded (Zacks et al., 1998). For example, alpha=0.25 means the selected dose has only a 25%

- 43 -

chance of being higher than the true MTD, γ. This dose (for the j+1^th patient) is identified from:

� (𝑎𝑙𝑝ℎ𝑎)⁻¹

𝑗 ( 2.14 )

Stopping rules can be applied as for CRM and at the end of the trial, an appropriate quantity can be derived from the marginal probability density for γ, for example 50^th percentile (median), mode or mean (expected value). Tighiouart et al. (2005) show that the dose 𝑥_[𝑗+1] selected for the j+1^th patient corresponds to the asymmetric loss function:

𝑙_{𝑎𝑎𝑎ℎ𝑎}(𝑥, 𝛾) = �𝑎𝑙𝑝ℎ𝑎(𝛾 − 𝑥) 𝑙𝑖 𝑥 ≤ 𝛾 i. e. underdose

(1 − 𝑎𝑙𝑝ℎ𝑎)(𝑥 − 𝛾) 𝑙𝑖 𝑥 > 𝛾 i. e. overdose ( 2.15 )

As such, any loss incurred by treating a patient above the MTD is penalised (1-alpha)/alpha times greater than the loss from equivalent under-dosing.

In the (common) situation when a clinical trial only has a discrete set of doses available then Tighiouart and Rogatko (2010) argue that the dose 𝑥[𝑗+1] should be selected based on defining a tolerance criteria:

𝑥[𝑗+1] = max𝑥_𝑖�(𝑥𝑖− 𝑥[𝑗+1]≤ 𝑇1) 𝑎𝑎𝑎 (∏ (𝑥𝑗 𝑗+1) − 𝑎𝑙𝑝ℎ𝑎 ≤ 𝑇2)� ( 2.16 )

where 𝑇₁and 𝑇₂ are non-negative numbers (referred to as tolerances).

In non-mathematical terms, the trialist needs to pre-specify a tolerance for:

• discrepancy between the recommended and available doses

• discrepancy between pre-specified overdose probability (alpha) and the corresponding overdose probability from an available dose

- 44 -

Tighiouart and Rogatko (2010) provide no recommendations on reasonable values for 𝑇₁ and 𝑇₂ and further evidence of the implications for such choices is warranted.

In contrast to the dose allocation rule of O’Quigley et al. (1990) in which the probability of overdose is a component of the rule when only discrete doses are available (see Equations ( 2.4 ) and ( 2.5 )), there may be clinical trial settings in which the investigator is willing to base the choice of next dose only on the difference between the dose increments. Hence, consider an alternative estimator which is a special case of Equation ( 2.16 ) i.e. setting 𝑇1= {half of the dose-spacing} and 𝑇2= 1-alpha, which can be expressed equivalently as:

𝑥[𝑗+1] = min𝑥_𝑖��𝑥𝑖− 𝑥_[𝑗+1]�� ( 2.17 )

Since little attention has been paid to the choices of 𝑇1 and 𝑇2 , the properties of the estimator in Equation ( 2.17 ) have also not been widely studied in the literature.

Practical use of this estimator can be demonstrated by the example in Figure 4 for which a target response rate of 0.4 would imply the next patient should be allocated dose 1.5766 (see Figure 4(a)), however, in the relatively common situation in which only discrete doses are available (in this example 0, 1, 2 and 3) the estimators from Equations ( 2.4 ) and ( 2.5 ) would recommend Dose=1 for the next patient (see Figure 4(b)) since the difference in probability of a response=0.4 is smaller on Dose=1 than Dose=2 (i.e. 0.4-0.0474 is smaller than 0.8176-0.4). As argued above, if the targeted response is an efficacy parameter and the investigator is satisfied with the observed safety of the new treatment thus far, then he/she may wish to administer Dose=2 since it is not too much of a dose increase and it is mathematically closer to the recommended dose.

- 45 -

Figure 4(a) Target Response Rate=0.4 Figure 4(b) Estimated Response Probabilities for Next Available Dose Allocation

Figure 4 Estimated Response Probabilities for Available Doses