Controlled amplification in oncology dose-finding trials
Introduction In parallel with the development of non-cytotoxic agents in oncology, two key evolutions have taken place in the design of dose-finding trials in recent years. Firstly, expansion cohorts have become customary. These tend to take place after the maximum tolerated dose (MTD) is determined, which is the dose that is associated with a certain maximum rate of significant side effects (dose-limiting toxicities (DLT)) . Their objectives vary, including (but not limited to) assessing efficacy of an investigational agent or with a combination of agents, evaluating different treatment schedules, assessing whether the drug’s efficacy depends on specific biomarkers, or evaluating two (or more) dose levels further. Secondly, expansion of lower dose levels during dose-escalation, known as backfilling , a relatively new concept, is now included in study protocols in order to spread experimentation at lower levels to collect additional endpoints such as efficacy and/or pharmacokinetic(PK)/pharmacodynamic(PD). The main objective of backfilling is to refine the selection of the recommended phase II dose (RP2D) by annotating the dose-efficacy curve further, as discussed in our previous publication . Given the mechanism of action of modern anticancer agents, in particular molecularly-targeted therapies and immunotherapies, higher dose levels might no longer necessarily be associated with greater efficacy . Moreover, these higher levels might be associated with additional toxicity, potentially leading to reduce compliance and/or treatment interruptions. As a case in point, the FDA has recently approved (May 2021) sotorasib for metastatic KRAS G12C-mutated non–small-cell lung cancers (NSCLC), conditional on the sponsor conducting a post-marketing trial to evaluate a lower dose level of 240 mg compared to the approved level of 960 mg . Several reasons were given by the FDA, including that the company did not provide evidence of a relationship between dose and efficacy. Progressing as fast as possible towards, and identifying, the dose level most likely to be the MTD is the guiding principle of model-based dose-finding algorithms for cytotoxic agents, such as the the Continual Reassessment Method (CRM) . This was well justified by the assumption of surrogacy between toxicity and efficacy, which implies that a higher dose will result in higher toxicity and higher efficacy. When this relationship is less immediate, carrying out experimentation at a dose (or doses) other than the one deemed to be most likely to be the MTD, in the light of all available evidence at any given time in the study, might be justified. This is a key feature of backfilling and expansion cohorts, which can be seen  by comparing the allocation of patients (at study end) between a conventional CRM-based study and a study that includes backfilling, for a given sample size. For , the total sample size was 63 patients, and the underlying dose-toxicity curve was such that the MTD was dose level 5. This level gathered 62% of patients under the CRM design and 25% with a backfill design. The challenges raised by non-cytotoxic agents do not imply that the relatively restrictive structure around which most dose-finding studies are built must be abandoned. Quite the opposite, actually certain parts of that structure can be broadened or relaxed in order to address these challenges. The ultimate goal of dose-finding designs in the era of non-cytotoxic agents is to collect the relevant information for final RP2D determination in a pragmatic way. While some information is collected on dose levels below and around the MTD via the dose-escalation design (such as the CRM), designs that include expansion cohorts or backfill patients effectively amplify that information by allocating patients slightly differently. The objective of this paper is to study more deeply the concept of amplification. Amplification is the umbrella term that we propose to refer to the objective of collecting more information (i.e. amplifying) on the lower dose levels than would have been achieved if a conventional escalation strategy (e.g. CRM) without backfill or expansion cohorts had been followed. The rest of the manuscript is organised as follows. In we describe amplification in more details. We consider in multiple ways by which controlled amplification can be achieved using randomisation. Randomisation  can play an important part as a tool to achieve amplification in a controlled manner. Until recently, randomization was not considered needed nor helpful in early phase trials but this viewpoint has evolved markedly in recent years, including among regulators . In we propose a metric to quantify the degree of amplification that a design may offer. We propose a trial design built around amplification in , and demonstrate its applicability in hypothetical trials in . We provide concluding remarks in . 

Amplification For the last few decades the main challenge asked of the statistical methodology in dose-finding studies for cytotoxics was to achieve the following two goals: locate the MTD with as great a precision as the sample size allows and, (2) to treat as many patients in the trial as possible at or very close to the MTD. The idea has been to reconcile two apparently conflicting goals: efficient estimation from the trial while treating those patients entered into the trial at the best possible dose, given our available knowledge at a given time. Model-based designs, such as the CRM, accomplish these goals well and, indeed, enjoy certain optimality properties . Treating as many patients as we possibly can at and adjacent to the MTD is the underlying goal of the CRM. In many ways, and in particular for trials with small samples, this appears to be very desirable. Learning more in terms of toxicity and efficacy at levels other than the MTD can be of value, and may be a legitimate aim of a dose-finding study in certain circumstances with non-cytotoxic agents. For example, lower doses may lose little in the way of efficacy but offer much in the way of reduced toxicity. We will have some information at levels removed from the MTD using a model-based design such as the CRM, but it may not be enough to make reliable inferences about the dose-efficacy curve, or perform an informative PK/PD analysis over the dose range, and identify the RP2D, hence the need for amplification. Achieving amplification can be done in multiple ways. For example, as a result of the sample size being reached, or the implementation of some stopping rule, investigators may wish to take the best estimate of the MTD as established and, then, pursue further experimentation at this level and one or two levels below via dose expansion cohorts. We will call this retrospective amplification as we amplify the information below the estimated MTD. In contrast to retrospective amplification, prospective amplification also can take place in a study. By prospective amplification we mean the amplification of doses whose relationship to the MTD we do not know. The reason we do not know is that our estimation process of the MTD is ongoing and, simultaneous to our estimate of the MTD, is our amplification of other doses. In some sense, we are learning about the MTD and about other doses at the same time, which is effectively what backfilling achieves . Amplification can be achieved using randomisation, which we discuss in the next section. 

Amplification induced by randomization The following three approaches use randomisation to achieve amplification. 3.1. Dose escalation using dose pairs Instead of assigning the same dose level to all patients in the same cohort as traditional phase I designs do, one can allow multiple dose levels in a given cohort of patients. This idea has been suggested using triplets of doses (i.e. three adjacent dose levels) before . More generally, the number of adjacent dose levels could be any number equal or greater than two. Blocked randomisation can be used to allocate the incoming patients of the same cohort to the adjacent levels of interest. If the study uses pairs/triplets of dose levels, then the block size would be 2/3. Weighted randomisation based on the posterior probabilities of being the MTD could also be used . We exploit the concept of dose pairs in our proposed design in . 3.2. Controlled backfill Randomisation can be used to assign backfill patients to the dose levels that are available in the backfill set, which may be dynamically reviewed based on accumulating data. Our previous work has developed methodology to define sequentially which dose levels belong to an acceptable backfill set and assess at study completion whether there might be a plateau on the dose-efficacy curve . 3.3. Randomised dose expansion cohorts Recent phase I trials include dose expansion cohorts that compare various schedules, or dose levels. Patients can be randomized to these cohorts and model-based designs can help inform the randomization scheme. At the end of a traditional dose-finding study with a modest sample size comprised between 20–40 patients, the posterior probability of the estimated MTD of being the true MTD does not exceed 50% in most cases . The remaining of the probability mass will mostly be distributed on the two adjacent dose levels either side of the best estimate of the MTD. This implies that there remains considerable uncertainty on the most appropriate dose to recommend for further study. For this reason, additional information may need to be collected on these levels to make the selection , which may be achieved by randomising patients to these dose levels that combine between them more than 50% (in some instances up to 70–80%) of the probability mass of being the true MTD. The sample size that is required for the randomised comparison, and the principles by which the final selection may be made between adjacent dose levels, can be derived from the principles of selection designs . 

Dose escalation using dose pairs Instead of assigning the same dose level to all patients in the same cohort as traditional phase I designs do, one can allow multiple dose levels in a given cohort of patients. This idea has been suggested using triplets of doses (i.e. three adjacent dose levels) before . More generally, the number of adjacent dose levels could be any number equal or greater than two. Blocked randomisation can be used to allocate the incoming patients of the same cohort to the adjacent levels of interest. If the study uses pairs/triplets of dose levels, then the block size would be 2/3. Weighted randomisation based on the posterior probabilities of being the MTD could also be used . We exploit the concept of dose pairs in our proposed design in . 

Controlled backfill Randomisation can be used to assign backfill patients to the dose levels that are available in the backfill set, which may be dynamically reviewed based on accumulating data. Our previous work has developed methodology to define sequentially which dose levels belong to an acceptable backfill set and assess at study completion whether there might be a plateau on the dose-efficacy curve . 

Randomised dose expansion cohorts Recent phase I trials include dose expansion cohorts that compare various schedules, or dose levels. Patients can be randomized to these cohorts and model-based designs can help inform the randomization scheme. At the end of a traditional dose-finding study with a modest sample size comprised between 20–40 patients, the posterior probability of the estimated MTD of being the true MTD does not exceed 50% in most cases . The remaining of the probability mass will mostly be distributed on the two adjacent dose levels either side of the best estimate of the MTD. This implies that there remains considerable uncertainty on the most appropriate dose to recommend for further study. For this reason, additional information may need to be collected on these levels to make the selection , which may be achieved by randomising patients to these dose levels that combine between them more than 50% (in some instances up to 70–80%) of the probability mass of being the true MTD. The sample size that is required for the randomised comparison, and the principles by which the final selection may be made between adjacent dose levels, can be derived from the principles of selection designs . 

Controlled amplification and precision During the course of the dose-escalation study, our running estimate of the most likely dose to be the MTD evolves. Increased precision is obtained with increasing experimentation. However, the typical sample size in the dose-finding part of early phase trials is not large. Consequently, the inherent uncertainty about which dose to choose for the next cohort of patients gives us information that we can use to experiment at levels other than, say, the current best estimate of the MTD, if amplification is desired. This being said, we cannot treat patients in an unstructured way and experiment anywhere we wish. Our goal in this Section is to provide some robust statistical structure within which we can effectively and, mindful of ethical concerns, carry out amplification. How far away from the MTD, and how frequently, is it acceptable to undertake the additional experimentation to achieve amplification is not a straightforward question to answer, and is context-dependent. In order to help address the issue we consider the probability of being the MTD as specific metric to guide where amplification might be performed. There are two main ways of deriving the probability of being the MTD for each dose level: numerical integration  or via Markov Chain Monte Carlo (MCMC) sampling. With MCMC, the (posterior) probability of being the MTD can be derived during the iterations  by recording the proportion of time that each dose is the dose closest to the target toxicity level. The concept of probability of being the MTD is illustrated in . Suppose then that we have k doses, d j , j = 1 , … , k , and, after the inclusion of n patients, the probability of dose d j being the MTD (or some other goal such as the MSD) is given by π n j , j = 1 , … , k . Let F n correspond to the probability function over the doses, so that F n ( d j ) = π n j . For large samples, i.e. allowing n to increase without bound, we can replace F n by F and π n j by π j , corresponding to convergence in distribution and almost sure convergence respectively. For the dose level closest to the target toxicity level, π n j will tend to 1, and π n j will tend to 0 for the other levels. Large sample theory is not without interest, but we are mostly interested in finite sample results. gives an illustration showing what we might anticipate for F n ( d j ) , j = 1 , … , 6 , and 24 included patients in a hypothetical study aiming to identify the dose with TTL 30%. Next, we have: Definition 1 . the empirical cumulative probability of toxicity at d j or lower is given by A ( d j ) = ∑ 𝓁 = 1 j F ( d 𝓁 ) and, in practice, we replace this by A n ( d j ) = ∑ 𝓁 = 1 j F n ( d 𝓁 ) = ∑ 𝓁 = 1 j π n 𝓁 . Simple elimination rules for the amplification set (i.e. the set of dose levels where amplification might be considered) can be based on A n ( d j ) . We might decide that any value less than 10% would be too small to justify further amplification at such levels. Applying this rule to the example of would lead to the elimination of d 1 as a dose to which amplification can be further applied. A less restrictive choice, e.g. 5%, would mean that we are at liberty to amplify at d 1 . Schemes that include randomisation are no more difficult to implement. Returning to , if the current level is level 4 for the 25th patient, we may wish to consider amplifying either levels 1, 2, or 3. We could randomise additional backfill patients to levels 1, 2 and 3 for amplification purposes. 

A design built around amplification The design that we are proposing in this Section considers the situation where efficacy may not be markedly different for some adjacent dose levels because the dose-efficacy curve would include a plateau or be shallow . Let us consider a design with two components: a dose escalation followed by a randomised expansion . In order to spread the experimentation, instead of escalating as quickly as possible to the MTD using the conventional version of the CRM, we propose to work with pairs of adjacent dose levels in the escalation part (see ). This controlled and prospective approach is such that experimentation is less concentrated at one or two levels at the end of the escalation phase. Effectively, each cohort of patients is randomly divided between adjacent dose levels in order to identify the pair (with respect to a toxicity target level (TTL)) to take forward to the expansion part (part 2) where the randomisation takes place, in order to further refine the selection based on efficacy and ultimately identify the RP2D. A key consideration is the required sample size for the randomised part. Importantly, if the objective is to recommend one dose level for further study (i.e. identifying the RP2D), the sample size of the randomised expansion part can be calculated as if it was a selection trial using a margin of practical equivalence , as described in . This Section is divided in two, according to the two parts of our proposed design (see ). The first part of the study is the dose-escalation, and the second part consists of an expansion where two adjacent dose levels are compared in a randomised manner. The objective of the first part is to select the pair whose toxicity profile is closest to the TTL denoted by α . In the second part, the objective is to compare efficacy, measured either as binary endpoint (e.g. objective response rate) or time-to-event endpoint (e.g. progression-free survival), to determine which dose level of the pair should be the RP2D. 5.1. Dose-escalation using pairs of adjacent dose levels The dose-escalation part of the study includes m dose levels and a maximum of n patients. The goal is to locate the pair of dose levels to take forward to the randomised expansion part in an efficient and ethical way. The j th patient, j = 1 , … , n is allocated to one of the m dose levels. This level is denoted X j ∈ { d 1 , … d m } . We define Y j to be a binary variable (0, 1) where 1 denotes that patient j suffered a DLT and is zero otherwise. The probability of DLT at dose level d i , i = 1 , … , m , is denoted R ( d i ) so that, R ( d i ) = Pr ( Y j = 1 ∣ X j = d i ) and where R ( d i ) < R ( d k ) when i < k . We lean on a one-parameter model-based design, ψ ( x j , a ) where a is the model parameter, to identify the pair to take forward. In particular, we develop an extension of the Bayesian CRM  that uses pairs of adjacent dose levels. This idea was initially proposed by Huang and Chappell  but with doses in triplets instead of pairs. The escalation part starts at the first dose level, and dose-skipping is not allowed. Within these constraints, the allocation of the next cohort of patients (divided equally between the two levels of the pair) is to the pair with the smallest estimated distance to α , the TTL. To model the dose-DLT relationship, we assume a working model of the form ψ ( d i , a ) = β i a , where β i are the standardised doses (skeleton) representing the discrete dose levels d i and a follows an exponential distribution with λ parameter equal to 1. Given that the prior mean of the model parameter is equal to 1, the standardised dose is equal to the prior estimate of DLT probability at each dose level. The pair P that minimises the distance to the TTL is determined via the following criterion: P = argmin i ( β i a − α ) 2 + ( β i − 1 a − α ) 2 , i = 2 , … , m . We employ the posterior plug-in mean estimate of DLT probability for dose recommendations  using the above criterion. As the study starts at the first dose level, i.e. i = 1 , there is no dose level i − 1 = 0 . However we use β 0 = 0 and only allocate half of the cohort size to dose level 1. For example, if 6 patients is the expected cohort size in general (divided equally between the two levels of the pair), when the study is at dose level 1 only 3 patients are given dose level 1 because there is no dose level 0. 5.2. Expansion part The sample size of the randomised expansion part is calculated according to the principles of selection trials . One assumes that one of the treatment options (e.g. one of the two dose levels, usually the higher dose level) is superior to the other(s). A threshold is then defined, for example 80% or 90%, and the sample size is numerically evaluated so that the chance of selecting the superior treatment is greater or equal to the threshold. A sample size calculator for selection trials with binary endpoints (for example objective response rate) is available online at https://hakdehbi.shinyapps.io/randomised_phase_2_margin_equiv/ . It is common to use binary endpoints in early phase trials, in particular tumour response, as the timeframe for it tends to be relatively short such as one or two cycles of treatment. However, in certain advanced cancers, given that median overall survival times may be short (e.g. a few months) early phase studies may use time-to-event endpoints such as progression-free survival (PFS) or overall survival (OS). The sample size can be calculated in the same way as binary endpoints. Assuming the exponential distribution as parametric distribution for the survival time greatly facilitates this calculation . The selection trial design includes a margin of practical equivalence. This margin quantifies the amount of additional efficacy that is required so that one of the options can be selected on efficacy grounds alone, assuming that the toxicity profile remains acceptable. In our context, if the two dose levels are within the margin of practical equivalence at study completion, then additional dimensions such as toxicity, quality of life or cost, can be considered to make the final recommendation. Indeed a lower dose level with comparable efficacy may avoid late toxicities and/or might be more practical in terms of number of tablets. 

Dose-escalation using pairs of adjacent dose levels The dose-escalation part of the study includes m dose levels and a maximum of n patients. The goal is to locate the pair of dose levels to take forward to the randomised expansion part in an efficient and ethical way. The j th patient, j = 1 , … , n is allocated to one of the m dose levels. This level is denoted X j ∈ { d 1 , … d m } . We define Y j to be a binary variable (0, 1) where 1 denotes that patient j suffered a DLT and is zero otherwise. The probability of DLT at dose level d i , i = 1 , … , m , is denoted R ( d i ) so that, R ( d i ) = Pr ( Y j = 1 ∣ X j = d i ) and where R ( d i ) < R ( d k ) when i < k . We lean on a one-parameter model-based design, ψ ( x j , a ) where a is the model parameter, to identify the pair to take forward. In particular, we develop an extension of the Bayesian CRM  that uses pairs of adjacent dose levels. This idea was initially proposed by Huang and Chappell  but with doses in triplets instead of pairs. The escalation part starts at the first dose level, and dose-skipping is not allowed. Within these constraints, the allocation of the next cohort of patients (divided equally between the two levels of the pair) is to the pair with the smallest estimated distance to α , the TTL. To model the dose-DLT relationship, we assume a working model of the form ψ ( d i , a ) = β i a , where β i are the standardised doses (skeleton) representing the discrete dose levels d i and a follows an exponential distribution with λ parameter equal to 1. Given that the prior mean of the model parameter is equal to 1, the standardised dose is equal to the prior estimate of DLT probability at each dose level. The pair P that minimises the distance to the TTL is determined via the following criterion: P = argmin i ( β i a − α ) 2 + ( β i − 1 a − α ) 2 , i = 2 , … , m . We employ the posterior plug-in mean estimate of DLT probability for dose recommendations  using the above criterion. As the study starts at the first dose level, i.e. i = 1 , there is no dose level i − 1 = 0 . However we use β 0 = 0 and only allocate half of the cohort size to dose level 1. For example, if 6 patients is the expected cohort size in general (divided equally between the two levels of the pair), when the study is at dose level 1 only 3 patients are given dose level 1 because there is no dose level 0. 

Expansion part The sample size of the randomised expansion part is calculated according to the principles of selection trials . One assumes that one of the treatment options (e.g. one of the two dose levels, usually the higher dose level) is superior to the other(s). A threshold is then defined, for example 80% or 90%, and the sample size is numerically evaluated so that the chance of selecting the superior treatment is greater or equal to the threshold. A sample size calculator for selection trials with binary endpoints (for example objective response rate) is available online at https://hakdehbi.shinyapps.io/randomised_phase_2_margin_equiv/ . It is common to use binary endpoints in early phase trials, in particular tumour response, as the timeframe for it tends to be relatively short such as one or two cycles of treatment. However, in certain advanced cancers, given that median overall survival times may be short (e.g. a few months) early phase studies may use time-to-event endpoints such as progression-free survival (PFS) or overall survival (OS). The sample size can be calculated in the same way as binary endpoints. Assuming the exponential distribution as parametric distribution for the survival time greatly facilitates this calculation . The selection trial design includes a margin of practical equivalence. This margin quantifies the amount of additional efficacy that is required so that one of the options can be selected on efficacy grounds alone, assuming that the toxicity profile remains acceptable. In our context, if the two dose levels are within the margin of practical equivalence at study completion, then additional dimensions such as toxicity, quality of life or cost, can be considered to make the final recommendation. Indeed a lower dose level with comparable efficacy may avoid late toxicities and/or might be more practical in terms of number of tablets. 

Illustration shows the true DLT probabilities and median PFS curve underlying the the hypothetical trial that we use as illustration. The cohort size was six patients (divided equally between the two levels of the pair), except at dose level 1 for which it was 3 patients (see ). In this study, we allocated 10 cohorts of patients for dose-escalation purposes, for a total of 57 patients (the first cohort at dose level 1 was of size 3 and the remaining nine cohorts of 6 patients each). The chosen TTL that 33%. In the randomised part, there were 30 patients per dose level. Progression-free survival was assumed to be exponentially distributed. We used a censoring probability of 20% for both levels. A margin of practical equivalence of one month was pre-specified to select the dose level to take forward. 30 patients per arm provide 86% chance of identifying the superior dose level, assuming 4 months median PFS at the inferior level and an increase of two months at the other level. In practice, the reference median survival time may be estimated from the data of the dose-escalation part, if homogeneous patient eligibility criteria are used between the two parts of the design, as well as external data. and show the evolution of the dose-escalation part and the randomised selection part respectively. shows that the pair that was chosen for the randomised part was the dose pair 5–6. One de-escalation took place, after the first 3 patients on dose level 6 each experienced a DLT. Overall the observed toxicity proportion at dose level 6 was 33% (four out of twelve patients experienced a DLT). In the randomised part, results indicated a median progression-free survival time of 3.34 months (95% confidence interval (CI) 2.56–5.42) at dose level 5, and 6.93 months (95% CI 3.90–12.46) at dose level 6. The difference between the observed median PFS was greater than one month, which was the margin of practical equivalence. Consequently, in this specific hypothetical trial dose level 6 was recommended for further testing. In the , we present an illustration of this design with a binary endpoint (overall response rate) for the expansion part. We also present the results of a simulation study in the . We considered 6 different scenarios with different dose-efficacy curves, to assess the performance of the design under various circumstances. 

Discussion Traditionally, phase I trials in oncology have been investigating experimental treatments under the assumption of monotonic dose-toxicity and dose-efficacy curve. The goal was to give patients as much drug as possible in order to shrink the tumor, as long as the drug was safe and tolerable. However, recent phase I data have supported the hypothesis that less drug could be more beneficial on the basis of several endpoints . In order to test the hypothesis that lower levels can be of similar benefit, we need to obtain information at these lower levels. In this paper we propose an integrated study design that allows spreading experimentation during the escalation part, and that uses randomisation subsequently to compare efficacy between two levels before identifying the RP2D. The main limitation of this approach is that the design is not adaptive between the two parts. In other words, it does not include the possibility of adjusting the design of the randomised part depending on the accumulated data. In some circumstances, instead of randomising between two dose levels, it might be more appropriate to randomise between three levels. Similarly, there might be circumstances where expanding on one level only is more appropriate if there is an obvious “winner” during the escalation part. The randomised part of the design includes a sample size calculation that ensures a high probability that if there is a superior dose level efficacy-wise it will be selected. However, there might be circumstances where there is a long plateau on the efficacy curve, such that it would be more appropriate to recommend a level that is two or three levels lower than the higher dose levels of the selected pair. With our design that only includes the selected pair, this possibility is not covered. This would be a missed opportunity. Nonetheless recommending one level lower remains an improvement. Amplification of information collected over the dose range can be achieved in multiple ways. By using pairs of dose levels, we ensure that the toxicity information we obtain is more spread than if we were experimenting at one dose level at a time. More generally one can think of multiple levels being associated with specific weights that can help guide allocation of patients within a cohort. Such weights might even be dynamically adjusted during the course of the study depending on the accrued evidence. This being said, in this paper, we chose a pragmatic approach with two adjacent levels and weights of 50% for each. An alternative to pairs of adjacent dose levels is to backfill patients via randomisation to dose levels below the level where the study is currently experimenting. This approach was described in detail in our recent paper . Regardless of the approach, spreading the experimentation allows collection of additional information such as PK, PD and efficacy response data to better inform the choice of the dose levels to randomise to in the second part of the study. In a case like Scenario F in our simulations (see ), backfilling might have increased the likelihood of selecting a pair of doses to randomise to at the beginning of the range, as enough evidence would have potentially been accumulated to support the hypothesis that the efficacy curve was flat. There is evidence that non-cytotoxic agents can present a plateau on the dose-efficacy curve . Randomisation is strongly recommended by the FDA in their recent communication , where the authors argue that in addition to toxicity, PK and PD markers should be taken into account to select the dose levels to randomise to. It is therefore essential to compare potentially active levels in a randomised fashion to ensure that if there is evidence of a plateau, then the lower level is recommended as the RP2D. This has the advantage of avoiding additional toxicity, short and long-term, that may lead to reduced compliance or treatment interruptions. Moreover, lower dose levels may be easier to administer and provide better quality of life for patients, while preserving efficacy. In our proposed design, we use the framework of selection designs to calculate the required sample size in the expansion part to make an informed selection. The margin of practical equivalence is a key design feature, as it allows researchers to define in advance what should be the gain in efficacy, measured as an increase in response rate or increase in median PFS, that would justify the choice of the higher dose level on the basis of efficacy alone. Our simulations showed that the randomised component of the study allowed a satisfactory selection ability of the lower dose level of the pair for the RP2D in situations where the dose-efficacy curves were flat or shallow. The final choice will always be based on the totality of the accrued data, including PK, PD, safety and efficacy data. We fully support the recent FDA communication  that for non-cytotoxic drugs there should be a paradigm shift towards testing the hypothesis that less may be more, or at least less dose may be equally beneficial, when it comes to dose determination in oncology. In this context our paper provides a methodological contribution in the form of a specific two-part trial design: firstly experimentation is spread using pairs of dose levels and subsequently the second step randomises patients between candidate dose levels, allowing an informed selection of the RP2D. This approach falls under the umbrella term of amplification in early phase dose-finding trials. 

