Phase I clinical trials in adoptive T-cell therapies

Abstract

We develop three approaches to phase I dose finding designs for engineered T cells in oncology. Our goal is to address a very particular difficulty in this clinical setting: an inability to fully administer the dose allocated to some patients. Current designs can be biased as a result of this incomplete information being ignored or discarded from the analysis. The performance of the three proposed solutions is largely similar, and all offer an advantage over the currently used design. One of the three methods is supported by theoretical study, and we provide some new results on this approach.

1 |. INTRODUCTION

Adoptive cell therapy is a therapeutic area in oncology that has seen considerable development over the last few years. One type of adoptive cell therapy engineers the T cells of a patient to express a specific chimeric antigen receptor (CAR). The CAR focuses the specificity of the T cells, such that once they are infused back into the patient, the T-cells identify the tumour cell antigens, resulting in both the elimination of the tumour cells and the proliferation of CAR and non-CAR T-cells (June & Sadelain, 2018).

The CAR selected for a cancer patient depends on the underlying malignancy and its associated tumour-specific antigens. The most successful CAR therapy to date used CD19 as a target, as CD19 is expressed on certain B-cell-associated leukaemias and lymphomas, and multiple products have now gained FDA approval (June & Sadelain, 2018). Additional CAR targets are now under development in other blood and solid tumours, such as the bb2121 CAR for multiple myeloma and the IL13Rα2 CAR for glioblastoma (Brown et al., 2016; Raje et al., 2019). There are also new generations of CAR products that add co-stimulatory domains to improve in vivo persistence. Collectively, these equate to many CAR T-cell clinical trials in early drug development.

Unlike typical chemotherapy agents, CAR therapy involves a complex and individualized manufacturing process. As a first step for autologous CAR T-cells, a patient undergoes leukapheresis, and the T cells are then selected, activated and transduced with a transgene (Feins et al., 2019). The subsequent step expands the product ex vivo to achieve a set number of viable T-cells, which defines the ‘dose level’ taken forward in drug development. However, this dose level cannot always be achieved in the manufacturing process. Therefore, early phase clinical trials needed to specify a window or range around the target dose and include evaluability criteria if that window/range is not achieved. For example, an early phase trial of CAR T cells in diffuse large B-cell lymphoma set a target dose of 5 × 10⁸ viable cells but additionally set a minimum allowable dose of 1 × 10⁸; patients who received cells below this dose were deemed inevaluable. Overall, the median dose received among evaluable patients was 3.5 × 10⁸ viable T-cells (range: 1 to 5 × 10⁸ cells) (Schuster et al., 2019).

Once generated, the product is infused into the patient. As the modified T-cells interact with the underlying tumour and activate, an inflammatory response occurs in some patients, deemed cytokine release syndrome (CRS), which clinically manifests as flu-like symptoms and can lead to hypotension and hypoxia (June & Sadelain, 2018; Park et al., 2016). Typically either concurrent or subsequent to CRS, immune effector cell-associated neurotoxicity syndrome (ICANS) can occur resulting in decreased alertness in the patient but ultimately may progress (Neelapu, 2019). Both CRS and ICANS are associated with T-cell expansion, and worse toxicity has been observed with higher infused dose levels. Severe CRS and ICANS are typically counted in the dose-limiting toxicities (DLT) definition in phase I protocols when identifying the maximum tolerated dose (MTD) for subsequent studies.

CAR phase I studies typically investigate four to five dose levels, and the majority of trials use the 3 + 3 design or a modification of the 3 + 3 design to identify a safe dose (Park et al., 2018; Raje et al., 2019). A unique challenge, however, in these phase I studies is how to handle the scenario when the ex vivo expansion of the T-cells fails to achieve the assigned dose level. Designed phase I studies to date handled reduced cell doses differently. Many allow for a window around the assigned dose, such that patients treated with 50–80% of the dose are deemed to have received the full assigned dose (Lee et al., 2015; Park et al., 2018; Raje et al., 2019). For further reductions in the generated product, protocols have various approaches on how to count these patients in the dose escalation algorithm. Some of these approaches include the following:

1. allow the treatment of the patient with the reduced cell dose but deem the patient ‘inevaluable’ as part of the dose escalation design and replace with a new patient for that specific cohort,

2. consider the patient in the algorithm as if treated at the closest lower dose level, or

3. abort treatment altogether for the patient (Park et al., 2018; Raje et al., 2019) (examples: NCT01583686; NCT00924326; NCT02215967).

As investigators attempt to fit the clinical scenario at hand into a standard dose escalation design, the subsequent modifications to the study design and participant evaluability criteria present present new limitations in the conduct and interpretation of the trial. For one, a core assumption is that the toxicity associated with 50–80% of the assigned cell dose is the same as if the patient had received 100% of the dose. However, as an example, one study of CAR T-cells showed that patients who received doses lower than the assigned dose had less high-grade CRS (Bachanova et al., 2019). Therefore, if this assumption is not met in the phase I study, a dose level with an unacceptable level of toxicity could be recommended by the design as the MTD. Additionally, patients treated with the CAR T-cells but deemed inevaluable in the dose escalation design is a complete loss of toxicity information and replacing the patient in the trial comes at a considerable financial cost and a delay in the study duration. While the production costs of a CAR product during drug testing is not publicly available, the cost of one approved product for leukaemia is $475,000, which reflects in part the high expense of the manufacturing process. Hence, replacing patients in the dose escalation algorithm is not a viable option.

As an illustration of the problem, the top row of Figure 1 shows a simulated phase I trial where patients are assigned to one of five dose levels; however, some patients receive only a fraction of the assigned dose. Central to this manuscript is how to efficiently incorporate all toxicity information observed in a trial and accurately estimate the location of the MTD among the assigned dose levels. To this end, this manuscript presents three solutions to account for the information obtained from patients receiving a fraction of the assigned cell dose. We build on the framework of the CRM design of O’Quigley et al. (1990), and modify it to allow the toxicity information for a patient who received a reduced dose to be included in the dose escalation as opposed to being excluded and replaced. This allows all patients to contribute information to the identification of the MTD location.

FIGURE 1.

One simulated trial with five dose levels. The first row shows the observed dose-limiting toxicities (DLTs) based on the assigned and received cell doses. The second and third rows show the sequential posterior probabilities that each dose level is the true MTD after the first 6, 12, 16, and 20th patient accrued using FDLA-CRM. The true toxicity rate is shown at the top of each posterior probability plot (Toxicity Scenario 2)

1.1 |. Examples from two published clinical trials

Two phase I studies were recently published that examined the safety of CAR T-cell products. The first study reported the safety of CD19-specific CAR T cells in patients with acute lymphoblastic leukaemia (Park et al., 2018). The study examined three different dose levels (1 × 10⁷, 3 × 10⁷, 1 × 10⁸) with one fallback level (3 × 10⁶) if excessive toxicity was observed at the initial level. The study utilized a 3 + 3 design to identify the MTD. The protocol defined evaluability such that patients who received less than 50% of the planned dose would be treated but would be removed and replaced as part of the DLT-evaluable population (i.e. inevaluable for dose escalation).

The second phase I study reported the safety of B-cell maturation antigen (BCMA)-targeting CAR T cells for patients with multiple myeloma (Raje et al., 2019). The study was designed to evaluate the safety of five different doses (5 × 10⁷, 15 × 10⁷, 45 × 10⁷, 80 × 10⁷, 120 × 10⁷) with one fallback dose (2.5 × 10⁷). The study similarly used a 3 + 3 design to identify the MTD. Patients were considered evaluable if they received at least 80% of the cell dose; when less than 80% of the dose was generated, patients would still receive the cell dose but would be replaced for MTD determination.

Further details about evaluability and dose escalation are provided in the study protocols; both studies provided their study protocol as supplemental material at the time of publication.

2 |. METHODOLOGICAL APPROACH

We divide this section into two parts: a subsection that describes the main goals and the overall approach and a subsection that focuses on how to adapt this approach to the specific problems raised by dose finding for adoptive cell therapies.

2.1 |. Main goal and overall approach

Our goal is to identify the dose level among k ordered levels, d₁, d₂, …, d_k, that has a toxicity rate closest to some prespecified target rate θ. As the trial progresses, pairs of random variables (X_j, Y_j) are observed, where the binary variable Y_j is the drug-related toxicity for individual j, equal to 1 if a dose-limiting toxicity (DLT) is observed after the individual is treated at X_j = x_j, ideally one of the levels under study. In practice we may fail to achieve the aimed for dose level, x_j, and we return to this below. Suppose for now that x_j ∈ {d₁, d₂, …, d_k}. For completeness, we add to this set the dose d₀ = 0. At the different dose levels, the probabilities of toxicity, Pr(Y_j = 1|X_j = x_j), satisfy: 0 < R(d₁) <, …, < R(d_k) < 1, translating the requirement that the function R(·) is invertible on the interval (0,1). More than one definition of the maximum tolerated dose, d_M(θ) is possible and the most common one is the dose closest to some target, θ, specifically,

$$d_M(\theta )=\underset{i}{\arg \min }\left|R\left(d_i\right)-\theta \right|.$$ (1)

The function R(d_i), i = 1, …, k, is not known and we replace it by a working model Ψ(d_i, a) where a is a parameter. We need for ψ(x, a) to respect certain minimal properties, described in several papers, and we recall these briefly below. For now, suppose we have consistent estimates $\widehat{\psi }\left(d_i,\cdot \right)$, of R(d_i) then, as a consequence we have a running estimate of d_M(θ) given by:

$${\tilde{d}}_M(\theta )=\underset{i}{\arg \min }\left|\widehat{\psi }\left(d_i,\cdot \right)-\theta \right|.$$ (2)

This is the basic idea underlying model-based designs, the CRM in particular, where the overall goal of the study and the goal of providing the best treatment for each entered patient are reflected in these two equations.

While we follow this basic idea, it turns out that we can relax the assumption that our model generates the observations—a strong assumption that would provide consistency—as long as the conditions outlined in Section 3.1 are satisfied.

The objective then is to estimate the MTD as accurately as possible while maximizing the number of patients treated at and around the MTD itself. We let Ω_j = {(x_ℓ, y_ℓ), ℓ = 1, …, j} denote the dose-toxicity information after the first j patients are treated. In the case where all patients fully receive the dose that they were meant to receive, the likelihood can be written as

$$L\left(a:{\mathrm{\Omega }}_j\right)=\prod _{\ell =1}^j{\psi }^{y_{\ell }}\left(x_{\ell },a\right){\left\{1-\psi \left(x_{\ell },a\right)\right\}}^{\left(1-y_{\ell }\right)}.$$ (3)

Some authors prefer a Bayesian approach, bringing on board a prior density g(a), for a. In this case, the posterior distribution for the parameter a, after observing j patients, is

$$f\left(a\mid {\mathrm{\Omega }}_j\right)=H^{-1}\left({\mathrm{\Omega }}_j\right)\times g(a)L\left(a:{\mathrm{\Omega }}_j\right)$$

where $H\left({\mathrm{\Omega }}_j\right)={\int }_{u\in [A,B]}g(u)L\left(u:{\mathrm{\Omega }}_j\right)du$. Estimates for a are sequentially updated, either via maximum likelihood or by working with the posterior distribution whereby ${\widehat{a}}_j={\int }_{a\in [A,B]}af\left(a\mid {\mathrm{\Omega }}_j\right)da$. The main idea of alternating between estimation and allocation with the goal of treating an increasing number of patients at and close to the MTD follows through without difficulty for the cell therapy dosing problem. The difficulty comes from our inability to obtain the dose indicated by the model—a dose likely not one of those included in the design—so that the form for L(a: Ω_j) given in Equation (3) is not applicable. We consider this next.

2.2 |. Solutions for the adoptive therapy dosing problem

There are three intuitive solutions to adapting Equation (3) in order for it to be amenable to our particular problem. The first two of these solutions were suggested by reviewers of an earlier version of this paper. The third solution offers some theoretical and interpretational advantages discussed below. Extensive simulations, across relatively small or modest sample sizes, show the three approaches have very comparable performance. The dose, $x_j^A$, assigned to the jth patient may not be the dose received, x_j, and it is likely that x_j ∉ {d₁, …, d_k.} The dose x_j will lie between two adjacent doses, $x_j^{-}$ and $x_j^{+}$, $(x_j^{-}<x_j^{+})$, where, in this case, $x_j^{-}$, $x_j^{+}\in \left\{d_0,d_1,\dots ,d_{k\cdot }\right\}$. The three solutions differ in their treatment of the observed x_j and to make this transparent we can re-write Equation (3) as

$$L\left(a:{\mathrm{\Omega }}_j\right)=\prod _{\ell =1}^j{\psi }^{y_{\ell }}\left(x_{\ell }^{*},a\right){\left\{1-\psi \left(x_{\ell }^{*},a\right)\right\}}^{\left(1-y_{\ell }\right)},$$ (4)

where the only difference between Equations (3) and (4) is that x_j has been replaced by a pseudo dose observation $x_j^{*}$. If we first let, $w_j=(x_j^{+}-x_j)/(x_j^{+}-x_j^{-})$, and take ${\mathcal{I}}_A$ to be 1 when A holds, zero otherwise, then two of our three solutions can be written:

• Solution 1: In Equation (4) we take $x_j^{*}=x_j^{+}+{\mathcal{I}}_{\left[w_j<0.5\right]}(x_j^{-}-x_j^{+})$

• Solution 2: In Equation (4) we take $x_j^{*}={\psi }^{-1}(\cdot ,0)\{w_j\psi (x_j^{-},0)+\left(1-w_j\right)\psi (x_j^{+},0)\}$

• Solution 3: This is the solution studied in greater depth in this paper. It does not immediately fit in with Equation (4) via a simple definition for $x_j^{*}$. Instead, we use w_j and 1 − w_j to re-assign the contribution x_j to $x_j^{+}$ and $x_j^{-}$ respectively.

We call w_j the dose attribution coefficient for subject j. A fourth solution was also proposed—analogous to Solution 2 but replacing the ‘0’ by ‘a’—and, for this proposal, our simulations showed behaviour comparable to the other solutions. However, we did not pursue this idea further since the theoretical problems here are not negligible. Essentially this fourth solution provides a model that does not belong to the class described by Cheung (2011) and would require development of a new theoretical background.

Solution 1 amounts to simply taking the dose level that is the closest to the one achieved. Depending upon what we are able to achieve, this dose will either be one dose higher or one dose lower than that indicated by x_j. We might imagine that estimates higher and lower than what they ought to be may roughly balance out, and this is indeed not far from what we observe, at least in modest and small samples. On the other hand we can imagine situations in which the higher (or lower) level is used far more often than the lower (or higher) one, in which case there would be clear bias. We would under or overestimate the probability of a DLT at the estimated MTD.

Solution 2, re-expressed above in terms of the dose, has a similar motivation to Solution 1 and interpolates between two modelled doses based on the size of w_j. This solution would be easier to justify under the assumption of continuous dosing but, even there, we would need make stronger assumptions than we usually make concerning the working, approximate model. For a mis-specified model, which is our usual assumption, it is not clear how to proceed theoretically for either Solution 1 or Solution 2.

In consequence, most of our focus has been on Solution 3. We call this fractionated dose level attribution. Before looking at this more closely, we might repeat that, as far as our simulations show, there is nothing really to choose between the three solutions in terms of performance. Since small samples are the norm in this context, it could be argued that large sample theory is not relevant. However, good large sample behaviour does provide some reassurance and, given the tendency for early phase trials to increase in size, the more solid theoretical base to Solution 3 is not without value. Further to the appealing large sample behaviour is a finite sample physical interpretation described next.

2.3 |. Fractionated dose level attribution

Solution 3, fractionated dose level attribution (FDLA), can be described from a slightly different angle. Suppose, for the sake of argument, that all patients are treated at precisely one of the doses, d₁, …, d_k. There are no intermediary doses and we denote by π_j(d_i), i = 1, …, k, the percentage of the j patients treated at d_i. Keeping j fixed and repeating the whole experiment an unbounded number of time, the law of large numbers tells us that π_j(d_i) → Eπ(d_i|j)a.s. This observation helps our motivation and, under the further restrictive conditions of Shen and O’Quigley (1996), as j increases without bound, then Eπ(d_i| j) also has an almost sure limit. Under these conditions, Eπ(d_i|j) → 0 at all doses apart from the dose d_M(θ) at which it is equal to 1. This fails in our setting. However, again under added conditions, we will be able to identify d_M(θ) even though we cannot guarantee settling at that dose. Indeed, not only do we fail to obtain that Eπ(d_i|j) → 0 at all doses other than the MTD, to establish the existence of the limit would require conditions not only on the unknown dose-DLT relationship but also on the mechanisms, including any time trends, underlying our experimentation away from the attempted dose. We do not study this and our large sample theory is based on different assumptions.

Returning to the case of interest here, we will assume that each patient will receive a dose d, lying between adjacent available doses, say d_h and d_h+1 and, for those patients receiving the full dose, this is simply a particular case of the more general set-up. The monotonicity assumption tells us that the probability of DLT at d_h is strictly less than that at d which, in turn, has a probability of DLT less than that at d_h+1. Suppose we were to sample a large set of individuals to be treated at either dose level, d_h or d_h+1. If Pr(d_h) is the probability of sampling at dose d_h and we condition at being either at dose d_h or d_h+1^, then, Pr(d_h) = 1 − Pr(d_h+1) = π(d_h)∕{π(d_h) + π(d_h+1)}. Suppose that

$$D_h^2(w)=R\left\{(1-w)d_h+w{d}_{h+1}\right\}-(1-w)R\left(d_h\right)-wR\left(d_{h+1}\right),w\in (0,1)$$

and that, for the purposes of this section, ${\sup }_w{\max }_h{D}_h^2(w)=0$, meaning that, between adjacent doses, we are approximating linearly the increase in the probability of DLT, as we move from d_h to d_h+1, the slopes themselves being free to depend on h. We do not assume such in our simulations, and, indeed, we study later the impact of strong violations of this assumption. Viewed as a function of w, R(·) maps one-to-one on the interval (0,1), assuming the values R(d_h) and R(d_h+1), respectively, when w = 0 and when w = 1.

In summary, the probability of DLT at the received dose, indexed by w, is precisely the same probability of DLT for a mixed group, π(d_h)∕{π(d_h) + π(d_h+1)} of which have been treated at d_h and π(d_h+1)∕{π(d_h) + π(d_h+1)} treated at d_h+1. So, in a large sample sense, if the dose-toxicity function is linear between d_h and d_h+1, then weighting the likelihood based on fractionated doses will provide the correct likelihood. Of course, this needs to be studied in practice because of two things: (1) the linearity assumption and (2) the working model which may not extrapolate well beyond the assigned dose. These approximations need to be looked at critically and there may be room for improvement. Note that the linearity assumption would only be taken to hold between adjacent doses and not over the whole dose-toxicity function. So, it may well be a reasonable first step. In a similar way, our working model although only valid locally, may be good enough as long as we stray no further than the doses adjacent to the current estimate of d_M.

Some further comments: the preceding requires us to know π(d_i), i = 1, …, k. Not only are these not known, but a key large sample property underlying convergence will fail (Shen & O’Quigley, 1996). This property is that, apart from π(d_M) all of the π(d_i), i ≠ M, will tend to zero. The first difficulty is overcome by using the attribution coefficient, described just below, and we can obtain the correct equivalent to π_n(d_i), i = 1, …, k. We are unable to achieve the property described in Shen and O’Quigley (1996), since, however large n, the physical constraints of the experimental design will inevitably lead us to experiment away from the MTD. That said, under some restrictive conditions, we are still able to estimate d_M(θ) consistently. This is considered below.

3 |. LARGE SAMPLE BEHAVIOUR OF FDLA

We outline in this section how established theoretical results can be adapted to FDLA-CRM. Following Shen and O’Quigley (1996) we assume a dose-toxicity working model, ψ(x, a) with the following properties: the parameter a belong to a finite interval [A, B]; for a fixed a, ψ(x, a) is continuous and strictly increasing in x; for a fixed x, ψ(x, a) is continuous and strictly decreasing in a. These properties are not restrictive, enabling us to avoid problems to do with singularities, as well as guaranteeing the existence and uniqueness of the dose and model parameter given the other. Under these conditions, we have k constants a₁, …, a_k ∈ [A, B] such that for 1 ≤ i ≤ k, Ψ(d_i, a_i) = R(d_i), Ψ(d_i, B) < θ < Ψ(d_i, A), and, for a unique a_M ∈ (a₁, …, a_k), Ψ(d_M, a_M) = R(d_M) = θ_M. We anticipate θ_M being close to θ although, in the practical setting, we do not expect them to exactly coincide. Some immediate consequences can be deduced from these properties and they will help us, for instance, when it comes to estimating which dose is the most likely candidate to be the MTD. We have:

Lemma 1

For each 1 ≤ i ≤ k − 1, there exists a unique constant κ_i such that

$$\theta -\psi \left(d_i,{\kappa }_i\right)=\psi \left(d_{i+1},{\kappa }_i\right)-\theta >0.$$

These constants allow us to partition the interval [A, B] into a union of non-overlapping intervals so that;

$$[A,B]=\underset{i=1}{\overset{m}{\cup }}{S}_i,$$

where

$$S_1=\left[A,{\kappa }_1\right),S_2=\left[{\kappa }_1,{\kappa }_2\right),\dots ,S_m=\left[{\kappa }_{m-1},B\right].$$

As an illustration, suppose that $\psi \left(d_i,a\right)={\alpha }_i^a$, and α₁ = 0.04, α₂ = 0.07, α₃ = 0.20, α₄ = 0.35, α₅ = 0.5, α₆ = 0.7. Then: S₁ = [0, 0.55), S₂ = [0.55, 0.79), S₃ = [0.79, 1.26), S₄ = [1.26, 2.08), S₅ = [2.08, 3.56), S₆ = [3.56, 5]. Note that this breakdown opens the way to a simple construction of any prior, including a non-informative one, by putting a piecewise uniform on each segment so that the prior probability associated with each interval is just 1/k (O’Quigley, 2006). The importance of this partition of the parameter space is reflected in the following corollary.

Corollary 1

Suppose that ${\widehat{a}}_j$ is the estimate of the parameter a after the inclusion of j patients. If ${\widehat{a}}_j\in S_i$ then d_i is the level recommended to patient j + 1.

The above lemma and corollary were shown in O’Quigley (2006) and form the basis for the main finite sample behaviour of the continual reassessment method. With these results in mind we consider their implications on the behaviour of FDLA. The linearity implications of the previous section were based on assuming that ${\sup }_w{\max }_i{D}_i^2(w)=0$. This will not generally be true and, in order to extend the above lemma and corollary, we would like for this quantity to be an order of magnitude smaller than the gaps between adjacent dose levels. This enables us to approximate the unknown dose toxicity function by a piecewise linear one, having a total of k parameters. This makes it very flexible although our only purpose for this extra modelling is to help with theoretical study. We make no use of it otherwise and it is not assumed to hold in any of the simulations.

Conditions required for studying the large sample behaviour of Solutions 1 and 2 may be found; however, further study is required and is beyond the scope of this work. While large sample theory is not developed in this manuscript, we return to these solutions in the simulations below.

3.1 |. Likelihood

The received dose for the jth patient, x_j will lie between two adjacent doses, $x_j^{-}$ and $x_j^{+}(x_j^{-}<x_j^{+})$, where, in this case, $x_j^{-}$, $x_j^{+}\in \left\{d_0,d_1,\dots d_{k\cdot }\right\}$ This is because, in the setting of adoptive T-cell studies, manufacturing the assigned number of cells is not always achieved and only a reduced quantity can be generated. It is assumed that this reduced quantity falls between two of the dose levels under consideration, $x_j^{-}$ and $x_j^{+}$, and we introduce the attribution coefficient, w_j so that $x_j=\left(1-w_j\right)x_j^{-}+w_j{x}_j^{+}$. When w_j = 1, the patient receives their full dose. Otherwise, we assume they receive so much, $w_j=(x_j^{+}-x_j)/(x_j^{+}-x_j^{-})$ of $x_j^{+}$ and the rest, 1 − w_j, is attributed to $x_j^{-}$. Individual j now makes a contribution that is shared between two levels. The likelihood can be written,

$$L\left(a:{\mathrm{\Omega }}_j\right)=L^{+}\left(a:{\mathrm{\Omega }}_j\right)\times L^{-}\left(a:{\mathrm{\Omega }}_j\right)$$ (5)

where;

$$L^{+}\left(a:{\mathrm{\Omega }}_j\right)=\prod _{\ell =1}^j{\psi }^{w_{\ell }{y}_{\ell }}\left(x_{\ell }^{+},a\right){\left\{1-\psi \left(x_{\ell }^{+},a\right)\right\}}^{w_{\ell }\left(1-y_{\ell }\right)}.$$

and where L⁻(a: Ω_j) is of the same form as L⁺(a: Ω_j) but with 1 – w_ℓ in place of w_ℓ and with $x_{\ell }^{-}$ in place of $x_{\ell }^{+}$.

Convergence arguments obtain from considerations of the likelihood. The same arguments apply to Bayesian estimation as long as the prior is other than degenerate, that is, all the probability mass is not put on a single point. Usual likelihood arguments break down since our models are mis-specified. We can define $U_n(a)=U_n^{+}(a)+U_n^{-}(a)$ where,

$$U_n^{+}(a)=\frac{1}{n\overline{w}}\sum _{j=1}^n{w}_j\left[y_j\frac{{\psi }^{\prime }}{\psi }\{x_j^{+},a\}+\left(1-y_j\right)\frac{-{\psi }^{\prime }}{1-\psi }\{x_j^{+},a\}\right]$$

and

$$U_n^{-}(a)=\frac{1}{n(1-\overline{w})}\sum _{j=1}^n\left(1-w_j\right)\left[y_j\frac{{\psi }^{\prime }}{\psi }\{x_j^{-},a\}+\left(1-y_j\right)\frac{-{\psi }^{\prime }}{1-\psi }\{x_j^{-},a\}\right]$$

in which $x_j^{+}$ denotes the dose immediately above x_j, $x_j^{-}$ the dose immediately below x_j (when x_j < d₁ then $x_j^{-}=0$) and w_j denotes the percentage of x_j that is attributed to $x_j^{+}$ while 1 − w_j indicates the percentage attributed to $x_j^{-}$. Finally, we denote the mean of the w_j by $\overline{w}=n^{-1}{\sum }_j{w}_j$. Closely related to $U_n^{+}$ and $U_n^{-}$ are:

$${\tilde{U}}_n^{+}(a)=\frac{1}{n\overline{w}}\sum _{j=1}^n{w}_j\left[R(x_j^{+})\frac{{\psi }^{\prime }}{\psi }\{x_j^{+},a\}+(1-R(x_j^{+}))\frac{-{\psi }^{\prime }}{1-\psi }\{x_j^{+},a\}\right]$$

and

$${\tilde{U}}_n^{-}(a)=\frac{1}{n(1-\overline{w})}\sum _{j=1}^n\left(1-w_j\right)\left[R(x_j^{-})\frac{{\psi }^{\prime }}{\psi }\{x_j^{-},a\}+(1-R(x_j^{-}))\frac{-{\psi }^{\prime }}{1-\psi }\{x_j^{-},a\}\right].$$

Analogous to U_n(a) we also define ${\tilde{U}}_n(a)={\tilde{U}}_n^{+}(a)+{\tilde{U}}_n^{-}(a)$. Once we have a solution to the equation U_n(a) = 0 then the maximum likelihood estimate, $\widehat{R}\left(d_i\right)=\psi \left(d_i,{\widehat{a}}_j\right)$, exists. Further properties follow in much the same way as those obtained in O’Quigley (2006). The following lemmas and corollaries make this more precise. For each n, let a(n) be the unique vanishing point of U_n(a).

Lemma 2

For each ε > 0 there exists N_ϵ such that |a(n) − a(n + 1)| < ε, whenever n > N_ϵ.

Proof Let $J_n^{+}(a)=U_n^{+}(a)-U_{n+1}^{+}(a)$. The continuity of $J_n^{+}(a)$ follows from that for $U_n^{+}$ and $U_{n+1}^{+}$. Also, in view of condition 6 from O’Quigley (2006), $J_n^{+}(a)$ is strictly monotonic in a. The same holds for $J_n^{-}$ where $J_n^{-}(a)=U_n^{-}(a)-U_{n+1}^{-}(a)$. Given continuity, we can find large enough $N_{ϵ}^{-}$ and $N_{ϵ}^{+}$ and, if we let $N_{ϵ}=\max \left(N_{ϵ}^{+},N_{ϵ}^{-}\right)$, the result follows.

Lemma 3

If _yn = 1 then a(n) < a(n − 1), otherwise a(n) > a(n − 1).

Cheung (2005) argued in favour of the desirability for certain operating characteristics, among which the property of being coherent. By coherence, we mean that, following a DLT the design will not recommend escalation, following a non-DLT the design will not recommend a de-escalation. More formally, we can say

Corollary 2

Escalation and de-escalation are coherent in the sense of Cheung (2005).

The corollary follows since both ψ and 1 − ψ are less than one, and both ${\psi }^{\prime }(x_j^{+},a)$ and ${\psi }^{\prime }(x_j^{-},a)$ are strictly greater than zero. Note also that the coherence condition applies to $x_i^A$, and not to the received dose x_j which will be restricted by our ability to achieve the dose indicated by $x_i^A$. In addition to the conditions described at the beginning of this section we require a further restrictive condition. First we need an open set of values of a and this is defined by:

$$S\left(d_M\right)=\{a:\left|\psi \left(d_M,a\right)-\theta \right|+{\mathrm{\Delta }}^2<\left|\psi \left(d_i,a\right)-\theta \right|,\text{for}\text{all}{d}_i\ne d_M\}.$$ (6)

We know that our working model is too poor to be able to reproduce the correct rate of DLT at all levels via a single parameter a₀. The set, S(d_M), gives us a way to quantify how far away the true situation can be from that represented by the model. The model is poor and yet good enough to correctly model the DLT rate at any level taken in isolation. At levels adjacent to the MTD, Δ² allows us to decide how far from the model we can be and yet still work, that is, the greater the value of Δ², the more difficult it will be to obtain convergence. We can view Δ² as a kind of penalty putting a bound on how far away can the unknown curve be from our assumed model. The added condition we require is

Condition 1

For i=1, …, k, a_i ∈ S(d_M) and ${\sup }_w{\max }_i{D}_i^2(w)\le {\mathrm{\Delta }}^2$.

Recall that in Shen and O’Quigley (1996), Δ² = 0, which means that D²(w) = 0, w ∈ (0, 1), that is, between doses we suppose linearity. In simulations we have studied the impact of these assumptions and we discuss them further later. They provide a restrictive structure within which we can obtain large sample properties. Certainly it would be possible to relax these restrictions to some degree, as in Cheung (2011), and our purpose is more to provide some theoretical support to our approach than to lean heavily upon things that cannot be tested. No restrictive assumptions are used as part of the simulations.

Lemma 1 shows that the parameter space breaks down in a logical way into a partition, the members of which indicate the level to be used. This result hints that we do not expect to wildly jump around the dose levels and that we can anticipate reasonable behaviour, as long as we choose a sensible partition. Also, if we impose a prior distribution on this partition, we are able to calculate the posterior distribution of any level being the MTD. This is looked at below.

Lemma 2 indicates that, with increasing sample size, dose allocation will stabilize as it we will eventually settle at some level. In our context, by settle, we mean that we will recommend the same dose and will not deviate from this recommendation with a probability tending to one. In practice we may use other doses because we fail to achieve that dose for any given patient. Unless we have many dose levels under consideration—which is unlikely to be the case in CAR T-cell trials—then experimentation will be relatively cautious. There is no need for extra rules to control escalation and de-escalation to avoid potential erratic behaviour until we have good estimates. The underparametrization greatly contributes towards creating this stability (Iasonos et al., 2016).

Lemma 3 and the subsequent corollary show that dosing is coherent. Following a DLT we will not recommend escalation and, following a non-DLT, we will not recommend de-escalation from that dose. Again, this only applies to the recommended dose and, of course, if we fail to achieve the recommended (assigned) dose, then, superficially, behaviour might be viewed as being not coherent.

3.2 |. Almost sure convergence to the MTD

In addition to the property of coherence, Cheung (2005) also argued that two further desirable properties of a good design are ε-sensitivity and the impossibility of getting stuck at some dose regardless of the data (rigidity). The first of these means that, in the long run, we will concentrate experimentation within a distance ε of the MTD (measured on the probability of DLT scale). The second property means that all of the doses have the potential to be explored. This is made formal in Lemma 4 below.

Given the model structure, for each 1 ≤ i ≤ k, the definition of a_i and the basic conditions outlined at the beginning of the section on the dose toxicity function, then a_i is the unique solution to the equation

$$\left\{R\left(d_i\right)\frac{{\psi }^{\prime }}{\psi }\left(d_i,a\right)+\left(1-R\left(d_i\right)\right)\frac{-{\psi }^{\prime }}{1-\psi }\left(d_i,a\right)\right\}=0.$$ (7)

Lemma 4

Following Cheung (2011) and Clertant and O’Quigley (2017), the design based on FDLA is ε-sensitive and, in consequence, is not rigid.

This follows from the above equation and means that dose exploration will be unhindered and will not get stuck at a level as can occur with two-parameter CRM. Our next step is to consider the finite interval S₁(d_M) = [a₍₁₎, a_(k)] in which a₍₁₎ = min{a₁, …, a_k} and a_(k) = max{a₁, …, a_k}. Again, referring to our working assumptions and, following the outline of Shen and O’Quigley (1996), we have that S(d_M) is an open and convex set and that S₁(d_M) ⊂ S(d_M). Rewrite ${\tilde{U}}_n(a)$ as

$${\tilde{U}}_n(a)=\sum _{i=1}^k{\pi }_n\left(d_i\right)\left\{R\left(d_i\right)\frac{{\psi }^{\prime }}{\psi }\left(d_i,a\right)+\left(1-R\left(d_i\right)\right)\frac{-{\psi }^{\prime }}{1-\psi }\left(d_i,a\right)\right\},$$ (8)

and, with $\tilde{a}$ defined by ${\tilde{U}}_n(\tilde{a})=0$, we have:

Lemma 5

Under the model conditions described above,

$$\underset{a\in [A,B]}{\sup }\left|U_n(a)-{\tilde{U}}_n(a)\right|\to 0,almostsurely.$$ (9)

This convergence result follows intuitively and can be demonstrated rigorously in a number of ways. For instance, observe that for each dose level d_i, (ψ′∕ψ)(d_i, ·) and { ψ′∕(1 − ψ)}(d_i, ·) are uniformly continuous in a over the finite interval [A, B]. As n becomes large, ${\tilde{a}}_n$ will fall into the interval S₁(a₀). Since ${\widehat{a}}_n$ solves U_n(a) = 0, and uniform continuity ensure that, almost surely, ${\widehat{a}}_n\in S\left(a_0\right)$ for n sufficiently large. Hence, for large n, ${\widehat{a}}_n$ satisfies Equation (6) and, in particular,

$$\left|\psi \left(d_M,{\widehat{a}}_n\right)-\theta \right|<\left|\psi \left(d_i,{\widehat{a}}_n\right)-\theta \right|,\text{for}i=1,\dots ,k,d_i\ne d_M.$$

Thus for n large enough $x_{n+1}^A\equiv d_M$ and we can summarize this via:

Corollary 3

Under the conditions described above, $x_n^A\to d_M$ almost surely as n increases without bound.

We do not have the more standard case where at this dose ${\widehat{a}}_n$ converges almost surely to a₀; instead, ${\widehat{a}}_n$ will converge to a point inside S₁(d_M). This point will depend on the experimental conditions and how closely, on average, is x_j to $x_j^A$.

3.3 |. Predictive probabilities for the true MTD

Using the likelihood L(Ω_j) in Equation (5), the posterior probability p_i that dose level d_i is the true MTD (d_m) given the data observed after the first j patients can be estimated as

$$p_i=P\left(d_m=d_i\right)=H^{-1}\left({\mathrm{\Omega }}_j\right){\int }_{u\in [A,B]}{\mathcal{I}}_{u\in S_i}L\left(u:{\mathrm{\Omega }}_j\right)g(u)du$$ (10)

where $H\left({\mathrm{\Omega }}_j\right)={\int }_{u\in [A,B]}L\left(u:{\mathrm{\Omega }}_j\right)g(u)du$ is a normalizing measure over [A, B] and where ${\mathcal{I}}_C=1$ when C holds and is zero otherwise. We can use the p_i as we make progress through the study to decide which level, or levels, appear to be the most likely to be the actual MTD (Iasonos & O’Quigley, 2016).

4 |. SIMULATION STUDY

The performance of the model-based designs were compared under various simulation configurations. These include the model-based approach where all fractionated patients were assigned to the closest dose level (Solution 1: ‘CRM Closest Level’); the model-based approach that weights the skeleton values $\psi (x_j^{-},0)$ and $\psi (x_j^{+},0)$ for fractionated patients (Solution 2: ‘Weighted Skeleton CRM’); lastly, the model-based approach using fractionated dose level attribution (Solution 3: ‘FDLA-CRM’). These three approaches are compared against the design most frequently used in this setting: the 3 + 3 algorithm-based approach, where patients are considered evaluable if they received at least 50% of the intended dose.

Five different toxicity scenarios were evaluated with either five or four total dose levels. Table 1 provides the toxicity rates for each scenario. When a patient received a fraction of the assigned cell dose, the true toxicity (DLT) rate assigned to the patient was a weighted combination of the toxicity rates for the two dose levels $x_j^{+}$ and $x_j^{-}$ that bound the dose the patient received (i.e. using the weights w_j and 1 − w_j described in Section 2). The target toxicity rate θ for the model-based approaches was set to 0.20, and the skeleton was selected based on Lee and Cheung (2009), which is also provided in Table 1.

TABLE 1.

Simulation parameters: assigned cell doses, skeleton values and true toxicities rates for the simulation scenarios with four or five dose levels

	Level 1	Level 2	Level 3	Level 4	Level 5
4 dose levels
Cell dose (× 106)	100	200	400	800
Skeleton	0.111	0.200	0.308	0.423
Toxicity rates
Scenario 1	0.05	0.10	0.15	0.25
Scenario 2	0.10	0.20	0.45	0.60
Scenario 3	0.05	0.10	0.20	0.30
Scenario 4	0.10	0.20	0.30	0.45
Scenario 5	0.00	0.05	0.10	0.20
5 Dose levels
Cell dose (× 106)	50	100	200	400	800
Skeleton	0.049	0.111	0.200	0.308	0.423
Toxicity rates
Scenario 1	0.10	0.20	0.40	0.55	0.60
Scenario 2	0.05	0.10	0.20	0.40	0.60
Scenario 3	0.12	0.20	0.30	0.40	0.55
Scenario 4	0.07	0.12	0.20	0.33	0.40
Scenario 5	0.01	0.05	0.10	0.15	0.25

In the main simulation configuration, the proportion of patients with a fractionated cell dose increased on average for higher dose levels. This is based on the combination of two random variables drawn from a binomial and beta distribution. The first random variable Bin(p) determined whether the patient received the full cell dose, and p decreased from 0.9 to 0.5 over four or five equally spaced probabilities depending on the number of dose levels. If the first random variable equalled 0, indicating a fractionated dose, then the random fraction of the assigned cell dose received was drawn from a Beta(5,5). Sample sizes of 16 and 20 total patients were evaluated for the model-based design. The methods were also compared when all patients received the full dose—therefore all model-based approaches were equivalent—and when all patients received a fractionated dose.

4.1 |. Example simulated trials

To illustrate FDLA-CRM as one approach and how information accumulates as patients accrue, we selected a single trial from each of the five toxicity scenarios (Figures S1 and S2). Figure 1 shows one of these trials for toxicity scenario 2 with five dose levels. Of the 20 patients accrued on trial, eight received a reduced cell dose. The four plots of posterior probabilities, p_i = P(d_m = d_i) using Equation (10), show that after the first six patients accrued, the probabilities were nearly uniform that each dose level is the true MTD; however, after an additional 6–10 patients, the highest probability was dose level 3, which has a true toxicity rate of 0.20 aligning with our target θ.

4.2 |. Evaluation of operating characteristics

The three model-based designs along with the 3 + 3 were compared in terms of the percentage of trials selecting each dose level as the MTD, the average distribution of patients assigned to each dose level, and the total sample size as an aggregate of full dose and fractionated dose patients.

Figure 2 shows the percentage of 10,000 simulated trials that selected each of the dose levels as the MTD for the main configuration when the average cell fraction received depended on the dose level. Comparing the model-based approaches, all three had nearly identical performance in MTD selection. These methods selected the dose level closest to the target of θ=0.20 in over 40% of the simulated trials. These methods had a gain in performance of approximately 20–30% over the standard 3 + 3 design.

FIGURE 2.

Percentage of simulated trials that selected each of the five dose levels (D1–D5) as the maximum tolerated dose (MTD) for the three model-based approaches and the 3 + 3 design. Model-based designs included 20 total patients

Figures 3 and 4 show the average patient allocation per assigned dose level and the average sample size for the same five dose level simulation study. Across the scenarios, all model-based methods treated the most patients closest to or at the target of 0.20. The relative allocation of patients for the 3 + 3 was more spread across the levels for some scenarios. In terms of sample size, the 3 + 3 had fewer evaluable patients than the 20-patient model-based methods, although the gains in terms of total sample size when aggregating inevaluable and evaluable patients varied by scenario.

FIGURE 3.

Percentage of patients allocated across the five dose levels (D1–D5) for the three model-based approaches and the 3 + 3 design. Model-based designs included 20 total patients

FIGURE 4.

Total sample sizes based on patients receiving a full dose, patients receiving a fractionated dose but considered evaluable, and patients receiving a fractionated dose but considered inevaluable for the three model-based approaches and the 3 + 3 design for the scenarios with five dose levels. Model-based designs included 20 total patients

The percentage of trials selecting each level as the MTD, the average allocation of patients per level, and the total sample size for the simulations with four total dose levels is provided in Figures S3, S4 and S5. Overall, these results largely mirror the simulations with five dose levels.

Section S3 provides simulation results for a total sample size of 16 patients for the model-based approaches. While there was a relative drop in performance compared to the 20-patient model-based designs, gains in performance over the 3 + 3 design remained. The model-based designs had a similar or smaller total sample size compared to the 3 + 3.

The model-based designs and the 3 + 3 were additionally compared under two configurations either when all patients received the assigned dose or when all patients received a random fraction of the assigned dose. In the latter configuration, the fraction received was randomly drawn from a Beta(5, 5) distribution for all patients. Figure S12 shows the MTD selection for five dose levels under both simulation configurations. Contrasting these two configurations side-by-side illustrates the clear limitations of the 3 + 3. Allowing patients who receive 50–90% of the assigned dose to remain evaluable under this design can result in dose levels with high rates of toxicity being selected as the MTD more often. This is due to the monotonicity of the dose-dependent toxicity rates. In contrast, while there was a decrease in performance compared to the full dose simulation, the model-based approaches under the all fractionated simulation still selected the dose level closest to the target θ most often across the scenarios and assigned the most patients to this level (Figure S13).

We further evaluated the performance of FDLA-CRM when the linearity assumption introduced in Section 2.3 no longer holds between dose levels. Simulations considered various deviations for the true toxicity rate, including the deviation as far as possible from this assumption (i.e. a step function) while respecting the requirement of lying between the rates of d_h and d_h+1. There was little impact on the performance of FDLA-CRM (Figure S14).

Lastly, we evaluated the methods for different toxicity targets, such as θ=0.15. Similar performance was observed. Results are available upon request from the authors.

5 |. DISCUSSION

This paper introduces three new model-based designs that are able to flexibly incorporate toxicity information from patients who receive a reduced cell dose in the setting of adoptive T-cell therapy. While there are key differences on how each of these three designs update their respective likelihoods when fractionated dose patients accrue on study, the broad simulation scenarios indicate largely similar performance for the methods. Therefore, from a practical perspective in the small sample size setting typically observed in phase I trials, any one of three methods may be suitable for implementing a new adoptive T-cell study.

However, one of the model-based designs, FDLA-CRM (Solution 3), is supported by large sample theory as detailed in Section 3. Additional work is required to provide a similar theoretical foundation to Solution 1 or Solution 2 under the common assumption made in this setting that the working model is mis-specified. Therefore, given the similar performance of the three methods across simulations, the solid theoretical framework to support FDLA-CRM leads us to currently recommend this approach for future phase I trials in this setting.

The model-based approaches were directly compared to the design most frequently used in this clinical setting: the 3 + 3 algorithm. Forcing this algorithmic design to the scenario at hand requires two dependent decisions at the design stage: what reduction of the assigned cell dose is allowed to still be considered as treated at the full assigned dose level, and two, how to handle patients who do not fulfill this criterion. A balancing act ensues. Depending on the number of dose levels (e.g. two to three levels), including only patients who received the full assigned dose under this design might be reasonable. However, this decision would considerably drive up costs due to replacing inevaluable patients, whose treatment with the manufactured cells is either aborted or continues outside of the primary dose escalation study. Reducing this level of stringency by creating a window around the assigned dose reduces the cost but can impact the accuracy of the design. This balancing act is ideally avoided altogether when designing a study and is what motivated the methodology proposed in this paper.

The methods were evaluated throughout the simulation studies for two sample sizes: 16 and 20 total patients. As expected, gains in performance were observed for the larger sample size. However, the 16-patient model-based designs performed well, outperforming the 3 + 3 in terms of accurately identifying the MTD at a similar average sample size. Therefore, the 16-patient model-based designs are recommended over the 3 + 3 when constrained by the total costs of the trial or anticipated accrual.

There are other designs with important parallels to our proposal. Wages and Fadul (2019) introduced a model-based design that identifies a feasible MTD by independently modelling the toxicity of cell dose levels along with the feasibility of manufacturing each level. A key difference is that patients under this design are treated at one of the predefined dose levels that is strictly less than the number of manufactured cells, while our methods allow for patients to be treated with all manufactured cells by attributing partial or full toxicity to adjacent levels. Which design is selected for a future phase I study will depend on the clinical question and objectives of the phase I study.

Piantadosi et al. (1998) and Hu et al. (2013) proposed modifications to CRM allowing for either continuous dosing falling between a minimum and maximum range or a dose-insertion method that inserts a dose level if one level has a toxicity probability much lower than the target and the subsequent level has a probability much higher. While both are flexible to accommodate additional intermediary doses, the objectives of their methods are different as they can recommend any cell count that falls between the predefined range. In our setting, planning for any possible dose may not be logistically feasible from a manufacturing perspective, and ultimately, the goal is to identify one of the predefined dose levels as the MTD, not an intermediary dose.

[Lastly, Ji et al. (2012) evaluated a Bayesian CRM model in the setting of phase I trials of adoptive T-cells. The motivation of their proposal was the limited sample size in some trials due to the high cost and challenging manufacturing process. While a similar setting, their proposal does not address reduced cell dose patients, and instead focuses on a design with good operating characteristics when constrained by a relatively small sample size. Therefore, the goals and challenges addressed under these designs are ultimately different.

There are potential extensions to FDLA-CRM that can be considered. Fractionated dose level attribution could be extended to the bivariate endpoints of toxicity and efficacy when considering phase I/II studies or expansion cohorts of adoptive cell therapy (Iasonos & O’Quigley, 2017; O’Quigley et al., 2001). Another possibility is combination dose finding studies of two therapies, which pose different constraints as the monotonicity assumption—central to the methodology proposed in this manuscript—no longer holds when two therapies are escalated simultaneously. One possibility is the conditional likelihood approach based on partial ordering as proposed by Wages et al. (2011), which could be extended to account for fractionated dose levels. Lastly, it may be possible to go one step further and investigate bivariate outcomes in the setting of combination therapies, such as proposed by Huang et al. (2007). These extensions require further study to investigate the corresponding statistical properties of the models in the specific clinical context of adoptive T-cell therapy.