DEMO: Dose Exploration, Monitoring, and Optimization Using a Biological
Mediator for Clinical Outcomes

Cheng-Han Yang1, Peter F. Thall2, Ruitao Lin2∗

1Department of Biostatistics, The University of Texas Health Science Center at
Houston,

Houston, TX 77030, USA

2Department of Biostatistics, The University of Texas MD Anderson Cancer
Center,

Houston, Texas 77030, U.S.A.

∗Corresponding author<EMAIL_ADDRESS>
Abstract. Phase 1-2 designs provide a methodological advance over phase 1
designs for dose finding by using both clinical response and toxicity. A phase
1-2 trial still may fail to select a truly optimal dose. because early
response is not a perfect surrogate for long term therapeutic success. To
address this problem, a generalized phase 1-2 design first uses a phase 1-2
design’s components to identify a set of candidate doses, adaptively
randomizes patients among the candidates, and after longer follow up selects a
dose to maximize long-term success rate. In this paper, we extend this
paradigm by proposing a design that exploits an early treatment-related, real-
valued biological outcome, such as pharmacodynamic activity or an
immunological effect, that may act as a mediator between dose and clinical
outcomes, including tumor response, toxicity, and survival time. We assume
multivariate dose-outcome models that include effects appearing in causal
pathways from dose to the clinical outcomes. Bayesian model selection is used
to identify and eliminate biologically inactive doses. At the end of the
trial, a therapeutically optimal dose is chosen from the set of doses that are
acceptably safe, clinically effective, and biologically active to maximize
restricted mean survival time. Results of a simulation study show that the
proposed design may provide substantial improvements over designs that ignore
the biological variable.

KEY WORDS: Bayesian adaptive design, Biomarker, Dose optimization,
Immunotherapy, Targeted agents

##  1 Introduction

The current paradigm for early-phase oncology trials is changing.
Conventionally, a phase 1 oncology trial chooses a maximum tolerated dose
(MTD) based on a binary indicator, $Y_{T}$, of dose-limiting toxicity (DLT).
Phase 1 designs originally were developed for cytotoxic agents, for which a
higher dose is more likely to provide a tumor response, indicated by $Y_{R}$,
because it kills more cancer cells, but also is more likely to cause toxicity
because it kills more normal cells. Some commonly used phase 1 designs include
a variety of 3+3 algorithms, versions of the continual reassessment method
(CRM) (Cheung, 2011), a Bayesian logistic regression model (BLRM) based method
(Neuenschwander et al., 2015), and Bayesian optimal interval (BOIN) designs
(Liu and Yuan, 2015; Yuan et al., 2022). A well known problem with Phase 1
designs is that they tend to select doses that are higher than necessary for
targeted agents or immunotherapies having biological mechanisms of action for
which increasing administered dose beyond a certain level, say $\tau_{B}$,
does not increase efficacy (Postel-Vinay et al., 2016; Wages et al., 2018;
Brock et al., 2021). This may occur if the metabolized agent reaches a
saturation level in the patient, so that increasing the administered dose
beyond $\tau_{B}$ does not increase the clinical response rate, but may cause
a higher risk of toxicity. Because phase 1 trials are small with DLT defined
based on short term follow up, such undesirable effects may not be seen until
much later in the treatment evaluation process, possibly during a phase 3
trial or in a large patient population following regulatory approval (Shah et
al., 2021), which forces practicing physicians to make ad hoc dose
adjustments.

By using both response and toxicity for screening and identifying an optimal
dose, phase 1- 2 designs provide a substantial improvement over phase 1
designs. Early phase 1-2 designs were proposed by Gooley et al. (1994), Thall
and Russell (1998), Braun (2002), and Thall and Cook (2004), who gave Bayesian
rules for sequential dose selection and monitoring safety and futility.
Utility-based phase 1-2 designs have been proposed by Thall and Nguyen (2012),
Lin and Yin (2017), Lin et al. (2020), and many others. Reviews are given by
Yuan et al. (2016) and Yan et al. (2018).

Phase 1-2 designs still may fail to identify truly optimal doses because
short-term response, $Y_{R}$, and toxicity, $Y_{T},$ are not perfect
surrogates for a later clinical outcome that characterizes long-term treatment
success. In oncology, long-term success may be defined in terms of a time-to-
event variable, $Y_{S}$, such as remission duration (RD) among early
responders (Hamada et al., 2018; Ritchie et al., 2018), progression-free
survival (PFS) time, or overall survival (OS) time, evaluated over longer
follow up than what is used to evaluate $Y_{R}$ and $Y_{T}$. For example, an
immunotherapy may extend survival by controlling cancer progression without
causing significant early tumor shrinkage, and consequently its efficacy can
only be established in terms of long-term PFS or OS time. To address this
problem, Thall et al. (2023) proposed the family of Generalized Phase 1-2 (Gen
1-2) designs, which use $Y_{R}$ and $Y_{T}$ for screening and long term
therapeutic success defined in terms of $Y_{S}$ evaluated over longer follow
up to optimize dose or schedule.

In this paper, we propose an extension of the Gen 1-2 paradigm that
incorporates a real-valued biological outcome, $Y_{B}$, measured shortly after
dose administration, that may be related to dose effects on the clinical
outcomes $Y_{T}$, $Y_{R}$, and $Y_{S}$. We assume that $Y_{T}$ and $Y_{B}$ are
evaluated over a short time period, corresponding to one or two courses of
therapy. This is followed by evaluation of $Y_{R}$, and later $Y_{S}$, subject
to administrative censoring. We will exploit the role of $Y_{B}$ as a mediator
between dose $d$ and $(Y_{T},Y_{R},Y_{S})$. This is illustrated by the
directed acyclic graph (DAG) in Figure 1, which shows that a dose $d$ may
influence each of the outcomes $Y_{T}$, $Y_{R}$, and $Y_{S}$ directly, and
also through indirect effects mediated by $Y_{B}.$ The DAG also allows
interactions between $Y_{T}$ and $Y_{R}$. The rationale for incorporating
$Y_{B}$ into the dose-outcome model and design is that, if $Y_{B}$ mediates
dose effects on the clinical outcomes, the additional information provided by
$Y_{B}$ may increase the probability of identifying a truly optimal dose. To
simplify the exposition, in the sequel we will refer to $Y_{S}$ as survival
time.

The proposed design, which we call DEMO, does Dose Exploration, Monitoring,
and Optimization. A DEMO design has three stages, (1) sequential dose
exploration, including elimination of unsafe or biologically inactive doses,
(2) randomized dose screening, and (3) randomized dose optimization, with
successively more intensive monitoring as the trial progresses. Because DEMO
incorporates $Y_{B}$, it is similar in spirit to the dose-finding designs of
Ursino et al. (2017) and Gerard et al. (2022), who explored methods utilizing
PK information, Takeda et al. (2018), who developed a PK exposure-toxicity
model based on the area under the concentration (AUC) curve, and Su et al.
(2022), who proposed a semi-mechanistic design based on dynamic PK/PD
modeling.

In addition to incorporating $Y_{B}$, another important difference between
DEMO and a Gen 1-2 design is the criterion used for optimal dose selection. In
DEMO, an optimal therapeutic dose (OTD) is defined as the dose that maximizes
restricted mean survival time (RMST), among doses that are acceptably safe,
have an acceptable clinical response rate, and are biologically active. In
contrast, a Gen 1-2 design relies on mean survival time (MST) or
$\Pr(Y_{S}>t_{S}\,|\,d)$ for given fixed $t_{S}$, such as six months, as a
criterion for selecting an optimal dose. An advantage of RMST over MST is that
RMST may be evaluated at any meaningful time point. Early-phase trials often
have limited follow-up time that may not be long enough to allow MST to be
estimated reliably. In such cases, RMST still can be computed. This
flexibility facilitates analysis of $Y_{S}$ for a variety of different
timeframes. In practice, the estimated RMST often is more reliable than the
estimated mean or median survival time (Royston and Parmar, 2013).

##  2 The DREAMM Studies

To motivate the DEMO design, we review a sequence of clinical trials conducted
to evaluate the safety and effectiveness of belantamab mafodotin (GSK2857916)
for patients with relapsed or refractory multiple myeloma (Trudel et al.,
2018; Lonial et al., 2020). Belantamab mafodotin is an immunoconjugate that
targets the B-cell maturation antigen (BCMA), a tumor necrosis superfamily of
cell-surface receptors required for cell survival. In the DREAMM-1 trial
(NCT02064387), 79 patients were assigned sequentially to doses in the set
{0.03, 0.06, 0.12, 0.24, 0.48, 0.96, 1.92, 2.5, 3.4, 4.5} mg/kg, using the
BLRM, with a target toxicity rate of 25%. No MTD was reached, and 3.4 mg/kg
was identified as the recommended phase 2 dose (RP2D). The bioactivity of
belantamab mafodotin may be quantified by the free soluble BCMA level measured
after infusion, defined as $Y_{B}$ = $-$log([post-infusion BCMA]/[baseline
BCMA]) evaluated at day 7, with larger $Y_{B}$ corresponding to greater
biological activity. The overall response rate (ORR) and PFS time were
evaluated to characterize clinical efficacy.

While 3.4 mg/kg achieved the highest ORR and no DLTs were observed in
DREAMM-1, a substantial proportion of patients needed subsequent dose
reductions to manage later adverse events (AEs). This motivated the
investigators to conduct the DREAMM-2 trial (NCT03525678), in order to
generate additional safety and efficacy data at the RP2D and compare it to a
lower dose. In DREAMM-2, patients were randomized fairly between 2.5 mg/kg
($n$ = 97) and 3.4 mg/kg ($n$ = 99). The DREAMM-2 data showed no clinically
meaningful efficacy differences, with ORR rates 31% at 2.5 mg/kg and 34% at
3.4 mg/kg, and estimated median PFS 11.0 months for 2.5 mg/kg and 13.7 months
for 3.4 mg/kg. Since the 2.5 mg/kg cohort had a slightly smaller severe AE
rate of 40% versus 47% for 3.4 mg/kg, the RP2D was chosen to be 2.5 mg/kg
because it showed comparable anti-myeloma activity and a more favorable safety
profile. However, if PFS had been used as the primary efficacy endpoint 3.4
mg/kg would have been selected because it had estimated median PFS 2.7 months
longer than that of 2.5 mg/kg. A more recent trial was DREAMM-3, a phase 3
randomized trial comparing belantamab mafodotin monotherapy given at 2.5 mg/kg
to the combination of pomalidomide and dexamethasone. Using PFS time as the
primary endpoint, the outcome of the DREAMM-3 trial was negative.

Considered together, the DREAMM trials illustrate what can go wrong in
settings where dose exploration and optimization are carried out in separate
trials, and a key biological outcome that is related to clinical outcomes is
not incorporated into the study design. In general, organizing and conducting
a sequence of trials in an ad hoc manner may lead to increased overall trial
duration, higher costs, and suboptimal decisions. For example, the choice of
the two doses compared in DREAMM-2 was not formally justified, and cannot
serve as a model for designing future trials. The DREAMM-2 study also showed a
disagreement between estimates of the intermediate outcome OR and long-term
PFS, which in general is not uncommon.

The DEMO design addresses practical and technical challenges that arose in the
DREAMM studies. DEMO is a seamless three-stage design that eliminates the gap
between separate trials and, ideally, might have been used to construct a
single trial to replace the two DREAMM trials. Throughout its three stages,
DEMO employs multiple endpoints, including short-term bioactivity and
toxicity, an intermediate tumor response indicator, and long-term survival.
Together, the three stages of DEMO provide a coherent basis for making better
informed decisions based on the available data, and increasing the probability
that the chosen dose will be safe and maximize long-term survival. Because
DEMO selects an optimal dose based on long-term survival data, it aligns with
the conventional goal of randomized confirmatory trials.

##  3 Dose Acceptability and Optimality Criteria

###  3.1 Notation

While the final objective of DEMO is to select an optimal dose based on RMST,
during the trial undesirable doses are dropped using acceptability criteria
defined in terms of $Y_{T},$ $Y_{B},$ $Y_{R},$ and $Y_{S}.$ Dropping doses
determined to be unacceptable based on interim data provides a basis for
enriching the sample sizes for acceptable doses that are potentially optimal,
and also benefits future patients enrolled in the trial. Denote the doses by
$d_{1}<d_{2}<\cdots<d_{J}$ and, for brevity, temporarily suppress notation for
model parameters. For simplicity, and to focus on the key elements of DEMO, we
will assume that $Y_{R}$ and $Y_{T}$ are binary indicators, and denote
$\pi_{k}(d_{j})$ = $\Pr(Y_{k}=1\,|\,d_{j})$ for $k=R,T.$ If desired, the DEMO
design may be generalized to accommodate ordinal $Y_{R}$ and $Y_{T},$ with
appropriate model extensions. The density, survival probability function, and
hazard function of $Y_{S}$ for a patient treated with dose $d_{j}$ are
denoted, respectively, by $f_{S}(y\,|\,d_{j}),$
${\mathcal{F}}_{S}(y\,|\,d_{j})$, and $h_{S}(y\,|\,d_{j})$ =
$f_{S}(y\,|\,d_{j})/{\mathcal{F}}_{S}(y\,|\,d_{j})$, for $y>0.$ Given a fixed
follow up time $t_{S}$, the RMST of $d_{j}$ is $\mu_{S}(d_{j})$ =
$\int_{0}^{t_{S}}{\mathcal{F}}_{S}(y\,|\,d_{j})dy.$ The DEMO design includes
dose acceptability criteria defined in terms of fixed limits elicited from the
clinical investigators, including $\overline{\pi}_{T}$ = the maximum
acceptable $\pi_{T}(d_{j})$, $\underline{\pi}_{R}$ = the minimum acceptable
$\pi_{R}(d_{j})$, and $\underline{\mu}_{S}$ = the minimum acceptable
$\mu_{S}(d_{j})$.

###  3.2 Biological Endpoints

The main extensions of a Gen 1-2 design provided by DEMO arise from
incorporating a biological endpoint, $Y_{B}$. This is used to screen out
biologically inactive doses, and to inform the distributions of the clinical
variables by regression on $Y_{B}.$ Depending on the trial’s setting, there
are many well-established biological endpoints that are known to be related to
clinical outcomes, and thus they may be used as $Y_{B}$ in a DEMO design.
Examples of $Y_{B}$ include pharmacokinetic (PK) or pharmacodynamic (PD)
variables such as the maximum serum concentration (Cmax), drug exposure
characterized by pharmacologic area under the curve (AUC) that characterizes
systemic exposure of a metabolized agent, or minimum inhibitory concentration
(MIC). Other examples of $Y_{B}$ include the expression level of a protein
involved in a signaling pathway, a targeted biomarker of programmed cell
death-ligand such as PD-L1 expression level for a checkpoint inhibitor, a B
cell surface glycoprotein CD19, the inflammatory cytokine IL-15, and cell
level following infusion of engineered CAR-NK or T-cell therapies. In the
motivating DREAMM studies, $Y_{B}$ would be BCMA level for belantamab
mafodotin. Well-established biomarkers exist that are associated with response
rate or long-term survival, such as the level of circulating tumor DNA (cfDNA)
for lung cancer (Cisneros-Villanueva et al., 2022). For a cell therapy trial,
Walter et al. (2012) reported that T-cell responses were associated with
better disease control and longer survival time for renal cell cancer
patients. Selitsky et al. (2019) found that B cell characteristics in skin
cutaneous melanoma play an important role in predicting $Y_{R}$ and $Y_{S}$
following immunotherapy.

Unlike design parameters used for clinical endpoints, such as a maximum
probability of toxicity or minimum probability of response, it typically is
not feasible to establish a target or threshold value for $\mu_{B}(d_{j})$ =
$E(Y_{B}\,|\,d_{j})$. This is because, in practice, what may be desirable or
undesirable values of $\mu_{B}(d_{j})$ in terms of effects on clinical
outcomes typically have not been established, especially in first-in-human
trials. Below, we will present a data-adaptive approach to identify and drop
biologically inactive doses based on $Y_{B}$. In the sequel, we will use the
terms “biological activity”, “bioactivity”, and “biological endpoint”
interchangeably to refer to $Y_{B}$.

Rather than assuming that $Y_{B}$ is a surrogate endpoint for either $Y_{R}$
or $Y_{S},$ we make the more realistic assumption that $Y_{B}$ may act as a
mediator between $d$ and the clinical variables $(Y_{R},Y_{T},Y_{S})$, as
shown in Figure 1. While the distribution of $Y_{B}$ may be constructed to
reflect a particular biological effect, for tractability we make the practical
assumption that

$Y_{B}\,|\,d_{j}\ \sim\ \mathcal{N}(\mu_{B}(d_{j}),\ \sigma_{B}^{2}),$ (3.1)

where $\mu_{B}(d_{j})$ = $E(Y_{B}\,|\,d_{j})$ is assumed to increase with
$d_{j}$. This monotonicity assumption reflects the underlying biology of the
dose effect on $Y_{B}$ in most real-world applications. For example, a larger
number of infused T-cells should lead to a higher level of targeted biomarker
engagement, quantified by $Y_{B}$. In contrast, a response indicator $Y_{R}$
or survival time $Y_{S}$ each may be impacted by many different factors apart
from dose. If the underlying biology does not imply monotonicity of
$\mu_{B}(d_{j})$, however, the model for $Y_{B}$ and the methodology should be
modified accordingly. In the DEMO design, at successive stages of the trial,
different models for $\mu_{B}(d_{j})$ are assumed for different purposes. The
first model, used in stage 1, is constructed to provide a basis for screening
out biologically inactive doses. The second model is used to exploit causal
relationships between $Y_{B}$ and the clinical outcomes (Figure 1) when using
the trial’s final data to identify the optimal dose.

We identify a set of biologically inactive doses by formulating the first
model for $\mu_{B}(d_{j})$ as a simple step function on the interval from 0 to
$d_{J}$ that may have at most one increasing step at one of the $d_{j}$’s. We
denote these step functions by $\mu_{B}^{*}(d)$ for
$d\in\\{d_{1},\ldots,d_{J}\\}$, to distinguish them from the general class of
mean functions $\mu_{B}(d).$ Using the set of $\mu_{B}^{*}(d_{j})$’s, we focus
on the problem of identifying the dose in $\\{d_{1},d_{2},\cdots,d_{J}\\}$
where the increasing step occurs, and we denote this dose by $\tau_{B}$. We
define the set of step functions to include the constant function having no
increase, with $\mu_{B}^{*}(d_{1})=\cdots=\mu_{B}^{*}(d_{J})$, and in this
case we define $\tau_{B}$ = $d_{1}$ and do not declare any doses to be
biologically inactive. The next step function has
$\mu_{B}^{*}(d_{1})<\mu_{B}^{*}(d_{2})=\cdots=\mu_{B}^{*}(d_{J})$ with
$\tau_{B}$ = $d_{2},$ and so on, up to the last step function where
$\mu_{B}^{*}(d_{1})=\mu_{B}^{*}(d_{J-1})<\mu_{B}^{*}(d_{J})$ and $\tau_{B}$ =
$d_{J}.$ For each of the $J-1$ step functions with $\tau_{B}$ = $d_{j}$ for
$j=2,\cdots,J,$ we define the lower doses $\\{d_{1},\cdots,\tau_{B}-1\\}$ to
be biologically inactive and the upper doses $\\{\tau_{B},\cdots,d_{J}\\}$ to
be biologically active.

Using this structure, we formulate estimation of $\tau_{B}$ as a Bayesian
model selection problem, with $J$ possible models for $\mu_{B}^{*}(d),$ where
each model corresponds to one of the step functions. For each $j=1,\cdots,J,$
we define the $j^{th}$ model, $\mathcal{M}_{j},$ to be the step function for
which $\tau_{B}=d_{j}$, and denote its prior probability by
$\Pr(\mathcal{M}_{j})$. To determine whether each dose is biologically
inactive ($-$) or active ($+$) using this construction, the distribution of
$Y_{B}$ under $\mathcal{M}_{j}$ is assumed to be normal with common variance
$\sigma^{2}_{B}$ and mean that is one of two values, $\mu_{-}$ or
$\mu_{+}>\mu_{-}$. Denoting the observed data by ${\mathcal{D}}$ and priors
for $\mu_{-}$, $\mu_{+}$, and $\sigma^{2}_{B}$ by $\pi(\mu_{-})$,
$\pi(\mu_{+})$, and $\pi(\sigma_{B}^{2})$, respectively, the posterior of
$\mathcal{M}_{j}$ is

$\displaystyle{\displaystyle\Pr(\mathcal{M}_{j}\,|\,\mathcal{D})}$
$\displaystyle\propto$
$\displaystyle\Pr(\mathcal{M}_{j})\int\pi(\sigma_{B}^{2})\left\\{\int\prod_{r=1}^{j-1}\phi_{r}(\mu_{-},\sigma_{B}^{2}\
\,|\,\ \mathcal{D})\pi(\mu_{-}){\rm d}\mu_{-}\right.$
$\displaystyle\left.\times\int\prod_{r=j}^{J}\phi_{r}(\mu_{+},\sigma_{B}^{2}\
\,|\,\ \mathcal{D})\pi(\mu_{+}){\rm d}\mu_{+}\right\\}{\rm d}\sigma_{B}^{2}.$

For each dose index $r=1,\cdots,J$ in this expression, $\phi_{r}$ denotes the
product likelihood of normal pdfs of the observed $Y_{B}$ values of all
patients treated at dose $d_{r}$. We assume priors

$\pi(\mu_{+})=\mathcal{N}(m_{\mu_{+}},v^{2}_{\mu}),\ \
\pi(\mu_{-})=\mathcal{N}(m_{\mu_{-}},v^{2}_{\mu}),\ \
\pi(\sigma_{B}^{-2})={\rm Gamma}(a_{\sigma},b_{\sigma}),$

where $m_{\mu_{-}}<m_{\mu_{+}}.$ In the Supplementary Materials, we give
elicited values of the hyperparameters $m_{\mu_{-}}$, $m_{\mu_{+}}$,
$v^{2}_{\mu}$, $a_{\sigma}$ and $b_{\sigma}$, and derive a closed form for
$\Pr(\mathcal{M}_{j}\,|\,\mathcal{D})$. For simplicity, we assume a discrete
uniform prior on the model probabilities, $\Pr(\mathcal{M}_{j})=1/J$ for all
$j$.

Using a model selection framework, a cutoff $c_{B}$ is specified so that a
dose $d_{j}$ is considered biologically inactive based on the data
$\mathcal{D}$ if $\Pr(\mathcal{M}_{r}\,|\,\mathcal{D})>c_{B}$ for a higher
dose $d_{r}>d_{j}$. We estimate the dose where the upward step occurs as
$\hat{\tau}_{B}$ = $\hat{\tau}_{B}(\mathcal{D})$ = the smallest dose $d_{j}$
such that $\Pr(\mathcal{M}_{j}\,|\,\mathcal{D})>c_{B}$. If no dose satisfies
this posterior criterion then $\hat{\tau}_{B}$ = $d_{1}$, and no doses are
declared biologically inactive. If $\hat{\tau}_{B}>d_{1}$ then doses
$\\{d_{1},\cdots,\hat{\tau}_{B}-1\\}$ are declared biologically inactive and
dropped.

###  3.3 Dose Acceptability and Optimality

The OTD, which is the criterion used by DEMO for final dose selection, is
defined using all four endpoints $Y_{B},Y_{T},Y_{R},$ and $Y_{S}$. The
definition is formulated to accommodate settings where the $RMST(d)$ curve
increases to a plateau at a dose $\tau_{B}$, and no longer increases for
higher doses. In such as case, more than one dose in
$\\{d_{1},\cdots,d_{J}\\}$ may maximize $RMST(d)$.

Definition The Optimal Therapeutic Dose (OTD) is the lowest dose $d_{j}$ in
$\\{d_{1},\ldots,d_{J}\\}$ that is biologically active ($d_{j}\geq\tau_{B}$),
safe ($\pi_{T}(d_{j})<\overline{\pi}_{T}$), and clinically effective in terms
of both response probability ($\pi_{R}(d_{j})>\underline{\pi}_{R}$) and
survival time ($\mu_{S}(d_{j})>\underline{\mu}_{S}$), that maximizes the RMST
$\mu_{S}(d_{j})$.

The following four statistical criteria are used during the trial to determine
which doses are acceptable. Given observed data ${\mathcal{D}}$ and the
decision cutoffs, $c_{B},$ $c_{T},$ $c_{R},$ and $c_{S}$, a dose $d_{j}$ is an
acceptable OTD candidate if it satisfies all of the following posterior
inequalities:

$\displaystyle{\rm Biologically\ Active}$ $\displaystyle\hskip
20.00003ptd_{j}\geq\hat{\tau}_{B}(\mathcal{D})$ (3.2a) $\displaystyle{\rm
Safe}\ $ $\displaystyle\hskip
20.00003pt\Pr\\{\pi_{T}(d_{j})\geq\bar{\pi}_{T}\,|\,\mathcal{D}\\}\leq c_{T},$
(3.2b) $\displaystyle{\rm Acceptable\ Clinical\ Response\ Rate}$
$\displaystyle\hskip
20.00003pt\Pr\\{\pi_{R}(d_{j})\leq\underline{\pi}_{R}\,|\,\mathcal{D}\\}\leq
c_{R},$ (3.2c) $\displaystyle{\rm Acceptable\ Restricted\ Mean\ Survival\
Time}$ $\displaystyle\hskip
20.00003pt\Pr\\{\mu_{S}(d_{j})\leq\underline{\mu}_{S}\,|\,\mathcal{D}\\}\leq
c_{S}.$ (3.2d)

These statistical requirements reflect the definition of OTD. They say that,
given current data $\mathcal{D},$ to be an acceptable candidate for being
selected as the OTD, $d_{j}$ must be biologically active (2a), and unlikely to
have an unacceptably high toxicity probability (2b), a low response
probability (2c), or a short RMST (2d). Under the model selection framework
given above, requiring $d_{j}\geq\hat{\tau}_{B}$ is equivalent to requiring
$\max_{d_{r}>d_{j}}\\{\Pr(\mathcal{M}_{r}\,|\,\mathcal{D})\\}\leq c_{B}$. In
each stage of the DEMO design, some combination of the four dose acceptability
criteria is applied, with each quantity computed from the most recent data. At
the end of the trial, the estimated OTD using the final data $\mathcal{D}$ is
the dose in $\\{d_{1},\cdots,d_{J}\\}$ that satisfies the acceptability
conditions (2a) – (2d) and maximizes the posterior mean RMST,
$E\\{\mu_{S}(d_{j})\,|\,{\mathcal{D}}\\}$, for $j=1,\cdots,J$. The decision
cutoffs $c_{B},c_{T},c_{R},c_{S}$ in (2a) - (2d) are calibrated by computer
simulation to obtain a design with good operating characteristics (OCs).

To compare the DEMO design to existing dose-finding designs, we give
conventional definitions of targeted doses, including the maximum tolerated
dose (MTD) and optimal biological dose (OBD). These definitions will be used
in the simulation scenarios discussed in Section 6. We will consider various
scenarios, including cases where MTD $=$ OBD $=$ OTD, MTD $>$ OBD $=$ OTD, MTD
$>$ OBD $\not=$OTD, and so on. It is important to clarify some logical
inconsistencies with terminology used conventionally to define these target
doses, however. When a 3+3 algorithm is used in a phase l trial, the MTD is
defined statistically as the dose in $\\{d_{1},\cdots,d_{J}\\}$ selected by
the 3+3 algorithm. That is, there is no optimality criterion used by a 3+3
algorithm, other than the algorithm itself. If, instead, a version of the CRM
(Cheung, 2011; Yuan et al., 2022) is used in phase 1, the MTD is defined as
the dose for which the estimate of the true toxicity probability
$\pi_{T}(d_{j})$ is closest to a fixed value ${\pi}_{T}^{target}$, such as
0.25 or 0.30.

It is common practice to refer to the dose that maximizes a phase 1-2 design’s
optimality criterion defined in terms of $(Y_{R},Y_{T})$ as an optimal
biological dose (OBD), despite the fact that no biological variable such as
$Y_{B}$ is used. This definition of OBD relies on an implicitly assumed
surrogacy of $Y_{R}$ for biological activity, which is not true in general.
This definition of OBD actually identifies what more properly may be called an
optimal phase 1-2 dose. In the present setting, we will consider phase 1-2
trials run using an optimality criterion defined in terms of a utility
function $U(a,b)$ that quantifies the desirability of each possible pair of
outcomes $(Y_{R}=a,Y_{T}=b)$, for $a,b=0,1.$ In practice, to establish
$U(a,b)$, it is convenient to first set $U(1,0)$ = 100 for the best and
$U(0,1)$ = 0 for the worst possible outcomes, and then elicit the two
intermediate values $U(0,0)$ and $U(1,1)$, possibly presenting the four
numerical values of $U(a,b)$ in a $2\times 2$ utility table (Lin et al., 2020;
Msaouel et al., 2023). The mean utility of dose $d_{j}$ is defined as
$\bar{U}(d_{j})=E\\{U(Y_{R},Y_{T})\,|\,d_{j}\\}$. This construction may be
generalized easily to accommodate bivariate ordinal outcomes (Thall et al.,
2017). While this convention for defining an OBD is illogical in the present
setting because we observe and make explicit use of $Y_{B},$ we will define
OBD following this common practice. Thus, given a utility, we define the OBD
as the lowest dose in $\\{d_{1},\cdots,d_{J}\\}$ that is safe
($\pi_{T}(d_{j})<\bar{\pi}_{T}$), clinically effective
($\pi_{R}(d_{j})>\underline{\pi}_{R}$), and maximizes $\bar{U}(d_{j})$.

Our definition of the OTD is far more stringent than the definitions of the
MTD or OBD. By aiming for the OTD and assuming a multivariate dose-outcome
model for $p(Y_{R},Y_{T},Y_{B},Y_{S}\,|\,d)$ in which $Y_{B}$ may exert
mediated effects on each of the clinical variables $Y_{R},$ $Y_{T},$ and
$Y_{S}$, the DEMO design can be more effective in identifying a truly optimal
dose.

##  4 Stages of the DEMO Design

###  4.1 Design Overview

The DEMO design proceeds in three stages, assuming stage-specific models to
evaluate the dose acceptability criteria given above. Figure 2 gives a
schematic plot to illustrate the three stages, and additional details will be
provided below in subsequent sections. If at any point of the trial there are
no admissibly safe, biologically active, and clinically effective doses, the
trial is stopped and no dose is selected. Stage 1 uses $Y_{T},$ $Y_{B},$ and
possibly $Y_{R}$ to assign successive cohorts of patients to doses in
sequentially adaptive steps, while eliminating doses that are unsafe or
biologically inactive. Stage 1 may use either a phase 1 or a phase 1-2
design’s rules to assign acceptable doses to patient cohorts, while also
applying the biological acceptability rule (2a) and the safety rule (2b). In
the example given in Figure 2, based on $Y_{T}$ and $Y_{B}$ collected during
stage 1, the acceptable dose set
$\mathcal{A}_{1}=\\{d_{2},d_{3},d_{4},d_{5}\\}$ is identified, since $d_{1}$
is dropped due to lack of bioactivity (criterion 2a) and $d_{6}$ is dropped
due to excessive toxicity (criterion 2b). In Stage 2, patients are randomized
among acceptable doses, screening for safety and biological activity is
continued, and the design also screens doses for unacceptably low response
rates (2c). In Figure 2, dose $d_{2}$ is eliminated in stage 2 due to
insufficient efficacy, leading to the acceptable dose set
$\mathcal{A}_{2}=\\{d_{3},d_{4},d_{5}\\}$. At the end of stage 2, if necessary
to obtain a feasible design, the set of acceptable doses is further reduced to
include at most $K$ doses, for $K\geq 2$. Stage 3 continues to randomize
patients among the acceptable doses, adds the RMST criterion (2d) for
restricting the set of acceptable doses, and chooses a final OTD based on
RMST. In Figure 2, dose $d_{5}$ is dropped due to lack of sufficient survival
benefit. At the end of stage 3, the RMST is estimated for each remaining
acceptable dose based on the final data to determine the OTD.

###  4.2 Stage 1: Sequential Exploration

A maximum of $N_{1}$ patients are treated in Stage 1, applying the safety
monitoring rule (2b), using either a phase 1 or phase 1-2 design to choose
acceptable doses for successive cohorts of patients. When $N_{1}/2$ patients
have been treated and their biological variables $Y_{B}$ evaluated, the
acceptability rule (2a) is applied to drop biologically inactive doses. If
feasible, $Y_{B}$ may be monitored more intensively, such as when $N_{1}/3$
and $2N_{1}/3$ patients have been evaluated.

To apply the monitoring rules (2a) and (2b) feasibly in stage 1, since the
rules focus on the marginal parameters, we model $(Y_{T},Y_{B})$ data in stage
1 assuming a simple independence model, with
$p(Y_{T},Y_{B}|d)=p(Y_{T}\,|\,d)\times p(Y_{B}\,|\,d)$. We assume that
$p(Y_{B}\,|\,d)$ follows the above step function model, and that toxicity
follows a Bayesian logistic regression model,
$\text{logit}\left\\{\pi_{T}(d_{j})\right\\}=\alpha_{0}+\alpha_{1}d_{j},$
where $\alpha_{0}$ is real-valued and $\alpha_{1}>0$. The priors are assumed
to be $\alpha_{0}\sim{\mathcal{N}}(m_{\alpha_{0}},\nu_{\alpha_{0}}^{2})$ and
$\log(\alpha_{1})\sim{\mathcal{N}}(m_{\alpha_{1}},\nu_{\alpha_{1}}^{2})$.
Elicitation of the hyperparameters
$m_{\alpha_{0}},\nu_{\alpha_{0}}^{2},m_{\alpha_{1}}$, and
$\nu_{\alpha_{1}}^{2}$ is illustrated in the Supplementary Materials.

###  4.3 Stage 2: Randomized Monitoring

In stage 2, a maximum of $N_{2}$ additional patients are randomized among the
acceptable doses identified in stage 1, applying the three acceptability
criteria (2a), (2b), and (2c). Because in practice $Y_{R}$ may take a longer
time to evaluate than $Y_{T}$ and $Y_{B}$, the acceptability rule (2c) is
added to take advantage of the fact that more response data will become
available as the trial progresses. Either simple or adaptive randomization can
be applied in stage 2.

To monitor dose acceptability while also taking advantage of the biological
effects of the agent when analyzing the stage 2 data, we assume that $Y_{B}$
is a mediator for $Y_{R}$ and $Y_{T}$. We exploit the likelihood factorization
$p(Y_{R},Y_{T},Y_{B}\,|\,d)=p(Y_{R},Y_{T}\,|\,Y_{B},d)\times p(Y_{B}\,|\,d),$
and assume that

$\displaystyle\mu_{B}(d_{j})$ $\displaystyle=$
$\displaystyle\gamma_{0}+\gamma_{1}d_{j}^{\gamma_{3}}/(\gamma_{2}^{\gamma_{3}}+d_{j}^{\gamma_{3}}),$
$\displaystyle\pi_{T}(Y_{B},d_{j})$ $\displaystyle=$
$\displaystyle\Pr(Y_{T}=1\,|\,Y_{B},d_{j})={\rm
logit}^{-1}(\alpha_{0}+\alpha_{1}d_{j}+\alpha_{2}Y_{B}),$ (4.1)
$\displaystyle\pi_{R}(Y_{B},d_{j})$ $\displaystyle=$
$\displaystyle\Pr(Y_{R}=1\,|\,Y_{B},d_{j})={\rm
logit}^{-1}(\beta_{0}+\beta_{1}d_{j}+\beta_{2}d_{j}^{2}+\beta_{3}Y_{B}).$

The Emax model for $\mu_{B}(d)$ in expression (4.3), which now replaces the
much simpler step function model used in stage 1, is assumed for its
flexibility. The baseline parameter $\gamma_{0}$ is real-valued,
$\gamma_{1}>0$ is the maximum possible bioactivity, $\gamma_{2}>0$ is the dose
that produces half of $\gamma_{1}$, and $\gamma_{3}>0$ is the Hill factor that
controls the steepness of $\mu_{B}(d)$. The logistic regression model for
$\pi_{T}$ as a function of $d$ alone in stage 1 is elaborated in stage 2 to
include $Y_{B}$ as a covariate. We assume real-valued $\alpha_{0}$, with
$\alpha_{1}>0$ and $\alpha_{2}>0$ to ensure that $\pi_{T}(Y_{B},d_{j})$
increases with both $d$ and $Y_{B}$. A similar model $\pi_{R}(Y_{B},d_{j})$ is
assumed for $Y_{R}$, but including a quadratic term to allow a more flexible
dose effect. We do not make monotonicity assumptions for
$\pi_{R}(Y_{B},d_{j})$, and allow $\beta_{0},\ldots,\beta_{3}$ to take on any
real values.

As tractable operational priors for the parameters in the joint model (4.3),
we assume

$\displaystyle\gamma_{0}\sim\mathcal{N}(m_{\gamma},\nu_{\gamma}^{2}),\quad\gamma_{l}\overset{\rm
i.i.d.}{\sim}{\rm
Gamma}(a_{\gamma_{l}},b_{\gamma_{l}}),l=1,2,3,\quad\sigma_{B}^{-2}\sim{\rm
Gamma}(a_{\sigma},b_{\sigma}),$ $\displaystyle\log\alpha_{l}\overset{\rm
i.i.d.}{\sim}\mathcal{N}(m_{\alpha_{l}},\nu_{\alpha_{l}}^{2}),l=1,2,\quad\beta_{l}\overset{\rm
i.i.d.}{\sim}\mathcal{N}(m_{\beta_{l}},\nu_{\beta_{l}}^{2}),l=1,2,3,$
$\displaystyle(\alpha_{0},\beta_{0})\sim\mathcal{BN}\left((\mu_{\alpha_{0}},\mu_{\beta_{0}}),(\nu_{\alpha_{0}}^{2},\nu_{\beta_{0}}^{2},\rho_{0})\right),$

where $\tilde{\bm{\theta}}$ = $(m_{\gamma},\nu_{\gamma}^{2}$,
$a_{\gamma_{l}},b_{\gamma_{l}}$, $a_{\sigma},b_{\sigma}$,
$m_{\alpha_{l}},\nu_{\alpha_{l}}^{2}$, $m_{\beta_{l}},\nu_{\beta_{l}}^{2}$,
$\mu_{\alpha_{0}},\mu_{\beta_{0}}$, $\nu_{\alpha_{0}}^{2}$,
$\nu_{\beta_{0}}^{2}$,$\rho_{0})$ is the vector of all model hyperparameters,
and $\mathcal{BN}$ denotes the bivariate normal distribution with correlation
$\rho_{0}$. Guidelines for specifying $\tilde{\bm{\theta}}$ are given in the
Supplementary Materials. We use a Gibbs sampler to obtain posterior samples of
the parameters in the joint model (4.3). To avoid numerical integration of
$\pi_{T}(Y_{B},d_{j})$ and $\pi_{R}(Y_{B},d_{j})$ over the distribution of
$Y_{B}$ when computing the marginals $\pi_{T}(d_{j})$ and $\pi_{R}(d_{j})$, we
use a plug-in approach by substituting the posterior mean
$\hat{\mu}_{B}(d_{j})$ in place of $Y_{B}$ to approximate $\pi_{T}(d_{j})$ and
$\pi_{R}(d_{j})$ for each $d_{j}.$ This facilitates computing the
acceptability rules (2a)–(2c).

###  4.4 Stage 3: Randomized Optimization

In stage 3, a maximum of $N_{3}$ additional patients are randomized among the
current set of acceptable doses, applying the four acceptability criteria
(2a), (2b), (2c), and (2d). If at any point there are no acceptable doses, the
trial is stopped and no dose is selected. Otherwise, the dose with largest
posterior mean RMST based on the final data is selected as the OTD.

The set of doses to be studied in stage 3 is determined as follows. First, for
a small fixed integer $L\geq 2$, adaptively select the best $L$ acceptable
doses, denoted by $\mathcal{C}_{1}$, in terms of the posterior mean response
rate, or the mean utility. To account for plateau scenarios where multiple
doses have similar response rates, we define set $\mathcal{C}_{2}$ to be the
best $K$ ($K\geq L$) acceptable doses with posterior tail probabilities of
$\pi_{R}(d_{j})$ at least a specified fraction $(1-\kappa)\in(0,1)$ of the
maximum value, that is,

$\Pr\\{\pi_{R}(d)>\underline{\pi}_{R}\,|\,{\mathcal{D}}\\}\geq(1-\kappa)\max_{d}\\{\Pr(\pi_{R}(d)>\underline{\pi}_{R}\,|\,{\mathcal{D}})\\}.$

The set of doses selected for stage 3 evaluation then is defined as
$\mathcal{C}_{1}\cup\mathcal{C}_{2},$ and the number of doses is denoted by
$W=|\mathcal{C}_{1}\cup\mathcal{C}_{2}|$. Intuitively, suppose that the top
$L$ acceptable doses are advanced to the final stage. If there are many
promising doses demonstrating nearly equivalent maximum response rates, at
most an additional $K-L$ doses will be added for stage 3. When $\kappa=0$,
this gives a smaller set with $\leq L$ doses, while $\kappa=1$ gives a larger
set with $\leq K$ doses.

To obtain a feasible sample size, depending on the number of doses $J$
specified at the beginning of the trial, as a heuristic rule we suggest using
$(L,K,\kappa)=(2,3,0.3)$ for $J\leq 5$ and $(L,K,\kappa)=(3,4,0.3)$ for $J\geq
6$. If $W$ is impractically large, one can further reduce $L$ and $K$ to
obtain a manageable stage 3 dose set. It also is important to note that the
number of selected doses $W$ may be smaller than $L$ due to the three
acceptability rules (2a)–(2c) imposed through stages 1 and 2. According to our
sensitivity analyses reported below, in Section 6, the design yields robust
performance using the default specifications of $(L,K,\kappa)$.

In stage 3, patients are randomized unequally among the selected doses,
restricted to ensure that each dose has a maximum sample size of $M$. As an
illustration, if $M$ = 20, and the $W$ = 3 selected doses have respective
sample sizes 12, 9, 14 at the start of stage 3, then the dose-specific sample
sizes in stage 3 would be $M-n_{1}=8,M-n_{2}=11,$ and $M-n_{3}$ = 6. Another
randomization strategy involves using response-adaptive randomization, which
allocates more patients to the better-performing arms, while capping the
maximum sample size at a reasonable value.

To evaluate the dose acceptability rules (2a)–2(d), we construct the stage
3-specific full model using the likelihood factorization

$p(Y_{S},Y_{R},Y_{T},Y_{B}\,|\,d)=p(Y_{S}\,|\,Y_{R},Y_{T},Y_{B},d)\times
p(Y_{R},Y_{T}\,|\,Y_{B},d)\times p(Y_{B}\,|\,d).$

Building on the stage 2-specific joint model for
$p(Y_{R},Y_{T}\,|\,Y_{B},d)\times p(Y_{B}\,|\,d)$, we assume that
$p(Y_{S}\,|\,Y_{R},Y_{T},Y_{B},d)$ is a Weibull regression model with hazard
function

$h(Y_{S}\,|\,d_{j},Y_{T},Y_{R},Y_{B})=\rho
Y_{S}^{\rho-1}\lambda_{j}\exp\left\\{\eta_{1}Y_{T}+\eta_{2}Y_{R}+\eta_{3}Y_{B}\right\\}$
(4.2)

where $\rho\geq 1$ is the shape parameter, and each $\lambda_{j}>0$ is a dose-
dependent scale parameter that characterizes the direct effect of dose $d_{j}$
on survival. The real-valued parameters $\eta_{1},\eta_{2},\eta_{3}$ quantify
indirect dose effects on survival mediated through $Y_{T}$, $Y_{R}$ and
$Y_{B}$, respectively (see Figure 1). For priors of the parameters in (4.2),
we assume $\rho\sim{\rm Gamma}(a_{\rho},b_{\rho})$,
$\log(\lambda_{j})\sim\mathcal{N}(0,\nu_{\lambda}^{2})$, and
$\eta_{1},\eta_{2},\eta_{3}\overset{\rm
i.i.d.}{\sim}\mathcal{N}(0,\nu_{\eta}^{2})$, where $a_{\rho}$, $b_{\rho}$,
$\nu_{\lambda}^{2}$, and $\nu_{\eta}^{2}$ are hyperparameters included in
$\tilde{\bm{\theta}}$, with suggested values provided in the Supplementary
Materials.

Based on the final data $\mathcal{D}$, we jointly estimate models (4.3) and
(4.2) using a Gibbs sampler. To obtain posterior estimates of RMST, at each
sampling iteration, we first compute the posterior mean of ${\mu}_{B}(d_{j})$
under the Emax model, denoted by $\hat{\mu}_{B}(d_{j})$, and the posterior
mean of the joint probability of toxicity and response at dose $d_{j}$,
denoted by $\hat{\pi}_{j}(u,v),$ for $u,v=0,1.$ The RMST over a fixed period
$t_{S}$ at dose $d_{j}$ then can be estimated by the plug-in method,

$\hat{\mu}_{S}(d_{j})=\sum_{u=0}^{1}\sum_{v=0}^{1}\hat{\pi}_{j}(u,v)\int_{0}^{t_{S}}\exp\\{-\hat{\lambda}_{j}Y_{S}^{\hat{\rho}}\exp(\hat{\eta}_{1}u+\hat{\eta}_{2}v+\hat{\eta}_{3}\hat{\mu}_{B}(d_{j}))\\}{\rm
d}Y_{S}.$

The OTD then is selected as the dose that satisfies (2a)–(2d) and maximizes
$\hat{\mu}_{S}(d_{j})$.

##  5 Illustration

To illustrate DEMO, we apply it to redesign and conduct hypothetical versions
of the DREAMM-1 and DREAMM-2 studies, investigating six doses,
$(d_{1},\ldots,d_{6})$ =(0.48, 0.96, 1.92, 2.5, 3.4, 4.5) mg/kg. For each
patient, we simulate data using the estimated event rates and survival times
based on the published trial data. This gives estimated $\pi_{R}(d)$ rates
$(0.06,\,0.11,\,0.18,\,0.31,\,0.33,\,0.34)$. Since no DLT was observed in
DREAMM-1, we assume that $\pi_{T}(d)$ rates are the negligible values
$(0.01,\,0.02,\,0.03,\,0.04,\,0.05,\,0.06)$. To generate the biological
outcome $Y_{B}$, which is the logarithmic change in BCMA concentration, we use
equation (3.1) assuming $(\mu_{B}(d_{1}),\ldots,\mu_{B}(d_{6}))$ =(3.88, 5.50,
5.93, 5.97, 5.99, 6.00) and $\sigma_{B}^{2}=1$, which gives $d_{1}$ as
biologically inactive, and $\\{d_{2},\cdots,d_{6}\\}$ as active. The published
results only provide median PFS times for doses $d_{4}$ and $d_{5}$, given by
11.0 and 13.7 months, respectively. We thus assume that the median PFS times
for the remaining doses $d_{1},d_{2},d_{3}$, and $d_{6}$, are 5.7, 6.5, 6.9,
and 10.2 months, respectively. Based on the specified median times, we
simulated the PFS time from the Weibull regression model (4.2) by fixing
$(\lambda_{1},\ldots,\lambda_{6})=(0.11,\,0.10,\,0.10,\,0.07,\,0.06,\,0.08)$,
$\rho=1.1$, $\eta_{1}=3$, and $\eta_{2}=-2$. Each patient was followed up to
36 months, and only administrative censoring was considered. The underlying
RMSTs over two years for the six doses were then 7.7, 8.7, 9.5, 12.8, 13.7,
and 12.2 months, respectively. To implement the DEMO design, we set the limits
$\bar{\pi}_{T}=0.25$, $\underline{\pi}_{R}=0.15$, and $\underline{\mu}_{S}=9$,
and, based on preliminary simulations. specified acceptability cutoffs
$c_{B}=0.5$, $c_{T}=0.6$, $c_{R}=0.7$, and $c_{S}=0.8$. We also adopted the
default specification of the hyperparameters as described in the Supplementary
Materials, by setting $(L,K,\kappa)=(3,4,0.3)$. In this hypothetical setting,
$d_{5}$ was the OTD.

Summary data at each decision-making time for the hypothetical trial are given
in Table 1. In stage 1, $27$ patients were allocated among the six doses using
the BOIN design with a cohort size of three. Based on the $Y_{T}$ data, all
doses satisfied (2a). Analysis of the $Y_{B}$ data showed that doses no lower
than $\hat{\tau}_{B}=2$ were likely to exhibit promising bioactivity, and the
acceptable dose set at the end of stage 1 was
$(d_{2},d_{3},d_{4},d_{5},d_{6})$. In stage 2, three new cohorts of patients
were assigned to each acceptable dose, with three patients in each cohort.
Dose $d_{2}$ was eliminated by the clinical response rate futility criterion
(2c), so the acceptable dose set after stage 2 was
$(d_{3},d_{4},d_{5},d_{6})$.

In stage 3, doses $d_{4}$, $d_{5}$, and $d_{6}$ were selected to treat further
patients, because $d_{3}$ falied to satisfy the selection rule, with
$\Pr(\pi_{R}(d_{3})>0.15\,|\,\mathcal{D})<0.7\Pr(\pi_{R}(d_{6})>0.15\,|\,\mathcal{D})$.
In the final stage, no further dose elimination was triggered, and 12, 12, and
6 additional patients were randomized to $d_{4}$, $d_{5}$, and $d_{6}$ to
achieve a total per-dose sample size of 24 patients. After completion of the
trial, the estimated two-year RMST values were 13.9, 14.9, and 12.1 months for
$d_{4}$, $d_{5}$ and $d_{6}$, respectively. Consequently, based on the
acceptability criteria (2a)–(2d) and the final estimated RMST values, the DEMO
design identified the OTD to be $d_{5}$ (3.4 mg/kg).

##  6 Simulation Study

###  6.1 Simulation Setting

We conducted a simulation study to evaluate the performance of the DEMO
design, considering six doses
$(d_{1},\ldots,d_{6})=(0.05,0.10,0.20,0.45,0.65,0.85)$. For simulated trials,
during stage 1, a maximum of 10 cohorts of size three patients each were
assigned to doses using the BOIN design with target toxicity rate
${\pi}_{T}^{target}=0.30$ and the default parameter settings for the BOIN
design. Stage 1 would terminate if the maximum sample size of 30 patients was
reached or if a maximum of nine patients had been treated at a particular
dose, whichever came first. In stage 2, a maximum of nine patients were
assigned to each acceptable dose, with interim monitoring rules applied after
every three patients. Stage 3 randomized patients until each acceptable dose
reached a maximum of 24 patients. One interim analysis was performed midway
through stage 3. We specified $\bar{\pi}_{T}=0.30$,
$\underline{\pi}_{E}=0.20$, and $\underline{\mu}_{S}=3$. Based on preliminary
simulations, the probability cutoffs for acceptability rules (2a)–(2d) were
calibrated to be $c_{T}=0.60$, $c_{B}=0.50$, $c_{R}=0.70$, and $c_{S}=0.80$.
Prior hyperparameter values of the DEMO design are provided in the
Supplementary Materials, and we set $(L,K,\kappa)=(3,4,0.3)$.

We included two other designs as comparators. The first comparator, referred
to as DFCE, utilized a dose finding plus randomized cohort expansion strategy
by employing BOIN to estimate the MTD during the dose-finding stage, followed
by expansion of the three dose levels MTD, MTD-1, and MTD-2, with a maximum
sample size of 24 at each dose. The DFCE design uses the same BOIN
configuration and trial monitoring procedures as DEMO, and selects the
acceptable dose with highest RMST as the OTD. The two differences between DFCE
and DEMO are that DFCE does not utilize the endpoint $Y_{B}$ (i.e., the
acceptability rule (2a)) to determine an acceptable dose, and the selection of
randomized doses by DFCE depends solely on $Y_{T}$ and the estimated MTD.

The second comparator was the two-stage utility-based U-BOIN design (Zhou et
al., 2019). Similarly to DEMO, the first stage of U-BOIN utilizes BOIN to
explore the dose space, which terminates when a particular dose has
accumulated nine patients. and then identifies acceptable doses based on
binary $Y_{T}$ and $Y_{R}$ data. In the second stage, U-BOIN randomizes
patients equally among acceptable doses until the maximum sample size of 120
is reached, or if the number of patients treated at any dose reaches 30. The
utility values for the joint outcomes
$(Y_{R},Y_{T})=\\{(0,1),(0,0),(1,1),(1,0)\\}$ were set to (0, 5, 95, 100). The
results of U-BOIN were obtained using the software available on
www.trialdesign.org, without changing any other design parameters. The key
difference between U-BOIN and DEMO is that U-BOIN only considers binary
$Y_{T}$ and $Y_{R}$ data for identifying the OBD, while DEMO incorporates
additional $Y_{B}$ and $Y_{S}$ data to determine the OTD.

Ten scenarios were considered, with varying configurations of underlying
assumed true $Y_{B}$ means, toxicity and response rates, and one-year RMST
values (Table 2). The $Y_{B}$ data were generated from (1) with variance
$\sigma_{B}^{2}=1$. The $Y_{S}$ data were generated from (4.2) by fixing
$\rho=1.5$, $\eta_{1}=3$, $\eta_{2}=-2$, and a two-year trial duration. The
parameters $\lambda_{1},\ldots,\lambda_{J}$ were obtained by back-solving to
ensure that the specified RMST values could be achieved. In scenarios 1 and 4,
the three target doses (i.e., MTD, OBD, and OTD) align at the same dose level.
In scenarios 2 and 3, the MTD and OBD are at the same dose level and higher
than the OTD. In scenarios 5, 8, and 9, the OTD coincides with the OBD, but
they are lower than the MTD. In scenarios 6 and 7, all three target doses are
at different levels. In scenario 10, all the six dose levels are either futile
or overly toxic.

###  6.2 Simulation Results

A total of 1000 trials were simulated for each scenario and design. The
results are presented in Table 3. Overall, across the ten scenarios, the DEMO
design consistently gives superior performance in terms of the percentage of
correct OTD selections (PCS) compared with both the DFCE and U-BOIN designs.
DEMO outperforms U-BOIN primarily due to its utilization of $Y_{S}$ as one of
the endpoints. Compared to DFCE, DEMO exhibits superior performance in most
scenarios when the OTD is not one of the doses MTD, MTD-1, and MTD-2, such as
in scenario 9. Even when the OTD equals the MTD, as in scenario 1, DEMO also
outperforms DFCE, because using $Y_{B}$ in later stages improves efficiency.

Table 3 also summarizes the average number of patients required for each
design in each scenario. Because of its termination rule, U-BOIN has the
smallest sample sizes in scenarios 1 to 9. In scenario 10, where no doses are
acceptable, DEMO correctly terminates all doses and leads to a smaller sample
size than U-BOIN. The results also show that DEMO requires fewer patients than
DFCE in most scenarios, mainly because incorporation of early $Y_{B}$ data
helps to reduce the acceptable dose set. Among the scenarios where DEMO
requires more patients, a typical example is scenario 9. This is due to DEMO’s
greater flexibility in selecting stage 3 doses, in contrast with DFCE, which
is limited to only three doses. Consequently, despite the larger sample size
required by DEMO in scenario 9, a substantial improvement of 77.9% in the PCS
is observed compared to DFCE.

###  6.3 Sensitivity Analysis

In the Supplementary Materials, we provide sensitivity analysis results to
demonstrate the robustness of the DEMO design across various prior
specifications. We also report an additional simulation study to evaluate
design OCs for different values of $(L,K,\kappa)$.

##  7 Discussion

In this paper, we have proposed a novel three-stage seamless generalized phase
1-2 clinical trial design for identifying an optimal dose that includes
information from a biological variable, toxicity, response, and survival time.
The DEMO design is structured to follow the actual process of outcome
evaluation and decision making in an early-phase dose-finding trial, where the
data are sparse at the beginning of the trial and become richer as the trial
proceeds. The statistical model and decision-making process underlying DEMO
are elaborated during successive stages of the trial to make use of
accumulating data.

The simulations showed that the DEMO design is superior to traditional
designs, in terms of both correct decision percentages and sample sizes, while
treating a greater proportion of subjects at the optimal dose. Incorporating a
biological variable provided greater efficiency for both screening out
ineffective doses and choosing an OTD. Sensitivity analyses confirmed that the
design is robust to changes in prior specifications. Selection of the design
parameters $(L,K,\kappa)$ may reflect causal relationships between the
bioactivity, toxicity, response, and survival time. While smaller values for
$(L,K,\kappa)$ may be appropriate if there are strong causal relationships
with long-term survival a priori. Since, in practice, such an association is
uncertain, we recommend setting $(L,K,\kappa)=(2,3,0.3)$ for trials with
$J\leq 5$ doses and $(L,K,\kappa)=(3,4,0.3)$ for trials with $J\geq 6$ doses,
respectively. Further generalizations of DEMO may include incorporating
multiple biological endpoints, longitudinally observed biomarker processes,
and optimizing subgroup-specific doses for trials that enroll patients with
different tumor types.

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Figure 1: Causal diagram for dose $d$, biomarker $Y_{B}$, phase 1-2 early
toxicity and response outcomes $Y_{T}$ and $Y_{R}$, and survival time $Y_{S}$.
Figure 2: Schematic of the three stages of the proposed dose
exploration–monitoring–optimization (DEMO) design. Circles, rectangles, and
triangles represent patient cohorts enrolled in stages 1, 2, and 3,
respectively. The absence of rectangles or triangles at certain doses
indicates that these doses do not meet at least one of the four acceptability
criteria (2a)–(2b). RMST stands for the restricted mean survival time.  Table
1: Data summary, including number of patients, sample mean of biological
variable $Y_{B}$, number of toxicities $Y_{T}$, number of responses $Y_{R}$,
24-month restricted mean survival time (RMST), median progression-free
survival time (PFS, in months), and decision-making statistics for the
hypothetical trial described in Section 5.

|  | Dose level |  | Dose level
---|---|---|---|---
|  | 1 | 2 | 3 | 4 | 5 | 6 |  | 1 | 2 | 3 | 4 | 5 | 6
|  | End of Stage 1 |  | End of Stage 2
Number of patients |  | 3 | 3 | 3 | 3 | 3 | 9 |  | 3 | 12 | 12 | 12 | 12 | 18
Sample mean of $Y_{B}$ |  | 4.1 | 6.1 | 5.7 | 5.6 | 5.2 | 5.6 |  | 4.1 | 5.3 | 5.6 | 5.8 | 6.1 | 5.6
Number of $Y_{T}$ |  | 0 | 0 | 0 | 0 | 0 | 0 |  | 0 | 0 | 0 | 1 | 0 | 1
Number of $Y_{R}$ |  | 0 | 0 | 2 | 1 | 1 | 3 |  | 0 | 1 | 2 | 3 | 3 | 6
RMST |  | 2.5 | 1.6 | 19.4 | 8.4 | 15.6 | 14.9 |  | 2.5 | 9.4 | 9.0 | 11.0 | 15.3 | 14.2
Median PFS |  | 3.1 | 1.9 | NA | 6.3 | 18.7 | 13.4 |  | 3.1 | 6.9 | 6.7 | 8.2 | 16.2 | 14.0
$\Pr(\mathcal{M}_{j}|\mathcal{D})$ |  | 0.00 | 0.89 | 0.04 | 0.03 | 0.02 | 0.02 |  | 0.00 | 0.84 | 0.11 | 0.04 | 0.01 | 0.00
$\Pr({\pi}_{T}(d_{j})>0.25|\mathcal{D})$ |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.16 |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$\Pr({\pi}_{E}(d_{j})<0.15|\mathcal{D})$ |  | NA | NA | NA | NA | NA | NA |  | 0.86 | 0.85 | 0.48 | 0.22 | 0.04 | 0.03
acceptable? |  | No | Yes | Yes | Yes | Yes | Yes |  | No | No | Yes | Yes | Yes | Yes
|  | 1st interim of Stage 2 |  | 1st interim of Stage 3
Number of patients |  | 3 | 6 | 6 | 6 | 6 | 12 |  | 3 | 12 | 12 | 18 | 18 | 21
Sample mean of $Y_{B}$ |  | 4.1 | 5.9 | 5.6 | 6.0 | 6.0 | 5.7 |  | 4.1 | 5.3 | 5.6 | 5.8 | 5.9 | 5.7
Number of $Y_{T}$ |  | 0 | 0 | 0 | 0 | 0 | 0 |  | 0 | 0 | 0 | 1 | 0 | 1
Number of $Y_{R}$ |  | 0 | 0 | 2 | 1 | 2 | 4 |  | 0 | 1 | 2 | 4 | 5 | 6
RMST |  | 2.5 | 7.9 | 11.8 | 7.9 | 13.5 | 15.1 |  | 2.5 | 9.4 | 9.0 | 13.0 | 14.5 | 12.7
Median PFS |  | 3.1 | 4.3 | 8.3 | 6.4 | 12.4 | 14.3 |  | 3.1 | 6.9 | 6.7 | 10.1 | 14.9 | 12.8
$\Pr(\mathcal{M}_{j}|\mathcal{D})$ |  | 0.00 | 0.95 | 0.02 | 0.01 | 0.01 | 0.01 |  | 0.00 | 0.83 | 0.12 | 0.04 | 0.01 | 0.00
$\Pr({\pi}_{T}(d_{j})>0.25|\mathcal{D})$ |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$\Pr({\pi}_{E}(d_{j})<0.15|\mathcal{D})$ |  | NA | NA | NA | NA | NA | NA |  | 0.92 | 0.88 | 0.48 | 0.17 | 0.02 | 0.05
acceptable? |  | No | Yes | Yes | Yes | Yes | Yes |  | No | No | Yes | Yes | Yes | Yes
|  | 2nd interim of Stage 2 |  | End of Stage 3
Number of patients |  | 3 | 9 | 9 | 9 | 9 | 15 |  | 3 | 12 | 12 | 24 | 24 | 24
Sample mean of $Y_{B}$ |  | 4.1 | 5.5 | 5.5 | 6.0 | 6.2 | 5.5 |  | 4.1 | 5.8 | 6.0 | 6.2 | 5.8 | 6.2
Number of $Y_{T}$ |  | 0 | 0 | 0 | 0 | 0 | 0 |  | 0 | 0 | 0 | 1 | 0 | 1
Number of $Y_{R}$ |  | 0 | 1 | 2 | 3 | 3 | 5 |  | 0 | 1 | 2 | 6 | 7 | 7
RMST |  | 2.5 | 10.0 | 10.5 | 10.8 | 15.8 | 15.4 |  | 2.5 | 9.4 | 9.0 | 12.9 | 13.9 | 12.5
Median PFS |  | 3.1 | 7.4 | 7.5 | 6.5 | 18.7 | 15.3 |  | 3.1 | 6.9 | 6.7 | 9.9 | 13.6 | 11.9
$\Pr(\mathcal{M}_{j}|\mathcal{D})$ |  | 0.00 | 0.92 | 0.04 | 0.03 | 0.01 | 0.00 |  | 0.00 | 0.81 | 0.14 | 0.05 | 0.00 | 0.00
$\Pr({\pi}_{T}(d_{j})>0.25|\mathcal{D})$ |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |  | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$\Pr({\pi}_{E}(d_{j})<0.15|\mathcal{D})$ |  | NA | NA | NA | NA | NA | NA |  | 0.93 | 0.88 | 0.36 | 0.08 | 0.05 | 0.04
acceptable? |  | No | Yes | Yes | Yes | Yes | Yes |  | No | No | Yes | Yes | Yes | Yes

Note: “NA” means either the data were not available or the median survival
time was not reached. Reported RMST values were obtained based on the
estimated Kaplan-Meier survival curve. At the end of stage 3, the RMSTs
estimated under the proposed Weibull regression model were (3.7, 9.8, 9.3,
13.9, 14.9, 12.1) months.

Table 2: Assumed true means of the biological outcome $\mu_{B}(d)$, toxicity probabilities $\pi_{T}(d)$, response probabilities $\pi_{R}(d)$, and one-year restricted mean survival times $\mu_{S}(d)$ for each dose under each simulation scenario. The optimal therapeutic dose (OTD) is given in boldface.  Dose level | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---|---|---|---|---|---
| scenario 1 | scenario 6
$\mu_{B}(d)$ | 2.00 | 2.01 | 2.08 | 2.76 | 3.75 | 4.73 | 2.24 | 4.00 | 5.77 | 5.99 | 6.00 | 6.00
$\pi_{T}(d)$ | 0.01 | 0.02 | 0.03 | 0.06 | 0.13 | 0.26 | 0.01 | 0.02 | 0.05 | 0.10 | 0.27 | 0.55
$\pi_{R}(d)$ | 0.04 | 0.05 | 0.08 | 0.20 | 0.35 | 0.47 | 0.07 | 0.14 | 0.32 | 0.41 | 0.42 | 0.44
$\mu_{S}(d)$ | 1.15 | 1.42 | 1.51 | 3.14 | 4.04 | 5.35 | 1.95 | 4.86 | 5.70 | 3.66 | 3.12 | 2.20
| scenario 2 | scenario 7
$\mu_{B}(d)$ | 2.01 | 2.09 | 2.24 | 4.17 | 5.29 | 5.95 | 5.04 | 5.83 | 5.98 | 6.00 | 6.00 | 6.00
$\pi_{T}(d)$ | 0.01 | 0.03 | 0.04 | 0.07 | 0.14 | 0.28 | 0.01 | 0.06 | 0.18 | 0.29 | 0.51 | 0.54
$\pi_{R}(d)$ | 0.04 | 0.06 | 0.09 | 0.23 | 0.37 | 0.44 | 0.22 | 0.35 | 0.41 | 0.42 | 0.44 | 0.45
$\mu_{S}(d)$ | 1.15 | 1.87 | 2.37 | 2.94 | 4.10 | 2.91 | 4.90 | 5.97 | 4.13 | 3.05 | 2.35 | 2.27
| scenario 3 | scenario 8
$\mu_{B}(d)$ | 2.02 | 2.18 | 3.14 | 5.86 | 6.55 | 6.79 | 3.50 | 5.71 | 5.98 | 5.99 | 6.00 | 6.00
$\pi_{T}(d)$ | 0.03 | 0.04 | 0.07 | 0.09 | 0.15 | 0.29 | 0.01 | 0.10 | 0.11 | 0.25 | 0.45 | 0.56
$\pi_{R}(d)$ | 0.04 | 0.08 | 0.17 | 0.38 | 0.46 | 0.46 | 0.15 | 0.42 | 0.43 | 0.46 | 0.48 | 0.50
$\mu_{S}(d)$ | 1.13 | 1.94 | 2.67 | 4.35 | 3.18 | 2.74 | 3.86 | 6.28 | 6.33 | 4.32 | 3.50 | 3.13
| scenario 4 | scenario 9
$\mu_{B}(d)$ | 2.00 | 2.01 | 2.13 | 3.98 | 5.70 | 6.47 | 3.50 | 5.85 | 5.99 | 6.00 | 6.00 | 6.00
$\pi_{T}(d)$ | 0.01 | 0.02 | 0.04 | 0.08 | 0.20 | 0.56 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06
$\pi_{R}(d)$ | 0.04 | 0.05 | 0.10 | 0.30 | 0.43 | 0.44 | 0.08 | 0.24 | 0.42 | 0.43 | 0.43 | 0.43
$\mu_{S}(d)$ | 1.40 | 1.85 | 2.41 | 3.98 | 5.71 | 5.65 | 2.38 | 3.88 | 6.34 | 4.81 | 4.77 | 4.73
| scenario 5 | scenario 10
$\mu_{B}(d)$ | 2.24 | 2.80 | 4.00 | 5.34 | 5.65 | 5.79 | 2.90 | 4.25 | 5.60 | 6.29 | 6.40 | 6.44
$\pi_{T}(d)$ | 0.01 | 0.02 | 0.04 | 0.08 | 0.26 | 0.52 | 0.32 | 0.36 | 0.41 | 0.42 | 0.48 | 0.52
$\pi_{R}(d)$ | 0.04 | 0.06 | 0.14 | 0.40 | 0.38 | 0.33 | 0.02 | 0.03 | 0.10 | 0.12 | 0.13 | 0.17
$\mu_{S}(d)$ | 1.40 | 1.87 | 3.26 | 5.97 | 3.67 | 2.49 | 1.99 | 2.07 | 2.36 | 2.59 | 2.63 | 2.89
Table 3: Selection percentages (Sel%) and number of patients (Pts) at each dose for each design under scenarios 1 to 10 given in Table 2. The optimal therapeutical dose (OTD) under each scenario is given in boldface. “None” is the percentage of trials without selecting any dose, and “N” is the average total sample size. DEMO is the proposed design for dose exploration, monitoring, optimization; DFCE is the design that integrates dose finding (DF) with cohort expansion (CE) strategies; and U-BOIN is the utility-based Bayesian optimal interval design. | DEMO | DFCE | U-BOIN |  | DEMO | DFCE | U-BOIN
---|---|---|---|---|---|---|---
Dose | Sel% | Pts | Sel% | Pts | Sel% | Pts |  | Sel% | Pts | Sel% | Pts | Sel% | Pts
| Scenario 1 |  | Scenario 6
1 | 0.0 | 3.8 | 0.0 | 3.4 | 0.4 | 3.3 |  | 0.0 | 3.2 | 0.1 | 3.6 | 0.5 | 3.6
2 | 0.0 | 4.0 | 0.0 | 3.9 | 0.6 | 3.5 |  | 0.7 | 5.5 | 5.1 | 7.2 | 1.3 | 4.4
3 | 0.0 | 4.2 | 0.0 | 6.4 | 0.5 | 4.0 |  | 94.0 | 23.4 | 72.2 | 19.5 | 15.0 | 10.3
4 | 1.3 | 6.9 | 2.9 | 21.2 | 3.7 | 6.5 |  | 2.3 | 23.8 | 15.3 | 23.0 | 43.7 | 17.7
5 | 16.5 | 19.5 | 21.6 | 22.3 | 27.9 | 13.8 |  | 2.3 | 19.4 | 5.9 | 19.3 | 35.1 | 16.8
6 | 78.8 | 20.7 | 68.0 | 17.6 | 66.9 | 23.0 |  | 0.0 | 6.2 | 0.3 | 5.0 | 4.4 | 7.3
(None, N) | 3.4 | 59.0 | 7.5 | 74.7 | 0.0 | 54.1 |  | 0.7 | 81.6 | 1.1 | 77.7 | 0.0 | 60.1
| Scenario 2 |  | Scenario 7
1 | 0.0 | 3.2 | 0.0 | 3.6 | 0.3 | 3.4 |  | 1.7 | 5.7 | 9.8 | 10.9 | 3.8 | 5.5
2 | 0.0 | 3.5 | 0.0 | 4.4 | 0.4 | 3.9 |  | 77.9 | 20.6 | 71.7 | 20.4 | 20.7 | 12.2
3 | 0.0 | 3.6 | 0.0 | 7.1 | 0.8 | 4.2 |  | 16.6 | 23.1 | 16.2 | 22.5 | 36.0 | 16.6
4 | 3.1 | 15.4 | 6.7 | 21.4 | 6.7 | 8.0 |  | 2.7 | 17.5 | 1.5 | 15.7 | 31.2 | 15.2
5 | 82.6 | 23.1 | 78.8 | 21.6 | 38.9 | 17.1 |  | 0.1 | 6.0 | 0.8 | 5.3 | 5.1 | 7.6
6 | 9.4 | 19.8 | 5.6 | 16.2 | 52.9 | 20.1 |  | 0.1 | 2.3 | 0.0 | 0.4 | 3.2 | 6.0
(None, N) | 4.9 | 68.5 | 8.9 | 74.2 | 0.0 | 56.7 |  | 0.9 | 75.2 | 0.0 | 75.2 | 0.0 | 63.1
| Scenario 3 |  | Scenario 8
1 | 0.0 | 3.4 | 0.0 | 4.1 | 0.0 | 3.5 |  | 0.4 | 3.5 | 0.5 | 8.5 | 0.8 | 4.2
2 | 0.0 | 3.6 | 0.1 | 5.1 | 0.3 | 3.9 |  | 48.9 | 22.4 | 41.9 | 18.4 | 23.2 | 12.8
3 | 0.2 | 3.9 | 0.9 | 9.1 | 0.7 | 4.8 |  | 49.9 | 23.7 | 53.9 | 20.9 | 28.0 | 13.9
4 | 90.0 | 23.5 | 87.0 | 22.1 | 19.9 | 11.5 |  | 0.6 | 20.2 | 2.3 | 18.2 | 34.3 | 16.1
5 | 3.9 | 23.1 | 2.1 | 20.2 | 45.7 | 17.9 |  | 0.2 | 8.4 | 1.4 | 7.9 | 11.3 | 9.3
6 | 4.7 | 19.3 | 2.4 | 14.8 | 33.3 | 15.4 |  | 0.0 | 2.9 | 0.0 | 1.0 | 2.4 | 4.8
(None, N) | 1.2 | 76.8 | 7.5 | 75.5 | 0.1 | 57.0 |  | 0.0 | 81.1 | 0.0 | 74.8 | 0.0 | 61.1
| Scenario 4 |  | Scenario 9
1 | 0.0 | 3.2 | 0.0 | 3.5 | 0.3 | 3.5 |  | 0.0 | 3.3 | 0.1 | 3.5 | 2.2 | 5.6
2 | 0.0 | 3.4 | 0.1 | 4.9 | 0.3 | 3.7 |  | 0.2 | 13.5 | 0.4 | 3.9 | 30.6 | 14.6
3 | 0.0 | 3.7 | 0.1 | 13.9 | 1.0 | 4.6 |  | 83.2 | 23.2 | 5.3 | 4.3 | 47.3 | 19.4
4 | 12.4 | 14.3 | 9.2 | 22.3 | 23.7 | 12.5 |  | 2.4 | 24.0 | 29.1 | 23.4 | 10.7 | 10.1
5 | 80.1 | 21.7 | 80.2 | 20.6 | 71.4 | 24.3 |  | 4.9 | 23.9 | 27.4 | 22.9 | 5.4 | 8.3
6 | 3.6 | 7.7 | 2.8 | 7.0 | 3.2 | 7.9 |  | 9.3 | 23.0 | 37.0 | 22.4 | 3.5 | 7.7
(None, N) | 3.9 | 54.0 | 7.6 | 72.2 | 0.1 | 56.5 |  | 0.0 | 110.9 | 0.7 | 80.4 | 0.3 | 65.7
| Scenario 5 |  | Scenario 10
1 | 0.0 | 3.4 | 0.0 | 3.6 | 0.3 | 3.4 |  | 0.0 | 9.2 | 0.0 | 13.0 | 0.7 | 10.6
2 | 0.0 | 3.8 | 0.0 | 5.6 | 0.7 | 3.8 |  | 0.0 | 6.2 | 0.1 | 5.5 | 1.9 | 8.7
3 | 0.3 | 8.7 | 0.7 | 15.9 | 1.5 | 5.1 |  | 0.0 | 3.5 | 0.1 | 2.1 | 10.5 | 10.9
4 | 94.3 | 23.3 | 92.1 | 22.8 | 53.6 | 20.3 |  | 0.2 | 1.6 | 0.3 | 0.6 | 13.4 | 11.7
5 | 3.4 | 20.3 | 1.7 | 19.9 | 40.6 | 17.8 |  | 0.0 | 0.9 | 0.1 | 0.1 | 7.7 | 10.0
6 | 0.1 | 7.5 | 0.2 | 5.9 | 3.3 | 7.0 |  | 0.0 | 0.7 | 0.0 | 0.0 | 6.1 | 9.1
(None, N) | 1.9 | 67.0 | 5.3 | 73.6 | 0.0 | 57.4 |  | 99.8 | 22.0 | 99.4 | 21.3 | 59.7 | 61.0

Supplementary Materials for “DEMO: Dose Exploration, Monitoring, and
Optimization Using a Biological Mediator for Clinical Outcomes”

Cheng-Han Yang1, Peter F. Thall2, Ruitao Lin2∗

1Department of Biostatistics, The University of Texas Health Science Center at
Houston,

Houston, TX 77030, USA

2Department of Biostatistics, The University of Texas MD Anderson Cancer
Center,

Houston, Texas 77030, U.S.A.

∗Corresponding author<EMAIL_ADDRESS>
##  S1 Posterior Model Probability for $Y_{B}$

For patient $i$, $i=1,\ldots,n$, the probability model for an individual
outcome $Y_{B}(i)$ at dose $d(i)\in\\{d_{1},\ldots,d_{J}\\}$ under
$\mathcal{M}_{j}$, $j=1,\ldots,J$ can be described as follows.

$\displaystyle Y_{B}(i)\mid d(i)$ $\displaystyle\sim$
$\displaystyle\mathcal{N}(\mu_{B}(d(i)),\sigma_{B}^{2}),$
$\displaystyle\mu_{B}(d(i))$ $\displaystyle=$
$\displaystyle\mu_{-}I(d(i)<d_{j})+\mu_{+}I(d(i)\geq d_{j}),$
$\displaystyle\mu_{-}$ $\displaystyle\sim$
$\displaystyle\mathcal{N}(m_{\mu_{-}},\nu_{\mu}^{2})$ $\displaystyle\mu_{+}$
$\displaystyle\sim$ $\displaystyle\mathcal{N}(m_{\mu_{+}},\nu_{\mu}^{2})$
$\displaystyle\zeta_{B}=\sigma_{B}^{-2}$ $\displaystyle\sim$
$\displaystyle{\rm Gamma}(a_{\sigma},b_{\sigma}).$

Without loss of generality, we take $\nu_{\mu}^{-2}$ as $n_{0}\zeta_{B}$. By
defining $\tau_{B}=d_{j}$, we partition the observations of the biological
endpoint into inactive (-) and active (+) groups known as $L_{j}$ and $R_{j}$,
respectively. Specifically, $L_{j}$ is the collection of all $Y_{B}(i)$ values
for which $z(i)$ is less than $d_{j}$, while $R_{j}$ comprises all $y_{B}(i)$
values for which $d(i)$ is greater than or equal to $d_{j}$ under
$\mathcal{M}_{j}$.

We use the symbol $|\cdot|$ to denote the cardinality of a set. The sample
means of the biological endpoints in $L_{j}$ and $R_{j}$ are represented by
$\bar{y}_{L_{j}}$ and $\bar{y}_{R_{j}}$, respectively. Additionally, we denote
the sample variances of the biological endpoints in $L_{j}$ and $R_{j}$ as
$s_{L_{j}}^{2}$ and $s_{R_{j}}^{2}$, respectively. Specifically,
$s_{L_{j}}^{2}$ is calculated as
$\sum_{i:d(i)<d_{j}}(y_{B}(i)-\bar{y}_{L_{j}})^{2}/|L_{j}|$, and
$s_{R_{j}}^{2}$ is calculated as $\sum_{i:d(i)\geq
d_{j}}(y_{B}(i)-\bar{y}_{R_{j}})^{2}/|R_{j}|$. Then, the posterior model
probability $\text{Pr}(\mathcal{M}_{j}\mid\mathcal{D})$ can be calculated as

$\text{Pr}(\mathcal{M}_{j}\mid\mathcal{D})=\Pr(\mathcal{M}_{j})\sqrt{\frac{1}{|L_{j}|+n_{0}}}\sqrt{\frac{1}{|R_{j}|+n_{0}}}\frac{\Gamma(\tilde{a}_{\sigma})}{{\tilde{b}_{\sigma}}^{\tilde{a}_{\sigma}}},$
(S1.1)

where

$\displaystyle\tilde{a}_{\sigma}$ $\displaystyle=a_{\sigma}+\frac{N_{1}}{2}$
$\displaystyle\tilde{b}_{\sigma}$
$\displaystyle=b_{\sigma}+\frac{1}{2}|L_{j}|s_{L_{j}}^{2}+\frac{1}{2}|R_{j}|s_{R_{j}}^{2}+\frac{1}{2}\frac{|L_{j}|n_{0}}{|L_{j}|+n_{0}}(\bar{y}_{L_{j}}-m_{\mu_{-}})^{2}+\frac{1}{2}\frac{|R_{j}|n_{0}}{|R_{j}|+n_{0}}(\bar{y}_{R_{j}}-m_{\mu_{+}})^{2}.$

The detailed derivation steps of the closed form of
$\text{Pr}(\mathcal{M}_{j}\mid\mathcal{D})$ is given as follows. Let
$\phi(\mu,\xi\mid Y)$ denote the likelihood of observing $Y$ from a normal
distribution mean $\mu$ and precision $\xi$. Under model $\mathcal{M}_{j}$,
the joint posterior distribution can be calculated by

$\displaystyle p(\mu_{-},\mu_{+},\zeta_{B}\mid\mathcal{D},\mathcal{M}_{j})$
(S1.2)
$\displaystyle=\prod_{\\{i:d(i)<d_{j}\\}}\phi(\mu_{-},\zeta_{B}|Y_{B}(i))\pi(\mu_{-})\prod_{\\{i:d(i)\geq
d_{j}\\}}\phi(\mu_{+},\zeta_{B}|Y_{B}(i))\pi(\mu_{+})\pi(\zeta_{B})$
$\displaystyle=$
$\displaystyle\frac{\zeta_{B}^{\frac{|L_{j}|}{2}}}{(2\pi)^{\frac{|L_{j}|}{2}}}\exp\left\\{-\frac{1}{2}|L_{j}|s_{L_{j}}^{2}\zeta_{B}-\frac{1}{2}|L_{j}|\zeta_{B}(\bar{y}_{L_{j}}-\mu_{-})^{2}\right\\}\frac{(n_{0}\zeta_{B})^{\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}}\exp\left\\{-\frac{1}{2}n_{0}\zeta_{B}(\mu_{-}-m_{\mu_{-}})^{2}\right\\}\times$
$\displaystyle\frac{\zeta_{B}^{\frac{|R_{j}|}{2}}}{(2\pi)^{\frac{|R_{j}|}{2}}}\exp\left\\{-\frac{1}{2}|R_{j}|s_{R_{j}}^{2}\zeta_{B}-\frac{1}{2}|R_{j}|\zeta_{B}(\bar{y}_{R_{j}}-\mu_{+})^{2}\right\\}\frac{(n_{0}\zeta_{B})^{\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}}\exp\left\\{-\frac{1}{2}n_{0}\zeta_{B}(\mu_{+}-m_{\mu_{+}})^{2}\right\\}\times$
$\displaystyle\frac{b_{\sigma}^{a_{\sigma}}}{\Gamma(a_{\sigma})}\zeta_{B}^{a_{\sigma}-1}\exp(-b_{\sigma}\zeta_{B}),$

By integrating the first term in the joint posterior distribution (S1.2) with
respect to $\mu_{-}$, we have

$\displaystyle\int\exp\left\\{-\frac{1}{2}|L_{j}|\zeta_{B}(\bar{y}_{L_{j}}-\mu_{-})^{2}-\frac{1}{2}n_{0}\zeta_{B}(\mu_{-}-m_{\mu_{-}})^{2}\right\\}d\mu_{-}$
$\displaystyle=\int\exp\left\\{-\frac{\zeta_{B}}{2}(|L_{j}|+n_{0})\left(\mu_{-}-\frac{|L_{j}|\bar{y}_{L_{j}}+n_{0}m_{\mu_{-}}}{|L_{j}|+n_{0}}\right)^{2}\right\\}\times$
$\displaystyle\hskip
28.45274pt\exp\left\\{-\frac{\zeta_{B}}{2}(|L_{j}|\bar{y}_{L_{j}}^{2}+n_{0}m_{\mu_{-}}^{2})+\frac{\zeta_{B}}{2}(|L_{j}|+n_{0})\left(\frac{|L_{j}|\bar{y}_{L_{j}}}{|L_{j}|+n_{0}}\right)^{2}\right\\}d\mu_{-}$
$\displaystyle=\int\exp\left\\{-\frac{\zeta_{B}}{2}(|L_{j}|+n_{0})\left(\mu_{-}-\frac{|L_{j}|\bar{y}_{L_{j}}+n_{0}m_{\mu_{-}}}{|L_{j}|+n_{0}}\right)^{2}\right\\}d\mu_{-}\times$
$\displaystyle\hskip
28.45274pt\exp\left\\{-\frac{\zeta_{B}}{2}\frac{|L_{j}|n_{0}}{|L_{j}|+n_{0}}(\bar{y}_{L_{j}}-m_{\mu_{-}})^{2}\right\\}$
$\displaystyle=\left(2\pi\cdot\frac{1}{(|L_{j}|+n_{0})\zeta_{B}}\right)^{\frac{1}{2}}\exp\left\\{-\frac{\zeta_{B}}{2}\frac{|L_{j}|n_{0}}{|L_{j}|+n_{0}}(\bar{y}_{L_{j}}-m_{\mu_{-}})^{2}\right\\}.$

Similarly, by integrating the second term in joint posterior distribution
(S1.2) with respect to $\mu_{+}$, we have

$\displaystyle\int\exp\left\\{-\frac{1}{2}|R_{j}|\zeta_{B}(\bar{y}_{R_{j}}-\mu_{+})^{2}-\frac{1}{2}n_{0}\zeta_{B}(\mu_{+}-m_{\mu_{+}})^{2}\right\\}d\mu_{+}$
$\displaystyle=\left(2\pi\cdot\frac{1}{(|R_{j}|+n_{0})\zeta_{B}}\right)^{\frac{1}{2}}\exp\left\\{-\frac{\zeta_{B}}{2}\frac{|R_{j}|n_{0}}{|R_{j}|+n_{0}}(\bar{y}_{R_{j}}-m_{\mu_{+}})^{2}\right\\}.$

As a result, the integral of the joint posterior distribution (S1.2) with
respect to $\mu_{-}$ and $\mu_{+}$ can be rewritten as

$\displaystyle p(\zeta_{B}\mid\mathcal{D},\mathcal{M}_{j})$ (S1.3)
$\displaystyle=\frac{1}{(2\pi)^{\frac{N_{1}+2}{2}}}\frac{b_{\sigma}^{a_{\sigma}}}{\Gamma(a_{\sigma})}\zeta_{B}^{\frac{N_{1}+2}{2}+a_{\sigma}-1}\exp\left(-\frac{1}{2}|L_{j}|s_{L_{j}}^{2}\zeta_{B}-\frac{1}{2}|R_{j}|s_{R_{j}}^{2}\zeta_{B}-b_{\sigma}\zeta_{B}\right)\times$
$\displaystyle\hskip
2.84544pt\exp\left\\{-\frac{\zeta_{B}}{2}\frac{|L_{j}|n_{0}}{|L_{j}|+n_{0}}(\bar{y}_{L_{j}}-m_{\mu_{-}})^{2}-\frac{\zeta_{B}}{2}\frac{|R_{j}|n_{0}}{|R_{j}|+n_{0}}(\bar{y}_{R_{j}}-m_{\mu_{+}})^{2}\right\\}\times$
$\displaystyle\left(2\pi\cdot\frac{1}{(|L_{j}|+n_{0})\zeta_{B}}\right)^{\frac{1}{2}}\left(2\pi\cdot\frac{1}{(|R_{j}|+n_{0})\zeta_{B}}\right)^{\frac{1}{2}}$
$\displaystyle=\frac{1}{(2\pi)^{\frac{N_{1}}{2}}}\frac{b_{\sigma}^{a_{\sigma}}}{\Gamma(a_{\sigma)}}\sqrt{\frac{n_{0}}{|L_{j}|+n_{0}}}\sqrt{\frac{n_{0}}{|R_{j}|+n_{0}}}\zeta_{B}^{\tilde{a}_{\sigma}-1}\exp\left\\{-\tilde{b}_{\sigma}\zeta_{B}\right\\}.$

The probability $\text{Pr}(\mathcal{M}_{j}\mid\mathcal{D})$ can be obtained by
Bayes theorem; that is,

$\displaystyle\text{Pr}(\mathcal{M}_{j}\mid\mathcal{D})$
$\displaystyle\propto\text{Pr}(\mathcal{M}_{j})\int
p(\zeta_{B}\mid\mathcal{D},\mathcal{M}_{j})d\zeta_{B}$
$\displaystyle=\text{Pr}(\mathcal{M}_{j})\frac{1}{(2\pi)^{\frac{N_{1}}{2}}}\frac{b_{\sigma}^{a_{\sigma}}}{\Gamma(a_{\sigma)}}\sqrt{\frac{n_{0}}{|L_{j}|+n_{0}}}\sqrt{\frac{n_{0}}{|R_{j}|+n_{0}}}\int\zeta_{B}^{\tilde{a}_{\sigma}-1}\exp\left\\{-\tilde{b}_{\sigma}\zeta_{B}\right\\}d\zeta_{B}$
$\displaystyle=\text{Pr}(\mathcal{M}_{j})\frac{1}{(2\pi)^{\frac{N_{1}}{2}}}\frac{b_{\sigma}^{a_{\sigma}}}{\Gamma(a_{\sigma)}}\sqrt{\frac{n_{0}}{|L_{j}|+n_{0}}}\sqrt{\frac{n_{0}}{|R_{j}|+n_{0}}}\frac{\Gamma(\tilde{a}_{\sigma})}{{\tilde{b}_{\sigma}}^{\tilde{a}_{\sigma}}}$
$\displaystyle\propto\text{Pr}(\mathcal{M}_{j})\sqrt{\frac{1}{|L_{j}|+n_{0}}}\sqrt{\frac{1}{|R_{j}|+n_{0}}}\frac{\Gamma(\tilde{a}_{\sigma})}{{\tilde{b}_{\sigma}}^{\tilde{a}_{\sigma}}}.$

##  S2 Prior Specification

Step function for $Y_{B}$

For the first model of $Y_{B}$, the hyperparameters are set as
$m_{\mu_{-}}=0,m_{\mu_{+}}=0.5,a_{\sigma}=0.01,b_{\sigma}=0.01$, and
$n_{0}=0.1$. Note that the hyperparameter $n_{0}$ is explicitly defined in
Section S1 and can be treated as the prior sample size based on our prior
knowledge.

Logistic regression model for $Y_{T}$

To calibrate the prior parameters for the logistic regression model for
$Y_{T}$ in stage 1, we ensure that the toxicity rate $\pi_{T}(d)$ lies between
0.001 and 0.999 with a prior probability greater than 0.90. As a result, the
hyperparameters are set as
$m_{\alpha_{0}}=-2,\nu_{\alpha_{0}}^{2}=10,m_{\alpha_{1}}=-0.693$ (i.e., the
prior mean of the slope is $\exp(-0.693)=0.5$), and $\nu_{\alpha_{1}}^{2}=5$.

Emax model for $Y_{B}$ We specify a non-informative prior for the the Emax
model for $Y_{B}$, where we take
$m_{\gamma}=0,\nu_{\gamma}^{2}=10,a_{\gamma_{1}}=1,b_{\gamma_{1}}=0.25,a_{\gamma_{2}}=0.1,b_{\gamma_{2}}=0.25,a_{\gamma_{3}}=0.75,b_{\gamma_{3}}=0.25,a_{\sigma}=0.1$,
and $b_{\sigma}=0.1$. Under this prior, $\gamma_{2}$ ranges from 0 to 50,
while $\gamma_{3}$ ranges from 0 to 10.

Logistic regression models for $Y_{T}\mid Y_{B}$ and $Y_{E}\mid Y_{B}$

Following the approach used in specifying hyperparameters for the marginal
logistic regression model of $Y_{T}$, we select hyperparameter values to
ensure that the prior probabilities of both $\pi_{T}(d)$ and $\pi_{R}(d)$
falling within the 0.001 to 0.999 range exceed 0.90. As a result, the
hyperparameters are specified as follows: $m_{\alpha_{1}}=-0.693$,
$\nu_{\alpha_{1}}^{2}=5$, $m_{\alpha_{2}}=-2.302$, $\nu_{\alpha_{2}}^{2}=5$,
$m_{\beta_{1}}=0$, $\nu_{\beta_{1}}^{2}=5$, $m_{\beta_{2}}=0$,
$\nu_{\beta_{2}}^{2}=5$, $m_{\beta_{3}}=0$, $\nu_{\beta_{3}}^{2}=5$,
$\mu_{\alpha_{0}}=-2$, $\mu_{\beta_{0}}=0$, $\nu_{\alpha_{0}}^{2}=10$,
$\nu_{\beta_{0}}^{2}=5$, and $\rho_{0}=0.2$.

Weibull regression model for $Y_{S}\mid Y_{T},Y_{B},Y_{R}$

We specify non-informative priors for the parameters associated in the Weibull
regression model (4). In particular, we take $a_{\rho}=0.1$ and
$b_{\rho}=0.1$, $\nu_{\lambda}^{2}=100$, and $\nu_{\eta}^{2}=100$.

## Sensitivity Analysis

We conducted a sensitivity analysis to evaluate the performance of DEMO under
various hyperparameter configurations. Keeping other hyperparameters unchanged
as in our main simulation study, we explored five additional settings, each
involved changing the values of the hyperparameters for a single model at a
time.

  * •

Hyperparameter setting 1: In the Emax model for $Y_{B}$, we took
$m_{\gamma}=0,\nu_{\gamma}^{2}=10,a_{\gamma_{1}}=0.1875,b_{\gamma_{1}}=0.125,a_{\gamma_{2}}=0.075,b_{\gamma_{2}}=0.125,a_{\gamma_{3}}=0.3125,b_{\gamma_{3}}=0.125,a_{\sigma}=0.1$,
and $b_{\sigma}=0.1$.

  * •

Hyperparameter setting 2: In the stage 1 logistic regression model for
$Y_{T}$, we enlarged the prior variance and took
$m_{\alpha_{0}}=-2,\nu_{\alpha_{0}}^{2}=50,m_{\alpha_{1}}=-0.693$, and
$\nu_{\alpha_{1}}^{2}=50$.

  * •

Hyperparameter setting 3: In the logistic regression model for $Y_{T}\mid
Y_{B}$, we enlarged the prior variance and set $m_{\alpha_{2}}=-2.302$ and
$\nu_{\alpha_{2}}^{2}=50$.

  * •

Hyperparameter setting 4: In the logistic regression model for $Y_{R}\mid
Y_{B}$, we enlarged the prior variance and set $\mu_{\beta_{0}}=0$,
$\nu_{\beta_{0}}^{2}=50$, $m_{\beta_{l}}=0$ and $\nu_{\beta_{l}}^{2}=50$ for
$l=1,2,3$.

  * •

Hyperparameter setting 5: In the Weibull regression model for $Y_{S}\mid
Y_{T},Y_{B},Y_{R}$, we took $a_{\rho}=0.01$, $b_{\rho}=0.01$,
$\nu_{\lambda}^{2}=1000$, and $\nu_{\eta}^{2}=1000$.

The results, presented in Table S1 and S2, demonstrate that the operating
characteristics of DEMO are generally robust against various settings of the
hyperparameters. This suggests that provided the prior distributions are non-
informative, characterized by a reasonably large variance, the DEMO design is
capable of achieving robust performance.

##  S3 Impact of $(L,K,\kappa)$

In this section, we examine the performance of DEMO by investigating multiple
choices of $(L,K,\kappa)$. The purpose of $L$ is used to decide the minimum
number of acceptable doses that should be studied in stage 3 (given that there
are more than $L$ acceptable doses at the end of stage 2). However, in some
cases, there are multiple better performing doses when efficacy plateaus at a
lower dose. In such situations, $(K,\kappa)$ takes effect. Specifically, the
parameter $\kappa$ determines the strength of the evidence required, and at
most $K$ dose levels are chosen into stage 3. When $\kappa=0$, it means that
at most $L$ doses will be selected, with no further consideration given to the
remaining $K-L$ doses.

As shown in Table S3 and S4, we considered different combinations of $L$, $K$,
and $\kappa$. Generally speaking, increasing $L$, $K$, and $\kappa$ results in
a higher probability of correctly selecting the OTD, but this comes at the
cost of enrolling a larger number of patients. However, the increase in the
average total sample size due to a larger $(L,K,\kappa)$ is not substantial.
When the patterns of $Y_{R}$ and $Y_{S}$ are more closely aligned, meaning the
OBD is equal to or nearly equal to the OTD, employing smaller values of $L$,
$K$, and $\kappa$ can nearly achieve optimal performance while requiring a
smaller sample size. This is evident in scenarios 1–5 and 7–8. On the other
hand, when $Y_{R}$ lacks strong predictive power for $Y_{S}$ – that is, when
the OTD and OBD significantly differ, as demonstrated in scenario 9 – opting
for smaller values of $L$, $K$, and $\kappa$ may result in inferior
performance. From our extensive simulation studies, we found that setting
$(L,K,\kappa)=(2,3,0.3)$ for $J\leq 5$ doses and using
$(L,K,\kappa)=(3,4,0.3)$ for $J\geq 6$ doses typically achieves a good balance
between a robustly high correct selection percentage and a reasonable sample
size.

Table S1: Selection percentages (Sel%) and number of patients (Pts), under scenarios 1 to 5, for the DEMO design using different hyperparameter settings. The optimal therapeutical dose under each scenario is given in boldface. “None” is the percentage of trials without selecting any dose, and “N” is the average total sample size. Hyperparameter Setting | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Dose | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts
Scenario 1
1 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8
2 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0
3 | 0.0 | 4.2 | 0.0 | 4.3 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2
4 | 1.3 | 7.0 | 1.3 | 7.4 | 1.7 | 6.9 | 0.5 | 6.5 | 1.3 | 6.9
5 | 15.9 | 19.4 | 20.0 | 19.5 | 19.7 | 19.5 | 16.0 | 19.1 | 16.5 | 19.5
6 | 79.3 | 20.7 | 73.7 | 19.1 | 73.8 | 20.5 | 79.1 | 20.7 | 78.9 | 20.7
(None, N) | 3.5 | 59.1 | 5.0 | 58.2 | 4.8 | 58.8 | 4.4 | 58.3 | 3.3 | 59.0
Scenario 2
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 |  1 3.2
2 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5
3 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.5 | 0.0 | 3.6
4 | 4.2 | 15.6 | 4.4 | 15.6 | 3.0 | 15.4 | 2.2 | 14.2 | 3.0 | 15.4
5 | 81.1 | 23.1 | 80.6 | 22.9 | 84.0 | 23.1 | 77.4 | 22.8 | 82.9 | 23.1
6 | 9.6 | 19.8 | 9.2 | 18.4 | 8.5 | 19.6 | 15.6 | 19.8 | 9.3 | 19.8
(None, N) | 5.1 | 68.6 | 5.8 | 67.2 | 4.5 | 68.4 | 4.8 | 67.0 | 4.8 | 68.5
Scenario 3
1 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 |  1 3.4
2 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6
3 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9 | 0.1 | 3.9 | 0.2 | 3.9
4 | 90.4 | 23.4 | 89.4 | 23.3 | 89.5 | 23.4 | 83.7 | 23.1 | 90.0 | 23.5
5 | 3.5 | 23.1 | 4.2 | 22.8 | 4.1 | 23.1 | 6.0 | 23.1 | 3.7 | 23.1
6 | 4.7 | 19.3 | 4.7 | 18.5 | 5.1 | 19.1 | 9.0 | 19.3 | 4.8 | 19.3
(None, N) | 1.2 | 76.8 | 1.5 | 75.6 | 1.1 | 76.6 | 1.2 | 76.4 | 1.3 | 76.8
Scenario 4
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2
2 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4
3 | 0.0 | 3.7 | 0.0 | 3.7 | 0.0 | 3.7 | 0.0 | 3.6 | 0.0 | 3.7
4 | 13.2 | 14.4 | 13.9 | 14.4 | 12.8 | 14.3 | 10.9 | 13.9 | 12.4 | 14.3
5 | 77.4 | 21.6 | 78.6 | 21.2 | 81.0 | 21.7 | 79.9 | 21.6 | 80.3 | 21.7
6 | 5.4 | 7.8 | 2.9 | 7.0 | 2.5 | 7.6 | 3.6 | 7.7 | 3.6 | 7.7
(None, N) | 4.0 | 54.1 | 4.6 | 52.8 | 3.7 | 53.8 | 5.6 | 53.5 | 3.7 | 54.0
Scenario 5
1 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4
2 | 0.0 | 3.8 | 0.0 | 3.9 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8
3 | 0.5 | 8.9 | 0.2 | 8.7 | 0.3 | 8.7 | 0.3 | 8.2 | 0.3 | 8.7
4 | 93.5 | 23.3 | 93.4 | 23.2 | 94.7 | 23.4 | 90.8 | 23.0 | 94.3 | 23.3
5 | 3.8 | 20.3 | 3.5 | 19.3 | 3.2 | 20.3 | 6.1 | 20.3 | 3.4 | 20.3
6 | 0.1 | 7.5 | 0.1 | 7.0 | 0.1 | 7.4 | 0.1 | 7.4 | 0.1 | 7.5
(None, N) | 2.1 | 67.2 | 2.8 | 65.6 | 1.7 | 67.0 | 2.7 | 66.1 | 1.9 | 67.0
Table S2: Selection percentages (Sel%) and number of patients (Pts), under scenarios 1 to 5, for the DEMO design using different hyperparameter settings. The optimal therapeutical dose under each scenario is given in boldface. “None” is the percentage of trials without selecting any dose, and “N” is the average total sample size. Hyperparameter Setting | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Dose | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts
Scenario 6
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2
2 | 1.2 | 5.6 | 0.6 | 5.5 | 0.7 | 5.5 | 0.6 | 5.5 | 0.7 | 5.5
3 | 93.3 | 23.3 | 94.3 | 23.4 | 94.0 | 23.4 | 87.8 | 22.7 | 94.0 | 23.4
4 | 2.2 | 23.8 | 2.0 | 23.6 | 2.4 | 23.8 | 5.4 | 23.7 | 2.6 | 23.8
5 | 2.3 | 19.4 | 2.3 | 18.5 | 2.2 | 19.4 | 4.9 | 19.4 | 2.0 | 19.4
6 | 0.0 | 6.3 | 0.0 | 5.9 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2
(None, N) | 1.0 | 81.6 | 0.8 | 80.1 | 0.7 | 81.5 | 1.3 | 80.8 | 0.7 | 81.6
Scenario 7
1 | 1.8 | 5.7 | 1.3 | 5.6 | 1.7 | 5.7 | 2.0 | 5.7 | 1.7 | 5.7
2 | 77.8 | 20.6 | 76.1 | 20.3 | 77.9 | 20.6 | 77.3 | 20.6 | 77.9 | 20.6
3 | 16.5 | 23.1 | 16.8 | 22.7 | 16.5 | 23.1 | 16.4 | 23.0 | 16.4 | 23.1
4 | 2.7 | 17.5 | 3.5 | 17.1 | 2.7 | 17.5 | 2.8 | 17.5 | 2.8 | 17.5
5 | 0.1 | 6.0 | 0.6 | 6.5 | 0.1 | 6.0 | 0.1 | 6.0 | 0.1 | 6.0
6 | 0.1 | 2.3 | 0.0 | 2.8 | 0.1 | 2.3 | 0.1 | 2.2 | 0.1 | 2.3
(None, N) | 1.0 | 75.2 | 1.7 | 75.0 | 1.0 | 75.2 | 1.3 | 75.1 | 1.0 | 75.2
Scenario 8
1 | 0.4 | 3.5 | 0.3 | 3.5 | 0.4 | 3.5 | 0.4 | 3.5 | 0.4 | 3.5
2 | 50.3 | 22.5 | 49.0 | 22.3 | 47.8 | 22.4 | 50.1 | 22.4 | 48.9 | 22.4
3 | 48.6 | 23.7 | 49.5 | 23.6 | 50.9 | 23.7 | 48.5 | 23.7 | 49.9 | 23.7
4 | 0.5 | 20.2 | 0.5 | 19.8 | 0.7 | 20.2 | 0.8 | 20.2 | 0.6 | 20.2
5 | 0.2 | 8.4 | 0.1 | 8.7 | 0.2 | 8.4 | 0.2 | 8.4 | 0.2 | 8.4
6 | 0.0 | 2.9 | 0.0 | 3.4 | 0.0 | 3.0 | 0.0 | 2.9 | 0.0 | 2.9
(None, N) | 0.0 | 81.2 | 0.6 | 81.3 | 0.0 | 81.2 | 0.0 | 81.1 | 0.0 | 81.1
Scenario 9
1 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3
2 | 0.2 | 13.6 | 0.2 | 13.6 | 0.1 | 13.5 | 0.1 | 13.6 | 0.2 | 13.5
3 | 82.6 | 23.2 | 83.4 | 23.2 | 82.7 | 23.2 | 78.9 | 22.8 | 83.0 | 23.2
4 | 2.6 | 24.0 | 2.3 | 23.9 | 2.5 | 24.0 | 3.5 | 24.0 | 2.4 | 24.0
5 | 5.2 | 23.9 | 4.8 | 23.9 | 5.1 | 23.9 | 6.5 | 23.9 | 5.0 | 23.9
6 | 9.4 | 22.9 | 9.2 | 22.9 | 9.6 | 23.0 | 11.0 | 22.9 | 9.4 | 23.0
(None, N) | 0.0 | 110.9 | 0.1 | 110.8 | 0.0 | 110.9 | 0.0 | 110.4 | 0.0 | 110.9
Scenario 10
1 | 0.0 | 9.2 | 0.1 | 9.7 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2
2 | 0.0 | 6.2 | 0.0 | 5.1 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2
3 | 0.0 | 3.5 | 0.0 | 3.3 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5
4 | 0.0 | 1.6 | 0.1 | 1.6 | 0.2 | 1.6 | 0.1 | 1.6 | 0.2 | 1.6
5 | 0.0 | 0.9 | 0.2 | 1.2 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9
6 | 0.0 | 0.7 | 0.0 | 1.1 | 0.0 | 0.7 | 0.1 | 0.7 | 0.0 | 0.7
Not found | 100.0 | 22.0 | 99.6 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0
Table S3: Selection percentages (Sel%) and number of patients (Pts) at each dose for different settings of $(L,K,\kappa)$ under scenarios 1 to 5. The optimal therapeutical dose under each scenario is given in boldface. “None” is the percentage of trials without selecting any dose, and “N” is the average total sample size. | ($L,K,\kappa$)
---|---
| (2,3,0.3) | (2,4,0.3) | (2,6,0.3) | (3,4,0.0) | (3,4,0.2) | (3,4,0.3) | (3,6,0.3)
Dose | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts
Scenario 1
1 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8
2 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0 | 0.0 | 4.0
3 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2 | 0.0 | 4.2
4 | 1.3 | 6.6 | 1.3 | 6.6 | 1.3 | 6.6 | 1.3 | 6.9 | 1.3 | 6.9 | 1.3 | 6.9 | 1.3 | 6.9
5 | 16.4 | 19.5 | 16.4 | 19.5 | 16.4 | 19.5 | 16.5 | 19.5 | 16.5 | 19.5 | 16.5 | 19.5 | 16.5 | 19.5
6 | 79.0 | 20.7 | 79.0 | 20.7 | 79.0 | 20.7 | 78.8 | 20.7 | 78.8 | 20.7 | 78.8 | 20.7 | 78.8 | 20.7
(None, N) | 3.3 | 58.7 | 3.3 | 58.7 | 3.3 | 58.7 | 3.4 | 59.0 | 3.4 | 59.0 | 3.4 | 59.0 | 3.4 | 59.0
Scenario 2
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 |  1 3.2
2 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5
3 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6
4 | 2.8 | 14.2 | 2.8 | 14.2 | 2.8 | 14.2 | 3.1 | 15.4 | 3.1 | 15.4 | 3.1 | 15.4 | 3.1 | 15.4
5 | 83.5 | 23.1 | 83.5 | 23.1 | 83.5 | 23.1 | 82.6 | 23.1 | 82.6 | 23.1 | 82.6 | 23.1 | 82.6 | 23.1
6 | 8.9 | 19.8 | 8.9 | 19.8 | 8.9 | 19.8 | 9.4 | 19.8 | 9.4 | 19.8 | 9.4 | 19.8 | 9.4 | 19.8
(None, N) | 4.8 | 67.3 | 4.8 | 67.3 | 4.8 | 67.3 | 4.9 | 68.5 | 4.9 | 68.5 | 4.9 | 68.5 | 4.9 | 68.5
Scenario 3
1 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 |  1 3.4
2 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6
3 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9 | 0.2 | 3.9
4 | 86.8 | 23.0 | 86.8 | 23.0 | 86.8 | 23.0 | 90.0 | 23.5 | 90.0 | 23.5 | 90.0 | 23.5 | 90.0 | 23.5
5 | 5.3 | 23.1 | 5.3 | 23.1 | 5.3 | 23.1 | 3.9 | 23.1 | 3.9 | 23.1 | 3.9 | 23.1 | 3.9 | 23.1
6 | 6.7 | 19.2 | 6.7 | 19.2 | 6.7 | 19.2 | 4.7 | 19.3 | 4.7 | 19.3 | 4.7 | 19.3 | 4.7 | 19.3
(None, N) | 1.0 | 76.3 | 1.0 | 76.3 | 1.0 | 76.3 | 1.2 | 76.8 | 1.2 | 76.8 | 1.2 | 76.8 | 1.2 | 76.8
Scenario 4
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2
2 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4
3 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.6 | 0.0 | 3.7 | 0.0 | 3.7 | 0.0 | 3.7 | 0.0 | 3.7
4 | 12.5 | 14.3 | 12.5 | 14.3 | 12.5 | 14.3 | 12.4 | 14.3 | 12.4 | 14.3 | 12.4 | 14.3 | 12.4 | 14.3
5 | 80.1 | 21.7 | 80.1 | 21.7 | 80.1 | 21.7 | 80.1 | 21.7 | 80.1 | 21.7 | 80.1 | 21.7 | 80.1 | 21.7
6 | 3.6 | 7.7 | 3.6 | 7.7 | 3.6 | 7.7 | 3.6 | 7.7 | 3.6 | 7.7 | 3.6 | 7.7 | 3.6 | 7.7
(None, N) | 3.8 | 53.9 | 3.8 | 53.9 | 3.8 | 53.9 | 3.9 | 54.0 | 3.9 | 54.0 | 3.9 | 54.0 | 3.9 | 54.0
Scenario 5
1 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4 | 0.0 | 3.4
2 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8 | 0.0 | 3.8
3 | 0.3 | 8.2 | 0.3 | 8.3 | 0.3 | 8.3 | 0.3 | 8.6 | 0.3 | 8.7 | 0.3 | 8.7 | 0.3 | 8.7
4 | 92.9 | 23.1 | 92.9 | 23.1 | 92.9 | 23.1 | 94.3 | 23.3 | 94.3 | 23.3 | 94.3 | 23.3 | 94.3 | 23.3
5 | 4.3 | 20.3 | 4.3 | 20.3 | 4.3 | 20.3 | 3.4 | 20.3 | 3.4 | 20.3 | 3.4 | 20.3 | 3.4 | 20.3
6 | 0.3 | 7.4 | 0.3 | 7.4 | 0.3 | 7.4 | 0.1 | 7.5 | 0.1 | 7.5 | 0.1 | 7.5 | 0.1 | 7.5
(None, N) | 2.2 | 66.2 | 2.2 | 66.3 | 2.2 | 66.3 | 1.9 | 67.0 | 1.9 | 67.0 | 1.9 | 67.0 | 1.9 | 67.0
Table S4: Selection percentages (Sel%) and number of patients (Pts) at each dose for different settings of $(L,K,\kappa)$ under scenarios 6 to 10. The optimal therapeutical dose under each scenario is given in boldface. “None” is the percentage of trials without selecting any dose, and “N” is the average total sample size. | ($L,K,\kappa$)
---|---
| (2,3,0.3) | (2,4,0.3) | (2,6,0.3) | (3,4,0.0) | (3,4,0.2) | (3,4,0.3) | (3,6,0.3)
Dose | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts | Sel% | Pts
Scenario 6
1 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2 | 0.0 | 3.2
2 | 0.2 | 5.3 | 0.7 | 5.5 | 0.6 | 5.5 | 0.2 | 5.3 | 0.7 | 5.5 | 0.7 | 5.5 | 0.6 | 5.5
3 | 80.5 | 21.7 | 85.0 | 22.3 | 85.1 | 22.3 | 89.5 | 22.8 | 93.1 | 23.3 | 94.0 | 23.4 | 94.1 | 23.4
4 | 11.0 | 23.8 | 7.3 | 23.8 | 7.3 | 23.8 | 6.0 | 23.8 | 2.9 | 23.8 | 2.3 | 23.8 | 2.3 | 23.8
5 | 7.0 | 19.4 | 6.1 | 19.4 | 6.1 | 19.4 | 3.2 | 19.4 | 2.6 | 19.4 | 2.3 | 19.4 | 2.3 | 19.4
6 | 0.4 | 6.1 | 0.0 | 6.2 | 0.0 | 6.2 | 0.4 | 6.1 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2
(None, N) | 0.9 | 79.6 | 0.9 | 80.4 | 0.9 | 80.4 | 0.7 | 80.7 | 0.7 | 81.5 | 0.7 | 81.6 | 0.7 | 81.6
Scenario 7
1 | 0.6 | 5.0 | 1.7 | 5.7 | 2.1 | 5.9 | 0.6 | 5.0 | 1.7 | 5.7 | 1.7 | 5.7 | 2.1 | 5.9
2 | 73.0 | 19.9 | 76.4 | 20.4 | 77.3 | 20.6 | 74.5 | 20.1 | 77.8 | 20.6 | 77.9 | 20.6 | 78.8 | 20.8
3 | 21.0 | 23.0 | 18.0 | 23.1 | 16.8 | 23.1 | 19.6 | 23.0 | 16.7 | 23.1 | 16.6 | 23.1 | 15.4 | 23.1
4 | 3.8 | 17.4 | 2.8 | 17.5 | 2.7 | 17.5 | 3.7 | 17.4 | 2.7 | 17.5 | 2.7 | 17.5 | 2.6 | 17.5
5 | 0.6 | 5.9 | 0.1 | 6.0 | 0.1 | 6.1 | 0.6 | 5.9 | 0.1 | 6.0 | 0.1 | 6.0 | 0.1 | 6.1
6 | 0.1 | 2.3 | 0.1 | 2.3 | 0.1 | 2.3 | 0.1 | 2.3 | 0.1 | 2.3 | 0.1 | 2.3 | 0.1 | 2.3
(None, N) | 0.9 | 73.5 | 0.9 | 75.0 | 0.9 | 75.4 | 0.9 | 73.7 | 0.9 | 75.2 | 0.9 | 75.2 | 0.9 | 75.6
Scenario 8
1 | 0.4 | 3.4 | 0.4 | 3.5 | 0.4 | 3.5 | 0.4 | 3.4 | 0.4 | 3.5 | 0.4 | 3.5 | 0.4 | 3.5
2 | 43.6 | 21.4 | 48.9 | 22.4 | 50.6 | 22.7 | 43.6 | 21.4 | 48.9 | 22.4 | 48.9 | 22.4 | 50.6 | 22.7
3 | 51.7 | 23.3 | 49.9 | 23.7 | 48.3 | 23.7 | 51.7 | 23.3 | 49.9 | 23.7 | 49.9 | 23.7 | 48.3 | 23.7
4 | 3.1 | 20.2 | 0.6 | 20.2 | 0.5 | 20.2 | 3.1 | 20.2 | 0.6 | 20.2 | 0.6 | 20.2 | 0.5 | 20.2
5 | 1.2 | 8.2 | 0.2 | 8.4 | 0.2 | 8.4 | 1.2 | 8.2 | 0.2 | 8.4 | 0.2 | 8.4 | 0.2 | 8.4
6 | 0.0 | 2.9 | 0.0 | 2.9 | 0.0 | 2.9 | 0.0 | 2.9 | 0.0 | 2.9 | 0.0 | 2.9 | 0.0 | 2.9
(None, N) | 0.0 | 79.4 | 0.0 | 81.1 | 0.0 | 81.4 | 0.0 | 79.5 | 0.0 | 81.1 | 0.0 | 81.1 | 0.0 | 81.4
Scenario 9
1 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3 | 0.0 | 3.3
2 | 0.0 | 12.9 | 0.2 | 13.5 | 0.3 | 21.9 | 0.0 | 12.9 | 0.2 | 13.5 | 0.2 | 13.5 | 0.3 | 21.9
3 | 17.9 | 14.3 | 83.2 | 23.2 | 80.1 | 23.2 | 17.9 | 14.3 | 79.3 | 22.7 | 83.2 | 23.2 | 80.1 | 23.2
4 | 18.8 | 23.9 | 2.3 | 23.9 | 2.5 | 23.9 | 18.9 | 24.0 | 3.1 | 24.0 | 2.4 | 24.0 | 2.6 | 24.0
5 | 25.0 | 22.9 | 4.8 | 23.9 | 5.7 | 24.0 | 25.1 | 22.9 | 5.6 | 23.9 | 4.9 | 23.9 | 5.8 | 24.0
6 | 38.3 | 22.7 | 9.5 | 23.0 | 11.4 | 23.8 | 38.1 | 22.7 | 11.8 | 23.0 | 9.3 | 23.0 | 11.2 | 23.8
(None, N) | 0.0 | 100.0 | 0.0 | 110.9 | 0.0 | 120.1 | 0.0 | 100.1 | 0.0 | 110.3 | 0.0 | 110.9 | 0.0 | 120.1
Scenario 10
1 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2 | 0.0 | 9.2
2 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2 | 0.0 | 6.2
3 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5 | 0.0 | 3.5
4 | 0.2 | 1.6 | 0.2 | 1.6 | 0.2 | 1.6 | 0.2 | 1.6 | 0.2 | 1.6 | 0.2 | 1.6 | 0.2 | 1.6
5 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9 | 0.0 | 0.9
6 | 0.0 | 0.7 | 0.0 | 0.7 | 0.0 | 0.7 | 0.0 | 0.7 | 0.0 | 0.7 | 0.0 | 0.7 | 0.0 | 0.7
(None, N) | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0 | 99.8 | 22.0