# Bayesian Framework for Multi-Source Data Integration - Application to Human
Extrapolation From Preclinical Studies

Sandrine Boulet1, Moreno Ursino1,2, Robin Michelet3, Linda B.S. Aulin3,
Charlotte Kloft3, Emmanuelle Comets4 and Sarah Zohar1,∗

( 1Inserm, Centre de Recherche des Cordeliers, Sorbonne Université,
Université Paris Cité, F-75006 Paris, France;
Inria, HeKA, F-75015 Paris, France
2Unit of Clinical Epidemiology, Assistance Publique-Hôpitaux de Paris,
CHU Robert Debré, Inserm CIC-EC 1426, F-75019 Paris, France
3Department of Clinical Pharmacy & Biochemistry, Institute of Pharmacy,
Freie Universitaet Berlin, 12169 Berlin, Germany
Inserm, Univ Rennes, EHESP, Irset (Institut de recherche en santé,
environnement et travail) - UMRS 1085, F-35000 Rennes, France
4Inserm, Université Paris Cité, IAME, F-75018 Paris, France
∗ email<EMAIL_ADDRESS>
)

###### Abstract

In preclinical investigations, e.g. in in vitro, in vivo and in silico
studies, the pharmacokinetic, pharmacodynamic and toxicological
characteristics of a drug are evaluated before advancing to first-in-man
trial. Usually, each study is analyzed independently and the human dose range
does not leverage the knowledge gained from all studies. Taking into account
the preclinical data through inferential procedures can be particularly
interesting to obtain a more precise and reliable starting dose and dose
range.
We propose a Bayesian framework for multi-source data integration from
preclinical studies results extrapolated to human, which allow to predict the
quantities of interest (e.g. the minimum effective dose, the maximum tolerated
dose, etc.) in humans. We build an approach, divided in four main steps, based
on a sequential parameter estimation for each study, extrapolation to human,
commensurability checking between posterior distributions and final
information merging to increase the precision of estimation.
The new framework is evaluated via an extensive simulation study, based on a
real-life example in oncology inspired from the preclinical development of
galunisertib. Our approach allows to better use all the information compared
to a standard framework, reducing uncertainty in the predictions and
potentially leading to a more efficient dose selection.

Keywords: Commensurability; Hellinger distance; Posteriors conflict;
Posteriors merging.

##  1 Introduction

At the beginning of the development of a new element, object, compound, etc.
(depending on the field), preliminary knowledge on its properties are
estimated through several experiments involving small sample sizes. For
example, the clinical development of a novel drug molecule is always preceded
by numerous preclinical studies. They include in vitro studies (studies in
subcellular fractions, cell cultures, micro-organisms, organoid models, etc.),
in vivo studies (animal testing in species such as mouse, rat, dog, monkey,
etc.) and in silico studies (simulations and synthetic data). These
preclinical studies generate an abundance of knowledge regarding the safety
and efficacy of the compound, including pharmacokinetics/pharmacodynamics
(PK/PD) on a cellular, tissue, organ or organism level, which is then used for
go/no go decisions and influences the design of clinical trials in humans.
However, preclinical trials usually involve small sample sizes, for ethics and
budget constraints.

The Food and Drug Administration (2005) (FDA) provides guidelines regarding
the use of preclinical knowledge to compute the Maximum Recommended Starting
Dose (MRSD) of the first-in-human (FIH) trial, with healthy volunteers and
drug products for which systemic exposure is intended. These guidelines
outline an empirical algorithmic approach to compute the FIH dose in four
steps; (1) for each animal species used in the in vivo studies, the Human
Equivalent Dose (HED) is computed based on the No-Observed-Adverse-Effect
Level (NOAEL) and on the body surface area; (2) the HED corresponding to the
most sensitive animal species is selected and (3) used for the calculation of
the MRSD by applying a safety factor accounting for the expected variability
coming from animal-to-human toxicity extrapolation; (4) the MRSD is then
adjusted based on the predicted pharmacological mechanism. Although this
approach is a valuable starting point, it shows several drawbacks. First,
there is no precise recommendation for choosing the safety factor and thus
ensuring the safety at the starting dose. Second, the dose selection is
primarily based on the minimization of toxicity risk, rather than on efficacy.
While this approach may be the only option when there is no comparable marker
in healthy volunteers, for example because they do not express the target
molecule for the tested drug, it is suboptimal when efficacy can be evaluated.
In this situation, to improve the clinical development process, the European
Medicines Agency (2017) (EMA) suggests to take under consideration toxicity
but also efficacy by calculating the Minimal Anticipated Biological Effect
Level (MABEL) based on all in vitro and in vivo information available. Other
approaches facilitating preclinical to clinical translation are mentioned in
Shen et al. (2019) and include PK-driven or PK/PD-driven approaches.
Translational PK/PD models extrapolate concentration and drug effect over time
rather than dose and account for differences in both PK and PD parameters when
moving from animal species to humans.

However, all of these frameworks use only a part of the collected preclinical
data (e.g., from the most appropriate animal species based on an empirical
estimation), thus ignoring a vast majority of the available data.
Additionally, each of the preclinical analyses is conducted independently,
without fully using the results already accrued in previous studies. Recently,
as part of the European project on Flagellin aerosol therapy as an
immunomodulatory adjunct to the antibiotic treatment of drug-resistant
bacterial pneumonia (FAIR), Michelet et al. (2021) have proposed to consider
the FIH clinical trial as a continuum of serial preclinical studies. The
authors propose an approach based on the “learn-predict-confirm” paradigm in
which mathematical models are updated at each step and the updated versions
are used to optimize the next study (Sheiner, 1997).

This type of approach can be easily placed under a Bayesian framework that can
update posterior knowledge each time new data becomes available. For example,
La Gamba et al. (2019) sequentially introduce knowledge in preclinical
investigations within a Bayesian PK/PD setting, using the posterior
distributions resulting from one trial to build the prior distributions for
the following trial. More widely, the Bayesian approach has also been used to
use information obtained from one population for the analysis of another
population. For instance, Zheng and Hampson (2020) use preclinical data from
animal to inform the design and prior distributions of a phase I clinical
trial via a Bayesian decision-theoretic approach and Zheng et al. (2020) via a
meta-analytic approach. Another example is that of Petit et al. (2018) who
propose a method allowing extrapolation and bridging of adult data in early-
phase dose-finding pediatric studies.

The main issue in these standard Bayesian frameworks for a sequence of
preclinical studies, where the posterior distributions of the previous study
are used to set prior distributions for the new study, is that they require
the use of the same mathematical models (or sub-models) for each criterion
(outcome, marker) between studies. Moreover, the process ignores possible
differences between species which are not taken into account by extrapolation
formulas, and not all previous information can be included in the new study,
since, as a general rule in a Bayesian approach, the amount of information of
the prior should not exceed the information of the current study. Therefore,
if at least one study has any parameter not consistent with all other studies,
the information chain can be broken, and information gathered before this
study may be lost.

To deal with these challenges and use all relevant information, we propose a
Bayesian framework accounting for differences and similarities between
preclinical studies at each extrapolation (up-dating) step. More precisely,
our approach needs to consider multi-source data (cells, mouse, …) and be able
to predict the quantities of interest (for example, MABEL, NOAEL, Minimal
Effective Dose - MED, Maximum Tolerated Dose - MTD, etc.) in humans. This new
framework was evaluated via an extensive simulation study. The simulation
setting was inspired by the development of an oncology drug, galunisertib, an
inhibitor of TGF-$\beta$ signaling which reduces tumor growth, where
extrapolation approaches were applied during the preclinical studies based on
markers of toxicity and efficacy (Gueorguieva et al., 2014).

This manuscript is organized as follow. Methods are presented in Section 2.
Section 3 describes the simulation study used to assess the proposed
methodology, and the simulation results are given in Section 4. Key findings
and recommendations are discussed in Section 5.

##  2 Methods

We assume that the development plan includes $K$ preclinical studies to be run
or, at least, analyzed sequentially before the FIH trial, involving the same
outcomes that can be analysed in different ways depending on the study. Our
objective is to determine, using appropriate extrapolation-transformation or
link functions, the quantities of interest (doses in our case), for example,
the MABEL dose or an appropriate range of doses for a FIH dose-ranging trial.
We propose a Bayesian framework in four steps (shown in Figure 1): (1) first,
we estimate the parameters of the appropriate model for each outcome; (2)
second, we apply extrapolation via pre-specified formulas to obtain parameter
distribution in humans for each criterion; (3) third, we check the
coherence/commensurability of posterior distributions via a divergence-based
measure; (4) finally, we merge the selected posterior distributions using an
extension of the Bayes formula. We also investigated an alternative hybrid
frequentist-Bayesian approach in Web Appendix A, where the estimation is
performed using a frequentist method for faster run-times and a probabilistic
sensitivity approach is then used to build parameter distributions. In the
following, each of the four proposed steps is detailed.

Figure 1: The Bayesian framework in four steps: (1) Sequential Bayesian
estimation, for each outcome, with weakly informative prior distributions; (2)
Extrapolation to human for each criterion (via pre-specified formulas); (3)
Coherence/commensurability checking of posterior distributions via a
divergence-based measure; (4) Merging the selected posterior distributions
using an extension of the Bayes formula. This figure appears in color in the
electronic version of this article.

###  2.1 First Step: Parameters Estimation

For $k\in\\{1,...,K\\}$, let $\mathbf{y}_{k}$ denote the outcomes measured in
the $k$th study. We suppose that $\mathbf{y}_{k}$ are related to each other
(they could be exactly the same outcome for each study), or that, if
different, they can be used to compute the doses of interest, $d_{r}$,
$r\in\\{1,\ldots,R\\}$ with $R$ the number of doses of interest (for example,
MABEL, NOAEL, MED, MTD, etc.). For each study $k$, Bayesian fixed- or mixed-
effect models, $\mathbf{f}_{k}(\mathbf{y}_{k},\boldsymbol{\theta}_{k}$),
(possibly nonlinear) are fitted, via Markov chain Monte Carlo (MCMC) methods,
on the dose-outcome relationships (PK/PD indices describing efficacy or
toxicity) with weakly-informative prior distributions on models parameters
$\boldsymbol{\theta}_{k}$. The functions $\mathbf{f}_{k}(.)$ can differ
between studies, as well as the vector $\boldsymbol{\theta}_{k}$ can have
different length or being study specific. For example, a fixed effect model
can be used in a study with only one outcome per subject while a mixed effect
model, with the same structure for the population effects but with random
effects on a few parameters, can be adopted for longitudinal studies. For the
first study, prior distributions are chosen based on prior knowledge external
to this process. When no prior knowledge is available, non-informative prior
distributions can be used. For the following studies, if any component of
$\boldsymbol{\theta}_{k-1}$ can be related to an element of
$\boldsymbol{\theta}_{k}$, the prior distributions are computed using the
estimated posterior means of the model parameters of the previous study,
$k-1$, (extrapolated between species, if necessary, using similar approaches
as described in the next step) and using appropriately large standard
deviations. In this way, prior distributions act as stabilization tools.

To summarise, at the end of the process, each study $k\in\\{1,...,K\\}$ yields
a posterior distribution for each element of each $\boldsymbol{\theta}_{k}$.

###  2.2 Second Step: Extrapolation to Human

While in the previous step, extrapolation was used between species/studies, at
this step we focus on human extrapolation. For each study $k\in\\{1,...,K\\}$,
we extrapolate the preclinical results (the estimated parameters) to humans
using predetermined formulas $\mathbf{g}_{k}(.)$ (transformations),
$\boldsymbol{\theta}_{h,k}=\mathbf{g}_{k}(\boldsymbol{\theta}_{k})$, where
$\boldsymbol{\theta}_{h,k}$ represent the vector of extrapolated human
parameters from the $k$th study. An example of such formulas is the allometric
scaling used to extrapolate pharmacokinetic parameters across species by
taking into account the size of the organisms (West, Brown, and Enquist,
1997). As stated in the previous section, these formulas could also be used in
the previous step to extrapolate parameter values between species/studies. In
Bayesian setting, the transformation is applied directly on posterior
distributions to obtain the posterior distributions of the extrapolated
parameters (via the random variable transformation theorem). Then, the doses
of interest, $d_{r}=d_{r,k}(\boldsymbol{\theta}_{h,k})$, for human, where
$d_{r,k}(.)$ represent the function that links the vector
$\boldsymbol{\theta}_{h,k}$ of extrapolated human parameters from the $k$th
study to the dose of interest $d_{r}$, are derived either analytically or from
Monte Carlo simulations. Finally, the associated posterior distributions of
$d_{r}$, $\pi^{post}_{k}(d_{r})$, are deduced (via the random variable
transformation theorem or approximated via Monte Carlo methods).

###  2.3 Third Step: Commensurability Checking and Posterior Distributions
Selection

In this step, we aim to compare the $K$ predicted human dose distributions,
$D_{k,r}\sim\pi^{post}_{k}(d_{r})$, for each $r$, and select the ones which
seems the most similar to each others. However, posterior distributions cannot
be directly compared since they bring a different amount of information
depending on the corresponding study sample size. One way to make two
posteriors comparable consists on discounting the likelihood of the study with
the highest sample size, as proposed by Ollier et al. (2020). This method
requires the re-analysis of one of the two experiences and the decision on the
discounting factor: it could be straightforward in case of fixed effect
models, but not really for mixed effects model involving longitudinal studies.
Inspired by the effective sample sizes (ESS) notion (Morita, Thall, and
Müller, 2008), we propose to transform the posterior distributions to move to
a situation where they could be assumed as Gaussian, and then we standardize
them at the highest variance,

$\displaystyle D^{*}_{k,r}$
$\displaystyle=\dfrac{\underset{h\in\mathcal{K}}{\max}~{}S_{h,r}}{S_{k,r}}\log(D_{k,r})+\left(1-\dfrac{\underset{h\in\mathcal{K}}{\max}~{}S_{h,r}}{S_{k,r}}\right)M_{k,r}$
(1) $\displaystyle\sim\pi^{post}_{k}(d^{*}_{r}),$

for $k\in\\{1,...,K\\}$, where $M_{k,r}=E(\log D_{k,r})$ and
$S_{k,r}^{2}=Var(\log D_{k,r})$ are respectively the mean and the variance of
$\log D_{k,r}$. This transformation allows the distributions to have the same
variance across studies but to keep their own expectations. Indeed, the
logarithmic transformation of the dose is first used to move to a situation
where the distribution could be assumed as Gaussian. According to Morita et
al. (2008), the ESS of a normal distribution, with a normal model and known
variance, depends only on the prior variance. Thus, the ESS of our transformed
distributions are equal (if we consider them as prior distribution on a
“Gaussian dose model”). Furthermore, for $k\in\\{1,...,K\\}$,
$E(D^{*}_{k,r})=M_{k,r}$ and
$Var(D^{*}_{k,r})=\underset{h\in\mathcal{K}}{\max}~{}S^{2}_{h,r}$.

Once the posteriors are comparable, the commensurability of the modified
distributions $\pi^{post}_{k}(d^{*}_{r})$ between studies
$k_{1}\in\\{1,...,K\\}$ and $k_{2}\in\\{1,...,K\\}$, $k_{2}\neq k_{1}$, is
checked using the Hellinger distance for each $r$,
$H_{r}(D_{k_{1}},D_{k_{2}})=\Big{[}\dfrac{1}{2}\int\big{(}\sqrt{\pi^{post}_{k_{1}}(d^{*}_{r})}-\sqrt{\pi^{post}_{k_{2}}(d^{*}_{r})}\big{)}^{2}\text{d}d^{*}_{r}\Big{]}^{\frac{1}{2}}$
(Ollier et al., 2021). We proposed this distance since it is bounded between 0
and 1, symmetric and it makes easier to define the consistency notion of
results between studies. In fact, the closer the Hellinger distance is to 0,
the more consistent the predicted dose distributions in humans are between
studies. On the contrary, a distance close to 1 indicates inconsistent
results.

An algorithm, along with a threshold, based on Hellinger distance results
should be defined to decide which studies will be selected for the next step.
When only a small number of studies are available, ad hoc algorithms can be
easily developed, as shown in our example with simple decision rules (see
subsection 3.4). In more complex settings, clustering methods can be used. We
suggest to run simulations of selected/relevant scenarios to optimize the
algorithm and to choose a threshold basing on the results accuracy. For each
selected scenario, we define the studies that we consider consistent and that
we wish to retain for the fourth and final step. Then, via simulation, we
compute the Hellinger distances and define a binary variable “true response”
related to each comparison between studies, that is equal to 1 if the two
studies are considered similar (theoretically, in the scenario) and 0
otherwise, and, for each possible threshold, a variable “predicted response”
that is equal to 1 if the computed distance is below the threshold and 0
otherwise. The value of the threshold is then chosen based on the curve of the
accuracy versus the Hellinger distance threshold. The accuracy is defined as
the proportion of correct predictions (that is, true positives and true
negatives) among the total number of cases,
$\text{acc}=\dfrac{TP+TN}{P+N},\text{~{}where~{}}P=TP+FN\text{~{}and~{}}N=TN+FP$,
where $TP$, $TN$, $FN$, $FP$, $P$ and $N$ are respectively the numbers of true
positives, true negatives, false negatives, false positives, positives and
negatives. Examples are given in subsection 4.2 and in Web Appendix C.

At the end of this step, consistent studies are selected. When no studies are
“clustered together”, only the results of the most relevant study are
considered for the following step.

###  2.4 Fourth Step: Merging the Selected Posterior Distributions

Let $\mathcal{K}=\\{1,2,\dots,K\\}$ be the set of indices of the preclinical
studies. Denoting $\mathcal{K}^{\prime}\subset\mathcal{K}$ the subset of
studies selected at the previous step, we merge the extrapolated dose
distributions $\pi^{post}_{k^{\prime}}(d_{r})$
($k^{\prime}\in\mathcal{K}^{\prime}$) using an extension of the Bayes formula
to obtain the final predicted dose distribution $d_{r}$ in humans:

$\pi(d_{r})\propto\prod_{k^{\prime}\in\mathcal{K}^{\prime}}\mathcal{\pi}^{post}_{k^{\prime}}(d_{r}).$
(2)

Eq. 2 exists when the $\text{Card}(\mathcal{K}^{\prime})$ selected
distributions share the same support, or at least part of it. It excludes
situations where the product term is 0 almost surely on the domain, that is
when at least one distribution is 0 almost surely in the part of the domain
where the others are different from zero and vice versa. However, this
phenomenon should be avoided by the previous steps. Moreover, the second term
of eq. 2 is Lebesgue integrable if the $\text{Card}(\mathcal{K}^{\prime})$
distributions (or at least all but one) are bounded. This condition is
sufficient but not necessary, since a product of U-shaped (or J, or inverse
J-shaped) beta distribution density functions is still a beta distribution (as
shown in Web Appendix A). In case of sequential Bayesian analyses (when the
posterior of the previous study is used as prior for the next one) of the same
study design repeated $K$ times, the final posterior distribution is given by
$\pi^{post}_{K}(\boldsymbol{\theta})\propto\pi^{\textit{prior}}(\boldsymbol{\theta})\prod_{k=1}^{K}\mathcal{L}(\boldsymbol{\theta}|\mbox{data}_{k})$.
If $\pi^{\textit{prior}}(\boldsymbol{\theta})$ is highly non-informative and
if we write $K-1$ other highly non-informative improper distributions close to
each likelihood term, we can see each couple terms of the product (the $k$th
likelihood and the associated highly non-informative improper distributions)
as a posterior distribution. This is why eq. 2 can be seen as an extension of
the Bayes formula. Because the information from informative prior
distributions will be counted $\text{Card}(\mathcal{K}^{\prime})$ times, we
suggest using non-informative prior distributions, as the amount of prior
information can be considered negligible also after the application of eq. 2.
Eq. 2 then gives the final dose distribution, which takes into account all
relevant information and from which inference can be drawn; that is, the
expected value (or the median) can be used as point estimator and credible
intervals can be extracted. To note, if only the most relevant study was
considered at the previous step, eq. 2 becomes its posterior distribution.

##  3 Simulation Settings

To illustrate and evaluate our approach, we use the preclinical and clinical
development of galunisertib (LY2157299) as a case-study to build different
simulation scenarios. In the following section, we describe how to define the
MTD using pharmacokinetics results. Therefore, only for the sake of
simplicity, $r=1$. However, simulation focusing on MED estimation (along with
the MTD, therefore $r=2$) using pharmacokinetics/pharmacodynamics modelling
are given in Web Appendix B.

###  3.1 Galunisertib Case Study

Galunisertib is a transforming growth factor (TGF)-$\beta$ receptor I kinase
inhibitor, developed for patients with glioma (tumor that starts in the glial
cells of the brain or the spine) and advanced cancers. It inhibits the
transforming growth factor (TGF)-$\beta$ receptor, blocking TGF-$\beta$
signaling which is an important growth regulator in advanced cancer. This
pathway is a potential target to simultaneously inhibit tumor growth and neo-
angiogenesis. Semi-mechanistic modeling was used during drug development to
predict a safe dosing regimen for phase I study and orient dose selection,
building a model from data in mouse, rat and dog (Bueno et al., 2008;
Gueorguieva et al., 2014). Studies in mouse described by Bueno et al. (2008)
showed that the inhibition of phosphorylated Smad proteins, a transducer in
the signaling cascade, could be used as a surrogate marker for tumor growth
inhibition (pSMAD); pSMAD was modelled using an indirect response model
depending on the PK of galunisertib. Studies in rat described by Gueorguieva
et al. (2014) suggested a toxicity limit on the exposure, measured by the area
under the curve (AUC), to prevent cardiovascular side-effects. Finally,
allometric scaling combining data from mouse, rat and dog was used to
extrapolate PK parameters to humans, assuming the same PD relationship with
the biomarker. The predicted dose proved safe and showed clinical benefits in
the FIH trial reported by Rodon et al. (2015).

We used this setting to design the simulation study, simplifying the PK model
to a one-compartment model as in Lestini, Dumont, and Mentré (2015). For a
given study, let $\mathcal{D}=(d_{1},d_{2},...,d_{J})$ be the set of $J$
administered doses.

###  3.2 Toxicity Data Simulation

In our example, surrogate toxicity data for a given study occurs if the value
of a function of the AUC of the drug in blood plasma exceeds a given
threshold.

The concentration $C(t)$ at time $t$ after oral administration of the dose
$d_{j}$, $j\in\\{1,...,J\\}$, at time 0, is first simulated using a first-
order absorption linear one compartment model

$C(t)=\dfrac{d_{j}}{V}\times\dfrac{k_{a}}{k_{a}-\frac{CL}{V}}(e^{-\frac{CL}{V}t}-e^{-k_{a}t})$
(3)

where $k_{a}$ is the absorption rate constant for oral administration, $CL$ is
the clearance of elimination, and $V$ is the volume of distribution. Both the
clearance of elimination and the volume of distribution are assumed to follow
log-normal distributions within a population: $CL\sim
LN(\log(\mu_{CL}),\omega_{CL})$ and $V\sim LN(\log(\mu_{V}),\omega_{V})$.

Then, $\text{AUC}=\int_{0}^{+\infty}C(t)dt=d/CL$, and binary toxicity is
generated for the subject at the given dose if $s(\text{AUC})\geq\tau_{T}$
where $s(\text{AUC})=\alpha\text{AUC}$, $\alpha\sim
LN(\log(\mu_{\alpha}),\omega_{\alpha})$ and $\tau_{T}$ is a threshold. In
summary, the probability of toxicity for dose $d$ is calculated as

$\displaystyle p_{T}(d)$ $\displaystyle=P(s(\text{AUC})\geq\tau_{T}|d)$
$\displaystyle=\Phi\Big{(}\dfrac{\log(d)-\log(\tau_{T})-\log(\mu_{CL})}{\sqrt{\omega_{CL}^{2}+\omega_{\alpha}^{2}}}\Big{)}$
(4)

where $\Phi$ is the cumulative function of a standard normal distribution
(Ursino et al., 2017). An unacceptable risk was defined as the probability of
exceeding an AUC of $\tau_{T}$ mg.L-1.h being greater than $p_{T}$. This
formula allows in the second step (see subsection 2.2) to deduce the
distribution of the MTD from the chosen values of the probability of toxicity
of this dose $p_{T}(\text{MTD})$ and the threshold $\tau_{T}$, as well as the
posterior distributions of $\mu_{CL}$, $\omega_{CL}$ and $\omega_{\alpha}$.
For this paper, we consider that $\alpha$ is constant, and thus
$\omega_{\alpha}=0$.

In our specific example, the toxicity threshold is related to the total AUC
(i.e. cumulative drug exposure) and therefore directly to the
pharmacokinetics. Other toxicity measures could be considered instead, either
continuous (high temperature, elevated liver damage marker, maximum
concentration, etc.) or of a different nature (occurrence of an adverse event,
etc.) which we would then model through the probability of occurrence.

We assume a proportional error of 20% for the measurements of the
concentration of the drug in blood plasma defined as $\tilde{C}\sim
LN(\log(C),0.2)$ where $C$ represents the true values of the measurements and
$\tilde{C}$ the measured values.

###  3.3 Scenarios

We simulate galunisertib (LY2157299) data corresponding to the dose ranges
used in the vivo studies. To keep the simulation study simple, we only
simulated three sequential studies, with mice, rats and dogs, respectively.
Mice ($k=1$), rats ($k=2$) and dogs ($k=K=3$) are assumed to be respectively
exposed to oral doses of either 10 ($n=15$), 30 ($n=15$), 50 ($n=15$), 75
($n=15$), 100 ($n=15$), 150 ($n=15$), 300 ($n=15$) mg/kg (for mice), 10
($n=8$), 30 ($n=8$), 50 ($n=8$), 100 ($n=8$), 300 ($n=8$) mg/kg (for rats) or
2 ($n=6$), 10 ($n=6$), 30 ($n=6$), 50 ($n=6$), 300 ($n=6$) mg/kg (for dogs).
For each of their doses, 3 mice are assumed to be sacrificed at the following
times: 15 min, 45 min, 2h, 5h and 15h; while all rats and dogs are
respectively sampled at either 15 min, 1h, 2h, 3h40 and 10h (for rats) or 10
min, 1h40, 2h, 5h30 and 15h (for dogs).

Toxicity data for human in scenario 1 (baseline scenario) are first simulated
from models 3 and 4; the parameters were chosen so that the simulated data
follow similar distributions to those produced in Lestini et al. (2015) (see
Figure 2). Then, $\mu_{\text{CL}}$ and $\mu_{\text{V}}$ parameters for dog,
rat and mouse in scenario 1 are calculated from human parameters using the
following allometric scaling formulas:

$\mu_{CL}^{sp2}=\mu_{CL}^{sp1}\times\Big{(}\dfrac{W^{sp2}}{W^{\text{sp1}}}\Big{)}^{0.75}$
(5)
$\mu_{V}^{sp2}=\mu_{V}^{{sp1}}\times\Big{(}\dfrac{W^{sp2}}{W^{{sp1}}}\Big{)},$
(6)

where $sp1$ and $sp2$ are respectively the human and the animal species
considered, and $W$ is the weight of the human or animal species. Note that
these formulas hold for conversions between all animal species. All the other
parameters (absorption rate constant $k_{a}$, variances of the PK parameters)
for dog, rat and mouse’s scenario 1 are chosen equal to those of human (see
Table 1). For scenario 1, using the extrapolated parameters of PK model from
dog, rat and mouse to human allows to predict accurately the MTD to be
approximately equal to 502 mg. Scenario 2 is based on an inaccurate
extrapolation of the parameter clearance $CL$, affecting toxicity (via AUC),
for one species. Other two scenarios (scenario 3 and 4) accounting for MED
estimation along with the MTD are given in Web Appendix B. Figure 2 gives the
probability of toxicity according to the extrapolated dose for scenarios 1 and
2 and for all species.

Figure 2: (a) Mean concentration of the drug versus time for several doses for human in scenario 1; (b) Probability of toxicity according to the extrapolated dose for scenarios 1 and 2 and for all species. Red horizontal lines depict the probability of toxicity $p_{T}$ threshold. The probability of toxicity for a dose should be lower than the threshold $p_{T}$. This figure appears in color in the electronic version of this article. Table 1: Simulation parameters for PK model and approximate extrapolated MTD for all species and scenarios 1 and 2. $W$: Weights; $k_{a}$: Absorption rate constant for oral administration; $(\mu_{CL},\mu_{V})$: Exponentials of the respective means of the logarithms of the distributions of the clearance elimination and the volume of distribution; $(\omega_{CL},\omega_{V})$: Respective standard deviations of the logarithms of the distributions of the clearance elimination and the volume of distribution; $\sigma_{C}$: Standard deviations of the logarithm of the distribution of the measured concentration of the drug in blood plasma; $\tau_{T}$: AUC threshold; $p_{T}$: Probability of toxicity threshold; MTD: Maximum tolerated dose. Parameters | Human | Dog | Rat | Mouse
---|---|---|---|---
|  |  | Sc.1 | Sc.2 |
$W$ (kg) | 70 | 10 | 0.15 | 0.15 | 0.025
$k_{a}$ (h-1) | 2 | 2 | 2 | 2 | 2
$\mu_{CL}$ (L.h-1) | 40 | 9.3 | 0.40 | 1.59 | 0.11
$\mu_{V}$ (L) | 100 | 14 | 0.21 | 0.21 | 0.04
$\omega_{CL}$ | 0.7 | 0.7 | 0.7 | 0.7 | 0.7
$\omega_{V}$ | 0.7 | 0.7 | 0.7 | 0.7 | 0.7
$\sigma_{C}$ | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
$\tau_{T}$ (mg.L-1.h) | 22.6 | 22.6 | 22.6 | 22.6 | 22.6
$p_{T}$ (%) | 20 | 20 | 20 | 20 | 20
$\mu_{\alpha}$ | 1 | 1 | 1 | 1 | 1
$\omega_{\alpha}$ | 0 | 0 | 0 | 0 | 0
Extrapolated | 502 | 502 | 504 | 2002 | 531
MTD (mg) |  |  |  |  |

###  3.4 Implementation

For each scenario, 500 datasets are simulated. We assume that equations 3, 4,
5 and 6 describing toxicity data generation and allometric scaling formulas
are known as well as $\tau_{T}$, $p_{T}$, $\mu_{\alpha}$, $\omega_{\alpha}$
and $W$. The other model parameters are estimated. At step 1 of our
methodology, for each simulated dataset and for each animal species, we fit
the previous mixed-effects model using Hamiltonian Monte Carlo (Betancourt,
2018; Thomas and Tu, 2021). For the dog and the rat, random effects
(intraindividual variability - IIV) are estimated on $CL$ and $V$, therefore
$\mathbf{y}_{k,i}=\mathbf{\tilde{c}}_{i}$, the vector of concentrations
measured of the $i$th subject of study $k$, $k=2,3$, and
$\mathbf{f}_{k}(.):\tilde{c}_{ij}=\log\left(C(t_{ij},\boldsymbol{\theta}_{k}=\\{k_{a,k},\mu_{CL,k},\mu_{V,k},\omega_{CL,k},\omega_{V,k}\\})\right)+\epsilon$
with $\epsilon\sim N(0,\sigma_{C})$ where $k=2,3$ and $j$ indexes the number
of time points for each subject, while for the mouse, random effects are only
estimated on $CL$ since there is only one data point per mouse (that is
$\mathbf{f}_{1}(.):\tilde{c}_{i}=\log\left(C(t_{i},\boldsymbol{\theta}_{1}=\\{k_{a,1},\mu_{CL,1},\mu_{V,1},\omega_{CL,1},\\})\right)+\epsilon$
and $\mathbf{y}_{1}=\tilde{c}_{i}$). In this situation,
$\boldsymbol{\theta}_{1}$, $\boldsymbol{\theta}_{2}$ and
$\boldsymbol{\theta}_{3}=\boldsymbol{\theta}_{2}$ share several parameters
with the same role, therefore extrapolation formulas (eq. 5 and 6 are adopted
as $\mathbf{g}_{k}$) can be used to set prior distributions when stepping from
one study to the next one. For the sake of simplicity, in the following we do
not use the sub-notation $k$ for the parameters, since it will be clear from
the contest to which study the parameters are linked to.

For the next steps of the analysis based on mouse data, we make the assumption
that $\omega_{V}=0.7$ for mouse. As sensitivity analysis, we also consider the
assumptions that $\omega_{V}=0.4$ or $\omega_{V}=1$ for mouse. In this
example, $d_{1,k}(\boldsymbol{\theta}_{h,k})$ is the solution of eq. 4 when
solving for $d$ while setting $p_{T}(d)=0.2$. We choose to perform $L=1000$
MCMC runs for steps 2 to 4 of the methodology when approximating $D_{k}$ and
$D^{*}_{k}$ distributions. In step 2, each one of the $L$ values of clearance
and volume is extrapolated from animal species to human using the previous
formula, while other parameters of the models are kept identical.

In step 3, in practice, the Hellinger distance is approximated using the
rectangle method:

$\left[0.5(b-a)/M\times\sum_{m=0}^{M-1}\left(\sqrt{\pi^{post}_{k_{1}}(d^{*}_{r,m})}-\sqrt{\pi^{post}_{k_{2}}(d^{*}_{r,m})}\right)^{2}\right]^{\frac{1}{2}},$

where $a=\underset{m\in\\{0,...,M-1\\}}{\min}d^{*}_{r,m}$,
$b=\underset{m\in\\{0,...,M-1\\}}{\max}d^{*}_{r,m}$. To compute the accuracy,
as proposed in section 2.3, for each scenario, the animal species that should
be kept for the final step (because they are consistent) have to be defined.
The extrapolated MTDs to humans from Table 1 are approximately equal to 502 mg
(the true MTD value for humans) for all animal species (mouse, rat and dog) in
scenario 1, and only for mouse and dog in scenario 2. Therefore, for the MTD,
the binary variable “true response” is set to 1 for all comparisons between
animal species in scenario 1, and only for the comparison between mouse and
dog in scenario 2; Otherwise, for all comparisons including rat in scenario 2,
it is set to 0. To note, in the case of MTD, all scenarios are considered as
relevant for choosing the Hellinger distance threshold. It is not the case for
the MED, as described in Web Appendix B. We used the ad hoc following
algorithm. If the three computed Hellinger distances are lower or equal to the
selected threshold, the three studies are selected. All three studies are
selected also if at least two measures are lower than the threshold. If only a
value is lower the threshold, the two corresponding studies are selected.
Finally, if all three Hellinger distances exceed the threshold the results of
the most relevant animal species are considered for the following step. In
this work, we assume that the most relevant animal species for galunisertib is
the dog.

For step 4, posterior distributions can be computed using the kernel density
estimator,
$\pi^{post}_{k^{\prime}}(d_{r})=(Lh_{k^{\prime}})^{-1}\sum_{l=1}^{L}\kappa\Big{(}(d_{r}-d_{{r,k^{\prime}},l})/h_{k^{\prime}}\Big{)},$
where $\kappa$ is the kernel (the standard normal density in this paper) and
$h_{k^{\prime}}>0$ is the bandwidth.

All analyses are performed using R software version 4.04 with Stan (Stan
Development Team, 2021) package rstan version 2.21.2. In rstan, 3 chains, a
burn-in of 3000 and 6000 other iterations are used and, as a convergence
criterion, Gelman and Rubin (1992)’s potential scale reduction factor Rhat =
1.

##  4 Simulation Results

###  4.1 Illustration

To illustrate the steps of our methodology, let us first focus on the results
from a single simulated dataset. Figure 3 shows the concentration of the drug
versus time for each animal of scenarios 1 and 2 for this dataset.

Figure 3: Simulated individual concentration of the drug versus time for one
simulated dataset for mouse (a), rat in scenario 1 (b), rat in scenario 2 (c)
and dog (d). This figure appears in color in the electronic version of this
article.

In step 1 of our methodology, the Bayesian models for mouse data are first
fitted using the non-informative prior distributions given in Web Appendix C.
The estimated posterior means for mouse of $\mu_{CL}$ and $\mu_{V}$ are equal
to 0.0961 L.h-1 and 0.0508 L (results shown in Web Appendix C). Using
extrapolation formulas 5 and 6 from mouse to rat, we obtain the values of
0.368 L.h-1 and 0.305 L that are used as means of the prior distributions of
the clearance and the volume for rat. Similarly, the estimated posterior means
for rat of $\mu_{CL}$ and $\mu_{V}$ are equal to 0.464 L.h-1 and 0.216 L for
scenario 1 (respectively 1.45 L.h-1 and 0.581 L for scenario 2) and we use the
extrapolated values of 10.8 L.h-1 and 14.4 L (respectively 33.8 L.h-1 and 38.7
L) as means of the prior distributions of the clearance and the volume for
dog.

The extrapolated MTD distributions from mouse, rat and dog to human resulting
from step 2 are drawn in the first line of Figures 4 for these scenarios and
each animal species as dotted lines.

Figure 4: Maximum tolerated dose (MTD) distributions for one simulated dataset
in scenarios 1 and 2, for the Bayesian approach. Extrapolated (animals to
human) MTD distributions (from Step 2) and final predicted MTD distribution
(from Step 4) in the dataset from scenario 1 (a) and from scenario 2 (b);
transformed extrapolated MTD distributions and Hellinger distances (from Step
3) in the dataset from scenario 1 (c) and from scenario 2 (d). The Hellinger
distances equal to 1 are due to approximation in computation.

In step 3, we transform the distributions from step 2 according to equation 1
so that the Hellinger distances between animal species do not depend on the
difference between sample size in each study. The transformed extrapolated MTD
distributions and the corresponding Hellinger distances are given in the
second line of Figures 4 for scenarios 1 and 2 and each animal species. For
the fourth and final step, we set the threshold for the Hellinger distance at
0.5 for MTD based on the sensitivity analysis described in subsection 4.2. For
scenario 1, Hellinger distances are equal to 0.51 for mouse versus rat, 0.36
for mouse versus dog and 0.19 for rat versus dog. Two of these distances are
less than 0.5 so all animal species are selected. For scenario 2, the two
distances including rat are greater than 0.5 (mouse versus rat: 1; rat versus
dog: 1) but the last distance is less than 0.5 (mouse versus dog: 0.36) so
only mouse and dog are selected.

In step 4, the final predicted MTD distributions are computed by merging the
extrapolated dose distributions between the animal species selected in the
previous step, and overlayed on the plots in the first line of Figures 4 as a
solid line. These final distributions have smaller variances than the
distributions from each animal species separately. The MTDs are finally
estimated to 465 mg and 437 mg for scenarios 1 and 2 respectively, as the
posterior means of these distributions.

###  4.2 Using Hellinger Distance to Check Extrapolability from Animal Species
to Human

We then applied our methodology to the scenarios with 500 replications each.
Figures 5 and 5 show boxplots of the Hellinger distances between two species
for the Bayesian approach in the different scenarios, for the MTD. As seen in
these figures, the Hellinger distance easily distinguishes animal species
between those that show similarities in extrapolated toxicity and those that
show dissimilarities.

Figure 5: Hellinger distance of the transformed predicted MTD distributions in
humans between animal species for scenario 1 (a) and scenario 2 (b) for the
Bayesian approach

over 500 replications (under the assumption that $\omega_{V}=0.7$ for mouse);
Accuracy by Hellinger distance threshold for MTD in scenarios 1 and 2 for the
Bayesian approach over 500 replications (c). MTD: Maximum tolerated dose. This
figure appears in color in the electronic version of this article.

Indeed, for scenario 1 for which the extrapolation is correct from all animal
species to human, the Hellinger distances between animal species for the MTD
distributions are lower than 0.5 in more than 73% of cases. This indicates
consistent results across all animal species and therefore the possibility of
using them all to derive human MTDs.

For scenario 2 that uses an inaccurate extrapolation of toxicity model
parameters for rat, the Hellinger distances between mouse and rat, and between
rat and dog are close to 1 but again lower for mouse versus dog. This
indicates MTD distributions are consistent between mouse and dog, but not for
rat.

As shown in Figure 5, as Hellinger distance threshold increases, so does
accuracy for MTD. For MTD, we set the threshold at 0.5. We decided to stop the
accuracy evaluation at the threshold of 0.5 since the Hellinger distance is
bounded between 0 and 1, and going beyond the median value did not seem
relevant to us.

###  4.3 Final Estimations of MTD in Humans

As shown in Figure 6, for scenario 1 for which data from all animal species
are often used to estimate the final MTD, the MTD is correctly estimated to
515 mg (standard deviation - sd - equal to 48 mg) by the Bayesian approach
(close to the true value of 502 mg). For scenario 2, the MTD is slightly
overestimated to 561 mg (sd: 296 mg).

The length of the equal tails 95% credible interval (CrI95) is considerably
greater when only the dog results are used (that is the standard approach) to
calculate the MTD than when using the proposed method (see Figure 6).
Actually, the mean of the CrI95 lengths is around 300 mg (sd: 47 mg) using
only dog results compared to the values of 165 mg and 242 mg (sd equal to 26
mg and 153 mg) for respectively scenarios 1 and 2 using the proposed approach.

Figure 6: Estimated MTD in humans (a) and the length of the 95% credibility
interval (CrI95) (b) for scenarios 1 and 2 for the Bayesian approach and using
the standard approach (that is, only dog data) over 500 replications, under
the assumption that $\omega_{V}=0.7$ for mouse. MTD: Maximum tolerated dose.
This figure appears in color in the electronic version of this article.

The results of the sensitivity analysis for $\omega_{V}=0.4$ and
$\omega_{V}=1$ for mouse, given in Web Appendix D, do not differ from those
obtained for $\omega_{V}=0.7$ .

##  5 Discussion

We proposed a Bayesian approach for multisource data integration, that was
tailored, in this work, to dose extrapolation from preclinical to clinical
research. In particular, steps 3 and 4 require methodological innovations such
as a way to normalize the quantity of information of a distribution come from
longitudinal data and the modified Bayes formulas. The new framework allows to
better use all available information compared to the standard methods,
reducing uncertainty in the predictions and potentially leading to a more
efficient dose selection. In preclinical setting, this new framework can be
seen as a Bayesian generalisation of the Food and Drug Administration (2005)
(FDA) guidelines. Furthermore, the framework is very flexible. Submodels
(linear, generalised, mixed-effect, etc.) can be specified according to the
study outcomes and could be different between studies.

To evaluate the performance of the approach, we used a case study inspired by
the preclinical development of galunisertib. In this example, only data from
in vivo studies were simulated. However, it is possible to include data from
in vitro studies as well, by simply considering the appropriate extrapolation
formulae to humans, that is working on areas or cells number. In our example,
only some estimates of PK fixed effects are extrapolated from animals to
humans using allometric formulas. Indeed, to our knowledge, there is no
analytical formula to extrapolate the parameters of the PD models, nor to
anticipate interindividual variability between species in this setting.
However, mouse or rat, for example, which are specially bred for preclinical
studies (controlled lines, similar age, same diet, same physical exercises, …)
could be more homogenous than healthy volunteers or patients. In our
simulations, a sensitivity analysis with lower variances for these species did
not indicate that the uncertainty on the variabilities was an issue, but we
could also incorporate expected changes across species in the extrapolation
step, for instance to integrate differences in disease course or mechanistic
pathways.

In our simulation setting, the theoretical MTD (i.e. the MTD calculated from
the true values of the model parameters) extrapolated to humans shown in Table
1 are not exactly the same between animal species, even for scenario 1, but,
in general, very close each other. Therefore, this imperfection of the
simulation scenarios should be kept in mind when interpreting the results.
However, in real life, the extrapolation of a dose between two species is
always subject to a margin of error.

In the third step of our methodology, a rule/algorithm must be defined to
check the commensurability between distributions and to select the ones to
carry to the next step. In our example, since we have three studies, we
proposed a simple algorithm to do it. However, when more than three studies
are available, clustering algorithms based on a distance matrix can be
investigated to select studies for the last step. Accuracy computation, as
done in this work, or ROC (Receiver Operating Characteristic) curves can be
adopted to calibrate algorithm thresholds and could be applied directly on
distances results (as in this work) or on the final results from the
algorithm. Moreover, another distribution distance could be used instead of
the Hellinger distance. We suggested it due to its symmetric propriety and,
since it is bounded, it is easily interpretable.

To run simulations, we decided to keep the results involving the more numerous
cluster, that is, in this case, if at least two species were selected at the
coherence step. In reality, if it is well-known that the results on a species
are really closed/related to human ones, this species can be selected as
default and only species clustered with it will be carried on at the merging
step.

Some additional extensions will be investigated. It could be interesting to
include external sources of information, such as expert opinion or literature
data. Such data could be integrated into the weakly informative prior
distributions of the parameters in step 1, or as additional dose distributions
in the final step. External information should be introduced wisely, since if
used multiple times it can impact the results of eq. 2. Indeed, when only
weakly informative prior distribution, carrying on a small and negligible ESS,
are adopted, the final impact on eq. 2 could be neglected. Otherwise, further
studies on how discounting embedded prior distribution are needed. Another
perspective would be to handle weighted posterior distributions through the
Hellinger distance and the merging formula, in order to give more weight to
certain results, for example to account for larger sample sizes or on previous
knowledge on human similarity. Following this idea, several doses can be
proposed regarding the number of posterior distribution modes. Then, novel
dose-finding methods should be developed to take into account a possible
change in the dose panel evaluated in the trial. In addition, in the
methodology, no uncertainty is attributed to the extrapolation approach
itself. However, it is possible that a lack of consistency is observed because
of inappropriate extrapolation rather than due to mismatch between preclinical
data and human. Thus, our framework could be expanded with different
extrapolation strategies (allometry, PBPK, etc.) that then will be used at the
coherence step, or distribution can be assumed for extrapolation parameter
formulas, carrying on this uncertainty during the whole extrapolation process.

Another straightforward extension is using this framework to compute prior
distributions for human model parameters used in the future (human) dose-
finding trial. Indeed, while our work was focused on doses estimation, we can
work at parameter level, that is clustering and merging parameter posterior
distributions. This will give, as results, a distribution for each parameter,
that could be used as prior distribution on human model parameters. ESS
discounting or dynamic borrowing may worth to be studied.

In conclusion, we proposed a new framework to facilitate preclinical to
clinical extrapolation in four main steps. Each step could be customized
(models, algorithms, extrapolation formulas and hypotheses, etc.) according to
the studies the researchers are working on.

#### Acknowledgements

This work is part of the European FAIR project that has received funding from
the European Union’s Horizon 2020 research and innovation program under grant
agreement N${}^{\text{o}}$ 847786\. Sandrine Boulet and Moreno Ursino made
equal contributions and are co-first authors. Emmanuelle Comets and Sarah
Zohar made equal contributions and are co-last authors.

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#### Supporting Information

Web Appendices, Tables and Figures, referenced in Sections 2, 3 and 4, are
available with this paper.