Bayesian framework for multi-source data integration-Application to human extrapolation from preclinical studies

Abstract

In preclinical investigations, for example, 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 all preclinical data through inferential procedures can be particularly interesting in obtaining a more precise and reliable starting dose and dose range. Our objective is to propose a Bayesian framework for multi-source data integration, customizable, and tailored to the specific research question. We focused on preclinical results extrapolated to humans, which allowed us to predict the quantities of interest (e.g. maximum tolerated dose, etc.) in humans. We build an approach, divided into four 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. Our approach allows us 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 translational

1. Introduction

At the beginning of the development of a new element, object, compound, etc. (depending on the field), preliminary knowledge of its properties is 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.

When studies are performed sequentially and the endpoints can be linked (via mathematical transformations, for example), the inference can be easily placed under a Bayesian framework that can update posterior knowledge each time new data becomes available. For example, in the preclinical and clinical context, La Gamba et al.¹ sequentially introduced 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 Hampson² used 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.³ via a meta-analytic approach. Another example is that of Petit et al.⁴ who propose a method allowing extrapolation and bridging of adult data in the early-phase dose-finding pediatric studies.

Application When stepping from preclinic to the clinic, several studies have usually been performed, and one of the key questions regards the computation of the starting dose (or the regime) in humans. The Food and Drug Administration (FDA)⁵ 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; and (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. Indeed, poorly estimating a dose range or starting dose for FIH studies may increase the time, the cost and the number of healthy volunteers or patients required. For instance, too low starting doses may lead to hardly detectable exposures during the initial escalation steps and potential difficulties in achieving maximum tolerated dose (MTD) while too large starting dose may result to safety concerns. In this situation, to improve the clinical development process, the European Medicines Agency (EMA)⁶ 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 by Shen et al.⁷ 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.⁸ 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.⁹

Aim The objective of this article is to propose a Bayesian framework accounting for differences and similarities between preclinical studies, to use all relevant information. More precisely, our approach needs to consider multi-source data (cells, mouse, etc.) and be able to better use all available information compared to the standard methods to predict the quantities of interest (e.g. MABEL, NOAEL, minimal effective dose-MED, MTD, etc.) in humans.

A classic Bayesian framework for a sequence of preclinical studies, where the posterior distributions of the previous study are used to set prior distributions for the new study, requires the use of the same mathematical models (or sub-models) for each criterion (outcome, marker) between studies. Moreover, in the simple process, not all previous information should 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. Furthermore, when robustification approaches¹⁰ are used to deal with possible differences between species, which are not taken into account by extrapolation formulas, 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.

Instead, our framework, in its sequential way and final Bayesian posteriors meta-analysis, preserves the individual studies in terms of results, endpoints end model used. A fully joint model could be adopted only in specific cases, when the same model (or submodel) is shared between studies, that is, usually unfeasible. Moreover, it could come with potential computation instabilities. Instead, using this framework, we expect to reduce uncertainty in the predictions and potentially leading to a more efficient dose selection while keeping the full process feasible. Within the framework, we propose some methodological innovations, such as a way to normalize the quantity of information of a distribution coming from longitudinal data and a Bayesian formula that encompasses the Bayes theorem in special cases. 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- $β$ signaling which reduces tumor growth, where extrapolation approaches were applied during the preclinical studies based on markers of toxicity and efficacy.¹¹

This article is organized as follows. 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 analyzed 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 and in Algorithm 1 in Section 1.1 in the Supplemental Material): (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 the Supplemental Material (Section 1.2), 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.

2.1. First step: Parameters estimation

For $k \in {1, \dots, K}$ , let $y_{k}$ denote the outcomes measured in the $k$ th study. We suppose that $y_{k}$ are related to each other between studies (they could be exactly the same outcome for each study), or that, if different, they can be used to compute the doses of interest, $d_{r}$ , $r \in {1, \dots, R}$ with $R$ the number of doses of interest (e.g. MABEL, NOAEL, MED, MTD, etc.).

For each study $k$ , Bayesian fixed- or mixed-effect models, $f_{k} (y_{k}, θ_{k}$ ), (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 $θ_{k}$ . The functions $f_{k} (.)$ can differ between studies, as well as the vector $θ_{k}$ 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 $θ_{k - 1}$ can be related to an element of $θ_{k}$ , the prior distributions are computed using the estimated posterior means of the model parameters of the previous study, $k - 1$ , (extrapolated between studies, 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. The question of how to specify a more sophisticated prior distribution for $θ_{k}$ using information contained in $θ_{k - 1}$ parameter(s) of the precedent study, especially when the studies collect data on different outcomes, and/or had been conducted following different experimental designs, is out of the scope of this paper. However, for instance, the interested reader could be inspired by the work of Duputel et al.¹² that, in one proposed model, developed a “mixed” likelihood in the case of different but related endpoints, that is, parametrizing the survival binary rate at a fixed time point using a survival parametric curve, or by Alt et al.¹³ where a scale transformed power prior approach is proposed when the endpoint can be related via parametric formulas. In any case, non-informative prior or weakly informative prior are necessary at this stage to not count multiple times the same information in the last framework step. Indeed, for such prior distributions, the effective samples sizes (ESSs),¹⁴ and hence the weights of each preclinical study, are similar. An example using beta-binomial model is given in Section 1.3 in the Supplemental Material.

To summarize, at the end of the process, each study $k \in {1, \dots, K}$ yields a posterior distribution for each element of each $θ_{k}$ .

2.2. Second step: Extrapolation to human

While in the previous step, extrapolation was used between preclinical studies, at this step, we focus on human extrapolation. For each study $k \in {1, \dots, K}$ , we extrapolate the preclinical results (the estimated parameters) to humans using predetermined formulas $g_{k} (.)$ (transformations), $θ_{k}^{human} = g_{k} (θ_{k})$ , where $θ_{k}^{human}$ represents the vector of extrapolated human parameters from the $k$ th study. As stated in the previous section, these formulas could also be used in the previous step to extrapolate parameter values between studies.

In the Bayesian setting, the transformation is applied directly on posterior distributions of parameters to obtain the posterior distributions of the extrapolated parameters (via the random variable transformation theorem). Then, the doses of interest, $d_{r} = d_{r, k} (θ_{k}^{human})$ , for human, where $d_{r, k} (.)$ represents the function that links the vector $θ_{k}^{human}$ 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 $d_{r}$ , $π_{k}^{p o s t} (d_{r})$ , are deduced (via the random variable transformation theorem or approximated via Monte Carlo methods). Note that taking account for the uncertainty associated with the fixed parameter used in extrapolation scaling is not the main goal of this article, but could be the topic of further investigations.

2.3. Third step: Commensurability checking and posterior distributions selection

In this step, we aim to compare the $K$ predicted human dose distributions, $D_{k, r} \sim π_{k}^{p o s t} (d_{r})$ , for each $r$ , and select the ones which seems the most similar to each other. In real-life practice, it can be more appropriate, for instance, to select first the predicted human dose distribution(s) from the most relevant study (or studies), and then to select the other distributions that are the most similar to them. This step can easily be adapted to this particular case, so we do not develop this case in this article.

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.¹⁵ This method requires the re-analysis of one of the two experiences and the decision on the discounting factor: it could be straightforward in the case of fixed effect models, but not really for mixed effect model involving longitudinal studies. Inspired by the ESS notion,¹⁴ 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. For example, in our case study,

D_{k, r}^{*} = \frac{max_{h \in K} S_{h, r}}{S_{k, r}} \log (D_{k, r}) + (1 - \frac{max_{h \in K} S_{h, r}}{S_{k, r}}) M_{k, r} \sim π_{k}^{p o s t} (d_{r}^{*})

(1)

for

k \in {1, \dots, K}

, where

M_{k, r} = E (\log D_{k, r})

and

S_{k, r}^{2} = V a r (\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, for our specific example, the logarithmic transformation of the dose is first used to move to a situation where the

D_{k, r}^{*}

distribution could be assumed as Gaussian; however, in other cases, other transformations could be more suitable. According to Morita et al.,¹⁴ 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 is equal (if we consider them as prior distribution on a “Gaussian dose model”). Furthermore, for

k \in {1, \dots, K}

E (D_{k, r}^{*}) = M_{k, r}

and

V a r (D_{k, r}^{*}) = max_{h \in K} S_{h, r}^{2}

Once the posteriors are comparable, the commensurability of the modified distributions $π_{k}^{p o s t} (d_{r}^{*})$ between studies $k_{1} \in {1, \dots, K}$ and $k_{2} \in {1, \dots, K}$ , $k_{2} \neq k_{1}$ , is checked using the Hellinger distance for each $r$ , $H_{r} (D_{k_{1}}, D_{k_{2}}) = {[\frac{1}{2} \int (\sqrt{π_{k_{1}}^{p o s t} (d_{r}^{*})} - \sqrt{π_{k_{2}}^{p o s t} (d_{r}^{*})})^{2} d d_{r}^{*}]}^{\frac{1}{2}}$ .¹⁶ 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 Section 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. More precisely, we 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. Then, via simulation, we compute the Hellinger distances between the studies, and, for each possible threshold on the Hellinger distance, 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 (i.e. true positives and true negatives) among the total number of cases, $acc = T P + T N / P + N, ~where~ P = T P + F N ~and~ N = T N + F P$ , where $T P$ , $T N$ , $F N$ , $F P$ , $P$ , and $N$ are, respectively, the number of true positives, true negatives, false negatives, false positives, positives, and negatives. Examples are given in Section 4.1 and, in the Supplemental Material (Section 3.2).

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 $K = {1, 2, \dots, K}$ be the set of indices of the preclinical studies. Denoting $K^{'} \subset K$ the subset of studies selected at the previous step, we merge the extrapolated dose distributions $π_{k^{'}}^{p o s t} (d_{r})$ ( $k^{'} \in K^{'}$ ) using an extension of the Bayes formula to obtain the final predicted dose distribution $d_{r}$ in humans:

π (d_{r}) \propto \prod_{k^{'} \in K^{'}} π_{k^{'}}^{p o s t} (d_{r})

(2)

Equation (2) exists when the

Card (K^{'})

selected distributions share the same support, or at least part of it. It excludes situations where the product term is zero almost surely on the domain, that is when at least one distribution is zero 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 equation (2) is Lebesgue integrable if the

Card (K^{'})

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 the Supplemental Material, Section 1.3). 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

π_{K}^{p o s t} (θ) \propto π^{p r i o r} (θ) \prod_{k = 1}^{K} L (θ | {data}_{k})

. If

π^{p r i o r} (θ)

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 equation (2) can be seen as an extension of the Bayes formula. Because the information from informative prior distributions will be counted

Card (K^{'})

times, we suggest using non-informative prior distributions, as the amount of prior information can be considered negligible also after the application of equation (2). Equation (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, equation (2) becomes its posterior distribution. Here we use the original dose posterior distributions, not the transformed ones by equation (1), consequently, no predicted human dose distributions is assumed to be Gaussian.

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 PK results. Therefore, only for the sake of simplicity, $r = 1$ . However, simulation focusing on MED estimation (along with the MTD, therefore, $r = 2$ ) using PK/PD modeling are given in the Supplemental Material (Section 2).

3.1. Galunisertib case tudy

Galunisertib is a transforming growth factor (TGF)- $β$ 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 TGF- $β$ receptor, blocking TGF- $β$ 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.^17,11 Studies in mouse described by Bueno et al.¹⁷ 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 modeled using an indirect response model depending on the PK of galunisertib. Studies in rat described by Gueorguieva et al.¹¹ 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.¹⁸

We used this setting to design the simulation study, simplifying the PK model to a one-compartment model as by Lestini et al.¹⁹ For a given study, let $D = (d_{1}, d_{2}, \dots, d_{J})$ 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 $d_{j}$ , $j \in {1, \dots, J}$ , at time 0, is first simulated using a first-order absorption linear one compartment model

C (t) = \frac{d_{j}}{V} \times \frac{k_{a}}{k_{a} - \frac{C L}{V}} (e^{- \frac{C L}{V} t} - e^{- k_{a} t})

(3)

where

k_{a}

is the absorption rate constant for oral administration,

C L

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:

C L \sim L N (\log (μ_{C L}), ω_{C L})

and

V \sim L N (\log (μ_{V}), ω_{V})

Then, $AUC = \int_{0}^{+ \infty} C (t) d t = d / C L$ , and binary toxicity is generated for the subject at the given dose if $s (AUC) \geq τ_{T}$ , where $s (AUC) = α AUC$ , $α \sim L N (\log (μ_{α}), ω_{α})$ and $τ_{T}$ is a threshold. In summary, the probability of toxicity for dose $d$ is calculated as

p_{T} (d) = P (s (AUC) \geq τ_{T} | d) = Φ (\frac{\log (d) - \log (τ_{T}) - \log (μ_{C L})}{\sqrt{ω_{C L}^{2} + ω_{α}^{2}}})

(4)

where

Φ

is the cumulative function of a standard normal distribution.²⁰ An unacceptable risk was defined as the probability of exceeding an AUC of

τ_{T}

mgL⁻¹h being greater than

p_{T}

. This formula allows in the second step (see Section 2.2) to deduce the distribution of the MTD from the chosen values of the probability of toxicity of this dose

p_{T} (MTD)

and the threshold

τ_{T}

, as well as the posterior distributions of

μ_{C L}

ω_{C L}

, and

ω_{α}

. For this article, we consider that

α

is constant, and thus

ω_{α} = 0

In our specific example, the toxicity threshold is related to the total AUC (i.e. cumulative drug exposure) and, therefore, directly to the PK. 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 $\tilde{C} \sim L N (\log (C), 0.2)$ , where $C$ represents the true values of the measurements and $\tilde{C}$ represents 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 (study $k = 1$ ), rats (study $k = 2$ ), and dogs (study $k = K = 3$ ), respectively. Mice, rats, and dogs 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, three mice are assumed to be sacrificed at the following times: 15 min, 45 min, 2 h, 5 h, and 15 h; while all rats and dogs are, respectively, sampled at either 15 min, 1 h, 2 h, 3 h40, and 10 h (for rats) or 10 min, 1 h40, 2 h, 5 h30, and 15 h (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 by Lestini et al.¹⁹ (see Figure 2). Then, $μ_{CL}$ and $μ_{V}$ parameters for dog, rat and mouse in scenario 1 are calculated from human parameters using the following allometric scaling formulas²¹:

\begin{aligned} μ_{C L}^{s p 2} & = μ_{C L}^{s p 1} \times (\frac{W^{s p 2}}{W^{sp1}})^{0.75} \end{aligned}

(5)

\begin{aligned} μ_{V}^{s p 2} & = μ_{V}^{s p 1} \times (\frac{W^{s p 2}}{W^{s p 1}}) \end{aligned}

(6)

where

s p 1

and

s p 2

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 the PK model from dog, rat, and mouse to human allows to predict the MTD to be equal to 502, 504, and 531 mg, respectively, that is near to the true value of 502 mg. Scenario 2 is based on an inaccurate extrapolation of the parameter clearance

C L

, affecting toxicity (via AUC), for one species. Other two scenarios (scenarios 3 and 4) accounting for MED estimation along with the MTD are given in the Supplemental Material (Section 2). Figure 2(b) gives the probability of toxicity according to the extrapolated dose for scenarios 1 and 2 and for all species.

Table 1.

Simulation parameters for PK model and approximate extrapolated MTD for all species and scenarios 1 and 2.

			Rat
Parameters	Human	Dog	Scenario 1	Scenario 2	Mouse
$W$ (kg)	70	10	0.15	0.15	0.025
$k_{a}$ (h⁻¹)	2	2	2	2	2
$μ_{C L}$ (Lh⁻¹)	40	9.3	0.40	1.59	0.11
$μ_{V}$ (L)	100	14	0.21	0.21	0.04
$ω_{C L}$	0.7	0.7	0.7	0.7	0.7
$ω_{V}$	0.7	0.7	0.7	0.7	0.7
$σ_{C}$	0.2	0.2	0.2	0.2	0.2
$τ_{T}$ (mgL⁻¹h)	22.6	22.6	22.6	22.6	22.6
$p_{T}$ (%)	20	20	20	20	20
$μ_{α}$	1	1	1	1	1
$ω_{α}$	0	0	0	0	0
Extrapolated	502	502	504	2002	531
MTD (mg)

$W$ : weights; $k_{a}$ : absorption rate constant for oral administration; $(μ_{C L}, μ_{V})$ : exponentials of the respective means of the logarithms of the distributions of the clearance elimination and the volume of distribution; $(ω_{C L}, ω_{V})$ : respective standard deviations of the logarithms of the distributions of the clearance elimination and the volume of distribution; $σ_{C}$ : standard deviations of the logarithm of the distribution of the measured concentration of the drug in blood plasma; $τ_{T}$ : AUC threshold; $p_{T}$ : probability of toxicity threshold; MTD: maximum tolerated dose; PK: pharmacokinetics; AUC: area under the curve.

For each scenario, 500 datasets are simulated. We assume that equations (3) to (6) describing the toxicity data generation and allometric scaling formulas are known as well as $τ_{T}$ , $p_{T}$ , $μ_{α}$ , $ω_{α}$ , and $W$ . The other model parameters are estimated.

All analyses are performed using R software version 4.04 with Stan²² package rstan Version 2.21.2. In rstan, three chains, a burn-in of 3000 and 6000 other iterations are used and, as a convergence criterion, Gelman and Rubin’s²³ potential scale reduction factor Rhat = 1.

3.4. Illustration on a single simulated dataset

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.

3.4.1. First step: Parameters estimation

At step 1 of our methodology, for each animal species / study $k$ , $k \in {1, 2, 3}$ , the mixed-effects models are fitted using Hamiltonian Monte Carlo.^24,25

First, for the mouse (study $k = 1$ ), to link the dose to the concentrations $y_{1, i} = {\tilde{c}}_{i}$ measured of the $i$ th subject, we use as mixed-effects model the first-order absorption linear one compartment model $f_{1} (.) : {\tilde{c}}_{i} = \log (C (t_{i}, d_{i}, θ_{1} = {k_{a, 1}, μ_{C L, 1}, μ_{V, 1}, ω_{C L, 1}})) + ϵ$ , where random effects are only estimated on $C L$ since there is only one data point per mouse. This model is fitted using the non-informative prior distributions given in the Supplemental Material (Section 2.4). For this illustration, the estimated posterior means for mouse of $μ_{C L, 1}$ and $μ_{V, 1}$ are equal to 0.0961 Lh⁻¹ and 0.0508 L (results shown in Table 4 in the Supplemental Material, Section 2.5).

Then, for the rat (study $k = 2$ ) and the dog (study $k = 3$ ), random effects (intraindividual variability-IIV) are estimated on $C L$ and $V$ , therefore, $y_{k, i} = {\tilde{c}}_{i}$ , the vector of concentrations measured of the $i$ th subject of study $k$ , $k \in {2, 3}$ , and $f_{k} (.) : {\tilde{c}}_{i j} = \log (C (t_{i j}, d_{i}, θ_{k} = {k_{a, k}, μ_{C L, k}, μ_{V, k}, ω_{C L, k}, ω_{V, k}})) + ϵ$ with $ϵ \sim N (0, σ_{C})$ , where $k \in {2, 3}$ and $j$ indexes the number of time points for each subject. In this situation, $θ_{1}$ , $θ_{2}$ , and $θ_{3} = θ_{2}$ share several parameters with the same role, therefore, extrapolation formulas (equation (5) and (6) are adopted as $g_{k}$ ) can be used to set prior distributions when stepping from one study to the next one. Using extrapolation formulas from mouse to rat, we obtain the values of 0.368 Lh⁻¹ and 0.305 L that are used as means of the prior distributions of $μ_{C L, 2}$ and $μ_{V, 2}$ for rat. Similarly, the estimated posterior means for rat of $μ_{C L, 2}$ and $μ_{V, 2}$ are equal to 0.464 Lh⁻¹ and 0.216 L for scenario 1 (respectively, 1.45 Lh⁻¹ and 0.581 L for scenario 2) and we use the extrapolated values of 10.8 Lh⁻¹ and 14.4 L (respectively, 33.8 Lh⁻¹ and 38.7 L) as means of the prior distributions of $μ_{C L, 3}$ and $μ_{V, 3}$ for dog.

For the next steps of the analysis based on mouse data, we make the assumption that $ω_{V, 1} = 0.7$ for mouse. As sensitivity analysis, we also consider the assumptions that $ω_{V, 1} = 0.4$ or $ω_{V, 1} = 1$ for mouse. We choose to perform $L = 1000$ MCMC runs for steps 2 to 4 of the methodology when approximating $D_{k}$ and $D_{k}^{*}$ distributions.

3.4.2. Second step: Extrapolation to human

In step 2, for each study $k \in {1, 2, 3}$ , we apply extrapolation formulas (5) and (6) to the $L$ values of the estimated parameters of $μ_{C L, k}$ and $μ_{V, k}$ to humans, which means that, if $(μ_{C L, k, l})_{1 \leq l \leq L}$ and $(μ_{V, k, l})_{1 \leq l \leq L}$ are the MCMC estimated parameters from the $k$ th study, we apply the transformations $μ_{C L, k, l}^{human} = μ_{C L, k, l} \times (W^{human} / W^{k})^{0.75}$ and $μ_{V, k, l}^{human} = μ_{V, k, l} \times (W^{human} / W^{k})$ for $l \in {1, \dots, L}$ . Other parameters of the models are kept identical, to obtain the posterior distributions of the vector of human parameters $θ_{k}^{human}$ from the $k$ th study.

Next, for each study $k \in {1, 2, 3}$ , the dose of interest ${MTD}_{k} = d_{1, k} (θ_{k}^{human})$ is the solution of equation (4) when solving for $d$ while setting $p_{T} (d) = 0.2$ . The extrapolated MTD distributions from mouse, rat and dog to human resulting from step 2 are drawn in the first line of Figure 4 for these scenarios and each animal species as dotted lines.

3.4.3. Third step: Commensurability checking and posterior distributions selection

In step 3, we first transform the dose 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.

Then, in practice, the Hellinger distance between the transformed dose distributions is approximated using the rectangle method: ${[0.5 (b - a) / M \times \sum_{m = 0}^{M - 1} {(\sqrt{π_{k_{1}}^{p o s t} (d_{r, m}^{*})} - \sqrt{π_{k_{2}}^{p o s t} (d_{r, m}^{*})})}^{2}]}^{\frac{1}{2}},$ where $a = min_{m \in {0, \dots, M - 1}} d_{r, m}^{*}$ , $b = max_{m \in {0, \dots, M - 1}} d_{r, m}^{*}$ . The transformed extrapolated MTD distributions and the corresponding Hellinger distances are given in the second line of Figure 4 for scenarios 1 and 2 and each animal species.

Thereafter, to compute the accuracy and then to choose a threshold, 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 very close 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 one 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 zero. 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 the Supplemental Material (see Section 2.5.2). Next, based on the sensitivity analysis described in Section 4.1, we set the threshold for the Hellinger distance at 0.5 for MTD.

Then, 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 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 <0.5 so all animal species are selected. For scenario 2, the two distances including rat are >0.5 (mouse vs. rat: 1; rat vs. dog: 1) but the last distance is <0.5 (mouse vs. dog: 0.36) so only mouse and dog are selected.

3.4.4. Fourth step: Merging the selected posterior distribution

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. These posterior distributions can be computed using the kernel density estimator, $π_{k^{'}}^{p o s t} (d_{r}) = (L h_{k^{'}})^{- 1} \sum_{l = 1}^{L} κ ((d_{r} - d_{r, k^{'}, l}) / h_{k^{'}}),$ where $κ$ is the kernel (the standard normal density in this article) and $h_{k^{'}} > 0$ is the bandwidth. The final predicted MTD distributions are overlayed on the plots in the first line of Figure 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 464 and 436 mg for scenarios 1 and 2, respectively, as the posterior means of these distributions.

4. Simulation results

We then applied our methodology to the scenarios with 500 replications each.

4.1. Using Hellinger distance to check extrapolability from animal species to human

Figure 5(a) and (b) shows the 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.

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 one 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(c), as Hellinger distance threshold (mentioned in Section 2.3) 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 zero and one, and going beyond the median value did not seem relevant to us.

4.2. Final estimations of MTD in humans

As shown in Figure 6(a), 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 (i.e. the standard approach) to calculate the MTD than when using the proposed method (see Figure 6(b)). Actually, the mean of the CrI95 lengths is around 300 mg (sd: 47 mg) using only dog results compared to the values of 165 and 242 mg (sd equal to 26 and 153 mg) for, respectively, scenarios 1 and 2 using the proposed approach.

The results of the sensitivity analysis for $ω_{V, 1} = 0.4$ and $ω_{V, 1} = 1$ for mouse, given in the Supplemental Material (Section 4), do not differ from those obtained for $ω_{V, 1} = 0.7$ .

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 on a distribution coming 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 generalization of the Food and Drug Administration (FDA)⁵ guidelines. An advantage of the Bayesian framework is that allows us to obtain the posterior distribution of the extrapolated doses, whatever the shape of this distribution. Therefore, other metrics than the mean value or the credible interval could be taken into account, such as the asymmetry or the peak of the distribution or other possible summary values. Furthermore, the framework is very flexible. Submodels (linear, generalized, mixed-effect, etc.) can be specified according to the study outcomes and could be different between studies. Our approach can be used either prospectively, that is when studies are done sequentially and there is enough time to perform the analysis in between studies, or retrospectively, at the end of all studies. If the preclinical studies are conducted simultaneously, it is always possible to use non-informative prior distributions.

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 cell numbers. 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, etc.) 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 context, a normal distribution on the logarithm transformed value of the concentration is used, that is, a quite standard choice in PKPD modeling. In other contexts, in the presence of studies with a small sample size, alternative distributions to the so common normal distribution for error term could be considered, such as the $t$ -distribution or other ones with heavier tails.

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 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.

In our specific case study, since the transformed distributions have identical variances across studies, the Hellinger’s distance simply reflects differences in means, at the given variance, and a simpler criteria could be used for commensurability checking. For simplicity and for the coherence of the algorithm, we decided to standardize all distribution moving to the lowest information level available, that is to the highest variance. We suggest it since it is more comfortable to reduce existing information than creating information by decreasing the variance of a posterior distribution. However, further investigation may be needed investigating other ways, for example, the two by two adjusted distributions or moving towards the medium/lowest variance.

To run simulations, we decided to keep the results involving the more numerous cluster, that is, in our case, if at least two species were selected in the coherence step. We used this decision to check the performance of the algorithm in its less supervised implementation. However, in real application, if it is well-known that the results in a species are really closed/related to human ones, and, therefore, more relevant in a clinical point of view, 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 equation (2). Indeed, when only weakly informative prior distribution, carrying on a small and negligible ESS, are adopted, the final impact on equation (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 of human similarity. For instance, in our illustration, if it is known that the extrapolated results from dogs to humans will be more useful, more weight can be given to the dog study. 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 a mismatch between preclinical data and humans. 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, hypotheses, etc.) according to the studies the researchers are working on.

Supplemental Material

sj-pdf-1-smm-10.1177_09622802241231493 - Supplemental material for Bayesian framework for multi-source data integration-Application to human extrapolation from preclinical studies

Supplemental material, sj-pdf-1-smm-10.1177_09622802241231493 for Bayesian framework for multi-source data integration-Application to human extrapolation from preclinical studies by Sandrine Boulet, Moreno Ursino, Robin Michelet, Linda BS Aulin, Charlotte Kloft, Emmanuelle Comets and Sarah Zohar in Statistical Methods in Medical Research

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is part the European FAIR project that has received funding from the European Union's Horizon 2020 research and innovation program [grant number 847786].

ORCID iDs

Sandrine Boulet

Moreno Ursino

Sarah Zohar

Supplemental material

Supplemental material for this article is available online. Web appendices, tables and figures, referenced in Sections 2 to 4, are available with this article at the Statistical Methods in Medical Research website.

References

La Gamba

Jacobs

Geys

, et al. Bayesian sequential integration within a preclinical pharmacokinetic and pharmacodynamic modeling framework: lessons learned. Pharmaceutical Statistics page pst.1941, 2019.

Zheng

Hampson

. A Bayesian decision-theoretic approach to incorporate preclinical information into phase I oncology trials. Biometrical J 2020; 62: 1408–1427.

Zheng

Hampson

Wandel

. A robust Bayesian meta-analytic approach to incorporate animal data into phase I oncology trials. Stat Methods Med Res 2020; 29: 94–110.

Petit

Samson

Morita

, et al. Unified approach for extrapolation and bridging of adult information in early-phase dose-finding paediatric studies. Stat Methods Med Res 2018; 27: 1860–1877.

Food and Drug Administration (FDA). Guidance for Industry: Estimating the Maximum Safe Starting Dose in Initial Clinical Trials for Therapeutics in Adult Healthy Volunteers, 2005.

European Medicines Agency (EMA). Guideline on strategies to identify and mitigate risks for first-in-human and early clinical trials with investigational medicinal products, 2017.

Shen

Swift

Mamelok

, et al. Design and conduct considerations for first-in-human trials. Clin Transl Sci 2019; 12: 6–19.

Michelet

Ursino

Boulet

, et al. The use of translational modelling and simulation to develop immunomodulatory therapy as an adjunct to antibiotic treatment in the context of pneumonia. Pharmaceutics 2021; 13: 601.

Sheiner

. Learning versus confirming in clinical drug development*. Clin Pharmacol Ther 1997; 61: 275–291.

10.

Schmidli

Gsteiger

Roychoudhury

, et al. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics 2014; 70: 1023–1032.

11.

Gueorguieva

Cleverly

Stauber

, et al. Defining a therapeutic window for the novel TGF-

β

inhibitor LY2157299 monohydrate based on a pharmacokinetic/pharmacodynamic model: early oncology development based on a pharmacokinetic/pharmacodynamic model. Br J Clin Pharmacol 2014; 77: 796–807.

12.

Duputel

Stallard

Montestruc

, et al. Using dichotomized survival data to construct a prior distribution for a bayesian seamless phase II/III clinical trial. Stat Methods Med Res 2023; 32: 963–977.

13.

Alt

Nifong

Chen

, et al. The scale transformed power prior for use with historical data from a different outcome model. Stat Med 2023; 42: 1–14.

14.

Morita

Thall

Müller

. Determining the effective sample size of a parametric prior. Biometrics 2008; 64: 595–602.

15.

Ollier

Morita

Ursino

, et al. An adaptive power prior for sequential clinical trials—application to bridging studies. Stat Methods Med Res 2020; 29: 2282–2294.

16.

Ollier

Zohar

Morita

, et al. Estimating similarity of dose–response relationships in phase I clinical trials—case study in bridging data package. Int J Environ Res Public Health 2021; 18: 1639.

17.

Bueno

de Alwis

Pitou

, et al. Semi-mechanistic modelling of the tumour growth inhibitory effects of LY2157299, a new type I receptor tgf-beta kinase antagonist, in mice. Eur J Cancer 2008; 44: 142–150.

18.

Rodon

Carducci

Sepulveda-Sánchez

, et al. First-in-human dose study of the novel transforming growth factor-

β

receptor I kinase inhibitor LY2157299 monohydrate in patients with advanced cancer and glioma. Clin Cancer Res 2015; 21: 553–560.

19.

Lestini

Dumont

Mentré

. Influence of the size of cohorts in adaptive design for nonlinear mixed effects models: an evaluation by simulation for a pharmacokinetic and pharmacodynamic model for a biomarker in oncology. Pharm Res 2015; 32: 3159–3169.

20.

Ursino

Zohar

Lentz

, et al. Dose-finding methods for phase I clinical trials using pharmacokinetics in small populations: dose-finding methods for phase I clinical trials using pharmacokinetics. Biometrical J 2017; 59: 804–825.

21.

West

Brown

Enquist

. A general model for the origin of allometric scaling laws in biology. Science 1997; 276: 122–126.

22.

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual, 2021.

23.

Gelman

Rubin

. Inference from iterative simulation using multiple sequences. Stat Sci 1992; 7: 457–472.

24.

Betancourt

. A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv:1701.02434 [stat] arXiv: 1701.02434, 2018.

25.

Thomas

. Learning Hamiltonian Monte Carlo in R. Am Stat 2021; 75: 403–413.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.49 MB