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Introduction
Why Model Based Bioequivalence?
Traditional bioequivalence (BE) study design and statistical methods
are well established (1,2) and are based on non compartmental analysis
(NCA). There are, however, a number of clinical scenarios where a
traditional BE study is impractical (3,4). Examples include:
Very long half life drugs.
Clinical scenarios were rich sampling is not feasible, for
example, pediatrics or critically ill patients.
Clinical scenarios were a multi-period cross over study is not
feasible, e.g., cytotoxic drugs that cannot be given to healthy
volunteers.
There is value in developing method to meet the BE regulatory
requirements for such drugs. Model Based Bio Equivalence (MBBE) is one
method to achieve this.
Algorithm
The key parameter for implementing MBBE is the power of simulated
studies. The power is key, rather than the outcome of any single
simulated study, as the outcome of any single study will depend on the
random seed chosen. A more general and robust answer regarding the
likelihood BE can be obtained by simulating many studies, and examining
the distribution of outcomes, and therefore, the predicted distribution
of outcomes of actual studies, were such studies to be performed. The
use of nonlinear mixed effect models and Monte Carlo simulation is well
established for describing such distributions. Typically the data used
for development of a population PK model do not come from a BE study.
They may come from other, e.g., parallel design studies and/or sparse
sampling. If data from a well designed BE study were available, Virtual
BE studies would not be needed.
Model uncertainty and model averaging
“All models are wrong, some models are useful” [5]. We interpret this
well known aphorism to mean that if a single model (structure and
parameters) is used to describe the data, the prediction will be wrong
(or at least not exactly correct). But, if “some” models (e.g., multiple
models averaged together) are used, it may be useful, or perhaps at
least less wrong. Aoki[8] showed that for simulated studies, model
averaging in most cases increases the accuracy of predictions. In Monte
Carlo simulation we typically include parameter uncertainty for a single
model. With model averaging, we extend model uncertainty from a single
model with uncertain parameters to uncertainty about the model structure
as well. Rather than a single (incorrect) model, a number of “adequate”
models are examined. Adequate models are models that meet some set of
minimal requirements in describing the data. The method described by
Aoki [8] is used for model averaging. A set of bootstrap samples from
the original data set are generated. NONMEM is then used to estimated
parameters for each adequate-model/bootstrap-data set combination is
performed. For each sample, the model with the best (lowest) Bayesian
Information Criteria (BIC [9]) is selected. The Monte Carlo simulation
is then done with the model structure and parameter estimates selected
for each bootstrap data set, thus capturing uncertainty in both the
model structure and parameters.
The simulations can be performed using a data set different from the
source data, as, typically, the reason for doing MBBE is that there are
no formal BE studies available. In this way, non-BE study data can be
used for model definition and then a formal BE study (e.g., 4 period,
single dose cross over study) can be simulated. The NCA parameters for
each (simulated) subject in each study are then calculated using and
algorithm derived. The workflow is:
Develop model(s) of the data. Model development and criteria for
an “adequate” model are not discussed, but typically would include
separate estimation by formulation for all absorption parameters (e.g,
F, Ka, Lag time, zero order infusion duration, transit compartment rate
constants etc.). Different structural models for the absorption of
refernce vs test formulation may also be considered. An adequate model
may also include between occasion variability on Volume and elimination
terms, if multiple periods are available in the observed data.
For each bootstrap sample, select the “best” model. In the
current implementation, the best model is selected based on Bayesian
information criteria (BIC). That is, parameters for each adequate model
are estimated for each bootstrap sample. For each bootstrap sample, the
model and associated parameter estimates with the best (lowest) BIC is
selected. In this way, the uncertainty of both the model structure and
the model parameters is described.
From the selected models, perform Monte Carlo simulation, with a
reasonable study design (e.g., 4 period, cross over, rich sampling, 50
subjects).
Determine power of the analysis by calculating two one side t
test statistics [6] on each simulated study.
Two options exist for how to combine those different models/parameter
estimate. First, the data from each of the simulations can be used as
simulated, with a single model per simulated study (the
“model_averaging_by”: “study” option). This represents the possibility
that any given model may be “correct” for all subjects, we’re just not
certain which one it is. Alternatively, the study data can be recombined
such that each study has a population of individuals based on the
representation of overall models. (the “model_averaging_by”: “subject”
option). For this option, any given study will be comprised of the same
set of subjects (demographics, sample time etc.), but have different
structure models. The interpretation here is that there is a
distribution of models within the population (similar to the
distribution of parameters). So, for the population we have a set prior
probabilities for the structural model, but uncertainty about which
model any given subject follows. This is analogous to a prior
distribution (THETA and OMEGA in NONMEM) for parameters, but uncertainty
about the “true value” parameters.
An overall diagram (from Aoki [8]) for model averaging is shown
below:
Nyberg et. al [10] described using a SADDLE_RESET as a check for
local non-identifiability. MBBE included an option
"use_check_identifiable": true,
to instruct the algorithm to determine whether any model resulting
from the bootstrap is identifiable, based on the largest absolute
fractional difference between pre and post saddle reset parameters of
delta_parms. delta_parms can be set in the json argument file e.g:
"delta_parms": 0.2,
Power:
Traditional bioequivalence is assessed in a single study population
(N=1, drawn from a universe of possible study populations). Thus there
is some degree of “luck” associated with whether or not the formulae are
deemed bioequivalent. MBBE permits a more robust approach, not dependent
on a single draw from a distribution, that is, the power of the given
study design to find that the formulae are bioequivalent. The success of
each study is then calculated based on the method described by Schütz (11) in
the replicateBE package in R. The resulting power then is simply the
fraction of simulated studies that successfully demonstrate
bioequivalence for each NCA endpoint.
Implementation:
MBBE is an R package available on CRAN. Details of implementation are
provide in the Step-by-Step Vignette. Several files are required,
include observed and simulation data sets, control file and an options
file. Sample files with explanation is given below.
run_dir - run directory where bootstrap samples will be
run
model_source - a json formatted list of NONMEM control files to
be used for model averaging
num_parallel - number of parallel NONMEM models to run for
bootstrap and Monte Carlo Simulation
crash_value - Value assigned for model “goodness” (BIC plus any
other penalty) if the model fails to generate output. Should be larger
than any “goodness” value for a model that completes
nmfe_path - path to nmfe??.bat file
delta_parms - fractional difference in any parameter before and
after a SADDLE_RESET that will be deemed not identifiable
use_check_identifiable - logical (true|false) for whether to test
for identifability
NCA_end_time - end time for NCA parameter calculation. Note that
all NCA intervals will start at 0 (likely requiring a reset event in the
simulation data set for each period)
rndseed - random seed for Bootstrap and Monte Carlo
Simulation
simulation_data_path - path to data file to be used for
simulation
ngroups - number of groups in simulation data set
samp_size - sample size to be used for Bootstrap and Monte Carlo
Simulation
reference_groups - GROUP value indicating reference formulation
in the simulation data set
test_groups - GROUP value indicating test formulation in the
simulation data set. Note that the total number of reference_groups and
test_groups must equal ngroups
plan - parallel plan to be used for bootstrap and Monte Carlo
Simulations. Options are
sequential
multisession
multicore
alpha_error” - type 1 error for the two one sides tests, default
= 0.05
NTID - Is this a Narrow Therapeutic Index Drug? (12)
model_averaging_by - should model averaging be by study or by
subject (experimental)
user_R_code - will R code be used to calculate a penalty to be
added to the Bayesian Information Criteria?
R_code_path - if user_R_code, path to the R source file
The Analysis data set.
The bootstrap analysis data set can have any form required for the
NONMEM model. However, several data items are required. These are:
PERIOD
GROUP
SEQ (Sequence)
TRT (treatment)
The analysis data set need not have multiple periods or sequences (in
which case all values can be 0, or missing). However the $INPUT record
is copied from the analysis control file into the simulation control
file, and the simulation control file MUST include GROUP, PERIOD, SEQ
and TRT data items for statistical testing.
The Simulation data set
Typically, the simulation data set will be comprised of:
A least two periods, cross over, single dose study, for a
randomized sequence of test and reference. A four period study, with
random sequence may be preferred.
Rich sampling out to at least 3 half-lives in the terminal phase
for nearly all subjects.
The data set must include:
PERIOD - sequential number of periods, must be at least
2
GROUP - group assigned, e.g., Groups 1 and 2 are reference,
Groups 3 and 4 are test
TRT (treatment) - usually 2, test and reference
SEQ (sequence) sequence of treatment, usually at least 2
sequence, e.g., AB or BA.
As these will be tested in the two one side T tests. These data items
need not be used in the bootstrap analysis control files.
The bootstrap analysis NONMEM control file will be the template
for the simulation control file. Therefore, the $INPUT must be the same
between the bootstrap analysis data set and the Monte Carlo simulation
data set. Basically, the “best” model control file for each bootstrap
sample is edited by replacing any $TABLE records with those needed for
the two one side T test, the $EST replaced with $SIM. The final
parameters copied from the selected model .xml output file into the
simulation control file.
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These binaries (installable software) and packages are in development. They may not be fully stable and should be used with caution. We make no claims about them.
