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\begin{document}

\title{Using Prior Parameter Knowledge in Model-Based Design of Experiments for
Pharmaceutical Production}



\author[1]{Ali Shahmohammadi}%
\author[2]{Kimberley McAuley}%
\affil[1]{University of Texas at Austin}%
\affil[2]{Queens University}%


\vspace{-1em}



  
  \date{June 02, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Sequential model-based design of experiments (MBDoE) uses information
from previous experiments to select run conditions for new experiments.
Computation of the objective functions for popular MBDoE can be
impossible due to a non-invertible Fisher Information Matrix (FIM).
Previously, we evaluated a leave-out (LO) approach that design
experiments by removing problematic model parameters from the design
process. However, the LO approach can be computationally expensive due
to its iterative nature and some model parameters are ignored. In this
study, we propose a simple Bayesian approach that makes the FIM
invertible by accounting for prior parameter information. We compare the
proposed Bayesian approach to the LO approach for designing sequential
A-optimal experiments. Results from a pharmaceutical case study show
that the Bayesian approach is superior, on average, to the LO approach
for design of experiments. However, for subsequent parameter estimation,
a subset-selection-based LO approach gives better parameter values than
the Bayesian approach.%
\end{abstract}%



\sloppy


\section*{Introduction}

{\label{introduction}}

Mathematical models are used in chemical and pharmaceutical industries
for analysis, design and control of chemical processes and for
maximizing product quality and profit.\textsuperscript{1,2} Especially
in pharmaceutical industries, models are important for Quality by Design
and development of continuous manufacturing processes, which are
becoming more widespread.\textsuperscript{3--5} Mathematical models for
pharmaceutical product development can be either empirical or
mechanistic.\textsuperscript{5--7} Although empirical models are
commonly used for pharmaceutical processes, they cannot reliably predict
the system behavior outside the range of operating conditions used for
model development.\textsuperscript{8} Therefore, fundamental models,
based on underlying chemistry and physics, are
preferred.\textsuperscript{9} These models usually contain unknown
parameters that require estimation using experimental
data.\textsuperscript{10} To obtain informative data, it is advantageous
to carefully plan the experiments aimed at parameter estimation using
design of experiment (DoE) techniques.\textsuperscript{11} As shown in
Table 1, optimal model-based design-of-experiments (MBDoE) techniques
select experiments to minimize uncertainties in parameters estimates or
model predictions.\textsuperscript{12--14} MBDoE techniques are
effective because they account for the structure of the model as well as
parameter and measurement uncertainties when selecting new run
conditions.\textsuperscript{13,15} Other benefits of MBDoE techniques,
compared to traditional factorial designs, are that they can be readily
used to design any number of experiments, e.g., one, three or seven
experiments, depending on available resources for
experimentation.\textsuperscript{15,16} MBDoE techniques have been
developed to satisfy a variety of objectives including minimizing total
variances of parameter estimates, minimizing the average variance of
model predictions, and designing experiments for model
discrimination.\textsuperscript{16}

Table 1 shows several MBDoE objective functions that have been used for
development of chemical and pharmaceutical production
models.\textsuperscript{15--17} If modelers are interested in obtaining
accurate parameter estimates for their model, A-, D- or E- optimal
designs can be selected.\textsuperscript{15--17} Alternatively, G- and
V-optimal designs focus on obtaining accurate model predictions at
specified operating conditions of interest to the
modeler.\textsuperscript{18--21} All of these MBDoE techniques in Table
1 require computation of the inverse of the Fisher Information Matrix
(\textbf{FIM} ) when selecting experimental
settings.\textsuperscript{21--23} The \textbf{FIM} carries information
about how changes in parameter values can affect the model predictions
and is therefore crucial for both MBDoE calculations and parameter
inference.\textsuperscript{24} For nonlinear models, which are common in
chemical and pharmaceutical applications, computation of the\textbf{FIM}
requires linearizing the model around some nominal parameter
values.\textsuperscript{17,25} If these nominal parameter values are
significantly different from the corresponding true values, the selected
MBDoE settings may lead to experimental data that are not very
informative.\textsuperscript{17,25} Sequential design approaches are
appealing because they enable updating of the parameter values, as well
as the experimental strategy, as more data become
available.\textsuperscript{26} Using sequential experimental designs,
valuable information from old experimental data can be used, which might
have been collected for other objectives than model
development.\textsuperscript{27,28}

Computation of the objective functions for sequential MBDoE is
problematic if the \textbf{FIM} is noninvertible or ill-conditioned.
Typical causes are limited experimental data, strongly correlated
influences of different parameters, and parameters with little or no
influence on the model predictions.\textsuperscript{29} In chemical,
biochemical and pharmacological systems, models often contain a large
number of kinetic and transport parameters (e.g., 10-80 parameters)
which may result in noninvertible/ill-conditioned\textbf{FIM}
s.\textsuperscript{30--34} To avoid this problem, several approaches
have been considered during sequential MBDoE calculations including
parameter subset selection,\textsuperscript{14,29,35}pseudoinverse
methods,\textsuperscript{21,36} Tikhonov
regularization,\textsuperscript{37--40} and Bayesian
approaches.\textsuperscript{13,41,42}

The parameter-subset-selection approach uses a model-reduction
perspective.\textsuperscript{35,43,44} In one methodology, parameters
are ranked from most-estimable to least-estimable so that problematic
(low-ranked) parameters can be recognized and fixed at their nominal
values.\textsuperscript{35,45} In this way, experiments can be designed
using a well-conditioned reduced \textbf{FIM} that ignores problematic
parameters. Alternatively, pseudoinverse methods approximate the inverse
of the \textbf{FIM} (e.g., using the Moore-Penrose pseudoinverse) during
MBDoE calculations.\textsuperscript{21,36,46} In Tikhonov
regularization, a penalty is added to diagonal elements of
the\textbf{FIM} to make it invertible.\textsuperscript{29,38--40}
Bayesian MBDoE using linear models results in Tikhonov penalties that
account for prior knowledge about parameters. However, for nonlinear
models, the situation can be considerably more complex, depending on how
the nonlinearity is treated.\textsuperscript{13,41,42} There is little
information in the literature regarding which approach is most
effective. In two previous articles, we considered pharmaceutical case
studies involving noninvertible \textbf{FIM} s. Two different approaches
were compared: i) a subset-selection-based approach that leaves out
problematic parameters (LO approach) and ii) a simpler approach that
uses a Moore-Penrose pseudoinverse in place of\(\mathbf{\text{FI}}\mathbf{M}^{\mathbf{-1}}\)(PI
approach).\textsuperscript{21,46} These case studies suggest that the LO
approach is often superior to the PI approach for designing both A- and
V-optimal experiments.\textsuperscript{21,46} A shortcoming of the LO
approach is that it can be complicated and computationally expensive due
to changes in the subset of parameters that is left out during MBDoE
calculations. This complication motivates us to find a more convenient
approach to deal with singular \textbf{FIMs} during MBDoE.

The focus of the current study is on a simplified Bayesian approach for
dealing with singular/ill-conditioned \textbf{FIMs} during MBDoE.
Bayesian approaches have been used in several past MBDoE studies for
chemical and biochemical systems.\textsuperscript{47--49} The main
benefit of the Bayesian MBDoE framework is that it accounts for prior
knowledge about plausible values of the model
parameters.\textsuperscript{13} However, many researchers raise concerns
about the use of Bayesian approaches in practical engineering
systems.\textsuperscript{13,50} Disadvantages of the Bayesian approach
include uncertainty about the reliability of assumptions made when
specifying prior information.\textsuperscript{49--51} Undesirable
computational complexity can also arise, depending on the assumptions
that are made. As a result, Bayesian MBDoE has not enjoyed widespread
applications in chemical process modeling.

\textbf{Table 1.} Optimality criteria for model-based design of
experiments \textsuperscript{21,46}\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\textbf{Optimality Criterion} & \textbf{Description} & \textbf{Eq.
\#}\tabularnewline
\midrule
\endhead
\(J_{A}=trace\left(\left(\mathbf{\text{FIM}}\right)^{-1}\right)\) & \textbf{A-optimal} design minimizes total parameter
variance. & (1.1)\tabularnewline
\(J_{D}=det\left(\left(\mathbf{\text{FIM}}\right)^{-1}\right)\) & \textbf{D-optimal} design minimizes the volume of
the joint confidence interval for the parameters. & (1.2)\tabularnewline
\(J_{E}=\lambda_{\max}(\mathbf{\text{FIM}}^{-1})\) & \textbf{E-optimal} design minimizes the largest
eigenvalue of the \textbf{FIM}, thereby minimizing the uncertainties in
the worst-case direction in the parameter space. & (1.3)\tabularnewline
\(J_{G}=max\ \left(\text{diag}\left(\mathbf{W}\left(\mathbf{\text{FIM}}\right)^{-1}\mathbf{W}^{\mathbf{T}}\right)\right)\) & \textbf{G-optimal} design minimizes the maximum
variance of model predictions at user-specified operating conditions of
interest, specified using a matrix\(\ \mathbf{W}.\) This is equivalent
to minimizing the largest value of the diagonal elements of
\(\mathbf{W}\left(\mathbf{\text{FIM}}\right)^{-1}\mathbf{W}^{\mathbf{T}}\)\textbf{.} & (1.4)\tabularnewline
\(J_{V}=trace\left(\mathbf{W}\left(\mathbf{\text{FIM}}\right)^{-1}\mathbf{W}^{\mathbf{T}}\right)\) & \textbf{V-optimal} design minimizes the total
variance of model predictions at user-specified operating conditions of
interest, which are specified using matrix\(\ \mathbf{W}.\) This is
equivalent to minimizing the trace of \(\mathbf{W}\left(\mathbf{\text{FIM}}\right)^{-1}\mathbf{W}^{\mathbf{T}}\). &
(1.5)\tabularnewline
\bottomrule
\end{longtable}

The objective of the current article is to formulate and test a simple
Bayesian MBDoE approach that is readily usable by model developers. The
effectiveness of the proposed Bayesian approach is compared to that of
the LO approach for designing A-optimal experiments when the\textbf{FIM}
is noninvertible. We use the pharmaceutical case study of Domagalski et
al., (2015), which is of interest to our industrial
sponsor.\textsuperscript{6,21} The associated dynamic model uses
Michaelis-Menten kinetics and enzyme-catalyzed reactions to describe the
production of a pharmaceutical agent.\textsuperscript{52} The remainder
of this article is organized as follows. First, background on
the\textbf{FIM} and sequential A-optimal design is presented. Next,
details of the Bayesian and LO approaches for parameter estimation and
experimental design are presented. A simple Bayesian approach is
proposed and a pharmaceutical case study is presented. Results obtained
using Monte Carlo (MC) simulations are provided, revealing that the
proposed Bayesian approach is superior to the LO approach for this case
study.

\section*{Background Information}

{\label{background-information}}

\subsection*{Fisher Information Matrix for Nonlinear
Models}

{\label{fisher-information-matrix-for-nonlinear-models}}

Consider the nonlinear model:

\(\mathbf{Y=g}\left(\mathbf{d,\theta}\right)\mathbf{+\varepsilon}\)(1)

where \(\mathbf{Y\ \in\ }\mathbf{R}^{N}\) is a vector of stacked measured responses,
\textbf{g} is the solution of equations that describe the system,
\(\mathbf{d}\in\mathbf{R}^{r\times D}\) is a matrix of experimental settings (for
\(r\) runs with \(D\) decision variables
specified for each), \(\mathbf{\theta}\in\mathbf{R}^{p}\) is the vector of model
parameters and\(\mathbf{\varepsilon}\mathbf{\in}\mathbf{R}^{N}\) is a vector of a measurement noise with
diagonal covariance matrix\(\mathbf{\Sigma}_{\mathbf{y}}\in\mathbf{R}^{N\times N}\). For dynamic multi-response
models with \(n\) sample times per run and
\(v\)response variables, the total number of data values is
\(N=nvr\). The\textbf{FIM} is computed using a parametric
sensitivity matrix\textbf{S} \({\in\mathbf{R}}^{N\times p}\) with elements:

\(S_{\text{ij}}=\left.\ \frac{\partial g\left(\mathbf{d},\mathbf{\theta}\right)}{\partial\theta_{j}}\right|_{{\hat{\theta}}_{k\neq j}}\)(2)

computed by linearizing the model around the best currently-available
parameter values:\textsuperscript{53}

The elements of \(\mathbf{S}\) should be scaled using parameter
uncertainties \(s_{\theta_{j}}\) and measurement
uncertainties\(s_{y_{i}}\) to reflect the modeler's prior
knowledge:\textsuperscript{54}

\(Z_{\text{ij}}=S_{\text{ij}}\frac{s_{\theta_{j}}}{s_{y_{i}}}\) (3)

resulting in a scaled sensitivity matrix \textbf{Z} . The \textbf{FIM}
is related to \textbf{Z} by:

\(\mathbf{FIM=}\mathbf{Z}^{T}\mathbf{Z}\) (4)

When performing sequential MBDoE calculations, \textbf{Z} contains two
parts:\textsuperscript{21,46}

\(\mathbf{Z=}\par
\begin{bmatrix}\mathbf{Z}_{\mathbf{\text{old}}}\\
\mathbf{Z}_{\mathbf{\text{new}}}\\
\end{bmatrix}\) (5)

where \(\mathbf{Z}_{\mathbf{\text{old}}}\) corresponds to experimental settings and data
from old experiments. The elements of\(\mathbf{Z}_{\mathbf{\text{old}}}\) are fixed
during sequential MBDoE and elements of \(\mathbf{Z}_{\mathbf{\text{new}}}\) are
determined by the optimizer. After each sequential design, elements
of\(\mathbf{Z}_{\mathbf{\text{old}}}\) are updated based on the new parameter values and
the number of rows in\(\mathbf{Z}_{\mathbf{\text{old}}}\) increases due to the recent
experiments.

\subsection*{Parameter estimation with a noninvertible
FIM}

{\label{parameter-estimation-with-a-noninvertible-fim}}

When estimating parameters, the \textbf{FIM} should be invertible,
otherwise unique estimates for the parameters cannot be
obtained.\textsuperscript{22,29} Several regularization approaches have
been used to overcome this problem.\textsuperscript{38,39,55} One
popular approach is to estimate a subset of the model parameters that
are estimable, with the remaining parameters fixed at nominal
values.\textsuperscript{23,45,56} Table 2 shows computational steps for
a commonly used orthogonalization-based approach that ranks parameters
from the most-estimable so problematic (unranked) parameters that lead
to a noninvertible \textbf{FIM} can be
determined.\textsuperscript{45,54} The ranking starts by computing the
magnitude of each column of the scaled sensitivity matrix
\textbf{Z}(Step 1). The parameter corresponding to the column with the
highest magnitude is selected as the most-estimable parameter (Step 2).
The columns of \textbf{Z} are then regressed onto columns
of\(\mathbf{X}_{k}\), a matrix that contains columns from \textbf{Z}
that correspond to the ranked parameters (Step 3). Residual
matrix\(\mathbf{R}_{k}\) is then computed to remove correlation between
columns for the unranked parameters and columns for the parameters that
have already been ranked (Step 4). The next-most-estimable parameter is
the one with the largest magnitude among columns of\(\mathbf{R}_{\mathbf{k}}\).
In Step 5, the column corresponding to the next-most-estimable parameter
is selected from the original\(\mathbf{Z}\) matrix and included in
\(\mathbf{X}_{k}\), resulting in matrix \(\mathbf{X}_{k+1}\). Steps two
to five are repeated to produce a ranked list with up to
\(p\) parameters. The ranking stops when all of the
parameters are ranked or at the iteration where\(\mathbf{X}_{k}^{T}\mathbf{X}_{k}\) (the
reduced \textbf{FIM} ) becomes noninvertible. The remaining unranked
parameters are categorized as problematic. They either have very little
influence on the predicted responses or highly correlated effects with
parameters on the ranked list.\textsuperscript{45,57} Using this
orthogonalization-based ranking approach prior to parameter estimation
helps to avoid numerical problems that would arise due to a
noninvertible \textbf{FIM} .

\textbf{Table 2.} Orthogonalization algorithm \textsuperscript{45,54}\selectlanguage{english}
\begin{longtable}[]{@{}l@{}}
\toprule
Compute the magnitude (i.e., the Euclidean norm) of each column in the
\(\mathbf{Z}\) matrix. Select the column with the largest magnitude
as the most estimable parameter. Set \(k\ \ =\ 1\). Construct the
matrix \(\mathbf{X}_{k}\) by including the \(k\) selected
columns from \(\mathbf{Z}\) that correspond to parameters that have
been ranked. Use \(\mathbf{X}_{k}\) to predict columns in
\(\mathbf{Z}\) using ordinary least squares: \({\hat{\mathbf{Z}}}_{k}=\left(\mathbf{X}_{k}^{T}\mathbf{X}_{k}\right)^{-1}\mathbf{X}_{k}^{T}\mathbf{Z}\)
(2.1)\tabularnewline
\bottomrule
\end{longtable}

and calculate the residual matrix:\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{R}_{k}=\mathbf{Z}-{\hat{\mathbf{Z}}}_{k}\) & (2.2)\tabularnewline
\bottomrule
\end{longtable}



\section*{Michaelis-Menten Case Study}

{\label{michaelismenten-case-study}}

\subsection*{Reaction scheme and dynamic
model}

{\label{reaction-scheme-and-dynamic-model}}

The case study considered in the current article uses a nonlinear
kinetic model based on a Michaelis-Menten batch reaction for the
production of a pharmaceutical agent. Domagalski et al. (2015) used this
case study to develop empirical models based on conventional DoE and
response surface methodology.\textsuperscript{6} We used the same case
study to develop and test the LO approach for V-optimal MBDoE in
previous work.\textsuperscript{21} The reaction starts with reagent SM1
reacting with catalyst D and generating intermediate SM1.D via
reversible reaction (1) in Figure 1. Next, intermediate SM1.D reacts
with reagent SM2 to make the product P and release the catalyst (i.e.,
reaction (2)). There is also a possibility of generating several
impurities: SM2 can react with P to generate impurity I1, SM1 can be
hydrolyzed to form impurity I2, D can be deactivated with water to make
I3, and P can degrade to generate I4. Table 3 provides a fundamental
dynamic model for the Michaelis-Menten batch reaction system. Equations
(3.11) to (3.14) show that the concentrations of SM1, D, SM2, and P are
measured, and these measurements have experimental errors. We assume
that the water concentration \(C_{H2O}\) and the solution volume
\(V\)are constant at 0.10 M and \(1.0\ L\),
respectively.

In the study by Domagalski et al., 3 rounds of simulated experiments
were performed. In each round, they conducted 16 fractional-factorial
runs + 4 center-point-runs (i.e., 20 experiments in each round and 60
overall). Table 4 shows Domagalski's center-point settings for their
first round of experimentation. We assume that data for the 4 replicated
center-point runs are available for initial parameter estimation and
construction of \(\mathbf{Z}_{\mathbf{\text{old}}}\). Step-by-step computation of
\(\mathbf{Z}_{\mathbf{\text{old}}}\) using these runs is described in the Supplementary
Information. The duration of each simulated batch experiment is 6.0 h
with measurements taken every 45 minutes, resulting in sampling at 9
times including the initial time\(t=0\). As a result, each
run involves\(\ 36\) measured values (i.e., 9 values each for
\(y_{SM1}\), \(y_{D}\), \(y_{SM2}\) and
\(y_{P}\)).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure . Reaction scheme for Michaelis-Menten case study
\textsuperscript{6}

\textbf{Table 3.} Dynamic kinetic model for the Michaelis-Menten batch
reaction system\textsuperscript{21}\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\frac{dC_{SM1}}{dt}=-k_{1f}C_{SM1}C_{D}+\frac{k_{1f}}{K_{\text{eq}}}C_{SM1D}-k_{4}C_{SM1}C_{H_{2}O}\) & (3.1)\tabularnewline
\midrule
\endhead
\(\frac{dC_{D}}{dt}=-k_{1f}C_{SM1}C_{D}+\frac{k_{1f}}{K_{\text{eq}}}C_{SM1D}+k_{2}C_{SM2}C_{SM1D}-k_{5}C_{D}C_{H_{2}O}\) & (3.2)\tabularnewline
\(\frac{dC_{SM1D}}{dt}=k_{1f}C_{SM1}C_{D}-\frac{k_{1f}}{K_{\text{eq}}}C_{SM1D}-k_{2}C_{SM2}C_{SM1D}\) & (3.3)\tabularnewline
\(\frac{dC_{SM2}}{dt}={-k}_{2}C_{SM2}C_{SM1D}-k_{3}C_{SM2}C_{P}\) & (3.4)\tabularnewline
\(\frac{dC_{P}}{dt}=k_{2}C_{SM2}C_{SM1D}-k_{3}C_{SM2}C_{P}-k_{6}C_{P}\) & (3.5)\tabularnewline
\(\frac{dC_{H_{2}O}}{dt}=-k_{4}C_{SM1}C_{H_{2}O}-k_{5}C_{D}C_{H_{2}O}\) & (3.6)\tabularnewline
\(\frac{dC_{I1}}{dt}=k_{3}C_{SM2}C_{P}\) & (3.7)\tabularnewline
\(\frac{dC_{I2}}{dt}=k_{4}C_{SM1}C_{H_{2}O}\) & (3.8)\tabularnewline
\(\frac{dC_{I3}}{dt}=k_{5}C_{D}C_{H_{2}O}\) & (3.9)\tabularnewline
\(\frac{dC_{I4}}{dt}=k_{6}C_{P}\) & (3.10)\tabularnewline
\(y_{SM1}=C_{SM1}+\varepsilon_{SM1}\) & (3.11)\tabularnewline
\(y_{D}=C_{D}+\varepsilon_{D}\) & (3.12)\tabularnewline
\(y_{SM2}=C_{SM2}+\varepsilon_{SM2}\) & (3.13)\tabularnewline
\(y_{P}=C_{P}+\varepsilon_{P}\) & (3.14)\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 4.} Initial conditions for center-point batch reactor
operation at\(T=\ 40\ \)\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
State variable & Units & Initial condition\tabularnewline
\midrule
\endhead
\(C_{SM1}\) & M & \(1\)\tabularnewline
\(C_{D}\ \) & M & \(0.05\ \)\tabularnewline
\(C_{SM2}\ \) & M & \(1.15\)\tabularnewline
\(C_{SM1D}\) & M & \(0\)\tabularnewline
\(C_{P}\ \) & M & \(0\)\tabularnewline
\(C_{H_{2}O}\ \) & M & 0.\(10\)\tabularnewline
\(C_{I1}\ \) & M & \(0\)\tabularnewline
\(C_{I2}\ \) & M & \(0\)\tabularnewline
\(C_{I3}\ \) & M & \(0\)\tabularnewline
\(C_{I4}\ \) & M & \(0\)\tabularnewline
\bottomrule
\end{longtable}

Table 5 shows the true kinetic coefficients used by Domagalski et al.
for generating simulated data. These parameter values were used in the
current study to compute true kinetic and equilibrium coefficients via
Arrhenius expressions:

\(k_{i}(T)=k_{i,ref}\exp\left(-\frac{E_{a,i}}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{ref}}}\right)\right)\)(17)

\(K_{\text{eq}}(T)=K_{eq,ref}\exp\left(-\frac{\Delta H_{1}}{R}\left(\frac{1}{T}-\frac{1}{T_{\text{ref}}}\right)\right)\)(18)

where \(k_{i}\) is the \(i\)th kinetic
coefficient, \(R\) is the universal gas constant,
\(T\) is the temperature in \(K\),
and\(T_{\text{ref}}\ \)= 313.15 K = 40 \selectlanguage{ngerman}°C is a reference temperature. In
equation (18), \(K_{\text{eq}}\ \)is the equilibrium coefficient for
reaction (1), and \(\Delta H_{1}\) is the reaction enthalpy. Table 6
provides measurement noise variances used in this study for generating
simulated data.\textsuperscript{21} Figure 2 shows one set of simulated
old data generated using the values in

Table \textbf{5} and Table 6. As shown in the simulated true response in
Figure 2, consumption of catalyst D is initially very fast and then the
catalyst gets released via reaction (2) as the product is formed.

\textbf{Table 5.} True values of the kinetic coefficients and
equilibrium constant\textsuperscript{6}\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
& Units & Value at 40 \selectlanguage{ngerman}°C & \(E_{a,i}\) or
\(\Delta H_{1}\ \left(\text{J.}\text{mol}^{-1}\right)\)\tabularnewline
\midrule
\endhead
\(k_{1f}\ \) & \(M^{-1}s^{-1}\) & \(1.09\times 10^{-01}\) &
\(5.00\times 10^{4}\)\tabularnewline
\(K_{\text{eq}}\ \) & \(M^{-1}\ \) & \(9.33\) &
\(-1.00\times 10^{4}\)\tabularnewline
\(k_{2}\ \) & \(M^{-1}s^{-1}\) & \(3.39\times 10^{-03}\) &
\(4.25\times 10^{4}\)\tabularnewline
\(k_{3}\) & \(M^{-1}s^{-1}\) & \(1.09\times 10^{-06}\) &
\(9.00\times 10^{4}\)\tabularnewline
\(k_{4}\ \) & \(M^{-1}s^{-1}\) & \(1.17\times 10^{-05}\) &
\(9.50\times 10^{4}\)\tabularnewline
\(k_{5}\) & \(M^{-1}s^{-1}\) & \(1.93\times 10^{-06}\) &
\(9.75\times 10^{4}\)\tabularnewline
\(k_{6}\ \) & \(s^{-1}\) & \(1.87\times 10^{-08}\) &
\(8.00\times 10^{4}\)\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 6.} Measurement variances used for generating simulated
data\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Measured response (M) & \(\sigma_{y_{i}}^{2}\ \ \ (M^{2})\)\tabularnewline
\midrule
\endhead
\(y_{SM1}\) & \(4.8\ \times 10^{-2}\)\tabularnewline
\(y_{D}\ \) & \(2.1\ \times 10^{-4}\)\tabularnewline
\(y_{SM2}\) & \(5.2\ \times 10^{-2}\)\tabularnewline
\(y_{P}\ \) & \(5.0\ \times 10^{-2}\)\tabularnewline
\bottomrule
\end{longtable}

The initial case study assumes that the system operates
at\(T=T_{\text{ref}}=40\ \), which results in seven parameters requiring
estimation (i.e.,\(\ \mathbf{\theta}=\left[k_{1f},\ K_{\text{eq}},\ k_{2},\ k_{3},\ k_{4},k_{5},\ k_{6}\right]^{T}\)). These seven parameters lead to
seven columns in the scaled sensitivity matrix \(\mathbf{Z}\).
Decision variables for the new experiments are initial concentrations
for the reactants \(SM1,\ \ D\), and
\(SM2\)(i.e.,\(\mathbf{d}=\left[C_{\text{SM}1_{0}},C_{D_{0}},C_{\text{SM}2_{0}}\right]^{T}\)). Lower and upper bounds for
these decision variables are provided in Table 7. Note that, the four
approaches (i.e., LO-LO, Bayes-LO, LO-Bayes and Bayes-Bayes) that are
compared in this initial case study also considered in an expanded case
study (reported in the Supplementary Information) where temperature is
an additional decision variable (\(35\leq T\leq\ 45\)). Considering
\(T\) as an additional decision variable results in a model
with 14 parameters.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure 2.} Simulated data (symbols) and noise-free concentration
profiles (curves) obtained at 40 \selectlanguage{ngerman}°C using experimental settings in Table
4, kinetic coefficients in

Table \textbf{5} and measurement errors in Table 6: a) SM1; b) D; c)
SM2; d) P.

In the initial case study, three new sequential experiments are designed
one-at-a-time. In the first step, one A-optimal experiment is designed
and the parameters are estimated using both the old and the new data.
Next, these parameter estimates and all of the data obtained are used to
design a second sequential experiment. Finally, a third experiment is
designed using parameter estimates and data from previous steps. For
comparison, three new experiments are designed all-at-once based on the
old data in Figure 2 and corresponding parameter estimates.

\textbf{Table 7.} Lower and upper bounds for the decision variables\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
Decision variables \((M)\) & Lower bound \((M)\) &
Upper bound \((M)\)\tabularnewline
\midrule
\endhead
\(C_{\text{SM}1_{0}}\) & \(0.10\) &
\(1.5\)\tabularnewline
\(C_{D_{0}}\) & \(0.01\) &
\(0.5\)\tabularnewline
\(C_{\text{SM}2_{0}}\) & \(0.10\) &
\(1.5\)\tabularnewline
\bottomrule
\end{longtable}

Table 8 provides information about the user-specified prior parameter
information used in three different Cases. The prior parameter guesses
and corresponding standard deviations were used in the Bayesian
objective function for parameter estimation (equation (6)). This prior
information is also used in LO parameter estimation to obtain scaled
sensitivity coefficients in \textbf{Z} (see equations (2) and (3)). As
result, prior assumptions about parameters influence which parameters
are estimated and which remained fixed at their initial values. The
three cases described in Table 8 were used to investigate the influence
of the prior parameter information on the quality of parameter estimates
and experimental settings. For all three cases, parameter initial
guess\({\hat{\theta}}_{j0}\) were selected randomly from normal distributions
with true mean \(\theta_{j}\) and true standard
deviation\(s_{\theta_{j}}\) (see Table 5). In Case I, the modeler
specifies prior information that is quite accurate (i.e., prior
parameter standard deviations are 1/5 of the true value), whereas in
Case II, the modeler is less certain about the initial parameter
guesses. The selection rules in the third column of Table 8 for Cases I
and II prevent random selection of unrealistic negative parameter values
and parameter values more than 3 standard deviations from the true
parameter values. Case III is used to investigate whether the Bayesian
or LO approach to MBDOE and parameter estimation is more robust to
misinformed prior information (i.e., when modelers mistakenly believe
that they know more about the plausible parameter values than is
warranted). In Case III, initial parameter guesses are further from the
true values than the modeler believes.

\par\null

\textbf{Table 8.} Selection of parameter initial guesses from normal
distributions\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
& \textbf{Prior Distribution for the} \(\mathbf{j}\)\textbf{th
parameter} & \textbf{Rules for selection of parameter initial guess}
\({\hat{\mathbf{\theta}}}_{\mathbf{j}\mathbf{0}}\)\tabularnewline
\midrule
\endhead
\textbf{Case I:} Informative initial guess & \(N(\mu:\ \theta_{j},\ \sigma:\ s_{\theta_{j}}=\frac{1}{5}\theta_{j})\) &
Discard any parameter initial guesses beyond \(\theta_{j}\)
\(\pm\ \)3\({s_{\theta}}_{j}\) and select again.\tabularnewline
\textbf{Case II:} Moderately informative initial guess &
\(N(\mu:\ \theta_{j},\ \sigma:\ s_{\theta_{j}}=\frac{1}{2}\theta_{j})\) & Discard any negative parameter initial guesses and
any values beyond \(\theta_{j}\)
\(\pm\ \)3\({s_{\theta}}_{j}\) and selected again.\tabularnewline
\textbf{Case III:} Misinformed initial guess & \(N(\mu:\ \theta_{j},\ \sigma:\ s_{\theta_{j}}=\frac{1}{5}\theta_{j})\) &
Select initial guess all \textbf{from the tails} of the distribution,
beyond \(\theta_{j}\) \(\pm\)3 \(s_{\theta_{j}}\).
Discard any negative value and select again.\tabularnewline
\bottomrule
\end{longtable}

\section*{Monte Carlo Simulation Results and
Discussion}

{\label{monte-carlo-simulation-results-and-discussion}}

\subsection*{Case I: Results when informative parameter initial guesses
are
used}

{\label{case-i-results-when-informative-parameter-initial-guesses-are-used}}

In this Case, 100 initial guesses for the seven parameters were selected
as described in the first row of Table 8. Using each set of initial
guesses and the simulated old data in Figure 2, preliminary values of
the model parameters were estimated using both Bayesian and LO
approaches. All seven parameters were estimated using the Bayesian
approach, whereas subsets of parameters were estimated using the LO
approach, with remaining parameters fixed at their initial guesses.
Using the LO approach, parameter \(k_{2}\) was always ranked as
the most-estimable parameter, followed by \(K_{\text{eq}}\),
\(k_{1f}\) and\(k_{3}\) (using the ranking algorithm in
Table 2). Parameters\(k_{4}\), \(k_{5}\) and
\(k_{6}\) were always left out of the ranked list. Using Wu's
\(r_{\text{cc}}\) criterion, the parameter subset\(\mathbf{\theta}_{\mathbf{\text{sub}}}=\left[k_{2},\ K_{\text{eq}}\right]^{T}\)was
selected for estimation in all 100 simulated old data sets.
Parameters\(k_{1f}\), \(k_{3}\), \(k_{4}\),
\(k_{5}\) and \(k_{6}\) were fixed at their initial
values.

The preliminary parameter estimates obtained via Bayesian and LO
estimation were then used to design sequential A-optimal experiments
using Bayesian and LO approaches. Details concerning how many and which
parameters tended to be estimated after each stage of sequential
experimentation are provided in the Supplementary Information.

Figure 3 provides boxplots for 100 values of the scaled sum of squared
deviations between the estimated and true parameter values:

\(\text{SS}D_{\theta}=\left(\ \hat{\mathbf{\theta}}-\mathbf{\theta}^{\text{true}}\right)^{T}\mathbf{\Sigma}_{\mathbf{0}}^{\mathbf{-1}}\left(\ \hat{\mathbf{\theta}}-\mathbf{\theta}^{\text{true}}\right)\)(19)

for all four approaches when selecting three new A-optimal experiments,
one at a time. The Bayes-LO approach is the superior approach on
average, resulting in the smallest mean and median
for\(\text{SS}D_{\theta}\) after each round of experimentation. The results
in Figure 3 indicate that designing experiments using the proposed
modified Bayesian approach (i.e., with equation (16) as the objective
function) is superior to designing experiments using the LO approach
(i.e., the Bayes-Bayes results are better than LO-Bayes, and the
Bayes-LO results are better than LO-LO). In addition, parameter
estimation using the LO approach is superior to the Bayesian approach
(i.e., Bayes-LO is better than Bayes-Bayes, and LO-LO is better than
LO-Bayes).

Figure 4 shows boxplots for 100 values of \(\text{SS}D_{\theta}\) when
designing three new A-optimal experiments all at once for Case I.
Bayes-LO is the superior approach and LO-Bayes is the worst approach.
Comparing these results with the results in Figure 3, it can be
concluded that designing experiments one-at-a-time resulted in better
final parameter values than designing all three new experiments at once.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 3} . Boxplots for 100 values of \(\text{SS}D_{\theta}\) for
Case I, when designing three sequential A-optimal experiments one at a
time using LO-LO, Bayes-LO, LO-Bayes, and Bayes-Bayes approaches\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 4.} Boxplots for 100 values of \(\text{SS}D_{\theta}\) for
Case I, when designing three sequential A-optimal experiment all at once
using Bayes-Bayes, LO-LO, Bayes-LO, and LO-Bayes approaches

Computation times for both Bayesian and LO approaches to MBDOE were
compared using MC simulations for the Bayes-Bayes and LO-Bayes
approaches. These two approaches use the same approach for parameter
estimation, but different methods for MBDOE, making it possible to
isolate the effects of Bayesian and LO approaches for MBDoE. Using a
core i5 laptop with 8 GB RAM, the average computations time for each
Bayes-Bayes run was 51.2 s, which is faster than 89.9 s on average for a
LO-Bayes run. Although this difference is relatively small for the
current case study, we anticipate that larger differences could occur
for larger models with more parameters and decision variables.

\subsection*{Case II: Results when moderately-informative initial guesses
are
available}

{\label{case-ii-results-when-moderately-informative-initial-guesses-are-available}}

Figure 5 compares boxplots for the 100 values of \(\text{SS}D_{\theta}\)for
four approaches when designing three A-optimal experiments one at a time
using the prior parameter guesses in Case II. Similar patterns are
observed compared to Case I: as more experiments are designed and more
data become available, the mean and median values of boxplots
for\(\text{SS}D_{\theta}\) becomes smaller. However, since the parameter
initial guesses are not as good as in Case I, the parameter estimates in
Case II are less accurate than in Case I. As in Case I, the Bayes-LO
approach provides the best parameter values on average. A key difference
between the results in Case I and Case II is that the LO approach tended
to estimate more parameters in Case II, due to higher initial parameter
uncertainties. Details concerning the frequency with which different
parameters were estimated are provided in the Supplementary Information.

\subsection*{Case III: Results when misinformed initial guesses are
used}

{\label{case-iii-results-when-misinformed-initial-guesses-are-used}}

Figure 6 shows boxplots for 100 values of \(\text{SS}D_{\theta}\)obtained for
Case III, with the misinformed parameter initial guesses described in
the third row of Table 8. As expected, because parameter initial guesses
were worse than in Case I and II, the mean and median for
\(\text{SS}D_{\theta}\) are larger than Case I and II for all four
approaches. The Bayes-LO approach to designing experiments and
estimating parameters was the best approach, even though the modeler
believed he or she had better prior knowledge about the parameters than
was justifiable. These results suggest that the Bayes-LO approach is
somewhat robust to specification of misinformed prior information. Note
that the settings in Case II and Case III were also used to design three
experiments all-at-once instead of one at a time. As in Case I, the
Bayes-LO approach was the best and the parameter estimations resulting
from the three-at-once experiments were not as good as those obtained
using the one-at-a-time approach. Details are provided in the
Supplementary Information.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 5.} Boxplots for 100 values of \(\text{SS}D_{\theta}\) for
Case II, when designing three sequential A-optimal experiment one at a
time using LO-LO, Bayes-LO, LO-Bayes, and Bayes-Bayes approaches

Cases I, II and III were also repeated using the extended 14 parameter
model that arises (due to activation energies and reaction enthalpy)
when the temperature is included as an additional decision variable. The
same trends were observed as for the 7 parameters model, indicating that
the Bayes-LO is the best of the four approaches studied. Details are
provided in the Supplementary Information.

\section*{Conclusions}

{\label{conclusions}}

A simple Bayesian approach is proposed for the sequential model-based
design of experiments (MBDoE) when the \textbf{FIM} is noninvertible.
The results for the proposed Bayesian approach were compared with a
leave-out (LO) approach developed in previous studies. In addition, the
effectiveness of Bayesian and LO approaches for parameter estimation
were also compared, so that four different approaches were investigated
(i.e., Bayes-Bayes, LO-LO, Bayes-LO, and LO-Bayes) were investigated.
These approaches were tested using simulated data generated from a
7-parameter isothermal pharmaceutical production model and a
corresponding 14-parameter non-isothermal model. Three different cases
were considered wherein the modeler specified different prior
information about the parameters. The results indicate that the Bayes-LO
approach (i.e., a Bayesian approach for MBDoE combined with a LO
approach for parameter estimations) is superior to the three other
approaches. The proposed Bayesian approach for designing experiments
consistently provided superior experiments for use in parameter
estimation compared with the LO approach. However, after new
experimental data had been obtained, the LO approach for parameter
estimation consistently provided parameter values that were closer, on
average, to their true values than parameter estimates obtained from
Bayesian estimation. Promising simulation results obtained using
misspecified prior parameter knowledge indicated that the Bayes-LO
approach was somewhat robust to misinformation for the current case
study.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 6.} Boxplots for 100 values of \(\text{SS}D_{\theta}\) for
Case III, when designing three sequential A-optimal experiment one at a
time using Bayes-Bayes, LO-LO, Bayes-LO, and LO-Bayes approaches

\section*{ACKNOWLEDGMENT}

{\label{acknowledgment}}

Authors acknowledge financial support from Eli Lilly and Company and
technical advice from Dr. Salvador Garcia-Munoz and Dr. Brandon Jacob
Reizman. An Ontario Trillium Scholarship for A. Shahmohammadi from
government from the province of Ontario is also gratefully acknowledged.

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