Mechanistic modelling of product degradation Henrique Reis Sardinha Thesis to obtain the Master of Science Degree in Biological Engineering Supervisor(s): Prof. José Monteiro Cardoso de Menezes Dr. Edward Close Examination Committee Chairperson: Prof. Duarte Miguel de França Teixeira dos Prazeres Supervisor: Prof. José Monteiro Cardoso de Menezes Member of the Committee: Prof. Carla Isabel Costa Pinheiro October 2016
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Mechanistic modelling of product degradation...developed in gPROMS ModelBuilder to calculate species concentration and precipitated species in a parenteral solution. It was possible
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Mechanistic modelling of product degradation
Henrique Reis Sardinha
Thesis to obtain the Master of Science Degree in
Biological Engineering
Supervisor(s): Prof. José Monteiro Cardoso de MenezesDr. Edward Close
Examination Committee
Chairperson: Prof. Duarte Miguel de França Teixeira dos PrazeresSupervisor: Prof. José Monteiro Cardoso de Menezes
Member of the Committee: Prof. Carla Isabel Costa Pinheiro
October 2016
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Aos meus avos que nao conseguiram estar presentes neste momento tao especial
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Acknowledgments
I would like to begin to show my deepest gratitude to prof. Carla Pinheiro, prof. Costas Pantelides
and Dr. Sean Bermingham for giving me the opportunity to take this internship at Process Systems
Enterprise Ltd. I would like to thank my PSE and IST supervisors for guiding me in this final bit of my 5
year journey. Edd Close I would like to show you my deepest gratitude, especially for the support you
gave me throughout this internship and for always putting me on the redline. Aos meus pais, obrigado
por fazerem de mim o homem que sou hoje e por me terem guiado sempre pelo melhor caminho. Joana,
obrigado por teres estado sempre ao meu lado durante estes 4 anos.
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Resumo
A estabilidade fısica e quımica de novos produtos farmaceuticos representam dois factores cruciais para
a industria farmaceutica. Todos os anos estas empresas gastam milhoes de dolares em programas de
estabilidade de novos produtos farmaceuticos. Caso estes produtos sejam considerados instaveis sob
as regras das agencias reguladoras, uma nova formulacao tera de ser proposta e todo o processo de
aprovacao tera de ser reiniciado, levando a custos adicionais. Foi desenvolvido um modelo preditivo
em gPROMS ModelBuilder que ao analisar diversos ensaios em condicoes aceleradas, estabelece um
tempo de prateleira plausıvel para o farmaco em estudo. O modelo foi validado com dados da industria e
recorrendo a analises de sensibilidade e de incerteza dos inputs foi possıvel verificar o grau de incerteza
da data de validade. Foi ainda criado um modelo adicional que preve a degradacao destes produtos
na sua emabalgem (garrafas de HDPE) e na presenca de disecantes (silica gel). Concluiu-se que a
incerteza de factores externos (temperatura e humidade relativa) e de factores cineticos (Ea, A e B)
tem grande impacto na vida do produto e o emabalmento estende o seu tempo de prateleira devido a
proteccao da humidade.
A adicao de especies quımicas a produtos parentericos pode levar a fenomenos de precipitacao de-
vido as baixas solubilidades dos compostos formados. Na segunda parte do trabalho foi desenvolvido
um modelo matematico em gPROMS ModelBuilder que calcula a concentracao de especies quımicas
em solucao e precipitadas numa formulacao parenterica. Verificou-se que a precipitacao de ferro(III)
ocorre a pH 2.5 em concentracoes de 1mM deste metal e que o citrato e um bom agente quelante do
ferro ao impedir a sua precipitacao quando utilizada uma concentracao de 0.448M. O estudo dos resul-
tados com analises de incerteza permitiu concluir que quando a concentracao de ferro numa solucao
apresenta uma distribuicao probabilıstica, a concentracao de citrato previamente encontrada deixa de
ser eficaz, tendo de se aumentar esta concentracao para 0.6M de forma a garantir que nao se formam
quaisquer precipitados.
Palavras-chave: Estabilidade de produtos farmaceuticos, gPROMS, modelacao, solucoes
parenterais.
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Abstract
Physical and chemical stability of new pharmaceutical products represent two crucial factors for the phar-
maceutical industry. Manufacturers spend millions of dollars in stability programs every year to study the
stability of new products. If these formulations are considered unstable under regulators guidelines, a
new product needs to be designed and all of the approval process needs to be done once more, thus in-
creasing costs. A predictive model was developed in the gPROMS ModelBuilder platform. By analyzing
experiments from accelerated conditions, the model outputs a likely shelf-life for the studied drug. The
model was validated with industry data and by implementing uncertainty and sensitivity analysis on the
results, shelf-life was estimated. An additional model that predicts degradation of pharmaceutical prod-
ucts in its initial packaging (HDPE bottles) and in the presence of desiccants (silica gel) was developed.
It was possible to conclude that uncertainty in external parameters (temperature and relative humidity)
and kinetic parameters (Ea, A and B) have a great impact in product’s shelf-life and that packaging
provides additional protection from moisture, thus increasing shelf-life.
The addition of chemical species to parenteral solutions might induce precipitation phenomena given
the low solubility of the formed species. In the second part of this work, a mathematical model was
developed in gPROMS ModelBuilder to calculate species concentration and precipitated species in a
parenteral solution. It was possible to note that iron(III) precipitation starts at pH 2.5 for 1mM concentra-
tion and that citrate is a good chelator for this system when a 0.448M concentration is used. Although,
when uncertainty of model inputs is considered (metal concentration), the previous citrate concentration
is no longer effective. A concentration of 0.6M guarantees that all iron remains dissolved.
Keywords: Drug stability, gPROMS, modelling, parenteral solutions.
gPROMS ModelBuilder is an environment that has powerful custom modelling capabilities that allow the
user to create first principle model of virtually any type of process. The user can then validate these
models against experimental data using built-in parameter estimation techniques.
3.2.1 Performed experiments
Experiments are used to improve the understanding of processes and create accurate models. The
quality of information generated by experiments depends strongly on the experimental conditions as
well as what is measured and when it is measured. In gPROMS we can consider the processing of data
from experiments to estimate the values of unknown model parameters through Parameter Estimation.
When using a model, one may have experimental data acquired in laboratorial or industrial experi-
ments. It is possible to associate this measured data with the model variables and use it as an input for
parameter estimation.
In performed experiments, a control is a variable that is adjusted from one experiment to another
and/or during an experiment. The user can specify the variation in a variable value using one of three
different mechanisms:time-invariant controls that provide a single variable value throughout the duration
of the experiment; piecewise constant controls that provide a value per control interval. This value will
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apply throughout the duration of the corresponding control interval; piecewise linear controls that provide
a start value and an end value per control interval. The variable value will be varied linearly over time
from the start value to the end value.
3.2.2 Parameter estimation tool
A detailed gPROMS model is constructed from equations describing the physical and chemical phe-
nomena that take place in the system. These equations usually involve parameters that can be adjusted
to make the model predictions match observed reality. Examples of model parameters include reaction
kinetic constants, heat transfer coefficients, distillation stage efficiencies, constants within physical prop-
erty correlations, and so on. The more accurate these parameters are, the closer the model response is
to reality.[35]
In gPROMS, the fitting of these parameters to laboratory or industrial data is called Parameter Es-
timation. This estimation is based on the Maximum Likelihood formulation which in gPROMS accounts
for the physical model of the process and the variance model of the measuring instruments. In the latter,
three variance models are considered such as constant variance (e.g thermocouple with an accuracy of
±1K), constant relative variance (e.g HPLC with an error of ± 2%) or an heteroscedastic variance, com-
bining both of the above. When solving a Maximum Likelihood Parameter Estimation problem, gPROMS
attempts to determine values for the uncertain physical and variance model parameters, θ , that max-
imise the probability that the mathematical model will predict the measurement values obtained from the
experiments. Assuming independent, normally distributed measurement errors, εijk, with zero means
and standard deviations, σijk, this maximum likelihood goal can be captured through the following ob-
jective function (Eq.3.1) [35]:
Φ =N
2ln(2π) +
1
2minθ
(NE∑i=1
NVi∑j=1
NMij∑k=1
[ln(σ2
ijk) +(zijk − zijk)2
σ2ijk
])(3.1)
Table 3.1: Parameter estimation symbol definitions
N Total number of measurements taken during all the experimentsθ Set of model parameters to be estimated. The acceptable values may be subject to given
lower and upper bounds, i.e θt ≤ θ ≤ θu
NE Number of experiments performedNVi Number of variables measured in the i th experimentNMij Number of measurements of the jth variable in the ith experimentσ2ijk Variance of the kth measurement of variable j in experiment i. This is determined by
the measured variable’s variance modelzijk kth measured value of variable j in experiment izijk kth predicted value of variable j in experiment i
Experimental measurements are taken using sensors:the uncertainty of the measurement is a prop-
erty of the measurement technique associated with the sensor. When solving a model validation prob-
lem, all measured variables are associated with a sensor. Some of the variables can also be grouped
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with the same sensor. Each sensor group is associated with a variance model. The variance model
of a given sensor comprises information associated with the variance of the error in the measurement
produced by the sensor. The errors of the measurements are assumed to be statistically independent
and normally distributed with zero mean. There are several types of sensor variance models. These can
be considered to take the general form:
Figure 3.1: Parameter estimation tool: Experiments and measurements
σ2 = σ2(z,B) (3.2)
where z is the model prediction of the measured quantity and B is a set of parameters.
Table 3.2: Variance models for parameter estimation
Variance model Mathematical description
Constant variance σ2 = θ2
Constant relative variance σ2 = θ2 × (z2 + ε)
Heteroscedastic variance σ2 = θ2 × (z2 + ε)γ
Linear variance σ2 = (α ∗ z + β)2 + ε
The set of parameters B comprise the parameters σ and γ as appropriate. ε is a small but non-zero
constant that ensures that variance is still defined for predicted values that are zero or very small. If
γ = 0 in the heteroscedastic model, then this reduces to the constant variance model. If γ = 1 in the
heteroscedastic model, then this reduces to the constant relative variance model.
3.2.3 Analysis of results
Upon completion of a parameter estimation run, a ”Results” folder will be created within the parameter
estimation case file. In the report, the final optimal values, initial guesses, lower and upper bounds
for the estimated parameters are shown. The table also describes the confidence intervals, 95% t-
values and standard deviations for each estimated parameter. This section can be utilized to gauge
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the confidence on the optimal value of a given parameter. The sensor section details the variance
models applied to measured variables for the parameter estimation. A correlation matrix is also shown
in the report providing an indication of the correlation between the estimated kinetic parameters. Values
near 1 indicate strongly correlated parameters. The lack of fit test table provides a comparison of the
weighted residuals and chi-squared value. A weighted residual less than the χ2(95%) indicates a good
fit between the model and the measured data. It also indicates whether the model sufficiently describes
the behaviour data. For cases where the weighted residual is greater than the χ2(95%) , the model may
need modifications to better describe the experimental data.
3.2.4 Optimisation tool
gPROMS can be used to optimise the steady-state and/or the dynamic behaviour of a continuous or
batch process. Both plant design and operational optimisation can be carried out. The form of the
objective function and the constraints can be quite general. Moreover, the optimisation decision variables
can be either functions of time or time-invariant quantities. By default, gPROMS treats optimisation
problems as dynamic ones, optimising the behaviour of a system over a finite non-negative time horizon.
However, in some cases, it is desired to optimise a system at a single time point–performing a so-
called ”point” optimisation. From the mathematical point of view, this is equivalent to solving a purely
algebraic problem in which a generally nonlinear objective function is maximised or minimised subject
to generally nonlinear constraints by manipulating a set of optimisation decision variables that may be
The new gPROMS platform features a new capability of performing uncertainty analysis with Monte
Carlo methods on the studied models through a tool called ”Global System Analysis” or GSA. This
Monte Carlo method will simulate the model in deterministically or probabilistically ways in order to
evaluate the uncertainty of the model itself. This method allows the user to assess the uncertainty of
the input variables. When choosing these variables and their range(called factors in GSA), the user
must have a deep knowledge of the model due to the large impact these variations might have on the
output variables under consideration (called responses in GSA). GSA is particularly useful to clarify
some aspects of the problem, by flagging models used out of context or to a degree of complexity not
sustained by available information
Setting up the experiment
GSA allows the user to perform uncertainty analysis or sensitivity analysis. This tool allows the user
to choose from two sample generation methods: Quasi-random (Sobol) sampling and pseudo-random
sampling. Let us assume the model under consideration can be represented by a function of the form
Y = f(X1, X2, ..., Xk) (3.3)
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where Y is the model prediction or output and X1, X2, ..., Xk is a set of uncertain input variables, defined
over ω, the k-dimensional unit hypercube
ω = (X|0 ≤ xi ≤ 1; i = 1, ..., k) (3.4)
Using pseudo-random sampling is much simpler than using a complex design because samples can
be generated at will. Although one problem with pseudo-random sampling is that generated samples
tend to have clusters and gaps. Therefore, discrepancy characterizes the lumpiness of a sequence
of points in a multidimensional space. Random sequences of k-dimensional points tend to have high
discrepancy [36]. Nonetheless, infinite sequences of k-dimensional points tend to have much lower
discrepancies, and they are called low-discrepancy sequences. When the number of samples is very
large, the discrepancy shrinks at the theoretical optimal rate. As a result, an estimated mean of function
3.3 will converge much more quickly than would an estimated mean based on the same number of
random points.
Quasi-random sequences are specifically designed to generate samples of X1, X2...Xk in the most
uniform possible way over the unit hypercube. Unlike random numbers, quasi-random points know about
the position of previously sampled points and fill the gaps between them [37]. For that reason they are
called quasi-random despite not being random at all.
3.3.1 Uncertainty analysis
Monte Carlo analysis is based on performing multiple model evaluations with randomly selected model
input, and then using the results to determine the uncertainty in model predictions. This procedure
involves 4 steps:
1. Select a range and probability input distribution function for each input. In gPROMS it supports
normal, uniform and discrete distributions or grided sets.
2. Generate a samples using a sampling method (Quasi-random Sobol sampling or pseudo-random)
3. Evaluating the model at each sample point Xk, obtaining the output Y . Model evaluation is the
most expensive in terms of computational time being its total, the product of the average time
needed to make one model run and the sample size used in the analysis.
4. Take the output values Y as the basis for the uncertainty analysis. The expected value and vari-
ance for the output value can be estimated by:
E(Y ) =1
N
N∑k=1
Yk (3.5)
V (Y ) =1
N − 1
N∑k=1
Yk − E(Y )2 (3.6)
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The results of an uncertainty analysis in gPROMS assume 3 different forms:an histogram, a scatter
2-D and a table with expected value, standard deviation, minimum value and maximum value.
3.3.2 Sensitivity analysis
Variance-based Sensitivity Analysis in gPROMS is the study of how the variance of the model output
depends on the input factors that are affected by uncertainty. Variance-based sensitivity indices mea-
sure the influence of individual factors on the model output. Based on Sobol’ [38] there are two types
of variance-based sensitivity indices: first-order effect index,Si and total effect index,STi. gPROMS
variance-based sensitivity analysis method is based on Saltelli’s method [? ] and the formulae esti-
mating the sensitivity indices are the ones proposed in [? ]. In variance based sensitivity analysis,
factors may be independent or correlated. The first-order effect index (Si) represents the main effect
contribution of each input factor to the variance of the output. The same quantity is also known as ”im-
portance measure”. The higher the value of Si, the higher the influence of the i-th factor on the output.
In other words, a high value of Si signals an important factor. If Si = 0, then the i-th factor has no direct
influence on the output; however, it may still be an important factor through its interactions with other
factors. Hence, a small value of Si does not necessarily imply a non-influential factor. For this reason,
its total effect must also be examined; a significant difference between Si and STi indicates an important
interaction involving that factor. The sum of all Si is always lower than or equal to 1. If it is equal to 1,
then there are no interactions between the factors and this implies that the model is additive. The total
effect index (STi) accounts for the total contribution to the output variance of the i-th factor, including
its individual contribution (first-order effect) plus all higher-order effects due to its interactions with other
factors. STi must be higher or equal to Si. If it is equal, then the factor has no interactions with the other
factors. If STi = 0, the i-th factor has no influence on the model output and the factor can be fixed at
any value within its range of uncertainty. The sum of all STi is always higher than or equal to 1. If it is
equal to 1, then there are no interactions between the factors. Variance-based sensitivity analysis is a
powerful technique that can be applied in several different settings, for example in factor prioritisation.
This setting is used to identify a factor which, when fixed to its true value, leads to the greatest reduction
in the variance of the output. In other words, the identified factor is that which accounts for most of the
output variance. Therefore, this setting allows the analyst to detect and rank those factors which need
to be better measured in order to reduce the output variance. It also allows the identification of factors
to be estimated in a subsequent numerical or experimental estimation process. Factor fixing is very im-
portant to identify factors in the model which, if left free to vary over their range of uncertainty, make no
significant contribution to the variance of the output. The identified factors can then be fixed, anywhere
in their range of variation, without affecting the output variance. Detecting and fixing the non-influential
factors may result in significant model simplification. Saltelli and Tarantola [39] also showed that the
total effect index provides the answer to the Factor Fixing setting. Recall that STi = 0 is a necessary
and sufficient condition for an input factor to be non-influential. The main drawback of variance-based
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sensitivity analysis is its high computational cost, which means that it becomes prohibitive to apply to
computationally expensive models. Estimating the sensitivity indices requires a large number of model
evaluations. In particular, with Saltelli’s method (2002) implemented in gPROMS Global System Anal-
ysis, N*(k+2) model evaluations are required (for each deterministic scenario) in order to estimate the
entire set of first-order and total effects, where N is the number of probabilistic samples requested by the
user. For example, for a model with 15 (probabilistic) factors, at least 17000 simulations (taking N=1000)
need to be executed per scenario. [35]
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Chapter 4
Solid dosage stability
This chapter presents the Solid dosage stability model developed in the gPROMS ModelBuilder platform
and the results obtained with the implementation of a case study. The developed model was validated
and its usage was tested under different scenarios. In the end of the chapter, the Packaging model is
also introduced and a case study is also explored. Uncertainty analysis and sensitivity analysis were
performed in this chapter to assess the statistical relevance of the results.
4.1 Model development
A lot of interest has been shown by pharmaceutical companies to have a user-friendly tool that allows
the prediction of degradation on an API formulation. With this in mind, a mathematical model based on
the humidity corrected Arrhenius equation referred in equation 2.2 in subsection 2.1.6 was developed in
gPROMS platform (figure 4.1).
To support this, a workflow is proposed to study degradation on solid dosages (figure 4.1)
The user starts by building the flowsheet model of the experiment containing the modified Arrhenius
equation proposed by [6] as shown by the step 1 in figure 4.1. The model is of the type
dX
dt= A.e
−EaRT +B(RH).f(X) (4.1)
where f(X) represents the chosen kinetic model that better explains degradation data acquired in the
ASAP. For this specific case study, the zero order model was used given the linearity of the data. These
models have been previously explained in section 2.1. Then, experimental data must be inserted in
the performed experiments tool (step 2 in figure 4.1). This data corresponds to accelerated conditions
and consists of degradation % vs time for different temperatures and relative humidities. After this
step, the user will use this data to estimate parameters for the model (Ea, Ln(A) and B) (step 3 in
figure 4.1). Taking these parameters extracted for accelerated data, the user can then insert normal
storage conditions in the model (e.g 298K, 60%RH) and the model will show the predicted shelf-life for
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Figure 4.1: Degradation workflow; 1) Flowsheet of the model in the gPROMS platform; 2) Accelerateddata input; 3) Parameter estimation tool; 4)Shelf life analysis with the solid dosage stability model; 5)Global System Analysis for uncertainty and sensitivity analysis of the results obtained in step 4);
those same conditions in the built-in report (step 4 in figure 4.1) Since these results do not show any
uncertainty of the parameters, a GSA must be used to assess those boundaries (step 5 in figure 4.1).
This analysis may include normal distributions in external parameters (RH and T) or normal distributions
in kinetic parameters (Ea, Ln(A), B) having as limits, the 95% confidence intervals provided by the
parameter estimation tool.
When building the model, all of the equations were implemented in the gPROMS platform and the
user interface was created within the gPROMS interface language in HTML (figure 4.2).
In figure 4.2 (a) the user can specify the kinetic model to be used (default zero order) and the modified
Arrhenius parameters can be inserted after parameter estimation. In (b) the user specifies simulation
time, temperature and RH to perform the shelf-life analysis while in (c) initial degradant percentage can
be inserted. This has to do with degradant that is often found in excipients coming from the suppliers or
degradant that was formed during the product formulation process. Finally in (d) the user specifies the
maximum limit for the studied degradant.
In figure 4.3 it is possible to note in 1) a table with degradant percentage and time. The user is able
to see in real time the value of this variable for each time point just by scrolling from left (day 1) to right
(simulation time chosen in figure 4.2 (b)) a button on top of the report (not shown in the figure 4.3). In
section 2) of the report, a plot of degradation % vs. time is generated along with the specification limit.
This model was built having in mind possible end users that are not familiar with gPROMS. Therefore,
an easy to use interface in a drag and drop flowsheet associated with automatic reporting of results
reporting was of high importance to minimize possible misinterpretations.
Table 4.2: Parameter estimation report: Weighted residual vs. χ2;
Weighted residual χ2 square confindence interval
7.9 22.3
As seen in table 4.2, the weighted residual is lower than the χ2 value which means that the model has
a good data fit. Nonetheless, it was seen that if the variance in the error measurement was increased
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Figure 4.4: gPROMS parameter estimation report with 0.02 variance error in the measurements; Finalvalues of the parameters, confidence intervals and standard deviations are presented in the report;
to 0.04 (figure 4.5) standard deviation of the kinetic parameters and 95% confidence intervals would
almost double when compared to the results presented in figure 4.4. Performing experiments that are
exact and reproducible is of high interest for this model and any imprecision in the measurements may
lead to greater uncertainties on the results.
Figure 4.5: gPROMS parameter estimation report with 0.04 variance error in the measurements; Finalvalues of the parameters, confidence intervals and standard deviations are presented in the report;
Applying the estimated parameters in figure 4.4 to the developed model Degradation regressor and
implementing the storage conditions of 298K and 60%RH (storage conditions recommended by the ICH
guideline Q1A for long term stability testing[3]) allowed the estimation of a possible shelf-life for this
drug. As seen in 4.6, the degradant percentage surpasses the specification limit by the day 3070, what
corresponds to approximately 8.4 years. This was accomplished while using the zero-order model (as
shown in figure 4.2 (a)) given the linearity of the provided data.
4.2.1 Global System Analysis-Uncertainty analysis
Nonetheless, this value is meaningless given that there were no uncertainties associated in the cal-
culation. In a real case scenario, the temperature and RH will not be constant (external parameters)
neither will the kinetic parameters (Ea, Ln(A), B). At a given time point, a higher temperature and RH
may cause the drug to temporary lose its crystallinity in a given point of the lattice making its molecules
become more mobile. This means that less energy will be required to activate one of the chemical
degradation reactions occurring in the formulation, thus decreasing Ea. In order to test the relevance of
these values, the global systems analysis tool was used. An uncertainty analysis was performed with
a simulation time of 730 days, corresponding to two years of the shelf life. This is the typical shelf-life
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Figure 4.6: Predicted shelf life by the Degradation model; Specification limit (orange line), ICH long termstorage condition 25◦, 60% RH (blue line);
value for most pharmaceutical drugs [8]. A 500 sample size was used in the uncertainty analysis. Figure
4.7 shows the moving average degradation % of the two sampling methods present in gPROMS Global
System Analysis. It is possible to verify that a Quasi-random Sobol sampling converges faster than a
pseudo-random sampling due to it’s location memory properties as explained in section 3.3.
Figure 4.7: Average convergence of degradation % with the two sampling methods available in gPROMSGSA (Quasi-random Sobol and Pseudo-random);
Figure 4.8 depicts the design space on the two external factors for the uncertainty analysis. A normal
distribution for temperature was used with an average of 298K and a standard deviation of 12K. For the
RH, a normal distribution with an average of 60% and a STD of 20% was used. In this case study, Monte
Carlo simulation demonstrates that there is a strong correlation between temperature and degradation
rate. On the other hand, RH does not seem to be correlated with degradation since there is not a trend
in the scatter of figure 4.9. The latter was already expected by parameter estimation since the B value
(sensitivity to relative humidity) was close to zero (B=0.0037±0.0018), meaning that this formulation is
not very sensitive to RH.
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Figure 4.8: External factors (RH and temperature) design space with quasi-random (Sobol) sampling(500 samples);
(a) RH vs. %Degradation (b) Temperature vs. % Degradation
Figure 4.9: Uncertainty analysis on the external parameters; Quasi-random Sobol sampling was usedon 500 samples;
Table 4.3: Distribution statistics table extracted from gPROMS GSA tool on day 730;
Degradation %
Expected value 0.647Standard deviation 0.114
Minimum value 0.525Maximum value 0.998
As seen in figure 4.4, gPROMS estimated a 95% confidence interval for each one of the kinetic
parameters. An uncertainty analysis was performed between the boundaries of these parameters to test
their impact on degradant % (figure 4.10). There is an exponential relationship between activation energy
Ea and degradant % for the confidence intervals considered. The percentage of degradant may rise as
high as 8% for the lower value of Ea which was 16.73Kcal/mol. The same relation is observed for the
logarithm of the collision frequency leading to a degradation of 5% on the upper end, at Ln(A) = 24.64.
This behavior was expected given the relation of all of these parameters in the Arrhenius equation. On
the other hand, there is a linear correlation between degradation and the sensitivity to relative humidity.
Although, the variation in this parameter leads to much lower degradations when compared with the two
other parameters. Another feature of gPROMS Global System Analysis is it’s capability of generating
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(a) Ea correlation with degradant % (b) Ln(A) correlation with degradant %
(c) B correlation with degradant %
Figure 4.10: Uncertainty analysis on the 95% confidence intervals of kinetic parameters; Quasi-randomSobol sampling was used on 500 samples;
Figure 4.11: Histogram generated with gPROMS GSA; Normal distribution was used on the externalparameters with simulation run on day 730; 12.4% of the samples are over the limit (red area);
histograms(absolute and normalized) and a distribution statistics table. Table 4.3 shows that even though
the expected value of degradation for a sample at the 2-year mark is 0.647%, the maximum value is
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0.998% which is already over the specification limit. Figure 4.11 allows the extraction of the percentage
of samples that were already over the specification limit at the 2 year shelf life. This value was equal to
12.4%.
(a) Normal distribution on external parameters; kinetic parameters with fixedvalues;
(b) Normal distribution on external and kinetic parameters
Figure 4.12: Uncertainty analysis with normal distributions on model parameters; 500 samples usingQuasi-random Sobol sampling
Figure 4.12 shows degradation % vs. time while using uncertainty on the model input parameters.
In (a) that uncertainty is only accounted on the external parameters (RH and T) in the form of a normal
distribution while kinetic parameters (Ea, Ln(A), B) are fixed. If it is known that in the second year of
shelf life there are 12.4% of samples over the limit, figure 4.12 (a) allows one to see when it is most likely
that there is going to be one sample over the limit, which should be at about day 500. In (b), a normal
distribution in both external and kinetic parameters is considered, thus increasing model uncertainty. For
the kinetic parameters with normal distribution, the mean value was the one estimated by the gPROMS
parameter estimation tool (figure 4.4 and standard deviation 10% of those values. It is possible to note
by figure 4.12 (b) that there will be samples over the limit since day one of the experiment, given the blue
dots over the red line (specification limit). This is associated with the increased uncertainty of the model
inputs and since degradation is related exponentially with Ea, any small variance in this value will have
tremendous impact on degradation.
To test the relevance of these results, another Global system uncertainty analysis was performed in
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two scenarios: 1) using normal distributions only on external parameters; 2) Using normal distributions
in all the parameters. This analysis was performed over 720 days and over 100000 samples. In the first
scenario, the probability of failing specification was 12.4% (as seen before) while for the second scenario
the probability of failing specification limit was 35.8%. This reveals the importance that uncertainty on
kinetic parameters might have in degradation models. Additionally, it is an important factor for manage-
ment personnel to decide whether this is an acceptable failure rate to keep going with the degradation
experiments or if it is a better move to invest in understanding more on the certain values for the kinetic
parameters that are responsible for the most variability in the results. Even though the model has a good
data-fit, if the uncertainty of the model parameters is not taken into account, the predicted shelf-life may
not be feasible. Therefore, a statistical analysis should always be performed.
4.2.2 Global System Analysis-Sensitivity analysis
In order to test the effects that each variable has on the output (degradation%) a sensitivity analysis was
performed on 350000 samples. This value was found by trial and error until convergence of the method
(b) Desiccant mass uncertainty analysis (c) Number of tablets uncertainty analysis
Figure 4.15: GSA on packaging model parameters; Quasi-random Sobol sampling of 500 samples;
not have a great impact on degradation.
Figure 4.16: Time until first sample over the spec. limit
By comparing figure 4.16 and figure 4.12 (a) it possible to note that packaging has extended the time
until one sample is over the limit (from 500 to 750 days). Since this experiment was adapted from Case
study 1 and [11], the reached conclusions with this experiment are not realistically meaningful but they
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can show the importance of packaging in drug degradation. Moreover, as stated in 4.2.1, the tablets of
this case study were not very sensitive to RH (due to the low value of B) and yet the validation with the
packaging model showed that it would extend its shelf-life. It is possible to draw a conclusion that if the
drug was more sensible to RH (higher values of B), the packging would have much more importance in
extending the drug shelf-life.
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Chapter 5
Parenteral solution stability
This chapter presents the Parenteral Solution stability model developed in the gPROMS ModelBuilder
platform and the results obtained with the implementation of a case study. The developed model was val-
idated and its usage was tested under different scenarios. Uncertainty analysis and sensitivity analysis
were performed in this chapter to assess the statistical relevance of the results.
5.1 Model development
A model for parenteral stability analysis was build on the gPROMS ModelBuilder platform. The challenge
for building this model has to do with the lack of understanding of metal speciation occurring in parenteral
solutions given the different sources of metal contamination explained in section 2.2. With that in mind,
the model also aimed to determine minimal concentrations of chelating agents that prevent the formation
of any precipitates, thus reducing costs for the pharmaceutical companies. The Parenteral stability
model built in the gPROMS platform is the only one available to date that is capable of integrating all the
available information and empower the user with a robust tool.
5.2 Parenteral stability workflow
To effectively use this model, the user starts by building the flowsheet with the template model that
contains the chemical equilibrium equations by drag and drop (block 1 of figure 5.1). The user then
selects options such as the minimum and maximum pH for the experiment and the number of conditions.
The available number for the latter parameter vary between 10 and 50, meaning that if the user chooses
50 pH conditions, gPROMS will divide pH range in 50 portions. Choosing 10 pH conditions lowers
resolution of the results thus decreases running time of the model (block 2 of figure 5.1 and figure 5.2).
After this step, the user proceeds to the equilibrium reactions tab where he will choose metals from the
database by clicking the three dotted button in figure 5.3. Clicking this button pops up a new window
with all the available metals from the database as shown in figure 5.4. After they have been selected,
the metals will then appear as chosen metals and the model will automatically create new boxes for the
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Figure 5.1: Parenteral stability workflow; 1)Flowsheet of the model with model equations for chemicalequilibrium; 2)Option selection within the model; 3) Simulation and results observation; 4)Global Sys-tems Analysis for uncertainty and sensitivity analysis; 5) Optimization tool;
metal molar concentration. As seen in figure 5.3, because only Fe3+ is selected, there is only one box in
the metal concentration variable. Solution volume is also needed in order to close all degrees of freedom
of the model. The user then proceeds to the chelating reactions tab (figure 5.5) where he will choose
a chelating agent from the database by clicking the three dotted button on the right. This action pops
up the chelating agents database as shown in figure 5.6. Again, this action creates new boxes in the
chelating agent liquid molar concentration according to the number of chosen chelating agents. Finally,
in the solubilization tab, the user will have to manually input the Log(Ksp) for the solubilization reactions
occurring in the created system. Because this variable is an array of the chosen metals variable, the
model will automatically create new boxes for the Log(Ksp) according to the number of chosen metals.
This means that if the user had chosen iron and aluminium, he would be asked to assign two Ksp’s in
this tab.
After all the options have been filled in, the user can validate the model observing the speciation
graphs, supersaturation and mass of metal in liquid and solid phases (block 3 of figure 5.1). Af-
ter the model has been validated, one can use gPROMS GSA for factor prioritization. By doing a
sensitivity analysis, the user finds out what are the chelators that contribute the most for model out-
puts(supersaturation) (block 4 of figure 5.1) and may use afterwords gPROMS optimisation tool (block
5 of figure 5.1) to answer the question ”What is the minimum amount of chelating agent I need to add
to the system to avoid any precipitation?”. Nonetheless, given the uncertainty of the parameters (for
example normal distributions in metal concentration) the user can go back to Global System Analysis to
further understand the probability of failure of the optimised value.
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Figure 5.2: pH tab in Parenteral Stability model GUI
Figure 5.3: Equilibrium reactions tab in Parenteral Stability model GUI
Figure 5.4: Metal database in the Parenteral Stability model GUI
Figure 5.5: Chelation reactions tab in Parenteral Stability model GUI
Although for the purpose of this work, the technical aspect of the implementation in the gPROMS
platform is not relevant, it was a challenging and time-consuming step. At one point, the gPROMS
solver was already dealing with over 12000 variables as shown in figure 5.8. Moreover, the conception
of user interfaces in HTML language was also a a huge barrier in this model development as shown in
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Figure 5.6: Chelating agents database in the Parenteral Stability model GUI
Figure 5.7: Solubilization reactions tab in Parenteral Stability model GUI
figures 5.2 to 5.7.
Figure 5.8: gPROMS structural analysis
5.2.1 Model validation
The model was validated for 5 different chelating species (Succinate, citrate, acetate, histidine) as shown
in figures 5.9 to 5.12.
This validation was performed by comparison with other tools available for metal speciation in par-
enteral solutions. In Appendix A it is possible to note the reactions that are computed by the model when