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.

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 5

2.1 Solid dosage stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Product stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Chemical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Chemical stability models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Physical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.5 Stability programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.6 Accelerated ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.7 Current industrial situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.8 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Parenteral solutions stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Implementation 19

3.1 gPROMS R© Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 gPROMS R© ModelBuilder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Performed experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Parameter estimation tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Analysis of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.4 Optimisation tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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3.3 gPROMS R© Global Systems Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Solid dosage stability 27

4.1 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Case Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Global System Analysis-Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Global System Analysis-Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Application of the packaging model on Case Study 1 . . . . . . . . . . . . . . . . . 37

5 Parenteral solution stability 41

5.1 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Parenteral stability workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusions 53

6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Bibliography 57

A Parenteral solution equations 61

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List of Tables

2.1 Solid-state reaction models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Parameter estimation symbol definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Variance models for parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Accelerated degradation data (degradant %) imported from PSE’s partners at different

temperatures and relative humidities; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Parameter estimation report: Weighted residual vs. χ2; . . . . . . . . . . . . . . . . . . . 30

4.3 Distribution statistics table extracted from gPROMS GSA tool on day 730; . . . . . . . . . 33

4.4 gPROMS Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 gPROMS Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 gPROMS Sensitivity analysis for factor prioritisation; . . . . . . . . . . . . . . . . . . . . . 47

5.2 Distribution statistics table for citrate concentration of 0.448M . . . . . . . . . . . . . . . . 49

5.3 Distribution statistics table for citrate concentration of 0.6M . . . . . . . . . . . . . . . . . 49

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List of Figures

2.1 Pharmaceutical R& D expenses; Source: Thomson Reuters . . . . . . . . . . . . . . . . . 10

2.2 Degradation study workflow used in industry; Steps needed to achieve a product formu-

lation over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Moisture equilibration in different packaging arrangements; Points represent the experi-

mental data and lines the fitted model [4]; . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Different packaging systems [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Parameter estimation tool: Experiments and measurements . . . . . . . . . . . . . . . . . 21

4.1 Degradation workflow; 1) Flowsheet of the model in the gPROMS platform; 2) Accelerated

data 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); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Degradation regressor model user interface; . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 HTML report automatically generated by gPROMS; 1) Degradation table; 2) Degradation

plot: degradant percentage (red line), specification limit (yellow line); . . . . . . . . . . . . 30

4.4 gPROMS parameter estimation report with 0.02 variance error in the measurements; Final

values of the parameters, confidence intervals and standard deviations are presented in

the report; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 gPROMS parameter estimation report with 0.04 variance error in the measurements; Final

values of the parameters, confidence intervals and standard deviations are presented in

the report; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 Predicted shelf life by the Degradation model; Specification limit (orange line), ICH long

term storage condition 25◦, 60% RH (blue line); . . . . . . . . . . . . . . . . . . . . . . . . 32

4.7 Average convergence of degradation % with the two sampling methods available in gPROMS

GSA (Quasi-random Sobol and Pseudo-random); . . . . . . . . . . . . . . . . . . . . . . . 32

4.8 External factors (RH and temperature) design space with quasi-random (Sobol) sampling

(500 samples); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.9 Uncertainty analysis on the external parameters; Quasi-random Sobol sampling was used

on 500 samples; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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4.10 Uncertainty analysis on the 95% confidence intervals of kinetic parameters; Quasi-random

Sobol sampling was used on 500 samples; . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.11 Histogram generated with gPROMS GSA; Normal distribution was used on the external

parameters with simulation run on day 730; 12.4% of the samples are over the limit (red

area); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.12 Uncertainty analysis with normal distributions on model parameters; 500 samples using

Quasi-random Sobol sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.13 Degradation regressor model with packaging upgrade (user interface); . . . . . . . . . . . 37

4.14 gPROMS packaging model results; Relative humidity inside the bottle (dotted line), spec-

ification limit (orange line) and degradant % (blue line); . . . . . . . . . . . . . . . . . . . . 38

4.15 GSA on packaging model parameters; Quasi-random Sobol sampling of 500 samples; . . 39

4.16 Time until first sample over the spec. limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Parenteral stability workflow; 1)Flowsheet of the model with model equations for chemical

equilibrium; 2)Option selection within the model; 3) Simulation and results observation;

4)Global Systems Analysis for uncertainty and sensitivity analysis; 5) Optimization tool; . 42

5.2 pH tab in Parenteral Stability model GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Equilibrium reactions tab in Parenteral Stability model GUI . . . . . . . . . . . . . . . . . . 43

5.4 Metal database in the Parenteral Stability model GUI . . . . . . . . . . . . . . . . . . . . . 43

5.5 Chelation reactions tab in Parenteral Stability model GUI . . . . . . . . . . . . . . . . . . . 43

5.6 Chelating agents database in the Parenteral Stability model GUI . . . . . . . . . . . . . . 44

5.7 Solubilization reactions tab in Parenteral Stability model GUI . . . . . . . . . . . . . . . . 44

5.8 gPROMS structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.9 Fe(III) 1mM - Citrate 20mM speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.10 Fe(III) 1mM - Succinate 20mM speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.11 Fe(III) 1mM - Acetate 20mM speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.12 Fe(III) 1mM - Histidine 20mM speciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.13 Iron speciation- Fe=1mM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.14 Fe(III) 1mM - Citrate 20mM; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.15 Mass fraction Fe(III) 1mM - Citrate 0.4488M . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.16 Supersaturation Fe(III) 1mM - Citrate 0.4488M . . . . . . . . . . . . . . . . . . . . . . . . 48

5.17 Convergence of the supersaturation average over 300 samples . . . . . . . . . . . . . . . 49

5.18 Uncertainty analysis on 0.4488M Citrate; . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.19 Uncertainty analysis on 0.6M Citrate; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.20 Multiple metals- Al 1mM +Fe 1mM + Succinate 20mM species . . . . . . . . . . . . . . . 50

5.21 Multiple metals: Al 1mM + Fe 1mM + Succinate 20mM; . . . . . . . . . . . . . . . . . . . 51

5.22 Multiple chelating agents-Fe 1mM + Succinate 20mM + Citrate 20mM species . . . . . . 52

5.23 Multiple chelating agents: Fe 1mM + Succinate 20mM + citrate 20mM; . . . . . . . . . . . 52

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Nomenclature

Greek symbols

A Collision frequency

B Sensitivity to relative humidity

Ci Concentration of species i

Ea Activation energy

k Degradation rate

R Perfect gases constant

Roman symbols

GSA Global System Analysis

GUI Graphical user interface

HTML HyperText Markup Language

NCE New chemical entity

RH Relative humidity

T Temperature

WVTR Water vapor transmission rate

Superscripts

R© Registered brand

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Glossary

ASAP Accelerated stability assessment program. It is

a program that defines temperatures, relative

humidities and oven times in which samples

must be placed to simulate accelerated condi-

tions of degradation.

PN Parenteral Nutrition is a multi component solu-

tion that aims to feed the patient with its daily

needs of both macro and micronutrients. Phar-

maceutical drugs can also be added to this so-

lution for a specific treatment.

gPROMS general Process Modelling System, is the pro-

prietary software developed by Process Sys-

tems Enterprise. It is among the most ad-

vanced general purpose process modelling and

simulation software available today.

gSAFT gSAFT stands for statistical associated fluid

theory. It is a gPROMS tool that predicts physi-

cal properties and reaction kinetics of chemical

species based initial inputs such as partition co-

efficients

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Chapter 1

Introduction

Determining the stability of drug substances and drug products is one of the most critical steps in the

drug development process. Stability refers to the storage time allowed before any degradation product

in the dosage form achieves a sufficient level to represent a risk to the patient. Based on this time, the

expiration date or shelf-life of a product is determined. The allowable level of any degradant depends on

its toxicity and quantity on the drug but generally, the allowable levels are well below 1%m/m. Formation

of degradants during an anticipated shelf-life (usually 2-3 years) may result in significant product intro-

duction delays, especially if such instability is uncovered late in the development process. Once clinical

trials have begun, a change in formulation may necessitate additional clinical trials to assure bioequiv-

alence of the new formulations which leads to increased development costs. Consequently, there is a

strong incentive to predict accurately any instability in pharmaceutical formulations as early as possible

in the development process, thereby enabling remedies to be applied. It is of high importance that these

methods are effective enough to predict slow rates of degradant formation and yet, remain accurate

enough so that degradation problems are addressed [1].

1.1 Motivation

The development of adequate pharmaceutical manufacturing process understanding is required by

health regulators on a global scale. Mechanistical and statistical modelling of the manufacturing pro-

cess can be used to demonstrate this understanding. This kind of approach to address manufacturing

issues is becoming common to expectation for pharmaceutical products, evident in the FDA Validation

Guidance [2], ICH Guidance [3]. Such modelling is needed to inter-relate the various manufacturing

sources of variability to show how changes in any of these affect the overall drug substance and drug

product quality. In addition, such modeling makes business sense, as it: 1) reduces the number of out of

specification (OOS) events and defects; 2) aids in the ability to define transfer functions that will be used

in process control; 3) reduces development time; 4) ensures patients have a continuous supply of safe,

efficacious, and quality product; 5) eliminates post-approval submissions, if the change is within design

space; 6) enables continuous improvement. Variability of the process and variability of analytical meth-

1

ods used to test the product are the main key drivers in model uncertainties regarding product stability.

Understanding each of these separately is not enough. The understanding of how they inter-relate and

influence each other is of high importance to ensure product quality.

Consequently, pharmaceutical companies invest a lot of money and time in drug stability programs

to ensure that the final product is stable over its shelf-life and throughout that time, degradation products

cannot cause adverse effects on the patients. Therefore, a solid dosage degradation model and a

parenteral stability model were created to address these issues and provide the end user with a robust

tool to make better decisions.

All of the present work was done at Process Systems Enterprise, a leading supplier of Advanced

Process Modelling software and model-based engineering and innovation services to the process in-

dustries.

1.2 State of the Art

Drug stability programs and methodologies to assess degradation products in pharmaceutical drugs

have been thoroughly described in [4], [5], [6], [7], [8], [9].

Packaging systems to prevent drug degradation and mathematical models are available in literature

[10], [11], [12], [13], [14].

Although state of the art work regarding parenteral solutions stability is described in this work, a lot

of information can be found in [15], [16], [17], [18], [19] and [20].

1.3 Thesis Outline

Firstly, in chapter 2, a literature review was performed over the current methodologies and approach

to solid dosage stability. The main drivers in degradation are explained such as chemical and physical

sources of instability and the concept of accelerated aging is also thoroughly reviewed. Packaging of

pharmaceutical and models for moisture permeation are also reviewed. It was also intended to give an

insight of how parenteral solutions may become unstable, despite the lack of literature in this field of

expertise.

Chapter 3 describes the software platform in which all the modelling present in this work was based

on. Additionally, all the important tools and capabilities used in the models implementation are thor-

oughly discussed such as parameter estimation, sampling and uncertainty and sensitivity analysis.

In chapter 4, a mathematical model for solid dosage stability on pharmaceutical drugs was imple-

mented. Additionally, a workflow to assess pharmaceutical drug stability was proposed. Further in this

chapter, a case study with real data was implemented and the proposed workflow was applied.

Chapter 5 represents the second part of this work, where a mathematical model for parenteral solu-

tion stability was developed. A workflow for these studies is proposed and a case study was implemented

to validate the model.

2

Finally, chapter 6 summarizes the main conclusions with the present work and discusses all the

future work that needs to be addressed.

3

4

Chapter 2

Background

The drug development process is a time-consuming process. Usually it takes over 10 years to bring

a new chemical entity (NCE) to market [5]. The drug development process generally consists of three

periods: discovery/toxicology, clinical development and commercialization. In the first step, studies are

conducted on animals with the purpose of understanding the safety and biological activity of the NCE.

In the clinical development there are four phases: Phase I, determination of safety and dosage on 20-

100 healthy volunteers; Phase II, evaluate effectiveness and side-effects on 100-500 patient volunteers;

Phase III, confirm effectiveness and monitor adverse reactions for long term use on 1000-5000 patient

volunteers; Phase IV, additional testing and post-approval changes. Stability testing plays an essecial

role in the drug development process. The safety and efficacy of drug products are established during

development via clinical studies. If the drug product stability profile changes beyond established ac-

cepted criteria, the agreed safety and efficacy are no longer applicable, and thus, safety and efficacy

need to be re-established.

Stability is a critical quality attribute of pharmaceutical products; therefore, stability testing plays a

crucial role in the drug development process [16]. The purpose of stability testing is to provide evidence

on how the quality of a drug substance or drug product varies with time under the influence of a variety

of environmental factors such as temperature, humidity and light and to establish a retest period for the

drug substance or a shelf-life for the drug product and recommended storage conditions.[3] Therefore,

it encompasses all the phases of the drug development process. A testing program for stability samples

requires a tremendous amount of resources and expertise. However, it is important to note that these

studies support decision-making activities of management personnel and may represent the difference

between profit or loss for pharmaceutical manufacturers.

2.1 Solid dosage stability

2.1.1 Product stability

One of the major challenges while creating a new pharmaceutical dosage form is to assure the stability

of the product under different environmental conditions.

5

Drug product shelf-life is determined based on the time a product remains within specifications

agreed upon with regulatory agencies [4]. These specifications can be divided into two dimensions:

physical stability and chemical stability. In the latter, one must assure that the drug product, in its pack-

aging, still has the proper potency (expressed as m/m% of the API, tipically no less than 95% of its

initial content [1]); On the other hand, degradants have specification limits agreed upon with regulatory

agencies (ICH and FDA) due to their toxicity to humans or the uncertainty of their effects in the human

body.

2.1.2 Chemical Stability

Active pharmaceutical ingredients (API’s), be they biological (i.e protein or nucleic acids) or small chem-

ical molecules are prone to chemical degradation processes. To mantain the safety and efficacy of

pharmaceutical products, regulatory agencies demand that these degradation products must be deter-

mined in order to establish a shelf-life [5]. The shelf life of a pharmaceutical product is set based on the

time it takes for any degradation product to reach a level that it becomes a safety concern or, for the

potency of the API to drop below a critical level. It should be noted that in some cases, a degradant is

also a process impurity that is present in the initial formulation.

Hydrolytic reactions are amongst the most common processes for drug degradation [1]. In addition

to rate dependencies on temperature and moisture, hydrolysis rates can depend on the concentration

of the catalytic species, usually acids or bases. With many hydrolytic reactions, such as those involving

esters, reactions are reversible, making degradation products reform the drug. Under these conditions,

it may be necessary to use a more complex kinetic model. Another type of reaction common in drug

degradation are oxidation reactions. Oxidative degradation of pharmaceuticals can be divided in two

types: 1) reactions with molecular oxygen; 2)reaction with other oxidizing agents present in the formula-

tion. Shelf-life of oxidizable drugs can be extended by the use of antioxidants. Nonetheless, since many

antioxidants are themselves consumed as they act to stabilize the drug, the shelf-life of the drug will

depend on the time before the antioxidant is depleted. Another common process in which drugs may be

degraded is through reactions with excipients. Reactivity of drugs with excipients often involve reaction

of nucleophilic drugs (amines, sulfides) with electrophilic excipients (e.g esters, carboxilic acids). Assum-

ing the reactive excipient is present in excess, many of of the reactions with drugs will depend linearly on

excipient concentration. Additionally, excipient impurities and degradants can react either directly with

drugs or act as catalysts for other drug degradation process e.g hydrolysis or oxidation [21]. This prob-

lem is very serious since excipient impurity level may vary from supplier to supplier. Light exposure can

also induce chemical degradation in susceptible molecules. Although a discussion of the various mech-

anisms involved in photochemical reactions is beyond the scope of this work, such reactions can be

divided in oxygen dependent processes (photo-oxidations) or oxygen independent (dehydrogenations

or dimerizations) [1].

6

2.1.3 Chemical stability models

Chemical reactions are hard to classify, but one can use a set of characteristics to accurately classify

them. Reaction rate is the rate at which a chemical species loses its chemical identity per unit of volume

expressed in a rate constant k. Current pharmaceutical methodologies focus on the rate of formation of

degradants to assess the shelf-life of a drug. Although drugs may develop a variety of degradants over

time, regulators only focus on specific products that may harm patients through their toxicity levels.[1]

The rate of a solid-state reaction can be globally described by

dX

dt= A.e

−EaRT .f(X) (2.1)

where, A is the collision frequency factor, Ea is the activation energy, T is the absolute temperature, R

the gas constant,f(X) is the kinetic model and dX/dt is the rate of formation of a certain degradant.[22]

Reaction kinetics in the solid-state are often studied by thermogravimetry or other analytical methods

such as differential scanning calorimetry, power X-ray diffraction and nuclear magnetic resonance. Ki-

netic analysis can be performed by either model-fitting or model-free (isoconversional) methods. A

model is a theoretical, mathematical description of what occurs experimentally. In solid-state reactions,

a model can describe a particular reaction type and translate that mathematically into a rate equation.

Many models for solid state kinetics are found in literature. Some of them are based in mechanistic

assumptions and others are empirically extracted [22]. Models are generally classified based on the

graphical shape of their isothermal curves or in their mechanistic approach. Based on their shape,

kinetic models can be grouped into acceleratory, deceleratory, linear or sigmoidal. Based on the mecha-

nistic approach, models are divided into reaction-order,geometrical contraction, nucleation and diffusion

categories as seen in table 2.1.

Order based models are the simplest models due to their similarity with homogeneous kinetics. In

these models, the reaction rate is proportional to concentration, amount or fraction remaining of reac-

tant(s) raised to a particular power which is the reaction order[22]. Nucleation and nuclei growth models

include crystallization, decomposition, adsorption hydration and desolvation phenomena. Because crys-

tals have fluctuating local energies from surface imperfections due to cracks and defects, these sites are

ideal for nucleation reactions. Nucleation involves the formation of a new product phase (gas) at the nu-

cleation sites in the lattice of the reactant due to minimized activation energy in these spots. Power law

models are used when nucleation rate follows a power law and nucleation growth is assumed constant.

Avrami Erofeev models are based on the total number of possible nuclei-forming sites due to restric-

tions in nuclei growth. In homogeneous kinetics, products can catalyse the reaction in an auto-catalytic

process. In solid-state kinetics, auto-catalysis occurs when nuclei growth catalyses the reaction due to

the continuous formation of cracks with lower activation energies in crystal lattices. The Prout-Tomkins

model is based on this phenomena.[23] Geometrical contraction models are based on the assumption

that nucleation occurs rapidly on the surface of the crystal and there is a reaction progress towards the

center of the crystal depending on the crystal shape.[24] The difference between homogeneous and

7

Table 2.1: Solid-state reaction modelsModel f(X)

Reaction-order models

Zero-order k

First-order X

Second-order X2

Third-order X3

Parabolic 1X

Geometrical contraction models

Contracting area 2X1/2

Contracting volume 3X2/3

Nucleation models

Power law (quadratic) 2(1−X)1/2

Power law (cubic) 3(1−X)2/3

Power law (quartic) (1−X)3/4

Avrami-Erofeev (quadratic) 2X(−ln(X))1/2

Avrami-Erofeev (cubic) 3X(−ln(X))2/3

Avrami-Erofeev (quartic) 4X(−ln(X))3/4

Prout-Tomkins X(1−X)

Diffusion models

1-D Diffusion 12(1−X)

2-D Diffusion −1ln(X)

3-D Diffusion 3X2/3

2(1−X1/3

Ginstling-Brounshtein 32(X(−1/3)−1)

heterogeneous kinetics lies on the mobility of system particles. In the first case, reactant molecules are

readily available to one another at all times while in the second case, reactions occur between different

crystal lattices where particle travelling is restricted and depend on environmental factors such as tem-

perature and RH. Therefore, diffusion within the crystal lattices of the solid plays a major role in reaction

rates. The 1-D diffusion model is for a flat plane that does not involve a shape factor while the 2-D

model is used for particles that are assumed to be cylindrical and diffusion occurs through a cylindrical

shell. On the other hand, if particles have a spherical shape and diffusion occurs through its radius,

the 3-D model or the Ginstling-Brounshtein model can be used. Solid state kinetics can be studied

through experimental methods or computational methods. For experimental methodologies, data can

be acquired in an isothermal or non-isothermal way. Isothermal methods maintain samples at several

constant temperatures while non-isothermal methods employ a heating rate, usually linear (β), to raise

the temperature. In the present work, all of the data studied in case study 1 is isothermal. When studying

solid-state kinetics through computational methods, different models are fit to the data and the one with

the best statistical fit is chosen. Model free methods compute kinetic parameters (Ea and A) without

modelistic assumptions, based only on the linearity of the data[25], [22].

8

2.1.4 Physical Stability

Sometimes, a drug product shelf-life may be limited due to physical changes rather than in chemical

changes. It becomes important when the bioavailability of the API in the drug is altered and therefore, a

decrease in drug performance is observed. For solid drugs this can be seen as a decrease in the API’s

dissolution and absorption in the human tract. Solid dosage forms possess quite often disintegrants.

These agents are responsible for breaking the pill apart in the stomach due to a rapid expansion when

in contact with water. Since solid dosage forms are often manufactured with low percentages of water,

the pills when in contact with a humid environment will pick up moisture from the air and the disintegrant

might break the pill when it is stored. Another possible outcome is that the slow adsorption of moisture

throughout shelf life may cause a slow expansion of the disintegrant and when the patient uses it, it will

not have the ability to disintegrate in the stomach, compromising the performance of the drug.

A solid drug substance may exist in different forms and have different internal physical arrangements

such as a crystalline form and an amorphous form. In terms of thermodynamics, a crystalline form is

characterized with a lower energy state and in general, chemically and physically more stable than an

amorphous form [26]. These crystalline forms often show multiple packing forms in their crystal lattices

(polymorphs). Drugs may adsorb water molecules reversibly in an hydration process as function of

temperature and RH. Although, solid dosages may also experience polymorph conversion which are

irreversible. The kinetics and mechanisms of these transitions are complicated and are quite often

affected by the storage conditions. There are two possibilities when developing the polymorphic API:

a) it will be developed in the most thermodynamical stable form throughout the normal range of storage

temperatures. This means that the drug will be in the lowest state of energy and one does not have to

concern for transitions to other states; b) a metastable polymorphic form (most energetic state) will be

used due to increased bioavailability when absorbed by the human tract. In this case, it is possible that

storage conditions will induce the transition to the preferred state of lower energy and higher stability.[8]

2.1.5 Stability programs

ICH stands for International Conference on Harmonisation of Technical Requirements for Pharmaceuti-

cal Human Use. This entity brings together all of the different regulatory authorities and pharmaceutical

industries to discuss scientific and technical aspects of drug registration. Its mission is to maintain a

forum of constructive dialogue between regulatory agencies and pharmaceutical industries, contribute

to the protection of public health and to ensure that safe, effective high quality medicines are developed

and registered in the most efficient manner. With this in mind, the most important guideline on stability

testing is the Q1A(R2)([3]). The scope of this guideline is to address the information to be submitted in

registration applications for new molecular entities and associated drug products regarding their stability.

A typical drug stability study consists of 6 months of accelerated data and at least 2 years of long term

stability testing. These studies must demonstrate that the product critical quality attributes remain within

specifications [3]. There has been an effort by the pharmaceutical companies to design accelerated

stability protocols to show the regulators that this period can be shortened and therefore, many costs

9

may be avoided. Nonetheless, any attempt to extrapolate beyond data generated by the stability testing

will introduce uncertainty in the results.

2.1.6 Accelerated ageing

When assessing the stability of drugs, the use of multiple methods can help in determining the mech-

anism of degradation. Accelerated ageing traditionally involves use of temperature and RH to increase

speed of reaction rates. Using this technique, one can estimate the reaction rates at higher external

conditions and then extrapolate the rates for the desired temperatures and relative humidities.

Once a pharmaceutical company applies for a new drug approval process, the shelf-life determination

regarding product consistency over time can be one of the slowest steps. Therefore, there has been a

significant interest in accelerating testing in order to cut costs associated with stability programs.

Figure 2.1: Pharmaceutical R& D expenses; Source: Thomson Reuters

Whether in solid state or in solution, in order for an API to go to a degradation product it must undergo

some combination of collisions and molecular reorganizations. Even for chemical degradation processes

that are exothermic, the initial API form is usually fairly stable. This means that it is at a local energetic

minimum with respect to collisions and rearrangements. The result is that for the majority of chemical

degradation processes, energy is needed to overcome the activation barrier. This leads to a relationship

known as the Arrhenius equation relating the rate of reaction k and temperature. This exponential

dependence is related to the distribution of molecules having different energy levels. Therefore, in order

to accelerate degradation, stability experiments can be conducted at elevated temperatures and RH.

Once a set of stability studies are conducted at elevated temperatures, an Arrhenius plot (lnk vs 1/T)

can be made. This in turn can be used to predict the rate of formation of a degradation product at normal

storage conditions. The shelf life will then correspond to the time needed to hit the specification limit of

a certain degradant in a pharmaceutical drug [27].

API degradation in some pharmaceutical systems do not show an Arrhenius behaviour (linear cor-

relation between lnk and 1/T) due to the combination of the following events: 1)Physical changes can

10

occur over the temperature range used. Such transition typically involve melting (glass transition), va-

porization and changes in solubility. With many physical changes, there can be an abrupt discontinuity

in the Arrhenius plots; 2) Buffers can change pH with temperature which can also have some impact

in degradation reactions; 3)Multiple chemical pathways can produce the same reaction products while

having different Arrhenius parameters; 4) Humidity can have a profound effect on degradation kinetics

for solid dosage forms, causing deliquescence. This is a process where liquid water is picked up by

the sample when it is stored above its critical relative humidity (CRH). When deliquescence occurs, a

dosage form will often display unacceptable changes in appearance and performance.A number of ex-

cipients commonly used in the pharmaceutical industry have relatively low CRH values. When they are

used, they can detrimentally affect the chemical and physical stability of a solid dosage form[5].

With that in mind, Ken Waterman has called this approach the Accelerated Stability Assessment

program or ASAP. It consists of four elements: 1) the concept of isoconversion to compensate for the

complexity of solid-state-kinetics; 2) a moisture corrected Arrhenius equation that explicitly takes into

account the effects of RH on reaction rates in solid state (equation 2.2); 3) a statistical analysis to both

provide reasonable estimation of parameters and determine uncertainty for the extrapolated shelf-lives;

4) Combining the effect of RH on stability with protection provided by packaging [6].

k = A.e−EaRT +B(RH) ↔ ln(k) = ln(A)− Ea

RT+B(RH) (2.2)

As discussed before in section 2.1.2, reaction rates in solid state are hard to understand due to the

existance of molecules in multiple states (crystal lattice, amorphous state) due to low mobility. Each

of these states can potentially react to form a degradant with its own kinetics, thus making reaction

rate determination an extremely complex process. The concept of isoconversion provides an alternative

approach. Rather then focusing on reaction rates, one considers only the time required for the reaction

to form a degradant until a specification limit is achieved. While in a chemical perspective it does not

provide an accurate chemical reaction rate constant, it still provides a rate constant that represents the

contribution of all the reactions occuring at that moment to form the undesired degradant [7],[5].

2.1.7 Current industrial situation

Current indutrial methods for stability testing are ilustrated in figure 2.2. It represents the actual workflow

used in pharmaceutical companies and available software to help scientists in the degradation assess-

ment.

1. Molecular interpretation

After the API molecule as passed the toxicology tests, an extensive analysis with Lhasa Zeneth R©

tool. Zeneth R© is an expert, knowledge-based software that provides an understanding of the

forced degradation pathways of organic compounds. This will help the scientists understand what

chemical compounds are more likely to be generated through the degradation reactions.

2. Stress Testing

11

Figure 2.2: Degradation study workflow used in industry; Steps needed to achieve a product formulationover time

Zeneth R© has provided the scientists with a list of potential degradants. In this step, a forced

degradation study is performed with hydrolytic and oxidative agents to stress test the API and ac-

curately define what degradants come up with the formulation. These experiments are performed

in solution and degradants are defined via HPLC.

3. API physical form studies

In this stage, solid stability of different crystals and salt forms is tested in order to define the key

properties of the most stable thermodynamical form. These properties include measurements

of hygroscopicity, glass transition temperatures, thermal effects on the physical structure, among

others.

4. Milling studies

When the final form of the API is chosen, it is micronized and by varying particle sizes, shapes and

crystal perfection, scientists study how degradation is affected in ASAP studies, making degrada-

tion a function of these variables: Degradation = f(Particlesize, crystalperfection)

5. Blend studies

The general rule is that the API is most stable before blending. This means that after excipients

are added to the formulation, cross reactions between API and excipients are frequent. Therefore,

ASAP studies with ASAPprime R© are performed to understand what combinations of excipients

cause less degradation. A parallel flow of studies is conducted at the same time to analyse the

physical changes in the tablet after compression. One of three phenomena can occur:

(a) A recrystallization with molecule rearrangement and water movement within the formulation;

(b) Degradation

(c) Nothing changes

12

6. Drug product

When the final product is obtained as a tablet, testing for any film coating degradation reactions

takes place and the ASAP studies on tablets are performed with ASAPprime R©. From these stud-

ies the team concludes if degradation is influenced by external conditions (temperature, relative

humidity) and based upon these conclusions the packaging is chosen accordingly. If the tablet

turns out to be stable enough through an acceptable shelf-life, a less expensive option of PVC

blister package is chosen. If not, the team might choose either from using bottles with desiccants

or aluminium foil blisters, which are impermeable to moisture. Accelerated degradation data of 3

consecutive batches and the corresponding shelf-life data is sent to the regulatory agency (ICH)

for validation.

2.1.8 Packaging

Many solid pharmaceutical products may adsorb moisture during long term storage as the commonly

used packaging materials are permeable to moisture. Moisture content can be used as a critical criteria

for judging the quality of products that are degraded by moisture. However, the determination of moisture

content of packaged products in real life is expensive and time consuming. Hence the ability to predict

the moisture content during storage under a variety of conditions is very important for reducing cost

and the cycle time of product development. Container moisture permeability is a common used criteria

for container ranking. However, permeability alone is not sufficient for predicting the rate of moisture

uptake by pharmaceutical drugs as the rate is also governed by environmental conditions as well as

water activity in the container[10].

Determination of relative humidity inside packaging

A moisture sorption isotherm is an essential component in modelling moisture equilibration and drug

product stability in pharmaceutical packaging. Water vapour can interact with drug products in four

ways: 1) it may be adsorbed into crystalline surfaces to form water layers or deliquescence; 2) water can

permeate into the material bulk; 3) it can condense in cappilary channels 4) or react with excipients or

API’s to form hydrates [4]. The extent of these reactions and the amount of water taken by the sample

is dependent on the activity of water in the gas phase surrounding the sample. For packaging selection,

the moisture sorption isotherm relates the water content of the dosage form to the solid’s water activity

(which corresponds to the RH the solid is exposed to) which is correlated with chemical and physical

stability.

Plastic containers, such as high-density polyethylene (HDPE), polypropylene(PP), and polyethylene

terephtalate (PET) bottles are widely used by the pharmaceutical industry for their products. One flaw

of these containers is that they are permeable to moisture vapour. Although these permeability rates

may be low, the overall sorption of moisture into these containers may lead to chemical or physical

instability as discussed in sections 2.1.2 and 2.1.4. Conversely, loss of moisture in liquid formulations

may also lead to losses in drug performance. Therefore, it is of high importance to characterize and

13

quantify moisture permeability of pharmaceutical containers [11]. To do so, investigators measure the

water vapour transmission rate (WVTR). This process consists of measuring the weight of sealed plastic

bottles filled with some kind of desiccant (anhydrous calcium chloride or silica gel) that were equilibrated

for a known low relative humidity. After that, they are exposed to high RH’s for a fixed time period in

specific RH controlled ovens . The weight of the bottles is measured again and the weight gain during

this period is the WVTR. These methods are described by the United States Pharmacopeial convention

[28].

Waterman et al. [4] showed the moisture equilibration inside 60-cc HDPE bottles stored at 40◦ /75%RH

of 500mg tablets depends on the presence of desiccant and number of tablets (figure 2.3).

Figure 2.3: Moisture equilibration in different packaging arrangements; Points represent the experimen-tal data and lines the fitted model [4];

Desiccants are materials that have high sorption capacities and thereby can reduce RH inside pack-

aging. The most commonly used desiccants in the pharmaceutical industry are silica gel, clay minerals

and molecular sieves. These materials are typically contained in canisters or cartridges.

Modelling of water uptake in drug products has been historically described by Brunauer-Emmett-

Teller (BET) models or by Zografi et al. using the Guggenheim-Anderson-de-Boer models (GAB)[12],[14],

[13]. Hence the model used in the present work is the one proposed by Y.Chen[10]

Under constant temperature and relative humidity, the rate of moisture permeation through a con-

tainer at quasi-steady state can be written as

dw/dt = k(RHout −RH) (2.3)

where dw/dt represents the rate of moisture permeation through the container, k represents the

apparent moisture permeability of the container, RHout and RH the relative humidity outside and inside

the package, respectively. Considering that the water content the components inside the container is a

function of RH through the moisture sorption isotherm:

wi = fi(RH) (2.4)

where wi is the amount of water of the i th component in the container expressed in mg and fi is the

moisture sorption isotherm of the i th component in the container.

14

The total amount of water WT in mg of the container can be expressed as :

WT =

n∑i=1

qifi(RH) (2.5)

where qi is the quantity of i th component in the container expressed in units (tablets) or mass

(desiccant).[10]

Relating packaging with drug stability

The initial RH of the components along with moisture sorption isotherms of the components and the

WVTR model model used, it is possible to compute an RH inside the package as a function of time.

Combining this RH with chemical degradation rate as function of RH, it is possible using iterative calcu-

lation to determine degradation of the drug product over time. The effectiveness of these calculations at

predicting drug-product stability in packaging is shown in figure 2.4.

Figure 2.4: Different packaging systems [4]

As can be seen, the fact that the degradant formation rate varies exponentially with RH results

in curvature in the degradant as a function of time graph: degradation rates in moisture-permeable

packaging will generally increase with time until the RH inside the package becomes relatively constant.

In figure 2.4 , the effect of higher tablet count on product stability is accounted for in the calculation due

to the lower RH as a function of time (greater moisture sorption capacity). It is also shown the impact of

desiccant on the drug stability based on the corresponding lowering of the RH as a function of time.

2.2 Parenteral solutions stability

A parenteral solution (PS) is a medicine that is administered directly in the patient blood stream. To

develop a parenteral product, the formulator must consider challenges such as drug solubility, product

stability, drug delivery and manufacturability. Clearly, a parenteral product should be formulated with a

pH close to physiological, unless stability or solubility considerations override this. Often, the pH se-

lected for the product is a compromise between the pH of maximum stability, solubility and physiological

15

acceptability. With that in mind, many products are formulated at slightly acidic pH because of solubility

and stability considerations, and the vast majority of licensed products have a pH between 3 and 9. Nev-

ertheless, there are products in the market with extreme pH values such as Dilantin (pH 12) or Robinul

(pH 2) injections [8], [15]. Many buffers such as citrate, phosphate or acetate are used to maintain

parenteral solution pH at physiological levels.

In liquid dosage forms, the physical stability may also be affected by environmental conditions that

can cause precipitation of the API or other components of the formulation. This may have an impact on

the bioavailability of the drug and overall performance [15]. Precipitation may occur due to sudden shifts

in the solution’s pH or to an increase in the API’s particle size due to agglutination with other molecules

[5]. Shifts in pH happen when a chemical reaction between the components of the drug generate an

acid or basic product, when carbon dioxide is absorbed from the environment or when the solution buffer

is consumed in a side reaction. [5] This chemical degradation can be described by Eq.2.6,2.7 and 2.8.

Chelation is a type of bonding between specific molecules and metal ions. Usually these ligands are

organic compounds and are called chelating agents, chelators or chelants. Their action prevents the

precipitation of metals at certain pH’s and their action depends also on their concentration.

KE,j =

NC∏i=1

(Csi )vij , i = 1, ..., NC; j = 1, ..., NR (2.6)

where NC is the number of chemical species taking part in NR equilibrium chemical reactions, vij the

stoichiometric coefficient of species i in reaction j, given concentration Csi and equilibrium constant

KE,j .

KA,j =

NC∏i=1

(Csi )vij , i = 1, ..., NC; j = 1, ..., NA (2.7)

where NC is the number of chemical species taking part in NA association reactions with chelating

agents, vij the stoichiometric coefficient of species i in association reaction j, given concentration Csi

and equilibrium constant KA,j .

Ksp,k =

NC∏i=1

(Csi )ζi , k = 1, ..., NS (2.8)

where ksp,k is the solubility product of salt k = 1, ..., NS, and ζi is the number of ions of species i that

are generated by the dissociation of one salt molecule.

In the current industrial situation, a parenteral solution as a finished product goes through three

stages: manufacturing, which corresponds to the formulation of the product; biopharmaceutics, where

scientists evaluate the action of the product in vivo; and stability assessment, where product is tested

to stay within specifications over its shelf-life. In the latter, it is well known that the combined action of

excipients, catalysts, packaging and manufacturing equipment causes to some extent the contamination

of the parenteral solution with metals. At certain pH ranges these metals may precipitate, thus compro-

mising stability of the product. Therefore, scientists elaborate on what the metals are present in solution

and estimate their concentrations. They then start an intensive work of screening one chelating agent at

16

a time at different concentrations based on initial guesses to assess the efficiency of those molecules to

prevent metal precipitation. This is a process that requires a lot time and effort and may cause significant

delays in the delivery of a finished product or even its failure to deliver. Moreover, this industrial problem

is associated with limited analysis tools that offer limited features, lack robustness and cannot consider

realistic systems (e.g multiple metals at the same time).

Parenteral nutrition (PN) includes IV administration of amino-acids, glucose, lipids, electrolytes and

trace elements. These nutrients are commonly admixed into 1 container, a term known in literature

as the all-in-one (AIO) concept. The AIO parenteral admixtures have been shown in literature to be

clinically and economically advantageous [29]. In addition to that, AIO systems show a lower risk of

contamination due to less manipulation, only need one IV pump and are easier to use at home [17].

Although, the benefits of an AIO admixture are limited by the physicochemical stability. The stability

of these solutions may be compromised in a number of ways. The mixture may show physical and

chemical incompatibilities, thus careful control is required during the compounding phase and in clinical

phase [29]. The main problems include the formation of large lipid globules, precipitation of insoluble

salts, lipid peroxidation and degradation of vitamins and amino acids [30]. The most critical parameter

of admixtures is particle diameter, which must be in the range of 0.4 to 1 micrometer. Patients require

protection from particles in the range of 5 to 10 micrometers as particles with this size might block

capillary vessels in the lungs[30].

The compounding of parenteral nutrition admixtures in large volume plastic containers leads in-

evitably to infusions that are less stable. Stability of parenteral solutions lays in the optimization of

manufacturing processes and packaging systems. Although in practice normal daily requirements for

adults can be included in PS without causing any precipitation, physical incompatibilities may arise in

attempting to achieve daily needs of children due to the low final volume typically delivered. These so-

lutions are therefore highly concentrated in some components and their precipitation might occur [31].

In the United States it has been reported a pulmonary deposition of calcium phosphate crystals in a

patient [18]. The patient developed diffuse granulomatus interstitial pneumonitis and calcium phosphate

was identified as the cause of blocking capillary vessels in the patient lungs. The FDA has now recom-

mended the use of in-line filters for PS administration of highly concentrated systems.

One of the main problems in parenteral solutions (or injectables) is the occurrence of precipitate

complexes with trace elements present in solution. For example, the hydrolysis of Fe3+ in the presence

of hydroxide ions. The authors from [32] saw that at 25◦ C in 3MNaClO4 using a calomel electrode

in combination with glass and Fe3+/Fe2+ electrodes, the formation of a gelatinous precipitate with

the addition of sodium hydroxide, which dissolved slowly. The species in solution are reasonably well

characterized [33] and the first hydrolysis step is described in equation 2.9

(Fe(H2O)6

)3++OH− ↔

(Fe(H2O)5(OH)

)2++H2O (2.9)

The association constant for this reaction, β , is defined as the quotient of concentrations powered to

the stoichiometric coefficients. Water molecules are ignored as their concentration can be assumed as

17

constant in aqueous solutions.

β1,1 =

[Fe(OH)2+

][Fe3+

][OH−

] (2.10)

In order to show the speciation as a function of pH, [OH] has to be replaced. According to self

ionization of water:

Kw =[H+].[OH−][OH] = Kw.

[H]−1 (2.11)

Substituiton of 2.11 in 2.9 gives 2.12:

β1,1.Kw = β∗1,−1 =

[Fe(OH)2+

][Fe3+

][H+]−1 (2.12)

Usually Log(β∗) are the published values in literature instead of the common dissociation constants.

The hydrolysis reactions of iron(III) are described in Appendix A.

Aluminum, a contaminant of commercial intravenous feeding solutions is potentially neurotoxic. It

has been seen in [34] that increased aluminum exposure has detrimental effects on the neurologic de-

velopment of infants fed with intravenous parenteral solutions. Therefore, manufacturers and regulators

have high interest in modelling aluminium final concentration in parenteral solutions given the different

sources of contamination from excipients, catalysts and manufacturing equipment.

A wide range of trace elements are necessary to meet the nutritional needs of patients receiving

PN. The list of recognized elements has increased due to the improved knowledge of micronutrient

functions in nutrition. Among these elements are chromium, selenium, copper or manganese. Although

daily requirements are poorly defined, maximum levels have been established by regulatory agency

due to the toxicity of these metals. Regarding stability of trace elements in parenteral solutions, two

aspects must be considered: 1) each trace element is chemically stable; 2) each element is physically

compatible with other molecules present in the parenteral solution [31]. As an example, it has been

shown that copper is incompatible with PS containing novamine, an amino acid used in the treatment of

severe bacterial infections [20]. The precipitate consisted of copper sulfide due to reaction with cysteine

[20]. Precipitation of iron added to PN mixtures has also been reported in [19].

18

Chapter 3

Implementation

3.1 gPROMS R© Platform

The gPROMS advanced process modelling platform is the powerful modelling tool and optimisation

framework developed by Process Systems Enterprise (PSE). It is the platform in which all PSE’s family

products such as gSOLIDS, gCRYSTAL and gCOAS are built. Advanced Process Modelling supports

more efficient decisions within process innovation and design through the employment of detailed high

fidelity mathematical models of process equipment and phenomena.

3.2 gPROMS R© ModelBuilder

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

19

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

20

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

21

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

either continuous or discrete.

3.3 gPROMS R© Global Systems Analysis

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)

22

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)

23

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

24

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]

25

26

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

27

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.

28

(a) Degradation kinetics tab

(b) Experimental conditions tab (c) Initital conditions tab

(d) Specification limit tab

Figure 4.2: Degradation regressor model user interface;

4.2 Case Study 1

The first case study comprises a series of experimental data under accelerated conditions acquired

from a PSE’s business partner. This data is characterized by the % of degradant product in total API

formulation over time. It was measured over 5 experimental conditions of temperature and relative

humidity with a maximum specification limit of 0.8%. It is important to notice that a degradation of

0.524% was already present in the initial measurements (Table 4.1).

These experiments were inserted in the gPROMS platform and the parameters were estimated, as

explained before in 3.2.2. The estimated parameters are summarized in figure 4.4. It is important to

note that this parameter estimation considered a 0.02 variance in the error measurement produced by

the sensor, while using a constant relative variance model. This is a typical value for chromatographic

methods found in literature [9].

29

Figure 4.3: HTML report automatically generated by gPROMS; 1) Degradation table; 2) Degradationplot: degradant percentage (red line), specification limit (yellow line);

Table 4.1: Accelerated degradation data (degradant %) imported from PSE’s partners at different tem-peratures and relative humidities;

Case Study 1

Time(days) 70 ◦ C/25%RH 70◦ C/75%RH0 0.524 -14 0.591 0.62721 0.628 0.63428 0.651 0.658

Time(days) 80◦ C/25%RH 80◦ C/75%RH 90◦ C/25%RH3 0.577 0.588 0.61214 0.692 0.742 0.79921 0.728 0.776 -28 0.799 0.84 -

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

30

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

31

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.

32

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

33

(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

34

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

35

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

was found.

Table 4.4: gPROMS Sensitivity analysis

Degradation (%)1st Order Total effect

External paramtersRH 0.00397 0.00662T 0.9935 0.99571

Kinetic parametersEa 0.6117 0.7630

Ln(A) 0.4731 0.6262B 0.00166 0.00300

As stated before in section 3.3.2, a parameter with a high 1st order index represents a parameter

responsible for great variations in model outputs. In addition to that, if 1st order index is equal to total

effect index, then there is no interaction between parameters to influence degradation. As seen in

table 4.4 temperature is the key driver in the response space amongst the external parameters. This

conclusion was seen before in figure 4.9 (b) where temperature showed an exponential correlation with

degradation. On the other hand, activation energy is the key driver in the response space amongst the

kinetic parameters. In the latter is it possible verify that the total effect is about 0.15 higher than the

1st order indicating that there are interactions between kinetic parameters that are also responsible for

some extent of variability in model outputs.

4.3 Packaging

An addition to the Degradation regressor model was done to account for the effects of packaging in

drug degradation through the Packaging model. This model consisted of the same user interface used

in the Degradation model with 3 added tabs. A packaging specification tab (figure 4.13 (a)) where the

36

user specifies the type of packaging to be used (HDPE bottles, PVC blisters, aluminium blisters) and

inserts the models parameters for the packaging in use (package volume, headspace volume, initial

internal RH and apparent moisture permeability). The latter is the most important parameter since

it is responsible for the permeation of moisture through the package walls by diffusion. The tablets

and desiccant tabs are very similar (figure 4.13 (b),(c)). The user just specifies the amount of each

component (number of tablets and mass of desiccant) and fills the others fields with the coefficients

obtained by a third order polynomial equation of the moisture sorption isotherm. It is important to mention

that the moisture sorption isotherm must be acquired experimentally and is a characteristic of each

formulation and desiccants.

(a) Packaging specification tab

(b) Tablets tab (c) Desiccant tab

Figure 4.13: Degradation regressor model with packaging upgrade (user interface);

4.3.1 Application of the packaging model on Case Study 1

An experiment was then conducted with the objective of validating the packaging model. This validation

was made with the kinetic parameters considered in Case study 1 and with operational parameters from

[11]. Considering an HDPE bottle containing 10 tablets of the drug studied in Case study 1 with 1.86g

of silica desiccant and considering the apparent moisture permeability of 0.0135mg/(day.%RH) stated in

[11], degradation and RH inside the bottles were computed with the packaging model.

As seen in figure 4.14 applying the same shelf-life of 8.4 years (approx.3000 days) predicted by the

model degradation regressor in section 4.2 to the packaging model, it is possible to see that degradation

37

Figure 4.14: gPROMS packaging model results; Relative humidity inside the bottle (dotted line), specifi-cation limit (orange line) and degradant % (blue line);

fails to achieve the specification limit. This means that the HDPE bottle provided some extent of protec-

tion to the drug on the inside thus increasing its shelf-life. In order to analyse the effects that apparent

moisture permeation, number of tablets and desiccant mass have in degradation, an uncertainty analy-

sis was performed with uniform distributions over 500 samples with quasi-random Sobol sampling for the

second year of shelf-life. Uniform distributions make different conditions have the same probability and

it is particularly useful in determining what happens to degradation when a different number of tablets,

desiccant or permeation rate is used. Through figure 4.15 it is possible to note that apparent moisture

permeation has the greatest impact on degradation. Both number of tablets and desiccant mass did not

appear to have a significant impact on degradation in this case study. Moreover, if a sensitivity analysis

is performed on these 3 parameters, it is possible to note through table 4.5 that only apparent moisture

permeation is responsible for the greatest variance in degradation. As seen before, 1st order index is

different from the total effect index. This means that some extent of model outputs are being effected by

interactions between different parameters.

Table 4.5: gPROMS Sensitivity analysis

Degradation %1st Order Total effect

Apparent moisture permeation 0.662 0.745Number of tablets 0.06 0.098Mass of desiccant 0.100 0.195

Different sources of literature stated that the number of tablets and amount of desiccant used in

pharmaceutical bottles may have tremendous impact in drug degradation ([11],[10]). Nonetheless, it has

been discussed before that the pharmaceutical product studied in this case study is not very sensitive

to humidity (low B parameter in the modified Arrhenius equation). Because packaging main purpose is

to protect drugs from moisture, table 4.5 confirmed that the number of tablets and desiccant amount do

38

(a) Apparent moisture permeation uncertainty analysis

(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

39

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.

40

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

41

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.

42

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

43

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

44

Figure 5.9: Fe(III) 1mM - Citrate 20mM speciation

Figure 5.10: Fe(III) 1mM - Succinate 20mM speciation

Figure 5.11: Fe(III) 1mM - Acetate 20mM speciation

the user selected a system containing Iron+Citrate. This is just an illustration for this specific case and

all of the reaction constants are stated in the model. If the user had selected other chelating agents,

other reactions would be computed although they are not shown in the present work.

5.2.2 Case study

For this case study, only iron speciation was considered and a concentration of 1mM was arbitrated.

This is a value thought to be relevant given the multiple sources of contamination a parenteral solution

45

Figure 5.12: Fe(III) 1mM - Histidine 20mM speciation

might be exposed to such as catalysts, excipients, manufacturing equipment and packaging.

After the Parenteral Stability model was dragged and dropped in the flowsheet and 1 mM of iron

was inputed in the model, one must first assess at what pH does iron precipitation starts for the chosen

concentration. As shown in figure 5.13precipitation of the solid form of iron hydroxide starts at pH 2.5.

Figure 5.13: Iron speciation- Fe=1mM

A chelating agent must then be chosen to avoid iron precipitation at normal pH ranges. Through

figure 5.10 it is possible to note that when using a concentration of 20mM of succinate, iron precipitation

only starts at pH 5.5. Although it ”pushes” iron precipitation a bit further to the right on the pH range, it is

not optimal. If then citrate (20mM) is taken into consideration as a chelating agent, it is possible to note

that for this case, iron speciation only starts at pH 7 with supersaturation values increasing above 1 and

all of the iron present in the liquid phase making the transition to the solid phase (figures 5.9, 5.14). In

this simple case, citrate was chosen based on experimental testing. Nonetheless, taking in consideration

that chelating agent database is still in development, one may want to screen all the chelating agents at

the same time to see which one is more effective in avoiding metal precipitation. To do so, the user may

perform a GSA sensitivity analysis on all chelating agents available and applying the factor prioritisation

methodology explained in section 3.3.2 to decide which one is more suitable for that specific case (table

5.1). The sensitivity analysis will rank all the chelating agents selected and calculate which one or ones

are responsible for the greatest variance in model outputs (supersaturation). it is possible to note in

46

table 5.1 that citrate had the highest total effect in supersaturation while the other chelating agents did

not had relevant indexes. For that reason and in this case study, citrate was the most relevant chelating

agent to be studied. Nonetheless, pricing and availability may also be a decisive factor for the choice of

a certain chelating agent so one must have that in consideration.

Table 5.1: gPROMS Sensitivity analysis for factor prioritisation;

Supersaturation %1st Order Total effect

Succinate 9.94E-6 3.6E-5Citrate 0.947 0.976Acetate 2.36E-5 7.58E-5

Histidine 1.25E-6 4.1E-6

(a) Supersaturation (b) Mass metal

Figure 5.14: Fe(III) 1mM - Citrate 20mM;

Following the workflow described in section 5.2, now that citrate is known to be the most relevant

chelating agent among others from the database, what is the minimal concentration that prevents any

precipitation for an iron concentration of 1mM?

A minimisation of its concentration was done using gPROMS optimisation tool. This prevents adding

more quantity than the one that is really necessary to prevent any precipitation. To do so, it was asked

the optimisation tool to solve the inequation 0 < Supersaturation < 1 and also that the variable mass

of metal in the solid phase must be equal to zero through all the pH range. The optimisation procedure

resulted in an optimized citrate concentration of 0.4488M. Running the Parenteral Stability model again

with this value on the chelating agent, resulted in figures 5.15 and 5.16. It possible to note that the solid

species of iron hydroxide is not present in the given pH range and supersaturation peaks at 0.7 for pH

10.

The next step was to incorporate uncertainty in the model by doing a gPROMS uncertainty analysis.

Metal concentration in a batch of parenteral solution bags or vials will have an associated uncertainty that

is most likely to follow a normal distribution. Assuming that the average of iron in a parenteral solution

is 1mM and STD is 10% of it (0.1mM), an uncertainty analysis on 500 samples with quasi-random

Sobol sampling was performed using GSA. Additionally, one can also consider a normal distribution on

chelating agent concentration. Although, in an industrial situation, the standard deviation for this value is

47

Figure 5.15: Mass fraction Fe(III) 1mM - Citrate 0.4488M

Figure 5.16: Supersaturation Fe(III) 1mM - Citrate 0.4488M

quite low given it is added in accordance with the needs of the batch. The optimized value was chosen

as mean of the normal distribution and a standard deviation of 0,1% was assumed.

As stated before, quasi random Sobol sampling is a faster converging method when compared with

pseudo-random sampling. As one can see through figure 5.17 supersaturation has converged after 300

samples, meaning that 500 samples was a suitable sampling size for the uncertainty analysis.

For the uncertainty analysis is is possible to note that on scatter of figure 5.18 (a) there is a probability

of supersaturation of 42%. In addition to that, the histogram of figure 5.18 (b) also shows that from the

500 samples, about 220 samples have a supersaturation value greater than 1. Table 5.2 also demon-

strates that the expected value of supersaturation is 0.971 with a STD of 0.176. So the optimized value

that was thought to minimize chelating agent concentration and at the same time guarentee that for the

chosen pH range there is no metal in the solid phase, fails when uncertainty of metal and chelating

agent concentration is incorporated. A further analysis is then needed to assure that, when uncertainty

of model parameters is incorporated, metal precipitation is not likely to occur.

Citrate was then optimized to 0.6M by trial and error. An uncertainty analysis was then performed

again with the same distributions on metal and chelating agent concentration. By figure 5.19 it is possible

48

Figure 5.17: Convergence of the supersaturation average over 300 samples

(a) Scatter 2-D (b) Histogram

Figure 5.18: Uncertainty analysis on 0.4488M Citrate;

Table 5.2: Distribution statistics table for citrate concentration of 0.448MExpected Value Std deviation Min. value Max. value

Supersaturation 0.971 0.176 3E-4 1.60

to note that only one sample was supersaturated giving a probability of failure of 0.2%.Moreover if table

5.3 is taken into consideration, the expected value for supersaturation went from 0.971 in table 5.2 to

0.569.

Table 5.3: Distribution statistics table for citrate concentration of 0.6MExpected Value Std deviation Min. value Max. value

Supersaturation 0.569 0.113 0.274 1.082

49

(a) Scatter 2-D (b) Histogram

Figure 5.19: Uncertainty analysis on 0.6M Citrate;

Multiple metals

In a real industrial situation, a parenteral solution will have multiple metals dissolved at the same time.

Their concentrations may vary given the different types of contamination sources. This work represents

the first attempt in literature to model multiple metals speciation in a parenteral solution. It was a much

more challenging concept to implement given that chelating agents will now scavenge for different metals

given the reaction rates for each chelator-metal reactions. While in theory this may seem an easy

concept to understand, translating it to a programmable language makes it much harder to implement.

Figure 5.20: Multiple metals- Al 1mM +Fe 1mM + Succinate 20mM species

In figure 5.20 it is possible to note the different chelated species of iron and aluminium and their

concentrations. It was not possible to show a mass fraction graph since mass fraction is given by the

amount of metal in a certain species divided by the total metal in solution and in this specific case we

have two metals, thus two mass fraction graphs would need to be generated.

In figure 5.21 it possible to see supersaturation of both metals and the mass of each metal in liquid

and solid phases. Precipitation of iron starts at pH 5 while aluminium precipitation starts at pH 7. In the

latter, supersaturation goes below 1 for pH’s above 8, interrupting precipitation and dissolving again the

50

(a) Al+Fe supersaturation

(b) Al+Fe mas of metal

Figure 5.21: Multiple metals: Al 1mM + Fe 1mM + Succinate 20mM;

metal that was already in the solid phase.

Multiple chelating agents

Another scenario would be the one where the scientist could use different chelating agents. This is par-

ticularly interesting because the scientist could study if there is some kind of synergy between chelating

agents and thus’ reducing the amount of each one to prevent precipitation at a given pH. Another reason

for using multiple chelators might have to do with their pricing or availability in the factory at a given time.

To implement this strategy, a system composed of 1mM of iron and 20mM of citrate and succinate

was generated as depicted in figure 5.22.

Although, as figure 5.23 depicts, when uncertainty in added to the system, results may be surprising.

In this specific case it was seen that when adding a normal distribution on iron concentration having

as average 1mM and as STD 0.1mM for pH’s above 2.5 the parenteral solution may or may not be

supersaturated depending on metal concentration. The same goes for the mass of metal in the liquid

and solid phase for the given pH range.

51

Figure 5.22: Multiple chelating agents-Fe 1mM + Succinate 20mM + Citrate 20mM species

(a) Fe(III): supersaturation

(b) Fe(III): mass of metal

Figure 5.23: Multiple chelating agents: Fe 1mM + Succinate 20mM + citrate 20mM;

52

Chapter 6

Conclusions

In this work, mathematical models were developed for the stability of solid dosages and parenteral

solutions.

In both parts of this work, an easy to use interface in a drag and drop flowsheet with automatic

reporting of results was built to give the end user a better overall experience and data interpretation.

Although this kind of architecture is not relevant in the present work, it was a great challenge to make

sure that the models were robust for a wide range of variable inputs.

In the Solid Dosage stability models it is known that the isoconversion approach fails for more com-

plex reaction mechanisms due to overly simplistic models. Model drawbacks may be overcome by

classifying the diverse degradation mechanisms and account them into the models, considering the

molecular and reactional aspects both of APIs and excipients used. Nonetheless, it has been shown

that incorporating packaging models in accelerated stability programs of solid dosage drugs is of high

importance. This addition allows the understanding of the level of protection a given type of package pro-

vides to pharmaceutical drugs against relative humidity. Solid dosage stability models along with GSA

capabilities allow a better understanding of degradation processes and help management personnel

take more confident decisions; They also reduce experimental cost and process time given its hability to

quickly analyse ”what-if” scenarios.

In the Parenteral Stability model it is possible to conclude that building cases where the user can

add multiple chelating agents and metals at the same time is of extreme importance. Another feature of

using the Parenteral stability model along with gPROMS features is that it enables the scientist to have

a broader understanding of what is happening in solution. Meaning that if the system is composed of

different metals and multiple chelators are also being used, he can start by studying the system in a

univariate way. Once he understands what are the main reactions for each metal and chelator, he can

then proceed to a multivariate analysis, incorporating groups or even all metals in the system and build a

layered knowledge of why precipitation might occur. This model empower the user to build an integrated

knowledge of their systems with an easy to use interface. Moreover, this model along with gPROMS

optimisation tool and GSA capabilities represents one way of submitting work to the regulatory agencies

with more confidence, resulting in more approved decisions and thus reduced costs.

53

It was also shown the importance of including uncertainty and sensitivity analysis on drug degra-

dation experiments. This is an implementation that was not found in literature and there is still a lot of

work to be done in this field of expertise. In addition to that, this work was the first one of its kind to

study multiple metal precipitation in parenteral solutions applied to pharmaceutical cases while applying

statistical analysis at the same time. For regulators, more than assuring that a pharmaceutical product

is stable, one must assure that it is statistically reliable. It has been demonstrated with the present work

that incorporating different sources of uncertainty in model inputs might affect dramatically model out-

puts. Therefore, this work also represents one step forward in providing regulators with meaningful data

to reduce risks for patients.

Finally, gPROMS is a powerful tool with integrated parameter estimation tools that allows the user

to easily extract and validate kinetic parameters. Moreover, the recent addition of the Global Systems

Analysis opened a new window of possible work to be done.

6.1 Achievements

This work opens a new perspective in stability testing of solid dosage forms with the incorporation of

packaging models and with the powerful gPROMS parameter estimation tool. Moreover, it represents

pioneer work in parenteral stability modelling given the built-in databases for different metals and chelat-

ing agents.

In addition to that, the uncertainty and sensitivity analysis performed in the two topics of the present

work is something that was never explored before and represent a new way of looking for stability testing.

6.2 Future Work

It is widely accepted that the Arrhenius fitting fails due to its inability to model more complex systems.

Future work in this field may require scientists to detach from this law-approach and focus on a more

mechanistic approach. This involves defining all the possible degradants that appear in a certain formu-

lation and hypothesize about cross-reactions that might occur between all the components in a drug. In

addition to that, incorporating physical degradation models is of high importance. These models account

for degradation processes occurring in the solid and will improve the understanding of overall product

stability.

On the other hand, in the Parenteral Stability models one might want to increase metals and chelat-

ing agents databases to have a full spectrum of options. This increase in databases would cover a more

realistic situation where parenteral solutions might have different metals in different residual concentra-

tions.

One might also want to consider cross-reactions with other molecules that can also form precipitates

such as phosphates or sulphates. Calcium phosphate precipitates have extensively described in litera-

ture as one of the most common precipitating molecules in parenteral nutrition and have even caused

severe diseases in patients being treated. These molecules can also affect chelating associations and

54

can form new species with chelating agents. Adding them to the database would also provide a broader

understanding of real situations.

In the present work, only the precipitation of insoluble salts was studied. Nonetheless as described

before, parenteral solutions show other stability problems such as lipid peroxidation and globulization,

vitamin degradation and amino acid interactions. These problems might be addressed in the future in a

modelling perspective to manufacture products that are more stable and safer for patients.

Another possibility would be to use gSAFT as a part of parenteral solution stability modelling. gSAFT

stands for statistical associated fluid theory and is a gPROMS tool that predicts physical properties and

reaction kinetics of chemical species based initial inputs such as partition coefficients. Various versions

of the SAFT have been successfully applied to the study of the properties of thermodynamically chal-

lenging systems, such as polymeric, hydrogen-bonding, and reactive systems. Details of the underlying

theory and the different applications of SAFT can be found in a number of comprehensive reviews ([40],

[41]). So if the user would input iron as a metal ion and succinate as a chelating agent, it would auto-

matically generate the most likely species to be formed in that considered system, assign equilibrium

constants for the reactions and generate solubilization constants for the solid species. This contrasts

with the current methodology where all of the species, reaction kinetics (stoichiometric coefficients and

equilibrium constants) were hard coded in the model itself.

55

56

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Appendix A

Parenteral solution equations

Equilibrium reactions

Fe3+ +H2O ↔ (Fe(OH))2+ +H+ β∗1,−1 =

[(Fe(OH))2+

][Fe3+

][H+]−1 (A.1)

Fe3+ + 2H2O ↔ (Fe(OH)2)+ + 2H+ β∗1,−2 =

[(Fe(OH)2)+

][Fe3+

][H+]−2 (A.2)

Fe3+ + 4H2O ↔ (Fe(OH)4)− + 4H+ β∗1,−4 =

[(Fe(OH)4)−

][Fe3+

][H+]−4 (A.3)

2Fe3+ + 2H2O ↔ (Fe2(OH)2)4+ + 2H+ β∗2,−2 =

[(Fe2(OH)2)4+

][Fe3+

]2[H+]−2 (A.4)

3Fe3+ + 4H2O ↔ (Fe3(OH)4)5+ + 4H+ β∗3,−4 =

[(Fe3(OH)4)5+

][Fe3+

]3[H+]−4 (A.5)

Chelating reactions

Fe3+ +H+ + C6H4O4−7 ↔ (FeH(C6H4O7)) β∗

1,1,1 =

[(FeH(C6H4O7))

][Fe3+

][H+][C6H4O

4−7

] (A.6)

Fe3+ + 2C6H4O4−7 ↔ (Fe(C6H4O7)2)5+ β∗

1,0,2 =

[(Fe(C6H4O7)2)5+

][Fe3+

][C6H4O

4−7

]2 (A.7)

Fe3+ +H+ + 2C6H4O4−7 ↔ (FeH(C6H4O7)2)4− β∗

1,1,2 =

[(FeH(C6H4O7)2)4−

][Fe3+

][H+][C6H4O

4−7

]2 (A.8)

Fe3+ + 2H+ + 2C6H4O4−7 ↔ (FeH2(C6H4O7)2)3− β∗

1,2,2 =

[(FeH2(C6H4O7)2)3−

][Fe3+

][H+]2[C6H4O

4−7

]2 (A.9)

61

2Fe3+ + 2C6H4O4−7 ↔ (Fe2(C6H4O7)2)2− β∗

2,0,2 =

[(Fe2(C6H4O7)2)2−

][Fe3+

]2[C6H4O

4−7

]2 (A.10)

Solubilization reaction

(Fe(OH)3)(s) + 3H+ ↔ Fe3+ + 3H2O Ksp =[Fe].[H+]−3 (A.11)

62