Dynamic Modeling of Bone Remodeling, TumorGrowth and Therapy

Simplifying and Diffusing with Variable Order Derivatives

Joana Catarina Lopes Pinheiro Carreira Neto

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Duarte Pedro Mata de Oliveira ValérioProf. Susana de Almeida Mendes Vinga Martins

Examination Committee

Chairperson: Prof. Paulo Jorge Coelho Ramalho OliveiraSupervisor: Prof. Duarte Pedro Mata de Oliveira Valério

Members of the Committee: Prof. José Carlos Fernandes PereiraDoutora Irina Margarida Pereira Alho Duarte

March 2017

Home is behind, the world aheadand there are many paths to tread

through shadows to the edge of night,until the stars are all alight.

Lord of The Rings - J. R. R. Tolkien

Acknowledgments

To Professors Duarte Valério and Susana Vinga, for guiding me through this newly discovered

world of science-making with an always open office door. Please let my profound appreciation be

recognized for all the encouragement and opportunities given. I could not have asked for better

advisors and mentors.

To Dominik Sierociuk, Wiktor Malesza, Michal Macias, Andrzej Dzielinski, and the joint Polish-

Portuguese project Modeling and controlling cancer evolution using fractional calculus (FCT-IDMEC

LAETA UID/EMS/50022/2013) and project PERSEIDS (PTDC/EMS-SIS/0642/2014), without whom

this work would have not been possible.

To Rui Coelho, for steering me into the bone remodeling models sea, and to Doctor Irina Alho, for

contextualizing these models.

To my family and friends and to my family of friends, for keeping up with me in this long, trouble-

some, immensely fulfilled and well-lived journey. A special full hearted thank you to my parents, for

everything that I’ve become; to my sister, for making me explore the world; to João and to Carolina,

for always being there (...correcting my English!).

Last but not least, to Instituto Superior Técnico, for a complicated relationship that made me grow

into a proud Engineer. Let the humbling ideology that we can always be and do better be perpetuated.

iii

Abstract

Bone undergoes a constant remodeling process that involves different cell types, in particular the

bone degrading osteoclasts and the bone forming osteoblasts. Their activity and biochemical reg-

ulation are typically represented through differential equations that represent the physiological phe-

nomena occurring within this organ. These mathematical models have been enriched with variables

and parameters related with disruptive pathologies of a tumor in the bone dynamics. Anti-cancer

treatments, as the pharmacokinetic/pharmacodynamic (PK/PD) of the most common ones, were also

included.

These models are expected to provide valuable insights about the bone complex system and

to support the development of clinical decision systems for bone pathologies with efficient targeted

therapies. Hence, a simplified model that mimics bone behavior, action of a tumor and therapy coun-

teraction is but an essential tool for a tailored treatment for each patient.

A new approach to the existing biochemical models is here proposed. Firstly, these physiological

mathematical models are here said to better explain an osteolytic metastatic bone disease environ-

ment, instead of the previously considered multiple myeloma. Second and most importantly, variable

order derivatives were introduced, for the first time in biochemical bone remodeling models, allowing

for a reduced set of parameters to describe similar results to those from the original formulations when

a tumor is acting in the bone micro-environment. A more compact model, that promptly highlights the

tumor interactions within bone, is here achieved with good qualitative simulations of what is known for

the bone dynamics.

Keywords

Bone remodeling; Bone metastases; Simplified mathematical models; Variable order derivatives;

Pharmacokinetics and Pharmacodynamics; PK/PD

v

Resumo

O osso é um órgão que se encontra em remodelações constantes. Este processo envolve di-

ferentes tipos de células, em particular os osteoclastos que degradam o osso e os osteoblastos

responsáveis pela sua formação. A sua actividade e regulação bioquímica é normalmente represen-

tada através de equações diferenciais que englobam os fenómenos fisiológicos que ocorrem neste

órgão. Estes modelos matemáticos têm sido enriquecidos com variáveis e parâmetros relacionados

com patologias na dinâmica óssea, como o desenvolvimento de um tumor. Tratamentos antican-

cerígenos, como a farmacocinética/farmacodinâmica (PK/PD) dos tratamentos mais comuns, foram

também adicionados.

É previsto que estes modelos providenciem não só informação relativa ao funcionamento deste

sistema complexo, como suportem o desenvolvimento de sistemas de decisão clínica para patolo-

gias ósseas com o aumento da eficiência da terapia personalizada. Assim, um modelo simplificado

que replica o comportamento ósseo, a acção de um tumor e do tratamento aplicado torna-se numa

ferramenta essencial para a aplicação de terapia ajustada a cada paciente.

Uma nova abordagem aos modelos existentes é aqui proposta. Primeiramente, considera-se que

estes modelos matemáticos explicam melhor uma doença de metástases ósseas osteolíticas, em vez

do mieloma múltiplo anteriormente proposto. Segundamente, foram aplicadas derivadas de ordem

variável, pela primeira vez neste campo, nos modelos existentes de forma a que um conjunto reduzido

de parâmetros descreva resultados semelhantes ao das formulações originais. É assim obtido um

modelo mais compacto, que realça as interacções tumorais no ambiente ósseo, com simulações

qualitativamente adequadas ao que é conhecido do comportamento ósseo.

Palavras Chave

Remodelação óssea; Metástases ósseas; Modelos matemáticos simplificados; Derivadas de or-

dem variável; Farmacocinética e Farmacodinâmica; PK/PD

vii

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Fractional and Variable Order Derivatives . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1.A Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1.B Varying Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Bone Remodeling Physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2.A Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2.B Tumor Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2.C Existing Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Fractional and Variable Order Derivatives 13

2.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Operator D for Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Basic Working Tool: the Gamma Function Γ . . . . . . . . . . . . . . . . . . . . . 15

2.2 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Matrix Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Relevant Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.5 Fractional Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.6 Fractional Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.6.A Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.6.B Frequency and Time Responses . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Derivatives with Varying Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Mathematical Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Matrix Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 Relevant Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix

2.3.5 Variable Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Mathematical Models for Bone Remodeling 31

3.1 A Pharmacokinetics and Pharmacodynamics Introduction . . . . . . . . . . . . . . . . . 32

3.1.1 Pharmacokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 Pharmacodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 A PKPD Control Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Healthy Remodeling Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Adding the Tumor Burden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Bone Metastases Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.3.A Osteoblastic Promotion & Anti-Cancer Therapy . . . . . . . . . . . . . . 41

3.2.3.B PKPD of Anti-Resorptive & Chemotherapy Treatment . . . . . . . . . . 42

3.3 Non-local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Healthy Remodeling Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Adding the Tumor Burden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.3 Bone Metastases Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3.A Osteoblastic Promotion & Anti-Cancer Therapy . . . . . . . . . . . . . . 47

3.3.3.B PKPD of Anti-Resorptive & Chemotherapy Treatment . . . . . . . . . . 47

4 Creating Variable Order Models: Methodology, Results and Discussion 51

4.1 From Multiple Myeloma to Metastatic Bone Disease - A New Starting Point . . . . . . . 52

4.2 Creating a Variable Order Model for Bone Remodeling . . . . . . . . . . . . . . . . . . . 53

4.2.1 Choosing and Shaping a Variable Order Formulation . . . . . . . . . . . . . . . . 53

4.2.2 Addressing the Tumor’s Dynamic for Therapy Purposes . . . . . . . . . . . . . . 54

4.2.3 Different Resorption Ratios R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.4 Non-local Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Simplified Bone Models With an Acting Tumor . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 Local Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2 Non-local Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Changing the Order, Extinguishing the Tumor — A Treatment Approach . . . . . . . . . 60

4.4.1 Osteoblastic Promotion & Anti-Cancer Therapy . . . . . . . . . . . . . . . . . . . 61

4.4.2 PKPD of Anti-Resorptive & Chemotherapy Treatment . . . . . . . . . . . . . . . 64

5 Conclusions and Future Work 69

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 73

x

Appendix A Fractional and Variable Order Calculus & Biochemical Models Support A-1

A.1 Integer Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.2 Variable Order Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.3 Healthy Bone Remodeling - Nontrivial Steady State . . . . . . . . . . . . . . . . . . . . A-3

A.4 Different Types of Steady-States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4

xi

List of Figures

1.1 Breast and Prostate Cancer 2012 Incidence Worldwide . . . . . . . . . . . . . . . . . . 2

1.2 Varying Order Derivatives: Simple-switching Scheme . . . . . . . . . . . . . . . . . . . . 5

1.3 BMU Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 BMU Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Osteolytic and Osteoblastic Pathways of Metastases to the Bone . . . . . . . . . . . . . 9

1.6 Tumor Growth Gompertzian Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Gamma Function Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Stability of Fractional Transfer Funcions . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Basic Fractional Transfer Function Bode Diagram . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Impulse and Step Response for sα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Frequency Behavior and Bode Diagram for Several α . . . . . . . . . . . . . . . . . . . . 23

2.6 Step Response and Bode Diagram for G1(s) . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Step Response and Bode Diagram for G2(s) . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 PK Model with a Single Drug Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 PK Model with Multiple Drug Doses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 PD Single Drug Effect, with Single and Multiple Doses . . . . . . . . . . . . . . . . . . . 34

3.4 PKPD: Identical Drugs PK Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 PKPD: Identical Drugs PD Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Healthy Bone Remodeling Simulation - Periodic Cycles . . . . . . . . . . . . . . . . . . 36

3.7 Bifurcation Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Local Tumor Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.9 Osteoclast, Osteoblast and Bone Mass with Tumor - Damped Case . . . . . . . . . . . . 40

3.10 Local Osteoclast and Osteoblast with Tumor and Treatment Effect . . . . . . . . . . . . 41

3.11 Local Tumor and Bone Mass Evolution with Treatment Effect . . . . . . . . . . . . . . . 42

3.12 PKPD Local Model Treatment Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.13 Initial Distributions of C(0, x) and T (0, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.14 Normal Bone One-dimensional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.15 Non-local Bone Simulation with Tumor Action . . . . . . . . . . . . . . . . . . . . . . . . 46

3.16 Non-local Bone Simulation with Tumor and Therapy . . . . . . . . . . . . . . . . . . . . 48

3.17 Non-local Bone Simulation with Tumor and Therapy . . . . . . . . . . . . . . . . . . . . 49

xiii

4.1 Type-D For Initial Conditions and For Continuity Between Orders . . . . . . . . . . . . . 53

4.2 Variable Order Simulink Implementation and Switching Between Integrator Types . . . . 55

4.3 Limiting the Tumor’s Innate Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Local Case: Variable Order Model for Bone Behavior with an Acting Tumor . . . . . . . 57

4.5 Local Case: Variable Order Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Non-local Case: Variable Order Model for Bone Behavior with an Acting Tumor . . . . . 59

4.7 Non-local Case: Variable Order Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.8 Local Case & Basic Treatment: Bone Micro-environment Simulation . . . . . . . . . . . 61

4.9 Local Case & Basic Treatment: Variable Order Evolution . . . . . . . . . . . . . . . . . . 62

4.10 Non-local Case & Basic Treatment: Bone Micro-environment Simulation . . . . . . . . . 64

4.11 Non-local & Basic Treatment: Variable Order Evolution . . . . . . . . . . . . . . . . . . . 65

4.12 Local, Anti-resorptive & Chemotherapy Treatment: Bone Micro-environment Simulation 65

4.13 Local, Anti-resorptive & Chemotherapy Treatment: Variable Order Evolution . . . . . . . 66

4.14 Non-local, Anti-resorptive & Chemotherapy Treatment: Bone Micro-environment . . . . 67

4.15 Non-local, Anti-resorptive & Chemotherapy Treatment: Variable Order Evolution . . . . 68

xiv

List of Tables

3.1 Local and Non-local Variables and Parameters Used for Simulations . . . . . . . . . . . 49

3.2 PKPD Treatment Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Resorption Ratios R Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xv

List of Symbols

N Natural numbers set

Z Integer numbers set

R Real numbers set

t Time — days

tstart Starting time of treatment — days

Ω One-dimensional diffusion region — Ω ∈ (0, 1)

x One-dimensional distance — x ∈ Ω

D Functional Operator

D1 Integer, first-order, derivative operator∂2

∂x2 Integer, secord-order, partial derivative operator

b t−ch c Floor of t−chdαe Ceiling of α

α Fractional derivative of order α

Dα Fractional order derivative operator

cDαt Fractional derivative of order α with terminal c and t

α(t) Local variable order derivative

α(t, x) Non-local variable order derivative

Dα(t) Local variable order derivative operator

Dα(t,x) Non-local variable order derivative operator

C(t) Zero-dimensional osteoclast cells

B(t) Zero-dimensional osteoblast cells

z(t) Zero-dimensional bone mass — %

T (t) Zero-dimensional bone metastases size — %

C(t, x) One-dimensional osteoclast cells

B(t, x) One-dimensional osteoblast cells

z(t, x) One-dimensional bone mass — %

T (t, x) One-dimensional bone metastases size — %

gCC

Osteoclasts autocrine regulator

gCB

Osteoclasts-derived osteoblasts paracrine regulator

gBB

Osteoblasts autocrine regulator

gBC

Osteoblasts-derived osteoclasts paracrine regulator

rCC

Tumor burden in osteoclasts autocrine regulator

rCB

Tumor burden in osteoclasts-derived osteoblasts paracrine regulator

rBB

Tumor burden in osteoblasts autocrine regulator

rBC

Tumor burden in osteoblasts-derived osteoclasts paracrine regulator

αC

Osteoclasts activation rate — %days−1

αB

Osteoblasts activation rate — %days−1

xvii

βC

Osteoclasts apoptosis rate — %days−1

βB

Osteoblasts apoptosis rate — %days−1

κC

Zero-dimensional bone resorption rate — %days−1

κB

Zero-dimensional bone formation rate — %days−1

κC

(x) One-dimensional bone resorption rate — %days−1

κB

(x) One-dimensional bone formation rate — %days−1

LT

Bone metastases maximum size — %

γT

Tumor growth rate — %days−1

σC

Osteoclast diffusion coefficient

σB

Osteoblast diffusion coefficient

σz Bone mass diffusion coefficient

σT

Tumor diffusion coefficient

C0 Zero-dimensional initial of osteoclasts cells

B0 Zero-dimensional initial of osteoblasts cells

z0 Zero-dimensional initial bone mass percentage — %

T0 Zero-dimensional initial tumor percentage — %

C0(x) One-dimensional initial of osteoclasts cells

B0(x) One-dimensional initial of osteoblasts cells

z0(x) One-dimensional initial bone mass percentage — %

T0(x) One-dimensional initial tumor percentage — %

Css Zero-dimensional steady-state osteoclast cells

Bss Zero-dimensional steady-state osteoblast cells

Tss Zero-dimensional steady-state tumor — %

Css One-dimensional steady-state osteoclast cells

Bss One-dimensional steady-state osteoblast cells

T ss One-dimensional steady-state tumor — %

V1(t) Treatment function for osteoblastic apoptosis — %days−1

V2(t) Treatment function for tumor destruction — %days−1

υ1 Intensity parameter for osteoblastic apoptosis treatment function

υ2 Intensity parameter for tumor destruction treatment function

Cg(t) Remaining drug concentration to be absorbed — g/L

Cp(t) Effective drug concentration in the plasma — g/L

κg Absoption rate of drug concentration in plasma — days−1

κp

Elimination rate of drug concentration in plasma — days−1

Cg0Initial concentration of C

g(t) — g/L

d(t) Pharmacodynamic drug effect

Cp50(t) Plasma concentration at 50% of its maximum drug effect — g/L

Cbasep50Initial value of Cp50

(t) — g/L

Lr Treatment concentration, in the plasma, threshold

xviii

Kr Tumor treatment resistance parameter

Ks Stimulatory control action of a drug

Ki Inhibitory control action of a drug

Rtumor Resorption ratio for an acting tumor

Rtreat Resorption ratio when treatment is applied

Rhealthy

Resorption ratio in a healthy environment

xix

Abbreviations

WHO World Health Organization

UN United Nations

DNA Deoxyribonucleic Acid

RANK Receptor Activator of NF-κB

cfms Macrophage Colony-stimulating Factor Receptor

CSF1 Colony-stimulating Factor 1

RANKL RANK-ligand

BMU Basic Multicellular Unit

BMP Bone Morphogenetic Protein

PTH Parathyroid Hormone

OPG Osteoprotegerin

IGF-I Insulin Growth Factor I

IGF-II Insulin Growth Factor II

TGF-β Transforming Growth Factor β

MM Multiple Myeloma

ODE Ordinary Differential Equations

PTHrP Parathyroid Hormone Related Protein

VOD Variable Order Derivatives

PDE Partial Differential Equations

FDE Fractional Differential Equations

PK/PD Pharmacokinetic and Pharmacodynamic

PK Pharmacokinetics

xxi

PD Pharmacodynamics

SISO Single Input Single Output

MIMO Multi-Input Multi-Output

xxii

1Introduction

Contents1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1

When I was in college, I wanted to be involved in things that would change the world.

Elon Musk, CEO at Tesla Motors and SpaceX

This chapter provides an introduction to the work developed in this thesis. The motivation for

this work (Section 1.1) overviews worldwide cancer incidence and the relevance of mathematical

models within this area. The context, introducing factional and variable order calculus and biochemical

bone remodeling models, follows (Section 1.2). Finally, the proposed original contributions and the

explanation of the remaining thesis outline completes this chapter (Sections 1.3 and 1.4, respectively)

1.1 Motivation

A neoplasm is an abnormal growth and proliferation of cells. Neoplasms of solid organs are often

referred to as tumors. Malignant tumors, also known as cancer, are characterized by the loss of

control of normal cell growth and by the ability to invade and spread into other tissues. Cancer cells

can spread from the original site to other parts of the body to form secondary tumors - a process

known as metastasization [40, 30].

According to the World Health Organization (WHO), cancer figures amongst the leading causes

of morbidity and mortality worldwide, with approximately 14 million new cases and 8.2 million cancer

related deaths in 2012. Predictions are that cancer incidence will rise by 70% over the next two

decades.

The United Nations (UN) estimates that lung cancer accounted for 13.0% of all diagnosed adult

cases worldwide (1,824,701 cases), closely followed by breast cancer (11.9%), colorectal cancer

(9.7%), and prostate cancer (7.9%) [70].

Figure 1.1: Worldwide incidence of (A) breast cancer and (B) prostate cancer, based on the GLOBOCAN2008project data: globalcancermap.com [International Agency for Research on Cancer - World Health Organization]

Malignant tumors, especially breast, prostate and lung cancers, are known to frequently metas-

tasize to the bone, which is associated with high levels of morbidity and worsening prognoses.

2

Metastatic, or secondary, tumors of the bone occur more commonly than primary bone tumors [29].

In females, malignant tumors of the breast and lungs are the most commonly associated with bone

metastases, accounting for approximately 80% of metastatic bone disease. Similarly, in males,

prostate and lung cancer are responsible for 80% of metastatic bone disease. In both males and

females, the remaining 20% of secondary tumors of the bone are attributable to kidney, colorectal,

and thyroid cancer, as well as tumors of unknown origin [10, 28]. Im Figure 1.1, the incidence of

breast and prostate cancer can be found.

In recent decades there have been major advances in cancer treatment research that includes

new host-targeted therapies, such as immunotherapy, and new combinations of chemotherapy, radio-

therapy, and hormone therapy [36].

Recent conceptual and technological advances have allowed for a better understanding of the

nature of cancer metastases. For example, interventions at the bone micro-environment level pose

a potential new therapeutic option. Cancer cells evolve towards metastasization through oncogenic

events that create genomic instability, including evasion of growth suppression and Deoxyribonucleic

Acid (DNA)-damage checkpoints [39].

How can the effects of oncogenic events affect a healthy bone environment? How can a therapy

combine minimum harm with optimized action for each individual patient? These are questions that

remain unanswered.

The interactions between a population of cells and their surrounding environment are complex.

From a mathematical point of view, understanding the large-scale dynamics of such interactions would

require a formal structure and predictive simulations. These tools, however, are inexistent thus far.

Mathematical modeling provides an invaluable tool for making assumptions explicit, highlighting

key factors, and providing quantitative indicators [2]. From an engineering standpoint, complex models

can be created to better study physiological and disease mechanisms. This approach to disease using

engineering systems is an area of growing interest with potential impact on the clinical sciences.

The goal is to promote the 4P approach to medicine: more predictive, personalized, preventive and

participatory [25].

Regarding bone micro-environment specifically, mathematical models can be used to reproduce

physiological and pathological states, including bone cancer. These models can even be used to

simulate and predict the effect of several therapies, allowing for the development of personalized

therapeutic regimens without excessive damage to the body [1, 31, 49]. These developments may

help in the creation of a road-map to guide future work and treatment.

1.2 Context

This section introduces the two essential backgrounds for the original work developed: fractional

and variable order derivatives and the essentials of bone remodeling physiology. The comprehension

of involved formulations and mechanisms is the first step into combining both strategies, under the

corresponding mathematical constructions presented in the following chapters.

3

1.2.1 Fractional and Variable Order Derivatives

But what if the differentiation order is not integer? Can any meaning be attached to dnf(t)dtn when

the order n is, say, equal to 12?

Duarte Valério and José Sá da Costa in An Introduction to Fractional Control [72]

1.2.1.A Fractional Calculus

Fractional calculus is more than 300 years old. The concept was created when Leibniz, inventor

of both ∂n

∂xn f(x) and∫f(x)dx notations, wrote to his friend L’Hôspital questioning him about fractional

exponents of successive differentials in a geometric progression [74].

It was born the idea to generalize differentiation and integration notions of order n ∈ N to that of

orders α ∈ R. The most natural and appealing generalization is based on the exponential function

f(x) = eax, with an nth derivative given simply by aneax. This suggests defining the derivative of order

α, which is not necessarily integer, as dα

dxα eax = aαeαx.

It is generally known that integer-order derivatives and integrals have clear physical and geometric

interpretations, even when it can be said that real objects are generally fractional (even if that frac-

tionality is very low) [46]. However, the same cannot be said of its fractional version [54]. For such

case, it can be shown that geometric interpretation of fractional integration is shadows on the walls

and its physical interpretation is shadows of the past [50]. Such conclusion can be extracted from the

mathematical study of fractional derivatives done in Chapter 2.

Since the conception of these derivatives, the number of their potential applications has rapidly

grown: they are used to formulate and solve different physical models allowing a continuous transition

from relaxation to oscillation phenomena; to predict the nonlinear survival and growth curves of food-

borne pathogens; to adapt the viscoelasticity equations (Hooke’s Law and the Newtonian fluids one)

[54, 46]. This has attracted a lot of attention in the Control society and, nowadays, fractional control

plays an important role in general physics, thermodynamics, electrical circuits theory and fractances,

mechatronic systems, signal processing, chemical mixing, chaos theory, and many others [46].

1.2.1.B Varying Order

The fractional calculus previously presented has allowed for operations of both integration and

differentiation to have any fractional order, of either real or imaginary value [32]. However, many

physical processes also appear to exhibit a fractional order behavior that varies with time or space.

On the field of viscoelasticity of certain materials, the temperature effect in small amplitude strains is

known to induce changes from an elastic to viscoelastic/viscous behavior, where real applications may

require a time varying temperature to be analyzed. The relaxation processes and reaction kinetics

of proteins, which are described by fractional differential equations, have been found to have an

order with a temperature dependence. The behavior of some diffusion processes in response to

4

temperature changes can be better described using variable order elements rather than time varying

coefficients, among other cases [32].

Such evidence allowed to consider the fractional order of integrals and derivatives as a function of

time or some other variable. This introduced variable order operators, where the operator’s order is

allowed to vary either as a function of the independent variable of integration or differentiation (t) or

as a function of some other variable (x) [32, 72]. However, the case of a derivative’s order changing in

time is a much more complex case to address than in its constant version. One of the main problems

is the number of existing definitions, nine of which have been given without a clear interpretation

associated [72, 35].

More recently, an interpretation of Variable Order Derivatives (VOD) in the form of switching strate-

gies was developed. It started with the simplest case of order switching, namely between two real

arbitrary constant orders, α1 and α2, and it was then generalized into a multiple-switching process.

This methodology can be interpreted as varying the order itself and can be seen schematically repre-

sented in Figure 1.2 [64]. Moreover, a numerical scheme was developed based on a matrix approach

that extended the analogous matrix form for the fractional constant order, ordinary and partial, differ-

ential equations to the one of variable order [35].

The study of fractional and variable order derivatives is detailed in Chapter 2.

Figure 1.2: Structure scheme of a simple-switching order derivative, from α1 to α2. Such strategy allows tophysically implement VODs in control applications. From Sierociuk, Malesza, and Macias [64].

1.2.2 Bone Remodeling Physiology

The basic physiology of the bone remodeling phenomena is here highlighted, accounting for the

effects and consequences of myeloma bone disease in this specific micro-environment. Existing

targeted therapies are also included.

1.2.2.A Basic Principles

Bone tissue is not static. Rather, it is a highly dynamic tissue that is constantly renewed at multiple

sites throughout the skeleton - a process known as remodeling. Bone remodeling is spatially hetero-

geneous. At any given point, regular asynchronous cycles of remodeling take place in 5 to 25% of the

total bone surface available. Remodeling is essential for the integrity of the skeleton [45].

The cells involved in the process include osteoblasts (bone-forming cells), osteoclasts (bone-

degrading cells), osteocytes (mature bone cells), and bone lining cells [19].

5

Osteoclasts are multi-nucleated cells that are uniquely adapted to remove mineralized bone.

Since their progenitor cells express Receptor Activator of NF-κB (RANK) and Macrophage Colony-

stimulating Factor Receptor (cfms), in the presence of Colony-stimulating Factor 1 (CSF1) and RANK-

ligand (RANKL) (which binds to cfms and RANK, respectively) these cells differentiate into osteoclasts

capable of bone resorption. The rate at which osteoclasts are generated determines the extension

of the Basic Multicellular Unit (BMU), while the life span of the cells determines the extent of bone

resorption. At the end of their life cycle, osteoclasts undergo apoptosis [57, 4].

Bone formation results from the activity of the mononucleated cells osteoblasts. Their differenti-

ation is controlled by Bone Morphogenetic Protein (BMP), Wnt-signaling and vitamin D, among other

factors. Osteoblasts can undergo apoptosis, differentiate into osteocytes or into bone lining cells [16].

Osteoblasts produce Osteoprotegerin (OPG), a soluble decoy receptor for RANKL which, by binding

to RANKL, inhibits osteoclastogenesis (the formation of new osteoclasts). When the secretion of OPG

is reduced, as a response to Parathyroid Hormone (PTH), osteoclastogenesis is promoted. Since os-

teoblasts also express PTH receptors, in response to this hormone, they upregulate the expression

of RANKL, which consequently binds to RANK expressed in osteoclasts precursors, also promoting

their activation and bone resorption. Hence, the RANK/RANKL/OPG pathway is of paramount im-

portance in regulating bone resorption and formation [19, 16, 57]. These relationships can be seen

schematically represented in Figure 1.3.

Figure 1.3: Schematic model of a bone cell population undergoing differentiation, proliferation, and apoptosis.Biochemical regulators (+/- biochemical activation/repression, respectively) are also represented. Adapted from

Scheiner, Pivonka, and Hellmich [63].

Bone remodeling is the result of the action of both osteoclasts and osteoblasts. Clusters of both

cell types are arranged in temporary anatomical structures known as BMU. In these structures, au-

tocrine and paracrine1 factors, produced by both osteoblasts and osteoclasts, regulate the formation

1Autocrine signaling is a form of cell signaling in which a cell secretes a hormone or chemical messenger (autocrine agent)that binds to receptors on that same cell, leading to changes in the cell. Paracrine signaling is a form of cell-cell communicationin which a cell produces a signal to induce changes in nearby cells, altering the behavior or differentiation of those cells.

6

and activation of these cells. The unique spatial and temporal arrangement of cells within the BMU

is critical to bone remodeling, ensuring coordination of the distinct and sequential phases of this pro-

cess: activation, resorption, reversal, formation, and termination [53, 59], as illustrated in Figure 1.4.

Figure 1.4: Schematic representation of a BMU and the associated bone-remodeling process, from Raggattand Partridge [53]

The remodeling process can be activated by either mechanical stimuli to the bone or through sys-

temic changes in homeostasis, which result in the production of estrogen or PTH [53, 45]. The action

of PTH on cells of the osteoblastic lineage results in the differentiation and activation of osteoclasts,

which form a cutting cone to degrade bone, initiating the resorption phase. This is followed by a re-

versal phase, where the lacunae created is prepared for the bone formation process, removing the

undigested demineralized collagen matrix. The beginning of the formation phase is coupled to the re-

sorption phase in a process not yet fully understood, as factors released from the bone matrix during

resorption (Insulin Growth Factor I (IGF-I), Insulin Growth Factor II (IGF-II) and Transforming Growth

Factor β (TGF-β)) may be involved in this coupling. Bone formation takes place even in the presence

of malfunctioning osteoclasts, which has led to the hypothesis that osteoclasts produce the coupling

factors responsible for attracting and up or down-regulating osteoblasts to the sites of bone resorption

[5, 31]. At the resorpted site, osteoblasts commence bone formation and replace the resorpted bone

by the same amount, ending the bone remodeling cycle [16].

An active BMU can travel across the tissue at a constant speed of 20-40 µm/day for up to 6 months

[59]. It comprises 10-20 osteoclasts that remove old and damaged tissue, and around 1.000-2.000 os-

teoblasts that secret and deposit unmineralized bone matrix, directing its formation and mineralization

into mature lamellar bone.

A mathematical model describing temporal changes in osteoblast and osteoclast populations and

the consequent change in bone mass at a single site of a bone remodeling BMU, was proposed in

Komarova et al. [31]. It was later extended to include bone remodeling dynamics with a physical

7

dimension, the effects of a tumoral burden for Multiple Myeloma (MM) disease and its respective

treatment [1]. These mathematical models, where cells have the ability to interact with each other via

effectors that are released or activated by bone cells and act in an autocrine or paracrine manner, are

studied in more detail in Chapter 3.

1.2.2.B Tumor Influence

During the formation of a primary lesion, tumor cells undergo a variety of molecular and epigenetic

events that eventually allow them to escape from the primary tumor site [69]. It is then said that the

tumor has metastasized. In the case of bone cancer, metastasization typically occurs during the late

stages of the disease, bringing significant morbidity and making the process virtually incurable [8].

Bone micro-environment is then altered since tumor-secreted factors stimulate bone cells, affecting

the osteoclasts/osteoblasts equilibrium. These cells, in turn, release growth factors that act back

on the tumor. Thus, the dynamic behavior of bone remodeling is severely disrupted, including its

biochemical regulation. Bone lesions arise due to the disturbed pathophysiology, resulting in the loss

of bone integrity [9].

Bone metastases can be osteolytic, when resorption is increased, or osteoblastic, when bone for-

mation is stimulated in an unstructured way. Both osteolytic and osteoblastic processes take place

in all types of metastases, though their relative proportion changes. The predominant process deter-

mines the type of metastasis. For instance, breast cancer metastases are known to develop osteolytic

metastasis, while prostate cancer usually produces osteoblastic metastases [69].

In osteolytic metastases, deregulation of the normal remodeling process occurs due to stimu-

lation of bone resorption by the metastatic cells [12]. TGF-β is released from the bone matrix and

stimulates Parathyroid Hormone Related Protein (PTHrP) production in metastatic cells. PTHrP then

binds to the PTH receptors on cells of the osteoblastic lineage, enhancing the secretion of RANKL.

This results is the subsequent activation of osteoclasts which, in turn, leads to increased bone re-

sorption. Active osteoclasts induce the release of TGF-β, giving rise to a vicious cycle [24, 16]. This

metastases, however, are rarely completely devoid of bone formation since typically there is some

reactive new bone around the new periphery of lytic lesions, which may represent an attempted re-

sponse to confine the tumor [5]. MM is a bone disease where little if any osteoblastic reaction exists

[68].

In osteoblastic metastases, tumoral cells grow as they stimulate osteoblasts into activating Wnt-

signaling. In addition, tumor-derived proteases contribute to the release of osteoblastic factors from

the extracellular matrix, including TGF-β and IGF-I. RANKL is increased due to tumor-induced os-

teoblast activity, leading to the release of PTH and the promotion of osteoclast activity [9]. Thus,

tumor microenvironment leads to the accumulation of newly formed bone.

The two pathways are represented in Figure 1.5. Typical tumor evolution is recognized to be of a

Gompertz2 form, regardless of osteolytic or osteoblastic origin, as represented in Figure 1.6.

2Type of mathematical model (sigmoid function) for a time series, where growth is slowest at the start and end of a timeperiod

8

Figure 1.5: The vicious cycle of bone metastases in the osteolytic pathway is represented on the left side,where PTHrP is set to induce osteoclastogenesis by upregulating the RANKL. The activated osteoclasts in turnproduce TGF-β, IGF-I and IGF-II which promote cancer cell growth. In the osteoblastic vicious cycle, cancercells produce osteoblast-stimulating factors, where PTHrP is also overexpressed.The activated osteoblasts in

turn produce several factors, which facilitate cancer cell colonization and survival upon arrival in the bonemicroenvironment. From Wong and Pavlakis [75].

Figure 1.6: Top/Left: tumor growth fraction, termed γT , declines exponentially over time, stabilizing at 1-4%.Bottom/Left: tumoral growth rate peaks before it is clinically detectable. Right: tumor size increases slowly,

goes through an exponential phase, and slows down again as it reaches its maximum size. Adapted fromKasper et al. [29].

For the mathematical models of bone remodeling presented in Chapter 3, the effect of tumoral cells

is added considering only the osteolytic phase, since the osteoblastic phase is defined by different

mechanisms. Such behavior was introduced in Ayati et al. [1] and detailedly study in Section 3.3 for

a starting point based in MM bone disease. This malignancy is associated with the least new bone

formation occurring, with little if any osteoblastic reaction [5].

1.2.2.C Existing Treatments

Several approaches to treating primary and metastatic bone disease have the potential to affect

both tumor affected and healthy cells from the bone micro-environment. Strategies can be oriented to

effectively inhibit tumor growth by targeting the bone and its micro-environment rather than the tumor

alone. In animal models, combined treatments have been shown to reduce the tumor burden and its

derived bone lesions [8].

Anti-resorptive therapy targets osteoclasts, when a osteolytic metastatic bone disease is present.

Biphosphonates such as alendronate and monoclonal antibodies like denosumab are effective treat-

9

ments currently being administrated. While bisphosphonates lodge in bone and poison osteoclasts

as they degrade bone, denosumab in turn binds exclusively to RANKL, increasing the OPG/RANKL

ratio and inhibiting osteoclast formation. For other diseases, such as multiple myeloma, therapies

include daily doses of PTH or endothelin, that act by targeting osteoblasts to recover bone mass. Of

course, anti-cancer agents that target metastatic and primary tumor cells directly (chemotherapy and

hormone therapy) should be used in combination with the aforementioned therapies in either case [8].

It is difficult to extrapolate the response to different drug treatments from in vivo studies, which

highlights the paramount importance of a framework that can accurately describe the complex rela-

tions involved [1]. An initial attempt was proposed in the models developed in Ayati et al. [1], where

anti-cancer proteasome inhibition in myeloma bone disease was used. This treatment has direct

anti-myeloma effects and stimulates osteoblasts differentiation and bone formation. More recently,

mathematical solutions have been proposed accounting for anti-resorptive therapy combined with

other different therapies, for an osteolytic metastatic bone disease. Such is achieved by applying

Pharmacokinetic and Pharmacodynamic (PK/PD)3 information in the models [17, 77, 38].

1.3 Original Contributions

A simpler mathematical model for the bone micro-environment is crucial to promptly highlight tu-

mor disrupted bone interactions and the effectiveness of different therapies. Clinical decision systems,

with personalized therapy schemes, are then made possible.

A new approach to the existing biochemical models for bone micro-environment dynamics ([1, 31])

is proposed and acts in two main categories: physiological context and mathematical simplification.

Firstly, a new starting point is considered by stating that the existing models ([1, 31, 17]) are

better used to describe an osteolytic metastatic bone disease instead of MM, as previously done

(Section 4.1).

Second and most importantly, the action of a tumor in the existing models is mathematically sim-

plified using VOD concepts to decrease the number of parameters used. Proposed therapies, as the

ones from Ayati et al. [1] and Coelho, Vinga, and Valério [16], are also encompassed in this simplifica-

tion. This type of differentiation method is here applied for the first time in bone remodeling, providing

more succinct equations with similar results of those from the original formulations (Sections 4.2-4.4).

Obtained results have already been used for submitted and on-the-work publications, as follows.

• Dynamic biochemical and cellular models of bone physiology: integrating remodeling

processes, tumor growth and therapy

To be published in 2017 for Springer’s Lecture Notes in Computational Vision and Biomechanics,

this chapter includes the simplified local mathematical model for a bone micro-environment

disrupted by a tumor’s action [15].3The pharmacokinetics (PK) of a drug describes its concentration evolution at the target tissue, whereas the effect of such

concentration is given by pharmacodynamics (PD).

10

• Variable Order Differential Models of Bone Remodeling

Accepted for the 20th IFAC World Congress, to be held in July 2017, in Toulouse, France. Pro-

vides a detailed mathematical analysis of the simplified local variable order bone model with

tumor [43].

• Simplifying Biochemical Tumorous Bone Remodeling Models Through Variable Order

Derivatives

To be submitted in the Computers & Mathematics with Applications Journal, this paper com-

prises both local and non-local formulations of the simplified models originally developed here

[42].

• Therapy Opportunities for Tumorous Bone Models Through Variable Order Derivatives

To be submitted, this on-going paper is the extension of the work developed in Neto et al. [42],

to include existing therapies in the variable order models developed [41].

1.4 Thesis Outline

For the better understanding of the work developed, a chapter hierarchy in growing complexity

was followed.

• Chapter 1: Introduction

This chapter contextualizes the work originally developed in this thesis. It provides a cancer

worldwide incidence overview (Section 1.1), followed by an introduction to fractional and variable

order calculus and to basic bone remodeling physiology, for healthy, tumor disrupted bone and

applied treatment cases (Section 1.2). It also details the original contributions made possible by

combining these last two differentiated subjects (Section 1.3).

• Chapter 2: Fractional and Variable Order Derivatives

Chapter 2 solely focuses on introducing fractional and variable order calculus, by highlighting

and explaining the formulations used in this theses. Basic working tools are provided in Sec-

tion 2.1, followed by a detailedly study of fractional derivatives in Sections 2.2, and variable

order constructions are encompassed in Section 2.3. For the last two sections, attention was

given in reviewing existing applications for each calculus type.

• Chapter 3: Mathematical Models for Bone Remodeling

Existing bone remodeling models are reviewed in this chapter, framing the obtained responses

to what is biological and physiological known. Parmacokinetics and pharmacodynamics theory

is initially introduced, in Section 3.1, followed by Sections 3.2 and 3.3 where existing local and

non-local models are reviewed. For each, healthy, bone with tumor and with applied treatment

bone micro-environment model are studied and stimulated.

11

• Chapter 4: New Variable Order Models: Methodology, Results and Discussion

Original variable order bone remodeling models, for a sole action of a tumor and with applied

treatment, are presented in this chapter. Justification for the new physiological starting point,

from MM to metastatic bone disease, can be found in Section 4.1. Some necessary previ-

ous considerations for the models constructions are highlighted in Section 4.2. In Section 4.3

simplified bone models with an acting tumor, with variable order derivatives, are presented in

their local and non-local one dimensional constructions, followed by Section 4.4 where a basic

treatment and a more elaborate PKPD approach were introduced to the original models.

• Chapter 5: Conclusions and Future Work

This chapter summarizes and highlights the results obtained in this thesis, providing a clear

interpretation within the physiological and mathematical state-of-the-art of the topic. It also

details the future work path that the work here developed made possible.

• Bibliography

All the publications referred in this thesis, that made the work possible, can be consulted here.

• Appendixes

Additional information, un-central do the thesis core but yet important to be added, can be found

in the Appendixes. More information on the fractional and variable order calculus formulations

and support to the biochemical models presented in Chapter 3 can be found in this section.

All presented results and simulations were obtained with Matlab and Simulink softwares.

12

2Fractional and Variable Order

Derivatives

Contents2.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Derivatives with Varying Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13

Harry: Excuse me sir, can you tell me where I might

find Platform Nine and Three-Quarters?

Station Guard: Nine and Three-Quarters? Think you’re being funny do ya?

J. K. Rowling, Harry Potter and the Philosopher’s Stone

This chapter solely focus on fractional and variable order calculus. The basic working tools are

presented in Section 2.1, followed by a detailedly study of fractional derivatives in Sections 2.2, and

variable order constructions in Section 2.3.

2.1 Introductory Concepts

Understanding either fractional or variable order derivatives begins with two fundamental concepts:

the functional operator D, and the Gamma (Γ) function. Both are crucial concepts for the underlying

mathematical definitions and formulations.

2.1.1 Operator D for Derivatives and Integrals

The infinite sequence of n-fold integrals and derivatives in Equation 2.1 can be formally defined as

presented in Equation 2.2. Here, the functional operator D is associated with an order n ∈ Z that not

only considers the natural differentiation case as it does with indefinite integrals of integer order [72].

...

∫ t

−∞

∫ t

−∞f(t)dtdt,

∫ t

−∞f(t)dt, f(t),

df(t)

dt,d2f(t)

dt2, ... (2.1)

cDnt f(t) =

dnf(t)dtn , if n ∈ N

f(t) = 0 if n = 0∫ tc cDn+1t f(t)dt if n ∈ Z−

tDnc f(t) =

(−1)n d

nf(t)dtn , if n ∈ N

f(t) = 0 if n = 0∫ ct tDn+1c f(t)dt if n ∈ Z−

(2.2)

For a function f(t), D can be numerically used to rewrite known definitions of orders n = 1 or

n ∈ N, as follows.

D1f(t) =df(t)

dt= limh→0

f(t)− f(t− h)

h

Dnf(t) =dnf(t)

dtn= limh→0

∑nk=0(−1)k

(nk

)f(t− kh)

hn

(2.3)

More formulations with operator D, as the Cauchy’s Formula for indefinite integral and the Rie-

mann Integral, can be found in Appendix A.1.

14

2.1.2 Basic Working Tool: the Gamma Function Γ

The Γ function is a component in various probability-distribution functions as well as combina-

torics. It finds a wide range of applications in areas such as quantum physics, astrophysics and fluid

dynamics [11]. Also, the gamma distribution, which is formulated in terms of the gamma function, is

used in statistics to model a wide range of processes (for example, the time between occurrences of

earthquakes) [55].

The function itself, and some relevant mathematical properties, can be defined as follows.

Γ(x) =

∫ +∞

0

e−yyx−1dy (2.4)

•Γ(1) =

∫ +∞

0

e−ydy = [−ey]+∞0 = 1 (2.5a)

•Γ(x+ 1) = xΓ(x)⇒ Γ(n) = (n− 1)!, n ∈ N (2.5b)• limx→0+

Γ(x) = +∞ (2.5c)

• limx→n

Γ(x) =∞, n ∈ Z (2.5d)

As it can be seen, Γ is an extension of the factorial function, with its argument shifted down by 1

unit when x = n ∈ N, for both real and complex numbers.

Figure 2.1: Gamma function a) representation in a part of the real axis, created with gnuplot for WikipediaCommons - Gamma Function and b) a hand-drawn graph of the absolute value of the complex gamma function,

from Jahnke, Losch, and Emde [27].

This special function, along with its properties, can also be used to rewrite the combination of a

things, taken b at a time. Such case is presented next.

(a

b

)=

a!

b!(a− b)!a, b ∈ Z+

0 (2.6)

Γ(x)⇒(a

b

)=

Γ(a+1)

Γ(b+1)Γ(a−b+1) if a, b, (a− b) /∈ Z−(−1)bΓ(b−a)Γ(b+1)Γ(−a) if a ∈ Z− ∧ b ∈ Z+

0

0 if [(b ∈ Z− ∨ (b− a) ∈ N) ∧ a /∈ Z−] ∨ (a, b ∈ Z− ∧ |a| > |b|)(2.7)

2.2 Fractional Derivatives

Factional calculus can be defined as a theory of integrals and derivatives of arbitrary order, that

unifies and generalizes the notions of integer-order differentiation and n-fold integration [50]. Simply,

15

it can be said that it extends the usual definitions, allowing differentiation and integration orders that

are not required to be positive integers [73].

This section presents how the operator D can be generalized to real orders, by using some of the

most common existing definitions for both fractional derivatives and transfer functions.

2.2.1 Formulations

The derivative of an arbitrary order α, given by Dα, can be considered as an interpolation of the

sequence of D operators presented in Section 2.1.1, and concisely referred as cDαt or cDα

t f(t) when

specifically applied to a function f [72].

There are several alternative definitions for fractional derivatives, of which the three main ones are

presented here [72]. This situation is similar to that of real-valued functions integrals of a real variable

that may be defined according to Rienmann or Lebesgue 1. However, for a large class of functions

f(t) well-behaved enough for necessary operations to take place, some definitions provide the same

result [73, 72].

Definition 1. Riemann-Liouville Construction

cDαt f(t) =

dαf(t)dtα , if α ∈ N

Ddαec D

α−dαet f(t), if α ∈ R+ \ N

f(t), if α = 0∫ tc

(t−x)−α−1

Γ(−α) f(x)dx, if α ∈ R−

(2.8)

Definition 2. Caputo Construction

cDαt f(t) =

dαf(t)dtα , if α ∈ N

cDα−dαet Ddαef(t), if α ∈ R+ \ N

f(t), if α = 0∫ tc

(t−x)−α−1

Γ(−α) f(x)dx, if α ∈ R−

(2.9)

Definition 3. Grünwald-Letnikoff Construction

cDαt f(t) = lim

h→0+

b t−ch c∑k=0

(−1)k(a

k

)f(t− kh)

hα(2.10)

(a

k

)=

Γ(α+1)

Γ(k+1)Γ(α−k+1) , if α, k, (α− k) ∈ R \ Z−(−1)kΓ(k−α)Γ(k+1)Γ(−α) , if α ∈ Z− ∧ k ∈ Z+

0

0, if (k ∈ Z− ∨ (k − α) ∈ N) ∧ α /∈ Z−(2.11)

1Riemann integration is based on subdividing the domain of an f function. This leads to the requirement of some “smooth-ness” of f for the Riemann integal to be defined: for x, y close, f(x) and f(y) need to have something to do with each other.Lebesgue integration is based on subdividing the range space of f : it is built on inverse images [6].

16

2.2.2 Matrix Constructions

The matrix form for the Grünwald-Letnikoff construction, in its recursive form (not presented here),

was detailedly described in Podlubny et al. [51]. The method, that allows a straightforward numerical

implementation of fractional-order derivatives, is based on a triangular strip matrix approach dis-

cretized to operators of differentiation and integration for an arbitrary real order. Such method takes

into account the whole time interval of interest at once, and not a step-by-step solution by moving from

the previous time layer to the next one. Besides triangular strip matrices, this construction also works

with a specific matrix product (Kronecker product, A⊗B) and a set of matrices called eliminators and

shifters.

The referred matrix form, as presented in Sierociuk, Malesza, and Macias [66], is given by Equa-

tions 2.12 and 2.13, where h > 0 is a step time, n = [t/h] andW (α, k) ∈ R(k+1)×(k+1), wα,i =(−1)j(αi)

hα .

0D

α0 f(t)

0Dαhf(t)

0Dα2hf(t)...

0Dαkhf(t)

= limh→0

W (α, k)

f(0)f(h)f(2h)

...f(kh)

(2.12)

W (α, k) =

h−α 0 0 · · · 0wα,1 h−α 0 · · · 0wα,2 wα,1 h−α · · · 0wα,3 wα,2 wα,1 · · · 0

......

... · · ·...

wα,k wα,k−1 wα,k−2 · · · h−α

(2.13)

2.2.3 Relevant Properties

• Non-local Operator

From all three definitions a common characteristic arises: operator D always depends on the in-

tegration limits c and t, called terminals. This makes D a non-local operator. Exception comes

when α ∈ Z+0 (the case of natural order derivatives and when α = 0). In this respect, fractional

derivatives look like the integrals and not to the standard calculus derivatives themselves (irre-

spective of the sign of α). Thus, cDαt f(t) depends on the values assumed by f between c and

t. If a variable t can be considered time, the non-local operator cDαt depends on the values of

f assumed before time t, Another way of putting this is saying that cDαt has a memory of past

values of f . Likewise, the non-local operator tDαc dependes on the values of f assumed after

time t [72].

• Linear Operator

As proven in Valério and Sá da Costa [72], D can be presented as a linear operator since the

previous fractional derivatives definitions are combinations of linear operators.

17

cDαt [αf(t) + bg(t)] = αcD

αt f(t) + bcD

αt g(t),a, b ∈ R

tDαc [αf(t) + bg(t)] = αtD

αc f(t) + btD

αc g(t)a, b ∈ R

(2.14)

• Relation Between Definitions

From Definitions 1, 2 and 3, it can be seen that the Riemann-Liouville and Caputo formulations

clearly differ for α > 0 but, on the other hand, Riemann-Liouville and Grünwald-Letnikoff defini-

tions provide the same result, as long as function f satisfies the conditions for the application

for both cases [72].

Theorem 1. If f(t) has a max [0, dαe] continuous derivatives and Dmax[0,dαe]f(t) is integrable,

then cDαt exists according to both Riemann-Liouville and Grünwald-Letnikoff definitions, which

provide the same result.

Theorem 2. In the conditions of the previous theorem, if cDαt f(t) = γ according to the Riemann-

Liouville and Grünwald-Letnikoff definitions, then the Caputo definition yelds:

cDαt f(t) = γ −

dαe−1∑k=0

(t− c)−α+kdkf(c)

dtk

Γ(k − α+ 1)(2.15)

• Law of Exponents

The exponential law for integer orders of D, given by Equation 2.16a, holds for the fractional

case by using using either the Riemann-Liouville or the Grünwald-Letnikoff definitions. However,

to be used, this law must be followed according to Equation 2.16b [72].

cDmt cD

nt f(t) = cD

m+nt f(t)

(m,n ∈ Z+0 ) ∨ (m,n ∈Z−0 ) ∨ (m ∈ Z+ ∧ n ∈ Z−)

(2.16a)

cDαt cD

βt f(t) = cD

α+βt f(t)

(α, β ∈ Z+0 ) ∨ (α, β ∈ Z−0 ) ∨ (α ∈ Z+ ∧ β ∈ Z−) ∨ (β ≤ 0 ∧ (α+ β ≤ 0))

(2.16b)

• Short-memory Principle

When used, approximations based on Riemann-Liouville or Grünwald-Letnikoff definitions have

an associated error, ε, that is limited by |ε| as presented next [72].

cDαt f(t) ≈ t−LD

αt f(t), t > (c+ L), α > 0 (2.17a)

tDαc f(t) ≈ tD

αt+Lf(t), t < (c− L), α > 0 (2.17b)

|ε| < M

Lα|Γ(1− α)|(2.17c)

18

2.2.4 Some Applications

From the wide range of applications made possible by fractional calculus, some illustrations are

presented next [54, 34, 67].

• Abel’s Integral Equation: this equation can be adapted to form a fractional-order system,

where 0 < α < 1 in Equation 2.18. It can be rewritten as in Equation 2.19.

1

Γ(α)

∫ t

0

φ(τdτ)

(t− τ)1−α = f(t), (t > 0) (2.18) 0D−αt φ(t) = f(t) (2.19)

• Viscoelasticity: the most extensive application field of fractional differential and integral opera-

tors is in viscoelasticity where, for instance, the relationship between stress and strain for solids

(known as Hooke’s Law with constant E) and for Newtonian fluids (with constant η) satisfy the

respective fractional differential equations presented.

σ(t) = Eε(t) σ(t) = ηdε(t)

dt(2.20)

Dασ(t) =Γ(1− α)t−α

Γ(1.2α)σ(t) Dαε(t) = Γ(1 + α)t−αε(t) (2.21)

In Magin [34], the shear stress in a uniform, homogeneous, Newtonian fluid is linked to a me-

chanical motion at the fluid’s interface through a fractional derivative approach, where the fluid

velocity distribution is coupled to the surface shear stress developed by a large rigid moving

plate. V (z, t) is the transverse fluid velocity, σ(z, t) the shear stress, ρ the fluid’s density and

µ its viscosity. The result, in Equation 2.23, gives the shear stress in therms of a fractional

derivative of the fluid velocity that succinctly describes the shear stress and the drag term on

the moving plate.

ρ∂

∂tV (z, t) = µ

∂2

∂z2V (z, t), (t > 0, z > 0) σ(z, t) = µ

∂

∂zV (z, t), (t > 0, z ≥ 0) (2.22)

σ(z, t) =

√ρµ

π

∫ t

0

1√t− τ

∂

∂τV (z, τ)dτ =

√ρµ

[1√πt∗ ∂∂tV (z, t)

](2.23)

• Food Science: a theory based in Fractional Differential Equations (FDE) involving 1st and 2nd

term equations was developed to predict the nonlinear survival and growth curves of foodborne

pathogens [54]. The solution of 1st-term equation leads to the Weibull model 2 and the 2nd-term

one is successful in describing the complex shapes of microbial survival and growth curves as

compared to the linear and Weibull models.

• Fractional Diffusion Equations: fractional calculus can be used to formulate and solve differ-

ent physical models allowing a continuous transition from relaxation to oscillation phenomena.

An application to an anomalous diffusion process demonstrates that the method used is also2Named after its inventor, Waloddi Weibull, this distribution is widely used in reliability engineering and elsewhere due to its

versatility and relative simplicity.

19

useful for more than one independent variable. For fractional diffusion equations, the following

example can be used.

Equation: 0Dαt ρ(x, t)− ρ0(x)

t−α

Γ(1− α)= Kαdxxρ(x, t)

Solution: ρ(x, t) =1√

2πKαtαH2,0

1,2

x2

2Kαtα[(1− α/2, α, (0, 1), (1/2, 1)]

(2.24)

In the work developed in Sierociuk et al. [67], a heat transfer process was adapted from an

integer order partial differential equation form, in a solid material, to that of an heterogeneous

media with a fractional order describing the anomalous sub or super-diffusion3 behavior of a

semi-infinite beam. In what follows, T (t, λ) is the temperature and H(t, λ) is the heat flux of

the beam at time instant t and space coordinate (distance) λ, and 1a2 is the beam material

conductivity.

Typical Diffusion Process

∂

∂tT (t, λ) =

1

a2

∂2

∂λ2T (t, λ)

H(t, λ) = − 1

a2

∂

∂λT (t, λ) =

1

a

∂0.5

∂t0.5T (t, λ)

(2.25)

Sub-Diffusion Process

∂α

∂tαT (t, λ) =

1

a2α

∂2

∂λ2T (t, λ)

H(t, λ) = − 1

a2α

∂

∂λT (t, λ) =

1

a2α

∂α2

∂λα2T (t, λ)

(2.26)

2.2.5 Fractional Laplace Transforms

When applying the Laplace transform to operator D, the two definitions presented next arise from

the fractional derivatives constructions. However, a more extensive study over these Laplace trans-

forms can be found in Chapter 2 of Valério and Sá da Costa [72].

• Riemann-Liouville and Grünwald-Letnikoff Laplace Transform Construction

L[0Dαt f(t)] =

sαF (s), if α ∈ R−

F (s), if α = 0

sαF (s)−dαe−1∑k=0

sk0Dα−k−1t f(0), if α ∈ R+

(2.27)

• Caputo Laplace Transform Construction

L[0Dαt f(t)] =

sαF (s), if α ∈ R−

F (s), if α = 0

sαF (s)−dαe−1∑k=0

sα−k−1Dkf(0), if α ∈ R+

(2.28)

3Anomalous diffusion is a diffusion process with a non-linear relationship to time: typical diffusion process have an order ofα = 1; super-diffusion is regulated by α > 1; and α < 1 represents sub-diffusion. Diffusion of colloidal particles in bacterialsuspensions and in disordered media are some examples.

20

2.2.6 Fractional Transfer Functions

For the integer-order Single Input Single Output (SISO) system, where the output and the input are

related by linear, time-invariant differential equations, its transfer function is the ratio of the Laplace

transforms of the output with the input, when all initial conditions are zero [72]. This definition is

also applicable when the differential equation involves fractional orders: L[Dαf(t)] = sαL[f(t)] [73],

following what was presented for fractional Laplace transforms in 2.2.5.

For the SISO system, the fractional transfer function is given by the following equation, where ak

are the denominator and bk are the numerator coefficients, the αk ≥ 0 are the denominator orders

and the βk ≥ 0 are the numerator orders.

G(s) =

m∑k=1

bksβk

n∑k=1

aksαk

(2.29)

Fractional Multi-Input Multi-Output (MIMO) transfer function matrices, on the other hand, are

possible to achieve for a system withm inputs and p outputs related by linear, time-invariant differential

equations. By using the Laplace transform of these equations, when all initial conditions are zero,

allows for m × p SISO transfer functions, each of them relating one input with one output when all

other inputs are zero. These transfer functions can be collected in a matrix that will, since the system

is said to be linear, relate all inputs to all outputs. This happens also when the differential equations

involve fractional orders. Formally, a fractional MIMO transfer function matrix of a plant with m inputs

and p outputs is a m× p matrix G, the elements of which are SISO fractional transfer functions.

2.2.6.A Stability

For integer transfer functions to be stable, it is known that all its poles must lie within the left-

hand complex half-plane. Similarly, a fractional system G(s) is stable if and only if Equation 2.30 is

respected [72].

For a stable commensurable4 system G(s), stability is verified as well by Matignon’s theorem [72]

and illustrated in Figure 2.2 with the classical stability criterion adapted to the influence of the order

value. It is also intuitive that, for orders α > 2, the system must be unstable [13].

∀s :

n∑k=0

aksak = 0, |∠s| > π

2if |∠s| ∈ [−π, π]rad (2.30)

4A commensurable transfer function occurs when all the orders ak and bk are integer multiples of at least one commondivisor a > 0.

21

Figure 2.2: Map for stability regions of fractional order systems of stable commensurable functions, in complexplane, where q represents the fractional order α used previously. From Petrás [46].

2.2.6.B Frequency and Time Responses

The frequency behavior of a fractional transfer function can be found, as usual, by replacing s with

jω, being ω the frequency. For a transfer function given by G(s) = sα, the Bode diagram response

can be found in Figure 2.3.

Figure 2.3: Bode diagram representation of the fractional transfer function G(s) = sα where υ is presented asthe fractional derivative order. From Valério [71].

For a generic transfer function, when an input given by u(t) = A sin(ωt) is applied to the stable

system G(s) (whose transfer function is given by 2.29) the output, after the transient response, will be

y(t) = |G(jω)|A sin(ωt+∠G(jω)). It can be seen that, for positive and negative values of the fractional

order α, the gain plot has a constant slope of 20α dB per decade; the gain is unity for ω = 1rad/s; and

the phase is constant for all frequencies and equal to α90o [73]. Based on such, the impulse and step

responses are mathematically given as follows, respectively, and its temporal representation can be

found in Figure 2.4 for a wide range of fractional orders α.

gimp(t) = t−α−1

Γ(−α) gstep(t) = t−α

Γ(−α+1) (2.31)

Since the impulse response translates the relationship between the input and the output of a

system, it can be seen in the first graphic of Figure 2.4 that the system comprised by G(s) = sα

tends to give a slower response as the fractional order becomes smaller, which suits what is known

for integer orders. This behavior is replicated in the step response for the same system.

22

5 10 15 20 25 30 35 40 45 50−16

−14

−12

−10

−8

−6

−4

−2

0

2

4x 10

4

Time t

Uni

t Im

puls

e R

espo

nse,

g(t

)

Unit Impulse Response for Several α

α =0.1α =0.2α =0.3α =0.4α =0.5α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

5 10 15 20 25 30 35 40 45 50−500

0

500

1000

Time t

Uni

t Im

puls

e R

espo

nse,

g(t

)

Unit Step Response for Several α

α =0.1α =0.2α =0.3α =0.4α =0.5α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

Figure 2.4: Unit impulse and unit step response, respectively, for the simple fractional transfer function given byG(s) = sα. These results were achieved by using FOMCON toolbox for MATLAB, for α varying between

[0.6; 1.5].

Studying the analogous of classical second order transfer functionG(s) =ω2

0

s2 + 2ξω0s+ ω02

, given

by G(s) =1

( sa )2α

+ 2ξ( sa )α + 1, it can be shown that the roots of such transfer function are given by

sα = aα(−ξ ±√ξ2 − 1). Hence, this transfer function stability will depend upon the value taken by

ξ. This coefficient can be interpreted similarly to the damping coefficient of an integer second order

system and here termed pseudo-damping coefficient [72], whose influence in the system’s response

is as follows.• ξ ≥ 1, the transfer function has absolute stability;• ξ ≤ −1 the system is always unstable;• ξ = 0 the roots are complex and equal to ±j. The system will be stable if α < 1;

• 0 < ξ < 1 and 0 < α ≤ 1 the roots are given by sα = aα(−ξ ±√ξ2 − 1) = aαe±j tan(

−1√

1−ξ2−ξ ), which

means the function transfer is always be stable;• − 1 < ξ < 0 and 0 < α < 1 it can be proven that the system is stable if − cos απ2 < ξ;• 0 < ξ < 1 and 1 < α < 2 the system is stable if ξ > − cos απ2 ;• − 1 < ξ < 0 and 1 ≤ α < 2 the system is unstable.

All of the previous cases can be summarized in the following: the system is stable if ξ > − cos απ2 .

Figure 2.5: Frequency behavior of function G(s) =1

( sα

)2α + 2ξ( sα

)α + 1, on the left, and gain plots of the Bode

diagram of functions F (s) = 1sα+1

for the specified α. From Valério and Sá da Costa [72] and Monje et al. [37],respectively.

The left side of the previous figure represents the location of resonance frequencies that can

be translated in none, one ou two local maxima on the gain Bode plot, according to the α and ξ

parameters. As expected, an integer first order transfer function will have none or only one resonance

frequency. Such can be seen on the right side of Figure 2.5 [13].

The following Figure ?? represents the unit impulse and step response of the system G1(s), which

can be said to have an absolute stability since its pseudo-damping coefficient is ξ ≥ 1 (left side of

23

Figure 2.5). Here, α is responsible for the dynamic response: for smaller orders it can be seen a

faster response but with a bigger overshoot value. From the Bode diagram, it is specially clear the

relationship presented for fractional transfer functions: the gain plot has a slope of 20α dB per decade

and the phase is constant and equal to α90o, according to the frequency associated with each pole.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

Time t

Uni

t Ste

p R

espo

nse,

g(t

)

Unit Step Response of G(s) for Several α

α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

−200

−150

−100

−50

0

Mag

nitu

de (

dB)

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

105

−135

−90

−45

0

Pha

se (

deg)

Bode diagram of G(s) for Several α

Frequency (rad/s)

α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

Figure 2.6: a) Unit step response and b) corresponding Bode diagram for the fractional transfer function givenby G1(s) =

s

sα + 3sα/2 + 1where ξ = 3

2and α varies between [0.6; 1.5]. These results were achieved by using

FOMCON toolbox for MATLAB.

For the following case, of G2(s), the pseudo-damping coefficient takes the value of ξ = − 12 and

the system response stability will largely depend on the value of α. Again, from the left side of Figure

2.5, it can be seen that for the value of ξ used it would be expected an unstable response for α > 0.75

and, before such value, a stable response with a resonance peak, which was verified by the growing

instability of the time response with a growing value of α (after 0.8).

In truth, from the Bode diagram of G2(s), the response has a unstable tendency as the magnitude

decays for larger frequencies and the phase, at the second pole frequency, exhibits a growing abrupt

behavior until it transitions to positive values. Such translates the zero/pole transition that occurs in

the overall system by changing its order.

0 100 200 300 400 500 600 700 800 900 1000−6

−4

−2

0

2

4

6

8

Time t

Uni

t Ste

p R

espo

nse,

g(t

)

Unit Step Response of G(s) for Several α

α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

−100

−50

0

50

Mag

nitu

de (

dB)

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

105

−90

0

90

180

270

360

Pha

se (

deg)

Bode diagram of G(s) for Several α

Frequency (rad/s)

α =0.6α =0.7α =0.8α =0.9α =1α =1.1α =1.2α =1.3α =1.4α =1.5

Figure 2.7: a) Unit step response and b) corresponding Bode diagram for the fractional transfer function givenby G2(s) =

s

sα − sα/2 + 1, where ξ = − 1

2and α varies between [0.6; 1.5]. These results were achieved by using

FOMCON toolbox for MATLAB.

For both G1(s) and G2(s), however, it’s important to highlight that the step response does not tend

to 1 with time, which can be confirmed through the origin of the magnitude plot of both Bode diagrams.

24

2.3 Derivatives with Varying Order

In the same way operator D was adapted to orders that are not required to be positive integers,

it can also be further extended to include a time, t, or space, for instance x, order dependence.

Based on the work developed in Valério and Sá da Costa [72] and extended in Sierociuk, Malesza,

and Macias [65], the existing definitions are here presented. Additionally, important properties are

introduced, such as approximations in terms of difference equations and corresponding matrix forms.

2.3.1 Mathematical Definitions

For an order α that is not constant with time there are three existing ways, according to [72], of

dealing with the branches α(t) ∈ R− and α(t) ∈ R+\N, when it comes to extending the fractional

order derivatives definitions to variable ones. It is noted that, however, only real orders are present

here.

In the first possibility, (1), the argument of the order is simply the current value of t. In the second,

(2), is to let the argument of α be the same as of f , for the negative branches of the order (α < 0).

Finally, the third case, (3), considers as argument the difference between the two previous possibilities

presented: the current time instance and the argument of f [73].

The three possible adaptations are presented only for the Grünwald-Letnikoff (GL) Variable-order

Construction (see Definitions 4-6), but for the remaining two fractional definitions the adaptation to

variable order can be found in Appendix A.2.

However, the work developed in Valério and Sá da Costa [72] was extended and comprised in

Sierociuk, Malesza, and Macias [65]. Here, three other definitions arose based on the duality principle

(see the following Equations 2.42) of the initial variable-order formulations (called the A − C types)

[66]. In total, this new nomenclature provided six equations raging from A to F type definitions, where

the first three are the same as in Valério and Sá da Costa [72].

Definition 4. GL: 1 or A-Type

Ac D

α(t)t f(t) = lim

h→0+

b t−ch c∑k=0

(−1)k(α(t)

k

)f(t− kh)

hα(t)(2.32)

Definition 5. GL: 2 or B-Type

BcD

α(t)t f(t) = lim

h→0+

b t−ch c∑k=0

(−1)k(α(t−kh)

k

)f(t− kh)

hα(t−kh)(2.33)

Definition 6. GL: 3 or C-Type

CcD

α(t)t f(t) = lim

h→0+

b t−ch c∑k=0

(−1)k(α(kh)k

)f(t− kh)

hα(kh)(2.34)

25

Ac D

α(t)tDc D−α(t)t f(t) = f(t) D

c Dα(t)tAc D−α(t)t f(t) = f(t)

BcD

α(t)tEcD−α(t)t f(t) = f(t) E

cDα(t)tBcD−α(t)t f(t) = f(t)

CcD

α(t)tFc D−α(t)t f(t) = f(t) F

c Dα(t)tCcD−α(t)t f(t) = f(t)

(2.42)

The following constructions can be obtained through difference equations applied to the fractional

constant order definitions of A to C types, as developed in Sierociuk, Malesza, and Macias [66]. For

the first case (Definition 4), the procedure is as follows.

∆nx(z) =

(1− z−1

h

)αX(z)⇔ ∆n

x(z)(1− z−1)−α = h−αX(z) (2.35)

Z−1

=⇒k∑j=0

(−1)j(−αj

)∆αxk−j = h−αxk (2.36)

Here, ∆nx(z) is the Z-transform of the signal difference of order α of variable x(t). This last equation

can be, in turn, rewritten in the form of Equation 2.37 and adapted to the variable order case as in

Equation 2.38.

∆αxk = h−αxk −k∑j=1

(−1)j(−αj

)∆αxk−j (2.37)

D∆αkxk =xkh−αk

−k∑j=1

(−1)j(−αkj

)D∆αk−jxk−j (2.38)

The continuous time domain of these equations provided with the D-type definition of variable

order derivatives as in Definition 7.

Definition 7. GL: D-Type

Dc D

α(t)t f(t) = lim

h→0

f(t)

hα(t)−

n∑j=1

(−1)j(−α(t)

j

)D0 D

α(t−jh)t−jh f(t− jh)

(2.39)

Definition 8. GL: E-Type

EcD

α(t)t f(t) = lim

h→0

f(t)

hα(t)−

n∑j=1

(−1)j(−α(t− jh)

j

)hα(t−jh)

hα(t)E0D

α(t)t−jhf(t)

(2.40)

Definition 9. GL: F-Type

Fc D

α(t)t f(t) = lim

h→0

f(b)

hα(b)−

n∑j=1

(−1)j(−α(jh)

j

)hα(jh)

hα(b)F0 D

α(t)b−jhf(t)

(2.41)

The referred duality principle, as the composition property of two types of derivatives for which the

variable-order of the opposite sign yields identity and that provided with D to F type definitions, is as

follows.

26

2.3.2 Matrix Constructions

The matrix construction for fractional constant-order derivatives, presented in Section 2.2.2, can

be extended to that of varying orders [66]. For the A type derivative, the matrix is as follows.

A0 D

α(t)0 f(t)

A0 D

α(t)h f(t)

A0 D

α(t)2h f(t)

...A0 D

α(t)kh f(t)

= limh→0

AW (α, k)

f(0)f(h)f(2h)

...f(kh)

(2.43)

AW (α, k) =

h−α(0) 0 0 · · · 0wα(h),1 h−α(h) 0 · · · 0wα(2h),2 wα(2h),1 h−α(2h) · · · 0wα(3h),3 wα(3h),2 wα(3h),1 · · · 0

......

... · · ·...

wα(kh),k wα(kh),k−1 wα(kh),k−2 · · · h−α(kh)

(2.44)

Considering their duality properties, the type D definition can also have a matrix form that is,

however, slightly more complex to attend than it’s dual. Based on the fractional difference definition,

type D construction is as follows.

D∆α0x0D∆α1x1D∆α2x2

...D∆αkxk

= Dk0

x0

x1

x2

...xk

where Dk0 = D(αk, k) · · ·D(α1, 1)D(α0, 0) (2.45)

D(α0, 0) =

[h−α0 01,k

0k,1 Ik,k

]and D(αr, r) =

Ir,r 0r,1 0r,k−rqr h−αr 01,k−r

0k−r,r 0k−r,1 Ik−r,k−r

∈ R(k+1)×(k+1)

(2.46)

For r = 1, · · · , k, where qr = (−v−αr,r, v−αr,r−1, · · · , v−αr,1) ∈ R1×r and v−αr,i = (−1)j(−αr

i

)for

i = 1, · · · , r. More detailed information can be found in Sierociuk, Malesza, and Macias [66] and, for

other constructions, in Malesza, Macias, and Sierociuk [35].

2.3.3 Relevant Properties

These properties are presented according to the work developed in Valério and Sá da Costa [72].

• Linear and Non-liner Operator Operator D can be seen as a functional that receives two

functions, f(t) and α(t), and returns one. Whatever the definition used, operator D a) remains

a linear operator regarding function f(t). However, b) operator D is non-linear relative to α(t).

For a constant function α(t) = p ∈ Z, where p = m+ n with m ∈ Z− ∧ n ∈ Z+, in can be verified

the following, for a general case.

27

cDα(t)t [κ1f1(t) + κ2f2(t)] = κ1cD

α(t)t f1(t) + κ2cD

α(t)t f2(t) (2.47a)

cDmt cD

nt f(t) 6= cD

m+nt f(t) (2.47b)

• Distributed Order Derivatives Functions α and f have been, so far, presented as dependent of

the same variable t. However, it is possible that the order itself (α) might depend on another vari-

able. It can be considered that: a) the order depends of the value of f(t) itself, then tDα(f(t))c f(t);

or b) an operator with an order that depends on its own output, g(t) = tDα(g(t))c f(t). Both cases

are sometimes known as distributed order derivatives.

• Memory of order α As said before, variable order derivatives have memory of past values of

f in similar way as the integral itself. However, when it comes to the order α(t) the derivative

may or may not have memory of its past values according to the variable order definition used.

Generally, if one follows Riemann-Liouville (c→ t or t→ c) or Caputo (c→ t only) constructions,

only the current value of α is used. For all other definitions, memory of past values of α exists.

2.3.4 Some Applications

Being a fairly recent concept, VODs play an important role in modeling complex phenomena that

so far have been beyond the reach of analysis and, thus, the existing applications are far from their

full wide impact capability. However, some examples have proven worthy of the inherent potential of

varying-order derivatives.

For instance, in Coimbra [18], an operator based on the Caputo variable order definition was

presented and applied to mechanical modeling of nonuniform viscoelastic frictional forces. Such

was done by providing the derivative’s operator with a very clear physical interpretation, used to

illustrate the dynamic behavior of variable-order frictional forces. By comparing the obtained results

with the fractional constant order counterpart, it was proven that the latter model is unable to capture

all the details of the variable order, particularly in the transition areas between dynamic modes.

Also, the dynamics of a modified Duffing equation was analyzed by including a variable order

derivative as the damping term of the system, through numerical simulation of nonlinear dynamic

oscillators systems and controllers [22].

More recently, the type D definition presented previously has been successfully used to model the

heat transfer process in a media where the structure is changing in time [60].

2.3.5 Variable Transfer Functions

A fractional time-varying transfer function is one much similar to the presented formula in Sec-

tion 2.2.6. It has, however, time-varying coefficients [72].

28

G(s, t) =

m∑k=1

bk(t)sβk(t)

n∑k=1

a(t)ksαk(t)

(2.48)

Fractional time-varying transfer functions can have orders varying continuously with time, since

αk(t) and βk(t) can only be continuous with t if they remain constant (if they change, they would have

to jump from an integer value to another one, in an integer-only case).

29

30

3Mathematical Models for Bone

Remodeling

Contents3.1 A Pharmacokinetics and Pharmacodynamics Introduction . . . . . . . . . . . . . 323.2 Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Non-local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

31

Any sufficiently advanced technology is indistinguishable from magic.

Sir Arthur Charles Clarke

Computational models for the bone remodeling dynamics, with metastases growth, are important

to understand the biochemical processes occurring within this specific micro-environment that could

promote the disease progression and, through such, propose targeted therapies with maximized effi-

ciency for the patient.

In light of such, this chapter is organized by primarily introducing pharmacokinetics and pharma-

codynamic theoretical concepts (Section 3.1). The following sections (Sections 3.2 and 3.2) present

the existing biochemical formulations in mathematical form for their local and non-local constructions.

In each the healthy, bone with metastasis and malignancy therapy cases are addressed.

In all that follows, D1 is a first order derivative in order to time, ddt and all models presented use

dimensionless variables and parameters, including the cell populations, except when said otherwise

in Tables 3.1 and 3.2 or in the List of Symbols.

3.1 A Pharmacokinetics and Pharmacodynamics Introduction

For the treatment of several tumors, chemotherapy has been shown to relieve symptoms and im-

prove survival in patients with various types of cancer. However, it lacks specificity along with narrow

therapeutic indexes and the occurrence of tumor resistance. This deficit can partly be attributed to

the traditional nature of cancer drugs, being highly toxic and used in maximum tolerated doses [77].

Progress in understanding the genetic basis of cancer coupled to molecular pharmacology of po-

tential new anticancer drugs calls for new approaches that are able to address key issues in the drug

development process, including Pharmacokinetics (PK) and Pharmacodynamics (PD) relationships

[77]. Here, PK of a drug describe its concentration evolution at the target tissue, whereas the effect

of such concentration is given by PD [17].

This allows for a translational research and individualized therapy to be placed [77], where a

methodology regarding prescribed medicine must be adequate to the patient’s clinical condition,

in light of the referred 4P approach to medicine: more predictive, personalized, preventive and

participatory [25]. Predictive PK/PD cancer treatment models are the utmost importance, when it

comes to understand disease mechanisms and to develop a personalized therapy schemes capable

of controlling cancer, without causing excessive damage to the body.

3.1.1 Pharmacokinetics

PK models are based on mass balance differential equations that characterize drug absorption

and disposition within the body. The most common PK modeling approach is known as a mammilliary

compartmental model, which conceptualizes the body as a series of compartments reversibly linked

to a central one [77], in which the prescribed drugs flow and were they are excreted from. Drug

32

transference between compartments is represented by zero or first order constant growth rates, withA

being the drug concentration and k∗ and k the zero and first order constant rate growth (Equation 3.1)

[21, 33].

Focusing solely in a one-compartment model for oral administration, the remaining drug concen-

tration to be absorbed (Cg(t)) and the effective drug concentration in the plasma (C

p(t)) are described

by the system of differential Equations 3.2 where κg and κp are the absorption and elimination rate,

respectively.

dA

dt= −k∗ or

dA

dt= −kA (3.1)

d

dtCg(t) = −κ

gCg(t)

d

dtCp(t) = κgCg (t)− κpCp(t) (3.2)

For a single dosage drug, of initial concentration Cg (0) = Cg0 , the plasma concentration can be

determined by Equation 3.3. For multiple doses, however, the plasma concentration of the nth dose

is given by Equation 3.4, assuming equal initial conditions for each dose administrated at equally

spaced intervals t′ = t− (n− 1)τ . Multiple dosage is govern by the steady-state Cpss

=Cg0τκp

.

Cp(s) = Cg0

κg(s+ κ

g)(s+ κ

p)

L−1

⇔ Cp(t) = Cg0

κgκg− κ

p

(e−κp t − eκg t) (3.3)

Cp(n, t′) = Cg0

κg

κg− κ

p

(1− e−nκpτ

1− e−κpτe−κp t

′− 1− e−nκg τ

1− e−κg τe−κg t

′)

(3.4)

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

Single Dosage Case − C(0)=0.008; κa=3; κ

d=0.5

Time t [days]

Rem

aini

ng D

rug

Con

cent

ratio

n to

be

Abs

orbe

d [n

g/m

m3 ] −

Cg(t

)

0 1 2 3 4 5 6 7 8 9 100

5

x 10−3

Pla

sma

Dru

g C

once

ntra

rion

[ng/

mm

3 ] − C

p(t)

Figure 3.1: PK model simulation results of Cg (t) andCp(t), for an oral administration of a single drug dose,

at t = 0 days, and with parameters Cg0 = 0.008, κg = 3and κp = 0.5.

0 5 10 15 20 250

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Time t [days]

Dru

g C

once

ntra

tion

[ng/

mm

3 ]

Multiple Dosage Concentrarion Response, for κa = 3 κ

d = 0.5 and for δ(t) = 0.008 at each τ = 1 day

δ(t)C

g(t)

Cp(t)

Cp

ss

Figure 3.2: PK model simulation results of Cg (t) andCp(t), for multiple drug doses applied at regular

intervals of τ = 1 day with intensity of δ(t) = 0.008.Steady state is given by Cpss = 0.16.

3.1.2 Pharmacodynamics

The interest and importance of pharmacodynamics PD to cancer related therapies has been rising

mainly due to technological advances that allow the measurement of multiple protein targets in cells

and tumors, and the explosion in translational and personalized medicine [77].

A PD mathematic model describes the effect of a drug as a function of its concentration. Various

models have been developed to quantify the time course of pharmacological effect of anticancer

drugs in relation to drug concentrations in plasma or target tissues, providing a theoretical framework

to understand experimental data and a quantitative basis to design and adjust dosing regimens [14].

In general, therapeutic responses to most anticancer drugs may be considered indirect in nature, and

thus, can be described by linking drug concentration to effect through an intermediate [77].

33

Mathematically speaking, a drug effect, here taken to be d(t), is described by a Hill1 function that

depends on the drug’s concentration according to the Equation 3.5 [48]. It varies between 0 and 1,

where Cp50(t) represents the concentration at 50% of its maximum effect, Cp50(t) = f(t)Cbasep50, and

the resistance to drug can be described by f(t) = 1 +Kr

∫ t0

max[0, Lr − Cp(λ)

]dλ [48].

d(t) =Cp(t)

Cp50(t) + Cp(t)(3.5)

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Single and Multiple Drug Effects for C50

=0.003

Time t [days]

Dru

g E

ffect

d(t

) [n

g/m

m3 ]

Multiple DoseSingle Dose

Figure 3.3: PD model results of d(t), for single and multiple doses with Cp50(t) = Cbasep50= 0.003. Single and

multiple cases correspond to those of Figures 3.1 and 3.2, respectively.

Different drugs can act in the same pathway and their combined action may differ from the isolated

administration of each drug. In order to quantify the synergistic or antagonistic effect of the combi-

nation of drugs, the so called Combination Index (CI) can be described in Equation 3.6, where C∗pi

is

the necessary concentration of drug i when combined with other drug to produce the same effect d

as concentration Cpi

of drug i when taken alone.

CI =(C∗

p)1

(Cp)1

+(C∗

p)2

(Cp)2

=(C∗

p)1

(C50

)1d/(1− d)+

(C∗p)2

(C50

)2d/(1− d)(3.6)

If CI = 1, the combined drugs have additive effect, which means that the effects of the drugs are

mutually exclusive but sharing similar modes of action. In case the drugs have different modes of

action, the effect of one drug is influenced by that of the others. When 0 < CI < 1, the combination

is said to exhibit synergistic effect. On the other hand, when CI > 1, the combination will give rise to

antagonism. Solving for d, the combined pharmacodynamics of drugs dc12 is given by Equation 3.7.

dc12=

[(Cp )1

(C50

)1+

(Cp )2

(C50

)2

]CI +

[(Cp )1

(C50 )1+

(Cp )2

(C50 )2

] (3.7)

Through the combination of pharmacokinetic and pharmacodynamic models one achieves a PK/PD

model for a drug, as presented in Figures 3.3, for one drug with single and multiple doses, and in Fig-

ures 3.4 and 3.5, for two identical administered drugs in multiple periodical doses.

1A Hill function, introduced by A.V. Hill in 1910, originally described the binding of oxygen to hemoglobin. Subsequently,they have been widely used in biochemistry, physiology, and pharmacology to analyze the binding equilibria in ligand-receptorinteractions [61].

34

0 5 10 15 20 25

7.94

7.96

7.98

8

8.02

x 10−3

Multiple Dosage Case, for Two Adminitred Druges with Cg

1

=Cg

2

and Cp

1

=Cp

2

Time t [days]

Rem

aini

ng D

rug

Con

cent

ratio

n to

be

Abs

orbe

d fo

r C

g 1, whe

re C

g 1=C

g 2 − [n

g/m

m3 ]

0 5 10 15 20 25

8x 10

−5

Pla

sma

Dru

g C

once

ntra

tion

Cp 1, w

here

Cp 1=

Cp 2 −

[ng/

mm

3 ]

Figure 3.4: PK simulation results of one of twoidentical administered drugs, for

Cp50(t) = Cbasep50= 0.002 at each τ = 1 day. κa1 = 0.01

and κd1 = 1.

0 5 10 15 20 250.7995

0.8

0.8005

0.801

0.8015

0.802

Combined Drug Effect for Two Identical Administred Drugs, dc12

(t), for CI = 1

Time t [days]

Com

bine

d D

rug

Effe

ct d

c12(t

)

Figure 3.5: PD simulation results of two identicaldrugs combined effect, dc12 , for a PK profile asdescribed in Figure 3.4and ruled Equation 3.7.

3.1.3 A PKPD Control Action

Since a drug pathway can have either an inhibitory or a stimulatory effect on a given metabolism,

such can be accounted for according to Equations 3.8, respectively, and provide a control action to

the presence of a tumor in the mathematical models for bone remodeling to follow (Sections 3.2 and

3.3). Constants Ki,Ks > 0 represent the maximum effect of a drug in a specific mechanism, where

d(t) is the pharmacodynamic response of a single or a combination of drugs.

Ie(t) = 1−Kid(t) Se(t) = 1 +Ksd(t) (3.8)

3.2 Local Models

In this section, existing local bone remodeling models are reviewed and simulated, considering

the work of an individual BMU.

3.2.1 Healthy Remodeling Dynamics

The simplest model for bone remodeling, proposed in Komarova et al. [31], involves a system of

three Ordinary Differential Equations (ODE) representing osteoclast and osteoblast densities, and a

normalized bone mass is introduced as a reflection of these cells activity.

In this early proposal, bone remodeling takes an S-system form [62] described by Equations 3.9.

Coupling of osteoclasts, C(t), and osteoblasts, B(t), behavior is done through biochemical autocrine

(gCC

, gBB

) and paracrine (gBC

, gCB

) factors expressed implicitly in the system’s exponents.

D1C(t) = αCC(t)gCCB(t)gBC − β

CC(t) (3.9a)

D1B(t) = αBC(t)gCBB(t)gBB − β

BB(t) (3.9b)

D1z(t) = −κC

max [0, C(t)− Css] + κB

max [0, B(t)−Bss] (3.9c)

For the osteoclasts population (Equation 3.9a) the autocrine parameter gCC

has a positive feed-

back on these cells production (gCC

> 0), whilst gBC

has the opposite effect (gBC

< 0). Both TGF-β

35

and the RANK/RANKL/OPG pathway effects are encoded in this paracrine regulation. In 3.9b, the os-

teoblast autocrine regulation, gBB

, is considered to have little effect on ostoblasts maturation. It has,

however, positive feedback on their production (gBB

> 0). The paracrine gCB

includes the positive

combined effects of such OPG and RANKL in osteoblast formation (gCB

> 0) [31, 1].

The production and death rate of the bone cells here considered are encompassed by αi and βi,

respectively, with i = C for osteoclasts and i = B for osteoblasts.

It is assumed that the bone mass, z(t), has a density determined by the extent to which normalized

values of C(t) and B(t) exceed nontrivial steady state levels, CSS

and BSS

, respectively. Below such

values, osteoclasts and osteoblasts populations are assumed to consist of less differentiated cells that

are unable to resorb or build bone, but are able to participate in autocrine and paracrine signaling.

The constants κC

and κB

represent the bone resorption and formation activities, respectively.

Simulations of the presented model can be found in Figure 3.6, for a periodical remodeling cycle.

As stated in [31], osteoclasts produce the coupling factors necessary for the bone remodeling dy-

namics, by attracting osteoblasts to the sites of bone resorption. Such behavior is achieved through

parameter gCC

, since it’s this autocrine action that triggers the different responses in the following

figures. Consequently, a single cycle response is possible to achieve by varying this parameter in the

osteoclasts/osteoblasts dynamic and adapting the remaining bone mass parameters [1] (gCC

= 0.5,

κC

= 0.24, κB

= 0.0017 C0 = 11.06, B0 = 213.13, z0 = 100, Css = 1.06 and Bss = 213.13).

0 100 200 300 400 500 600 700 800 900 10000

5

10

Osteoclast Population

t [days]

Ost

eocl

asts

0 100 200 300 400 500 600 700 800 900 10000

200

400

600

800Osteoblast Population

t [days]

Ost

eobl

asts

0 100 200 300 400 500 600 700 800 900 100090

95

100

105Bone Mass

t [days]

Bon

e M

ass

Figure 3.6: Simulation of oscillatory changes in the number of osteoclasts, osteoblasts and bone mass duringnormal bone remodeling for which the model solutions are periodic. Parameters were again set according to

Ayati et al. [1] and can be found in column 0D-H of Table 3.1.

In the previous figures, the resorption phase is triggered with the increase of osteoclasts numbers.

Such results in a decreasing bone mass while the number of the resorption cells is above their steady-

state value. Osteoblasts then lead bone formation, since their number rose as a response to the

increased osteoclastic activity. This phase ends when the number of osteoblasts goes below its

steady-state, meaning the remodeling cycle is over as the bone mass returns to its original value.

The system of Equations 3.9 has a unique nontrivial steady state. Such is proven by the Cauchy-

Lipschitz theorem: for a given initial condition, a system of differential equations has one and only one

36

solution, through which the system’s behavior can be completely determined [23]. The steady state

is given as follows (Equations 3.10a), as confirmed in Appendix A.3.

Css =

(βC

αC

) 1−gBBγ

(βB

αB

) gBCγ

(3.10a)

Bss =

(βC

αC

) gCBγ(βB

αB

) 1−gCCγ

(3.10b)

γ = gCBgBC− (1− g

CC)(1− g

BB)(3.10c)

J(Css, Bss) =

=

[βC

(gCC− 1) β

CgBC

CssBss

βBgCB

BssCss

βB

(gBB− 1)

](3.11)

The stability of this state can be analytically investigated through the system’s Jacobian, given by

Equation 3.11. For typical ODEs, it’s the real-part of this last matrix eigenvalues (eigenvalues of equi-

librium, λi) that is responsible for different stability-wise responses at the steady state (Css, Bss) [26].

The positioning of such values can be analyzed through the matrix’s trace, determinant and discrimi-

nant, tr(J(Css, Bss)), det(J(Css, Bss)) and ∆(J(Css, Bss)), respectively, as given by Equation 3.12.

tr(J(Css, Bss)) = βC

(gCC− 1) + β

B(gBB− 1) (3.12a)

det(J(Css, Bss)) = βCβB

((gCC− 1)(g

BB− 1)− g

BCgCB

) (3.12b)∆(J(Css, Bss)) = (β

C(gCC− 1) + β

B(gBB− 1))2 + 4β

CβBgBCgCB

(3.12c)

Since the non-linear system is composed by the two Equations 3.9a and 3.9b, with C(t) and B(t)

as variables, their Jacobian at the steady-state is a [2 × 2] matrix with two equilibrium eigenvalues

[23, 26]. Again, it is reminded that bone mass variations are but a reflection of osteoclasts and

osteoblasts activity. The effectors gij can be set as variables as to understand their implications in

the system’s dynamic, a study that results in the bifurcation diagram of Figure 3.7. In this image,

surfaces represent the regions where tr(J(Css, Bss)), det(J(Css, Bss)) and ∆(J(Css, Bss)) transition

from negative to positive values.

Figure 3.7: Bifurcation diagram, from Coelho, Vinga, and Valério [16], presenting the surfaces corresponding totr(J(Css, Bss)) = 0, det(J(Css, Bss)) = 0 and ∆(J(Css, Bss)) = 0 expressed in terms of the autocrine and

paracrine parameters gCC , gBB and gCBgBC .

37

When the Jacobian has negative trace and positive determinant (tr(J(C,B)) < 0, det(J(C,B)) >

0) the eigenvalues will be real and negative and, thus, the system will converge to its steady-state as

time goes to infinity. If combined with a positive discriminant (∆(J(C,B)) > 0), this steady-state will

be a stable node leading to an asymptotically stable solution. If the discriminant is negative, however,

the equilibrium point is a stable focus and the system will present damped oscillations around the

steady-state.

For a zero trace and a negative discriminant (tr(J(C,B)) = 0, ∆(J(C,B)) < 0), the system will

have periodic solutions for osteoclasts and osteoblasts, where the amplitude and frequency responses

depend only on the initial conditions given. Through simulation in [16], it was concluded that this

system’s steady-state is a center point and that the parameters in Table 3.1 induce a periodic behavior,

as represented in Figure 3.6 [16, 1].

In Appendix A.4, more information can be found over the different types of steady-states.

3.2.2 Adding the Tumor Burden

A novel model was presented in Ayati et al. [1] incorporating a the influence of a tumor in the

healthy bone remodeling micro-environment for a physiological behavior of MM disease. As a reflec-

tion of what was presented in Section 1.2.2.B, metastasis act through the autocrine and paracrine

regulation pathways and, hence, through the autocrine and paracrine parameters of the model.

A new variable T (t) represents the tumor cells density at time t, and its dynamic equation takes

a Gompertz form with a constant growth γT> 0. It’s considered to be independent of bone loss,

it translates a possible maximum tumor size of LT and has an initial condition of T (0) = T0. Such

evolution can be seen in Figure 3.8.

D1C(t) = αCC(t)

(gCC

(1+r

CCT (t)LT

))B(t)

(gBC

(1+r

BCT (t)LT

))− β

CC(t) (3.13a)

D1B(t) = αBC(t)

(gCB

1+rCB

T (t)LT

)B(t)

(gBB−r

BBT (t)LT

)− β

BB(t) (3.13b)

D1z(t) = −κC

max [0, C(t)− Css] + κB

max [0, B(t)−Bss] (3.13c)

D1T (t) = γTT (t) log

(LTT (t)

)(3.13d)

The tumor affects the autocrine and paracrine parameters through newly introduced rCC

, rCB

, rBC

and rBB

terms, all of them nonnegative.

As the tumor grows, the secretion of factors that promote osteoclast formation increases. This

leads to a reduction of the paracrine inhibition of osteoclasts in the form of(−g

BC> −g

BC

(1 + r

BC

T (t)LT

)), since g

BC< 0. This can be translated in the growing production

of PTHrP and RANKL by osteoblasts and the decrease of OPG, as the metastasis develop. RANKL-

independent mechanisms, that promote osteoclasts differentiation and activation in the presence of

metastatic cells, affect the osteoclast autocrine parameter, leading to its promotion:

gCC

(1 + r

CC

T (t)LT

)> g

CC.

38

The inhibitor effect of the tumor in the activation of osteoblasts is encoded in this cell’s autocrine

factor: gBB− r

BB

T (t)LT

< gBB

with gBB

> 0. The tumor’s action in the paracrine signaling from osteo-

clasts to osteoblasts is also accounted for: (gCB(

1 + rCB

T (t)LT

) < gCB

with gCB

> 0) [1, 16].

As a result, the tumor parameters yield perturbations in the healthy bone environment that result

in limit cycles. These can be either damped oscillations, that converge to the nontrivial steady state,

or undamped oscillations that diverge away from this state.

The equation for bone mass, z(t), is the same as in the normal case, but where the nontrivial

steady-states are now given by Equations 3.14 and 3.15:

Css = 0;Bss = 0;Tss = LT (3.14)

Css =

(βC

αC

)( 1−gBB−rBB

Λ

)(βB

αB

)(gBCΛ (1+r

BC))

(3.15a)

Bss =

(βB

αB

)( gCB

Λ

1+rCB

)(βB

αB

)( 1−gCC

(1+rCC

)

Λ

)(3.15b)

Tss = LT (3.15c)

Λ =

(gCB

1 + rCB

)(gBC

(1 + rBC

))− (1− gCC

(1 + rCC

))(1− gBB

+ rBB

) (3.15d)

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

100

110

t [days]

Tum

or

Figure 3.8: Tumor evolution, based on the Gompertz form of the tumor Equation 3.13d and according to whatwas described in Figure 1.6. Parameters are expressed in Table 3.1.

As the bone resorption becomes up-regulated, bone formation decouples from it in the sense that

the number of osteoblasts activated is decreasing while the number of osteoclasts increases. This

leads to a decreasing bone mass as the tumor grows. The change in dynamics of the system from

undamped to damped oscillations can be predicted by analyzing the stable cycle limit, φ, which is

given by the trace of the Jacobian of this new system at the new steady state (3.15), combined with

the determinant and discriminant of the same matrix.

φ = tr(J) = βC

(gCC

(1 + rCC

)− 1) + βB

(gBB− r

BB− 1) (3.16a)

det(J) = βCβB

(gCC

(1 + rCC

)− 1)(gBB− r

BB− 1)− (g

BC(1− r

BC))

(gCB

1 + rCB

)(3.16b)

∆(J) = βC

(gCC

(1 + rCC

)− 1) + βB

(gBB− r

BB− 1)2 + 4β

CβB

(gBC

(1− rBC

))

(gCB

1 + rCB

)(3.16c)

39

If φ < 0, then the system exhibits damped oscillations converging to the nontrivial steady state;

and if φ > 0, then the system exhibits unstable oscillations converging away from the nontrivial steady

state (where φ is sufficiently close to 0). The first case (φ < 0) simulation results can be seen in

Figure 3.9, whilst the undamped case is not presented since it caress physiological meaning. The

parameters used are specified in column 0D-T of Table 3.1.

0 200 400 600 800 1000 12000

5

10

15

20

t [days]

Ost

eocl

ast P

opul

atio

n

0 200 400 600 800 1000 12000

200

400

600

800

t [days]

Ost

eobl

ast P

opul

atio

n

0 200 400 600 800 1000 12000

20

40

60

80

100

120

t [days]

Bon

e M

ass

Figure 3.9: Osteoclast, osteoblast and bone mass response, in the presence of tumor, for an initial stimulationof the osteoclast population elevated above the nontrivial steady state by 10 units and with damped oscillatoryresponse. The model parameters were set according to Ayati et al. [1], with values specified in column 0D-T of

Table 3.1.

For φ < 0, the parameters expressed in Table 3.1 provide a positive determinant, while the dis-

criminant remains negative as the tumor grows (see equations 3.16). The trace decreases to negative

values, which translates in a stable focus for the new steady-state achieved [16]. Such characteristics

correspond to a damped oscillatory behavior of the system, as visible in the previous figure.

Interpreting this simulation, it can be seen an oscillatory damped response in the osteoclast and

osteoblast population, to the new nontrivial steady state Css, Bss, where an initial increase in these

populations reflect a system attempt to maintain the normal coupling of bone resorption and formation,

even in the presence of a tumor [1]. While the tumor grows to its maximum capacity, the bone mass

converges with periodic oscillations to 0.

As for the case where φ > 0, the only change necessary compared to the damped oscillations

parameters is the coefficient rCC

(taken to be 0.02, instead of 0.005, for an induced undamped be-

havior). As a response, the osteoclast and osteoblast populations tend to exhibit unstable oscillations,

whose amplitude grows with time, with a collateral decrease in bone mass and increased growth of

the tumor to its limiting size [1].

3.2.3 Bone Metastases Therapy

This sub-section is structured in two different approaches regarding treatment possibilities: ini-

tially, as done in Ayati et al. [1], chemotherapy and proteasome inhibitors account for the existing

treatment for MM, simulated with two basic step functions (Sub-Section 3.2.3.A); for an osteolytic

40

metastatic bone disease, chemotherapy is paired with anti-resorptive treatments through the adminis-

tration of bisphosphonates and denosumab, through their pharmacokinetics and pharmacodynamics

(Sub-Section 3.2.3.B).

3.2.3.A Osteoblastic Promotion & Anti-Cancer Therapy

The initial model to include a treatment possibility accounts for the effects of proteasome inhibition

and chemotherapy in MM bone disease [1]. The first treatment was chosen, as previously explained,

because proteasome inhibitors are known to have direct anti-myeloma effects and to have direct

effects on osteoblasts stimulation. Hence, two time-dependent treatment functions, V1(t) and V2(t),

were introduced in Equation 3.13.

V1(t) =

0.0, for t < tstart,

υ1, for t ≥ tstartV2(t) =

0.0, for t < tstart,

υ2, for t ≥ tstart(3.17)

The starting time of the treatment is given by tstart and υ1 and υ2 are the intensity parameters

of treatment. Therefore, the model equations are as before except that the treatment promotes os-

teoblast production and inhibits tumor growth.

D1C(t) = αCC(t)

(gCC

(1+r

CCT (t)LT

))B(t)

(gBC

(1+r

BCT (t)LT

))− β

CC(t) (3.18a)

D1B(t) = αBC(t)

(gCB

1+rCB

T (t)LT

)B(t)

(gBB−r

BBT (t)LT

)− (β

B− V1(t))B(t) (3.18b)

D1T (t) = (γT− V2(t))T (t) log

(LTT (t)

)(3.18c)

Since the novelty made to the model with tumor was just the introduction of functions V1(t) and

V2(t) on the already existing equations, the steady-state values for bone dynamics and tumor param-

eters remain the same as before (column 0D-T of Table 3.1). Only the initial conditions now differ,

taken to be C0 = 13 and B0 = 300 in the following simulation.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

2

4

6

8

10

12

14

16

18

t [days]

Ost

eocl

ast P

opul

atio

n

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

200

300

400

500

600

700

800

t [days]

Ost

eobl

ast P

opul

atio

n

Figure 3.10: Osteoclast and Osteoblast population evolution, for the zero-dimensional bone model with tumorand treatment. The tumor was triggered by an initial condition applied at t = 0, T0, and the treatment began at

time tstart = 600 days. The treatment intensities were set at υ1 = 0.001 and υ2 = 0.008, with steady-states nowbeing C0 = 13 and B0 = 300. Parameters were again set according to Ayati et al. [1], being the same as the

bone model with tumor accounted in column 0D-T of Table 3.1.

41

0 200 400 600 800 1000 1200 1400 1600 1800 200070

75

80

85

90

95

100

105

110

115

t [days]

Bon

e M

ass

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

t [days]

Tum

or

Figure 3.11: Tumor and Bone Mass evolution, for the zero-dimensional bone model with tumor and treatment,as presented in Figure 3.10. Parameters were again set according to Ayati et al. [1] ans presented in column

0D-T of Table 3.1.

The previous images correspond to the applied treatment of Equations 3.18 into the untreated

tumor simulations presented in previous section 3.2.2. After the treatment starts, at 600 days, the

tumor is gradually extinguished and the osteoclast and osteoblast populations recover regular cycles.

Also, the bone mass tends gradually to its normal and initial value.

3.2.3.B PKPD of Anti-Resorptive & Chemotherapy Treatment

Besides the direct anti-cancer agents of the previous model, given by Equations 3.18, the pro-

posed treatment considers specific osteoblast stimulation for bone metastasis recovery, a strategy

based in proteosome inhibitors for MM bone disease.

Considering an osteolytic metastatic bone disease, a new approach was proposed in Coelho,

Vinga, and Valério [16] to account treatment in the form of anti-resorptive therapy, as explained in Sec-

tion 1.2.2.C. The effects of denosumab and bisphosphonates are introduced in the models through

d1(t) and d2(t) variables, respectively, and chemotherapy remains as a anti-cancer tactic through the

combination of two distinct drugs, dc34(t). Equations 3.13 are then modified with a stimulatory or

inhibitory action of these drugs, in a PK/PD stance, as can be seen in Equations 3.19.

D1C(t) = αCC(t)

gCC

(1+r

CCT (t)LT

)B(t)

gBC

(1+r

BCT (t)LT

)(1+Ks1d1(t))

− (1 +Ks2d2(t))β

CC(t) (3.19a)

D1B(t) = αBC(t)

(gCB

1+rCB

T (t)LT

)B(t)

(gBB−r

BBT (t)LT

)− β

BB(t) (3.19b)

D1T (t) = (1−Ki34dc34(t)) γTT (t) log

(LTT (t)

)(3.19c)

As explained previously, in Section 1.2.2.C, denosumab acts as a decoy for RANKL, inhibiting the

RANK binding in the osteoclasts precursors, hence limiting their maturation. Taking into account that

gBC

paracrine parameter models the OPG/RANKL ratio and that gBC

< 0, the effect of this drug is

included increasing the absolute value the negative exponent. Such has an inhibitory action in the

activation of osteoclasts. Bisphosphonates, on the other hand, increase the apoptosis of osteoclasts

when absorbed by these cells, which can be included by introducing a promoting effect on the death

42

rate of osteoclasts, βC

. Chemotherapy destroys tumor cells directly, through the encoded inhibitory

term in the growth rate of the tumor.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30Bone Micro−environment with Tumor and PKPD Treatment

Ost

eocl

asts

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

500

1000

Ost

eobl

asts

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

Bon

e M

ass

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

t [days]

Tum

or

Figure 3.12: Osteoclasts, osteoblasts, bone mass and tumor local simulations according to Equations 3.19, fora bone remodeling environment with multiple stages: in the absence of a tumor (0 ≤ 1000 days); is the

presence of untreated tumor (1000 ≥ t < 1730 days); with treatment (1730 ≤ t < 3700 days) and after treatmentis interrupted (t ≥ 3700). Used parameters for PKPD terms can be found in Table 3.2 and the remaining

parameters were set according to column 0D-T of Table 3.1.

In Figure 3.12, simulation of the new model of Equations 3.19 can be found, for a sequence os

stages that replicates the bone micro-environment raging from a healthy dynamics to a tumor-treated

case. At the beginning, for t ≤ 1000 days, bone remodeling occurs in its healthy state as done and

seen in Equations 3.9 and Figure 3.6, respectively. Tumor starts evolving, at t > 1000 days, disrupting

the normal remodeling behavior where a PK/PD based treatment is introduced, for t ≥ 1730 days. As

the tumor is extinguished, treatment ceases at t = 3700 days and the bone remodeling returns to its

natural cycle. However, the lost bone mass is never recovered, an expected result of bone metastasis

and due to the inexistent targeted therapy for bone formation.

3.3 Non-local Models

The previous section addressed local models for bone remodeling represented by ODEs. Under

this framework, only the dynamic behavior of an individual BMU is taken into account. The present

section extends these models to one-dimensional geometries, thus modeling diffusion processes in

the bone through Partial Differential Equations (PDE), for a distance x ∈ Ω with a diffusion term of∂2

∂x2 . Tridimensional geometries can be achieved by simply adding the diffusion dependence ∂2

∂y2 and∂2

∂z2 onto the formulated models presented next.

43

3.3.1 Healthy Remodeling Dynamics

In Ayati et al. [1], the original healthy model for bone remodeling was extended to a new spatial

domain Ω = [0, 1], with x ∈ Ω, represented in Equations 3.20. Hence, diffusion over one dimension is

allowed by introducing σi ∂2

∂x2 into the osteoclasts, osteoblasts and bone mass equations, whose den-

sity is now C(t, x), B(t, x) and z(t, x). However, σz should not be interpreted as bone cells migration,

but as a term encoding stochasticity present in natural bone dynamics. Coefficients κC

(x) and κB

(x)

are depend on C0(x) and B0(x).

D1C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)

gCCB(t, x)

gBC − β

CC(t, x) (3.20a)

D1B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)

gCBB(t, x)

gBB − β

BB(t, x) (3.20b)

D1z(t, x) = σz∂2

∂x2z(t, x)− κ

C(x) max[0, C(t, x)− Css] + κ

B(x) max[0, B(t, x)−Bss] (3.20c)

The nontrivial steady states of the local model for a healthy bone, Css and Bss, are also steady

states of this one-dimensional construction. They can be seen as constant functions given by Css ≡

Css and Bss ≡ Bss, since both diffusion terms are zero for this type of state. The new Newmann’s

boundary and initial conditions are, respectively, given by Equations 3.21 and 3.22 as follows.

∂

∂xC(t, 0) = 0;

∂

∂xC(t, 1) = 0 (3.21a)

∂

∂xB(t, 0) = 0;

∂

∂xB(t, 1) = 0 (3.21b)

∂

∂xz(t, 0) = 0;

∂

∂xz(t, 1) = 0 (3.21c)

C0(x) = C0(x);B0(x) = B0(x); z0(x) = z0(x) (3.22a)

Simulation of the non-local healthy bone model, based on the parameter study made in Ayati et al.

[1], took into account an initial distribution of osteoclasts C(0, x) elevated above Css = 1.16 in several

sites within Ω, as it can be seen in 3.13. The solutions obtained sustain spatial and temporal cycles

about the steady states presented previously.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

x

C(0

,x)

0200

400600

8001000

12001400

16001800

2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

20

40

60

80

100

t [days]

x [distance ∈ [0,1]]

Tum

or E

volu

tion

Figure 3.13: On the left side, the shape of the initial distribution of C(0, x) = C0(x) for the bone model with anadditional space dimension, and on the left side tumor temporal and spatial evolution due to an initial condition

applied at t = 0.

44

Figure 3.14: Normal bone model simulation with an initial distribution of osteoclasts elevated above its steadystate by C(0, x) as in Figure 3.13, where a) represents the osteoclast evolution, b) the osteoblast evolution andc) the bone mass response. Parameters were again set according to Ayati et al. [1] and can be found in column

1D-H of Table 3.1.

The initial condition C(0, x) triggers the remodeling event, throughout Ω, by disrupting the steady-

state osteoclast population, soon followed by a response of the osteoblasts in Ω and t. As a conse-

quence, and according with Equation 3.20c, the bone mass suffers a heterogeneous remodeling that

tends to normalize in z0(x) = 100.

3.3.2 Adding the Tumor Burden

In the same way the previous model was created from the one initially introduced in Komarova et

al. [31] (Section 3.3.1), the same can be done accounting for the tumor burden in the bone remodeling

dynamics in the simple non-local model. For the spatial model with tumor it is assumed the tumor cells

are diffusing in Ω, with variables C(t, x), B(t, x) as before, and T (t, x) being the density of tumor cells

at time t with respect to x ∈ Ω, and initial condition of T (0, x) = T0(x). The diffusion coefficient

for the tumor is given by σT , which allows for its spatial growth [1]. The adapted model is given by

Equations 3.23.

D1C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)

(gCC

(1+rCC

T (t)LT

))B(t, x)

(gBC

(1+rBC

T (t)LT

))

−βCC(t, x) (3.23a)

D1B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)

(gCB

1+rCB

T (t)LT

)B(t, x)

(gBB−r

BBT (t)LT

)−β

BB(t, x) (3.23b)

D1T (t, x) = σT

∂2

∂x2T (t, x) + γ

TT (t, x) log

(LT

T (t, x)

)(3.23c)

This system is subject to the following boundary and initial conditions, of Equations 3.24,in addition

to those presented in Equations 3.21 and 3.22.

45

∂

∂xT (t, 0) =

∂

∂xT (t, 1) = 0 (3.24a)

T (0, x) = T0(x) (3.24b)

Regarding the bone mass equation, z(t, x), the expression is the same as in Equation 3.20c

subject to the same initial and boundary conditions expressions, although the nontrivial steady state

is now given by Equations 3.15 – the same state of the local model including tumor burden.

As for the initial conditions setup, C(0, x) = C0(x) had the same spatial distribution as applied

in the one-dimensional model without tumor. As for the tumor itself, however, its initially distribution

T (0, x) = T0(x) was small and located on the right side of Ω = [0, 1], but its density T (t, x) converges

to the maximum capacity T ss = LT

for all x ∈ Ω as time increases [1]. Both initial distributions are

represented in Figure 3.13.

0200

400600

8001000

12001400

16001800

2000

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

6

7

t [days]x [distance ∈ [0,1]]

Ost

eocl

ast P

opul

atio

n

0200

400600

8001000

12001400

16001800

20000

0.10.2

0.30.4

0.50.6

0.70.8

0.91

100

150

200

250

300

350

400

450

t [days]x [distance ∈ [0,1]]

Ost

eobl

ast P

opul

atio

n

0200

400600

8001000

12001400

16001800

2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

50

100

150

t [days]

x [distance ∈ [0,1]]

Bon

e M

ass

Evo

lutio

n

Figure 3.15: Bone model simulation with an initial distribution of osteoclasts and tumor by C(0, x) and T (0, x)respectively, as in 3.13, where a) represents the osteoclast evolution, b) the osteoblast evolution, and c) thebone mass response. Parameters were set according to the local model presented for damped remodeling

cycles as in Ayati et al. [1], adapted to the non-local case as in column 1H-T of Table 3.1.

The tumor acts in several areas of the spatial bone environment Ω, growing over time. Temporal

and spatial cycles of bone resorption and formation are consequently are disrupted, with the ostoclast

and osteoblast populations ultimately approaching the local tumor model nontrivial steady state Css =

5 and Bss = 316. It can be seen, from the tumor and osteoclast initial conditions, that a delayed

response from the osteoblast population occurs, which is according to the healthy bone remodeling

cycle while bone dynamics attempts to maintain a normal coupling. The bone mass sees an increased

loss in areas where the tumor has a more intense effect.

3.3.3 Bone Metastases Therapy

Similarly to what was done in Section 3.2.3, this section provides non-local treatment to the models

with tumor of Equations 3.23. Again, chemotherapy is initially applied with proteasome inhibitors

46

for MM (Section 3.3.3.A), followed by the PK/PD model of denosumab and bisphosphonates for an

osteolytic metastatic bone disease (Section 3.3.3.B). Treatment diffusion is not considered here, so it

is assumed an uniformly and constant distribution throughout the bone.

3.3.3.A Osteoblastic Promotion & Anti-Cancer Therapy

Applying therapy is done through two time dependent functions V1(t) and V2(t), as in Section 3.2.3,

resulting in the model of Equations 3.25.

D1C(t, x) = σC

∂2

∂x2C(t, x)

+αCC(t, x)

(gCC

(1+r

CCT (t)LT

))B(t, x)

(gBC

(1+r

BCT (t)LT

))− β

CC(t, x) (3.25a)

D1B(t, x) = σB

∂2

∂x2B(t, x)

+αBC(t, x)

(gCB

1+rCB

T (t)LT

)B(t, x)

(gBB−r

BBT (t)LT

)− (β

B− V1(t))B(t, x) (3.25b)

D1T (t) = σT

∂2

∂x2T (t, x) + (γ

T− V2(t))T (t, x) log

(LT

T (t, x)

)(3.25c)

This system is subject to the same initial and boundary conditions as the non-local tumor model

(Equations 3.24), with the same nontrivial steady-state. The bone mass expression, z(t, x), is, again,

the same as in Equation 3.20c.

The introduced treatment, applied at t = 1500 days for the following simulation, corresponds to

drugs such as proteasome inhibitors, which promote osteoblast production and inhibit tumor growth.

This effect can clearly be seen in the graphic c) of Figure 3.16, where the tumor is extinguished and the

osteoblast population returns to its normal case steady-state value. Both osteoclast and osteoblast

recover normal cycle coupling, and bone mass partially recovers by the end of the simulation: despite

the tumor disappearance, areas of the bone where it’s effect was stronger were unable to fully recover

its original density.

3.3.3.B PKPD of Anti-Resorptive & Chemotherapy Treatment

The treatment model, as proposed in Equations 3.19, is here extended to a non-local approach

resulting in Equations 3.26. The therapy has now anti-resorptive characteristics, where d1(t) and d2(t)

variables account for the PK/PD action of denosumab and bisphosphonates, respectively, and dc34(t)

represents the combination of two chemotherapy drugs effects.

D1C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)

gCC

(1+r

CCT (t)LT

)B(t, x)

gBC

(1+r

BCT (t)LT

)(1+Ks1d1(t))

− (1 +Ks2d2(t))β

CC(t, x) (3.26a)

D1B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)

(gCB

1+rCB

T (t)LT

)B(t, x)

(gBB−r

BBT (t)LT

)−β

BB(t, x) (3.26b)

D1T (t) = σT

∂2

∂x2T (t, x) + (1−Ki34dc34(t)) γ

TT (t, x) log

(LT

T (t, x)

)(3.26c)

47

Figure 3.16: Bone model simulation with an initial distribution of osteoclasts and tumor by C(0, x) = T (0, x), asin 3.13, where a) represents the osteoclast evolution, b) the osteoblast evolution, c) the bone mass responseand d) the tumor distribution. Treatment was introduced at t = 1500 days, with parameters υ1 = 0.0001 and

υ2 = 0.006. Parameters were again set according to Ayati et al. [1] and presented in column 1H-T of Table 3.1.

In Figure 3.17 simulation results can be found, for a tumor starting time of zero days. Until treat-

ment is applied, at t = 730 days, the same bone micro-environment response is seen when compared

to Figures 3.15 and 3.16. Osteoclasts see an increase in their number as the bone mass is diminished

by the action of these cells. However, after treatment begins, a severe reduction occurs in the cells

populations as in the tumor density. After the tumor is extinguished and treatment stops, osteoclasts

and osteoblasts return to their normal tightly coupled cycle. Bone mass resorption stops as its density

stabilizes, never returning to the full healthy value of 100% and being more severe in the areas where

the action of the tumor was more intense, since bone formation is not promoted when compared to

the resorption in the new healthy state.

48

Figure 3.17: Bone model simulation with an initial distribution of osteoclasts and tumor by C(0, x) = T (0, x), asin 3.13, where a) represents the osteoclast evolution, b) the osteoblast evolution, c) the tumor distribution, andd) the bone mass response. Tumor began at t = 0 days and treatment was introduced at t = 730 days. Modelparameters can be found in column 1D-T of Table 3.1, as presented in Ayati et al. [1], whereas PK/PD values

are in Table 3.2.

Table 3.1: Variables and parameters used for simulation of local and non local models, taken from Ayati et al.[1] and applied in Equations 3.9-3.23. For the local case, as seen in Figures 3.6 and 3.9, the healthy bone

remodeling parameters can be found in the following column 0D-H and its version with tumor took the valuesexpressed in 0D-T. For the non-local approach, as simulated in Figures 3.14 and 3.15, the healthy and tumor

disrupted bone evolutions parameters can be found in columns 1D-H and 1D-T, respectively.

Variables Description Unitst Time daysx Distance x ∈ [0, 1]C(t, x) Number of osteoclasts -B(t, x) Number of osteoblasts -z(t, x) Bone mass density %T (t, x) Bone metastases density %Parameters Description 0D-H 0D-T 1D-H 1D-T UnitsαC

OC activation rate 3 3 3 3 day−1

αB

OB activation rate 4 4 4 4 day−1

βC

OC apoptosis rate 0.2 0.2 0.2 0.2 day−1

βB

OB apoptosis rate 0.02 0.02 0.02 0.02 day−1

gCC

OC autocrine regulator 1.1 1.1 1.1 1.1 —gBC

OC paracrine regulator −0.5 −0.5 −0.5 −0.5 —gCB

OB paracrine regulator 1.0 1.0 1.0 1.0 —gBB

OB autocrine regulator 0 0 0 0 —rCC

OC tumor autocrine regulation — 0.005 — 0.005 —rBC

OC tumor paracrine regulation — 0 — 0 —rCB

OB tumor paracrine regulation — 0 — 0 —rBB

OB tumor autocrine regulation — 0.2 — 0.2 —σC

Diffusion coefficient for OC — — 10−6 10−6 day−1

σB

Diffusion coefficient for OB — — 10−6 10−6 day−1

σz Diffusion coefficient for bone mass — — 10−6 10−6 day−1

σT

Diffusion coefficient for metastases — — —- 10−6 day−1

κC

Bone resorption rate 0.0748 0.0748 0.45 0.45 day−1

κB

Bone formation rate 6.39× 10−4 6.39× 10−4 0.0048 0.0048 day−1

LT

Maximum size of bone metastases — 100 — 100 %γT

Metastases growth rate — 0.005 — 0.004 % day−1

C(0, x) Initial number of osteoclasts 11.16 15 Figure 3.13 Figure 3.13 -B(0, x) Initial number of osteoblasts 231.72 316 231.72 316 -z(0, x) Initial bone mass percentage 100 100 100 100 %T (0, x) Initial tumor mass percentage — 1 — Figure 3.13 %Css Steady-state OC number 1.16 5 1.16 5 -Bss Steady-state OB number 231.72 316 231.72 316 -

49

Table 3.2: Variables and parameters used for simulation of the local and non local models with PK/PDtreatment, for Equations 3.19 and 3.26c. The combined effect for both chemotherapy drugs is given by dc34(t).

Simulations can be found, for the local case, in Figure 3.12 and the non-local case in Figure 3.17.

Variables Denosumab Bisphosphonates Chemotherapy UnitsIdentifier d1(t) d2(t) d3(t) d4(t) —

τ 1 1 1 1 daysκg

0.01 0.01 0.01 0.01 days−1

κp

1 0.5 1 1 days−1

Cp50(t) 0.008 0.008 0.008 0.008 g/L

Cbasep500.2 0.2 5.9× 10−5 5.9× 10−5 g/L

Kr 0 0 0 0 —Lr 0 0 0 0 —CI − − 1 —Ks,i 2 2 2 —

50

4Creating Variable Order Models:

Methodology, Results andDiscussion

Contents4.1 From Multiple Myeloma to Metastatic Bone Disease - A New Starting Point . . . 524.2 Creating a Variable Order Model for Bone Remodeling . . . . . . . . . . . . . . . 534.3 Simplified Bone Models With an Acting Tumor . . . . . . . . . . . . . . . . . . . . 564.4 Changing the Order, Extinguishing the Tumor — A Treatment Approach . . . . . 60

51

Gentlemen, you are looking through a window into another world.

Walter Bishop in Fringe

Original work is presented in this Chapter. For the models of bone micro-environment with tumor,

the physiological starting-point transition from MM to a metastatic bone disease is explained in Sec-

tion 4.1. The following section, Section 4.2, addresses the prior considerations taken into account

when building the new variable order bone micro-environment formulations. Section 4.3 introduces

the new simplified bone model with tumor, where applied variable order derivatives allowed for a re-

duction in the number of parameters. The last Section 4.4 extends the newly simplified models to

include therapy approaches, based on chemotherapy combined with both proteasome inhibitores [1]

and PK/PD of bisphosphonates and denosumab [16].

4.1 From Multiple Myeloma to Metastatic Bone Disease - A NewStarting Point

In light of what is known nowadays, regarding the biochemical evolution of a tumor in healthy

bone micro-environment, osteolytic metastases are rarely without some degree of bone formation.

Typically, reactive new bone is being formed in the periphery of tumor induced lesions, which might

represent an initial attempt to confine the tumor [5]. Such is accounted for in the work developed by

Ayati et al. [1], and can be seen in Figures 3.9 and 3.15 for the local and non-local cases, respectively.

The referred work, however, was based on the known fact that the most typical bone malignancy with

the least new bone formation associated is MM. For such disease, there is little if any osteoblastic

reaction [68].

Here, a new guideline is proposed that changes the starting point from MM to a solely osteolytic

bone disease due to a primary tumor elsewhere (usually breast cancer) [69]. Such was mainly consid-

ered to account for the role that osteoblasts effectively play in a tumor disrupted bone mass evolution,

since these cells, in the existing mathematical models as in real life, still participate in the remod-

eling process (either the new models proposed in the following sections or the local and non-local

constructions presented in Equations 3.13 and 3.23).

Since it is not known, to this day, if the tumor-induced changes in osteoblasts act through effectively

reducing their number or by diminishing the existing cells bone formation capability [24], a mixed-

behavior for osteoblasts was considered in the new models to follow. This population responds to

variations in existing osteoclats numbers, as these cells couple bone resorption and formation [5],

due to the non-linear differential equations dependence on both osteoclasts and osteoblasts. They

also see their activity reduced as an increase of tumor action inhibits osteoblast activity, achieved by

adequately determining the bone resorption-formation ratio R based on a single osteoclasts cycle, as

done in Ayati et al. [1].

52

4.2 Creating a Variable Order Model for Bone Remodeling

Major prior considerations that had to be accounted for the new bone remodeling models are here

detailed. Section 4.2.1 not only presents the variable order formulation, chosen within the options

from Section 2.3.1, as the adapted mathematical construction of α(t) influenced by the tumor burden.

Numerical implementation considerations are also provided. Limits to the tumor dynamics, when

different treatments are applied, are detailed in Section 4.2.2. Used resorption ratios R, for the bone

mass behavior, are explained in Section 4.2.3. Finally, Section 4.2.4 provides the non-local Simulink

model considerations for a numerical and spatial implementation.

4.2.1 Choosing and Shaping a Variable Order Formulation

Remembering the fractional variable order definitions presented in Sub-Section 2.3.1, type-D

Grünwald-Letnikov definition was used throughout this work. This variable order construction cor-

responds to an input-reductive switching scheme, as presented in Sierociuk, Malesza, and Macias

[66], that assumes the rejection of the chain of differentiators from the input. This allows the effect of

order switching to start immediately after the switch and without output switches. Such can be verified

in the right side of Figure 4.1.

The recursive type-D construction, presented in Equations 2.37-2.38, allows also to define a length

of constant initial conditions, T , to be infinite. It can be reshaped as follows, with D−∞Dαl f(t) = c =

const, for l = (−Th, 0) with T = −∞, since only the results for an infinite length of initial conditions are

expected to be a constant function (left side of Figure 4.1). This will be necessary for the simulation

process of the bone remodeling models to be presented, as constant initial conditions are applied.

D−∞D

α(t)t f(t) ≈

f(t)

hα(t)−

n∑j=1

(−1)j(−α(t)

j

)(D−∞D

α(t−jh)t−jh f(t)− c

)+ c

(4.1)

Figure 4.1: On the left, results of a constant order α = 0.5 integrators output with initial conditions c = 0.5 anddifferent values of T , the length of initial conditions function (from [43]). On the right, comparison between

different variable order constructions for a linear-time function derivatives given by α3(t), where 1st, 2nd, 3rd and4th indications correspond to type A, B, C and D construction results, respectively (from [66]).

Numerical implementation of the recursive type-D construction followed the matrix approach pre-

sented in Sub-Section 2.3.2, available through the variable order derivatives Simulink -toolbox pre-

sented in Sierociuk, Malesza, and Macias [65].

53

When it comes to shaping the order itself, to better reflect the tumor behavior in the healthy bone

micro-environment models, some considerations have to be taken into account. Since the tumor

equations presented, either in their local or non-local versions (Equations 3.13d and 3.23c, respec-

tively), are said to be independent of the bone micro-environment [1], and that bone mass variations

are but a reflection of osteoclasts and osteoblasts activity [31], the new model is formulated by chang-

ing the derivative’s order of the osteoclasts and osteoblasts equations only. The variable order, α(t)

or α(t, x), influenced by the tumor dynamics, is now responsible for inducing in the original healthy

model (Equations 3.9 and 3.20) the same response as the tumor disrupted bone one (Equations 3.13

and 3.23). The new equation associated with the required order format is presented in Equations 4.2,

for local and non-local scenarios, where θ is a constant term experimentally determined. Time t is

related to the beginning of the tumor growth.

α(t) = 1− ρ(t)T (t) = 1− θ × t× T (t) (4.2a)α(t, x) = 1− ρ(t)T (t, x) = 1− θ × t× T (t, x) (4.2b)

It is noted that, being the osteoclats and osteoblasts formulated in differential equations, the order

will be subjected to an integration process represented by the 1 − ... in the following formulation and

translated in the integrator D−1, as explained in Equation 4.3 for a generic function f(t) when a

derivative’s order α(t) is applied. Simulink implementation can be seen Figure 4.2.

D1−α(t)f(t) = γ [f(t)]⇔ D1D−α(t)f(t) = γ [f(t)]⇔⇔ D1f(t) = Dα(t) [γ [f(t)]]⇔ f(t) = D−1 [Dα(t) [γ [f(t)]]] (4.3)

The last step in Equation 4.3 is given by f(t) = D−1 [Dα(t) [...]]. According to the law of exponents

of operator D, presented in Valério and Sá da Costa [72], even if the order is time-varying the referred

law is verified as long as the conditions of Equation 4.4 are achieved. For such case, coefficient

correspondence is β = −1 and α = α(t).

cDαtcD

βt f(t) =c D

α+βt f(t) for β ≤ 0 ∧ α+ β ≤ 0 (4.4)

The numerical Type-D formulation behavior, when the tumor is being extinguished, is said to pro-

vide inaccurate results by the authors Sierociuk, Malesza, and Macias [65]. To solve such numerical

behavior, when the tumor’s mass is approximately zero, a switching scheme is applied bypassing the

variable order Simulink block from the toolbox described in Sierociuk, Malesza, and Macias [65]. An

acceptable value for the switching is for an achieved tumor density of 10−3%.

4.2.2 Addressing the Tumor’s Dynamic for Therapy Purposes

Due to intrinsic characteristics of its differential equation, the tumor is constantly growing until it

reaches the maximum size of 100%. When treatment is applied, the tumor’s density decreases to

such small values it can be considered inexistent. However, when the referred therapy is halted,

tumor density tends to retake its Gompertz form and grow again until maximum capacity is achieved.

54

Figure 4.2: Simulink implementation of the operator D combination D−1[Dα(t)]. Also included the switchingscheme between integrators, to bypass numerical incoherences of type-D construction as the whole order

tends to the integer value of 1. Switching parameter was chosen based on the time the tumor density isapproximately zero (from ≈ 10−3).

Physiologically speaking, such situation is in accordance to a tumor relapse, which is not meant

to be mathematically replicated here. So, to solve such innate behavior when the tumor’s density

is approximatively null, after treatment, it should be maintained that way. Here, this was achieved

through a simple Simulink step function and the obtained result can be seen in Figure 4.3.

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

60

70

80

90

100Tumor Evolution With and Without Applied Step Restriction

With Restriction at 2000 days Without Restriction

Figure 4.3: Comparison between regular tumor dynamics, for a chemotherapy and proteasome inhibitors basedtherapy (Equations 3.18), with and without tumor growth restriction after a successful applied treatment.

4.2.3 Different Resorption Ratios R

In an effort to reduce parameters in the original models of Ayati et al. [1], the action of tumor

through coefficients rij

is replaced by the variable order derivative α(t) applied in the operator D of

C(t) and B(t) equations. However, since the resulting model’s steady-state, from a mathematical

standpoint, is the same as the local healthy one, for all contemplated cases here, and most pa-

rameters remain with the same value, the associated activity of osteoclasts and osteoblasts in bone

mass must differ from the original case. As such, the bone resorption-formation ratio, given by Equa-

tion 4.5, is determined between 0 and t, that corresponds to the completion time of a single cycle of

C(t) in the new model. Such is the same method as used originally in Ayati et al. [1]. Bone resorption

and formation activities are then given by κC

= rR and κB

= r, respectively.

R =

∫ t0

max [0, C(t)− Css]∫ t0

max [0, B(t)−Bss](4.5)

In Section 4.3 the simplified bone models with tumor are presented. Since the model’s evolution

depends only on the tumor growth, a single resorption ratio is determined. However, in Section 4.4,

55

when different treatments are applied, the duration of an individual remodeling cycle is changed when-

ever an introduction is done. Consequently, three resorption rates must be determined: when the

tumor begins, when treatment is applied and when the tumor is extinguished and the treatment is

consequently stopped. Each ratio is determined for the first complete cycle after the induced change.

4.2.4 Non-local Implementations

Following the variable order methodology just presented, some considerations were made for the

non-local models simulation. The spatial numerical method used was, in a way, a more rudimentary

construction than its temporal counterpart (Simulink ode45 variable-step explicit continuous solver

for nonstiff problems). Using a built-up for Simulink software, a model was created that replicates, at

each single discretized node, the original local model based only on the ddt terms, and whose diffusion

terms relate the influence of each neighbor node according to the numerical method used. Such

methodology was chosen to more promptly apply the variable order derivative toolbox in Sierociuk,

Malesza, and Macias [65], also Simulink -based.

Here, bone was discretized in N equidistant slices (i = N + 1 nodes for a slice size of h = 1N ),

for a dimension x ∈ [0, 1], using the central difference method for the first and second derivatives in

order to space [47], ∂∂x and ∂2

∂x2 , respectively. Such was due to the existing symmetric null Newmann

boundary conditions, that were imposed onto the non-local bone remodeling models at each extremity

of the space region. The technique, for an inner-bone area and for the null first derivative boundary

conditions nodes, is as follows.

Inner Bone Region Bone Boundaries

D1f(t, x) =f(t, x− h) + f(t, x+ h)

2hD1f(t, x)|x=0,1 = 0⇔ f(t, x+ h) = f(t, x− h)

D2f(t, x) =f(t, x− h)− 2f(t, x) + f(t, x+ h)

h2D2f(t, x)|x=0,1 = 2

[f(t, x+ h)− f(t, x)]

h2

(4.6)

For the boundary conditions applied at x = 0 and x = 1, two fictitious nodes were added, located

x = −h, and x = (N + 1)h, so that the derivatives at those two locations were possible to achieve.

This numerical construction was tested, before adding the variable order derivatives, by comparing

the healthy and tumor bone remodeling models simulated with the PDE-Matlab toolbox. Achieved

results were identical.

4.3 Simplified Bone Models With an Acting Tumor

Here, new local (Section 4.3.1) and non-local (Section 4.3.2) constructions, based on the work

developed by Ayati et al. [1], are presented. These new models were made simpler, in what concerns

the number of parameters, by introducing variable order derivatives to mimic the tumor’s behavior

56

within the bone micro-environment. This allows for a simpler model to represent the nefarious action

of the tumor in the bone dynamics.

In all that follows, type-D construction D−∞Dα(t)t f(t) is now addressed simply as Dα(t)f(t) or

Dα(t,x)f(t, x), where f(t)/f(t, x) represents either the osteoclasts or the osteoblasts population.

4.3.1 Local Model

The new model for the local case is presented in Equations 4.7, where α(t) represents the vari-

able order and Dα(t) its corresponding derivative. The tumor relates to the imposed order through

Equation 4.7e previously presented, where ρ(t) takes a form linear in time with a constant θ gradient.

Dα(t)C(t) = αCC(t)gCCB(t)gBC − β

CC(t) (4.7a)

Dα(t)B(t) = αBC(t)gCBB(t)gBB − β

BB(t) (4.7b)

D1z(t) = −κC

max [0, C(t)− Css] + κB

max [0, B(t)−Bss] (4.7c)

D1T (t) = γTT (t) log

(LT

T (t)

)(4.7d)

α(t) = 1− ρ(t)T (t) = 1− θ × t× T (t) (4.7e)

However, further adaptations are required. As presented in Sub-Section 4.2.3, a new resorption

ratio R must be determined, since the associated activity of osteoclasts and osteoblasts in bone mass

must differ from the original case. This results in a different period for an individualized remodeling

cycle and, hence, a different t to determine R.

0 200 400 600 800 1000 12000

5

10

15

20

25

Ost

eocl

asts

[cel

ls]

Integer vs. Variable Order Tumorous Models

Integer ModelVariable Order Model

0 200 400 600 800 1000 12000

200

400

600

800

1000

Ost

eobl

asts

[cel

ls]

0 200 400 600 800 1000 12000

20

40

60

80

100

Bon

e M

ass

[%]

Figure 4.4: Simulation results for the new model presented, with variable order derivatives inducing the tumordisrupted behavior on the bone micro-environment. Osteoclasts, osteoblasts and bone mass evolutions aredisplayed, where R = 238.43 was the resulting ratio for κC = 0.1548 and κB = 6.4924× 10−4. The order’s

constant gradient was θ = 4× 10−8, for a running time of 1200 days (approximately three years). Remainingparameters can be found in column 0D-H of Table 3.1. Simulation results for the original Equations 3.13 are

provided for comparison.

Simulation of Equations 4.7 can be found in Figure 4.4, where the results are compared with

the ones from the original model given in Section 3.2.2. As justified in Ayati et al. [1], the tumor

57

stimulates an increase in osteoclast numbers to which the osteoblast population increases in an initial

attempt to maintain the equilibrium in the bone remodeling process. In this new model, however,

the initial behavior is diminished and such phenomenon is not replicated. Nonetheless, as the tumor

continuously promotes the activity of osteoclasts and inhibits osteoblasts [69], bone loss gradual and

periodically occurs in the same manner presented in Sub-Section 3.2.2.

Mathematically speaking, as time passes and the tumor follows the same growing Gompertz form

(see Figures 1.6 and 3.8), the order of the two governing equations of this system continually de-

creases, after the integration process from the differential equations takes place. The evolution of the

θ× t× T (t) term, in the α(t) expression, is given in Figure 4.5 where a growing and almost linearized

shape is followed.

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

t [days]

Var

iabl

e O

rder

Evo

lutio

n 1−

α(t)

Figure 4.5: Order α(t) evolution, for the part θ × t× T (t) of its expression. Initially, since t is small, the shape issimilar to the one of the tumor. However, as time passes, the time effect in the expression almost linearizes its

original Gompertz form.

This described evolution of order α(t) induces the same qualitative response in the original healthy

model of Equations 3.9, as in its integer counterpart with tumor in Equations 3.13. An overall balance

comparing this last model with the new one presented in this section allows to say that, even though

another equation was added, three parameters were made redundant with good qualitative results.

4.3.2 Non-local Model

For the non-local model, the approach is similar to the one described in the previous section. The

order is now both time t and space x dependent, and the initial osteoclasts and tumor distribution

throughout the bone is the same as in Figure 3.13. Boundary conditions are the same as the integer

non-local model, as they obey null Newman constructions at both x = 0 and x = 1 borders, for C(t, x),

B(t, x), z(t, x) and T (t, x). Bone resorption and formation activities, κC

and κB

, remain the same as

in the variable order local case (κC

= 0.1548 and κB

= 6.4924×10−4). The order expression, however,

has a different θ coefficient since introducing diffusion parameters onto the equations slightly changes

the dynamic response given by the local model. The resulting model is given by Equations 4.8, and

simulations can be found in Figure 4.6.

58

Dα(t,x)C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)gCCB(t, x)gBC − β

CC(t, x) (4.8a)

Dα(t,x)B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)gCBB(t, x)gBB − β

BB(t, x) (4.8b)

D1z(t, x) = σz∂2

∂x2z(t, x)− κ

C(x) max[0, C(t, x)− Css] + κ

B(x) max[0, B(t, x)−Bss] (4.8c)

D1T (t, x) = σT

∂2

∂x2T (t, x) + γ

TT (t, x) log

(LT

T (t, x)

)(4.8d)

α(t, x) = 1− ρ(t)T (t, x) = 1− θ × T (t, x) (4.8e)

Osteoclasts

0200

400600

8001000

12001400

16001800

2000

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

6

7

t [days]x [distance ∈ [0,1]]

Ost

eocl

ast P

opul

atio

n

Osteoblasts

0200

400600

8001000

12001400

16001800

20000

0.10.2

0.30.4

0.50.6

0.70.8

0.91

100

150

200

250

300

350

400

450

t [days]x [distance ∈ [0,1]]

Ost

eobl

ast P

opul

atio

n

Bone Mass

0200

400600

8001000

12001400

16001800

2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

50

100

150

t [days]

x [distance ∈ [0,1]]

Bon

e M

ass

Evo

lutio

n

Figure 4.6: Comparison between the results for the original bone remodeling model with tumor (left), aspresented in Ayati et al. [1] in Section 3.3.2, with the non-local bone remodeling simulations with a time and

space dependent derivative order, α(t, x) (right). For the right side figures, the initial distribution of osteoclastsand tumor was the same as in Figure 3.13. Osteoclasts and ostoblasts activities are the same as the localvariable order model: R = 238.43, κC = 0.1548 and κB = 6.4924× 10−4. Tumor growth rate has now the

non-local parameter of γT = 0.004 and the order’s constant gradient is θ = 2.5× 10−7. All diffusion coefficientshave the value of 10−6. Remaining parameters can be found in column 0D-H of Table 3.1.

The results for the non-local case differ from the ones in Figure 3.15, mainly due to the differ-

ent steady-states involved in the tumor disrupted integer and variable order models. However, as

previously described, the existing literature is interpretative regarding the fluctuation of osteoblasts

populations when the action of a tumor is taking place [20, 76, 58, 24]. So, instead of a growing

number of these cells which is not adjusted to what is in known for MM, as done in Ayati et al. [1],

here osteoblasts vary in response to increasing osteoclasts populations, but tend to normalize in their

59

original healthy number, as their activity is gradually diminished and bone loss occurs. This repli-

cates general metastatic bone disease of osteolytic nature, in accordance to what was presented in

Section 4.1.

Regarding the new bone mass simulation results, it can be seen that it has a slower development

that in the integer order models. However, the nefarious action is reflected in the bone density earlier

which, by the end time of the simulation, replicates the same nefarious action of the tumor in the bone

mass.

Again, tumor evolution simulation is not presented as it is the same as Figure 3.13. However, the

variable order 1− α(t, x) evolution is presented in Figure 4.7, even though the influence of the tumor

in it is barely noticeable due to the coefficient θ minimizing action.

Figure 4.7: Order α(t, x) evolution, for the part θ × t× T (t) of its expression. Similarly to the local evolution of1− α(t) (Figure 4.5), 1− α(t, x) linearizes the original Gompertz evolution of the tumor. However, the spatial

distribution of the tumor now has its effect on the order, even if it is not easily seen in the results due to the smallvalue of θ considered.

4.4 Changing the Order, Extinguishing the Tumor — A TreatmentApproach

As presented in Section 1.2.2.C and mathematically developed in Sections 3.2.3 and 3.3.3, for dif-

ferential equations of integer order, two different therapeutic approaches were studied. In this section,

the simplified bone model with an acting tumor presented is extended to include such treatments. In

Section 4.4.1, an osteoblastic promotion based therapy was introduced in the models for an MM bone

disease, whereas in Section 4.4.2 the PK/PD dynamics for anti-resorptive therapy was considered for

an osteolytic metastatic bone disease. For both cases, chemotherapy was used as an anti-cancer

agent and local and non-local constructions of the models are presented.

Regardless of the therapy chosen, the structure of each model remains the same as presented in

Chapter 3. However, by introducing a variable derivative of order α(t) or α(t, x), coefficients rij

were

removed from the formulations. The order itself follows Equations 4.2, since treatment acts directly in

the tumor and bone cells expressions. Hence, the order varies due to chemotherapy action directly in

the tumor’s density.

For both treatments, in their local or non-local forms, the tumor evolution is not presented as the

treatment values used were the same as those referred throughout Chapter 3. Since the tumor is

60

mathematically said to be independent of the bone micro-environment characteristics, its different

evolutions remain equal to those of the indicated chapter.

4.4.1 Osteoblastic Promotion & Anti-Cancer Therapy

The therapy proposed in Ayati et al. [1] is here considered. The simplified bone models with tumor

of Equations 4.7 and 4.8 are adapted to include proteasome inhibitors and chemotherapy functions,

again with intensities V1(t) = υ1 and V2(t) = υ2, respectively.

For the local case, Equations 4.9 provide the resulting adaptation. Chemotherapy directly affects

the order evolution, by reducing the tumor burden through V2(t) function, and V1(t) proteasome in-

hibitors related function disrupts the dynamic response by diminishing the osteoblasts apoptosis rate.

Dα(t)C(t) = αCC(t)gCCB(t)gBC − β

CC(t) (4.9a)

Dα(t)B(t) = αBC(t)gCBB(t)gBB − (β

B− V1(t))B(t) (4.9b)

D1z(t) = −κC

max [0, C(t)− Css] + κB

max [0, B(t)−Bss] (4.9c)

D1T (t) = (γT− V2(t))T (t) log

(LT

T (t)

)(4.9d)

α(t) = 1− θ × t× T (t) (4.9e)

Simulation results can be found in Figure 4.8. The evolution of the integer model (Equations 3.13)

is also presented for comparison purposes. For either case, tumor acts since the beginning of the

simulation (t = 0), treatment is introduced at 600 days (t = 600) and is consequently stopped when

the tumor is considered extinct at 2000 days (t = 2000, when the tumor density is less than 10−3%).

0 500 1000 1500 2000 2500 30000

5

10

15

20

Ost

eocl

asts

[cel

ls]

VariableInteger

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

Ost

eobl

asts

[cel

ls]

VariableInteger

0 500 1000 1500 2000 2500 30000

50

100

t [days]

Bon

e M

ass

[%]

VariableInteger

Figure 4.8: Bone micro-environment simulation results, when the action of a tumor, starting at t = 0 days, iscounteracted with proteasome inhibitors and chemotherapy based treatment, at t = 600 days. Healthy

remodeling cycles are resumed after the tumor is extinguished and the treatment stopped, at t = 2000 days. Forsuch results, order α(t) constant parameter was θ = 4× 10−8. Treatment intensities were υ1 = 0.001 andυ2 = 0.008, like the integer model of Ayati et al. [1]. Resorption rates can be found in Table 4.1. Remaining

parameters can be found in column 0D-H of Table 3.1.

As stated in Section 4.2.3, the resorption ratio R had to be updated each time a major variation

was introduced in the model. Adding the prescribed medication, for instance, changed the previous

61

period of the remodeling cycles. The same when the tumor is considered extinguished and treatment

is halted. In Figure 4.8, this is clear when comparing variable order results with integer ones, as the

latter has approximately constant periods throughout the simulation. The bone direct resorption rate

was, like the case of untreated tumor, taken to be κC

= 0.1548 and each resorption ratio (Rtumor

,

Rtreat

, and Rhealthy

), as well as the order’s value, can be found in Table 4.1.

Regarding the bone micro-environment evolution itself, for t < 600 days, the simulation follows

Figure 4.4, as it corresponds to a simplified scenario of tumor without applied treatment. When the

therapy is introduced, however, the dynamic response changes with a stimulatory action towards

osteoblastic cells which, it turn, increases this population. Even though osteoclasts are thought to

regulate the coupling mechanisms between bone resorption and formation cells [5], osteoclasts pop-

ulation increases in response to the stimulatory action of osteoblasts. When the treatment is stopped,

at t = 2000 days, both populations stabilize with periodic cycles. Bone mass, however, reflects the

consequences of an osteoclastic-derived coupling mechanism between bone cell, as the amplitude

of its cycles continuously grows, even after the treatment is ceased. Consequently, an osteoblas-

tic promotion increases bone density but fails to reinstate a healthy equilibrium in the bone micro-

environment.

It is again stressed that the tumor evolution is not presented since the treatment intensity param-

eters used were the same as in the integer case (υ1 = 0.001 and υ2 = 0.008) and, hence, the tumor

evolution is the same as in Section 3.2.2.

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5x 10

−3

t [days]

Var

iabl

e O

rder

Evo

lutio

n 1−

α(t)

Figure 4.9: Order α(t) evolution, for the part θ× t× T (t) of its expression when θ = 4× 10−8. Until treatment isapplied, at t = 600 days, 1− α(t) evolution is equal to that of Figure 4.5. After therapy is introduced, the changein the tumor’s density is initially weak as not enough time has passed for the full treatment effect to take place.

This is reflected in the order’s behavior, as a small variation in the tumor’s density multiplied by the running time,slows down its growth, until 1− α(t) starts decreasing as T (t) is continuously diminished.

Analyzing the overall results, the bone remodeling model with tumor simplified with variable order

derivatives provides solid results, even when treatment modifies its original structure. Stimulatory

action in the osteoblasts population results in an increase of the bone mass, as the tumor density

decreases. Osteoclasts population also increases and healthy and stable periodical cycles are re-

sumed for the bone micro-environment, even if it is not possible to fully recover the original bone

mass density of 100% by the end of treatment time. These results are even in accordance to what is

known for the coupling mechanisms between bone cells, as stimulation of osteoblastic cells is not an

effective treatment that allows for the correct equilibrium between bone resorption and formation to be

62

achieved [5]. Bone micro-environment is then set in the direction of continuous growth (z(t) > 100%

as t→∞), which can be regarded as an osteoblastic lesion of the bone.

The next step is to extend Equations 4.9 to their non-local version. Variables are now both time t

and space x dependent, diffusion terms are added to each equation of the bone micro-environment

and therapeutic action is again achieved through V1(t) = υ1 and V2(t) = υ2 functions. Equations 4.10

provide the resulting model.

Dα(t,x)C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)gCCB(t, x)gBC − β

CC(t, x) (4.10a)

Dα(t,x)B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)gCBB(t, x)gBB − (β

B− V1(t))B(t, x) (4.10b)

D1z(t, x) = σz∂2

∂x2z(t, x)− κ

Cmax

[0, C(t, x)− Css

]+ κ

Bmax

[0, B(t, x)−Bss

](4.10c)

D1T (t, x) = σT

∂2

∂x2T (t, x) + (γ

T− V2(t))T (t, x) log

(LT

T (t, x)

)(4.10d)

α(t, x) = 1− θ × t× T (t, x) (4.10e)

Simulation for these equations can be found in Figure 4.10, where the integer model simulations

are also presented, for an easier comparison between results.

As in the local variable order model, the tumor starts developing with the simulation, for t = 0 days.

Therapy is then introduced at t = 600 days, and interrupted at t = 2000 days, as the tumor density is

T (2000, x) < 10−3%, for ∀x ∈ Ω.

Even though Equations 4.10 differ from their local version only due to diffusion terms, these tend

to add a cumulative effect that slightly changes the overall dynamic response. Unlike the simpler

tumor model, where the transition from local to non-local was achieved just by adjusting the order’s

θ parameter, this is not enough when treatment is present. Resorption rates were computed again,

for a distance where major oscillation values occur (x ≈ 0.2). For the first two cases, R remained

the same: Rtumor ≈ 239.12 for t < 600, and Rtreat ≈ 110.81 for 600 ≤ t < 2000 days. When the

tumor is extinguished, however, the system’s behavior differs and the amplitude of the osteoclasts

and osteoblats oscillations does not stabilize. The resulting ratio was, for t ≥ 2000, Rhealthy

= 94.33.

These values can be compared in Table 4.1.

Unlike what happened in the local case, and easily compared to Figure 4.8, the non-local tumor

stimulates an initial increase in the osteoclasts population, as it happened in Ayati et al. [1]. Os-

teoblasts rise due to coupling mechanisms [5], but their action is not enough to prevent the bone

mass loss that effectively occurs. As this phenomenon continues, the bone cells general popula-

tion decreases until the applied treatment starts to take effect. When such happens, osteoclasts,

osteoblasts and bone density increase until the tumor is destroyed and the treatment is interrupted.

After this, and due to the therapeutic characteristics of proteasome inhibitors already explained for

the local case, bone mass continuously increases. Bone cells, however, tend to stabilized in their

stationary state Css = Css and Bss = Bss, with oscillations of decreasing amplitude.

63

Osteoclasts

Osteoblasts

Bone Mass

Figure 4.10: On the left, non-local bone remodeling simulations with integer order, and on the right simulationsof a model with a time and space dependent derivative order, α(t, x). Dynamic responses of osteoclasts,

osteoblasts and bone mass, for an initial distribution of osteoclasts and tumor as in Figure 3.13. Resorptionratios can be found in Table 4.1. Tumor growth rate is is γT = 0.004, as in the integer non-local case, and the

order’s constant gradient is θ = 2× 10−6. All diffusion coefficients have the value of 10−6 and remainingparameters can be found in column 0D-H of Table 3.1.

4.4.2 PKPD of Anti-Resorptive & Chemotherapy Treatment

Models including anti-resorptive and chemotherapy treatment, as presented in Sections 3.2.3.B

and 3.3.3.B, can also be simplified for an osteolytic metastatic bone disease. The action of the tumor

in the bone cells populations, through coefficients rij

, can be replaced with a variable order derivative

that induces the same behavior.

For the local case, Equations 3.2.3.B can be transformed in the following Equations 4.11, through

the adaptation process followed in the previous section.

Dα(t)C(t) = αCC(t)gCCB(t)gBC (1+Ks1d1

(t)) − (1 +Ks2d2(t))β

CC(t) (4.11a)

Dα(t)B(t) = αBC(t)gCBB(t)gBB − β

BB(t) (4.11b)

D1z(t) = −κC

max [0, C(t)− Css] + κB

max [0, B(t)−Bss] (4.11c)

D1T (t) = (1 +Ki34dc34

(t)) γTT (t) log

(LT

T (t)

)(4.11d)

α(t) = 1− θ × t× T (t) (4.11e)

Again, PK/PD effects of denosumab and bisphosphonates are accounted in d1(t) and d2(t) vari-

ables, respectively, and chemotherapy remains as an anti-cancer tactic through the combination of

64

Figure 4.11: Evolution of the θ × t× T (t, x) component of α(t, x), for θ = 2× 10−6. The local behavior of α(t)(Figure 4.9) is here replicated, for each x ∈ Ω affected by the tumor’s spatial distribution in the corresponding

location.

two distinct drugs, dc34(t). Simulations can be found in Figure 4.12, for the treatment parameters of

Table 3.2.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

5

10

15

20

25

Ost

eocl

asts

[cel

ls]

Bone Micro−environment with Tumor and PKPD Treatment

VariableInteger

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

200

400

600

800

1000

1200

Ost

eobl

asts

[cel

ls]

VariableInteger

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

100

t [days]

Bon

e M

ass

[%]

VariableInteger

Figure 4.12: Bone micro-environment simulation, for a tumor developing at t = 0 days. Anti-resorptive andchemotherapy therapy was applied at t = 600 days. Healthy remodeling cycles are resumed after the tumor is

extinguished and the treatment stopped, at t = 2340 days. The order’s α(t) constant parameter wasθ = 4× 10−8. Resorption rations can be found in Table 4.1. Remaining parameters can be found in column

0D-H of Table 3.1 and treatment parameters in Table 3.2.

Simulation times were similar to those considered thus far, with the tumor appearing at t = 0 days

and treatment being applied at t = 600 days. However, due to the slower action of chemotherapy, the

tumor is considered extinct at t = 2340 days. Therapy was interrupted at the same time. Relevant

new parameters can be found in Table 4.1,

Comparing both integer and variable order results, in Figure 4.12, the latter can be said to present

far more soothing results. As the tumor action gains hold of the bone cells dynamics, their populations

decrease until the treatment is applied. After such a turn-mark, both population gradually increase

their numbers until treatment is halted. Then, they periodically oscillate with constant amplitude. Such

behavior is far more intuitively physiological correct than in its integer version, as a sudden increase

in the populations occurs, at t = 2340 days. However, since it is not known what happens, both results

65

are physiologically possible. It is noted that, like in the local osteoblastic promotion treatment case

(Figure 4.8), no initial increase in the osteoclasts and osteoblasts occurs.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5x 10

−3

t [days]

Var

iabl

e O

rder

Evo

lutio

n 1−

α(t)

Figure 4.13: Evolution of 1− α(t) = θ × t× T (t, x), for θ = 4× 10−8 and for a tumor growth according toFigure 3.12, considering the tumor starting time of 0 days.

For the non-local adaptation of this applied treatment to be formulated, it needs only to follow the

same logic of the previous section. The resulting model is presented in Equations 4.12.

Dα(t,x)C(t, x) = σC

∂2

∂x2C(t, x) + α

CC(t, x)gCCB(t, x)gBC (1+Ks1d1 (t))−

− (1 +Ks2d2(t))βCC(t, x) (4.12a)

Dα(t,x)B(t, x) = σB

∂2

∂x2B(t, x) + α

BC(t, x)gCBB(t, x)gBB − β

BB(t) (4.12b)

D1z(t, x) = σz∂2

∂x2z(t, x) +−κ

Cmax

[0, C(t, x)− Css

]+ κ

Bmax

[0, B(t, x)−Bss

](4.12c)

D1T (t, x) = σT

∂2

∂x2T (t, x) + (1 +Ki34

dc34(t)) γ

TT (t, x) log

(LT

T (t, x)

)(4.12d)

α(t, x) = 1− θ × t× T (t, x) (4.12e)

Obtained results can be found in Figure 4.14, with relevant new parameters presented in Table 4.1.

It can be seen, from the osteoclasts and osteoblasts evolution, that the tumor initially promotes an

increase in these populations. However, as bone mass is continuously destroyed, these populations

decrease in response to the monopolizing action of the tumor in the bone micro-environment. When

treatment starts to take effect, bone cells populations increase, and so does the bone mass, until such

treatment is ceased. After such time, cell population periodically tend to their stationary state as the

bone mass has stabilized in a value z(t, x) < 100% as the tumor is considered extinct.

Such results, when compared to the integer ones also provided in Figure 4.14, can be said to

provide far more soothing and realistic results, even if it is not known exactly was happens in the bone

micro-environment populations, being it a healthy or a pathology situation.

66

Osteoclasts

Osteoblasts

Bone Mass

Figure 4.14: On the left, non-local bone remodeling simulations with integer order, and on the right simulationsof a model with a time and space dependent derivative order, α(t, x). Presented results for osteoclasts,

osteoblasts and bone mass, for an initial distribution of osteoclasts and tumor as in Figure 3.13. The order’sα(t, x) constant parameter was θ = 4× 10−7. Resorption rates for each segment were Rtumor = 239.12

(t < 600 days), Rtreat = 93.21 (600 ≤ t < 2340 days) and Rhealthy = 98.56 (t > 2340), as seen in Table 4.1. Alldiffusion coefficients have the value of 10−6. Remaining parameters can be found in column 0D-H of Table 3.1

and treatment parameters in Table 3.2.

Table 4.1: Resorption ratios used for proteasome inhibitors based treatment (PI) and for anti-resorptive therapy(AR). R values are presented for three different remodeling scenarios: when a tumor is growing in the bone

micro-environment, Rtumor (t < 600 days); when treatment is initiated to counteract the tumor disruptive actionRtreat (600 ≤ t < 2000 days, for PI, or 600 ≤ t < 2340 for AR); and when healthy remodeling cycles are

resumed after the tumor is extinguished, Rhealthy (t > 2000, for PI, and t > 2340, for AR).

PI ARLocal Non-local Local Non-local

Rtumor 239.12 239.12 239.12 239.12Rtreat 110.81 110.81 110.81 93.21Rhealthy

112.74 94.33 112.74 98.56κC

0.1548 0.1548 0.1548 0.1548

67

Figure 4.15: Order α(t) evolution, for the part θ × t× T (t) of its expression, when chemotherapy is acting todestroy the tumor. Parameters θ was θ = 4× 10−7.

68

5Conclusions and Future Work

Contents5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

69

It always seems impossible until it’s done.

Nelson Mandela

Conclusions for the original work presented in Chapter 4 are here detailed (Section 5.1), followed

by the future work made possible by such developments (Section 5.2).

5.1 Conclusions

As previously stated, bone is in a constant state of turnover [5]. Cells termed osteoclasts are

responsible for bone resorption, whereas cells deemed osteoblasts are accountable for its formation.

Remodeling cycles originate from the tightly couple mechanism between these cells. However, when

a tumor metastasizes to the bone, this equilibrium is disrupted. For tumor-induced lesions of oste-

olytic nature, promotion of osteoclast activity and change in the number and/or activity of osteoblasts

occurs in a process not fully understood yet [24, 8]. Initially, chemotherapy was combined with protea-

some inhibitors to counteract MM disease [44], and bisphosphonates and denosumab are the primary

choices to treat an osteolytic metastatic bone disease [56, 52].

The biochemical process of bone remodeling can be replicated through mathematical and compu-

tational models. The comparison of an healthy bone behavior with pathological states [31, 1] was then

made possible, serving these models as clinical decision support systems for the implementation of

therapeutic regimes. They consist on a system of ordinary differential equations that relate the inter-

actions between osteoclasts and osteoblasts, by reproducing the effects of autocrine and paracrine

control mechanisms. The resulting calculation of the dynamic response of these cells populations, for

an acting MM disease, determines the changes in bone mass in the bone remodeling cycles.

In MM osteoblast activity is known to be severely suppressed or even absent [68], which results

in aggravated bone loss as osteoclasts are continuously promoted. However, in the results for the

biochemical mathematical model presented in Ayati et al. [1] and here in Chapter 3, the bone forming

cells still have some degree of action. Such behavior is here considered to better fit a metastatic

bone disease of osteolytic nature, such as a typical metastization to the bone of breast cancer [24],

as achieved results for the bone micro-environment simulation were physiologically better described

by such new approach.

Aiming for a more direct and easily applicable clinical decision support systems, variable order

derivatives were introduced in the system of equations. By doing so, for the first time in the bone

remodeling models area, a significant reduction of variables and parameters was made possible.

Hence, a simpler and equivalent mathematical model is provided with far more soothing results than

the original ones. These simplification techniques were applied to three scenarios: a simpler case

with a developing tumor; a tumor fought with chemotherapy and proteasome inhibitor for MM; and a

tumor with applied PK/PD action mechanism of chemotherapy with denosumab or bisphosphonates.

For each case, an overall counting returns 1 equation added, for the system’s order Dα(t)/Dα(t,x), and

3 parameters suppressed, as all 4 rij were removed and θ was added to the order’s equation.

70

For the three scenarios described, local and non-local one-dimensionalized results were pre-

sented in Chapter 4. Globally, it can be seen that the new approach here considered provides good

qualitative results when compared with the results of Ayati et al. [1] and to what is known for the bone

micro-environment under the effect of a developing tumor. The common behavior translates a severe

decrease in both osteoclasts and osteoblasts populations, as the tumor grows in density and bone

mass experiences an inverse evolution. However, the non-local results replicate two known phenom-

enas: firstly, the initial tumor growth promotes osteoclast activity; and secondly, since osteoclasts are

thought to regulate osteoblastic functions, these cells also rose in number in what is identified as an

attempt to confine the tumor [5]. Regarding the bone mass simulation results, either in its local and

non-local constructions, it can be seen that it has a slower development that in the integer order mod-

els. However, since the action of the tumor is reflected in the bone density earlier than in the original

models, the new bone loss results are quantitative similar at the simulation end time.

5.2 Future Work

Computational analysis of bone physiological models is expected to have an impact on the devel-

opment of clinical decision support systems in the future. Here are highlighted the road-map steps in

an effort to bring that desired reality closer.

• Adapt the type-D numerical variable order toolbox of Sierociuk, Malesza, and Macias [65] and

the non-local numerical guideline here followed, to provide better and faster non-local results;

• Add diffusion terms, and consequent boundary conditions, for the PK/PD treatments prescribed,

considering that, when a treatment is applied, it is done in a specific site and not uniformly in a

said region;

• Applying the techniques here developed to models that include a more detailed and accurate

description of the biochemical processes involved [17];

• Extend the models originally proposed in Komarova et al. [31] and Ayati et al. [1] to incorporate

the biomechanical effects in the bone, which translates in adding mechanical solicitations in the

original equations [3, 7]. Such work would be simplified by using variable order derivatives in

the resulting equations;

• Propose new models that should be constructed to allow the representation of a metastatic bone

disease of osteoblastic nature, adapting the autocrine and paracrine control mechanisms of the

models worked so far to account for the different physiological pathways disrupted [24];

• Finally, coefficients should be measured from experimental data, possibly data extrapolated

from experiments with animals, as the used parameters wee little more than educated guesses

by clinicians and oncobiologists at the magnitude of the values, leading to reasonable results

from the qualitative point of view. Finding actual experimental values is probably the biggest

challenge facing the model.

71

72

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76

AFractional and Variable Order

Calculus & Biochemical ModelsSupport

A-1

A.1 Integer Derivatives

For the previous definitions of the operator D, in Chapter 2.1.2, the n ∈ Z− branch can be trans-

lated in the following.

cDnt f(t) =

|n|integrations︷ ︸︸ ︷∫ t

c

...

∫ t

c

f(t)dt...dt =dnf(t)

d(t− c)n, n ∈ Z−

tDnc f(t) =

∫ c

t

...

∫ c

t

f(t)dt...dt︸ ︷︷ ︸|n|integrations

=dnf(t)

d(c− t)n, n ∈ Z−

(A.1)

The Cauchy’s Formula provides the indefinite integral of order n ∈ N of function f(t) given by the

following.

cD−nt f(t) =

|n|integrations︷ ︸︸ ︷∫ t

c

...

∫ t

c

f(t)dt...dt =

∫ t

c

(t− τ)n−1

(n− 1)!f(τ)dτ

tD−nc f(t) =

∫ c

t

...

∫ c

t

f(t)dt...dt︸ ︷︷ ︸|n|integrations

=

∫ c

t

(τ − t)n−1

(n− 1)!f(τ)dτ

(A.2)

However, since the Riemann Integral is given by∫ tcf(t)dt, as follows, cD−nt f(t) can be rewritten

in the following form.

∫ t

c

f(t)dt = limh→0+

b t−ch c∑k=0

hf(t− kh)

cDt−nf(t) = lim

h→0+

b t−ch c∑k=0

h(kh)n−1

(n− 1)!f(t− kh)

(A.3)

A.2 Variable Order Definitions

• Riemann-Liouville (RL) Variable-order Construction

Definition 10. RL: 1

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

DdαecDα(t)−dαet f(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(t)−1

Γ(−α(t))f(x)dx if α(t) ∈ R−

(A.4)

A-2

Definition 11. RL: 2

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

DdαecDα(t)−dαet f(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(x)−1

Γ(−α(x))f(x)dx if α(t) ∈ R−

(A.5)

Definition 12. RL: 3

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

DdαecDα(t)−dαet f(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(t−x)−1

Γ(−α(t− x))f(x)dx if α(t) ∈ R−

(A.6)

• Caputo (C) Variable-order Construction

Definition 13. C: 1

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

cDα(t)−dαet Ddαef(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(t)−1

Γ(−α(t))f(x)dx if α(t) ∈ R−

(A.7)

Definition 14. C: 2

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

cDα(t)−dαet Ddαef(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(x)−1

Γ(−α(x))f(x)dx if α(t) ∈ R−

(A.8)

Definition 15. C: 3

cDα(t)t f(t) =

dα(t)f(t)

dtα(t)if α(t) ∈ N

cDα(t)−dαet Ddαef(t) if α(t) ∈ R+\N

f(t) if α(t) = 0∫ tc

(t− x)−α(t−x)−1

Γ(−α(t− x))f(x)dx if α(t) ∈ R−

(A.9)

A.3 Healthy Bone Remodeling - Nontrivial Steady State

The solution, given by Css and Bss in equation 3.10a, can be proven to be a nontrivial steady state

of the system encompassed by 3.9a and 3.9b, considering d/dt = 0 onto the original equations, as

follows.

A-3

αCCss

gCCBss

gBC = β

CCss

αBCss

gCBBss

gBB = β

BBss⇔

Css =

(αC

βC

BgBCss

) 11−g

CC

Bss =

(αB

βB

CgCBss

) 11−g

BB

⇔

⇔

Css =

αC

βC

(1−g

BB−(g

CBgBC−(1−g

CC)(1−g

BB))

)αB

βB

(gBC

−(gCB

gBC−(1−g

CC)(1−g

BB))

)

Bss =αC

βC

(gCB

−(gCB

gBC−(1−g

CC)(1−g

BB))

)αB

βB

(1−g

BB−(g

CBgBC−(1−g

CC)(1−g

BB))

) ⇔

⇔

Css =

βC

αC

1−gBBγ β

B

αB

gBCγ

Bss =βC

αC

gCBγ β

B

αB

1−gBBγ

(A.10)

A.4 Different Types of Steady-States

• Node - This type of steady state has same sign real eigenvalues. The node is stable when the

eigenvalues are negative and unstable when they are positive.

• Saddle - When real eigenvalues have different signs, the system will leave its steady state as

time increases. Hence, its an unstable equilibrium.

• Center - In this case, the eigenvalues are complex and with zero real part, which imposes a

periodic behavior whose amplitude and frequency depends on the initial condition.

• Focus - This name is for complex-conjugate eigenvalues that, like nodes, are stable if their real

part is negative and unstable if positive. The imaginary part of the eigenvalue is responsible

for the oscillations around the steady state: when negative the oscillations will be damped and,

consequently, stable; if positive, the steady state is unstable.

A-4