bone remodelling model including overload simulation - ulisboa

Bone remodelling model including overload simulation

Bernardo Afonso Gonçalves Ferreira Rodrigues

Thesis to obtain the Master of Science Degree in

Biomedical Engineering

Supervisor(s): Prof. João Orlando Marques Gameiro FolgadoProf. Carlos Miguel Fernandes Quental

Examination Committee

Chairperson: Prof. Fernando Manuel Fernandes SimõesSupervisor: Prof. João Orlando Marques Gameiro Folgado

Member of the Committee: Prof. Paulo Rui Alves Fernandes

March 2018

Acknowledgments

First, I would like to thank Professor Joao Folgado and Professor Carlos Quental, my supervisors, for

allowing me to work on this Thesis with them. Their guidance and advices throughout this work were

crucial to the outcome of this Thesis. It was a privilege working under their supervision and I truly

appreciate their patience and help.

I would like to thank my family, especially my mother, my brother and my grandparents for supporting

me in life and in my academic journey from day one. Without their unconditional love and support I would

not have achieved this goal in my life and I would not have been the person I am today.

I would like to thank my beloved girlfriend Maria Joao Amaral for all the love and emotional support

not only throughout this work but also the past years of my life.

Finally, to all my friends, especially those who shared these five years at Instituto Superior Tecnico

with me a big thank you for your companionship and good times spent. Without you, this period of my

life would not have been this amazing.

iii

Resumo

O osso e um tecido dinamico que altera as suas caracterısticas consoante o ambiente mecanico a que

esta sujeito. Segundo Julius Wollf, quando o estımulo mecanico e elevado, o osso aumenta a sua densi-

dade; por outro lado, quando o estımulo e baixo, a densidade ossea diminui. Modelos de remodelacao

ossea tem vindo a ser desenvolvidos, a fim de modelar computacionalmente este comportamento. Con-

tudo, em situacoes mecanicas extremas, o osso pode perder densidade devido a sobrecarga, o que nao

esta presente na maioria dos modelos desenvolvidos.

Este trabalho tem como objectivo o desenvolvimento de um modelo matematico de remodelacao

ossea que tenha em conta a diminuicao de densidade devido a sobrecarga mecanica, utilizando a

deformacao como estımulo. O desempenho do modelo e avaliado e comparado com outros existentes

na literatura. Para tal, um modelo de elementos finitos de duas vertebras (em duas dimensoes) e de um

femur (em duas e tres dimensoes) sao utilizados, com e sem proteses implantadas. Ambos os modelos

matematicos sao testados com e sem sobrecarga incorporada e as diferencas obtidas analisadas.

Os resultados obtidos sao qualitativamente semelhantes para ambos os modelos matematicos e e

possıvel observar a diminuicao da densidade devido a sobrecarga, maioritariamente nas regioes do

osso em contacto com a protese. Tambem e possıvel concluir que quando a protese e implantada em

osso de menor densidade este fenomeno e mais proeminente.

Apesar de este trabalho ter permitido simular a sobrecarga ossea, testes mais complexos deverao

ser feitos ao modelo para garantir o seu bom desempenho em casos de estudo reais.

Palavras-chave: Modelos de remodelacao ossea, Sobrecarga mecanica, Metodo dos ele-

mentos finitos, Mecanica Computacional.

v

Abstract

Bone is a dynamic tissue that adapts its form, size and structure to the mechanical environment. Accord-

ing to Julius Wolff, bone density increases if the mechanical stimulus is high and it decreases if it is low.

This behavior has been modelled computationally, using mathematical models, called bone remodelling

models. However, under extreme mechanical conditions, bone density may decrease due to excessive

stimulus, which is not present in the majority of the mathematical models developed. This phenomenon

is called overload resorption.

In this work, a novel mathematical model is proposed for bone remodelling, using strain as mechan-

ical stimulus, which takes in consideration bone resorption due to overload. The performance of this

model is evaluated and compared to existing models in the literature. For this, finite element models of

two vertebrae (two dimensional) and of a femur (two and three dimensional) are used, with and without

prostheses implanted. The mathematical models are tested with and without overload incorporated and

the differences obtained are analyzed.

The results obtained using both mathematical models are qualitatively similar and it is possible to

observe bone resorption due to overload, mainly in bone regions contacting with the prosthesis. It

is also possible to conclude that the overload phenomenon is more prominent when the prosthesis is

implanted in low density bone.

The work developed allowed to simulate bone resorption due to overload. However, the mathe-

matical model should be further tested, using more complex finite element models to assure its good

performance in real case studies.

Keywords: Bone remodelling model, Bone overload, Finite element method, Computational

Mechanics.

vii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

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

1 Introduction 1

1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical Overview 3

2.1 Bone anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Bone remodelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Bone Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Literature Review 10

4 Computational model for bone remodelling 15

4.1 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Mathematical Models for bone remodelling . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Strain energy density model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.2 Strain-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.1 Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.2 Second order Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.3 Numerical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3.4 Step limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4.1 Strain energy density model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4.2 Strain-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5 Application cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

ix

4.5.1 Two-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.5.2 Three-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Results 27

5.1 Baseline comparison of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Vertebrae model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Femur model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.1 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.2 Standard Prosthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.3 Large prosthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.4 Three-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4 Sensitivity of the bone remodelling models . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4.2 Initial Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4.3 Parameter D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Discussion 43

7 Conclusions and Future Work 47

References 49

References 49

x

List of Tables

4.1 Forces used in the 2D femur model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Forces used in the 3D femur model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Parameters used in the strain energy density model for the 2D femur model. . . . . . . . . 27

5.2 Parameters used in the strain-based model for the 2D femur model. . . . . . . . . . . . . 28

5.3 Parameters used to simulate overload phenomenon in the vertebrae with a prosthesis

implanted using the strain energy density model. . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 Parameters used to study the influence of mesh size in the simulations. . . . . . . . . . . 37

xi

List of Figures

2.1 Structure of a long bone. Source [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Sagittal section of a lumbar vertebra. Adapted from [13] . . . . . . . . . . . . . . . . . . . 5

2.3 Three main parts of a vertebrae: body (purple), vertebral arch (green), and processes for

muscle attachment (brown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Hierarchical structural organization of bone. Adapted from [14]. . . . . . . . . . . . . . . . 6

2.5 Types of forces produced by different loads applied in a femur. Adapted from [16]. . . . . 7

2.6 Schematic representation of the bone remodelling process. Aadapted from [28]. . . . . . 9

3.1 Schematic representation of Frost Mechanostat Theory. Adapted from [9] . . . . . . . . . 11

3.2 Change in the bone density rate as a function of the mechanical stimulus for the bone

remodelling model of Huiskes et al. [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Piecewise linear relationship between the stress stimulus and the bone apposition/resorption

rate, according to the model of Beaupre et al. [34]. . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Flowchart of the bone remodelling model developed by Crupi et al. Adapted from [7]. . . . 13

3.5 Comparison between Huiskes et al. (dotted line) and Li et al. bone remodelling models

(solid line). Adapted from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Bone density rate as a function of the mechanical stimulus Uρ according to the bone re-

modelling model developed by Li et al. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Bone density rate as a function of the mechanical stimulus ψ(ε) according to the bone

remodelling model developed in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Method used to limit the density variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 2D model of the vertebrae: (a) intact bone and (b) implanted bone. . . . . . . . . . . . . . 23

4.5 2D model of the femur: (a) intact bone, (b) implanted bone and (c) the side plate used,

with thickness of 5mm, 3mm and 1mm from top to bottom. . . . . . . . . . . . . . . . . . . 24

4.6 3D model of the femur: (a) intact bone, (b) implanted bone and (c) the prosthesis used. . 25

5.1 Density distribution obtained in the 2D model of the femur: (a) strain energy density model,

(b) Strain-based model with ψ1, (c) Strain-based model with ψ2 . . . . . . . . . . . . . . . 28

5.2 Evolution of the bone remodelling simulation for the vertebrae model using the strain-

based law: (a) strain, (b) bone density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xiii

5.3 Bone density distribution in the vertebrae with prosthesis obtained using the strain-based

model: (a) without overload, (b) with overload . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.4 Zoom of the bone density distribution to highlight bone overloading. The solutions are

obtained for the models: (a) without overload, (b) with overload. . . . . . . . . . . . . . . . 30

5.5 Bone density distribution in the vertebrae with prosthesis obtained using the strain energy

density model: (a) without overload, (b) with overload . . . . . . . . . . . . . . . . . . . . 31

5.6 Bone density distribution in the femur with the standard prosthesis obtained using the

strain-based model: (a) without overload, (b) with overload. . . . . . . . . . . . . . . . . . 33

5.7 Bone density distribution in the femur with the standard prosthesis obtained using the

strain energy density model: (a) without overload, (b) with overload. . . . . . . . . . . . . 33

5.8 Bone density distribution in the femur with the large prosthesis obtained using the strain-

based model: (a) without overload, (b) with overload. . . . . . . . . . . . . . . . . . . . . . 34

5.9 Bone density distribution in the femur with the large prosthesis obtained using the strain

energy density model: (a) without overload, (b) with overload (D=60). . . . . . . . . . . . 34

5.10 Bone density distribution in the 3D femur obtained using the strain based model: (a) view

of a coronal cut, (b) medial view, (c) view of a transversal cut. . . . . . . . . . . . . . . . . 35

5.11 Bone density distribution in the 3D femur obtained using the strain energy density model:

(a) view of a coronal cut, (b) medial view, (c) view of a transversal cut. . . . . . . . . . . . 36

5.12 Bone density distribution in the 3D femur with a resurfacing prosthesis obtained using the

strain based model without overload (a) view of a coronal cut, (b) medial view and with

overload (c) view of a coronal cut, (d) medial view. . . . . . . . . . . . . . . . . . . . . . . 36

5.13 Bone density distribution in the 3D femur with a resurfacing prosthesis obtained using the

strain energy density model without overload (a) view of a coronal cut, (b) medial view

and with overload (c) view of a coronal cut, (d) medial view. . . . . . . . . . . . . . . . . . 37

5.14 Minimum principal strain distribution in the region of the vertebra in contact with the pros-

thesis obtained using the strain-based model and different mesh: (a) mesh with 957 ele-

ments (simulation 1), (b) mesh with 12000 elements (simulation 2). . . . . . . . . . . . . . 38

5.15 Strain energy density distribution in the region of the vertebra in contact with the prosthe-

sis obtained using the strain energy density model and different mesh: (a) mesh with 957

elements (simulation 1), (b) mesh with 12000 elements (simulation 2). . . . . . . . . . . . 38

5.16 Density distribution in the region of the vertebra in contact with the prosthesis obtained

using the strain-based model and different mesh: (a) mesh with 957 elements (simulation

1), (b) mesh with 12000 elements (simulation 2). . . . . . . . . . . . . . . . . . . . . . . . 39

5.17 Density distribution in the femur with the standard proshtesis implanted obtained using

the strain-based model and different initial densities, ρi: (a) ρi = 1 g · cm−3 , (b) ρi = 1.5

g · cm−3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


the strain energy density model and different initial densities, ρi: (a) ρi = 1 g · cm−3 , (b)

ρi = 1.5 g · cm−3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xiv


the strain energy density model and different values of D: (a) D = 50, (b) D = 73, (c) D = 74. 41

6.1 Representation of density distribution in a real femur. Dark areas represent high density

and bright areas low density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xv

Nomenclature

Greek symbols

σ Stress tensor.

ε Strain tensor.

ε Strain.

εmax Maximal principal strain.

εmed Medium principal strain.

εmin Minimal principal strain.

ν Poisson’s ratio.

ψ Strain stimulus.

ξ Limits of density variation.

ρ Bone density.

∂ρ∂t Change rate of bone density.

Roman symbols

D Direction of the density variation in the bone remodelling models.

E Young’s Modulus.

MES Non-mechanical mechanisms that control bone remodelling process, acording to Frost Theory.

s Half of the baseline length in the remodelling model developed by Huiskes et al.

U Strain energy density.

xvii

Chapter 1

Introduction

1.1 Motivation and objectives

Bone is a dynamic tissue that can adapt its form, size and structure according to the applied loads. This

phenomenon was first described by Julius Wolff in 1982 [1]. Wolff theorized that bone can either be

formed or absorbed in different regions, depending on the loads applied [2]. These changes contribute

to bone homeostasis. In a healthy condition, there is an equilibrium between bone absorption and bone

formation. However, if the mechanical environment is disturbed, bone integrity may be compromised.

For example, bone density decreases due to prolonged periods of inactivity, which makes the bone

weaker and more prone to injuries and fractures. On the other hand, when submitted to high loads

(within a certain range), the bone becomes denser [3].

Over the last decades, several mathematical models have been developed to simulate bone adapta-

tion to the mechanical environment. These models are, in most cases, locally regulated by a mechanical

stimulus (that vary between models) [4] and are composed of three regions: a stationary region or dead

zone, in which no bone absorption or formation occurs, a bone resorption region and bone formation

region. In most models, the bone formation region is not limited, i.e., the higher the stimulus, the higher

the bone formation rate [5], which is not physiological because under excessive loads, damage may

accumulate faster than the bone is able to recover, leading to bone absorption instead of bone forma-

tion [6]. Models that can simulate bone resorption due to overloading are needed to be able to predict

the bone adaptation process under more critical conditions, such as in the presence of implants. Some

work has been done in this area, such as the models developed by Crupi et al. [7] and Li et al. [8].

However, bone overloading phenomenon is no yet well comprehended and further studies need to be

done in this subject.

In this work, a novel mathematical model for bone remodeling that takes in consideration bone re-

sorption due to overload, based on the theory developed by Frost [9], is proposed. The performance

of this model is tested under normal conditions (without overload) and under overload conditions and

compared with the model developed by Li et al. [8].

1

1.2 Thesis Outline

Apart from the introduction, six main chapters compose this thesis.

Chapter 2 presents the main theoretical concepts that are fundamental to understand the work de-

veloped. Firstly a brief description of bone is made, including its functions, histology and mechanics.

After this, the bone remodelling process is addressed and bone overloading is introduced.

Chapter 3 consists of a literature review, in which several mathematical models of bone remodelling

are mentioned.

In Chapter 4, the mathematical models for bone remodelling and the finite element models studied

in this work are presented. A brief explanation of the finite element method and the numerical methods

used is provided.

Chapter 5 presents the results obtained in this work which are analyzed in Chapter 6.

Finally, Chapter 7 consists on the conclusions taken from this work and some suggestions for future

work.

2

Chapter 2

Theoretical Overview

This chapter elaborates on the main theoretical concepts about bone and the bone remodelling process,

essential to understand the work developed.

2.1 Bone anatomy

Along with cartilage, tendons and ligaments, bones are part of the skeletal system and have a variety of

functions in the human body. They serve as a framework where muscles and other tissues are attached,

providing support and maintaining the shape of the body, protect vital organs from injury, such as the

brain and heart, allow movement of the body by acting as levers between muscles and favoring the use

of forces generated by them, serve as a reservoir of minerals, such as calcium and phosphorus, release

them when needed and enable haematopoiesis, which is the production of red blood cells, in the bone

marrow [10].

Bones can be classified in five main categories, according to their shape and function [10]:

• Long bones, such as the femur and the radius, have as principal functions bodyweight supporting

and movement enabling;

• Short bones, for example the tharsals, are cube-shaped and provide stability and movement;

• Sesamoid bones, such as the patella, exist to reinforce tendons and to protect them from stress

and wear;

• Flat bones have as main function the protection of internal organs, such as the brain in the case of

the cranial bones;

• Irregular bones, such as the vertebrae, have complex shapes and therefore do not fit in any of the

other four categories. They mainly protect internal organs.

Considering that computational models of a femur and a vertebra are applied in this study, their struc-

tures are analyzed in further detail. Long bones are divided in three main regions: epiphysis, diaphysis

and metaphysis. The epiphysis is located in both ends of long bones. It is formed by spongy bone,

3

with red marrow filling its spaces. The extremities of the epiphysis are covered by articular cartilage

to optimize the connection and movement between bones. The diaphysis runs between the proximal

and distal ends of the bone, delimited by compact bone. It has a hollow region, called medullary cavity,

filled with yellow marrow. This cavity is delimited by the endosteum, a thin connective tissue membrane.

The periosteum, a dense connective tissue, covers the outer surface of bone. The metaphysis is the

connecting region between epiphysis and diaphysis [11]. A representation of this type of bone, more

precisely a femur, is shown in Figure 2.1.

Figure 2.1: Structure of a long bone. Source [12].

Irregular bones, including the vertebrae are formed by cancellous bone surrounded by a thin layer of

compact bone. A representation of this type of bone, more precisely a vertebra, can be seen in Figure

2.2.

There are 33 vertebrae in the human spinal collumn and are divided into five regions: cervical (neck),

thoracic (mid back), lumbar (low back), sacrum and coccyx. All vertebrae have three common main

parts: the body that supports the weight, an arch-shaped bone that protects the spinal cord and a

star-shaped processes where the muscles attach. These parts are shown in Figure 2.3.

4

Figure 2.2: Sagittal section of a lumbar vertebra. Adapted from [13]

Figure 2.3: Three main parts of a vertebrae: body (purple), vertebral arch (green), and processes formuscle attachment (brown).

Bone can be analysed at different scale levels, revealing different structures at each level, with dif-

ferent properties: macrostructure level, composed of cancellous and cortical bone; microstructure level,

formed by Haversian systems and single trabeculae; and lastly, nanostructure level that consists of

collagen and minerals [14]. These hierarchic structures are represented in Figure 2.4.

The macroscopic structure of bone consists of high density cortical bone (also called compact bone)

and low density trabecular bone (also called spongy or cancellous bone). Cortical bone is found in

the diaphysis of long bones and on the outer surface of flat bones. Its unit structure is called osteon

or Haversian system. In the center of each osteon there is the Haversian canal which contains blood

vessels to supply blood to the osteocytes. Trabecular bone is found in the inner layer of bones, mainly

in the ends of long bones and is composed of trabeculae, aligned along the load distribution [15].

At the microstructure level haversian systems and trabeculae can be identified. Haversian systems

form cortical bone and have a cylinder shape with a diameter of 200 to 250 µm and trabeculae form

5

Figure 2.4: Hierarchical structural organization of bone. Adapted from [14].

cancellous bone and are composed of trabecular rods which have a diameter of about 50–300 µm [14].

At the nanostructure level bone is a composite material, made of collagen fibers surrounded by

minerals, such as hydroxyapatite.

The mechanical properties of these structures define the unique mechanical properties of bone. Due

to the irregular, yet optimized arrangement of its components, bone is heterogeneous and anisotropic.

The anisotropy of bone means that its behavior changes depending on the direction of the load appli-

cation [16]. For example, in the femur, the elastic modulus is higher in the longitudinal direction (11-21

GPa) than in the transversal direction (5-13 Gpa), since the collagen fibres tend to be aligned with the

first one [14]. Bone is also a viscoelastic tissue, meaning that it has both viscous and elastic properties.

Because of this, bone will behave differently, depending on the speed and length of the applied load [16].

The mechanical properties of bone depend on three main characteristics: porosity, degree of miner-

alization and orientation of the collagen fibers and trabeculae [17]. These characteristics are influenced

by several factors, such as age, gender and location in the body and may also be altered by diseases.

With ageing, bone porosity tends to increase (meaning a smaller density), which results in a smaller

elastic modulus [18].

Depending on its direction, an applied load can produce different forces in the bone: compression,

tension, shear, torsion or bending, as depicted in Figure 2.5. Different bones support different forces,

which translates in a wide variability in strength between bones [16].

Bone injury can occur due to the application of a single load of high magnitude or the repeated appli-

cation of loads of low magnitude, called stress fracture. Considering the collagen fibers orientation, bone

injuries occur with lower magnitudes of shear stress than of compressive or tensile stress. Moreover,

bone can support higher compressive stress than tensile stress [19].

Overall, bone has a very complex structure which generates complicated mechanical properties,

very hard to mimic computationally.

6

Figure 2.5: Types of forces produced by different loads applied in a femur. Adapted from [16].

Concerning the histology of the bone, there are three types of cells: osteoblasts, osteoclasts and

osteocytes. Each of these cells has distinct functions and plays a different role in bone modelling and

remodelling processes [20, 21].

Osteoblasts come from osteoprogenitor cells and are bone forming cells – they are responsible for

synthesizing the organic components of the bone matrix, such as collagen type I and proteoglycans –

and are found in bone growing regions. They also synthesize the enzyme alkaline phosphatase, needed

for the mineralization of osteoid, the cover of bone matrix. Active and inactive osteoblasts have different

shapes: the active form has a columnar shape, whereas the inactive one has a flattened shape. When

bone formation reaches its final stage osteoblasts undergo apoptosis, developing into osteocytes or

bone-lining cells, which regulate ion flux in bone. [10]

When osteoblasts are surrounded by the osteoid matrix produced by themselves, they become os-

teocytes. They are the most abundant bone cells, representing approximately 95 % of all bone cells, and

are involved in the maintenance of bone matrix, by controlling the activity of osteoblasts and osteoclasts

within a basic multicellular unit (BMU). Each cell is in a lacuna that contact with adjacent cells via gap

junctions. The existence of empty lacunae suggests that osteocytes suffer apoptosis [22, 23].

Lastly, osteoclasts are multinucleated cells responsible for bone resorption and derive from mono-

cytes and macrophages. They have a much shorter life than osteoblasts, but their effect dominates over

that of osteoblasts because they absorb bone faster than osteoblasts synthesize it. The balance be-

tween the activity of these two cells is responsible for the ongoing modelling and remodelling processes

of the bone.

7

2.2 Bone remodelling

Bone modelling is the alteration of bone size and shape, related to bone growth. It consists of either

bone formation (osteoblast activity) or bone resorption (osteoclast activity), occurring independently at

different sites. On the other hand, bone remodelling is the result of sequential actions at the same site

of both bone-degrading osteoclasts and bone-forming osteoblasts, called the bone remodelling units. It

is crucial to maintain the integrity of the skeleton because it repairs the micro-damage caused by the

stress that bone is subjected in everyday activities and to regulate mineral homeostasis [24].

According to Wolff’s Law [1], bone will remodel in response to the loads that it is subjected. This

implies that if load increases in a particular region, the bone will remodel to become stronger and support

that load. The opposite also occurs: if the load decreases, the bone will remodel itself to become less

dense, given the lack of stimulus [25]. This last phenomenon may result in osteopenia, a frequent

problem when a prosthesis is implanted [26].

Chemical stimulus can also trigger bone remodelling, by means of hormones and growth factors, that

either activate or supress the bone remodelling units (e.g. fibroblast growth factor activates osteoclasts

and osteoblasts and insulin-like growth factors activate osteocytes) [27].

While the bone remodelling process is taking place, osteoblasts and osteoclasts are arranged within

anatomical structures, the basic multicellular units, delimited by cells. A BMU consists of osteoclasts

in front, followed by osteoblasts in the tail. Bone remodelling takes essentially five steps, illustrated in

Figure 2.6 [27]:

• Activation Phase - consists on the detection of the signal (mechanical or chemical);

• Resorption Phase - osteoclast precursors differentiate and osteoclasts resorb the bone, with the

creation of a “sealed zone”;

• Reversal Phase - preparation of the zone to osteoblast activity, by removal of the collagen matrix

covering the sealed zone;

• Formation Phase – bone formation by secretion of molecules that will form the bone (e.g. collagen

type I, proteoglycans and glycosylated proteins);

• Termination Phase – The conclusion of the remodelling cycle. Mature osteoblasts undergo apop-

tosis or differentiate into osteocytes.

The remodelling process is very complex and rely on the sequential and cooperative activity of os-

teoblasts and osteoclasts. Deregulation of this process may be connected to some diseases such as

osteoporosis and can bring serious problems [29].

8

Figure 2.6: Schematic representation of the bone remodelling process. Aadapted from [28].

2.3 Bone Overloading

According to Julius Wolff, high mechanical loads tend to promote bone formation and increase bone

density. However, it has been proved that, above a certain load, the damage accumulated in the bone

is higher than the recovery which will result in bone loss [2]. This way, in excessive load regions bone

degradation occurs, leading to bone density decrease, which is called bone overloading [8]. This phe-

nomenon can also alter the density distributions of the surrounding bone tissue which will impact the

whole bone. Moreover, excessive loads in an implanted bone may lead to large motions between pros-

thesis and bone, resulting in loosening of the implant [30].

The overload phenomenon is not yet well comprehended and is of crucial importance when analyzing

extraordinary load cases such as prostheses implants.

9

Chapter 3

Literature Review

The development of mathematical models that can explain bone remodelling due to mechanical stimuli

is of significant importance since they allow to study bone adaptation under different situations. This

chapter reviews some of the most significant contributions to this field.

Since Wolff’s observations that culminated in Wolff’s Law [1], around 1892, several attempts have

been made to describe the bone remodelling process. Most of these models combine finite element

models, to determine the internal mechanical environment, with mathematical descriptions of the bone

remodelling mechanism. However, neither the mechanical stimulus nor the communication process are

yet well defined.

Around 1960, Frost proposed a theory based on the existence of a mechanism that monitors the

bone mechanical usage and sends feedback to the bone cells so that mismatches between bone mass

and mechanical usage may be corrected [9]. This mechanism, which Frost called ”mechanostat”, is

composed by the bone itself, mechanisms that transform the mechanical usage into signals and another

mechanism that receives these signals and regulates bone mass. This regulation can occur by three

processes: longitudinal growth, bone modelling and bone mechanical usage based remodelling. Lon-

gitudinal growth adds new bone to the pre-existing one and it is promoted by high mechanical usage.

On the other hand, low mechanical usage decreases longitudinal growth. In high strains (> 1500-3000

microstrains), bone modelling occurs connected to bone mass increase, whilst in low strains (< 100-300

microstrains), it is connected to bone mass decrease. The proposed theory also takes into consid-

eration non-mechanical mechanisms that control bone model and remodel, called MES mechanisms.

These mechanisms regulate the signal transduction between bone’s mechanical usage and its cells.

They are influenced by four types of agents: hormones and nutrition, biochemical messengers, genetic

disorders and toxic agents. The architecture of this hypothesis can be seen in Figure 3.1.

In 1976, Cowin et al. [31] formulated a model, controlled by strain, to understand the bone remod-

elling process. According to their work, prolonged straining of a bone tends to make it denser, whereas

a bone subjected to less strain than the normal level will become less dense. In the model developed,

the bone matrix was considered to be a porous elastic solid containing a fluid and the bone remodelling

due to strain is represented by a chemical reaction which transfers mass, momentum, entropy and en-

10

Figure 3.1: Schematic representation of Frost Mechanostat Theory. Adapted from [9]

ergy to and from the bone matrix, altering its porosity. The authors also state that if the strain exceeds

a certain critical level, the bone will resorb and lose density to which they called “over-strain necrosis”.

This process is progressive, which means that it will propagate to adjacent bone tissues which may lead

to failure of the prosthesis.

In 1987, Huiskes et al. developed a bone remodelling model, considering the strain energy per unit

of bone mass, Uρ , the driving stimulus for bone remodelling [32]. Using the summation convention, the

strain energy density, U, is given by:

U =1

2σijεij (3.1)

where σij and εij are the components of the stress and strain tensors, respectively. The bone remod-

elling law is expressed as:

∂ρ

∂t=

B(

U

ρ− (1− S)k), if

U

ρ< (1− S)k

B(U

ρ− (1 + S)k), if

U

ρ> (1 + S)k

0, otherwise

(3.2)

where ρ is the density, B, s and k are constants that need to be determined.

If multiple forces are considered, U is substituted by Ua, an average of strain energy density calcu-

lated for the loads applied. Figure 3.2 illustrates the change in the bone density rate as a function of

the mechanical stimulus Uρ . The parameter k is a reference value for the threshold of the stimulus and s

is represented as half of the baseline length. As a result of the combination of the mathematical model

with a finite element model, the mechanical stimulus can be expressed per element, representing the

stimulus in each element of the bone. The density is also updated in each element. This is an iterative

model and it is considered to converge when the rate of density change reaches zero or the density

reaches its pre-set minimal or maximal values. The model was tested on a proximal femur, yielding

results consistent with real observations.

11

Figure 3.2: Change in the bone density rate as a function of the mechanical stimulus for the boneremodelling model of Huiskes et al. [32].

In 1990, Beaupre et al. presented a bone modelling and remodelling theory in response to the daily

loading history [33, 34]. The theory is based on the fact that bone needs an optimal level of mechanical

stimulus, that defines the dead or lazy zone, to maintain its density. If the stimulus is above that level,

bone density increases, if it is below that level, bone density decreases. In their theory, Beaupre et al.

defined the driving force as the difference between the tissue level stimulus and the tissue level attractor

state, which is the level at which bone remodelling is practically zero i.e., the stimulus corresponding to

the lazy zone. This tissue attractor state varies depending on genotype, adjacent tissue interaction and

metabolic state, i.e., it varies between people and between bones of the same person. The driving force

changes the activity of osteoblasts and osteoclasts, which then changes the model and remodelling

processes. Figure 3.3 illustrates the relationship between the activity rate of the bone cells and the

tissue stress stimulus, which is then used to update the bone geometry and density.

Figure 3.3: Piecewise linear relationship between the stress stimulus and the bone apposition/resorptionrate, according to the model of Beaupre et al. [34].

In 2004, Crupi et al. [7] developed an adaptation routine based on the theory of Beaupre et al. [34]. By

clinical observations it was proved that there is resorption of bone around an oral implant for high loads,

even if they are at the right-hand side of the dead zone, which was not included in the original model

of Beaupre et al. Accordingly, the model was modified to introduce bone resorption due to overload

i.e., a certain limit was implemented, above which bone density would decrease instead of increasing.

Although this limit is not well known, it was set to 35 MPa, using Taylor’s crack propagation theory to

calculate it [35]. It was determined as being equal to the stress value separating stable and unstable

propagation of cracks because, according to Taylor’s theory, the adaptation process could be initiated

12

by the propagation of cracks. Similarly to Beaupre’s theory, this is a time dependent model and an

iterative process. As shown in Figure 3.4, the new bone properties and geometry in each iteration serve

as input for the finite element method for the next iteration. The model converges when the difference

between two consecutive bone density distributions is lower than a predefined tolerance. This work

found resorption of bone around an oral implant that could lead to failure, which is observed clinically.

Moreover, in cases of implants fixed in the lower mandible, where the difference between the elastic

modulus of cortical bone and trabecular bone is high, the overload process is more prominent and

resorption occurs.

Figure 3.4: Flowchart of the bone remodelling model developed by Crupi et al. Adapted from [7].

Following the work of Crupi et al. [7], Li et al. [8] incorporated bone resorption due to overload into

the bone remodelling model proposed by Huiskes et al. [32]. They conducted the study in dental implant

treatments. Considering the original bone remodelling law expressed as:

dρ

dt= B(

U

ρ− k) (3.3)

the authors added a quadratic term to simulate the decrease in bone density under high loads. Mathe-

13

matically, the bone remodelling law of Li et al. is given by:

dρ

dt= B(

U

ρ− k)−D(

U

ρ− k)2 (3.4)

where D is a constant. For low loads, the model resembles the model of Huiskes et al. However, for high

loads, the quadratic term is dominant and the rate of density change becomes negative. The two zeros

of Equation 3.4 are called critical loads and divide the model into three regions: underload, growth and

overload. The bone density decreases in the underload and overload regions whereas it increases in

the growth region. Figure 3.5 illustrates the difference between the bone remodelling laws of equations

3.3 and 3.4. This model was tested in a dental implant and the notable increase of density of the bone

deeper into the mandible was supported by medical observations.

Figure 3.5: Comparison between Huiskes et al. (dotted line) and Li et al. bone remodelling models (solidline). Adapted from [8].

Several bone remodelling models have been developed in the last decades, using different mechan-

ical stimulus. However, few of them consider bone resorption due to overload. This is a phenomenon

that needs more study to be fully comprehended and assimilated in the new models developed.

14

Chapter 4

Computational model for bone

remodelling

In this work, two mathematical models are applied to replicate the bone remodelling process, including

bone resorption due to overload. The main difference between the models is the mechanical stimulus

considered. In parallel to the mathematical model, finite element analyses are performed, in the soft-

ware ABAQUS, to assess the mechanical environment of bone. Iterative procedures, based on Euler and

Runge-Kutta methods, are adopted to solve the differential equations of the bone remodelling models.

In each iteration, the mechanical stimulus obtained in the finite element analysis is used in the mathe-

matical model. Then, the density distribution obtained in the mathematical model is used in ABAQUS for

the finite elements analysis. The iterations are repeated until a certain convergence criteria is satisfied.

This chapter presents the computational framework for the simulation of the bone remodelling process

including overload.

4.1 Finite element method

The finite element method is a computational technique very commonly used in engineering to obtain

approximate solutions of boundary value problems [36]. It divides the domain of interest, also called field,

in smaller domains, elements, that are connected at nodes. The variables of interest, field variables, are

explicitly calculated at the nodes and interpolated in the interior of the elements. Considering an example

of a two dimensional case with a field variable, φ(x, y), and a triangular element, the field variable is given

by:

φ(x, y) = u1(x, y) · φ1 + u2(x, y) · φ2 + u3(x, y) · φ3 (4.1)

where u1, u2 and u3 are the interpolation or shape functions and φ1, φ2 and φ3 are the values of the field

variable at the nodes of the element.

A finite element analysis normally consists of three main steps: Preprocessing, in which the geometry

15

of the problem is defined, such as, element types, material properties, mesh and boundary conditions,

Solution, in which the primary field variables are calculated and then used to calculate other derived

variables and Post-processing, in which the results are sorted and printed [37].

4.2 Mathematical Models for bone remodelling

The mathematical formulation of the bone remodelling models applied in this work, along with the nu-

merical methods used, are presented next.

4.2.1 Strain energy density model

This model was developed by Li et al. [8] and uses strain energy density as the mechanical stimulus.

The differential equation that rules the model is expressed by:

dρ

dt= B(

U

ρ− k)−D(

U

ρ− k)2 (4.2)

where B, D and k are constants, with the following units: [k] = J · g−1, [B] = (g · cm−3)2(MPa ·

timeunit)−1 and [D] = (g · cm−3)3MPa−2(timeunit)−1. U is the strain energy density and ρ is the bone

density which is limited between 0.01 g · cm−3 and 1.74 g · cm−3. The second order term, in Equation

4.2 adds the overload phenomenon to the model, in high load regions. The constant k is the threshold

value for the stimulus, i.e., the strain energy density above which the density increases and below which

it decreases (without taking in consideration the overload region), whereas BD +k is the threshold for the

overload region, i.e., above this threshold overload resorption occurs and bone density decreases. It is

easy to conclude that for higher values of D, overload will occur for lower values of strain energy density.

Figure 4.1 illustrates the change in bone density rate expressed in Equation 4.2.

Figure 4.1: Bone density rate as a function of the mechanical stimulus Uρ according to the bone remod-

elling model developed by Li et al. [8].

16

4.2.2 Strain-based model

Based on the work of Wiskott et al. [38], a novel model was developed using strain as the mechanical

stimulus. According to the authors, five strain regions cause different bone density changes:

• strain < 100µε - Strains in this range cause bone resorption due to disuse;

• 100µε < strain < 2000µε - Strains in this range do not change bone density;

• 2000µε < strain < 4000µε - Strains in this range cause bone formation due to stimulation;

• 4000µε < strain < 20000µε - Strains in this range cause pathological overload due to excessive

load i.e., bone density decreases;

• strain > 20000µε - Strains in this range cause bone fracture.

Considering all these regions, the following bone remodelling law was defined:

dρ

dt=

B(ψ − ε1), if ε < ε1

0, if ε1 < ε < ε2

B(ψ − ε2), if ε2 < ε < ε3

B(ε4 − ψ), if ε3 < ε < ε5

Fracture, if ε > ε5

(4.3)

where ψ is the stimulus considered and εi, with 0 < i < 6, are the strains that delimit the different steps

of the model, as it can be seen in Figure 4.2. B is a constant.

The bone density change rate as a function of strain (representation of the model) is presented in

Figure 4.2.

Figure 4.2: Bone density rate as a function of the mechanical stimulus ψ(ε) according to the boneremodelling model developed in this work.

Considering strain as the driving force, two different stimuli are considered:

ψ1 = max(|εmax|, |εmed|, |εmin|) (4.4.a)

ψ2 = |εmax|+ |εmed|+ |εmin| (4.4.b)

17

where εmax is the maximum principal strain, εmed is the medium principal strain and εmin is the minimum

principal strain. In the two-dimensional case εmed=0 These two stimulus are chosen based on the work

developed by Ruimerman [39].

4.3 Numerical methods

To solve the differential equations that describe the bone remodelling process, two numerical methods

are used: the Euler method and a second order Runge-Kutta method. A description of these methods

is provided next.

4.3.1 Euler Method

The Euler method is the simplest to approximate the result of a differential equation, expressed by:

y = f(x, y) (4.4)

Considering the interval xn = n · h, divided into n parts of equal length h and yn = y(xn), a Taylor

series expansion of yn+1 results in:

yn+1 = y(xn + h)

= y(xn) + h · y(xn) +O(h2)

= y(xn) + h · f(xn, yn) +O(h2)

(4.5)

This way, the Euler method is represented by [40]:

yn+1 = y(xn) + h · f(xn, yn) (4.6)

This is a first order method and it advances in time from the initial value with a time step h.

4.3.2 Second order Runge-Kutta

Second order Runge-Kutta denotes a family of methods of second order of convergence [41].

Expanding yn in Taylor Series, the first two terms are given by:

yn+1 = yn + h · f(xn, yn) + h2 · 1

2!f ′(xn, yn) (4.7)

Runge and Kutta defined this equation as follows:

yn+1 = yn + h · (a1 · k1 + a2 · k2) (4.8)

where k1 = f(xn, yn) and k2 = f(xn + h · p1, yn + h · q11 · k1)

18

Assuming an arbitrary value for one of the constants, normally a2, the other three can be calculated

and different second order Runge-Kutta methods can be defined. In this work, the Midpoint method is

used, for which the constants hold the following values: a2 = 1, a1 = 0, p1 = 12 , q11 = 1

2 . The method is

given by:

yn+1 = yn + h · f(xn +1

2· h, yn +

1

2· h · f(xn, yn)) (4.9)

4.3.3 Numerical stability

When trying to solve mathematical models such as the ones studied in this work, numerical instabili-

ties may appear related to the convergence of the solutions. Three main types of instabilities can be

identified [42]:

• Checkerboard problem, related with the formation of regions of alternating high and low density

elements, ordered in a checkerboard pattern [43];

• Mesh dependence problem which consists on obtaining different solutions for the same problem

when using different meshes;

• Local minima, associated with obtaining different solutions for the same problem when using dif-

ferent algorithms;

Different methods are used in the present work to avoid these instabilities. Both the Euler method

and a Runge-Kutta method are used to solve the mathematical models and the solutions are compared.

To prevent mesh dependence problems and checkerboard problems, mesh independence filtering, pro-

posed by Sigmund [42], is used. It is a method that changes the sensitivity of the elements based on

an average of the surrounding elements. This way, the value of each element depends on the value of

other elements and a smoothing effect is obtained. The element sensitivity is modified as follows:

∂f

∂ρk= ρ−1

k

1∑Ni=1 Hi

N∑i=1

Hiρi∂f

∂ρi(4.10)

where ∂f∂ρk

is the filtered sensitivity used, Hi is the weight factor with Hi = rmin − dist < k, i >, rmin is

the radius chosen which delimits the filter area and dist < k, i > is the distance between the element

being smoothed, k, and the element i in the filtered area. If the element i is outside the radius chosen,

Hi is zero and the element will not take part in the smoothing process. It is also possible to notice that

the sensitivity, Hi, decays linearly with the distance between the elements being processed. This filter

does not increase the computational work since it requires little processing time.

19

4.3.4 Step limitation

To improve the convergence speed and avoid inconsistencies in the iterative process used to solve the

mathematical bone remodeling models, the variation of bone density in each iteration is limited [44],

according to Equation 4.11

(ρi)k+1 =

max[(1− ξ)(ρi)k, ρmin] if (ρi)k + step ·Dk ≤ max[(1− ξ)(ρi)k, ρmin]

min[(1 + ξ)(ρi)k, ρmax] if (ρi)k + step ·Dk ≥ min[(1 + ξ)(ρi)k, ρmax]

(ρi)k + step ·Dk, otherwise

(4.11)

Where (ρi)k is the value of bone density in the element i in iteration k, ρmin and ρmax are the minimum

and maximum density allowed in the mathematical model, 0.01 g ·cm3 and 1.74 g ·cm3, respectively. The

parameter ξ (0 < ξ < 1) define the superior and inferior limits of the variation, so that sharp variations

are avoided. In this work, a value of 0.5 is used for this parameter. Dk represents the direction of the

variation. The step is chosen by the user and keeps constant throughout the simulation.

This method is represented in Figure 4.3.

Figure 4.3: Method used to limit the density variation.

4.4 Mathematical formulation

The mathematical formulations of the models studied, using the Euler and Runge-Kutta methods are

presented in this section.

4.4.1 Strain energy density model

From equations 4.2 and 4.3, y corresponds to ρ and f(x, y) corresponds to B( Uρt − k) − D( Uρt − k)2.

Hence the Euler method formulation yields:

ρt+∆t = ρt + ∆t(B(Utρt− k)−D(

Utρt− k)2) (4.12)

The implementation of the second order Runge-Kutta method is more complex. First, an intermediate

step is performed to compute a new density, ρm, given by:

20

ρm = ρt +∆t

2(B(

Utρt− k)−D(

Utρt− k)2) (4.13)

Then, ρm is used to calculate ρt+∆t as follows:

ρt+∆t = ρt + ∆t(B(Umρm− k)−D(

Umρm− k)2) (4.14)

Because the second order Runge-Kutta method requires an additional density evaluation compared

to the Euler method, which implies an additional finite element analysis, it is more time consuming.

4.4.2 Strain-based model

Similarly to the strain energy density model, the Euler method formulation for the strain-based model is

expressed as:

ρt+∆t =

ρt + ∆t ·B(ψt − ε1), if ε < ε1

0, if ε1 < ε < ε2

ρt + ∆t ·B(ψt − ε2), if ε2 < ε < ε3

ρt + ∆t ·B(ε4 − ψt), if ε3 < ε < ε5


(4.15)

The Runge-Kutta formulation requires an intermediate step to calculate ρm. Then, the density ρm, is

used to calculate the density ρt+∆t as follows:

ρt+∆t =

ρt + ∆t ·B(ψm − ε1), if ε < ε1

0, if ε1 < ε < ε2

ρt + ∆t ·B(ψm − ε2), if ε2 < ε < ε3

ρt + ∆t ·B(ε4 − ψm), if ε3 < ε < ε5


(4.16)

4.5 Application cases

To obtain the mechanical stimuli used in the mathematical models and study the changes in bone,

finite element analyses are performed, both in 2D and 3D. Using the software ABAQUS, different finite

element models are developed considering geometric models of two vertebrae (an approximation of two

vertebrae) and of a femur.

For each model, healthy and prosthetic conditions are modelled, in which bone is intact or is im-

planted, respectively. The intact bone models are used to tune the bone remodelling parameters, by

applying the mathematical models without taking the overload phenomenon in consideration and ob-

taining the bone density distribution closest to the biological distribution. The models with prostheses

21

are analyzed using the tuned parameters and the density distribution obtained for the intact bone as

initial condition.

In all simulations performed, bone is considered isotropic. Poisson’s ratio, ν, is defined as 0.3 and

the Young’s modulus, E, is given by the following relationship between E and the bone density, ρ, [45]:

E = 3790× ρ3 (4.17)

where ρ, in g · cm−3, varies between 0.01 and 1.74 and E is given in MPa.

Because the prostheses are more rigid than the bone they are implanted in, they are chosen to be

the master surface. Consequently, the bone is the slave surface.

4.5.1 Two-dimensional models

Two-dimensional finite element models have some limitations compared to three-dimensional models,

concerning the study of the biomechanical behavior of complex structures [46]. Nonetheless, they are

extensively used and seen as a good alternative to three-dimensional models [47, 48].

Vertebrae Model

The intact model, presented in Figure 4.4a, is composed of three parts which represent the two vertebrae

and the intervertebral disc. The interaction between these parts is defined using a ”Tie” constraint, which

bonds the two parts. The model has 1092 four-node elements (CPS4).

The model of the implanted bone, presented in Figure4.4b has four parts: two vertebrae, the inter-

vertebral disc and the prosthesis. The prosthesis is attached to both vertebrae with a ”Tie” constraint.

This model has 12466 four-node elements (CPS4). The larger number of elements in the implanted

model is due to a more refined mesh, needed to be able to capture the rapid changes in the solution

around the prosthesis.

In both models, the node at the lower right corner is fixed in all directions. The intervertebral disc

has Poisson’s ratio of 0.3 and Young’s Modulus of 100 MPa, material properties equivalent to polyethy-

lene [49] and the prosthesis is considered to be made of steel (Poisson’s ratio of 0.3 and Young’s Mod-

ulus of 200 GPa [50]) . Both the bone and the prosthesis have 20 mm of thickness. Two symmetrical

forces, F1 and F2, are applied in two reference points (RP-1 and RP-2) that are coupled to the top and

bottom elements of the models, to simulate a distributed load along the vertebrae.

Femur Model

The model of the intact femur, presented in Figure 4.5.a, is composed of two parts: the femur and a side

plate. The side plate, shown in Figure 4.5.c, has three sections of thickness 1 mm, 3 mm and 5 mm

(from top to bottom) and it is considered to be made of cortical bone, with Young’s Modulus of 17 GPa

and Poisson’s ratio of 0.3 [51]. It is connected to the bone through a ”Tie” constraint in its outer region to

22

(a)(b)

Figure 4.4: 2D model of the vertebrae: (a) intact bone and (b) implanted bone.

simulate the three-dimensional connectivity of the cortex [52]. The femur has thickness of 40 mm and

3075 four-node elements.

The model of the implanted bone, shown in Figure 4.5.b, has three parts: the femur, the side plate

and the prosthesis. The prosthesis is bonded to the femur by a ”Tie” constraint and it is made of cobalt-

chromium alloy, CoCr, with Poisson’s ratio equal to 0.3 and Young’s Modulus equal to 230 GPa [53]. The

cut femur has 18525 four-node elements, to be able to capture the rapid changes in the solution.

In both models the bottom nodes are fixed in all directions. Two forces are applied (Fa and Fh) in

two reference points, RP-1 (Fa) and RP-2 (Fh), common to both models. In the intact model, the points

RP-1 and RP-2 are connected to the femur and in the implanted femur are connected to the upper left

region of the femur and to the head of the implant, respectively. The forces used, presented in Table 4.1,

simulate the forces during the gait cycle [44].

4.5.2 Three-dimensional models

To complement the two-dimensional finite element analyses, three-dimensional finite element models

are developed.

23

Figure 4.5: 2D model of the femur: (a) intact bone, (b) implanted bone and (c) the side plate used, withthickness of 5mm, 3mm and 1mm from top to bottom.

Table 4.1: Forces used in the 2D femur model.

Force Fx Fy

Fa 768 1210

Fh -224 -2246

Femur Model

A 3D model of a left femur with a resurfacing prosthesis is analyzed. This type of prosthesis is used

in patients with hip pathology, leading to osteoarthritis. The pathological bone in the femur’s head is

removed and it is shaped to receive the prosthesis, which is fixed to the bone [54]. The CAD model of

the intact femur was downloaded from GrabCad [55] and the prosthesis was designed in SolidWorks.

Then, both models were imported to the software ABAQUS and the prosthesis was implanted in the

femur model. The model of the intact femur, shown in Figure 4.6.a, has 28759 tetrahedral elements.

The model of the femur with the prosthesis, shown in Figure 4.6.b. is composed of two parts: the femur

and the prosthesis. The prosthesis, presented in Figure 4.6.c, is made of CoCr with Poisson’s ratio of

0.3 and Young’s modulus of 230 GPa [53] and is bonded to the bone with a ”Tie” constraint. The cut

femur has 30879 quadratic tetrahedral elements.

24

Similarly to the two-dimensional model, two forces, (Fa and Fh), are applied in two reference points,

RP-1 (Fa) and RP-2 (Fh). In the intact model, the points RP-1 and RP-2 are connected to the femur

and in the implanted femur are connected to the upper left region of the femur and to the prosthesis,

respectively. The forces have the same magnitude and are applied in the same position as in the

two-dimensional model. However, in 3D the z component of forces is no longer neglected. Moreover,

because in this case a left femur is modelled and in the 2D model a right one is studied, a 180o rotation

(along the traditional yy axis) is applied to the forces, which are represented in Table 4.2. In both models,

the nodes at the bottom surface are constrained in all directions.

Table 4.2: Forces used in the 3D femur model.

Force Fx Fy Fz

Fa -768 1210 -726

Fh 224 -2246 972

Figure 4.6: 3D model of the femur: (a) intact bone, (b) implanted bone and (c) the prosthesis used.

25

Chapter 5

Results

In this chapter, the results of the finite element analyses performed using the mathematical remodelling

models, described in Chapter 4, are presented. First, both remodelling models are compared using a

two-dimensional finite element model of an intact femur. Then, a simplified model of two vertebrae is

studied, followed by two models of a femur with a prosthesis (two and three-dimensional). Finally, a

study is done to determine some parameters that influence the solution of both remodelling models.

5.1 Baseline comparison of the models

First of all, the mathematical models are compared, without overload resorption, using the 2D femur

model. Several simulations were performed for different bone remodelling parameters to simulate the

bone density distribution of a real femur. Values between 1×10−3 and 1×10−4 were tested for parameter

k, considering tests done in previous works. Simulations were done with initial density of 1, 1.5 and 1.74

g · cm−3 and several tests were done with step between 1 and 500. The values of εi were chosen

according to the work of Wiskott et al. [38], except for ε1. This parameter was chosen by testing different

values because the original one did not yield acceptable results.

The parameters that yielded the best results are shown in tables 5.1 and 5.2 and the density distri-

butions obtained using both mathematical models and these parameters are presented in Figure 5.1.

Table 5.1: Parameters used in the strain energy density model for the 2D femur model.

ρi 1.5 g · cm−3

k 5× 10−4

B 1

D 0

step 100

iterations 75

27

Table 5.2: Parameters used in the strain-based model for the 2D femur model.

ρi 1.5 g · cm−3

ε1 4× 10−4

ε2 2× 10−3

ε3 3× 10−3

ε4 4× 10−3

ε5 2× 10−2

step 100

iterations 75

In the results obtained using the strain energy density model there is a low density column in the

centre of the body of the femur (the medullary canal), surrounded by high density bone. The femur’s

head, mainly composed of medium density bone, has a low density region that is called the Ward’s

triangle. In the strain-based model, a medium density column is present in the body of the femur,

surrounded by high density regions while the head of the femur has medium density bone. Considering

that both models based on strain have similar results, only the results obtained for ψ2 = |εmax|+ |εmed|+

|εmin| are presented and analyzed throughout this work. Comparing the strain-based models with the

strain energy density model, the latter shows lower density regions both in the medullary canal and head

of the femur.

Figure 5.1: Density distribution obtained in the 2D model of the femur: (a) strain energy density model,(b) Strain-based model with ψ1, (c) Strain-based model with ψ2 .

28

5.2 Vertebrae model

For further analysis, the 2D model of two vertebrae, described in Chapter 4, is used.

To begin, the strain-based model is tested, using the model of the intact bone. The force applied, F1,

is adjusted so that the final density obtained is 0.3 g · cm−3. This value is chosen because it is close to

the density of a real vertebra [56] and it has been used in previous works [57]. The force found to have

this impact in the bone is F1 = -125 N (and consequently F2 = 125 N). In Figure 5.2, the evolution of

density and strain in the intact vertebrae, using these forces and initial density equal to 0.25 g · cm−3,

is shown. In this case the density along the vertebrae is uniform and the results obtained for ψ1 and ψ2

are similar.

Figure 5.2: Evolution of the bone remodelling simulation for the vertebrae model using the strain-basedlaw: (a) strain, (b) bone density.

Using the -125 N force, the model of the vertebrae with a prosthesis is analyzed, so that bone

overloading can be studied. The mathematical model is tested without overload, i.e, the last branch of

the bone remodelling law is removed, which means that for high strains bone density increases, instead

of decreasing, and with overload. The density distributions obtained in the vertebrae are presented in

Figure 5.3.

Both models, with and without overload, yielded similar results, differing in the zone of contact be-

tween bone and prosthesis. There is low density bone throughout the vertebrae. Two vertical columns

of higher density form in the middle of the vertebrae, starting in the interface region between bone and

prosthesis.

In Figure 5.4, a zoomed view of the density distribution near the region of contact between the upper

vertebra and the prosthesis is presented. In the lower vertebra, the result is similar.

In the results obtained using the model without overload, there is a peak of high densities in the most

interior region of contact between bone and the prosthesis surrounded by medium density bone, while

the rest of the vertebra is made of low density bone. In the model with overload, in the region of contact

aforementioned there is low density bone surrounded by medium density bone. The rest of the bone is

composed of low density bone. There is a clear difference between the model with and without overload

in the region of contact between bone and the prosthesis vertex: the peaks of high density in the model

without overload, are of low density in the model with overload.

29

Figure 5.3: Bone density distribution in the vertebrae with prosthesis obtained using the strain-basedmodel: (a) without overload, (b) with overload

Figure 5.4: Zoom of the bone density distribution to highlight bone overloading. The solutions areobtained for the models: (a) without overload, (b) with overload.

To test the strain energy density model, the same forces (F1 = -125 N and F2 = 125 N) are used, so

that comparisons between both models can be done. The model with intact bone is first tested without

overload in the mathematical model (D = 0) to adjust the parameter k in order to obtain the density

distribution obtained for the strain-based model. The value of k that produces this result is 3.5× 10−4.

Using the parameteres shown in Table 5.3, the model of the vertebrae with a prosthesis implanted is

analyzed. Once more, the mathematical model is tested without overload (with parameter D=0) and with

overload. To simulate overload, several simulations with increasing D are performed. The parameter D

that yielded the best results was chosen (D=660). The results obtained are presented in Figure 5.5.

In both models it is possible to observe a higher density column in the center of the bone while the

30

Table 5.3: Parameters used to simulate overload phenomenon in the vertebrae with a prosthesis im-planted using the strain energy density model.

ρi 0.3 g · cm−3

k 3.5× 10−4

B 1

D 660

step 100

iterations 100

Figure 5.5: Bone density distribution in the vertebrae with prosthesis obtained using the strain energydensity model: (a) without overload, (b) with overload

rest of the vertebrae is composed of low density bone. In the model without overload a high density

zone is present in the most interior contact region between bone the prosthesis, while in the model with

overload the density in this area is reduced.

31

5.3 Femur model

In this section the models of the femur presented in Chapter 4 are studied. Firstly, the two-dimensional

model, with different types of prostheses implanted, is tested and then the three-dimensional model with

the head prosthesis is addressed.

5.3.1 Two-dimensional model

The initial density used in the simulations in the following subsections is the one presented in Figure 5.1,

that best mimics the real distribution in a healthy femur.

Apart from the prosthesis presented in Chapter 4, denominated as standard prosthesis, another

prosthesis, with larger stem, called large prosthesis, will be tested. Considering the initial density, the

large prosthesis is inserted in high density bone whereas the standard prosthesis is inserted in lower

density bone. This way, the overloading process can also be studied considering the density of the

surrounding bone.

5.3.2 Standard Prosthesis

In the standard model, the prosthesis is inserted in low density bone of approximately 1 g · cm−3. The

mathematical model is tested with and without overload. The density distribution obtained using the

strain-based model can be seen in Figure 5.6.

The results obtained are similar for both models. The outer region of the diaphysis is composed by

high density bone, while the inner region, that is in contact with the prosthesis, is made of medium and

low density bone. In the region of transition between methaphysis and diaphysis (metadiaphysis) there

are regions of low density bone surrounding the prosthesis. The epiphysis is made of medium density

bone with a region of low density. The major differences between both models are located in the region

of the diaphysis in contact with the prosthesis: in the model with overload, there is a thin layer of low

density bone in this region whereas, in the model without overload, there is medium density bone. The

different bone density in this region alters the surrounding regions.

The results obtained using the standard prosthesis and strain energy density model are presented in

Figure 5.7. The results with overload are obtained using D = 60.

In both models, high density bone is present in the the outer region of the diaphysis, while the

metaphysis and epiphysis are mostly made of low density bone. In the model without overload, the

region of the diaphysis connecting with the prosthesis is composed by medium density bone. In the

model with overload, there are some parts of low density bone present in this region.

5.3.3 Large prosthesis

To test the difference between both prosthesis, similar simulations are done for the large prosthesis. In

this model, the prosthesis is implanted in bone with high density of 1.74 g · cm−3.

The density distributions using the strain-based model are presented in Figure 5.8.

32

Figure 5.6: Bone density distribution in the femur with the standard prosthesis obtained using the strain-based model: (a) without overload, (b) with overload.

Figure 5.7: Bone density distribution in the femur with the standard prosthesis obtained using the strainenergy density model: (a) without overload, (b) with overload.

The results obtained with and without overload are very similar. There is high density bone in the

outer region of the lower part of the diaphysis of the femur, while in the middle and upper part of the

diaphysis, there is low density bone. The metaphysis and epiphysis are made of medium density bone

with a region of low density.

33

Figure 5.8: Bone density distribution in the femur with the large prosthesis obtained using the strain-based model: (a) without overload, (b) with overload.

The results obtained using the strain energy density model, with D=0 and D=60, are presented in

Figure 5.9.

These results are very similar to what was obtained using the strain-based model. The differences

are mostly located in the epiphysis, where there is more low density bone when using the strain energy

density model. In this model the results obtained with and without overload in the mathematical model

are also very similar.

Figure 5.9: Bone density distribution in the femur with the large prosthesis obtained using the strainenergy density model: (a) without overload, (b) with overload (D=60).

34

5.3.4 Three-dimensional model

In the following subsection the three-dimensional model of a femur with a resurfacing prosthesis is

studied, to complement the results already presented. The intact femur is first analyzed using both

mathematical models without overload, so that a proper density distribution can be obtained. The results

using the strain based model are presented in Figure 5.10.

Figure 5.10: Bone density distribution in the 3D femur obtained using the strain based model: (a) viewof a coronal cut, (b) medial view, (c) view of a transversal cut.

The diaphysis of the femur is composed of medium density bone in the inner region, with the center

made of low density bone - the medullary canal. The outer region of the diaphysis has high density

bone. There is a low density region in the neck of the femur - the Ward’s Triangle - and in the lower

region of the articular surface. The rest of the femur is composed of medium density bone.

The results obtained using the strain energy density model are presented in Figure 5.11 . The bone

density distribution obtained is similar to that of the strain based model, but the medullary canal is more

prominent, presenting a lower density. Also, the femur’s head has a higher density bone.

Using the density distributions presented in Figure 5.10 and Figure 5.11 as initial conditions, the

mathematical models are tested with and without overload, using the finite element model of a femur

with a resurfacing prosthesis. The results obtained using the strain based model and the strain energy

density models without and with overload are shown in Figure 5.12 and Figure 5.13, respectively. The

differences worth noting between these results and the ones obtained using the intact femur model

are located in the head of the femur, where the prosthesis is implanted. With the bone remodelling

model without overload resorption, the density distribution is close to the one obtained in the intact

femur, whereas with the model considering overload resorption the density in the head of the femur is

decreased, showing bone resorption in these sites.

35

Figure 5.11: Bone density distribution in the 3D femur obtained using the strain energy density model:(a) view of a coronal cut, (b) medial view, (c) view of a transversal cut.

Figure 5.12: Bone density distribution in the 3D femur with a resurfacing prosthesis obtained using thestrain based model without overload (a) view of a coronal cut, (b) medial view and with overload (c) viewof a coronal cut, (d) medial view.

36

Figure 5.13: Bone density distribution in the 3D femur with a resurfacing prosthesis obtained using thestrain energy density model without overload (a) view of a coronal cut, (b) medial view and with overload(c) view of a coronal cut, (d) medial view.

5.4 Sensitivity of the bone remodelling models

The mathematical models studied have parameters that need to be adjusted, which is not trivial. Be-

cause of the uncertainty present in their definition, further tests are made to evaluate the sensitivity of

the bone remodelling models to different parameters and conditions.

5.4.1 Mesh

Several simulations are performed, using meshes with different element sizes, to understand the interfer-

ence that mesh refinement has in the results. This is an important study because mesh size changes the

computational time of the finite element analyses, so simulations with less refined meshes are favored

as long as yielding good results.

Using the meshes described in Table 5.4, the model of the vertebrae with a prosthesis is applied

using the strain-based model. Because minimum principal strain is the one with the highest value in this

simulation and thus having a bigger contribution to ψ2, its distributions are presented in Figure 5.14

Table 5.4: Parameters used to study the influence of mesh size in the simulations.

Simulation Element size Number of nodes

1 2 957

2 0.5 12000

37

Figure 5.14: Minimum principal strain distribution in the region of the vertebra in contact with the prosthe-sis obtained using the strain-based model and different mesh: (a) mesh with 957 elements (simulation1), (b) mesh with 12000 elements (simulation 2).

In the model with the less refined mesh, the minimum principal strain around the implant oscillates

between values of around 1×10−3 and 2×10−3. These values are bellow the value of strain that causes

overload resorption. In the model with the more refined mesh, the minimum principal strain distribution

is similar to the one mentioned before, except in the most interior region of the bone in contact with the

prosthesis. In this area a peak of high minimal principal strain of around 4 × 10−3 is obtained, which

causes overload resorption in the bone. There is a clear difference between both results: the more

refined mesh is able to detect the peaks of high minimal principal strain, while the model with the less

refined mesh is not.

Figure 5.15 presents the density distribution obtained using the strain-based model and the meshes

described in Table 5.4. In the model with the less refined mesh, the density distribution around the

implant is uniform around the value of 0.4 g · cm−3. In the model with the more refined mesh, the density

distribution is uniform along the bone, except in the most interior region of the bone in contact with the

prosthesis. There is a region of low density (with the minimum allowed density) followed by a region of

higher density, of around 0.8 g · cm−3. There is a clear difference between both results in the region of

the bone in contact with the prosthesis.

Figure 5.15: Strain energy density distribution in the region of the vertebra in contact with the pros-thesis obtained using the strain energy density model and different mesh: (a) mesh with 957 elements(simulation 1), (b) mesh with 12000 elements (simulation 2).

38

Figure 5.16 presents the distributions of strain energy density obtained using the strain energy den-

sity model and the meshes presented in Table 5.4.

Figure 5.16: Density distribution in the region of the vertebra in contact with the prosthesis obtainedusing the strain-based model and different mesh: (a) mesh with 957 elements (simulation 1), (b) meshwith 12000 elements (simulation 2).

In the model with the less refined mesh, a distribution of low values of strain energy density (around

3 × 10−9) is present along the bone, with a column of higher values forming from the region of contact

with the prosthesis to the upper part of the vertebra (around 5 × 10−4). In the model with the more

refined mesh, the strain energy density magnitude along the bone is low (around 3×10−9) and a column

of higher values (around 7× 10−4) is also observed from the region of contact with the prosthesis to the

upper part of the vertebra. Comparing to the case with the less refined mesh, in this case, this collumn

is located more to the left. In the most interior area of the vertebra in contact with the prosthesis a peak

of high strain energy density is present (1× 10−3). A difference can be observed between both models,

especially in the magnitudes of strain energy density present in the region of contact between bone and

prosthesis.

5.4.2 Initial Density

To asses the influence of the initial density in the result, several simulations are performed using the

model of the femur with the standard prosthesis implanted with different initial densities.

Figure 5.17 presents the density distribution obtained using the strain-based model, with uniform

initial density of 1 g · cm−3 and 1.5 g · cm−3. With initial density of 1 g · cm−3, the density distribution

along the femur is uniform with value of approximately 0.8 g · cm−3, except in the lower part of the bone.

In this region, there is a high variation of densities between 0.01 g · cm−3 and 0.8 g · cm−3. With initial

density 1.5 g · cm−3 in the lateral extremities of the lower part of the body of the femur there is high

density bone. The rest of the femur is composed of uniform density of around 0.8 g · cm−3.

Figure 5.18 presents the density distribution using the strain energy density model and the same

initial densities used in the previous case. With initial density of 1 g · cm−3 the femur is mainly composed

of bone with the minimum allowed density, 0.01 g · cm−3. On the other hand, with initial density 1.5

g · cm−3 the femur is composed of uniform density of around 0.8 g · cm−3, except in the outer regions of

the diaphysis, where there is high density bone.

39

There is a clear similarity between the results obtained with both remodelling models using 1.5 g ·

cm−3 as initial density. Both cases in which the initial density is 1 g · cm−3 resulted in a region or several

regions of discontinuity in the bone.

Figure 5.17: Density distribution in the femur with the standard proshtesis implanted obtained using thestrain-based model and different initial densities, ρi: (a) ρi = 1 g · cm−3 , (b) ρi = 1.5 g · cm−3.

Figure 5.18: Density distribution in the femur with the standard proshtesis implanted obtained using thestrain energy density model and different initial densities, ρi: (a) ρi = 1 g · cm−3 , (b) ρi = 1.5 g · cm−3.

40

5.4.3 Parameter D

To asses the importance of the value of D in the strain energy density model, tests are performed with

different values of this parameter. Figure 5.19 presents the results obtained using the model of the femur

with the standard prosthesis and different values of the parameter D: D=50, D=73 and D=74. With the

parameter D=50 high density bone is present in the the outer regions of the diaphysis, with a column

of low density bone in the centre, while the metaphysys and epiphysis are made of low density bone.

The results obtained with D=73 are similar. With parameter D=74, the inferior part of the femur is made

of minimum density bone, without the outer regions of the diaphysis of high density. There is a clear

difference in the inferior part of the femur (where it is fixed) when using the parameter D=74.

Figure 5.19: Density distribution in the femur with the standard proshtesis implanted obtained using thestrain energy density model and different values of D: (a) D = 50, (b) D = 73, (c) D = 74.

41

Chapter 6

Discussion

Bone remodeling models are very important to study bone adaptation under different mechanical envi-

ronments. In more critical conditions, bone overloading may occur and influence the bone remodeling

process. Some effort has been made in the past few years to incorporate this phenomenon in math-

ematical models that can simulate bone remodeling [7, 8]. These models are needed to study bone

adaptation under more critical conditions, such as in the presence of implants.

This work focused on developing an alternative bone remodeling model, including overload resorp-

tion, using strains as the mechanical stimulus. The performance of this model, described throughout

this work as strain-based model, was evaluated under different conditions and it was compared with the

model developed by Li et al [8], referred as strain energy density model.

The results obtained confirmed the occurrence of bone overloading in critical conditions (when the

prosthesis is implanted in the bone), using both the strain-based model and the strain energy density

model. The results obtained applying the two bone remodelling models are similar in most cases.

The strain energy density model was more effective in reproducing the bone density distribution of

an intact femur (6.1). In fact, the result obtained with the strain energy density model (5.1) presents the

low density areas in the femur (Ward’s triangle and medullary canal), whilst with the strain-based model

these areas have a medium density.

Figure 6.1: Representation of density distribution in a real femur. Dark areas represent high density andbright areas low density.

43

The results obtained in the vertebrae model without overload, presented in figures 5.3(a) and 5.5 (a),

are supported by the work of Espinha et al. [57]. In the region compressed between the screws the

density decreases and in the regions above the superior screw and below the inferior screw it increases,

showing that the load passes from the superior to the inferior vertebrae through these regions and

the prosthesis. Adding the overload phenomenon to the mathematical model (figures 5.4(b) and 5.5

(b)) results in a drastic density decrease in the region in contact with the screws, where high strain

concentrations occur, leading to bone overload resorption. This is coincident with the work of Espinha

et al., where concentrations of high energy and strain were found in this region.

Using the two-dimensional femur model with different prostheses implanted, the influence of bone

density in the overloading phenomenon could be studied. Implanting the prosthesis in trabecular bone

(Figure 5.6 and Figure 5.7) resulted in a decrease in bone density in the regions adjacent to the pros-

thesis, when overload was included in the mathematical models. This shows that overload resorption

occurred in these regions due to the concentration of high strain and energy. On the other hand, the

results obtained when the prosthesis was implanted in high density cortical bone (Figure 5.8 and Figure

5.9) did not present any differences between the models without and with overload simulation. This sug-

gests that the overload phenomenon may be more prominent in low density bone, which has also been

addressed by Crupi et al. [7]. Taking this in consideration, the prosthesis design could be of extreme

importance to reduce overload resorption in some cases, that may lead to the prosthesis failure. Also,

when comparing the results obtained using both models with the two prosthesis, under similar circum-

stances, the strain-based model produced more uniform results in the head of the femur. The strain

energy density model produced several discontinuities of low density in this region. This phenomenon

could not be eliminated with mesh independence filtering which suggests that it may be a limitation of

the remodelling model.

The results obtained in the three-dimensional femur model, without the prosthesis, using both math-

ematical models, are similar and close to the density distribution of a real femur. The improvement of the

results compared to the two dimensional model, are more notable in the strain-based model which may

come from the fact that in three dimensions three strain components are used, instead of only two. Also,

the two-dimensional models, as referred earlier, have more limitations than three-dimensional models,

despite the simplicity of the three-dimensional model considered. In the model with the resurfacing pros-

thesis, without overload, bone resorption in the femoral head, which is in contact with the prosthesis, is

observed. According to Ong et al. [58], this result is due to superolateral stress shielding, mostly due to

the stress and strain transfered through the stem of the prosthesis. This phenomenon is more prominent

in fully bonded prosthesis. So, to reduce bone resorption, the stem may not be fully bonded to the bone.

Adding overload to the remodelling model results in a bigger decrease in bone density in the region

of the femoral head in contact with the prosthesis, which may result in aseptic loosening, according to

Perez et al. [59]. This decrease suggests the existence of overload in this region, which is consistent

with the results obtained by Rothstock et al. [60]. Although, their study did not focus on the overload

phenomenon, high strain concentrations were found in this particular region, which may cause overload.

During the simulations using the finite element method, the size of the elements of the mesh revealed

44

to be very important in the result. Less refined meshes were not able to detect strain and energy

peaks occurring in the contact regions between bone and prosthesis. Accordingly, when studying bone

overloading, refined meshes need to be used so that the phenomenon can be detected.

Other important factor that should be considered when carrying such studies is the initial condition

used. Different initial conditions lead to distinct results. In particular, low initial densities ended in

disruptive results. This has to due with the fact that, with low initial densities, the whole bone is under

overload in the beginning of the bone remodelling process. Moreover, the bone is not able to recover

from overload and loses bone throughout the whole remodelling process, especially in the region where

it is fixed, which makes the solution inconsistent.

In the strain energy density model, the parameter D revealed to be very sensitive to variation. This

may come from the fact that with increasing D, the parabola that represents the mathematical model

becomes narrower and the edges become steep, which means that the same mechanical stimulus

causes higher density variations. This is an important issue that needs attention when using this model

because it may lead to instability.

In this work a novel bone remodelling model including overload resorption was developed and tested.

Some limitations were faced during this process, mostly concerning the lack of information about the

overload process which made the validation process difficult. However, bone resorption due to overload

was seen and the results were consistent with previous studies.

45

Chapter 7

Conclusions and Future Work

The current work proposed a mathematical model for bone remodelling that takes overload resorption

in consideration, using strains as mechanical stimulus. This model was studied using a finite element

model of two vertebrae with a prosthesis implanted and a femur with two different prosthesis implanted

(resurfacing and hip implant). Comparisons with the bone remodelling model developed by Li et al [8]

were made. The influence of bone density surrounding the prostheses was also studied and some

parameters that may influence the models performance were tested.

The density distributions obtained in the intact bones are close to the real ones. Under critical

conditions (prosthesis implanted) both mathematical models revealed bone overload resorption near the

prostheses due to concentrations of high strains or energies which is in agreement with the literature.

Differences between implanting the prosthesis in cortical or trabecular bone were seen: in trabecular

bone, overload resorption is more prominent than in cortical bone.

Some parameters were concluded to be crucial to the results such as the size of the elements of the

mesh and the initial density used. The model developed by Li et al. revealed to be very sensitive to the

definition of the parameter D, which ultimately caused instabilities in the results.

Although the bone remodeling model developed was able to simulate bone overloading, further tests

should be done. Because of the fact that the finite element models used in the present work are not

complex, future tests should focus on using more complex models, such as considering more load

cases, and study the performance of the mathematical model in these cases. Also, cases of bone

overloading should be studied in patients and compared to the ones obtained computationally with these

models. The fact that the bone overloading phenomenon is not yet well studied which results in lack of

information, made the validation of the model hard. With advances in this area, the model should be

updated and further validated to follow the novel conclusions.

47

References

[1] J. Wolff. Das Gesetz der Transformation der Knochen. Deutsche Medicinische Wochenschrift, (47):

1222–1224, 1892.

[2] H. M. Frost. Wolff’s Law and bone’s structural adaptations to mechanical usage: an overview for

clinicians. Angle Orthodontist, 64(3):175–188, 1994.

[3] H. M. Frost. A 2003 Update of Bone Physiology and Wolff ’ s Law for Clinicians. Angle Orthodontist,

74(1):3–15, 2004.

[4] H. Rodrigues, C. Jacobs, J. M. Guedes, and M. P. Bendsøe. Global and local material optimiza-

tion models applied to anisotropic bone adaptation. IUTAM Symposium on Synthesis in Bio Solid

Mechanics, pages 209–220, 2002.

[5] P. R. Fernandes, J. Folgado, C. Jacobs, and V. Pellegrini. A contact model with ingrowth control for

bone remodelling around cementless stems. Journal of Biomechanics, 35(2):167–176, 2002.

[6] H. M. Frost. Bone’s mechanostat: A 2003 update. the Anatomical Record Part a, 275A(August):

1081–1101, 2003.

[7] V. Crupi, E. Guglielmino, G. La Rosa, J. Vander Sloten, and H. Van Oosterwyck. Numerical analysis

of bone adaptation around an oral implant due to overload stress. Proceedings of the Institution of

Mechanical Engineers, Part H: Journal of Engineering in Medicine, 218(6):407–415, 2004.

[8] J. Li, H. Li, L. Shi, A. S. L. Fok, C. Ucer, H. Devlin, K. Horner, and N. Silikas. A mathematical model

for simulating the bone remodeling process under mechanical stimulus. Dental Materials, 23(9):

1073–1078, 2007.

[9] H. M. Frost. Bone “Mass” and the “Mechanostat”: A Proposal. The Anatomical Record, 219(1):1–9,

1987.

[10] B. Clarke. Normal bone anatomy and physiology. Clinical journal of the American Society of

Nephrology : CJASN, 3 Suppl 3:131–139, 2008.

[11] M. Locke. Structure of long bones in mammals. Journal of Morphology, 262(2):546–565, 2004.

[12] Open Stax. Bone Structure · Anatomy and Physiology, 2016. URL http://philschatz.com/

anatomy-book/contents/m46281.html.

49

http://philschatz.com/anatomy-book/contents/m46281.html

http://philschatz.com/anatomy-book/contents/m46281.html

[13] C. Gray H, H. Gray, and C. Clemente. Anatomy of the Human Body, 1985.

[14] J. Y. Rho, L. Kuhn-Spearing, and P. Zioupos. Mechanical properties and the hierarchical structure

of bone. Medical Engineering and Physics, 20(2):92–102, 1998.

[15] A. K. G. Robert E. Marx. Bone Structure, Metabolism, and Physiology: Its Impact on Dental Im-

plantology. Implant Densitry, 7(4):267–276, 1998.

[16] A. Dalla Pria Bankoff. Biomechanical Characteristics of the Bone. Human Musculoskeletal Biome-

chanics, pages 61–86, 2012.

[17] M. B. Schaffler and D. B. Burr. Stiffness of compact bone: effects of porosity and density. Journal

of Biomechanics, 21(I):13–16, 1988.

[18] U. E. Schonau, E. Werhahn. Influence of muscle strength on bone strength during childhood and

adolescence. In Bone Biology and Growth, pages 63–66, 1996.

[19] X. Shi, X. Sherry Liu, X. Wang, X. Edward Guo, and G. L. Niebur. Type and orientation of yielded

trabeculae during overloading of trabecular bone along orthogonal directions. Journal of Biome-

chanics, 43(13):2460–2466, 2010.

[20] F. Saltel, A. Chabadel, Y. Zhao, M. H. Lafage-Proust, P. Clezardin, P. Jurdic, and E. Bonnelye.

Transmigration: a new property of mature multinucleated osteoclasts. J Bone Miner Res, 21(12):

1913–1923, 2006.

[21] P. A. Downey and M. I. Siegel. Bone Biology and the Clinical Implications for Osteoporosis. Physical

Therapy, 86(1):77–91, 2006.

[22] R. Florencio-silva, G. Rodrigues, E. Sasso-cerri, M. J. Simoes, P. S. Cerri, and B. Cells. Biology of

Bone Tissue : Structure , Function , and Factors That Influence Bone Cells. 2015, 2015.

[23] L. Xing and B. F. Boyce. Regulation of apoptosis in osteoclasts and osteoblastic cells. 328:709–720,

2005.

[24] J. C. Crockett, M. J. Rogers, F. P. Coxon, L. J. Hocking, and M. H. Helfrich. Bone remodelling at a

glance. Journal of Cell Science, 124(7):991–998, 2011.

[25] J. L. Ferretti, G. R. Cointry, R. F. Capozza, and H. M. Frost. Bone mass, bone strength, muscle-

bone interactions, osteopenias and osteoporoses. Mechanisms of Ageing and Development, 124

(3):269–279, 2003.

[26] R. Huiskes and H. Weinans. The Relationship Between Stress Shielding and Bone Resorption

Around Total Hip Stems and the Effects of Flexible Materials. Biomechanics Section, Institute for

Orthopaedics, 274:124–134, 1991.

[27] L. J. Raggatt and N. C. Partridge. Cellular and molecular mechanisms of bone remodeling. Journal

of Biological Chemistry, 285(33):25103–25108, 2010.

50

[28] Bone Remodeling Structure And Function Of The Muscu-

loskeletal System Minimalist. URL http://patantour.com/

bone-remodeling-structure-and-function-of-the-musculoskeletal-system-minimalist.

[29] T. J. Martin. Bone remodelling : its local regulation and the emergence of bone fragility. Best

Practice & Research Clinical Endocrinology & Metabolism, 22(5):701–722, 2008.

[30] W. F. Zimmerman, M. A. Miller, R. J. Cleary, T. H. Izant, and K. A. Mann. Damage in total knee

replacements from mechanical overload. Journal of Biomechanics, 49(10):2068–2075, 2016.

[31] S. C. Cowin and D. H. Hegedus. Bone remodeling I: theory of adaptive elasticity. Journal of

Elasticity, 6(3):313–326, 1976. ISSN 15732681.

[32] H. G. H. Weinans, R. Huiskes. The Behavior of Adaptive Bone-Remodeling. Journal of Biomechan-

ics, 25(12):1425–1441, 1992.

[33] G. S. Beaupre, T. E. Orr, and D. R. Carter. An Approach for Time Dependent Bone Modeling and

Remodeling- Theoretical Development. Journal of orthopaedic research, 8(3):651–661, 1990.

[34] G. Beaupre, T. Orr, and D. R. Carter. An approach for time-dependent bone modeling and remod-

eling—application A preliminary remodeling simulation. Journal of Orthopaedic Research, pages

662–670, 1990.

[35] D. Taylor and P. J. Prendergast. A model for fatigue crack propagation and remodelling in compact

bone. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in

Medicine, 211(5):369–375, 1997.

[36] R. Huiskes and E. Y. S. Chao. A survey of finite element analysis in orthopedic biomechanics: The

first decade. Journal of Biomechanics, 16(6):385–409, 1983.

[37] G. Dhatt, E. Lefrancois, and G. Touzot. Finite Element Method. Wiley, 2013.

[38] H. Wiskott and C. Belser. Lack of integration of smooth titanium surfaces. Clinical Oral Implant

Research, pages 429–444, 1999.

[39] R. Ruimerman. Modeling and Remodeling in Bone Tissue. PhD thesis, Technische Universiteit

Eindhoven, 2005.

[40] Kendall E. Atkinson. An Introduction to Numerical Analysis. Wiley, 1989.

[41] G. Dahlquist and A. Bjorck. Numerical Methods in Scientific Computing, Volume I. Society for

Industrial and Applied Mathematics, jan 2008.

[42] O. Sigmund and J. Petersson. Numerical instabilities in topology optimization: A survey on proce-

dures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization,

16(1):68–75, 1998. ISSN 0934-4373.

51

http://patantour.com/bone-remodeling-structure-and-function-of-the-musculoskeletal-system-minimalist

http://patantour.com/bone-remodeling-structure-and-function-of-the-musculoskeletal-system-minimalist

[43] A. Dıaz and O. Sigmund. Checkerboard patterns in layout optimization. Structural Optimization, 10

(1):40–45, 1995.

[44] J. Folgado. Modelos computacionais para Analise e Projecto de Proteses Ortopedicas. PhD thesis,

Instituto Superior Tecnico, 2004.

[45] A. Wall and T. Board. The compressive behavior of bone as a two-phase porous structure. In

Classic Papers in Orthopaedics, volume 59, pages 457–460. oct 2014.

[46] S. A. Romeed, S. L. Fok, and N. H. F. Wilson. A comparison of 2D and 3D finite element analysis

of a restored tooth. Journal of Oral Rehabilitation, 33(3):209–215, 2006.

[47] M. Martin, T. Lemaire, G. Haıat, P. Pivonka, and V. Sansalone. A thermodynamics framework to

describe bone remodeling: a 2D study. In 23eme Congres Francais de Mecanique, Lille, 2017.

[48] M. Mengoni and J. P. Ponthot. An enhanced version of a bone-remodelling model based on the

continuum damage mechanics theory. Computer Methods in Biomechanics and Biomedical Engi-

neering, 18(12):1367–1376, 2015.

[49] A. J. Peacock. Handbook of polyethylene: structures: properties, and applications. Marcel Dekker,

2000.

[50] P. D. Harvey. Engineering Properties of Steel. American Society for Metals, 1982.

[51] D. T. Reilly and A. H. Burstein. The Mechanical Properties of Cortical Bone. The Journal of Bone

and Joint Surgery, 56(5):1001–1022, 1974.

[52] C. Jacobs, M. Levenston, G. Beaupre, J. Simo, and D. Carter. Numerical instabilities in bone

remodling simulations: The advantage of a node-based finite element approach. J. Biomech., 28

(4):449–459, 1995.

[53] J. Black and G. Hastings. CoCr-based alloys. In Handbook of Biomaterial Properties, pages 167–

178. Springer US, Boston, MA, 1998.

[54] S. Karas. Outcomes of Birmingham hip resurfacing: A systematic review, 2012.

[55] GrabCAD. URL https://grabcad.com/.

[56] W. J. Anderst, E. D. Thorhauer, J. Y. Lee, W. F. Donaldson, and J. D. Kang. Cervical spine bone

mineral density as a function of vertebral level and anatomic location. Spine Journal, 11(7):659–

667, 2011.

[57] L. Espinha. Computational Analysis of Bone Remodeling during an Anterior Cervical Fusion. PhD

thesis, Instituto Superior Tecnico, 2009.

[58] K. L. Ong. Biomechanics of the Birmingham hip resurfacing arthroplasty. Journal of Bone and Joint

Surgery - British Volume, 88-B(8):1110–1115, 2006.

52

https://grabcad.com/

[59] M. A. Perez, P. A. Vendittoli, M. Lavigne, and N. Nuno. Bone remodeling in the resurfaced femoral

head: Effect of cement mantle thickness and interface characteristics. Medical Engineering and

Physics, 36(2):185–195, 2014.

[60] S. Rothstock, A. Uhlenbrock, N. Bishop, L. Laird, R. Nassutt, and M. Morlock. Influence of interface

condition and implant design on bone remodelling and failure risk for the resurfaced femoral head.

Journal of Biomechanics, 44(9):1646–1653, 2011.

53

bone remodelling model including overload simulation - ulisboa

Documents