Application of an Optimisation Algorithm to Configure an Internal …eprints.qut.edu.au/42517/1/Salma_Ibrahim_Thesis.pdf · 2011. 7. 8. · Application of an Optimisation Algorithm

Application of an Optimisation

Algorithm to Configure an Internal

Fixation Device

Salma Ibrahim, B.E. (Medical)

Submitted for the award of the degree of Master of

Engineering in School of Engineering Systems of the faculty

for Built Environment and Engineering, Queensland

University of Technology

2010

Optimisation of an Internal Fixation Device

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Keywords

Biomechanics, Optimisation Algorithm, Finite Element Modelling,

Fracture Healing, Internal Fracture Fixation


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Abstract

Project Title: Application of an optimisation algorithm to configure an internal fixation device

Author: Salma Ibrahim

Supervisors: Dr. Sanjay Mishra (Primary)

Dr. Gongfa Chen (Secondary)

Fractures of long bones are sometimes treated using various types of fracture

fixation devices including internal plate fixators. These are specialised plates which

are used to bridge the fracture gap(s) whilst anatomically aligning the bone

fragments. The plate is secured in position by screws. The aim of such a device is to

support and promote the natural healing of the bone.

When using an internal fixation device, it is necessary for the clinician to decide

upon many parameters, for example, the type of plate and where to position it; how

many and where to position the screws. While there have been a number of

experimental and computational studies conducted regarding the configuration of

screws in the literature, there is still inadequate information available concerning

the influence of screw configuration on fracture healing.

Because screw configuration influences the amount of flexibility at the area of

fracture, it has a direct influence on the fracture healing process. Therefore, it is

important that the chosen screw configuration does not inhibit the healing process.

In addition to the impact on the fracture healing process, screw configuration plays

an important role in the distribution of stresses in the plate due to the applied loads.

A plate that experiences high stresses is prone to early failure. Hence, the screw

configuration used should not encourage the occurrence of high stresses.

This project develops a computational program in Fortran programming language to

perform mathematical optimisation to determine the screw configuration of an


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internal fixation device within constraints of interfragmentary movement by

minimising the corresponding stress in the plate. Thus, the optimal solution suggests

the positioning and number of screws which satisfies the predefined constraints of

interfragmentary movements. For a set of screw configurations

the interfragmentary displacement and the stress occurring in the plate were

calculated by the Finite Element Method. The screw configurations were iteratively

changed and each time the corresponding interfragmentary displacements were

compared with predefined constraints. Additionally, the corresponding stress was

compared with the previously calculated stress value to determine if there was a

reduction. These processes were continued until an optimal solution was achieved.

The optimisation program has been shown to successfully predict the optimal screw

configuration in two cases. The first case was a simplified bone construct whereby

the screw configuration solution was comparable with those recommended in

biomechanical literature. The second case was a femoral construct, of which the

resultant screw configuration was shown to be similar to those used in clinical cases.

The optimisation method and programming developed in this study has shown that

it has potential to be used for further investigations with the improvement of

optimisation criteria and the efficiency of the program.


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Table of Contents

Keywords ........................................................................................................................................................... i

Abstract ............................................................................................................................................................ ii

Table of Contents ........................................................................................................................................ iv

Figures and Tables ................................................................................................................................... vii

Abbreviations used in the text .............................................................................................................. ix

Statement of Originality............................................................................................................................ x

Acknowledgements.................................................................................................................................... xi

1. Introduction ......................................................................................................................................... 1

1.1. Background .................................................................................................................................. 1

1.2. Problem .......................................................................................................................................... 1

1.3. Aims ................................................................................................................................................. 3

1.4. Significance of the Study ........................................................................................................ 3

1.5. Outline of Thesis ........................................................................................................................ 4

2. Literature Review and Background ............................................................................................... 6

2.1. Treatment of Long Bone Fractures ........................................................................................... 6

2.1.1. Internal Fixators .................................................................................................................... 7

2.1.2. Fracture Healing .................................................................................................................... 9

2.2. Factors Influencing the Strength of the Fixation Construct and Bone Healing11

2.2.1. Stiffness of Fracture Fixation ........................................................................................ 12

2.2.2. Physical Conditions for Fracture Healing ............................................................... 12

2.3. Influence of Working Length and Fracture Gap on Fixation Stability ............... 14

2.4. Screw Positioning ....................................................................................................................... 18

2.5. Limitations of previous studies ............................................................................................ 19

2.6. Summary ......................................................................................................................................... 20

3. Methods - Optimisation .................................................................................................................... 21

3.1. Mathematical Definition of Optimisation ........................................................................ 21

3.2. Types of Optimisation Problems and How to Solve Them ...................................... 22

3.2.1. Constrained/ Unconstrained Optimisation Problems ...................................... 22

3.2.2. Multi-modal Optimisation .............................................................................................. 24

3.2.3. Deterministic Methods .................................................................................................... 26


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3.3. Powell’s method ...........................................................................................................................27

3.3.1. Conjugate Directions .........................................................................................................27

3.3.2. The Algorithm .......................................................................................................................28

3.3.3. Golden Section Search – Search in One Direction. ..............................................32

3.4. Use of optimisation methods in medical engineering................................................33

3.5. Optimisation of Screw Configuration in Internal Fixators ......................................35

3.5.1. Objectives and Constraints.............................................................................................35

3.5.2. Optimisation Criteria ........................................................................................................36

3.5.3. Objective Function ........................................................................................................37

3.5.4. Calculation of Function Value (with the use of FE method)......................37

3.5.5. Data Transfer ...................................................................................................................41

4. Results........................................................................................................................................................44

4.1. Case 1: Simplified Model ..........................................................................................................44

4.1.1. Bone Geometry.....................................................................................................................44

4.1.2. Plate and Screws Geometry ...........................................................................................44

4.1.3. Material Properties ............................................................................................................46

4.1.4. Boundary and Loading Conditions .............................................................................47

4.1.5. Variables to be Optimised ...............................................................................................47

4.1.6. Selection of Values for Optimisation Criteria ........................................................47

4.1.7. Solution for Simplified Model .......................................................................................51

4.2. Case 2: Clinical Model ................................................................................................................55

4.2.1. Clinical Cases .........................................................................................................................55

4.2.2. Additional Cases ..................................................................................................................57

4.2.3. Femoral Bone Geometry ..................................................................................................57

4.2.4. Plate and Screws of Femoral Construct ...................................................................58

4.2.5. Assembly .................................................................................................................................58

4.2.6. Materials..................................................................................................................................58

4.2.7. Loading and Boundary Conditions .............................................................................59

4.2.8. Variables to be Optimised ...............................................................................................59

4.2.9. Selection of Optimisation Criteria...............................................................................62

4.2.10. Solution .................................................................................................................................64

5. Discussion ...........................................................................................................................................72

5.1. Limitations of this Study ......................................................................................................74


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5.2. Improvements to the Model................................................................................................... 76

5.3. Improvement to the Optimisation Criteria ................................................................ 77

5.4. Future Work - Improvements to the Optimisation Method ............................... 79

6. Conclusions ........................................................................................................................................ 81

7. References .......................................................................................................................................... 83

Appendix ....................................................................................................................................................... 87

Optimisation program including subroutines in Fortran................................................. 87

Python script file to read out values from FEA ...................................................................... 94


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Figures and Tables

Figure 1 LCP-combination hole allowing conventional plate fixation as well as application of

locked screws (Source: (Perren, 2002)).................................................................................................. 7

Figure 2 Internal fixator used with locked screws. Fixator barely touches the bone as screws

allow reliable maintenance of the initial distance between internal fixator and bone

(Source: (Perren, 2002)). ................................................................................................................................ 8

Figure 3 Direct healing from osteotomy of sheep tibia with compression stabilisation. The

bone fragments are close and compressed and there is no displacement at the site of

the osteotomy. The shape of the osteones do not change when crossing the fracture.

(Source: Perren, 2002) ..................................................................................................................................... 9

Figure 4 Histological images of secondary fracture healing in bone (Source: J. Bone Miner.

Res., 16, 1004– 1014, 2001). ...................................................................................................................... 11

Figure 5 Left: X-ray image of femoral fracture in 35 year old male with flexible fixation.

Right: X-ray image 4 months after fixation, showing obvious signs of callus growth.

Source: (Chen et al., 2010). .......................................................................................................................... 15

Figure 6 Seven week postoperative x-ray showing fracture fixation by placing several

locking screws in main fragments. The screw holes were occupied adjacent to the

fracture site resulting in high stress concentrations occurring in that section of the

plate. Source: (Sommer et al., 2003) ...................................................................................................... 16

Figure 7 Example of contours of an objective function (Source: Rao, S. S.; Engineering

Optimization-Theory and Practice, 3rd Ed. 1996, pp.363) ........................................................ 23

Figure 8 A multi-modal function. Source: (Singh et al., 2006) ............................................................. 25

Figure 9 Conjugate Direction (Source: Rao, S. S.; Engineering Optimization-Theory and

Practice, 3rd Ed. 1996, pp.363) ................................................................................................................. 28

Figure 10 Progress of Powell's Method (Source: Rao, S. S.; Engineering Optimization-Theory

and Practice, 3rd Ed. 1996, pp.363) ....................................................................................................... 31

Figure 11 Illustration of Golden Section Search .......................................................................................... 32

Figure 12 Showing data transfer between different software packages ........................................ 41

Figure 13 Screw positions (variables) to be optimised in the simplified model ........................ 45

Figure 14 Mesh of the simplified cylindrical model .................................................................................. 47

Figure 15 (a) Rigid simplified construct, (b) flexible simplified construct ................................... 48

Figure 16 Nodes used to calculate displacements ..................................................................................... 49

Figure 17 Showing sharp edge (a cause of FE errors) in screw holes of the locking

compression plate ............................................................................................................................................ 51

Figure 18 Optimised solution for simplified model................................................................................... 51

Figure 19 Maximum principal stress distribution in cylindrical construct .................................. 52

Figure 20 (a) treatment of transverse fracture of 73 yr old patient (b) X-ray image showing

failure of implant 7 weeks post-op (c) treatment of fracture of a 35 year old male (d) X-

ray showing successful healing of fracture ......................................................................................... 56

Figure 21 Shows 4 fixed screws (black, 2 at each end of plate) and 6 screw positions

(yellow) to be optimised............................................................................................................................... 60

Figure 22 (a) Fracture healing in patient after 4 months using a flexible screw configuration.

(Source: J Eng Med. Chen et al, 2010); (b) Simulation of the same combination used for

FE analysis ........................................................................................................................................................... 61

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Figure 23 (a) flexible construct, (b) rigid construct .................................................................................. 63

Figure 24 (a) Screw configuration used in clinical case from Chen et al (2010); (b) Resultant

screw configuration from optimisation algorithm .......................................................................... 65

Figure 25 Maximum principal stress distribution in femoral construct of the optimised

solution .................................................................................................................................................................. 67

Figure 26 (a) Flexible construct, (b) Construct with more rigidity due to shorter working

length ...................................................................................................................................................................... 68

Figure 27 Some screw configurations that were tried and tested by the optimisation

algorithm. White represents screws that were chosen by the optimisation algorithm

that were tested. Black represents screws that were fixed throughout the optimisation

process. .................................................................................................................................................................. 73

Figure 28 Illustration of concept of local versus global minimisation ............................................. 75

Figure 29 “Boundaries for optimal healing in the sheep model[s] that lead to timely healing”

(Source: Epari et al, 2007) ........................................................................................................................... 78

Table 1Interfragmentary displacement and maximum principal stress for the most rigid and

the most flexible cylindrical models ....................................................................................................... 49

Table 2 Comparison of displacements from solution and those from constraints for the

cylindrical model .............................................................................................................................................. 52

Table 3 Comparison of displacement and stress resulting from flexible and rigid construct

with that of solution construct for the cylindrical model ............................................................ 53

Table 4 Shear, axial displacement and stress in plate resulting from the configuration from

Chen et al (2010)............................................................................................................................................... 62

Table 5 Interfragmentary displacement and maximum principal stress for most rigid and

most flexible femur models ......................................................................................................................... 64

Table 6 Comparison of displacements from solution and those from constraints .................... 67

Table 7 Comparison of displacement and stress resulting from flexible and rigid construct

with that of solution construct .................................................................................................................. 68

Table 8 Axial and shear displacement resulting from the flexible and rigid constructs from

Figure 27 (a) and (b) ....................................................................................................................................... 69

Table 9 Axial and shear displacements resulting from the removal of pairs of screws from

each side of the fracture gap from the all screws in place construct ..................................... 70

Table 10 Comparison of displacement and stress from optimised solution with that from

screw configuration used in clinical case from Chen et al (2010) .......................................... 70

file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566220file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566220file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566221file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566221file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566221file:///C:/Users/Ibrahim/Desktop/BeginningThesisCloseRevisedDraft%5b1%5dWithoutTrackChanges%5b1%5d.docx%23_Toc279566221


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Abbreviations used in the text

FE = Finite Element

LCP = Locking Compression Plate

LISS = Less Invasive Stabilising System

DCP = Dynamic Compression Plate

TSP = Travelling Salesman Problem


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Statement of Originality

“The work contained in this thesis has not been previously submitted for a degree or

diploma at any other higher education institution. To the best of my knowledge and

belief, the thesis contains no material previously published or written by another

person except where due reference is made.”

Signature: ________________________________________

Date: ______________________________________________


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Acknowledgements

I would like to thank my supervisors, Dr. Gongfa Chen and Dr. Sanjay Mishra for

their help, support and guidance throughout this study. I am grateful to the trauma

team at IHBI, my fellow colleagues and friends making the research environment

more enjoyable, and Mr. Mark Barry and the HPC team for their assistance with the

supercomputer.


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1. Introduction

1.1. Background

Severe trauma to the extremities is the leading cause of disability during the wage-

earning period of life. Bone fractures cost the Australian healthcare system one

billion dollars a year. In addition to this cost, the functional loss of limbs impact

significantly on the patients’ quality of life. Studying the impact of fixation devices

on bone healing will fill knowledge gaps and enhance the usefulness of these

devices for the purpose of fracture healing. This will ultimately reduce costs and

improve quality of life for the patient.

High-energy collisions with long bones often result in fractures with significant

misalignments of bone fragments. In these cases it is difficult for the body to pursue

its natural healing course in order to produce a successful healing outcome. For

these instances, surgical fracture treatment is usually required. There are a number

of fracture fixation devices available, including external fixators, intermedullary

nails and internal plate fixators. The need to use any one of them depends on the

physical characteristics of the trauma. Ultimately, the purpose of using these

fixation devices is to restore functionality to the bone and limb.

1.2. Problem

To promote a successful fracture healing outcome, it is necessary to correctly

configure the fracture fixation device according to the physical condition of the

trauma. Some of the configuration parameters that should be decided upon are the


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type of plate, including the length; where to position the plate; how many and

where to position the screws. Each parameter contributes to the progress and

outcome of the healing fracture. If the fixation device is improperly configured, it

can hinder the fracture healing process, resulting in revision surgery. This

increases the burden of the healthcare system and decreases quality of life for the

patient involved.

In internal plate fixation, the screw configuration is one of the vital parameters

decided upon by the orthopaedic surgeon. If the surgeon uses too many screws, the

plate may prematurely fail during treatment of the fracture, in which case, revision

surgery may be required. Furthermore, there may not be sufficient motion at the

fracture gap required for healing. At the other extreme, in the case of using too few

screws, the stress in the plate is decreased at the expense of an increased amount

of motion of the bone fragments. Excess movement causes further complications,

such as a delayed or non-union of the bone fragments. Therefore, the goal is to find

the best screw configurations to be used following the requirement that the

fracture successfully heals, while the implant does not fail.

Previous studies (Tornkvist et al, 1996; Stoffel et al, 2003; Duda et al, 2002) have

used mainly experimental techniques and some finite element analyses to evaluate

the strength and stiffness of certain screw configurations, and to identify trends in

screw placement. The approach taken in this study is to optimise the screw

configuration of the fixation device using mathematical programming, with the

added advantage of simultaneously creating optimum conditions for healing.

Mathematical optimisation techniques have been used successfully for numerous

applications in various fields of engineering. However, they have not been applied

to the topic of fracture healing in the biomedical field. In this study, mathematical

programming is utilised to ultimately optimise the screw configuration with

respect to bone fragment movement constraints in certain directions, ensuring that

the stress in the plate is minimised.


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1.3. Aims

There are two main aims of this study.

1. Develop an optimisation tool.

This involves creating an interface between various softwares used to

develop the optimisation process. Finite element (FE) software is used to do

numerical analysis on a computational fracture model while a Python

program is used to extract information from the FE output database. The

mathematical optimisation algorithm itself is written in Fortran

programming language. It was used to create the software interface.

2. Investigate the potential for the optimisation tool to solve the clinical

problem.

Apply the developed optimisation process to various cases to determine the

optimal screw configuration that enhances bone healing and avoids

mechanical failure of the plate in internal fixation for a particular fracture.

1.4. Significance of the Study

By defining the requirements for timely fracture healing, fracture fixation devices

may be configured in a manner in which they support healing conditions.

In the field of fracture healing, researchers (Epari et al, 2007; Goodship and

Kenwright, 1985) have strived to define the precise conditions required for timely

fracture healing. Goodship and Kenwright applied rigid fixation to one group of


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fractures and controlled axial movement in another group in vivo, in an attempt to

determine the optimal parameters for fracture healing. The results showed that

controlled micro-movement significantly improved healing. Epari et al looked at

the association between strength of healed bones to the stiffness of their respective

fracture fixation configurations. It was found that optimising axial stability and

limiting shear movements was required for timely healing.

In a recent paper, Chen et al (2010) biomechanically analysed two cases presented

to them from the clinical environment from different orthopaedic surgeons. The

internal fixation device that was configured in a rigid manner failed due to a fatigue

fracture and did not heal. The other case which was configured in a more flexible

manner did heal. Although there has been progress in this research field over many

decades, there is still the knowledge gap of selecting the best screw configuration

for a fracture fixation device in a given situation. This project aims to further

research in this area using optimisation mathematical programming.

1.5. Outline of Thesis

Chapter 2 will discuss fixation stability regarding internal fixation devices and

fracture healing. Fracture fixator parameters such as screw positioning and

numbers, and their influence on the strength of the fixator and the stress in the

plate, as well as the influence of the size of the fracture gap will be examined.

Chapter 3 will provide a detailed explanation of the how the mathematical

programming method is interfaced with results from the FE calculations for

optimisation of the screw configuration.

Chapter 4 is the results section which addresses two computational model cases to

which the optimisation method was applied. One is a simplified cylindrical case,

while the other is a femoral ‘clinical’ case.


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Chapter 5 holds a discussion of various aspects of the optimisation method used

and improvements are suggested.

It should be noted that this thesis focuses on the method of optimisation used

rather than the final application.


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2. Literature Review and Background

Severe trauma to the extremities is the leading cause of disability during the wage

earning period of life (BJD, 1998). Over 150,000 Australians are hospitalised with

fractures each year (Welfare, 2006). The socio-economic burden of fractures is

substantial. Loss of working capacity represents over 60% of the total cost of bone

fractures, while less than 20% is due to the direct cost of medical treatment.

Optimal outcomes, therefore, require not only solid bone union but also early and

complete recovery of limb function.

This chapter describes the process of fracture healing and the mechanical

conditions necessary for healing. Fixation stability is vital for fracture healing and is

characterised by the mechanical configuration of the fracture fixator being used.

Although there are numerous mechanical parameters involved in fracture fixation,

this review will focus on one of the mechanical aspects, i.e., screw configurations

and its importance in fracture healing.

2.1. Treatment of Long Bone Fractures

A fracture occurs when a high amount of energy is absorbed by the bone until

failure occurs (Brighton, 1984). For these types of fractures, surgery is often

necessary. There are three main types of fracture fixation treatments involving

surgery. As previously mentioned, they are external fixation, intramedullary nailing

and internal fixation. All fixation devices are designed for the restoration of limb

function, anatomical reduction by stabilisation of bone fragments and promotion of

bone healing.

Internal fixators, i.e. plates and screws, are common in the treatment of shaft

fractures up to the metaphyseal area (Ruedi et al., 2001). A failure rate of 7% is

reported with plate failure, screw loosening or breakage being the causes of failure

(Riemer et al., 1992). In the incidence of failure, due to a large range of possible


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complications, revision surgery is often necessary which decreases the quality of

life of the patient, and increases costs to the healthcare system.

2.1.1. Internal Fixators

Locking plates are internal fracture fixation devices that have been designed to

allow maximal vascularisation to the damaged bones and achieve a minimal

implant-bone interface. Two methods of treatment are available using the Locking

Compression Plate (LCP). This is made possible by the screw combination holes in

which part of the hole allows the fitting of a locked screw, whereas the other part of

the hole allows screws to be positioned at different angles (Figure 1).

Figure 1 LCP-combination hole allowing conventional plate fixation as well as application of locked screws (Source: (Perren, 2002)).

In the compression treatment method, as in conventional plating, anatomic

reconstruction and absolute stability may be achieved. The other treatment method

is called locked splinting, in which the LCP is used to simply bridge the fracture gap,

leaving the defected zone untouched. This method is ideal for the fixation of

comminuted, diaphyseal and metaphyseal fractures (Wagner et al., 2007) (Figure

2). The LCP allows the combination of the compression method and the locked

splinting method.


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Figure 2 Internal fixator used with locked screws. Fixator barely touches the bone as screws allow reliable maintenance of the initial distance between internal fixator and bone (Source: (Perren, 2002)).

With a variety of plates and types of screws available for security of the fracture,

and the large number of different configurations possible, the orthopaedic surgeon,

based on his experience, has to decide upon many mechanical factors regarding

configuration, whilst taking into consideration the biological conditions of the

fracture. The determinants of the fixation method are: which type of plate,

including the length; where to position the plate; how many and where to position

the screws (Wagner et al., 2007).

Chen et al (2010) has undertaken a FE study of comparing the influence of different

numbers of screws on plate failure. The configurations of the screws that were

compared were those that had been used in a clinical case. One was a flexible

fixture (6 screws out of a possible 14) in which successful healing occurred. The

other was a rigid fixture (12 screws out of a possible 14) in which plate failure

occurred without healing of the fracture. In the FE study, it was found that under

physiological loading, the plate that was rigidly fixed experienced significantly

higher stresses than the one fixed in a more flexible manner. In the fatigue analysis

it was found that the plate under rigid fixation fractured at 20 days after surgery,

whilst the plate under flexible fixation was able to endure 2000 days. This study


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has highlighted the major impact of screw configuration for fixation stability for

fracture healing as well as its influence on plate failure.

It is important to describe the fracture healing process to better understand the

implications of mechanical stimulus due to fracture fixation.

2.1.2. Fracture Healing

Naturally, the body has two ways of healing bone fractures. One is primary healing,

which involves direct compression across the bone fragments. In this case there is

no displacement of fragments and there is absolute stability of fixation. Osteones

(functional unit of compact/cortical bone) are able to grow across the bone

fragments. The disadvantage of this process is that the fracture takes an extended

period of time to heal compared to the secondary healing.

Figure 3 Direct healing from osteotomy of sheep tibia with compression stabilisation. The bone fragments are close and compressed and there is no displacement at the site of the osteotomy. The shape of the osteones do not change when crossing the fracture. (Source: Perren, 2002)

The other type of healing is secondary healing, which involves the formation of a

callus around the fracture site. This usually occurs when there is high impact

trauma to the bone, and there is extensive soft tissue damage. Healing of a fracture


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of this calibre involves a number of stages and may take weeks until the formation

of bone is observed.

In such an open fracture, the local bone marrow, periosteum, adjacent soft tissue

and blood vessels are injured. The first course of action of the body is to clot the

vessels around the fracture site, and prevent or fight infection in the area of

trauma. Haematoma and haemorrhage formation results from disruption of

periosteal and endosteal blood vessels at the fracture site (Figure 4, one day after

fracture). Pain and swelling eventually decreases and primary soft callus then

forms (Figure 4, 7 days after fracture). At day 14 the soft callus becomes

mineralised to form new bone. Three weeks post-fracture, the bone fragments are

no longer moving. The stability at this stage is adequate to prevent shortening,

although angulation of the fracture site may still occur. The cells that are stimulated

and sensitised produce new blood vessels, fibroblasts and supporting cells.

Chondroblasts also appear in the callus between bone fragments. Following the

linkage between the bone fragments by the callus, the stage of hard callus begins

until they are firmly united by new bone (Figure 4, days 21 and 28). Bony bridging

of the callus usually occurs at the periphery of the periosteal callus and endosteal

bone preceding the remodelling phase, which continues for several years (Ruedi et

al., 2001).

The previous explanation is the ideal fracture process by secondary healing.

Because the fracture zone is sensitive to mechanical stimulus (Kenwright et al.,

1989) and the tissues differentiate accordingly, it is important to achieve or come

close to achieving adequate mechanical conditions for the stimulation of healing. To

one extreme, there may be too much movement, and the fracture is unstable. In this

case, bone healing will be delayed or will not occur. To the other extreme, there

may be insufficient movement to stimulate any healing. This condition is similar to

primary healing.


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It should be noted that in addition to mechanical stimulation, biological factors

such as hormones, growth factors and blood supply are required for healing.

However, this study will address some of the mechanical influences rather than the

biological aspects.

Figure 4 Histological images of secondary fracture healing in bone (Source: J. Bone Miner. Res., 16, 1004– 1014, 2001).

2.2. Factors Influencing the Strength of the Fixation

Construct and Bone Healing

Bone healing is known to be sensitive to mechanical stability of fixation (Yamagishi

et al., 1955). The strength and stiffness of the fracture callus is related to the degree

of stability of the fixation device (Goodship et al., 1985; Kenwright et al., 1989). The

maturation of the callus is related to the amount of motion between the fracture,


12

which depends on the applied loads and fixation stability (Claes et al., 1998; Duda

et al., 2002).

2.2.1. Stiffness of Fracture Fixation

Knowing the amount of stiffness required from a fixator to promote a successful

fracture healing outcome is vital. Epari et al (2007) have achieved this for a variety

of external fixators and intramedullary nails. The study measured firstly the

stiffness of the fixators in vitro, and secondly, the strength and stiffness of healed

tibiae after nine weeks that were treated using the various types of external

fixators and intramedullary nails. Using the experimental technique, a relationship

between the fixation stability and strength of the tibiae was found (Epari et al.,

2007).

A similar study conducted by Woo et al (1984) compared stiffness and strength of

healed femurs using flexible versus rigid internal fixator constructs. The purpose of

the study was to develop concepts for the ideal internal fixation plate, “based on the

mechanical demands of plate stiffness and strength in balance with the

physiological responses of the underlying bone” (Woo et al., 1984). It was found

that in the early stages of healing, plate stiffness in the bending and torsion must be

sufficient to promote union without bone angulation or implant failure. In the later

stages, plate stiffness should be low enough so that the bone may share the

physiological loads.

2.2.2. Physical Conditions for Fracture Healing

As quantitative measurements of the stiffness of internal fixators are unavailable in

the literature, an alternative method of defining the optimal conditions for healing

is required. As aforementioned, the mechanical conditions of the callus are related

to the movements between the fracture gap (interfragmentary movements).


13

Goodship and Kenwright (1989) studied the effects of applying 0.5 mm, 1 mm and

2 mm of axial displacement in a 3 mm fracture gap. It was found in the tibiae with

0.5 mm displacement and 1 mm displacement (with 200 N applied force),

increased rates of fracture stiffness and mineralisation was seen. A displacement of

2 mm was detrimental to healing in terms of mineralisation and fracture stiffness.

In the clinical investigation conducted by Goodship and Kenwright, movements

between 0.2 mm and 1 mm were permitted. Movements between these limits

supported healing (Kenwright et al., 1989).

Augat et al (2003) investigated the effects of shear movement at the fracture gap. It

was seen that, in a 3 mm gap size, displacement of 1.5 mm in a shear direction was

detrimental to healing, while that of the same magnitude in the axial direction

supported healing. Shear movements may induce delayed unions and

pseudoarthroses. The type of tissue produced is cartilage and fibrous tissue at the

fracture site (Yamagishi et al., 1955; Augat et al., 2003).

In summary, it is seen that for a 3 mm fracture gap, certain amounts of

displacements in their respective directions are required to promote healing.

Therefore, what is required is a fixation structure that when under an applied load,

creates sufficient motion that promotes healing. As previously mentioned, there are

a number of mechanical determinants contributing to the strength and stiffness of

the internal fixator that may be controlled. This includes which type of plate,

including the length; where to position the plate; how many and where to position

the screws (Wagner et al, 2007). However, from this point, the literature review

will focus mainly on the topic of screw configurations, which is of the scope of the

present study. The arrangement of screws strongly impacts on the loading of the

implant itself, as well as the healing outcome of the fracture.

The distance between the screws and the number of screws in a plate has influence

on the axial, bending and torsional stiffness’ of the fixation construct. Furthermore,

these aspects have great impact on the stress distribution in the plate which is


14

important to estimate in order to prevent early plate failure during clinical

treatment. Previous studies (Tornkvist et al, 1996; Duda et al, 2002; Stoffel et al,

2003) have been conducted to investigate the influence of screw arrangement on

the stresses and strains in the plate and screws, rather than their impact on

fracture healing.

2.3. Influence of Working Length and Fracture Gap on

Fixation Stability

In internal fixators, having a large working length (distance between the innermost

screws) greatly dissipates the stress along the length plate under applied loading.

By leaving a space of between 2 and 3 holes across the fracture gap, stress

concentrations may be avoided (Wagner et al., 2007). It was shown by Stoffel et al

(2003) that if a large working length is used (e.g., 10 hole spaces), for example, in

the case of bridging a comminuted fracture under dynamic loading tests, the

construct failed early. Using a large working length will also render the construct to

be too flexible, allowing excessive motion between bone fragments. This movement

will cause a non-union or a delayed union of the bone fragments (Kenwright et al.,

1989; Claes et al., 1998).

With sufficient fixation stability and blood supply, the fracture will heal

successfully. An example of this is from Chen et al (2010) which illustrates an X-ray

image (Figure 5) of a 35 year old male who suffered a femoral fracture treated with

a flexible construct. The surgeon used a moderate working length, with not more

than 3 screws on either side of the fragment.

By using too many screws, large stress concentrations are created in the plate

which lead to premature implant failure. In a study by Sommer et al (2003), this

phenomenon was demonstrated. A 73 year old woman suffered a periprosthetic

fracture in the middle to distal third part of her femoral shaft. The surgeons placed

screws immediately adjacent to the fracture site, inclusive of 12 out of a maximum


15

14 holes. The screw combination resulted in high stresses generated in that section

of the plate which led to early failure (7 week post-op) (see Figure 6). Because of

the extreme rigidity of the structure, there was insufficient interfragmentary

movement to promote callus formation. Therefore no healing occurred.

Figure 5 Left: X-ray image of femoral fracture in 35 year old male with flexible fixation. Right: X-ray image 4 months after fixation, showing obvious signs of callus growth. Source: (Chen et al., 2010).

Working length (length between the innermost screws) has been identified as a

major influence on the distribution of stress in the plate, and stiffness and strength

of the bone-fixator construct. Stoffel et al (2003) included in their study an FE

comparison of the stresses experienced in an LCP plate due to the working length,

using gap sizes of 1 mm and 6 mm. It was shown that as the working length


16

increased from the distance of 2 holes on the plate to 4 holes, for the 6 mm gap, the

Von Mises stress in the plate increased by 133 %. This was different for the 1 mm

gap model, in which it was demonstrated that the Von Mises stress in the plate

decreased by 10 % (Stoffel et al, 2003).

Figure 6 Seven week postoperative x-ray showing fracture fixation by placing several locking screws in main fragments. The screw holes were occupied adjacent to the fracture site resulting in high stress concentrations occurring in that section of the plate. Source: (Sommer et al., 2003)

In a similar study, Duda et al (2002) used the Less Invasive Stabilisation System

(LISS) plate to secure a ‘worst’ defect of 11 mm representing a comminuted


17

fracture. By doubling the working length, i.e. from 2 to 4 hole spaces across the

defect, there was a considerable reduction in the Von Mises stress of the internal

fixator (Duda et al., 2002). Thus, there is a direct contrast in the stresses generated

in the internal fixator due to an increase in working length, i.e. the results given by

Stoffel et al for the 6 mm gap and that of Duda et al for the 11 mm gap. However, for

both cases, the construct became less stiff in compression and bending and the

stresses in the implant were reduced.

In another study investigating the impact of fracture gap sizes, Ellis et al (2001)

used a Dynamic Compression Plate (DCP) to stabilise a no-gap model, 10 mm gap

model and a 40 mm gap model. Plate strain was calculated. For the 10 mm model

and the 40 mm model, placing the screws closest to the fracture site decreased the

strain in the plate. In the no-gap model, placing the screws farthest from the

fracture site minimised the strain in the plate (Ellis et al., 2001).

Claes et al (1998) studied the influence of fracture gap and interfragmentary

strains on biological healing of the fracture gap. Different interfragmentary strains

were applied to the various in-vivo fracture gap size models of 2 mm and 6 mm. As

mentioned previously, it was found that although a large callus formed in the small

gap model due to large interfragmentary strain (31 %), the tissue that was formed

was connective tissue rather than bone. When the 2 mm gap model was subjected

to a smaller strain (7 %), bony bridging occurred which resulted in successful

healing. For larger gap models (6 mm) regardless of the interfragmentary strain,

the tissue type that was found to be produced at the end of the 9 weeks in-vivo

study was connective tissue (Claes et al., 1998).


18

2.4. Screw Positioning

Screw positioning is important in determining the loading of the implant itself

(Duda et al., 2002). At least 3 screws should be placed either side of the fracture,

regardless of the quality of the bone (Wagner et al., 2007). More than 3 screws

either side of the fracture site does not increase the axial stiffness of the construct.

In the LCP, by placing additional screws towards the plate ends, the axial stiffness

decreased (Stoffel et al., 2003). This is in contrast to conventional plating where in

the stiffness would increase. Under torsional load, more than 4 screws per

fragment did not have an influence on the rigidity of the construct (Stoffel et al.,

2003).

In a study of compression plate fixation by Cheal et al (1983), in a 3 dimensional FE

model, it was found that in the presence of a fracture gap, the loads on the

innermost screws are increased and are more inclined to static failure during the

early stages of weight bearing. It was also found that the outermost screws are

more vulnerable to fail due to fatigue if the plate is left for a long period.

A study by Field et al (1999) concerned the influence of screw omission on bone

strain. It was found that “certain omission treatments provoked higher levels of

bone strain than would have been obtained if the plate were attached using all

screws” (Field et al., 1999). In an earlier study by Korvick et al (1988) it was shown

that the removal of the inner 2 to 4 screws from a screw-filled 8-hole plate resulted

in significantly higher levels of bone strain. Additionally, it was shown that by

replacing bi-cortical screws (screws that pierce both cortices) by mono-cortical

screws, the strain experienced by the bone was significantly reduced.

Shortening the plate by removing the end screws did not have any major effect on

the rigidity of the construct (Korvick et al., 1988). This is in contrast to the study by


19

Sanders et al (2002) who found that the length of the plate was more important

than the position of the screws in providing bending strength (Sanders et al., 2002).

Tornkvist et al (1996) used a dynamic compression plate (DCP) to investigate the

relationship between screw positions and number, and the strengths of the

constructs. It was found that under torsion, strength in the plate was dependent on

the number of screws. It was also found that under bending, as in conventional

plates, strength in the plate was improved by the wider spacing of screws rather

than the increase in the number of screws (Tornkvist et al., 1996).

2.5. Limitations of previous studies

Duda et al (2002) and Stoffel et al (2003) did not take into account the

interfragmentary movement at the fracture site which is important for healing. The

recommendations made by Stoffel et al were based on the maximum Von Mises

stress in the plate and screws disregarding the interfragmentary movements as

well as the stress and strain in the callus.

There is limited information in the literature on the influence of screw

configurations on the physiological responses of the bone, in terms of the stresses

and strains that occur at the callus site. Goodship and Kenwright (1985) did

investigate interfragmentary movements for a fracture gap of 3 mm. However, the

stresses and strains that occur in the callus were not measured.

Stoffel et al (2003) and Tornkvist et al (1996) both conducted experiments to test

screw configurations on the strength of the bone-plate-screw construct or the

stress in the plate and screws. These studies did not test the effect of screw

configurations on the callus stress and strains as it was not possible. By using the

finite element method, it is possible to create a callus material around the fracture

gap. This has been attempted (Claes et al, 1999). However, the problem lies in

defining and validating the callus material as this information is unavailable.


20

2.6. Summary

Previous studies (Stoffel et al, 2003; Field et al, 1999; Cheal et al, 1983) show that

the concept of working length cannot be generalised to all types of plates.

It can be observed that the stress in the implant under applied loads is not simply

influenced by the working length and the number of screws. The distribution of

stress in the plate is influenced by other mechanical aspects that have not been

highlighted and specifically addressed in the literature. The size of the fracture gap

plays an important role, as plate stress distribution varies with it. In addition, the

design of the plate influences the stress distribution. Further research needs to be

conducted in these areas. However, this study will not address these issues as it is

not in the scope of the project. A LCP plate will be used with a fracture gap size of 3

mm as there is more information in the literature about these parameters.

For a particular type of fracture (comminuted, oblique, spiral, etc), it is necessary to

find out what configuration of screws (i.e. number and placement) is required to

reduce the stresses in an internal fixator as well as to promote healing of the

fracture. Although some work (experimental and computational) has been done in

this area the most suitable screw configuration for a type of fracture is unknown.

This project attempts to approach the problem using mathematical programming

techniques.


21

3. Methods - Optimisation

In broad sense of the term, optimisation is the efficient allocation of limited

resources. The aim is to arrive at the best possible decision in any given set of

circumstances.

3.1. Mathematical Definition of Optimisation

In mathematics, the field of optimisation is dedicated to finding the minimum or

maximum of a function of n real variables, subject to one or more

constraints. Ultimately, the aim is to minimise the effort required or maximise the

benefit desired in a situation, which is often described by a function (Rao, 1996).

Mathematically, the optimisation problem may be stated as follows:

Find

which minimises

Subject to the constraints: , j = 1,2,...,m and

, j = 1,2,...,p

where is known as the objective function, which is the design parameter of

the problem that is wished to be minimised or maximised, with respect to other

design parameters. The constraints, and are inequalities and equalities

respectively. The problem described above is a constrained optimisation problem. A

problem without the constraints is known as an unconstrained optimisation

problem.


22

In its simplest form, the objective function will have one variable. This is called a

one-dimensional problem, for which there are a number of mathematical methods

available to solve. Brent’s Method and Golden Section Search are some examples

(Walsh, 1975). As more variables are added to the function, the problem becomes a

multi-dimensional case which is solved using more complex mathematical

procedures, which are described in the following sections.

To illustrate the complexity of the multi-dimensional optimisation problem, refer to

Figure 7. In this case, there are two variables, and . The ellipses represent the

contours of the objective function. The feasibility region, which is bounded by the

constraint functions, is presented. With the addition of more variables, the

objective function surfaces become harder to visualise and have to be solved purely

mathematically (Rao, 1996).

3.2. Types of Optimisation Problems and How to Solve

Them

Depending on the information available about the optimisation problem, there are

a variety of methods available to produce a solution. There are direct, indirect and

gradient methods which make use of different fundamental principles to ultimately

obtain an optimum. These methods are used to solve multi-dimensional problems.

3.2.1. Constrained/ Unconstrained Optimisation Problems

Optimisation methods to solve unconstrained optimisation problems fall in two

categories. One is direct search methods, in which derivatives of the objective

function are not required. The other is descent (gradient methods) methods that

require the derivatives of the function.


23

Figure 7 Example of contours of an objective function (Source: Rao, S. S.; Engineering Optimization-Theory and Practice, 3rd Ed. 1996, pp.363)

Constrained minimisation problems may be solved using direct search methods

and indirect methods. A constrained problem becomes replaced by a series of

unconstrained minimisation problems in which penalty functions are used. The

penalty terms represent a measure of violation of the constraint. This is the indirect

search method.

Constraint functions


24

The main principles of direct search methods are as follows. An initial guess point

must be selected as to where the location of the minimum (optimal value) is .

This point is checked to determine if it is the optimum. The next step is to generate

a new point .

Direct search methods are unique in the way that they select the new point, as well

as the way they subsequently test the point for optimality. Some examples of direct

search methods are Grid Search Methods, Pattern Directions, Hooke and Jeeves’

Method, Powell’s Method and Simplex Method.

Indirect search (descent) methods are those that utilise the gradient of the

function. Moving in the gradient direction from any point in space will increase or

decrease the function value at the fastest rate. Unfortunately this gradient direction

applies on a local level rather than a global one. Local versus global minima will be

further explained in Section 3.6. All descent methods make use of the gradient

direction to facilitate selection of search directions. Examples of optimisation

methods that use these principles are Steepest Descent (Cauchy) Method, Newton’s

Method, Quasi-Newton Methods and the Davidson-Fletcher-Powell Method.

The problem presented in this study is a constrained optimisation problem.

Although there are a number of optimisation methods available to solve it, there

are certain attributes of one algorithm over another that make it desirable to use.

The following section will discuss the attributes of different types of optimisation

methods used to solve constrained optimisation problems.

3.2.2. Multi-modal Optimisation

Usually the functions dealt with are multi-modal functions (multi-dimensional

problems), which are simply functions with a number of optimums. It may be

assumed that the function in this study is one with multiple optimums, of which the

location of the peak optimum is unknown. An example of a function with 20

optimums is shown in Figure 8. However, one of the prevalent problems with


25

optimisation algorithms is that they tend to look for a local optimum rather than

the global one. This means that the algorithm generally tends to find the closest

optimum from its initial point. Hence, there is no guarantee that the optimum

solution found is necessarily the best one.

Figure 8 A multi-modal function. Source: (Singh et al., 2006)

There are algorithms, generally called multi-modal algorithms that have been

created to overcome this problem.

An advantage of using multi-modal algorithms is that they are able to search a

population of points in parallel, rather than just a single point. Any starting point is

permitted as it would not make a significant difference to the number of iterations

necessary to find solutions. The algorithm can provide a number of potential

solutions, as opposed to a single one.

Evolutionary algorithms are examples of multi-modal algorithms that require a

probability distribution function to govern the generation of a new search point.

Unfortunately the present study does not have a probability distribution function,

which is a requirement of this method.

Fitness


26

Heuristics are effectively search procedures that move from one solution point to

another with the object of improving the value of the model criterion. They can be

used to develop good (approximate) solutions. This type of algorithm uses the rule

that given a current solution to the model, allow the search of an improved solution

(Taha, 1976).

Simulated Annealing (SA) and Genetic Algorithms (GA) are examples of heuristic

probabilistic methods which are multi-modal algorithms (Singh et al, 2006). The

disadvantage of using these methods is that they are impractical for the

optimisation of structures using the finite element method, which is used in this

study. These methods require a large number of iterations before they would be

able to converge.

3.2.3. Deterministic Methods

Deterministic heuristic methods such as the Simplex method and Powell’s method

are gradient-based mathematical programming methods. These methods have

been used in a number of engineering applications to find the optimal solution for

continuous variables.

They have been known to excel when the gradient of the objective function is

unavailable (Nelder et al., 1965; Del Valle et al., 1988). The Simplex method and

modified Simplex methods have been used in analytical chemistry optimisation

problems. It was observed that using these Simplex methods sometimes there was

lack of convergence, and therefore inefficient. Powell’s method was found to be

more efficient in that it converged quicker than compared to the Simplex method

(Del Valle et al., 1988).

In a study comparing the efficiency of Powell’s method and the Simplex method on

the application of flow injection systems, it was found that Powell’s method

reached optimal conditions with a lower number of experimental evaluations (Del

Valle et al., 1988).


27

The optimisation method that is used in this study is Powell’s method. This

algorithm has its advantages and disadvantages. The advantages are that it is a

widely used and tested algorithm which has been used extensively in engineering

and one of the most efficient of those not based on the estimation of the gradient of

the objective function (Del Valle et al., 1988). The disadvantage is that Powell’s

method searches for a local solution rather than a global one. However, the global

optimisation techniques that are available are not well tested and used, under-

developed and inefficient. To reduce the effects of the ‘global issue’ an educated

estimation of the starting point in the search space assists the algorithm in seeking

the optimum.

A description of Powell’s method is provided in the following section.

3.3. Powell’s method

Powell’s method makes use of the properties of conjugate directions. This is

advantageous as convergence is accelerated by minimising along each of a

conjugate set of directions.

3.3.1. Conjugate Directions

Mathematically, conjugate directions may be described as follows. Suppose a

system of linear equations,

Where A is a symmetrical positive definite n-by-n matrix (i.e. , Ax for

all non-zero vectors in and real). Two non-zero vectors u and v are conjugate

(with respect to A) if

Figure 9 is used to illustrate conjugate directions. If X1 and X2 are the minima of the

function, Q obtained by searching along the direction S from 2 different starting


28

points Xa and Xb, respectively, the line (X1 - X2) will be conjugate to the search

direction S (Rao, 1996).

Figure 9 Conjugate Direction (Source: Rao, S. S.; Engineering Optimization-Theory and Practice, 3rd Ed. 1996, pp.363)

3.3.2. The Algorithm

Powell discovered a direction set method that produces n mutually conjugate

directions (Walsh, 1975; Press, 1992; Mathews, 2004).

Let be the set of values of variables as the initial guess of the location of the

minimum of the function,


29

1. Approximate the minimum of the function to generate the next estimation,

, by proceeding successively to a minimum of f along each of the N

standard base vectors. The process generates a sequence of points,

.

2. Along each standard base vector the function f is a function of one variable.

To minimise each function f requires the application of a one dimensional

minimisation method, such as the Golden Ratio Search.

3. The vector PN – P0 represents the “average” direction moved during each

iteration. It is the average direction moved after trying all N possibilities.

The point X1 is determined to be the point at which the minimum of the

function f occurs along this vector and requires minimisation using, for

instance, the Golden Ratio Search.

4. Since PN – P0 is regarded as a good direction; it replaces one of the direction

vectors in the next iteration. The iteration is then repeated using the new set

of direction vectors to generate a sequence of points.

The algorithm for Powell’s method can be summarized in the following (Press,

1992).

Let be an initial guess at the minimum of the function .

Let be the standard base vectors,

, and let

1. Set , where is the initial guessed point.

2. For find the value of that minimises and set

3. Set .

4. Set for Set .

5. Find the value of that minimises Set

6. Repeat steps 1 to 5.


30

A more illustrative explanation of Powell’s method may be explained with

reference to Figure 10.

Powell’s method begins with an initial point and independent search

directions, which are initially the co-ordinate directions.

A search for the minimum is conducted along each of the directions (uni-directional

search) in turn. Successively, on each search, the minimum point obtained in the

previous search is the used as the new departure point. From Figure 10, search is

conducted along the directions then , delivering point 3 as the start for the

next minimisation.

When all directional searches are complete, the total displacement is used as the

new search direction, beginning with a two-fold distance point. If this direction of

expansion is sufficient, it replaces the direction that gave the lower improvement.

Suppose that gave the largest decrease. It is replaced by the new direction and

the unidirectional minimum is found at point 5.

By repeating this procedure, all the initial directions are replaced and a good

estimation of the conjugate directions is obtained. The solution converges when the

difference between the points is sufficiently small.

As mentioned in the previous section, the Powell’s optimisation algorithm includes

a one-dimensional (uni-directional) search method. The search method is used to

find the minimum in each direction initialised by the algorithm. There are a

number of uni-directional search methods available. The one used in this study was

the Golden Section Search method. It is among one of the most efficient region

elimination methods to optimise functions of a single dimension.


31

Figure 10 Progress of Powell's Method (Source: Rao, S. S.; Engineering Optimization-Theory and Practice, 3rd Ed. 1996, pp.363)


32

3.3.3. Golden Section Search – Search in One Direction.

The golden search method is implemented in Powell’s method. It is used to search

for the minimum in one direction. Each direction is not part of a set of ‘planned’

directions. Rather, it is formed after the evaluation of the result of search in a

previous direction.

Suppose the positions a, b, c and x (Figure 11) are points that lie sequentially on the

x-axis and is a fraction of the way between and . Therefore,

The next trial point is an additional fraction beyond

The next bracketing segment will be either of length relative to the current

one, or else of length . To minimise the worst case possibility, choose to

make these equal.

Figure 11 Illustration of Golden Section Search

b

z

x

w

a c

1-w

b


33

The new point is symmetric to point b in the original interval. is equal to

. This implies that the point lies in the larger of the two segments. The scale

similarity implies that x should be the same fraction of the way from b to c as was b

from a to c. In other words,

Therefore, the quadratic equation, , yields

This means that the optimal bracketing interval has its middle point a

fractional distance 0.38197 from one end, say, and 0.61803 from the other end,

say, c. These fractions are called golden-mean or golden section. This optimal

method of function minimisation, the analogue of the bisection method for finding

zeros, is thus called the golden section search.

In summary, the concept of golden search is as follows: given a bracketing triplet of

points, the next point to be tried is that which is a fraction 0.38197 into the larger

of the two intervals. This occurs until a point where the difference between the

current point and the next point is minimal or close to zero. Each new function

evaluation will bracket the minimum to an interval 0.61803 times the size of the

preceding interval (Press, 1996).

3.4. Use of optimisation methods in medical engineering

Mathematical optimisation programming techniques became widely developed in

the 1960s. Since then, optimisation techniques have been used for train scheduling,

optimising design parameters in structural engineering, design of aircraft for

minimum weight, design of wind turbines and pumps for maximum efficiency,


34

optimal design of electrical networks, analysis of statistical data and experimental

results to obtain the most accurate representation of the physical phenomenon,

design of optimum pipeline works for process industries, selection of a site for an

industry, and many more applications. In the medical field, there are two main

focuses. One is operating theatre scheduling time, using mainly stochastic dynamic

programming models, and the other is treatment planning in various fields. For

example, the treatment plan for stereotactic radio-surgery using conjugate

gradients and simulated annealing methods.

Many research groups have studied cancer therapy theoretically, clinically and

mathematically. Esen et al (2006) applied an optimisation model called Weapon-

Target Assignment problem (WTA) of military operations research to optimise

cancer therapy. It used mixed-integer nonlinear goal programming models. It had

three objectives: maximise the weighted damage of the cancer cells, minimise the

total weighted side effects and minimise the total dose therapy costs. The model

created facilitates cancer therapists to act in a multi-objective frame. However, the

model created must be clinically validated (Esen et al., 2008). This example alone

demonstrates the power of mathematical programming and its implications.

Maratt et al (2008) investigated the feasibility of an integer programming model to

assist in pre-operative reduction and internal fixation of a distal humerus fracture.

The model aimed at maximising the number of bicortical screws placed while

avoiding screw collisions and favouring screws of grater length over multiple

fracture planes (Maratt et al., 2008).

Rozema et al (1992) used a linear programming technique and a muscle

architecture model to minimise the strains in plate-osteosynthesis devices for

internal fixation of mandibular fractures. The objective was to minimise the strain

in the mandibular bone plate by optimising the position of the plate based on a

number of factors (Rozema et al., 1992). The objective function to be minimised

was an energy function, in which the variables included 3-D displacements and


35

rotations that occurred at given external forces and torques on the computational

model. In finding the minimum, the values of the variables became known and were

used as input for the next step, which was to minimise the maximum internal strain

by optimising the placement (co-ordinates) of the bone plates using a deterministic

method called the Simplex Method.

3.5. Optimisation of Screw Configuration in Internal

Fixators

This section explains the application of the optimisation programming tool to find

the best screw configuration in an internal fixation device. It will discuss the

optimisation criteria, the calculation of the function value and the interface

between the FE software and the algorithm itself.

3.5.1. Objectives and Constraints

In this study, the objective function is

Find

which minimises f(S) (maximum principal stress)

subject to the constraints

and

See definitions in the next section.


36

3.5.2. Optimisation Criteria

The optimum solution:

1. allows displacement, a, of between mm and mm in the axial direction,

2. allows displacement, d, of between mm and mm in a shear direction, and

3. minimises the maximum principal stress in the plate (objective)

Previous researchers (Stoffel et al, 2003; Duda et al, 2002) have calculated the Von

Mises stress rather than the maximum principal stress. Von Mises stress is the

criterion used to assess yielding of materials whereas maximum principal stress is

used as a failure criterion. Physiologically, the plate is subjected to cyclic loading,

from which fatigue failure results. As failure of the implant is of interest, maximum

principal stress is measured (Chen et al., 2010; Shipley et al., 2002).

There is information in the literature about the required interfragmentary

movements for a certain type and gap of fracture (Kenwright et al., 1989; Claes et

al., 1999). This project uses a transverse cut, distracted 3 mm, as in the

experimental work by Goodship and Kenwright (1989), but it does not take into

account many other biological and mechanical factors used to simulate their model.

Consequently, the quantitative values for interfragmentary movement described in

the abovementioned studies cannot be compared to those of the model used in this

study. Therefore, the range values of and (displacement in the axial and

shear directions) which are important optimisation criteria for this project, are

estimated by comparing the interfragmentary movements from the stiffest and

most flexible models for each computational model to be optimised using the

program.


37

3.5.3. Objective Function

The analytical function that is used in a classic optimisation problem is usually

formulated from the available information or data about the situation. The

equation, known as the objective function, is used to obtain a value that is

evaluated at each point in space. This is known as the function value, which is used

as feedback for the optimisation algorithm to evaluate for optimality. It is for

comparison with the previous function value to see if the difference is sufficiently

small in order to determine if a maximum or minimum value has been reached.

This is also known as convergence.

Unfortunately, there is no analyti

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