Application of an optimisation algorithm to configure an internal fixation device

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 availabl

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

Biomechanics, Optimisation Algorithm, Finite Element Modelling,

Fracture Healing, Internal Fracture Fixation

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

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

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

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

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

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

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 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

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

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:

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

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

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

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

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

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

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

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

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,

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

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

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,

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)

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

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

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

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

L)SS plate to secure a worst defect of mm representing a comminuted

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)

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 )t 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

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

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

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:

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

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

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

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 (ooke 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

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

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)

)n 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)

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

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