application of an optimisation algorithm to configure an ... · application of an optimisation...

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


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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,


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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).


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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 analytical function describing the relationship between

the stress in the internal fixator and the presence of screws. Therefore, it is

necessary to find an alternative way to evaluate the function value. To do this, the

output values, i.e., the interfragmentary displacements in the shear and axial

direction, as well as the maximum principal stresses in the plate, from finite

element analyses of computational bone models are combined in an equation to

create a function value. This will be further explained in the following section.

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

From the FE software, ABAQUS, the following output values are extracted and used

as feedback into the optimisation code which is written in the programming

language, Fortran 90/ 95.

1. Maximum principal stress in the plate (Name of variable = ).

2. Displacement in the axial direction (Name of variable = ).

3. Displacement in the shear direction (Name of variable = ).


38

For the axial direction, to calculate function value, is initialised as the

maximum stress in the plate.

IF THEN

END IF

Where function value

penalty

absolute value

minimum limit (mm)

maximum limit (mm)

calculated displacement in the axial direction

‘ideal’ displacement in axial direction

It is required that displacement in the axial direction be between mm and mm.

If it is not calculated to be within these limits, the function value will be punished

by being increased using the penalty value. Therefore, the smaller the difference

between the ideal displacement and the actual displacement ,

the less severe the punishment will be to the function value .

Similarly, to calculate function value in the shear direction

IF THEN

END IF

Where

minimum limit (mm)

maximum limit (mm)


39

calculated displacement in the shear direction

ideal’ displacement in shear direction

Similarly, for displacement in the shear direction, it should be between mm and

mm. If it is not calculated to be within these limits, the function value will be

punished by being increased using the penalty value. The smaller the difference

between the ideal displacement and the actual displacement ,

the less severe the punishment will be to the function value, .

The purpose of the penalties is to force the displacement values to the desired ones.

In this way, the displacement values are controlled to reach the defined optimal

range.

There is a second penalty that is introduced which is used to control the variables.

In pseudo code,

Where

value of variable (between 0 and 1) as calculated by Powell’s method

difference between value and extremity (0 or 1)

second penalty

function value


40

Powell’s method will select a number between 0 and 1. Because discrete values (0

or 1) cannot be immediately selected for testing, continuous variables (between 0

and 1) are calculated and need to be forced towards the extreme values in order to

represent the presence or absence of a screw. This is achieved by the application of

the penalty values.

Penalty functions play a major role in the optimisation process. It is important to

get this correct as a penalty that is too large will contribute in a major way to make

the solution bias. The purpose of the penalty is to avoid the function value from

straying far from the feasible region. Therefore it should be a small fraction (10-

30%) of the function value.

If the number selected by Powell’s method is less than 0.5 then the difference

between it and 0 (which represents the absence of a screw) is multiplied by the

second penalty and added to the function value . Similarly, if the value

from the optimisation algorithm is greater than 0.5 then the difference between it

and 1 (which represents the presence of a screw) is multiplied by the second

penalty and added to the function value.

For the purpose of finite element analysis, it is necessary that the , as

calculated by Powell’s method is multiplied by the Young’s modulus of a steel screw

(200 GPa) for computational purposes.


41

3.5.5. Data Transfer

Figure 12 Showing data transfer between different software packages

Figure 12 shows the flow of data between different stages of the optimisation

process. The optimisation process involves active transfer of data between

different softwares and text files. Therefore, it was required that an interface be

created to allow the smooth flow of data. The optimisation program was written in

Fortran programming language.

A brief overview of the process is as follows:

NO

YES

Optimisation

Algorithm

Finite

Element

Analysis

(FEA)

Outputs

Input

s

END

Optimisation

criteria met?


42

i. Input of initial screw configuration and constraints from user.

ii. Summoning of the finite element software, ABAQUS, to run the

numerical analysis using the input data.

iii. Reading out the required output data from the output database using

Python programming language.

iv. Reading the Python generated text files using Fortran and comparing the

values with the required optimisation criteria values.

v. Use of the optimisation algorithm (Powell’s method).

vi. Reading the solution from Powell’s method and translating it into input

for numerical analysis in ABAQUS.

1. Inputs

a. Initial screw configuration

At any time before the finite element analysis is conducted, the computational

models and loadings remain unchanged, except for the screw materials, which

determines the screw configuration. The Young’s modulus which shows the

stiffness of the material, defines the screw material property. A Young’s modulus of

200 GPa represents a material with high stiffness and hence the presence of a

screw. A Young’s modulus of 20 MPa represents a material with low stiffness and

hence the absence of a screw. To initialise the optimisation algorithm, a screw

combination must be input.

b. Specification of user-defined constraints, i.e. range of displacements

in axial and shear directions.

The interfragmentary displacements resulting from the most rigid or the most

flexible construct is different depending on the initial configuration of the variables.

Therefore the range in which the optimum displacement is assumed to be in varies

from model to model.


43

Epari et al (2007) showed that for optimum healing conditions, the fixator used

should not create very high or very low axial compressive stiffness. From

experimental work, it was concluded that the best healing conditions would occur

in the middle third of the range between the highest axial stiffness and the lowest

axial stiffness. Therefore, the constraints used in this study were estimated to be

within the middle third range of the largest displacement and the smallest

displacement that occurred from the stiffest construct and the least stiffest

construct respectively.

2. ABAQUS (Simulia) finite element software

The numerical analysis of the models due to the different screw combinations is

conducted in ABAQUS finite element software.

3. Outputs

From the FE analysis, it is possible to output the maximum principal stress for

groups of elements such as plate and screws, as well as the nodal displacement at

the fracture gap. This is done by requesting field output requests to the output

database file (.ODB file). The output database may be read using a python script file

(.py), and data may be output into text files (.txt), which in turn may be read by

Fortran programs. From these measurements, the interfragmentary movement at

the fracture gap, as well as the maximum principal stress in the plate may be

calculated using Fortran.

4. Optimisation Algorithm

The optimisation algorithm, Powell’s method, is used and comprises of a number of

sub-programs from Numerical Recipes in Fortran (Press, 1992).


44

4. Results

This chapter consists of an explanation of two different cases in which a number of

variables were optimised using the methods described in Chapter 3. The first case

is a simplified case, using a cylindrical model, while the other case is more complex,

using a femur model. For each case there is an explanation of the computational

models used, followed by a description of the selection of the values of the

optimisation criteria. Thereafter the solution resulting from the optimisation

algorithm is discussed.

4.1. Case 1: Simplified Model

This section describes the model used, material properties, loading and boundary

conditions used for the simplified model. It will be followed by a description of the

variables to be optimised using the optimisation algorithm, as well as the criteria

used to evaluate the screw configurations. The resulting configuration is discussed.

4.1.1. Bone Geometry

A number of research groups have used cylinders to represent the geometry of

bone (Stoffel et al, 2003; Tornkvist et al, 1996; Ellis et al, 2001; Field et al, 1999). In

this study, a hollow cylindrical model of length 400 mm, 8 mm thickness and 35

mm diameter was created using Solidworks (Dassault Systèmes SolidWorks Corp,

USA). A mid-shaft transverse cut was made and a fracture gap size of 3 mm was

used.

4.1.2. Plate and Screws Geometry

The fracture was treated using a locking compression plate (LCP 4.5/5.0, broad, 14

holes, 252 mm). The plate was positioned so that the fracture gap lay under the

middle of the plate. A screw was fixed in each of the outermost holes of the plate


45

(positions 1 and 14 of Figure 13). In the simulations, the threads of the screw holes

were removed to reduce the complexity of the finite element mesh.

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

01

Bone

Plate

Load

14

Bone

Callus

Fixed end

Variables to be optimised

Variables to be optimised

03

04

09

10

11

12

05

06


46

The screw was represented by a cylinder with a truncated cone head. The head of

the screw was modelled to fit the diameter of the locking part of the holes in the

plate. It was assumed that the head and body were rigidly connected to the plate

and bone respectively. The diameter of the body was 4.4 mm which was the core

diameter of the LCP locking screw (5.0 mm with threads). No screw threads were

modelled. Seebeck et al (2005) assumed that in real bone the influence of the

thread design might be less pronounced or even negligible (Seebeck et al., 2005).

The FE study by Hou et al (2004) supported this statement, finding the effects of

the screw threads more important when dealing with osteoporotic bone (Hou et al.,

2004).

In addition, by modelling threads in the screw, the number of elements would

increase and make it necessary to apply some contact algorithm between the bone

and screw thread surface, as well as between the plate-screw thread surfaces. With

the increase in computational burden and time, sharp edges of the thread may

result in distorted meshes which are prone to errors during FE analysis. All screws

were treated as uni-cortical screws (piercing one cortex).

4.1.3. Material Properties

The plate and screws were made of stainless steel and assigned a Young’s modulus

of 200 GPa (Muraca et al., 1972). The bone cylinder was assumed to be isotropic

and homogenous, with a Young’s modulus of 20 GPa (Rho et al., 1993). A Poisson’s

ratio of 0.3 was assigned to both the steel and the bone materials. The material of

the callus in the fracture gap had a Young’s modulus of 50 MPa and Poisson’s ration

of 0.3. This material represented a callus tissue that was differentiating from initial

connective tissue (Young’s modulus = 3 MPa; Poisson’s ratio = 0.4) to soft callus

(Young’s modulus = 1000 MPa; Poisson’s ratio = 0.3) (Claes et al., 1999).


47

4.1.4. Boundary and Loading Conditions

The distal end of the cylinder was constrained with no movement in the 3

translational directions. At the opposite end of the cylinder, 580N of compressive

force and 7 Nm of torque were applied. This was similar to the loading used to

mechanically test a human tibial fixation construct (Duda et al, 2002).

Figure 14 Mesh of the simplified cylindrical model

The simplified cylindrical model had a total of 26886 nodes and 102707 elements.

Tetrahedral elements were used to generate the mesh shown (Figure 14).

4.1.5. Variables to be Optimised

For the cylindrical model, there were eight screw positions to be optimised, that is,

four screws in the proximal region and four in the distal region (Figure 13).

4.1.6. Selection of Values for Optimisation Criteria

To determine the values of the optimisation criteria, it was essential to estimate a

range of values (axial and shear displacements) in which the optimal solution may

lie. Firstly, the minimum and maximum values of the axial and shear displacements

were to be known. Thereafter, theoretically, middle third range of values may be

used as the limits for optimal displacement. To do this, it was assumed that under

loading, the most rigid construct, i.e. all variables assigned screw material

properties (Figure 15a) would result in the minimum values for interfragmentary

displacements in the axial and shear direction. Likewise, it was assumed that under

loading, the most flexible construct, i.e. all screw variables assumed to be empty


48

except for two screws fixed permanently in the endmost positions (Figure 15b),

would result in the maximum values for interfragmentary displacements.

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

The models described above were run in the FE software to determine the

maximum and minimum displacements in the interfragmentary gap. From these

resulting values, the displacement ranges for the ‘optimum’ solution may be

decided.

For each model, under loading, the interfragmentary axial and shear displacement

were calculated. The resultant displacements were calculated from the subtraction

of the displacements at the nodes situated on the two bone fragments at the

position opposite the plate (Figure 16). Displacement was calculated in the axial

direction (Y direction) and in a shear direction (X direction) (Figure 16).

Displacement in the third direction (Z direction, normal to the plate surface) was

negligible and hence, not accounted for.

(a)

(b)

Screws


49

Figure 16 Nodes used to calculate displacements

The calculated displacements and stress values for the rigid and flexible constructs

are shown in Table 1.

Shear Displacement (mm)

Axial Displacement (mm)

Max. Principal Stress (MPa)

Rigid 1.09 8.80 42.6

Flexible 1.99 2.00 23.4

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

Nodes

Callus

Bone

Bone

Plate


50

Table 1 it can be seen that the displacement in the shear direction is almost double

in the flexible construct as compared to the rigid one. However, displacement in the

axial direction is multiple times smaller in the flexible construct compared to the

rigid one.

It is observed that the displacement and stress values calculated for these models

are unrealistically small compared to those that may be observed in a physiological

simulation. This is due to multiple factors such as the geometry, material

properties, material interaction properties, and so on. Because the purpose of the

project is not to simulate a ‘real-life’ model, it is acceptable to optimise the model as

they are distinguished percentage-wise rather than by specific values.

The following displacements constraints were used:

The values of the displacement limits in the axial and shear displacements above

exclude the extreme displacement values resulting from the most flexible and most

rigid screw configurations. Therefore, the axial and shear displacements resulting

from the solution screw configuration should be within or close to these limits.

To create the function value, which was described in Section 3.5.4. of the methods

chapter, the maximum principal stress in the plate was calculated.

The stress value was calculated by averaging the top 0.3% of maximum principal

stress values of the whole plate. This percentage was used as it was shown to be

large enough to reduce the effects of errors that may result from a high stress

concentration area, such as at the corner of the join between the two ellipses

forming the screw hole in the locking compression plate (Figure 17).


51

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

4.1.7. Solution for Simplified Model

The optimisation process took approximately 12 hours consisting of 388 iterations

to arrive at a solution on a processor of Intel(R) Xeon(R) CPU E5462 at 2.80GHz.

Eight screw positions were optimised by the algorithm. The optimised solution

showed that out of the 8 positions, screws should be placed at positions 4 and 12 to

satisfy the optimisation constraints (Figure 18).

Figure 18 Optimised solution for simplified model

These positions show a moderate working length of 7 screw hole spaces between

the closest screws to the fracture gap. From Figure 19, it can be seen that the plate

endures a higher stress than the bone. The maximum principal stress is between 7

and 14 MPa along the distance of the middle 6 holes in the plate. The largest stress

(27 to 40 MPa) occurs in the areas of stress concentrations which are around the

holes of the plate.

Sharp

edge

12 04


52

Figure 19 Maximum principal stress distribution in cylindrical construct

Table 2 compares the optimal displacements calculated from the solution screw

configuration (highlighted column) with the minimum and maximum constraint

values specified in the optimisation method.

Min. Constraint

Solution Max. Constraint

Shear Displacement (mm) 1.00 1.25 Axial Displacement (mm) 6.30

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

12

04


53

The solution value of the displacement in the axial direction was found to be

smaller than the value specified in the optimisation criterion, while the solution

value of the displacement in the shear direction was found to be within the limits.

As per Powell’s method, when the constraints of the displacement in the axial

direction were violated, the Penalty method would have attempted to resolve the

problem by forcing the value toward the feasible region. After many attempts, the

resulting solution value is as close as possible to the range of allowable axial

displacement.

Rigid Solution Flexible Shear Displacement (mm) 1.09 1.25 1.99 Axial Displacement (mm) 8.80 6.30 2.00 Max. Principal Stress (MPa) 42.6 39.5 23.4

Table 3 Comparison of displacement and stress resulting from flexible and rigid construct with that of solution construct for the cylindrical model

Table 3 compares the displacements and stress resulting from the solution screw

configuration to those resulting from the most rigid construct and the most flexible

construct. It is seen that the axial and shear displacements as well as the values of

the stress in the plate are skewed towards those resulting from the rigid construct.

The stress in the plate due the screw configuration of the solution construct is 7%

less than that of the rigid construct. Likewise, there is 28% and 15% more

movement in the axial and shear directions respectively than in the rigid construct.

It is seen that the solution screw configuration given by the optimisation solution is

acceptable compared to the objectives and constraints of the problem. Although

there are only two screws fixed on each side of the fracture gap, sufficient

adjustment to the optimisation criteria would reform the solution screw

configuration. To make the solution more practical for surgeons, it would be better

to refine the constraints. For example, if the surgeon recommended at least 3

screws on each side of the fracture gap this would force the program to search for


54

an appropriate position out of the variables for a third screw on each side of the

fracture.

The solution configuration is seen not to be symmetric. This is because the model

itself is asymmetric. The loads are applied at one end of the cylinder and fixed at

the other end. Hence it is acceptable that the solution be asymmetric.


55

4.2. Case 2: Clinical Model

The simplified case from the previous section is a preliminary one which was used

to test the optimisation method. This case presented in this section is based on the

paper by Chen et al (2010) who are part of the Traumatology Group at IHBI, QUT,

Brisbane. As this paper plays a large role in the motivation of this study, and is

frequently referred to, it will be briefly described here.

4.2.1. Clinical Cases

Chen et al studied two cases from the clinical realm. Both clinical cases were

fractures of the shaft and each was treated by a different orthopaedic surgeon.

Relying largely upon the screw configurations to promote facture healing, the

outcomes were different.

The first case was a transverse peri-prosthetic fracture of a 73 year old woman. The

fracture was fixed using a LCP with 12 bi-cortical screws. The plate broke at 7

weeks post operation. This is shown in Figure 20 a and b.

The second case was the treatment of a fracture of the left femur of a 35 year old

man. A flexible configuration consisting of 4 bi-cortical screws and 2 mono-cortical

screws were used (Figure 20 c and d). This combination was successful as it

promoted healing and the implant did not fail.


56

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

(b) (a)

(d) (c)


57

4.2.2. Additional Cases

To perform a biomechanical analysis on the different cases, Chen et al created a

rigid simulation, contrast to the one shown in Figure 20 a and b. Similarly, a more

flexible fixation was simulated, contrast to that of Figure 20 c and d. It was found

that the failure of the implant was due to fatigue resulting from the large stress

concentrations in the plate due to the rigid fixation. The longer working length of

the flexible fixation method considerably reduced the stress concentrations in the

plate. Therefore, the implant fixed in this manner was seen to last up to 2000 days.

The paper suggested that an estimation of the fatigue life of the implant is an

important indication to surgeons of when an implant should be replaced (Chen et

al, 2010). In order to prolong the life of the implant, the stress concentrations

should be reduced. This is achievable by carefully positioning the screws, which

evidently have a great influence on the loading of the implant itself. Therefore, the

optimisation algorithm seeks to find the combination of screws which minimises

the stress in the plate (with respect to the interfragmentary displacement

constraints).

The optimisation algorithm was applied to simulated femur models similar to those

used in Chen et al (2010). This was done so as to compare the results obtained

from the optimisation algorithm to that from Chen et al (2010). A description of the

model will be followed by an explanation of the optimisation criteria used and the

result from the optimisation program.

4.2.3. Femoral Bone Geometry

A 3-dimensional intact bone was constructed from the CT (computed tomography)

images of the left femur of a 74 year old male with a weight of 70 kg. The femur was

scanned with a clinical CT scanner at 120kVP and a spacing of 1 mm for the axial

slices. The planar pixel size was 0.5 x 0.5 mm. A model of cortical bone was


58

generated from the CT-image data using the segmentation technique in the

software package Amira (Mercury Computer Systems Inc., Chelmsford, USA).

To generate the surfaces of the model, the segmented model was imported into

Rapidform 3D Scanning Software (INUS Technology Inc., Seoul, South Korea). This

model was saved as a .stl file and was then imported into SolidWorks (Dassault

Systèmes SolidWorks Corp, USA).

4.2.4. Plate and Screws of Femoral Construct

Similar to the simplified model of Case 1, a locking compression plate (LCP 4.5/5.0,

broad, 14 holes, 252 mm) was used. In the simulations, the threads of the screw

holes were removed to reduce the complexity of the finite element mesh.

4.2.5. Assembly

The models of the bone, plate and screw were assembled in the CAD software,

SolidWorks. The plate was positioned as laterally as close as possible to the bone

(the tensile side). An oblique 3mm fracture gap was created close to the middle of

the bone and a material added to the gap to simulate callus tissue. The assembled

model was saved as a parasolid .X_T file and imported into the finite element

software package, ABAQUS.

4.2.6. Materials

Similar to the simplified model, the material of the plate and screws was stainless

steel, whose Young’s modulus was 200 GPa and Poisson’s ratio, 0.3 (Materials Data

Handbook, 1972). The material for the bone, like the steel, was assumed to be

isotropic and homogenous. It was assigned a Young’s modulus of 20 GPa and a

Poisson’s ratio of 0.3 (Rho et al, 1993). For the callus material, the elastic modulus

and Poisson’s ratio that was used was 50 MPa and 0.3 respectively. This material

represented a callus tissue that was differentiating from initial connective tissue


59

(Young’s modulus = 3 MPa; Poisson’s ratio = 0.4) to soft callus (Young’s modulus =

1000 MPa; Poisson’s ratio = 0.3) (Claes et al., 1999).

4.2.7. Loading and Boundary Conditions

A force of 1717 N (2.5 times body weight) axial compressive force was applied to

the femoral head of the model along the direction of the mechanical axis (centre of

the femoral head to the inter-condylar area). It was assumed that the load would be

the maximum applied to the human femur during the gait cycle. It was applied as a

constant static throughout the analysis. The distal end of the femur was

constrained in the 3 translational directions.

The mesh of the femoral construct consisted of 35354 nodes and 136215 elements

using tetrahedral elements.

4.2.8. Variables to be Optimised

The variables to be optimised included two screws in the proximal region and four

in the distal region. Two screws were permanently fixed at each end of the plate

(Figure 21Figure 21). There were no screws placed in the two holes adjacent of

each side of the fracture gap. Having at least 2 to 3 screw-free holes to the fracture

line reduces the effects of a delayed or non-union (Wagner et al, 2007).


60

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

Screw positions 09

08

Screw positions

05

04

03

02


61

As described earlier, Chen et al (2010) showed an example of a screw configuration

which was used clinically. Screws were placed in positions 11, 5 and 4 as shown in

the simulation of Figure 22b. The X-ray image of the femur 4 months after fracture

is shown in Figure 22a.

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

11

05

04

(a) (b)


62

For the purpose of comparing the results of the screw configuration used in the

clinical case described in Chen et al (2010) with the screw configuration found in

this study, the FE software, ABAQUS, was used to run the exact combination of

screws used in the clinical case described in Chen et al (2010) (Figure 22b) with

similar boundary and loading conditions used in this study. The maximum

principal stress in the plate and the interfragmentary displacements were

calculated and displayed in Table 4.

Configuration of Chen et al (2010)

Shear Displacement Axial Displacement Max. Principal Stress

0.195 mm 0.566 mm 390 MPa

Table 4 Shear, axial displacement and stress in plate resulting from the configuration from Chen et al (2010)

4.2.9. Selection of Optimisation Criteria

As mentioned in the methods section, the specifications for optimisation vary from

model to model. For each case, it was required to decide the range of acceptable

axial and shear displacements (as optimisation criteria).

Similar to the simplified case, to assist in defining these ranges, two ‘extreme’

models were taken into consideration. It was assumed that the most flexible

construct was one in which all the variables were made to represent the absence of

a screw. This meant that only two screws were fixed in the two outermost screw

positions of the plate (Figure 23). Likewise, it was assumed that the most rigid

construct was one in which all the variables represented a screw including those at

the ends of the plate (Figure 23).

For both models, the shear and axial displacements, and the maximum stress in the

plate were calculated. From

Table 5, it is seen that the rigid construct experiences 57% more stress than the

flexible construct. In the axial and shear direction respectively, the flexible


63

construct experiences 3.5% and 64% more displacement than the rigid construct.

From these figures it can be seen that there is a wider range in which the shear

displacement may be optimised compared to displacement in the axial direction.

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

Screws

(a) (b)


64

Shear Displacement (mm)

Axial Displacement (mm)

Max. Principal Stress (MPa)

Rigid 0.158 0.544 482

Flexible 0.260 0.563 307

Table 5 Interfragmentary displacement and maximum principal stress for most rigid and most flexible femur models

For the optimisation method, the following displacements constraints were used:

Comparing the constraints abovementioned with those in

Table 5, it may be seen that constraints exclude the displacement values resulting

from the most flexible and the most rigid constructs. It is important to avoid large

shear displacements and to allow some axial displacement to promote optimum

healing conditions.

The penalty values are also to be decided upon. As described in the methods

section, one penalty term contributes to forcing the variables to either a 1 or 0. This

is so that each variable represents either the presence of a screw or the absence of

one. The other penalty term is to minimise the function value. This value should be

a small percentage of the objective function value (stress value).

With the constraints and penalty values defined, the optimisation algorithm may be

run for the femur model.

4.2.10. Solution

The optimisation method took 79 hours consisting of 621 iterations to arrive at a

solution on a processor of Intel(R) Xeon(R) CPU E5462 at 2.80GHz. Figure 24b


65

shows the optimal solution wherein the screws should be placed in positions 5 and

8 in the plate. For the purpose of comparing the patterns of the screw

configurations used in the clinical case in Chen et al (2010) and the one found by

the optimisation algorithm in this study, their solid models are provided here for

convenience.

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

From Figure 24a it can be seen that the screw configuration used in the clinical case

has a larger working length than the one provided as the solution from the

05

08

03

04

08

(a) (b)


66

optimisation algorithm in this study. A larger working length means that the

stresses have a larger area over which to spread, hence reducing the overall

concentration of stress (Stoffel et al, 2003; Duda et al, 2002). The maximum

principal stress was calculated to be 390 MPa for the clinical case compared to 466

MPa from the solution screw configuration in this study, in which the working

length was found to be shorter, due to the screw in position 5 of Figure 24b

compared to positions 3 and 4 of Figure 24a. The high stress is acceptable as it

means that the algorithm has optimised the solution with respect to the constraints

of the interfragmentary displacements in their respective directions.

The maximum principal stress is more notable in the plate than the bone (Figure

25). The highest stress is in the middle of the plate between 440 MPa and 460 MPa.

The stress concentrations around the middle 4 holes of the plate vary between 200

MPa and 250 MPa. Failure would occur across the area of maximum stress, i.e. in

the middle of the plate. The axial and shear interfragmentary displacements that

were calculated during the optimisation process are listed against the constraints

used in the optimisation method in Table 6.


67

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

Min. Constraint

Solution Max. Constraint

Shear Displacement (mm) 0.1 0.166 0.19 Axial Displacement (mm) 0.48 0.552 0.53

Table 6 Comparison of displacements from solution and those from constraints

It can be seen that for the solution model, displacement in the shear direction was

found to be within the constraints, whereas the value of displacement in the axial

direction was seen to be outside its range (Table 6).


68

Rigid Solution Flexible Shear Displacement (mm) 0.158 0.166 0.260 Axial Displacement (mm) 0.544 0.552 0.563 Max. Principal Stress (MPa) 482 466 307

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

From Table 7 it can be observed that the shear and axial displacements in the

‘Solution’ column are all within the range of the displacements calculated between

the most flexible and the most rigid constructs (Figure 23 a and b). The value of the

stress calculated for the solution construct was found to be skewed towards that of

the rigid construct. The value of the axial displacement for the solution was

approximately halfway between the flexible and rigid construct. The value of the

shear displacement is also skewed towards that of the rigid construct, the

difference being 5%.

From preliminary studies of this project, investigations regarding the influence of

screw positioning on the displacements of the bone fragments were undertaken.

Two constructs were compared. One was a very flexible construct in which two

screws were fixed to each end of the plate, providing a large working length (Figure

26a), while the other was a construct in which additional screws were placed closer

to the centre of the plate, providing a shorter working length (Figure 26b). This is

similar to the solution of the optimisation method of the femoral construct (Figure

24b). The results are shown in Table 8.

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

(a)

(b)

3.5Nm and 580N Fixed end

3.5Nm and 580N Fixed end


69

Flexible (Figure 26 a) Rigid (Figure 26 b)

Shear Displacement (mm) 1.87 0.85

Axial Displacement (mm) 2.00 1.21

Table 8 Axial and shear displacement resulting from the flexible and rigid constructs from Figure 26 (a) and (b)

Flexible (Figure 26 a) Rigid (Figure 26 b)

Shear Displacement (mm) 1.87 0.85

Axial Displacement (mm) 2.00 1.21

Table 8 shows that placing screws towards the centre of the plate reduces the axial

displacement by 40% compared to the flexible construct.

From a mechanical point of view, the quickest method of reducing the axial and

shear interfragmentary displacements is by positioning a screw as close as possible

to the fracture gap. In doing so, the rigidity of the construct as well as the stress in

the implant is increased. In the case of the optimisation solution of the femoral

construct, the screws were placed as close as possible to the fracture gap on either

side. Therefore, this combination represented a rigid construct whose

interfragmentary displacement values were close to those of the most rigid

construct.

Although placing the screws immediately adjacent to the fracture gap would reduce

the displacement by a further 40% (Table 9, row 2), and the stress in the plate

would not increase drastically, for biological reasons it is advised not to place the

screws very close to the fracture.


70

Pairs of screws removed from all screws in place

Axial Displacement

(mm)

Shear Displacement

(mm)

Maximum Principal

Stress (MPa)

0 0.41 0.40 42.7

1 0.72 0.82 42.6 2 0.76 1.02 30.0 3 0.77 1.17 29.0

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

This example shows the strong influence of the constraints in optimisation. It is

guaranteed that the change in the values of the constraints will result in a different

solution of screw configuration.

Solution Chen et al (2010) Shear Displacement (mm) 0.166 0.195 Axial Displacement (mm) 0.552 0.566

Max. Principal Stress (MPa) 466 390 Table 10 Comparison of displacement and stress from optimised solution with that from screw configuration used in clinical case from Chen et al (2010)

Table 10 compares the displacements and stresses for the solution model from the

optimisation process and the model in the clinical case from Chen et al (2010). Both

the shear and axial displacement was found to be less than that calculated from the

clinical screw configuration case from Chen et al (2010) by 17% and 3%

respectively. The constraints used in the optimisation algorithm forced the

displacements to fall within the range of their limits or to come as close as it can

towards those limits. On the other hand, in comparing the stresses, it is seen that

the stress calculated from the solution model is 20% more than the stress

calculated in the plate of the clinical case.

The optimisation algorithm used in this study did not allow for the solution to have

higher displacements such as that calculated from the clinical screw configuration

from Chen et al (2010). Because the limits were set, and in this case, happened to


71

be skewed towards the displacements resulting from the rigid construct, the

solution screw configuration tended to converge to a rigid construct allowing

highly restricted movement. Therefore, due to the screw configuration, higher

stresses were calculated in the plate.


72

5. Discussion

The advantage of using the mathematical programming approach is that the

number and types of constraints that may be imposed on the optimisation problem

are unlimited, provided the computational resources are available to handle the

calculations.

The work of Duda et al (2002), Stoffel et al (2003) and Tornkvist et al (1996)

focused on the influence of screw configurations on the strength, stiffness and

stress of the implant. However, they did not take into account the interfragmentary

displacements, which should be controlled to optimise the healing process. Using

mathematical programming, this study investigated the influence of screw

configurations on the stress in the plate as well as the interfragmentary

displacements in an attempt to optimise fracture healing.

The result from the femoral model showed that the screws should be positioned as

close as possible to the fracture gap on either side. By doing this, the displacement

in the axial direction is minimised. This is in agreement with Stoffel et al (2003)

whereby placing a screw closer to the fracture gap increases the axial stiffness of

the construct. It is important to note that while axial stiffness is important for

fixation stability, an excessive percentage is detrimental for healing and for the

implant.

The displacement criteria used for the optimisation method were made to be

between the range and excluding the displacements resulting from the most rigid

construct and that of the most flexible construct. The range of the shear

displacement was seen to be wider than the range of the axial displacement.

However, the displacements resulting from the optimum solution screw

configuration were calculated to be within their respective limits for the solution

screw configuration.


73

Stoffel et al (2003) used fracture gap sizes of 1 mm and 6 mm without any callus

material. In the case where screws were placed close to the fracture gap (short

working length), the stresses were found to be lower in the plate compared to that

when placing the screws further apart (large working length). Contrastingly, Duda

et al (2002), Chen et al (2010) as well as this study, found that when a short

working length was used, stresses in the plate were increased, whereas when a

longer working length was used, the stress decreased. The difference may be

caused by the loading of the callus material used in these studies.

During preliminary work, a cylindrical model was created with no callus material

included in the fracture gap. For testing purposes, this screw configuration of the

model was optimised using the optimisation program. Using a different set of

variables and displacement limits to the ones presented in this study, the following

result was found.

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.

(c)

(d)

(e)

(f)

(g)

(b)

(a)


74

The diagram in Figure 27 shows some of the screw configurations that the

optimisation algorithm tested prior to arriving to the optimal configuration. Two

screws were fixed at each ends of the plate (shown as black). Screws shown in

white were the ones that were chosen by the optimisation algorithm to be tested

with the black fixed screws. The columns on the right denote whether the

corresponding screw configuration resulting in acceptable axial and shear

displacements. By observing the change in screw configuration and its associated

change in displacement, the influence of the positioning of the screw may be seen.

For example, in (c) of Figure 27, it is seen that by positioning only one screw in the

third position from the fracture gap, the axial displacement is calculated to be

within acceptable limits according to the set displacement criteria. In (e) of Figure

27, by the addition of another screw at the other side of the fracture gap, in the

second position, the shear displacement is also calculated to be within acceptable

limits. In (b) the configuration that included a screw in the 4th hole from the

fracture gap resulted in large axial and shear displacements and did not fall within

the acceptable displacement range. It must be noted that the optimised solution is

to a large extent dependent on the constraints set. Changing the limits will change

the optimal solution.

The description above suggests that as well as searching for the optimal screw

configuration, the screw configurations that do not satisfy the constraints may be

seen by observing the iterative screw configurations. This is another advantage in

using mathematical optimisation techniques.

5.1. Limitations of this Study

With any optimisation algorithm, there are limitations in terms of the accuracy of

the solution.

The basic concept is to search for a sequence of improved function values of the

model, leading to the optimal solution. Commencing with an initial point, the


75

program begins a search in a direction and attempts to improve the function value

in that direction. Failure to improve the function value results in the exploration of

improved values in another direction.

Once the procedure finishes, the solution that is derived is a local minimum, rather

a global minimum. The reason for this is as follows.

To satisfy the user-defined constraints, there may be a number of solutions

possible. With the improvement in function value as well as finding the first

solution that satisfies the optimisation problem, the optimisation algorithm will

continue in this direction of exploration. This will continue until the function value

ceases to improve, and hence the optimum solution is found.

However, the solution found may not necessarily be the best solution for the

problem. It is possible that there is a ‘smaller’ minimum further along the function

and the optimisation algorithm has not detected this. Although Powell’s Method is

extremely powerful, the solution that is derived is locally optimal with respect to

the given constraints (Walsh, 1975). Figure 28 illustrates the concept of a local

minimum being found as opposed to a global minimum.

Figure 28 Illustration of concept of local versus global minimisation


76

A few ways of overcoming the problem of finding only the local minimum are listed

below.

1. Use a more concise range of values of the constraints, i.e. axial and shear

displacement. To do this, the exact quantitative measurements of the shear

and axial displacements that must be achieved should be known.

2. With the use of a computational bone-callus model, the constraints may be

better defined in terms of stresses and strains as opposed to

interfragmentary displacements only.

3. The use of a global optimisation algorithm.

In order to obtain more clinically applicable solutions it is important to improve the

simulated model as well as to improve the optimisation criteria.

5.2. Improvements to the Model

It was assumed that all screw-plate, bone-screw and bone-plate interfaces were

rigidly bonded. Unlike the contact between the screw threads and plate which may

be assumed to be tightly bonded, the contact between bone and screw is not one

which is tightly bonded. At the commencement of fracture treatment, there would

be high screw-bone purchase. However, with compressive and tensile forces acting

on the screw during loading causing bending and screw loosening, contact between

the screw and bone gradually reduces. Modelling of this contact would contribute

to the accuracy of the solution as it would influence stress in the plate, which is the

parameter to be minimised in the optimisation process. In this study, however, for

simplicity, the contact between each of the materials was assumed to be perfect.

As mentioned in previous chapters, interfragmentary displacement values that

have been seen to promote fracture healing for a 3 mm mid-shaft gap in vivo, have

been measured and are available in the literature. However, interfragmentary

displacements have not been measured in vivo for other types of fractures, i.e.


77

fractures of different gap sizes and positions along the bone length. Therefore, the

optimisation criteria (interfragmentary displacements) that have been used in this

study cannot be used for the optimisation of screw configuration for all types of

fracture gaps. Hence, it is important to use optimisation criteria that are

independent of the fracture gap size and location of the fracture. Stresses and

strains occurring in modelled callus material may be measured, as different

amounts of stresses and strains experienced in the tissue translate to the different

types of tissues formed, i.e. connective tissue, fibrous tissue, cartilage, bone, etc.

With the modelling of a callus material around the fracture site, and the need to

accurately measure the stresses and strains occurring in it, more refined material

models should be used, such as elastic-plastic models (Cheal et al., 1983). A higher

element mesh density should be used in the gap region compared to the rest of the

model.

5.3. Improvement to the Optimisation Criteria

As abovementioned the optimisation criteria play an important role in the

formation of the solution.


78

Figure 29 “Boundaries for optimal healing in the sheep model[s] that lead to timely healing” (Source: Epari et al, 2007)

Figure 29 is an extraction from the work by Epari et al (2007) which shows the

predicted type of healing outcome for fixation constructs that support

corresponding amounts of axial compressive stiffness and shear stiffness. From the

plot, the best healing outcome would be a result of the combination of high shear

stiffness and moderate axial compressive stiffness.

An alternate to measuring the stresses and strains occurring in the callus material

to determine the type of tissue being formed (section 5.1), optimisation criteria

may be based on the suggested stiffness values (from Epari et al, 2007).

Alternatively, the shear displacement may be limited to that calculated due to the

rigid construct, while displacement in the axial direction may be limited to the


79

middle range between the axial displacement of the rigid construct and that of the

flexible construct.

5.4. Future Work - Improvements to the Optimisation

Method

For the femoral construct, the optimisation method took 79 hours to arrive at a

solution. At this stage of complexity of the model, it is necessary to reduce the

amount of time to arrive at a solution. In the future, the model is expected to

become more accurate, and with the addition of a fracture healing iterative process,

there will be an increase in its complexity. This means that the time taken to run a

single screw configuration simulation will increase drastically, and hence the time

taken to arrive at an optimal solution will also be escalated. Therefore, to remain

within reasonable time, it is necessary to study the optimisation method

thoroughly to find areas in the code where computational time may be reduced.

The optimisation method was seen to calculate many more iterations than

necessary. For example, in the case of the femoral construct, there were 6 screw

positions to be optimised. This means that there are combinations of screws

that are possible. The optimisation method was found to have done 557 more

iterations than necessary. This is because the optimisation method needs to

gradually force the value of the variable to either a represent a screw or the

absence of a screw. This was done using penalty methods which was a way to

minimise with respect to discrete variables.

There has been some work done in this area of programming whereby the number

of iterations may be reduced by trying to eliminate the continuous variables, and

focus on the discrete variables. A method called Design of Experiments may be used

to further reduce the number of iterations to be calculated. This method should be


80

used and implemented in the optimisation code to decrease computational effort

and save time.

The problem of finding the best screw configuration by mathematical programming

is similar to the famous travelling salesman problem (TSP) which is to find the

cheapest way to visit all the cities once and return to the starting point, given a set

of cities and the cost of travel between them.

In the biomechanical problem presented in this study, the city represents screw

positions and the distance represents the maximum stress in the plate which is to

be optimised. The money represents the constraints of displacement which make

the problem considerably harder.

TSP, which is similar to the screw configuration problem, has been described as a

NP-complete (non-deterministic polynomial) optimisation problem. This means

that there are known heuristics methods to solve the problem, however, it takes a

very large amount of resources to find a solution in polynomial time

(computational time taken to solve the problem). This means that if the

optimisation methods of Simulated Annealing and Genetic Algorithms (global

optimisation methods) were used, the time and resources needed to solve the

problem of screw configurations would be unreasonably large. Additionally, with

an increase in the number of screw positions the polynomial time would increase

exponentially. Hence, the TSP, till now, is one of the most intensely studied

problems in optimisation as it can be applied in many fields including planning and

logistics. Optimisation researchers are striving to find ways of reducing the time to

solution to a more reasonable one.


81

6. Conclusions

This project developed a computational program in Fortran to perform

mathematical optimisation to determine the best screw configuration for an

internal fixation device within constraints of interfragmentary movement by

minimising the corresponding stress in the plate. The algorithm involved the

interaction of a finite element software with an optimisation algorithm.

It was shown that mathematical programming is potentially advantageous in that it

allows for the testing of a number of variables provided the required computational

resources are available. While some research groups (Stoffel et al, 2003; Tornkvist

et al, 1996; Duda et al, 2002) have investigated the stresses produced in the

internal fixation device, the study of the effects of screw configuration on

interfragmentary movement (or a method of determining fracture healing

progress) is necessary when determining its influence on the fracture healing

process. The optimisation algorithm allows for this calculation.

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

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

the screw configuration solution 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.

It was seen that the displacement constraints play a critical role in determining the

ultimate screw configuration. Increasing or decreasing the range of values

impacted significantly on the solution. Therefore, it is vital that the constraints

chosen are close to representing the physiological situation.


82

Mathematical programming allows the observation of trends of screw

configurations that gradually progress towards the solution. In addition, it shows

the screw configurations that do not satisfy the requirements.

To improve the screw configuration solution, the model should be refined to be

more representative of the physiological fracture condition. The optimisation

criteria should be better defined in terms of the measurement of fracture healing

progress (stiffness of construct or the measurement of stress or strains in the

callus).


83

7. References

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Esen, Ö., et al. (2008). "A Mathematical Immunochemoradiotherapy Model: A Multiobjective Approach." Nonlinear Analysis: Real World Applications 9(2): 511-517. Field, J. R., et al. (1999). "The Influence of Screw Omission on Construction Stiffness and Bone Surface Strain in the Application of Bone Plates to Cadaveric Bone." Injury 30(9): 591-598. Goodship, A. E., et al. (1985). "The Influence of Induced Micromovement Upon the Healing of Experimental Tibial Fractures." The Journal of Bone and Joint Surgery (Br) 67-B(4): 650-655. Hou, S.-M., et al. (2004). "Mechanical Tests and Finite Element Models for Bone Holding Power of Tibial Locking Screws." Clinical Biomechanics 19(7): 738-745. Kenwright, J., et al. (1989). "Controlled Mechanical Stimulation in the Treatment of Tibial Fractures." Clinical Orthopaedics and Related Research 241: 36-47. Korvick, D. L., et al. (1988). "The Effects of Screw Removal on Bone Strain in an Idealised Plated Bone Model." Veterinary Surgery 17(3): 111-116. Maratt, J. D., et al. (2008). "An Integer Programming Model for Distal Humerus Fracture Fixation Planning." Computer Aided Surgery 13(3): 139 - 147. Mathews, J. H., Fink, Kurtis. D. (2004). Numerical Methods Using Matlab. New Jersey, Prentice-Hall, Inc. Muraca, R. F., et al. (1972). Materials Data Handbook. California, USA, Western Applied Research and Development Inc. Nelder, J. A., et al. (1965). "A Simplex Method for Function Minimization." The Computer Journal 7: 308-313. Perren, S. M. (2002). "Review Article: Evolution of the Internal Fixation of Long Bone Fractures." The Journal of Bone and Joint Surgery (Br) 84-B(8): 1093-1110. Press, W. H., Vettering, W. T., Teukolsky, S. A., Flannery, B. P. (1992). Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge, Cambridge University Press. Rao, S. S. (1996). Engineering Optimization: Theory and Practice. New York, Wiley. Rho, J. Y., et al. (1993). "Young's Modulus of Trabecular and Cortical Bone Material: Ultrasonic and Microtensile Measurements." Journal of Biomechanics 26(2): 111-119.


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Riemer, B. L., et al. (1992). "Immediate Plate Fixation of Highly Comminuted Femoral Diaphyseal Fractures in Blunt Polytrauma Patients." Orthopaedics 15: 907-916. Rozema, F. R., et al. (1992). "Computer-Aided Optimization of Choice and Positioning of Bone Plates and Screws Used for Internal Fixation of Mandibular Fractures." International Journal of Oral and Maxillofacial Surgery 21(6): 373-377. Ruedi, T. P., et al. (2001). A.O. Principles of Fracture Management. Stuttgart, AO Publishing and Thieme. Sanders, R., et al. (2002). "Minimal Versus Maximal Plate Fixation Techniques of the Ulna: The Biomechanical Effect of the Number of Screws and Plate Length." Journal of Orthopaedic Trauma. 16(3): 166-171. Seebeck, J., et al. (2005). "Mechanical Behavior of Screws in Normal and Osteoporotic Bone." Osteoporosis International 16(0): S107-S111. Shipley, R. J., et al. (2002). Fatigue Properties of Implant Materials, Asm Handbook, Vol.11, Fatigue Analysis and Prevention. Ohio, ASM International. Singh, G., et al. (2006). "Comparison of Multi-Modal Optimization Algorithms Based on Evolutionary Algorithms." Association for Computer Machinery: 1305-1312. Sommer, C., et al. (2003). "First Clinical Results of the Locking Compression Plate (Lcp)." Injury 34(Supplement 2): 43-54. Stoffel, K., et al. (2003). "Biomechanical Testing of the L.C.P- How Can Stability in Locked Internal Fixators Be Controlled?" Injury 34: S-B11-S-B19. Taha, H. A. (1976). "Operations Research: An Introduction." Tornkvist, H., et al. (1996). "The Strength of Plate Fixation in Relation to the Number and Spacing of Bone Screws." Journal of Orthopaedic Trauma. 10(3): 204-208. Wagner, M., et al. (2007). "Locked Plating: Biomechanics and Biology and Locked Plating: Clinical Indications." Techniques in Orthopaedics 22(4): 209-218. Walsh, G. R. (1975). Methods of Optimisation. New York, John Wiley and Sons Ltd. Welfare, A. I. o. H. a. (2006). "Australian Hospital Statistics 2004–05." Health Services Series no. 26: AIHW cat. no. HSE 41. . Woo, S. L.-Y., et al. (1984). "Less Rigid Internal Fixation Plates: Historical Perspectives and New Concepts." Journal of Orthopaedic Research 1(4): 431-449.


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87

Appendix

Optimisation program including subroutines in Fortran !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

OPTIMISATION OF STRESS IN INTERNAL FIXATION PLATE USING POWELL'S METHOD.

-------------------------------------------------------------

Using the screw materials for the screw holes as variables, this program

optimises the stress in the internal plate fixator.

The screw materials are interchanged according to the displacement calculated

between the fracture gap, and the stress in the whole plate.

The optimisation procedure used is POWELL'S METHOD. The object function is

calculated in the function, E_FUNC.

The DISPLACEMENT AND STRESS are calculated by ABAQUS (finite element method);

It is in functon E_FUNC.

The programs needed to run this optimisation procedure are:

1. Martin_01.py - to read stress and displacement from odb.

2. Martin_01.inp - input file for ABAQUS analysis.

3. PracticalExampleOptiVar.txt - file with variables to be optimised.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

PROGRAM test

IMPLICIT NONE

REAL, PARAMETER :: TOLER = 1

! the fractional tolerance in the function value such that failure to decrease

! by more than this amount on one iteration signals doneness.

INTEGER, PARAMETER :: NOPTI = 6

! number of variables

EXTERNAL E_FUNC

REAL,DIMENSION(6) :: OPV

! starting point (e.g.20000.,20000.,20000.,20000.) Young’s modulus of screws

! see first DO loop

INTEGER IOPTI

REAL ENERGY

DO 10 IOPTI=1, NOPTI

OPV(IOPTI) =0.0001

10 CONTINUE

WRITE(*,*) "OPVAR = ", (OPV(IOPTI), IOPTI=1, NOPTI)

CALL POWELL(NOPTI, OPV, TOLER, ENERGY, E_FUNC)

! call powell's method to minimise the stress.

WRITE (*, *) 'Energy = ' , ENERGY

WRITE (*, *) (OPV(IOPTI), IOPTI=1, NOPTI )


88

END PROGRAM

FUNCTION E_FUNC(NOPTI, OPV)

IMPLICIT NONE

REAL, PARAMETER :: Force=580,YOUNG=2.0e+05 !compressive load on bone in

Newtons.

REAL,DIMENSION(6):: OPVAR, OPV

INTEGER NOPTI,IOPTI,status, i

REAL penalty,Disfix,E_FUNC,X1,X2,DisplaX, Y1,Y2,DisplaY,

DisplacementX, DisplacementY

REAL Displacement, penalty2,ave2 ! Stiffness, MaxPrincipalStress,

REAL, DIMENSION(132056):: value

CHARACTER*50 AAA(6)

DATA AAA /'*Material, name=Screw_11',&

'*Material, name=Screw_10',&




'*Material, name=Screw_04'/


OPVAR(IOPTI) = ABS(OPV(IOPTI))* YOUNG

WRITE(*,*) 'OPV(IOPTI)*YOUNG = ',OPVAR(IOPTI)

10 CONTINUE

WRITE(*,*) "E-func OPV = ", (OPV(IOPTI), IOPTI=1, NOPTI)

WRITE(*,*) "E-func OPVAR = ", (OPVAR(IOPTI), IOPTI=1, NOPTI)

E_FUNC = 0.

Displacement=0.

! Write into a file, 'PracticalExampleOptiVar.txt', the materials of the screws

of the plate.

! These are the variables to be optimised in the optimisation procedure.

! The DATA above is cut out from the 'Assemb4Weak.inp' input file, and saved

into

! a new file called 'TrussOptiVar.txt'. For each iteration in the optimisation

! procedure, these cross-sectional areas are rewritten.

! In the 'Truss.inp' file, the keyword *INCLUDE,

input=PracticalExampleOptiVar.txt must be

! written in place of the cut-out variables. So the file

'PracticalExampleOptiVar.txt' will be

! included in ABAQUS input file 'Assemb4Weak.inp'.

OPEN (UNIT= 6, FILE='PracticalExampleOptiVar.txt')

loop1: DO IOPTI=1, NOPTI

! OPVAR(IOPTI) = ABS(OPVAR(IOPTI))

! roundingmodulus: IF (OPVAR(IOPTI).LT.99500) THEN

! OPVAR(IOPTI) = 1000.

! ELSE IF (OPVAR(IOPTI).GE.99500) THEN

! OPVAR(IOPTI) = 200000.

! END IF roundingmodulus

END DO loop1

DO IOPTI=1, NOPTI

WRITE (6, *) AAA(IOPTI)

WRITE (6, *) '*Elastic'

WRITE (6, 100) OPVAR(IOPTI), ', 0.3'

100 FORMAT (' ',ES10.3,A)

END DO


89

CLOSE (6)

OPEN (UNIT = 5, FILE="Cylinder_CompTors_6VarsOpti310709.lck",

IOSTAT=status, ACTION='READ', STATUS='OLD')

deletelockfile: IF (status == 0.) THEN

CALL SYSTEM ("echo y | del Cylinder_CompTors_6VarsOpti310709.lck")

END IF deletelockfile

CALL SYSTEM ("echo y | abaqus job=Cylinder_CompTors_6VarsOpti310709

interactive")

CALL SYSTEM ("abaqus python Cylinder_CompTors_6VarsOpti310709.py")

! Call ABAQUS to do the finite element analysis.

! Call python script file 'PracticalExampleOpti2.py'

! The 'PracticalExampleOpti2.py' is used to read from the ABAQUS

'Assemb4Weak.odb' file, the

! displacement and stress values and write them into a file called

! 'PracticalExampleDispRes.txt', so that Fortran is able to read the numbers.

OPEN (UNIT = 5, FILE='PracticalExampleDispRes.txt')

READ (5, *) X1,Y1,X2,Y2

CLOSE (5)

DisplaX = ABS(X1-X2)

DisplaY = ABS(Y1-Y2)

!WRITE(*,*) 'X1 = ', X1 !, X1, 'X2 = ', X2

!WRITE(*,*) 'Y1 = ', Y1, 'Y2 = ', Y2

!WRITE(*,*) 'DisplaX = ', DisplaX

!WRITE(*,*) 'DisplaY = ', DisplaY

!Displacement = SQRT((ABS(DisplaX)**2) + (ABS(DisplaY)**2))

WRITE(*,*) 'Displacement in x direction = ', DisplaX

WRITE(*,*) 'Displacement in y direction = ', DisplaY

DisplacementX = DisplaX

DisplacementY = DisplaY

!openif: IF (status == 0) THEN

OPEN

(UNIT=5,FILE='PracticalExampleStressRes.txt',ACTION='READ',STATUS='OLD',

IOSTAT=status)

DO i = 1, 132056

READ (5, *) value(i)

! reads the mises stress of all 517588 elements in element set, PLATE PLUS

SCREWS.

END DO

CLOSE (5)

! MaxPrincipalStress = (SUM(value, 1, value>100) / COUNT(value>100, 1))

! WRITE(*,*) 'MaxPrincipalStress = ', MaxPrincipalStress

WRITE(*,*) 'largest value max principal= ', MAXVAL(value,1)

! this is the average Von Mises stress in the element set, 'SET_MIDDLEOFPLATE'

!END IF openif

! on one side,i.e. 2 steel screws on screw 1 and 8, and No_Screw material in

the rest of

! the screws holes. The strongest screw combination is the one where all the

! steel screws are connected to the bone


90

penalty = 1.0e+06

Disfix = 1.2

E_FUNC = MAXVAL(value,1)

IF (DisplacementX.LT.0.9 .OR. DisplacementX.GT.1.4) THEN

E_FUNC = E_FUNC + penalty * ABS(Displacement-Disfix)

END IF

IF (DisplacementY.LT.0.9 .OR. DisplacementY.GT.1.45) THEN

E_FUNC = E_FUNC + penalty * ABS(Displacement-Disfix)

END IF

! Calculate the object function, using penalty to introduce the condition of

! the same displacement. The displacement at the loading point is fixed at:

! Disfix = 0.4. Penalty parameter = 1.e+06.


ave2 = OPV(IOPTI)

IF ( ave2.LT.0.5) THEN

penalty2=ABS(ave2-0.0001)

ELSE IF ( ave2.GE.0.5) THEN

penalty2=ABS(ave2-1.0)

END IF

E_FUNC = E_FUNC + (penalty2*penalty)

20 CONTINUE

! Penalty2 is to control the variables (Young's modulus). By forcing ave2

(Young's modulus)

! to go to 0.01 for the NoScrew part, and to go to 1 for the steel part.

! We want to reach the condition where (Displa - Disfix = 0). So when this

! condition is not true, it is multiplied by the penalty number.

! Gradually E_FUNC should become smaller.

WRITE(*,*) ' E_FUNC = ', E_FUNC

RETURN

END FUNCTION

! Subroutine POWELL uses the subroutines mnbrak and linmin,and functions

! f1dim and golden, from Numerical Recipes.

SUBROUTINE powell (n, p, ftol, fret, func)

IMPLICIT REAL (A-H, Q-Z)

DIMENSION p(n)

REAL, EXTERNAL :: f1dim, func

PARAMETER (NMAX=1225, ITMAX=50000)

DIMENSION xi(NMAX, NMAX), pt(NMAX), ptt(NMAX), xit(NMAX)

INTEGER IOPTI

WRITE(*,*) "From Powell Subroutine OPVAR = ", (p(IOPTI), IOPTI=1, n)

! produces matrix of direction wectors that are conjugate to each other

! e.g. if n=3 then matrix = [1 0 0]

! [0 1 0]

! [0 0 1]

do 1301 i=1,n

do 1301 j=1,n

xi(i,j)=0.

1301 continue

do 1302 i=1,n

xi(i,i)=1.


91

1302 continue

fret=func(n, p)

do 11 j=1,n

pt(j)=p(j)

11 continue

iter=0

1 iter=iter+1

fp=fret

ibig=0

del=0.

do 13 i=1,n

do 12 j=1,n

xit(j)=xi(j,i)

12 continue

fptt=fret

call linmin(p,xit,n,fret,func)

if (abs(fptt-fret).gt.del) then

del=abs(fptt-fret)

ibig=i

endif

13 continue

if (2.*abs(fp-fret).le.ftol*(abs(fp)+abs(fret))) return

do 14 j=1,n

ptt(j)=2.*p(j)-pt(j)

xit(j)=p(j)-pt(j)

pt(j)=p(j)

14 continue

fptt=func(n, ptt)

if (fptt.ge.fp) goto 1

t=2.*(fp-2.*fret+fptt)*(fp-fret-del)**2-del*(fp-fptt)**2

if (t.ge.0.) goto 1

call linmin (p, xit, n, fret,func)

do 15 j=1,n

xi(j,ibig)=xi(j,n)

xi(j,n)=xit(j)

15 continue

goto 1

END

SUBROUTINE LINMIN (p, xi, n, fret, func)

IMPLICIT REAL (A-H, Q-Z)

DIMENSION p(n), xi(n)

PARAMETER (NMAX=1225, TOL=1.E-03)

DIMENSION pcom(NMAX), xicom(NMAX)

COMMON /f1com/ pcom, xicom, ncom


ncom=n

do 11 j=1,n

pcom(j)=p(j)

xicom(j)=xi(j)

11 continue

ax=0.

xx=1.

call mnbrak (ax,xx,bx,fa,fx,fb,f1dim,func)

fret=golden(ax,xx,bx,f1dim,func,TOL,xmin)

! fret= brent(ax,xx,bx,f1dim,func,TOL,xmin)

do 12 j=1,n

xi(j)=xmin*xi(j)

p(j)=p(j)+xi(j)

12 continue


92

return

end

FUNCTION f1dim(x,func)

IMPLICIT REAL (A-H, O-Z)

PARAMETER (NMAX=1225)

DIMENSION pcom(NMAX), xicom(NMAX), xt(NMAX)

COMMON /f1com/ pcom, xicom, ncom

do 11 j=1,ncom

xt(j)=pcom(j)+x*xicom(j)

11 continue

f1dim=func(ncom, xt)

return

END

SUBROUTINE mnbrak (ax, bx, cx, fa, fb, fc, f1dim,func)

! Mnbrak: this subroutine is used to bracket a minimum in one dimensional

minimisation.

! This provides the initial guess for the search for the minimum in that

dimension.



PARAMETER (GOLD=1.618034, GLIMIT=100., TINY=1.E-15)

fa=f1dim(ax,func)

fb=f1dim(bx,func)

if (fb.gt.fa) then

dum=ax

ax=bx

bx=dum

dum=fa

fa=fb

fb=dum

endif

cx=bx+GOLD*(bx-ax)

fc=f1dim(cx,func)

1 if (fb.ge.fc) then

r=(bx-ax)*(fb-fc)

q=(bx-cx)*(fb-fa)

if ( abs(q-r) .lt. TINY ) then

WRITE(6,*) ax, bx, cx, fa, fb, fc

stop

endif

u=bx-((bx-cx)*q-(bx-ax)*r)/(2.*sign(max(abs(q-r), TINY), q-r))

ulim=bx+GLIMIT*(cx-bx)

if ((bx-u)*(u-cx).gt.0.) then

fu=f1dim(u,func)

if (fu.lt.fc) then

ax=bx

fa=fb

bx=u

fb=fu

return

else if (fu.gt.fb) then

cx=u

fc=fu

return


93

endif

u=cx+GOLD*(cx-bx)

fu=f1dim(u,func)

else if ((cx-u)*(u-ulim).gt.0.) then

fu=f1dim(u,func)

if (fu.lt.fc) then

bx=cx

cx=u

u=cx+GOLD*(cx-bx)

fb=fc

fc=fu

fu=f1dim(u,func)

endif

else if ((u-ulim)*(ulim-cx).ge.0.) then

u=ulim

fu=f1dim(u,func)

else

u=cx+GOLD*(cx-bx)

fu=f1dim(u,func)

endif

ax=bx

bx=cx

cx=u

fa=fb

fb=fc

fc=fu

goto 1

endif

return

END

FUNCTION golden(ax,bx,cx,f1dim,func,tol,xmin)


REAL,EXTERNAL :: f1dim, func

PARAMETER (R=.618034, C=1.-R)

x0=ax

x3=cx

if (abs(cx-bx).gt.abs(bx-ax)) then

x1=bx

x2=bx+C*(cx-bx)

else

x2=bx

x1=bx-C*(bx-ax)

endif

f1=f1dim(x1,func)

f2=f1dim(x2,func)

1 if (abs(x3-x0).gt.tol*(abs(x1)+abs(x2))) then

if (f2.lt.f1) then

x0=x1

x1=x2

x2=R*x1+C*x3

f1=f2

f2=f1dim(x2,func)

else

x3=x2

x2=x1

x1=R*x2+C*x0

f2=f1

f1=f1dim(x1,func)

endif

goto 1


94

endif

if (f1.lt.f2) then

golden=f1

xmin=x1

else

golden=f2

xmin=x2

endif

return

END

Python script file to read out values from FEA

# to print axial displacement from Step-1(compression)

from odbAccess import *

from abaqusConstants import *

odb = openOdb(path='Cylinder_CompTors_6VarsOpti310709.odb')

step1 = odb.steps['CompressionTorsion']

lastFrame = odb.steps['CompressionTorsion'].frames[-1]

#for fieldName in lastFrame.fieldOutputs.keys():

# print fieldName

myAssembly = odb.rootAssembly

#for instanceName in odb.rootAssembly.instances.keys():

# print instanceName

#for nodesets in odb.rootAssembly.nodeSets.keys():

# print nodesets

centre = odb.rootAssembly.nodeSets['NODE_DISP']

displacement = lastFrame.fieldOutputs['U']

centreDisplacement = displacement.getSubset(region=centre)

centreValues = centreDisplacement.values

dispFile = open('PracticalExampleDispRes.txt','w')

for v in centreValues:

# print v.data[1]

# print v.magnitude

dispFile.write('%10.4f\n %10.4f\n' % (v.data[0],v.data[1]))

dispFile.close()

odb.close()

# to print out the Von Mises stress in the middle of the plate

# in the set SET_MIDDLEOFPLATE

from odbAccess import *

from abaqusConstants import *

odb = openOdb(path='Cylinder_CompTors_6VarsOpti310709.odb')

step1 = odb.steps['CompressionTorsion']

myAssembly = odb.rootAssembly

lastFrame = odb.steps['CompressionTorsion'].frames[-1]

centre = odb.rootAssembly.elementSets['PLATE_PLUS_SCREWS']

stress = lastFrame.fieldOutputs['S']

centreStress = stress.getSubset(region=centre)

centreValues = centreStress.values

dispFile = open('PracticalExampleStressRes.txt','w')

for v in centreValues:

# print v # this gets all the values for the region(centreStress).

# from here we can see all the attributes or types of values

# that are present under the attribute, values.

# mises=v.mises

# elementlabel=v.elementLabel


95

# print elementlabel, mises

dispFile.write('%10.4f' % (v.maxPrincipal)+'\n')

dispFile.close()

odb.close()

application of an optimisation algorithm to configure an ... · application of an optimisation...

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