MATHEMATICAL SIMULATION OF BREAST CANCER GROWTH USING HIGH

PERFORMANCE COMPUTING

DOLLY SII TIEN CHING

UNIVERSITI TEKNOLOGI MALAYSIA

MATHEMATICAL SIMULATION OF BREAST CANCER GROWTH USING HIGH

PERFORMANCE COMPUTING

DOLLY SII TIEN CHING

A report submitted in partial fulfillment

of the requirements for the award of the degree of

Bachelor of Science and Computer and Education (Mathematics)

Faculty of Education

Universiti Teknologi Malaysia

APRIL 2006

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ACKNOWLEDGEMENT

My greatest gratitude goes out to my supervisor and mentor, Dr Norma Alias

for the invaluable advice, guidance and support given to me throughout the progress

of this thesis. I obtained a lot of help from her when doing the thesis. She also

provided me a lot of sources and benefit comments to complete my thesis.

In addition, I would like to acknowledge my family’s support, encouragement

and care. And I would like to thank to my entire dearest housemates who always

support me and provide me helpful suggestions during doing this thesis.

Last but not least, I would like to thank all those who are involved directly or

indirectly in helping me to complete this thesis.

THANK YOU.

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ABSTRACT

This thesis focuses on the implementation of parallel algorithm for the

simulation of breast cancer growth using one dimensional hyperbolic equation on a

distributed parallel computer system. The hyperbolic equation can be used as

mathematical models in science and engineering fields especially for Fluid Dynamic

Systems. The numerical finite-difference method is chosen as a platform for

discretizating the hyperbolic equations. The numerical solution is applied to solve a

mathematical model in medicine field. Breast cancer is the commonest female

malignancy in Malaysia and all over the world. The incidence of breast cancer in

Malaysia is estimated to be around 27 per 100,000 population, with close to 3,000

new cases annually. The mathematical model is a hyperbolic model that visualizes

the growth of breast cancer. The pressure inside the breast is increasing with the

tumor size. Parallel Virtual Machine (PVM) is emphasized as communication

platform in parallel computer system. Besides, the performance of the parallel

computing will be analyzed from the aspect of execution time, speedup, efficiency,

effectiveness and temporal performance.

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ABSTRAK

Tesis ini memberi fokus kepada penggunaan teknik algoritma selari dalam

menyelesaikan dan melakarkan pertumbuhan kanser payudara dengan menggunakan

persamaan hiperbolik satu dimensi. Persamaan hiperbolik boleh digunakan sebagai

model matematik dalam bidang sains dan kejuruteraan terutamanya dalam masalah

aliran haba, gelombang, pengiraan keupaya elektrik. Kaedah penghampiran beza

terhingga digunakan untuk mendiskretkan persamaan hiperbolik. Kaedah analisis

berangka ini telah diaplikasikan untuk menyelesaikan satu model matematik dalam

bidang perubatan. Kanser buah dada merupakan kanser yang paling kerap berlaku di

kalangan wanita di Malaysia dan seluruh dunia. Insiden bagi kanser buah dada

dianggarkan berlaku pada kadar 27 bagi setiap 100,000 penduduk, dengan 3,000 kes

baru dilaporkan berlaku setiap tahun. Justeru itu, model matematik tersebut dapat

menggambarkan pertumbuhan barah buah dada. Tekanan tumor buah dada akan

bertambah mengikut pertumbuhan saiz tumor. Mesin Selari Ingatan Maya iaitu

PVM digunakan sebagai pelantaraan komunikasi dalam sistem komputer selari.

Selain itu, prestasi algoritm selari dari aspek kecepatan, kecekapan, kebersanan dan

masa pelaksanaan telah dianalisiskan.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

TITLE i

DECLARATION ii

ACKNOWLEDGEMENT iii

ABSTRACT iv

ABSTRAK v

TABLE OF CONTENTS vi

LIST OF TABLES ix

LIST OF FIGURES x

LIST OF SYMBOLS xi

LIST OF APPENDINCES xii

1 INTRODUCTION

1.1 Introduction 1

1.2 Problem Formulation 3

1.3 Research Objectives 4

1.4 Scope of Research 5

1.5 Outline 5

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2 FINITE DIFFERENCE METHODS FOR HYPERBOLIC

EQUATION

2.1 Introduction of Partial Differential Equations (PDEs) 8

2.2 Finite Difference Approximations to Derivatives 15

2.3 Hyperbolic Partial Differential Equations 17

2.4 Iteration Point Methods for Solving the Finite

Difference Equations 21

2.4.1 Gauss Seidel Iterative Method 22

2.4.2 Red Black Gauss Seidel Iterative Method 24

3 PARALLEL VIRTUAL MACHINE (PVM)

3.1 The Parallel Computing 26

3.3.1 Parallel Architectures 27

3.2 A Brief History of PVM 30

3.3 Introduction to PVM 31

3.4 PVM System 33

3.5 PVM Programming 35

4 THE DETECTION OF BREAST CANCER

4.1 Introduction 40

4.1.1 What is the Breast? 41

4.1.2 What is Breast Cancer? 42

4.1.3 Types of Breast Cancers 43

4.2 The Mathematical Model 47

4.2.1 The Discretization of the Mathematical Model 49

4.3 The Visualization of the Breast Cancer Growth 51

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5 ANALYSIS OF THE PERFORMANCE OF PVM

5.1 Introduction 55

5.2 Performance Analysis 56

5.2.1 The Execution Time 57

5.2.2 The Speedup 59

5.2.3 The Efficiency 61

5.2.4 The Effectiveness 63

5.2.5 The Temporal Performance 65

6 CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions 68

6.2 Recommendations for Further Study 69

REFERENCES 71

Appendices A - F 73

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LIST OF TABLES

TABLE NO TITLE PAGE

4.1 The Pressure of Breast Cancer 51

4.2 Sound Velocity and Coefficients of Frequency-Dependent Power Law

Attenuation of The Breast Tissue and Tumor 53

5.1 Time, Convergence and Time step for Parallel Algorithm and

Sequence 56

5.2 The Execution Time of Parallel Computer 57

5.3 The Speedup of Parallel Computer 59

5.4 The Efficiency of Parallel Computer 61

5.5 The Effectiveness of Parallel Computer 63

5.6 The Temporal Performance of Parallel Computer 65

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LIST OF FIGURES

FIGURES NO. TITLE PAGE

2.1 The Grid System for Finite Difference Method 19

3.1 Flynn’s Taxonomy of Computer Architectures 28

3.2 PVM Computation Model 34

3.3 PVM program hello.c 37

3.4 PVM program hello_other.c 38

4.1 Normal Breast Structure 42

4.2 The Growth Rate of Breast Cancer 52

5.1 The Execution Time versus Number of Processors 58

5.2 The Speedup versus Number of Processors 60

5.3 The Efficiency versus Number of Processors 62

5.4 The Effectiveness versus Number of Processors 64

5.5 The Temporal Performance versus Number of Processors 66

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LIST OF SYMBOLS

yx, - The space at coordinate system

p - Pressure

x - Space variable

t - Time variable

c - Sound speed of the traversed tissue

γ - Damping or attenuation parameter

0α - Dependent material constant

f - Frequency

y - Frequency-power exponent varying from 0 to 2 and depending

on tissue

tp∂∂ - The change of the pressure at time, t

2

2

tp

∂∂ - Second order derivative for p at t

hΔ - Small interval for x

kΔ - Small interval for t

)( 4hO - Term containing fourth and higher powers of h RΩ - Domain at red grid BΩ - Domain at black grid

tΔ - Time step size

M - Mass matrices

K - Stiffness matrices

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LIST OF APPENDICES

APPENDIX TITLE PAGE

A The Heterogeneous Parallel Computer Architecture

Located at Computer Lab, Block C22, Mathematic

Department Level 4, Science Faculty, University

Technology of Malaysia 73

B The Sequential C Programming for Solving

the Mathematical Model 74

C Flow Chart to Show the Communication between

Master and Slaves in PVM 77

D The Parallel Programming for Solving the Mathematical

Model Using PVM under RedHat Linux 9.2 Operation 78

E Programming Model 82

F Message Passing Paradigm 83

CHAPTER 1

RESEARCH INTRODUCTION

1.1 Introduction

The behavior of scalar physical and mathematical quantities that can be

represented by an unknown function u of two or more variables can be often be

characterized by an equation that is related to some of the partial derivatives of u.

Partial differential equations (PDEs) can be classified as parabolic equation, elliptic

equation and hyperbolic equation. Partial differential equations are used commonly as

mathematical models for solving all of the science and engineering fields. For example,

the parabolic and elliptic equations can be used for steady and unsteady heat transfer in

solids, flow in porous media and diffusion problems, steady electrostatics of dielectric

and conductive media and potential flow. The hyperbolic PDE is used for transient and

harmonic wave propagations in acoustics, electromagnetic and transverse motions of

membranes.

This research will focus on the study of hyperbolic equations. An efficient finite

difference method is used to solve the hyperbolic equations.

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The implementation of parallel algorithms in solving the mathematical problems

using parallel computing system will be introduced in this research. Parallel computing

is the simultaneous use of multiple compute resources to solve a computational problem.

The parallel computing is done by its transition from sequential of parallel technique in

solving large scale problems. The heterogeneous PC cluster system contains 6 Intel

Pentium IV CPUs (each with a storage of 40GB, speed 1.8 MHz and memory 256 MB)

and 2 servers (each with 2 processors, a storage of 40 GB, speed AMD-Athlon (tm) MP

processor 1700++ MHz and memory 1042 MB) are connected with internal network

Intel 10/100 NIC under RedHat Linux 9.2 operation are used in this research. The

communication platform that is used is Parallel Virtual Machine (PVM). PVM is a

software that provides a unified framework within which parallel programs can be

developed in an efficient and straightforward manner using existing hardware. PVM

enables a collection of heterogeneous computer systems to be viewed as a single parallel

virtual machine. PVM transparently handles all message routing, data conversion, and

task scheduling across a network of incompatible computer architectures. Besides, the

performance of the parallel algorithm is analyzed from the aspect of execution time,

speedup, efficiency, effectiveness and the temporal performance.

In addition, a brief introduction for breast cancer is explained. Breast cancer is a

type of cancer in which cells in the breast become abnormal and grow and divide

uncontrollably. These extra cells form a mass of tissue, called a growth or tumor.

Tumors can be benign, which means not cancerous, or malignant, which means

cancerous. Breast cancer occurs when malignant tumors form in the breast tissue. A

malignant tumor is a group of cancer cells that may invade surrounding tissues or spread

to distant areas of the body. The application of the hyperbolic equation with numerical

finite-difference method is applied to solve a mathematical model in medical field.

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1.2 Problem Formulation

The hyperbolic equation governing ultrasound pressure fields in lossy attenuated

medium consists of a dissipative wave equation incorporating a frequency-dependent

attenuation

))t,x(pc.(t

)t,x(pt

)t,x(pc1

2

2

2 ∇∇=∂

∂+

∂∂ γ (1.1)

where

p – pressure

x – space variable

t – time variable

c – sound speed of the traversed tissue

γ – damping or attenuation parameter

y0 fc

2)f(α

γγ ==

0α – dependent material constant

f – frequency

y – frequency-power exponent varying from 0 to 2 and depending on tissue

The breast tissue is initialized with the atmosphere pressure and the condition

atm0 p)t,x(p = and 0)t,x(tp

0 =∂∂ (1.2)

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The transducer incident wave is implicitly specified as a Dirichlet boundary condition

for the wave equation. A homogeneous Neumann condition is set on the reflecting

boundary:

0np=

∂∂ (1.3)

While first-order absorbing conditions are set on the non-reflecting boundaries:

tp

c1

np

∂∂

−=∂∂ (1.4)

1.3 Research Objectives

The first objective of the research is to study the one-dimensional hyperbolic

equation as a partial differential equation (PDEs).

The second objective of this thesis is to study the parallel computing systems and

using parallel computing system based on the Parallel Virtual Machine (PVM)

communication platform and C programming to solve the complex and grand challenge

applications.

The third objective is using the hyperbolic partial differential equation in medical

field. A mathematical model for the diffusion of breast cancer will be emphasized. The

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growth of the breast cancer will be presented in a graph to visualize the pattern of the

cancer cells.

The fourth objective is to analyze the performance of the parallel computer from

the aspect of execution time, speedup, efficiency, effectiveness and temporal

performance using the relevant formula.