the development of two dimensional brain … filethe development of two dimensional brain tumors’...

THE DEVELOPMENT OF TWO DIMENSIONAL BRAIN TUMORS’ ALGORITHM

ON DISTRIBUTED PARALLEL COMPUTER SYSTEMS

SIEW YOUNG PING

UNIVERSITI TEKNOLOGI MALAYSIA

i

THE DEVELOPMENT OF TWO DIMENSIONAL BRAIN TUMORS’ ALGORITHM

ON DISTRIBUTED PARALLEL COMPUTER SYSTEMS

SIEW YOUNG PING

A report submitted in partial fulfillment

of the requirements for the award of the degree of Bachelor

of Computer Science with Education (Mathematics)

Faculty of Education

Universiti Teknologi Malaysia

APRIL 2006

iii

To my father, mother and my beloved sisters

iv

ACKNOWLEDGEMENT

I would like to express my sincere appreciation and gratitude to my thesis

supervisor, Dr. Norma Alias for her guidance, suggestions and encouragement. She

helped and provided me with a lot of sources and useful comments during this period of

doing my thesis. Besides, I would like to give my sincere thankfulness to Assoc. Prof.

Dr Shaharuddin Salleh as my internal examiner for his useful suggestions to improve my

thesis.

A special thanks to my family for being my great supporter. I would also like to

thank my dearest sisters who always supported me spiritually during this thesis.

v

ABSTRACT

This thesis focuses on the implementation of parallel algorithm using the

distributed parallel computer system in solving two-dimensional brain tumors growth

problem. The study of partial differential equations (PDEs) is a fundamental area of

mathematics, which links are important to pure mathematics for computational

mathematics application. The numerical finite-difference method is emphasized as

platform for discretization of two-dimensional parabolic equations. The result of a type

of finite difference approximation namely explicit will be presented graphically. The

numerical solution is applied in medical field by solving a mathematical model for the

diffusion of brain tumors. A parabolic model as mathematical model used to describe

and visualize the evolution of tumor from the avascular stage to the vascular one through

the angiogenic process. The parallel algorithm based on parallel computing systems is

used to capture the growth of brain tumors cells in two-dimensional visualization.

Parallel Virtual Machine (PVM) is software for communication platform in parallel

computer systems. The software system functions to allow a collection of heterogeneous

computers to be used as synchronize and flexible concurrent computational resource.

The performance measurements of algorithm from the aspect of speedup, efficiency,

effectiveness and temporal performance will be analyzed.

vi

ABSTRAK

Tesis ini adalah fokus kepada pelaksanaan algoritma selari dalam menyelesaikan

masalah pertumbuhan sel barah otak dua dimensi dengan menggunakan sistem komputer

selari ingatan teragih. Satu kajian terhadap persamaan pembezaan separa, PDE ialah

bidang asas kepada matematik dimana kaitannya adalah penting terhadap matematik

tulen untuk aplikasi dalam matematik berkomputer. Kaedah penghampiran beza

terhingga yang difokuskan sebagai pelantaraan digunakan untuk pendiskretan persamaan

parabolik dua dimensi. Hasil kajian bagi satu daripada kaedah penghampiran beza

terhingga, iaitu kaedah tak tersirat akan ditunjukkan dalam bentuk graf. Kaedah analisis

berangka ini diaplikasikan dalam bidang perubatan dengan menyelesaikan satu model

matematik yang melibatkan penyebaran sel barah otak. Satu model parabolik sebagai

model matematik digunakan untuk menghuraikan dan mengambarkan perkembangan sel

barah secara beransur-ansur daripada peringkat avascular ke vascular melalui proses

angiogenic. Tujuan pelaksanaan algoritma selari berdasarkan sistem komputer selari

adalah untuk mengambarkan pertumbuhan sel-sel barah otak dalam dua dimensi secara

visual. Mesin Selari Ingatan, PVM ialah perisian untuk pelantaraan komunikasi dalam

sistem komputer selari. PVM sebagai fungsi sistem perisian yang membolehkan

sekumpulan komputer heterogen digunakan sebagai satu sumber pengiraan yang

bekerjasama secara teratur dan fleksibel serta dihubungkan oleh satu sistem rangkaian.

Prestasi ukuran bagi algoritm telah dianalisiskan dari aspek kecepatan, kecekapan,

keberkesanan, dan masa perlaksaan.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF GLOSSARY xiii

LIST OF SIMBOLS xv

LIST OF APPENDICES xvii

1 RESEARCH INTRODUCTION

1.1 Introduction 1

1.2 Problem Formulation 5

1.3 Objectives of Research 8

1.4 Scope of Research 9

1.5 Outline 9

viii

2 FINITE DIFFERENCE METHODS FOR PARABOLIC

EQUATION

2.1 Introduction of PDEs 11

2.2 Finite Difference Approximations to Derivatives 13

2.3 Parabolic Partial-Differential Equations 15

2.3.1 Transformation to Standard Form 16

2.3.2 A Weighted Average Approximation 18

2.4 Two Numerical Finite Difference Method 20

2.4.1 Explicit Method 20

2.4.2 Crank Nicolson Method 23

2.5 Explicit Finite Difference Scheme for Brain Problem 25

2.6 Iteration Point Methods for solving the Finite

Difference Equations 27

2.6.1 Gauss-Seidel Iteration Method 27

2.6.2 Red Black Gauss-Seidel Iteration Method 29

3 PARALLEL VIRTUAL MACHINE (PVM)

3.1 The Parallel Computing 30

3.1.1 Parallel Architectures 31

3.2 Parallel Programming 33

3.2.1 Methodical Design 34

3.3 Introduction to Parallel Virtual Machine (PVM) 37

3.4 The PVM System 37

3.4.1 PVM Daemon 38

3.4.2 Library of PVM Interface Routines 39

3.5 Starting PVM 40

3.6 PVM Programming 40

3.6.1 Message Passing 41

ix

4 THE EVOLUTION AND MODELING OF

TUMOR GROWTH

4.1 The Evolution of Tumor Growth 44

4.1.1 Phenomenological Observation of the

Biological System 45

4.2 The Mathematical Model 48

4.2.1 The Derivation of the Mathematical Model in

Standard Form 50

4.2.2 The Discretization of the Model Equations 50

4.3 The Visualization of the Brain Tumor Growth Using

Explicit 54

5 ANALYSIS OF THE PERFOMANCE OF PVM

5.1 Introduction 60

5.2 Performance Analysis 61

5.2.1 The Speedup 62

5.2.2 The Efficiency 63

5.2.3 The Effectiveness 65

5.2.4 The Temporal Performance 66

6 CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions 69

6.2 Recommendations for Further Study 70

REFERENCE 73

APPENDICES A-D 75

x

LIST OF TABLES

TABLE NO. TITLE PAGE

4.1 The Number of Cancer Cells for 30 Days 54

5.1 Time, Convergence and Number of Iteration for

Parallel Algorithm and Sequence Algorithm 61

5.2 The Speedup of Parallel Computer 62

5.3 The Efficiency of Parallel Computer 64

5.4 The Effectiveness of Parallel Computer 65

5.5 The Temporal Performance of Parallel Computer 67

xi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 A Diagram of A Weighted Average Approximation 19

2.2 A computational Molecule for Explicit Method 21

2.3 Grid System for Explicit Scheme on a two dimensional 22

2.4 A computational Molecule for Level 1+n of Crank

Nicolson Method 24

2.5 A computational Molecule for Level of Crank n

Nicolson Method 24

3.1 Example data partitions of 2-D decomposition 34

3.2 Communication Example Jacobi Finite Difference Method 35

3.3 Agglomeration Example Jacobi Finite Difference Method 35

3.4 A Data Structure for Parallel Programs 36

3.5 PVM Program hello.c 42

3.6 PVM Program hello_other.c 43

4.1 Movement of tumor cells 46

4.2 The angiogenesis process 47

4.3 The Growth Rate of Brain Tumor in 6 days 56





4.7 The Growth Rate of Brain Tumor 58

5.1 The Speedup vs. Number of Processors 63

xii

5.2 The Efficiency vs. Number of Processors 64

5.3 The Effectiveness vs. Number of Processors 66

5.4 The Temporal Performance vs. Number of Processors 67

xiii

LIST OF GLOSSARY

Angiogenesis - The formation of new blood vessels, especially blood

vessels that supply oxygen and nutrients to cancerous

tissue.

Avascular - Not associated with or supplied by blood vessels.

Capillary - Relating to the capillaries or resembling a hair; fine and

slender.

Cellular - Consisting of, composed of, or containing a cell or cells.

Diffusion - The spontaneous intermingling of the particles of two or

more substances as a result of random thermal motion.

Drift - A variation or random oscillation about a fixed setting,

position, or mode of behavior.

Endothelial - A thin layer of flat epithelial cells that lines serous

cavities, lymph vessels, and blood vessels.

Extracellular - Located or occurring outside a cell or cells.

Extracellular matrix - Any material part of a tissue that is not part of any cell in

biology and it is the defining feature of connective tissue.

Intracellular - Occurring or situated within a cell or cells.

Intercellular - Located among or between cells.

Invasion - The act of invading, especially the entrance of an armed

force into a territory to conquer; an intrusion or

encroachment.

Macromolecules - A very large molecule, such as a protein, consisting of

many smaller structural units linked together.

Macroscopic - Relating to observations made by the unaided eye.

xiv

Metastasize - To be transmitted or transferred by or as if by metastasis.

Metastatic - Cancer that has spread from one area of the body

to another (metastasis).

Penetrate - To cause to be permeated or diffused; steep.

Proliferate - (cells) to reproduce rapidly; to multiply

Proliferation - The growth and reproduction of similar cells; a rapid

growth or increase in numbers.

Spheroid - Having a generally spherical shape.

Stimulus - That which can elicit or evoke an action or response in a

cell, an excitable tissue, or an organism.

Vascular - Of, relating to, or containing blood vessels.

Vasculature - Arrangement of blood vessels in the body or in an organ or

body part.

xv

LIST OF SIMBOLS

u - The temperature

t - Time

yx, - The space at coordinate system

)( 4hO - Term containing fourth and higher powers of h

tu∂∂ - The change of the temperature at time, t

)(uL - Second-order elliptic partial differential operator

u∇ - Spatial gradient of u

β , κ , - Constants c

U - The temperature at the distance X from one thin uniform rod

L - The length of the rod

0U - Some particular temperature

θ - The weight at the grid ( tjxi Δ+Δ )21(, )

Δ - Laplacian operator

A - Matrix represents the values of the left hand side of equation.

2

2

xu

∂∂ - Second order derivative for u at x

2

2

yu

∂∂ - Second order derivative for u at y

)(,njiu - The value of u at the grid point at time step n ),( ji

RΩ - Domain at red grid BΩ - Domain at black grid

xvi

)(N ji, t - A certain type of cells (or chemical factors) in the node (i, j) at the

time t

ji,V - The elementary volume centered in the node (i, j)

Γ - Generation (proliferation/production) coefficient

L - Death/decay coefficient

Q - Diffusion coefficient (a cell diffuses along x)

W - Drift velocity field

P - Drift velocity (a cell is transported along x)

R - Drift velocity (a cell is transported along y)

p - Number of processors

xvii

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 75

B The Sequential C Programming for Solving the

Mathematical Model Under RetHat Linux 9.2

Operation 77

C Flow Chart to Show the Communication Between

Master and Slaves in PVM 83

D The Parallel Programming for Solving the

Mathematical Model Using PVM Under RetHat

Linux 9.2 Operation 85

CHAPTER 1

RESEARCH INTRODUCTION

1.1 Introduction

In the last decade the deduction of the mathematical models addressed towards

the study of the evolution of biological systems related to tumor growth has become one

of the relevant fields of research activity in biomathematics. The type if cells involved in

this description, the interactions between cells and the effects to consider make the

problem very complex. This research will focus on the tumor growth of the brain tumor.

A brain tumor is a mass or growth of abnormal cells or normal cells in an

inappropriate place in the brain. The brain and the spinal cord are the main components

of the central nervous system (CNS). The brain is composed of a number of different

types of cells and tissues. Any of these can form tumors. Some of the tumors are a

combination of cell types. According to the National Cancer Institute (NCI),

(http://cancer.healthcentersonline.com/brainnervoussystemcancer/braintumors2.cfm,

2006) brain tumors account for 85 to 90 percent of all primary CNS tumors. A primary

CNS tumor develops from cancer cells within the central nervous system rather than

cancer cells that have spread (metastasized) to the CNS. Approximately 17,000 new

cases of primary brain tumors are treated each year in the United States

(http://www.medifocus.com/guide_detail.php?a=a&gid=OC009, 2006).

http://cancer.healthcentersonline.com/brainnervoussystemcancer/braintumors2.cfm

http://www.medifocus.com/guide_detail.php?a=a&gid=OC009

2

Brain tumors can either develop within the brain (primary) or develop from

cancer cells that spread to the brain (metastatic or secondary). Primary tumors can be

grouped into non–cancerous (benign) and cancerous (malignant). Benign brain tumors

usually grow slowly and can often be removed by surgery depending upon their specific

location in the brain. Almost half of all brain tumors are non-cancerous. Malignant brain

tumors are commonly called brain cancer tend to grow rapidly spreading into the

surrounding brain tissue and often cannot be entirely removed surgically. They are

usually invasive and life–threatening. Most of them have spread from other tumors in

the body to the skull. However, metastatic brain tumors can only be malignant.

According to the NCI, they are much more common that primary brain tumors. It is

estimated that metastatic brain tumors outnumber primary brain tumors by 10 to 1.

These tumors are formed from cancer cells that begin growing elsewhere in the body

and travel to the brain, usually through the bloodstream. Metastatic brain tumors are

always cancerous and commonly come from cancers of the lung, breast and colon.

According to the American Cancer Society (ACS), approximately 18,500


2006) malignant tumors of the brain or spinal cord will be diagnosed in the United States

in 2005. It is anticipated that about 12,690 deaths due to primary malignant brain tumors

in 2004 and about 12,760 deaths from CNS cancers are expected in 2005


2006). These types of cancer account for approximate 1.4 percent of all the cancers and

2.4 percent of all the cancer deaths. These statistics include adults and children with

CNS tumors. The average survival rate for patients with glioblastomas is approximately

12 months. Researchers at Cedars-Sinai Medical Center have found that brain tumors

account for one in every 100 cancers diagnosed annually in the United States

(http://www.cedars-sinai.edu/5192.html, 2006). It is estimated that one out of three

people will be diagnosed with cancer in their lifetime. About 500,000 people die from

cancer in the United States each year (http://www.cedars-sinai.edu/1120.html#top, 2006).



http://www.cedars-sinai.edu/5192.html

http://www.cedars-sinai.edu/1120.html#top

3

According to NCI, brain tumors are the leading cause of death from childhood

cancers among persons up to 19 years old (http://thinkaboutbraintumors.org/index2.cfm,

2006). 3,200 new childhood primary malignant and non-malignant and central nervous

systems tumors are expect to be diagnosed in 2004. Of those, 2,450 will be in children

under the age of 15 (http://thinkaboutbraintumors.org/index2.cfm, 2006). Five-year

survival rates are a mere 27.9% in males and 30.1% in females. Brain tumors are the

second leading cause of cancer related deaths in males ages 20-39 while there are the

fifth leading cause of cancer-related deaths in women ages 20-39

(http://thinkaboutbraintumors.org/index2.cfm, 2006). Because there is no known cause

of brain tumors, there is no way to clearly prevent them.

For the simulation of brain tumors growth using parabolic equations, it needs us

to see the definition of partial differential equations. Indeed, partial differential equations

(PDEs) are usually classified as elliptic, hyperbolic or parabolic according to the form of

the equation and the form of the subsidiary conditions, which must be assigned to

produce a well-posed problem. Elliptic equations generally arise from a physical

problem that involves a diffusion process that has reached equilibrium, for example a

steady state temperature distribution. Hyperbolic PDEs usually arise in connection with

mechanical oscillators, such as a vibrating string, or in convection driven transport

problems. Mathematically, parabolic PDEs serve as a transition from the hyperbolic

PDEs to the elliptic PDEs. Physically, parabolic PDEs tend to arise in time dependent

diffusion problems, such as the transient flow of heat in accordance with Fourier's law of

heat conduction.

This research will focus on the study of parabolic equation in two space

dimensions. An efficient finite difference discretization method is used to solve the

parabolic equations. The two finite-difference methods that are explicit and Crank

Nicolson have been studied. The explicit method is used to solve the parabolic equations

in this research. The finite-difference equations are converted into matrix forms and

solved by iterative methods. Gauss-Seidel iterative methods will also be discussed.

http://thinkaboutbraintumors.org/index2.cfm



4

The using of heterogeneous parallel computer system in solving the

mathematical problems by parabolic equations in two space dimensions will be

introduced. Parallel computing is a software of using heterogeneous or homogeneous

parallel computer system as a counting source, which coordinate together and been

connected with cluster system. Parallel computing is the simultaneous use of multiple

compute resources to solve a computational problem. The heterogeneous PC cluster

system contains 6 Intel Pentium IV CPUs (each with a storage of 40GB, speed 1.8MHz

and memory 256 MB) and two servers (each with 2 processors, a storage of 40GB,

speed AMD-Athlon (tm) MP processor 1700++ MHz and memory 1024 MB) are

connected with internal network Intel 10/100 NIC under RetHat Linux 9.2 operation are

used in this research. A software package that is used is Parallel virtual Machine (PVM).

PVM as a communication platform permits a heterogeneous collection of Linux or

Window computers hooked together by a network to be used as a single large parallel

computer. PVM is designed to link computing resources and provide with a parallel

platform for running their computer applications, irrespective of the number of different

computers are used and where the computers are located. Besides, the PVM model is a

set of message passing routine, which allows data to be exchanged between tasks by

sending and receiving messages. This research will analyze the performance of the

parallel computer from the aspect of speedup, efficiency, effectiveness, and temporal

performance.

A brief introduction for brain tumors also described in this research. It is about

the evolution of tumor growth from the avascular stage to the vascular one through the

angiogenic process. The application of the parabolic equation with numerical finite-

difference methods is applied to solve a mathematical model in medical field. The

mathematical model is converting to matrices form using the finite-difference methods.

Then, parallel computing system is chosen to solve mathematical problems.

5

1.2 Problem Formulation

The derivation of the mathematical model, which consists in an evolution

equation for the variable u will be done on the basis of a lattice schematization

procedure, mainly consisting in mass balances. This variable u can be referring to cells

and macromolecules. Between these two classes there is a profound difference as cells

are much larger than chemical factors and macromolecules. They are delimited by a

membrane and cannot penetrate each other. Cells, then, occupy part of the available

space. On the contrary, chemical factors are molecules which diffuse in the intercellular

space, attach to the cell membrane, or pass through it either to feed the cell or as a cell

product, so that they actually do not occupy space.

Consider, for simplicity, but without loss of generality, a plane square lattice and

identity each point by the pair if integers (i, j). The elementary volume centered in the

node (i, j) is denoted by and its volume by ji,V ji,VΔ . Finally, all cells in are

considered concentrated in the node (i, j). We want to study the evolution of the number

of a certain type of cells (Angelis and Preziosi, 2000) found in the node (i, j) at

the time t, which is related to the number density by

ji,V

)(N ji, t

),( xtu

(1.1) ∫=

jiVji dxxtutN

,

.),()(,

We assume that there exist two mechanisms, which govern the movement of

cells as:

(a) A diffusive-like phenomenon governed by a Brownian motion with

probability, which depends on the direction of the movement. We

indicate, for instance, with the transition probability density per unit

time that a cell diffuses from (i, j) to (i + 1, j).

jiiQ 1, +

6

(b) A transport-like phenomenon, which generates a drift of cells with a

probability, which again depends on the direction of motion where is

the transition probability density per unit time that a cell is transported

along x from (i, j) to (i + 1, j) and that it is transported along y from

(i, j) to (i, j + 1). We will assume that the drift velocity has positive

components. One then has the following scheme

jiiP 1, +

1, +jjiR

Number of cells in (i, j) at time t + dt

=Number of cells present in (i, j) at time t

+

Number of cells drifting in along x from (i −1, j) and along y from (i, j −1)

−

Number of cells drifting away along x to (i +1, j) and along y from (i, j+1)

(1.2)

Number of cells diffusing from neighbour-ing nodes

Number of cells diffusing away to neighbour-ing nodes

+ − +Generation of new cells −

Death of cells

Which leads to the following finite difference equation

)( ttN ij Δ+ = )()()()([)( 1,

1,1,,1

,1,1 tNRtNPtNRtNPttN ijjj

iijjiiji

jjiji

jiiij

++−

−−− −−+Δ+

)()()()( 1,,1

1,,1

,1,1,1,1 tNQtNQtNQtNQ jijj

ijijj

ijij

iijij

ii ++

−−

++−− ++++

(1.3) )],()()( ,1,1,

1,1, tNLtNQQQQ ijijjiijjj

ijj

ijii

jii −Γ++++− +−

+−

where and are the generation and death rates, respectively. ji ,Γ jiL ,

Notice that if at time t the cells occupy a finite number of nodes of the infinite

lattice, they will still occupy a finite number of nodes at any time. In addition, if

summing over all possible (i, j), the total number of cells is preserved.

Assuming symmetry of the transition probability density as , one can

rewrite Equation (1.3) as

jijiji NL ,,, =Γ

jii

jii QQ 1,,1 −− =

7

ttNttN ijij

Δ

−Δ+ )()(= )]()([)]()([ 1,

1,,1

1,,1,1 tNRtNRtNPtNP ijjj

ijijj

iijjiiji

jii

+−

−+−− −+−

)]()()()([ 11,1,,1,1,1 tNQtNQQtNQ jijiiij

jii

jiiji

jii +++−−− ++−+

)]()()()([( 1,1,1,,1

1,,1 tNQtNQQtNQ ji

jjiij

jji

jjiji

jji +

++−−

− ++−+

).(tNL ijijij −Γ+ (1.4)

Assuming now that the transition probability densities of the diffusion process

depend on some properties evaluated in the mean point of the path as , and

that the transition probability densities of the transport phenomenon depend on the

starting point as , one has the scheme

ji

jii QQ 1,1 −− =

ji

jii PP 1,1 −− =

ttNttN ijij

Δ

−Δ+ )()(= )]()([)]()([ 1,

1,11 tNRtNRtNPtNP ij

jiji

jiij

jiji

ji −+− −

−−−

)]()()()([ 1111,11 tNQtNQQtNQ jij

iijj

ij

ijij

i +++−−− ++−+

)]()()()([( 1,111

1,1 tNQtNQQtNQ ji

jiij

ji

jiji

ji +

++−−

− ++−+

).(tNL ijijij −Γ+ (1.5)

Under suitable regularity assumptions one can expand andQPN ,, R , use

and write ji,,, V),()( Δ≈ jiji xtutN

,)()()()( LuyuQ

yxuQ

xyRu

xPu

tu

−Γ+∂∂

∂∂

+∂∂

∂∂

+∂

∂−

∂∂

−=∂∂ (1.6)

with and where the indices (i, j) have been substituted with

the dependence of u and of all coefficients on the space variable. Equivalently, one can

write

jijiji Vtyxt ,, /)(),,( ΔΓ=Γ

.∇+∂∂

tu (Wu) = ,).( LuuQ −Γ+∇∇ (1.7)

8

where, in two dimensions, W = (P, R). The general advection-diffusion model (1.7)

requires the specification of the drift, diffusion, proliferation, and death coefficient in the

terms and in particular of their dependence of the state variables. LQW ,,, Γ

This result could have been expected as (1.5) represents an explicit finite

difference scheme of (1.6) in which the diffusive term is approximated with a second-

order central scheme and the convective term with a first-order upwind scheme.

1.3 Objectives of Research

The first objective of this thesis is to study parabolic equations in two spatial

dimensions. Before doing the discretization, the parabolic equations are transformed into

two-dimensional form.

The second objective is studying two finite-difference methods, which are

explicit and Crank Nicolson method. The result of a type of finite difference

approximation namely explicit methods will be presented and applied in a mathematical

method using parallel computing.

The third objective of this research is to study the parallel computing systems

with PVM communication platform and C programming to solve the great mathematic

problems in two-dimensional.

The fourth objective is applying the two dimensional parabolic partial

differential equation in medical field. A study will be on a mathematical model for

evolution of brain tumors. The growth of the brain tumor will be presented in a graph to

visualize the pattern of this cancer cell growth.

9

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

the aspect of speedup, efficiency, effectiveness and temporal performance.

the development of two dimensional brain … filethe development of two dimensional brain tumors’...

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