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MODELLING RADIATION DOSES TO CRITICAL ORGANS 
OF PATIENTS UNDERGOING INTRACAVITARY 
BRACHYTHERAPY TREATMENT AT KOMFO ANOKYE 
TEACHING HOSPITAL. 
 
 
 
 
John Owusu-Banahene 
         
 
 
 
 
 
 
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MODELLING RADIATION DOSES TO CRITICAL ORGANS 
OF PATIENTS UNDERGOING INTRACAVITARY 
BRACHYTHERAPY TREATMENT AT KOMFO ANOKYE 
TEACHING HOSPITAL. 
 
  
THIS DISSERTATION IS SUBMITTED TO THE UNIVERSITY OF 
GHANA, LEGON IN PARTIAL FULFILMENT OF THE 
REQUIREMENT FOR THE AWARD OF PhD MEDICAL PHYSICS 
DEGREE 
 
BY 
JOHN OWUSU-BANAHENE (10191759)   
BSc( HONS)( KNUST), MPHIL(UG) 
 
Department of MEDICAL PHYSICS 
SCHOOL OF NUCLEAR AND ALLIED SCIENCES 
UNIVERSITY OF GHANA, LEGON 
 
 
                                                                          JUNE, 2013 
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DECLARATION 
I hereby declare that, except for references to other people’s work, which have been duly 
cited, this dissertation is the result of my own research work and that it has neither in part 
nor in whole been presented for any degree elsewhere. 
 
 
………………………………                                            ………………………..…… 
JOHN OWUSU- BANAHENE      PROF. E. O. DARKO 
   (STUDENT)              (PRINCIPAL SUPERVISOR) 
 
 
 
..........................................                                                        
PROF. J. H. AMUASI  
(CO-SUPERVISOR)                
     
 
 
 
 
 
 
 
 
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DEDICATION 
This work is first dedicated to the Almighty God for helping me through this programme 
successfully and secondly to my wife Philomina Owusu-Banahene for her endless advice 
and encouragement.  
 
 
 
 
 
 
 
 
 
 
 
 
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ACKNOWLEDGEMENTS 
I would like to express my sincere thanks to my supervisors Prof. E.O. Darko and Prof. J. 
H. Amuasi for their professional guidance, encouragement and constructive criticisms. 
Prof. A. W. K. Kyere, Head of Department of Medical Physics of the School of Nuclear 
and Allied Sciences, University of Ghana- Ghana Atomic Energy campus is hereby 
acknowledged for his advice, assistance and encouragement during the study. 
I would also like to thank Prof Cyril Schandorf for his continuous advice and 
encouragement given to me all the time. 
I am also very grateful to Mr. E.K.T. Addison at the National Centre for Radiotherapy and 
Nuclear Medicine at Komfo Anokye Teaching Hospital and also to the Radiation 
Protection Institute of Ghana Atomic Energy Commission for using their facilities. 
Finally, my sincere thanks to all and sundry, especially the staff of the Radiation Protection 
Institute who contributed diversely to the success of this work. 
 
 
 
 
 
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ABBREVIATIONS 
 1D =  One dimensional 
 2D = Two dimensional 
 3D  = Three dimensional 
 keV =  Kilo electron volts 
 MC  = Monte Carlo 
 MeV = Mega electron volts 
 BTE  = Boltzmann transport equation 
 PTV  = Planning target volume 
x,y,z =  usual orthogonal axes 
r  = spatial coordinate, specified by coordinate (x,y,z) 
 =  direction of photon specified by angular coordinates 
4 =  angular domain 
 =Rx4 , phase space: A position in phase space as specified by (r,  ) 
t =  time 
(r,,t)  =  angular photon flux at (r,  ); the number of photons per unit area moving in 
the direction   about d  at r  at time t 
S(r,  ,t) = volumetric source of photons at ( r ,  ). The number of photons produced at 
r in dr  at   in d  at time t 
 (r) = total macroscopic cross section for collision at  r  t  
 (r) =  total macroscopic cross section for absorption at  r  a  
 (r, ' )  = differential macroscopic scattering cross section at  r  to scatter photons   s
from   to   '  
 (r) = total macroscopic scattering cross section at r  s  
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  H  =   space of trial functions which have square integrable gradients    
,  =   incoming (-) and outgoing (+) boundary inner  products 
P = mass matrix in time dependent finite element approximation 
   
(t)  = time dependent vector of nodal fluxes 

(t)  = time derivative of (t)  
A =  matrix of coefficients 
  
  K =  matrix due to collision terms 
  T =  matrix due to transport terms 
  B = matrix due to boundary terms 
 GTV = Gross Tumour Volume 
 CT = Computed Tomography 
 MRI = Magnetic Resonance Imaging 
 PET = Positron Emission Tomography 
 CTV = Clinical Target Volume 
OAR = Organs at Risk 
MTD = Minimum Target Dose 
KERMA = Kinetic Energy Released in Matter 
FEM = Finite Element Method 
LDR = Low Dose Rate 
HDR = High Dose Rate 
PTE = Photon Transport Equation 
PDE = Partial Differential Equation 
TPS = Treatment Planning System 
KATH = Komfo Anokye Teaching Hospital 
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LIST OF TABLES 
Table 2.1 Characteristics of some radionuclides used in brachytherapy…………….12 
Table 2.2 Stages in cancer development……………………………………………..52 
Table 2.3 Tumour Stage Grouping …………………………………………………..53 
Table 4.2  Comparison of the photon flux intensity from Green’s Function with that of   
the model (FEM)………………………………………….……………….90 
Table 4.3       Comparison of Monte Carlo method with this work…………………..…..91 
Table 4.4        Comparison of the FEM from this work with Published data and the TPS at 
the Oncology Centre KATH……………………………………………….92   
 
 
 
 
 
 
 
 
 
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                                        LIST OF FIGURES 
Figure 2.1 The reference points A and B according to the Manchester system……...14 
Figure 2.2 Anatomical Structure of the human cervix….............................................19 
Figure 2.3 Electron undergoing a Compton scattering event…………….…………...24 
Figure 2.4 Graphical representation of volume of interests as defined in ICRU…..….33 
Figure 2.5 A simplified schematic diagram of the physical simulation process showing 
FEM modeling terminologies…………………………………………...…47 
Figure 3.1   The Manchester system showing the reference points A and B and arrangement 
of the sources in the tandem and ovoids ...……………………………………57 
Figure 3.2     A Cs-137 LDR afterloading unit with tube arrangements similar to Amersham 
‘J-type tube…………………………………………………………..…….58 
Figure 3.3    Non-homogeneous photon flux distribution around a Cs-137 source in saggital 
plane…………………………………………………………………….…59 
Figure 3.4     Anatomical structure of the cervix showing the target points………………61 
Figure 3.5     Diagram of a physical domain (patient) represented by a slab phantom in 
plane geometry………………………………………………………….….67 
Figure 3.6 Particle trajectory from point r to the boundary V  along a straight line in 
the direction ………………………………………………………….....68 
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Figure 3.7 Polar coordinates ( r, , )  indicating the direction of the particle in the 
standard coordinate system x  (x, y, z)  with r 1……………………..…70 
Figure 3.8      Geometry for the generation of mesh…………………………………….…73 
Figure 4.1      Source strength distribution as a function of distance distance……………..83 
Figure 4.2 Photon Flux distribution for different cancer stages as a function of 
distance………………………………………………………………….…84 
Figure 4.3    Variation of photon flux with angles…………………………………….…...85 
Figure 4.4    Variation of the flux error distribution function with distance in the solution of 
the photon flux using the finite element method ……………………….…86 
Figure 4.5    Variation of dose as a function of distance for different cancer stages....…....87 
Figure 4.6    Cumulative dose as a function of time during treatment…………………..…88 
Figure 4.7    Variation of the number of cancer patients with cancer stage…….…………94 
 
 
 
 
 
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LIST OF PLATES 
Plate.2.1  Fletcher-Suit-Delcos applicator…………………………….……………..13 
Plate.2.2 Tandem-Ring applicator…………………………………………………...13 
Plate.2.3  Fletcher-Suit-Delcos system in gynaecology…………………….……….17 
  
 
 
 
 
 
 
 
 
 
 
 
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ABSTRACT 
The main aim of radiotherapy is to destroy cancerous tissues with ionizing radiation while 
the other normal parts of the body are saved or spared. Intracavitary brachytherapy is a 
procedure in which radioactive sources are placed in the body cavities close to or inside the 
target volume to deliver radiation at short distances. In this mode of treatment, high 
radiation dose can be delivered to the tumour volume with rapid dose fall-off into the 
surrounding normal tissues. In brachytherapy, the dosimetry in biological tissues is a 
complex process. Dosimetric parameters such as the dose to critical organs and the total 
dose to the reference points as in the case of Manchester system are critical for patients 
undergoing intracavitary brachytherapy treatment.  In this study, the Finite Element 
Method (FEM) has been utilized to solve Boltzmann Transport Equation (BTE) to 
determine the distribution of angular photon fluxes at various positions in the cervix of 
cancer patients and the dose distribution calculated for the organs of interest. The time-
dependent linear BTE was used to describe the passage of ionizing radiation taking into 
account tissue heterogeneities and other scattering phenomena before depositing the 
absorbed dose in a patient. The simulation was carried out to determine doses to the critical 
organs, namely the rectum and bladder. Results from the study indicate doses to the rectum 
and the bladder to be in the range of 10.13-85.67cGy and 21.32-78.81cGy respectively for 
stage I to stage IV cancer patients. Comparison of the results from the model with data 
from published articles and dose prescriptions from the treatment planning system of the 
Radiotherapy Center of the Komfo Anokye Teaching Hospital in Ghana for different 
cancer stages indicate good agreement with standard error of ±20% to ±34%.                       
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TABLE OF CONTENTS 
                               Page 
Declaration………………………………………………………………….... i 
Dedication………………………………………………………………………….. ………ii 
Acknowledgement…………………………………………………………………. ……...iii 
Abbreviations………………………………………………………………………. ……...iv 
List of tables………………………………………………………………………………..vi 
List of figures………………………………………………………………………. …….vii 
List of plates………………………………………………………………….…….. ……..ix 
Abstract……………………………………………………………………………............. x 
Table of Contents…………………………………………………………………………..xi 
CHAPTER ONE: INTRODUCTION                                                                   
1.1 Background………………………………………………………… ………………1 
1.2 Statement of the problem…………………………………………............... ………3 
1.3 Relevance and justification of the study…………………………………………… 4 
1.4 Scope and objectives of the study…………………………………………………. 5 
1.5 Structure of the dissertation………………………………………………………...6 
 
 
 
 
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CHAPTER TWO:  LITERATURE REVIEW 
2.1 General overview……………………………………………………………………7 
2.2 Principles of Brachytherapy……………………………………………………….. 8 
2.3 Importance of Low Dose Rate Brachytherapy…………………………….. ………9 
2.4  Clinical Brachytherapy sources……………………………………………. ……..11 
2.5 Intracavitary Brachytherapy in gynaecology………………………. ……………..12 
2.6 Organs at Risk (OAR)……………………………………………… ……………..15 
2.7 Critical organ dose monitoring in gynaecology……………………………. ……..16 
2.8 Anatomy of the cervix……………………………………………………………..17 
       2.8.1 Risk factors of cervical cancer……………………………………………..19 
       2.8.2 Symptoms of cervical cancer………………………………………. ……..20 
2.9    Physics of ionizing radiation interaction in radiotherapy……………. ……………..20 
        2.9.1 Photon interactions in biological media……………………………. ……..21 
         2.9.2 Photoelectric effect………………………………………………… ……..22 
         2.9.3 Coherent and Incoherent Scattering………………………………... ……..23 
         2.9.4 Pair Production……………………………………………………...……..25 
         2.9.5 Characteristic X-rays and Auger electrons……………………….... ……..27 
         2.9.6 Rayleigh Scattering………………………………………………… ……..27 
2.10 Linear Energy Transfer (LET)……………………………………………………..27 
2.11 Electron Interactions……………………………………………………….. ……..28 
2.12 Methods for calculation of radiation doses from acquired image data…….. ……..29 
2.13 International recommendations on target and Organs at Risk……………... ……..32 
           2.13.1 ICRU definitions…………………………………………………… ……..32 
2.14 The importance and use of numerical methods …………………………… ……..34 
2.15 Dose calculations in Brachytherapy……………………………………….............35 
2.16 Radiation Safety during Brachytherapy treatment…………………………. ……..38 
2.17 Quality Control and Quality Assurance in Brachytherapy………………………..38 
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2.18 The photon transport theory………………………………………………………..39 
 2.18.1  Advantages of using the transport equation……………………….. ……..43 
            2.18.2   Disadvantages of the transport equation………………………….. ……..43 
2.19 The Finite Element Method (FEM)………………………………………………..44 
 2.20 Photon dose deposition mechanisms………………………………………………48 
 2.20.1  The dose deposition step…………………………………… …….. ……..50 
 2.21    Cancer staging and Grouping………………… …………………………………50 
             2.21.1 Stage grouping……………………………………………………………..53 
 
CHAPTER THREE: MATERIALS AND METHODS 
3.1 Theoretical modeling of doses to the critical organs ……………………… ……..55 
 3.1.1 Governing equation………………………………………………… ……..55 
 3.1.2 Source term………………………………………………………… ……..56 
            3.1.3    Target Geometry…………………………………………………………..60 
            3.1.4   The Photon Transport Equation ……………………………………. ……..62 
            3.1.5   Finite Element Model………………………………………………………62 
                   3.1.5.1 Time dependent case…………………………………………………..64 
                   3.1.5.2 Time independent case in plane geometry…………………….. ……. 66 
                   3.1.5.3 Boundary condition for the photon flux………………………. ……..68 
                   3.1.5.4 Error correction and convergence…………………………………….70 
                   3.1.5.5 Photon dose calculation……………………………………….. ……..71 
 3.2 Model implementation and simulation………………………………………... ……..72 
     3.2.1 Assumptions………………………………………………………………. ……..72 
     3.2.2 Mesh generation…………………………………………………………... ……..72 
     3.2.3 System matrix ……………………………………………………………. ……..73 
     3.2.4 Matrix solution …………………………………………………………… ……..74 
     3.2.5 Time dependent case……………………………………………………… ……..74 
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    3.2.6 Simulation using Matlab…………………………………………………... ……..76 
3.3  Optimization of dose to organs at risk………………………………………… ……..77 
3.4  Experimental validation……………………………………………………….. ……..78
  
 
CHAPTER FOUR: RESULTS AND DISCUSSION 
 4.1 General………………………………………………………………………… ……..81 
    4.1.1 Source term………………………………………………………………... ……..82 
    4.1.2 Photon flux distribution…………………………………………………… ……..83 
    4.1.3 Error distribution function………………………………………………… ……..85 
    4.1.4 Dose profile analysis………………………………………………………………86 
4.2   Comparison with analytical reference method……………………………….. ……..88 
4.3   Comparison with Monte Carlo method………………………………………..……..91 
4.4    Comparison with Published data and Treatment Planning System (TPS) of Komfo 
         Anokye Teaching Hospital………………………………………………....... ……..92 
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS 
5.1 Conclusion……………………………………………………………………………..95 
5.2 Recommendations……………………………………………………………… ……..96 
   5.2.1 Oncology Centre…………………………………………………………… ……..96 
    5.2.2 Research Institutions………………………………………………………. ……..96 
   5.2.3 Regulatory Authority………………………………………………………. ……..97 
REFERENCES…………………………………………………………………….. ……..98 
APPENDICES……………………………………………………………………………109 
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CHAPTER ONE 
INTRODUCTION 
 This chapter presents the background, relevance and justification, statement of the 
problem, the scope and objectives of the research work and the structure of the thesis 
 
1.1 Background 
 
After X-rays were discovered by Wilhelm Conrad Roentgen in 1895, ionizing radiation 
has been used for the treatment of cancer.  Nowadays, the three main modalities for 
cancer treatment are surgery, radiotherapy and chemotherapy. Radiotherapy consists of 
teletherapy and brachytherapy. In this current study, only brachytherapy is considered. 
The word ‘brachytherapy’ originated from a Greek word which means ‘brachos’ and it 
refers to a short-range therapy.  Brachytherapy is the short distance treatment procedures 
of malignant diseases or tumours with radiations coming from encapsulated sources. The 
radiation sources are placed either directly into or near the treatment volume to deliver 
radiations at short distances (Girinsky et al, 1993). The purpose of brachytherapy 
treatment is to destroy the cancer cells with ionizing radiation by localizing high radiation 
dose in the tumour volume while the normal tissues and critical organs receive as low 
dose as possible (ICRU 1985; Williamson, 1988). 
 
Brachytherapy is important for the achievement of local disease control in the treatment 
of cervical cancer. In this mode of treatment, high radiation dose can be delivered to the 
target volume with rapid dose fall-off in the surrounding normal tissues. This type of 
treatment is always temporal and lasting only for some few days. It should therefore be 
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performed under strict rules of standard source-application systems such as the 
Manchester system or ICRU system (ICRU 1985).  These systems simply prescribe the 
target dose to a generic reference point or to a reference volume without identifying the 
target volume because the systems were established before the target volume could be 
readily identified geometrically. Geometric identification of the individual target volume 
and other regions of interest have become essential in the practice of radiotherapy, 
especially of intracavitary brachytherapy, for specifying treatment parameters such as 
clinical target volume, maximum and minimum target doses, and doses to organs at risks 
(Williamson, 1988). Intracavitary brachytherapy is another possibility of internal 
radiotherapy, in which the sources of ionizing radiation are implanted inside the tumour 
volume of the patient. This study considers the internal radiotherapy, in which the photon 
and electron beams are the most common sources of radiation used in intracavitary 
brachytherapy for most clinical applications. The heavy particles such as neutrons and 
protons are rarely used because of the enormous cost involved (Feng et al, 1995; Serkies 
et al, 2001, Boman et al, 2005). The treatment unit is typically an AMRA-Curietron in 
which photons are made to travel through special tubes called channels to deliver the 
treatment doses into the tumour volume of cancer patients. In this treatment unit, the 
sources usually emit photons during treatment, which is considered as a stream of 
photons moving into the treatment volume and consequently depositing dose into the 
tissue. Due to the complex track geometries and various types of photon interactions with 
matter, dose deposition is a complex process. 
Current tumour therapy techniques usually use complex optimization algorithms or 
calculation methods to deliver fast treatment planning and dose calculations in order to 
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achieve very accurate results. Some of the various methods used are Monte Carlo 
simulations, Sievert Integrals and AAPM-TG43 (Deshpande et al, 1997; Bobylev et al, 
1998; Favorite et al, 2005). Owing to the various cost of computational problems usually 
encountered in the aforementioned methods or algorithms there is the need for a faster 
but accurate method for dose calculation algorithms and the finite element method (FEM) 
has attracted much attention as a golden standard method of solving the photon transport 
equation for dose distributions in biological media.  
In this study, radiation doses to critical organs will be modelled using the FEM method in 
solving the photon transport equation in heterogeneous media (Santhamma et al, 1978, 
Schweiger et al, 1995; Arridge et al, 1993; 1995,). The photon transport in tissues is 
restricted to geometries that are cylindrically symmetric around the incident photon. The 
use of the FEM as a mathematical tool to solve the photon transport equation in an 
integro-differential form will therefore be a confirmatory determination of radiation doses 
to critical organs and the target volume. The FEM is very fast, accurate and therefore the 
results obtained will enhance the speed accuracy of the treatment planning system for 
cancer patients undergoing intracavitary brachytherapy treatment. 
 
1.2  Statement of the problem 
During intracavitary brachytherapy treatments, the specific doses to the target volumes 
are specified by the radiation oncologist based on clinical experience. Using the 
Manchester or the ICRU system and the generic reference points of protocol adopted at 
the oncology unit, the doses to the critical organs like pelvic wall, bladder and rectum are 
determined, consequently the time of implant of the radioactive source is also calculated. 
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The most frequent clinical complications usually encountered during intracavitary 
brachytherapy treatment is when portions of the critical organs like rectum and bladder 
receive some amount of radiation doses. Errors in calculating the treatment dose and 
inadequate procedures for placement of sources in the applicators have resulted in many 
accidents in intracavitary brachytherapy treatments.  As a result, a confirmatory 
determination of doses to critical organs and target volume during treatment planning is 
highly desirable for comparison of the results before delivery of radiation doses to 
brachytherapy cancer patients.  
 
1.3  Relevance and Justification of the study 
In intracavitary brachytherapy treatment planning, one seeks the best localization of the 
sources in both the tandem and ovoids to achieve the desired dose distribution in the 
tumour volume and as low radiation dose as possible to the surrounding healthy tissues 
and critical organs.  Source localization in both the tandem and ovoids is usually difficult 
and time consuming, and this poses a major problem leading to complications in the 
critical organs.  
Radiation treatment planning and dose calculations are the crucial steps needed before the 
actual treatment is delivered. Failure to correctly plan the treatment and precisely 
determine the amount of treatment dose deposited in the patient according to the plan 
could result in catastrophe. Nowadays, the use of three dimensional (3D) imaging 
techniques, such as computed tomography (CT) and magnetic resonance imaging (MRI), 
enable accurate determination of patient 3D imaging. This enables the use of conformal 
radiotherapy, in which the goals of the radiotherapy treatment are met ( i.e the high dose 
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volume conforms to the planning target  volume (PTV) and other healthy tissues and 
critical organs receive as low dose as possible) (ICRU 1985).  
From the mathematical point of view, the radiotherapy treatment planning is an inverse 
problem, (i.e. the problem is to find the best source arrangements to receive the desired 
dose distribution within the patient).  In solving the inverse problem, one needs a solution 
of the forward problem, (i.e. a solution for the model of the dose calculation). In the 
forward problem of intracavitary brachytherapy, one computes the dose distribution in a 
patient resulting from the localization of sources of the ionizing radiation due to both the 
tandem and ovoids configurations. Several models have been developed for the dose 
calculations in radiotherapy (Schweiger et al, 1979, Allen et al, 1986,), and one of the 
best models currently gaining attention is the FEM.  
 
1.4  The scope and objectives of the study  
The primary objective of this study is to model radiation doses to critical organs of cancer 
patients undergoing intracavitary brachytherapy treatment using the FEM. 
The specific objectives of the study are  
i)     to group tumour stages of patients according to the International Federation of 
Gynaecology and Oncology(FIGO) staging guidelines as stated or used by 
Radiation Oncologists 
ii) to specify the doses to the tumour volume based on the prescribed dose to the 
reference point  
iii)  to model radiation doses to critical organs of intracavitary brachytherapy patients 
using the Finite Element Method to solve the photon transport equation  
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iv) to assess the risk of radiation doses to both the rectum and bladder of patients 
undergoing intracavitary brachytherapy treatment  
v) to use the results obtained from the model to improve the speed and accuracy in 
treatment planning system of oncology centres. 
vi)  to minimize the radiation doses to critical organs of patients undergoing 
intracavitary brachytherapy treatment in clinical applications using the principle 
of optimization.  
 
1.5   Structure of the dissertation 
Chapter 1, describes the introduction relating to the background, statement of the 
problem, relevance and justification of the study, the scope and objectives of the study. 
In chapter 2, literature review on brachytherapy and mathematical models are discussed.   
Chapter 3, describes the materials and method used in the study. Details of the dose 
calculations to critical organs and target volume of patients undergoing intracavitary 
brachytherapy is presented and this is the solution of the Boltzmann transport equation 
using FEM.  The results and discussions are presented in chapter 4.  Concluding remarks 
and recommendations are presented in chapter 5. 
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CHAPTER 2 
LITERATURE REVIEW 
This chapter reviews the literature relevant to radiotherapy with particular emphasis on 
brachytherapy. A general overview of radiotherapy, clinical brachytherapy sources and 
various brachytherapy applicators, organs at risk and critical organ dose monitoring, 
physics of ionization radiation interaction in radiotherapy, photon transport and Finite 
Element Method (FEM) and cancer staging and groupings are also presented. 
   
  2.1 General Overview 
 Cancer is one of the most deadly diseases in the world killing about seven million people 
every year, with about 70% of the death occurring in the developing countries (IAEA 
2010). The number of cancer cases is growing globally but in developing countries they 
are worst hit by the cancer crisis since the resources needed to prevent, diagnose and 
ensure treatment are severely limited. It has been highlighted by the Programme of 
Action for Cancer Therapy (PACT) that these figures quoted for cancer rates are likely to 
double by the year 2020 and tripling nearly by year 2030 with a projection of seventeen 
million deaths(IAEA, 2010). Most of the developing countries have very fragile 
economies and budgets for health care systems are too small. Hence, any additional 
disease developing alongside with known disease like malaria, HIV and tuberculosis 
(TB), can lead to crisis. 
Although cancer is a devastating disease, it is highly preventable and also curable if 
detected early particularly for common cancers such as breast, colorectal, prostate, cervix, 
head and neck. While effective treatment could increase, patient survival reduce cancer 
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mortality in the short term, preventive measures such as tobacco control, reduction of 
alcohol consumption, increased physical activity; vaccination against liver and cervix 
cancers, screening and awareness programmes could have a great impact on reducing the 
global cancer burden (IAEA, 2010) 
 
2.2 Principles of brachytherapy 
  Over the last hundred years, ionizing radiation has been increasingly applied in 
medicine and is now firmly established as an essential tool in both diagnosis and 
therapy. Brachytherapy or radioactive implantation has been employed since the 
discovery of radioactive isotopes such as radium in 1903. In brachytherapy the sources 
are inserted into the treatment volume or close to the treatment volume. The different 
types of brachytherapy implants are intracavitary, intraluminal, interstitial, surface 
plaques and intravascular brachytherapy. The brachytherapy implant can either be 
temporal or permanent. For temporal implants, the dose is delivered over a period of 
time which is short as compared to the half-life of the sources (Beyer et al, 2000). The 
sources are also removed after the prescribed dose is reached. In the case of permanent 
implant, the dose is delivered over the life time of the source until the source undergoes 
complete radioactive decay. The main characteristic of brachytherapy is the rapid dose 
fall-off with distance from the radiation source whereby the tumour in contact or close 
to the source receive high amount of radiation dose but outside the tumour, normal 
tissues receive doses at considerable much lower levels (Stewart et al, 2006). The 
physical advantage of the effectiveness of brachytherapy as compared to teletherapy is 
when the tumour is small with well-defined geometry or sometimes it may be used as a 
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boost dose to more advanced or ill-defined tumours (Georg et al, 2008). During 
brachytherapy treatment, the following considerations are crucial and should always be 
considered; these are: 
(i) using a suitable dosimetric model for the calculation of dose distributions and 
treatment time. 
(ii) prevention or avoidance  of geometrical misses due to accurate positioning of 
sources. 
(iii) using a calibrated source with its calibration traceable to a standard laboratory.  
 
2.3   Importance of Low Dose Rate brachytherapy  treatment. 
There are three main advantages of Low Dose Rate (LDR) brachytherapy. These are  
(i)  positioning of the source at short distances from the tumour volume, this allows a 
very good dose distributions in the target volume; that is good radiation dose distribution 
is the ability to locate radioactive sources in or close to the tumour, either by topical 
mold,  intracavitary or interstitial implant, which represents the optimal conformal dose 
delivery system. Specifically, it is generally easier to compare with  external beam 
radiotherapy to deliver high doses of radiation to target tissues, while minimizing 
radiobiological damage to normal adjacent tissues hence, good dose distributions sparing; 
(1) early responding normal tissues, which in external beam radiotherapy, typically 
produce the complications that force treatments to be prolonged over more than one 
month; and (2) late-responding normal tissues, which, in external beam radiotherapy, 
often represent the dose-limiting endpoint (Han et al, 1999). 
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(ii) short overall treatment times, to counter tumour repopulation; It is now generally 
accepted that long overall treatment times can be a significant cause of local failure in 
radiotherapy, because accelerated repopulation during the treatment means that tumour 
cells start to divide more rapidly than they can be killed (Withers et al 1988; Slevin et al 
1992;  Brenner, 1993). However, generally speaking, the optimal strategy for any 
radiotherapeutic regimen requires:  
(a) short overall times to limit tumour repopulation; and  
(b) long overall times to reduce early normal-tissue sequelae, especially to the skin and 
mucosa. Prima facie, requirements (a) and (b) are mutually exclusive; therefore, in most 
radiotherapeutic situations, the overall treatment time represents a compromise between 
short treatment times to minimize tumour repopulation and long treatment times to 
prevent unacceptable early complications. On the other hand, brachytherapy, because of 
its good dose distribution, inherently produce less early normal tissue damage. Therefore, 
the compromise on overall time does not have to be made, and much shorter times are 
tolerable in brachytherapy than could conceivably be employed in external-beam 
radiotherapy for a comparable effective tumour dose and  
(iii) low dose rate, which results in an increased therapeutic advantage between tumour 
control and damage to late-responding tissues.  It has been known for many decades that 
lowering the dose rate generally results in a reduction in radiobiological damage. Also, 
decreasing the dose rate will decrease late effects much more than it will decrease tumour 
control. Thus, the therapeutic ratio (i.e the ratio of tumour control to complications) will 
increase as the dose rate decreases. Therefore, the lower the dose rate, the better the 
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differential response that can be achieved between tumour control and late sequelae 
(Withers, 1985). 
 
2.4     Clinical brachytherapy sources 
Different types of sealed sources are used in brachytherapy treatment. They all contain a 
certain amount of a radionuclide that is encapsulated in sealed layers of a metal such as 
platinum or stainless steel or some kind of thin foil in the case of  -emitters. Various 
types of sources such as tubes, needles, wires, pellets, seeds, and a single stepping source 
connected to a cable, are available. 
There are three different types of brachytherapy sources. These are photon, beta and 
neutron sources. For the photon sources, they emit gamma rays through  -decay and 
possibly characteristic X-rays through internal conversion and electron capture. Some of 
the photon sources normally used in brachytherapy treatments are Co-60, Cs-137, Ir-192, 
I-125 and Pd-103. The beta sources emit electrons following β-source decay. Examples 
are Sr-90 and Y-90. Neutron sources emit neutrons following spontaneous nuclear 
fission. A typical example is Cf-252. Due to the cost implications and the relevance of the 
sources to this study, consideration is given to only the photon sources. The dosimetric 
characteristics of the sources are photon energy, half-life, halve value layer (HVL) of 
shielding material, specific activity, source strength and inverse-square dose fall off. 
 
 
 
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Table 2.1:   Characteristics of some radionuclides used in brachytherapy 
Radionuclide Average photon Half-life (T )  HVL in Pb(mm) 1
2
energy(MeV) 
Co-60 1.25 5.2 years 11.0  
Cs-137 0.66 30 .0 years 6.5  
Au-198 0.41 2.7 days 2.5  
Ir-192 0.38 73.8 days 3.0  
I-125 0.028 60.0 days 0.02  
Pd-103 0.021 17.0 days 0.01  
  
2.5 Intracavitary brachytherapy in gynaecology 
This is the type of brachytherapy treatment in which the applicators are inserted into a 
body cavity to reach the volume of the tumour. It is mainly used for the treatment of 
cancer of the cervix, uterine body and the vagina. Various applicators are used to hold the 
sources in an appropriate configuration in the tumour volume.  These include the 
Fletcher-Suit-Delcos and the Tandem-Ring applicators as shown in Plates 2.1 and 2.2 
respectively. 
The most commonly used applicator for intracavitary brachytherapy treatment is the 
Fletcher-Suit-Delcos system which consists of a central tube called the tandem and lateral 
capsules also called ovoids or colpostats. The most widely used brachytherapy source for 
the treatment of gynaecological cancers is Cs-137. It is often necessary to use sources of 
different strength in order to achieve the desired dose distribution (Tanderup et al, 2010). 
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                                     Plate 2.1:  Fletcher-Suit-Delcos applicator 
 
 
                                              Plate 2.2:   Tandem-Ring applicator 
Different systems are used for dose specifications in brachytherapy treatment. These 
include; Stockholm, Memorial, Paris, Quimby, Manchester and the ICRU systems. The 
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two most commonly used systems in gynaecology are the Manchester and ICRU systems.  
The Manchester system uses the Fletcher-Suit-Delcos applicator and is characterised by 
doses to four points, namely point A, B, bladder and rectum as depicted in Figure 2.1. 
Ideally, point A represents the location where the uterine vessel crosses the urethra and 
also relates to the position of the sources. The dose at point A is very sensitive to the 
position of the ovoid sources relative to the tandem. Point A is defined to be 2cm superior 
to the external cervical end of the tandem and also 2cm lateral to the cervical canal. Point 
B is defined as 3cm laterally to point A or 5cm from the midline of the tandem or when 
the central canal is not displaced. The duration of the irradiation is based on the dose rate 
calculated at point A or B depending on the reference point adopted by the oncology unit.  
 
 
 
 
 
 
 
 
 
 
Figure 2.1: The reference points A and B according to the Manchester system 
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The diagram shows the reference points A and B used in intracavitary brachytherapy 
when using the Manchester system for the treatment of gynaecological cancers.  In the 
ICRU system, the dose distribution is related to the target volume rather than to a specific 
point. The ICRU report identifies a dose level of 60Gy as the appropriate reference point 
for a low dose rate brachytherapy source. This result is a requirement to specify the 
dimensions of the pear-shaped isodose reference volume of 60Gy. 
 
2.6 Organs at risk (OAR). 
Organs at risk are sometimes called critical normal structures. They are normal structures 
that because of their radiosensitivity and their location close to the target volume may 
significantly influence the treatment planning or the prescribed dose level (ICRU, 1999). 
Organs at risk such as the bladder and rectum often limit the prescribed dose to the target 
volumes in cervical cancer in brachytherapy treatment. Therefore it is very important to 
accurately report OAR doses and correlate them to clinical outcome. In intracavitary 
brachytherapy for cervical cancer treatment, the planned tumour volume is limited to the 
anterior-posterior (AP) direction by the presence of bladder and rectum. The treatment is 
frequently planned to the maximum tolerable dose to these organs at risk. Treatment 
plans based on the desired dose levels, are specified for the tumour and organ at risk. The 
desired dose levels are chosen so that a high tumour control probability is realized and 
also the probability of complications in any organ at risk is very low (upper bound on 
dose level in healthy tissue) (Haie-Meder et al, 2005). The desired dose levels are 
specified in any organ which has the same harmful or beneficial effect as the actual 
inhomogeneous dose that is given by the radiation treatment equipment. 
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2.7 Critical organ dose monitoring in gynaecology 
Brachytherapy of cervix carcinoma results in high doses to some surrounding critical 
structures such as rectum and bladder. Therefore, these structures should be closely 
monitored as in many cases complications are developed decreasing patient quality life 
and therapeutic ratio. The most related factors that could lead to the complications on 
these critical structures are the volume of irradiated rectum, the total dose to the rectum 
and the dose rate of the brachytherapy modality used (ICRU 1985; Chen et al, 2000). 
In gynaecology, the most critical organs associated with intracavitary brachytherapy 
treatment are bladder and rectum. The most frequent clinical complications of 
intracavitary gynaecological radiation treatments results from a high dose delivered to 
portions of either the rectum or the bladder. In clinical applications, surgical gauze is 
used to displace the sensitive critical structures away from the applicators so as to keep 
the doses to the critical organs as low as possible but in most cases complications are 
often encountered (Chen et al, 2000; Ferrigno et al, 2001; Nag et al, 2002). 
The diagram in Plate 2.3 shows the female anatomy and placement of applicators for 
intracavitary brachytherapy treatment in gynaecological cancers using the Manchester 
system. 
 
 
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                              Plate 2 .3:  Fletcher-Suit-Delcos system in gynaecology 
Direct measurements of rectal and bladder doses have been attempted using various 
miniature ionization chambers, scintillation detectors and MOSFET dosimeters. It has 
proven that, measured data give large variability and correlate poorly with calculated 
values. 
 
2.8 Anatomy of the cervix 
The cervix carries out many critical functions that contribute to the overall reproductive 
health and wellbeing of women. The cervix is the portion of the uterus which is 
connected to the vaginal canal and: 
1. allows the passage of menstrual fluid; 
2. promotes fertility; 
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3. protects the uterus, upper reproductive tract, and a developing fetus from 
pathogens; and 
4. it may play a role in women’s sexual pleasure. 
The cervix (Latin word meaning neck) is the lower-most portion of the uterus. 
Cylindrical in shape, the cervix consists of the following two major structures: the 
ectocervix and the endocervix. The ectocervix is the portion of the cervix which is 
touchable and visible through the vaginal canal. This is also the area which is swabbed 
during a pap smear. The endocervix is the internal, canal-like portion of the cervix which 
opens into the uterus. The size, shape and color of the cervix depend on a woman’s age, 
hormonal state and whether or not she has given birth. For women who are nulliparous, 
or have not given birth, the cervix appears to have a small circular opening (external os) 
at its center. In parous women, the cervix is bulkier and the external os has a more slit 
like appearance. 
The cervix is a small, cylindrical organ which is about several centimeters long but less 
than 2.5cm in diameter. It also forms part of the lower neck of the uterus. The cervix 
separates the body and cavity of the uterus from the vagina. There is a small canal 
running through the cervix which allows sperms to pass from the vagina into the uterus 
and during menstruation, blood passes through this same canal. 
The cervical canal which forms part of birth canal during childbirth dilates widely to 
allow the passage of the baby. The bulk of the cervix consists of fibrous tissue with some 
smooth muscle. This tissue makes the cervix to form a sphincter (circular muscle) and 
allows great adaptability in its size and shape during pregnancy and childbirth. 
 
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                  Figure 2.2:  Anatomical structure of the human cervix 
2.8.1 Risk Factors of cervical cancer  
A person is likely to develop cervical cancer if the following practices are often indulged 
in 
(i) smoking processes 
(ii) starting to have sexual acts at an early stage 
(iii)  having many sexual partners 
(iv)  having taken contraceptive pills for longer periods 
(v) having a weakened immune system 
 
 
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2.8.2   Symptoms of cervical cancer 
The following are some of the various associated symptoms of cervical cancer. These are 
(i) unusual bleeding between menstrual periods and also after sex 
(ii) chronic itching of the vulva 
(iii)  pain in the pelvis 
(iv) persistent vaginal bleeding 
(v) post menopausal bleeding 
(vi)  lower back pain 
 
2.9   Physics of ionizing radiation interaction in radiotherapy 
Ionizing radiation is used in radiotherapy treatments because of its ability to damage 
living tissue. The intended high local concentrations of absorbed energy can kill a cell 
either directly or indirectly. Ionizing radiation therefore excites and ionizes these medium 
atoms when traveling through the medium (Attix, 1986). It is classified as directly 
ionizing radiation when either electrons or positrons deliver energy to the medium 
directly. However, indirectly ionizing radiation, photons first transfer their energy to 
charged particles, which then deliver energy to the medium. In indirectly ionization 
radiation, there is formation of highly reactive chemical species such as free radicals in 
water, which are always present in a human tissue. A free radical is an atom or a 
compound with unpaired electrons.  
Photons consist of X-rays and gamma rays named according to their mode of origin, 
although their properties are almost the same (Attix, 1986). Electromagnetic radiation 
emitted by charged particles in changing atomic energy levels is called characteristic or 
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fluorescence X-rays. Continuous or bremsstrahlung X-rays are emitted from inelastic 
collisions of charged particles in which they slow down in Coulombic force field.  
Gamma rays are electromagnetic radiation emitted from the nucleus or in annihilation 
reactions. Annihilation radiation is emitted when there is re-combination of positron and 
electron.  β-rays (negative or positive) and δ-rays consist of electrons and positron.  β-
rays are emitted from the nucleus and δ-rays are resulted from the charged particle 
collisions (Attix, 1986).  
As photons are uncharged they travel quite a long distance in a medium without 
interacting with medium atoms. On the other hand, charged particles, such as electrons 
and positrons, travel only a small distance until they undergo interactions with medium 
atoms. Thus, photons undergo only few interactions until they attenuate and secondary 
electrons born, while electrons loose their energies in many small interactions with 
medium atoms along their trajectories (Attix, 1986). 
Radiation therapy is primarily used to treat cancer by locally targeting radiation to the 
diseased tissue or cancerous cells. 
 
2.9. 1 Photon interactions in biological media 
Actually, photons by themselves do not deposit energy to matter, but during different 
interactions, the electrons of medium atoms obtain energy from photons and these 
electrons then impart that energy to matter in many interactions along their tracks 
(Attix,1986). Photons are individual units of energy. As photon beam passes through a 
medium, there are three possibilities associated with the photon; these are 
(i) it may penetrate through the matter without interacting 
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(ii) it may interact with the matter and be completely absorbed hence depositing 
its energy 
(iii)  it may interact and be scattered or deflected from its original direction and 
deposit part of its energy in the medium. 
There are two main major interactions through which photons deposit their energy. These 
are photoelectric effect and Compton effect. Photon interactions are dominated by 
absorption and inelastic scattering collisions, in which the photon's energy is reduced. 
The most important interactions of photons are Photoelectric effect, Compton effect, Pair 
production and Rayleigh scattering.  
 
2.9.2 Photoelectric Effect 
In the photoelectric effect, an incident photon gives all its kinetic energy to a tightly 
bound electron, such as those in the inner shells of an atom (Attix,1986 ; 2004). The 
photoelectric effect can only occur when the kinetic energy of incident photon is greater 
than the electron binding energy. However, the photoelectric effect is more probable in 
lower energies, as long as the energy is more than the binding energy of an electron. The 
photoelectric effect is dominant at energies below about 200keV for low Z media or 
materials. The energy range increases as Z increases. Photoelectric effect is an event in 
which an incident photon disappears and the so called photo-electron is ejected. The 
affected gap in the atomic shell is then immediately filled with another electron from the 
outer less bound atomic shell resulting in an emission of the characteristic X-ray, 
sometimes called fluorescence X-ray or emission of an Auger electron with kinetic 
energy. 
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The photoelectric effect is the process by which a photon with energy hv  is absorbed by 
an atom which, as a result, emits an electron with energy hv  EB   where EB , is the 
binding energy of the ejected electron. This process leaves the atom in an excited state 
and it relaxes by emitting a characteristic X-ray or subsequent Auger electron. The 
photoelectric effect is most likely to occur when the photon's energy is slightly greater 
than the electron's binding energy and low energy interactions tend to produce photo-
electrons ejected at right angles while higher energy interactions will eject photo-
electrons in a forward direction. 
3
The mass attenuation coefficient for photoelectric interactions varies as Z  for high-Z 
3.8
media and as Z  for low-Z media. For low-Z materials the photoelectric effect is 
1
dominant at energies below 200 keV and varies approximately with energy as  
hv3
2.9.3 Coherent and Incoherent Scattering 
The most common photon interaction at therapeutic energies is scattering by electrons. 
There are two types of photon scatter: coherent and incoherent. Coherent, or Rayleigh 
scattering, involves the deflection of a photon by atomic electrons during which none of 
the photon's energy is converted to electron kinetic energy. Atomic electrons, subject to 
the electric field of an incident photon, are set into vibrational motion and the oscillations 
of each electron emit a particular wavelength of radiation that combine to form the wave 
of the incident photon, but travelling in a new direction. Rayleigh scattering is primarily 
forward directed and has little impact on photons with energy greater than 100 keV. 
  
 
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                 Figure 2.3: An electron undergoing a Compton scattering event. 
 
The most significant interaction for photons between 20 keV and 5 MeV in tissue-like 
material is incoherent scattering, or Compton scattering, during which a photon interacts 
with an atomic electron and is scattered at a new trajectory while the electron is ejected 
with kinetic energy imparted by the incident photon. Figure 2.3 shows  a schematic 
diagram  of a Compton scattering event. The photon has initial energy hv  before 
colliding with the electron. Because the electron's binding energy is greatly exceeded by 
the energy of the incident photon, it is treated as a free electron. The electron is ejected 
from its orbit at some angle   with energy E( )  while the photon is scattered at a 
corresponding angle   with energy hv  E( ) . 
The Klein-Nishina formula is used by EGSnrc Monte Carlo to determine the Compton 
cross section and is given per unit solid angle as 
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2
 r 2o 2  1   
2 (1 cos )2 
   (1 cos  )  1    (2.1) 
 2 1 (1 cos )
2
  1 (1 cos )(1 cos  )
hv
where     and m c2o  is the rest energy of an electron.  
moc
2
If the scattered electron is emitted in the direction of the incident photon, the scattered 
photon will be emitted at 180o  and the electron will have the maximum energy possible 
for a Compton electron. Conversely, a scattered electron will receive the least possible 
energy if the incident photon grazes by and continues nearly straight forward, emitting 
the scattered electron at nearly90o .  
Compton interactions are nearly independent of atomic number and decrease with 
increasing energy (Johns et al, 1983). The Compton effect is dominant at energies 
between 20keV and 30 MeV in mediums like human tissue or water (number of electrons 
per molecule Z = 10). The dominance region is narrower for high Z mediums. In 
Compton effect, the incident photon (energy E΄p, direction Ω΄p) collides with an electron 
and gives some of its energy to that electron. After the collision the incident photon 
energy and direction are changed to Ep and Ωp, respectively. Also the electron has now 
kinetic energy Ee and direction Ωe.  If the electron is assumed to be initially at rest and 
unbounded, then  
Ee = E΄p -Ep , and the relation between Ωp΄ and Ωe is obtained by kinematics.  
 
2.9.4  Pair Production 
In pair production, the incident photon with a minimum energy of 2Eo = 1.022MeV is 
absorbed and electron and positron are born or created (Attix,1986). Pair production is 
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more dominant at very high energies. It can only take place in a Coulomb force field, 
which is usually near an atomic nucleus. Sometimes, pair production can occur in the 
field of an atomic electron. Then the interaction is called triplet production, because the 
initial electron achieves also kinetic energy and thus two electrons and one positron has 
now a significant amount of kinetic energy. An opposite interaction to pair production is 
an annihilation process, in which electron and positron recombine leading to 
disappearance and emergence of two photons. 
Pair production is the conversion of energy to mass that results when a photon is subject 
to the strong field of an atomic nucleus and becomes an electron-positron pair. The 
photon must have energy of at least 1.022 MeV to comprise the rest masses of the 
charged particles. Any additional energy held by the photon is distributed between the 
electron and positron as kinetic energy. Because the atomic nucleus plays a part in this 
interaction it receives a very small portion of the photon's momentum; consequently, the 
momentum of the electron does not uniquely predict the momentum of the positron and 
vice-versa. If the incident photon has energy of at least 2.04 MeV, it may interact in the 
field of an orbital electron to the same end, with the addition of an ejected atomic 
electron. This less common process is called triplet production as it produces a total of 
three charged particles. A positron will propagate through and ionize matter in the same 
manner as an electron until it slows down enough to annihilate with a free electron and 
o
produce two 0.511MeV photons which are ejected at 180  from one another. If the 
positron has some remaining kinetic energy when it annihilates, the angle will be nearly 
o
180 . The incidence of pair-production interactions increases rapidly with increasing 
energy once the 1.022 MeV threshold has been met and the mass attenuation coefficient 
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increases approximately with atomic number. Pair production is the dominant photon 
interaction at energies in excess of 5 MeV.  
2.9.5  Characteristic X-rays  and Auger Electrons 
Characteristic X-rays, or fluorescence X-rays, are the photons emitted when an atomic 
orbital vacancy is filled by an electron from a higher orbital shell. The energy of a 
characteristic X-ray is equal to the difference in binding energies between the orbital 
levels and is unique to atomic number. If a characteristic X-ray is absorbed by an outer 
orbit electron, that electron is ejected with the energy of the characteristic X-ray less the 
binding energy of the electron and is called an Auger electron (Attix, 2004). 
 
2.9.6  Rayleigh scattering 
In Rayleigh scattering, the incident photon is scattered by the combined action of the 
whole atom (Attix, 1986 and 2004). That is why it is also called coherent scattering. 
Rayleigh scattering is elastic, which means that no energy is lost in the interaction, in 
which the atom moves just enough to conserve momentum and initial photon is redirected 
in a small angle. Rayleigh scattering is more significant at low energies, in which the 
photon scattering angle is much bigger.  
 
2.10  Linear Energy Transfer(LET) 
The total distance an electron travels in a material before losing all its energy is referred 
to as  its range. There are two main factors that determine the range, these are the initial 
energy of the electrons and the density of the material or medium. One important 
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characteristic of electron interactions is that all electrons of the same energy have the 
same range in a specific material. 
The rate at which an electron transfers energy to a material is known as LET and it is 
expressed in terms of the amount of energy transferred per unit distance travelled in a 
medium. 
Mathematically,  it can be expressed as 
dE
LET 
                                     dx        (2.2) 
The S.I unit of LET is  keV
m
 
In tissue, the LET depends on the kinetic energy or velocity of the electron. Generally, 
the LET is inversely proportional to electron velocity. As the electron looses energy, its 
velocity decreases and the value of LET increases until all its energy is dissipated. 
Particles can loose energy continuously along their path when interacting with the matter 
and continuously slowing down. 
When working out the impact of biological systems or matter, the amount of energy lost 
by the particle, then the LET is important. Particles with large LET can do a large amount 
of damage, but usually over a short range. 
 
2.11  Electron interactions 
Because electrons are charged particles their interaction probabilities are much higher 
than those of uncharged particles. For example, photons need only few interactions to 
dissipate all of its kinetic energy rather than electron would typically undergo several 
(about 105) interactions before losing all of its kinetic energy (Attix, 1986 and 2004). 
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Because photons may pass a slab of matter without any interaction or it may lose all of its 
kinetic energy in a few interactions, it is impossible to predict individual photon traveling 
distance. On the contrary, because electrons interact almost with every atomic electron or 
nucleus it passes and loses its kinetic energy gradually in small friction-like process, it is 
customary to describe electron traveling by stopping power, range and yield. These are 
expectation values for a charged particle energy loss per path length, for a path length and 
for an electromagnetic radiation production, respectively. Due to the nature of the 
charged particle, it is not customary to describe charged particle interactions by total 
cross sections and differential cross sections, which are needed in the Boltzmann 
transport equation and these, are found more rarely in the literature than those for photons 
(Zerby et al, 1967; Zheng et al, 1992). Electron interactions are elastic scattering, 
inelastic scattering and radiation processes, which take place in a form of bremsstrahlung 
and electron positron annihilation. The bremsstrahlung process dominates at high 
energies where as the inelastic scattering dominates at low electron energies in high Z 
media. In tissue-like materials, the inelastic scattering dominates at all energies and the 
influence of the bremsstrahlung process is only few percents (Attix, 2004).  
 
2.12  Methods for calculation of radiation doses from acquired image data 
There are various ways of radiation-dose determination in particular, computational 
methods and other systems for calculating radiation doses delivered to anatomical tissues 
and critical structures during radiotherapy treatments. Radiation transport plays an 
important role in many aspects of radiotherapy and medical imaging. In radiotherapy, 
radiation dose distributions are accurately calculated to both the treated regions and 
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neighbouring organs and other structures where the dose is to be minimized. Dose 
calculations are commonly used in radiotherapy treatment planning and verification 
(Daskalov et al, 2000). The dose calculations are often repeated numerous times in the 
development and verification of a single patient plan. Some modalities include external 
beam radiotherapy, brachytherapy, and targeted radionuclide therapies.  
In addition, many variations exist in beam delivery methods, including 3D conformal 
radiotherapy (3D-CRT), intensity modulated radiotherapy (IMRT), and stereotactic 
radiosurgery (SRS). Brachytherapy treatments include photon, electron and neutron 
sources with the use of a variety of applicators and other delivery mechanisms in 
delivering the required prescribed dose.  
The physical models that describe radiation transport in the anatomical structures are 
usually complex and difficult, hence accurate methods for radiation dose determination 
due to radiation transport in tissues are highly needed and also of great importance in 
brachytherapy. 
The various methods employed in radiotherapy for radiation-transport computations can 
be broadly grouped into three categories; these are stochastic Monte Carlo methods, 
deterministic and analytic or empirical methods.  
Monte Carlo methods, stochastically predict particle transport through media by tracking 
a statistically significant number of particles. While Monte Carlo methods are recognized 
as highly accurate, simulations are time consuming, limiting their effectiveness for 
clinical applications. In most clinical applications, this method of approach is based on 
simplifications of results which limit its accuracy and scope of applicability. 
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The phrase "deterministic radiation-transport computation," refers to methods which 
solve the Linear Boltzmann Transport Equation (LBTE), which is known to be the 
governing equation of radiation transport, by approximating its derivative terms of 
discrete volumes. Examples of such approaches include discrete-ordinates, spherical-
harmonics, and characteristic methods. However, the use of deterministic methods in 
radiotherapy and imaging applications has been limited to research in radiotherapy 
delivery modes such as neutron capture therapy and brachytherapy. This can be attributed 
to several factors, including methodic limitations in transporting highly collimated 
radiation sources, and the computational overhead associated with solving equations 
containing a large number of phase-space variables.  
The phrase "analytic or empirical radiation-transport computation methods," refers to 
approaches which employ approximate models to simulate radiation transport effects; for 
example, using pre-defined Monte Carlo scattering or dose kernels. Examples of analytic 
or empirical methods include pencil beam convolution (PBC) methods and collapsed 
cone convolution (CCC). Due to their relative computational efficiency, PBC approaches 
are widely used in radiotherapy, even though their accuracy is limited, especially in the 
presence of narrow beams or material heterogeneities.  
In radiotherapy, various uncertainties in dose delivery may translate into suboptimal 
treatment plans of cancer patients, and for imaging, a reduced reconstructed image 
quality may result. This has necessitated the need for accurate, fast, stable, generally 
applicable and computationally efficient method in both radiotherapy and diagnostic 
applications by solving the radiation transport equations using different approaches or 
methods. 
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2.13 International recommendations on target and organs at risk: ICRU-definitions   
The success of Boltzmann transport equation requires the delivery of a high radiation 
dose directly to the tumour volume or target whilst possibly sparing to some degree, the 
surrounding normal tissues.  The target volume must always be described, independently 
of the dose distribution, in terms of the patient’s anatomy and tumour volume.  
The gross tumour volume (GTV), may be evaluated by various diagnostic methods; 
clinical examination i.e. inspection and palpation, endoscopies, and image techniques 
such as radiography, Computer Tomography (CT), Magnetic Resonance Imaging (MRI), 
Positron Emission Tomography (PET), ultrasound, or other techniques, depending on the 
location and type of pathology.  
The clinical target volume (CTV), is the volume which contains the gross and subclinical 
disease. Clinically, it thus contains the GTV and a safety margin around the GTV to take 
into account (probable) subclinical involvement. The CTV may also include other 
anatomical areas, e.g., regional lymph nodes or other tissues with suspected (or proven) 
subclinical involvement.  
The planning target volume (PTV), is a geometrical concept, taking into consideration the 
net effect of all possible geometrical variations.  
The organ at risk (OAR) is normal tissue whose radiation sensitivity may significantly 
influence treatment planning or prescribed dose.  
   
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Figure 2.4:  Graphical representation of the volumes of interest, as defined by 
ICRU (ICRU,1999). 
 
For brachytherapy the planning target volume PTV is the same as the clinical target 
volume, CTV. The prescribed dose is defined as the dose which the oncologist or 
clinician intends to give and enters in the patient treatment chart. 
The minimum target dose (MTD) at the periphery of the CTV is the minimum dose 
decided upon by the clinician as adequate to treat the CTV (minimum peripheral dose). 
MTD is approximately equal to 90% of the prescribed dose in the Manchester system. 
A pear-shaped dose distribution with an extremely sharp dose gradient is usually 
achieved by using tandem and ovoid applicators with appropriate configuration of 
sources (ICRU Report 50). 
 
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2.14  The importance and use of numerical methods 
 Numerical methods have played important role both in science and engineering to solve 
and analyze problems. The solution to these problems can be achieved with the help of 
computers. The importance of numerical methods in analysis is due to several factors; 
some of them, in  most natural phenomena can best be described by differential equations 
with varying boundary conditions, whose solutions cannot be obtained by analytical 
means except only in very simple cases. Great improvements have been made in 
developing numerical techniques so that problems could be solved at low cost and within 
span of short times. With the arrival of high speed computers, scientists and engineers 
have succeeded in exploiting a lot of numerical methods. Numerical solution of the 
transport equation finds application in numerous areas like nuclear reactor design, 
radiation shielding, semiconductor device design, radiation oncology and radiation 
transfer in stellar atmospheres. 
Transport is the process by which particles distribute themselves within a host medium 
and by streaming off in various directions, interacting with other particles of the medium 
until they are either finally absorbed or scattered out of the system.  
The transport of photons is focused on this current study of research. The photon 
transport was governed by the Boltzmann transport equation which was derived by 
Ludwick Boltzmann in connection with the kinetic theory of rare gases. It describes the 
distribution of photons in the host medium as a function of their position, energy, 
direction of motion and time. 
There are two main classes of computational techniques that are used in solving the 
transport equation. In the first class, using deterministic methods, the transport equation is 
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discretized  by using a variety of methods and solved directly or iteratively. Different 
types of discretization give rise to different deterministic methods such as spherical 
harmonics, discrete ordinate method, finite element method and others like Monte Carlo 
methods which uses stochastic techniques. 
 
2.15   Dose calculations in brachytherapy 
 In brachytherapy, one or more localized radioactive sources are placed within or in close 
proximity to the treated region, and dose conformity is achieved by optimizing a 
brachytherapy source arrangement which can maximize dose to the treated region, while 
minimizing radiation doses to neighbouring healthy tissues and critical organs. Each of 
the brachytherapy sources may be represented as a point source, which is anisotropic in 
both angle and energy, representing the exiting photon flux on the surface of the 
brachytherapy source. In most brachytherapy applications spatial electron transport can 
be neglected, and the dose field can be obtained by a kinetic energy released in matter 
(KERMA) using the energy-dependent photon flux.  
There are various methods of calculating doses in brachytherapy. Some of the current 
methods are stochastic Monte Carlo simulations and deterministic methods like the 
discrete ordinate method, spherical harmonics and finite element methods (Jones et al, 
1994). 
Nowadays, the Monte Carlo (MC) method is assumed to be the most accurate method for 
the dose calculation in brachytherapy (Rogers, 2006). In MC, the paths of the scattering 
particles are followed by randomly selecting the directions and energies of the particles. 
In different media, there are scattering interactions whereby some particles loose their 
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energy and other particles are created and finally, energy is absorbed in matter depositing 
the dose. With several millions particle histories MC method gives an estimate for the 
dose distribution. The major drawback of the MC method is the long computation time 
and because of this, the pencil-beam models are the most common approach which is 
sometimes used in treatment planning systems (Ulmer et al, 1996). In pencil-beam 
models, the dose distribution in a patient is achieved as a superposition of appropriately 
weighted dose deposition kernels. The kernels can be modeled with Monte Carlo 
methods, using some approximations or using empirical beam data (Strochi et al, 1999). 
Most of the dose calculation models have their roots in Boltzmann transport equation 
(BTE). The previously presented MC method uses BTE to simulate particle transport in a 
medium.  In its original form, the BTE takes rigorously into account the patient 
inhomogeneity and scattering effects. It is an integro- differential equation and it is 
studied in many fields of physics. The linear BTE can be used in radiation therapy, 
because the high energy particles move nearly at the speed of light and the interactions of 
particles with each other are assumed to be negligible. The BTE describes particle 
traveling in a medium and it is based on the particle equilibrium in a small volume. The 
unknown function in the BTE based forward problem is the particle angular flux, from 
which the dose can be calculated. Instead of finding an analytical solution of the BTE, 
one can seek for a numerical solution, which can be stochastic or deterministic.   
The solution of the BTE using deterministic methods has been carried out by many 
authors and several production codes have been generated in the field of radiation therapy 
physics (Mordant, 1986 and Williams et al, 2003). In deterministic methods, the 
stationary BTE is usually solved using some grid-based numerical method, in which the 
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phase space is discretized in spatial, angular and energy domains. The energy 
discretization is often done by multi-group approximation, in which the energy range is 
divided into energy groups and the interaction cross sections are replaced by multi-group 
cross sections (Allen, 1986). Most of the methods often used for angular discretization 
are the discrete ordinates and spherical harmonic approximations (Manteuffel et al, 
2000). Finite difference and finite element methods (FEM) are usually used in spatial 
discretization  and also for angular discretization. The finite element discretization 
method may be used for all variables simultaneously (Tervo et al, 1999) but sometimes 
the discontinuous FEM is used. Other methods, which are successfully used to solve the 
BTE, are the method of characteristics, the method of moments, the electron multiple 
scattering theory, the collocation method and hybrid collocation-Galerkin-Sn method.   
Mainly due to computational problems, the BTE based on deterministic models are not 
often used in dose calculations in clinical applications. BTE describes particle traveling in 
a medium and it is based on the particle equilibrium in a small volume. The unknown 
function in the BTE based forward problem is the particle angular flux, from which the 
dose can be calculated. In the stationary case, it has six variables, three spatial, two 
angular and one energy variable. BTE can be solved analytically only in very simplified 
geometries, in which it does not have very much of practical use. Instead of finding an 
analytical solution of the BTE, one can seek for a numerical solution, which can be 
stochastic or deterministic (Tagziria, 2000).  
 
 
 
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2.16   Radiation safety during brachytherapy treatment 
There is the need for radiation safety precautions when patients are implanted with 
radioactive sources during brachytherapy. All temporary patients undergoing low dose 
rate (LDR) brachytherapy treatments are hospitalized for some few days and the radiation 
levels in and around their room should be monitored to assure low radiation levels for 
personnel attending to the patient. For high dose rate (HDR) brachytherapy treatments the 
staff should not be allowed in the treatment room during the short time of irradiation 
which lasts between five to sixty minutes (Lertsanguansinchai et al, 2004).  
The walls of the treatment room are designed to meet acceptable levels of exposure 
outside. Time, shielding and distance are common key factors used to limit exposure to 
staff and visitors. Sometimes, additional movable lead shields are used to lower 
excessively high radiation levels around implant patients. For LDR brachytherapy, 
visitors are allowed to see their patients. In the case of remote afterloading technique, the 
treatment will be interrupted but such an interruption should not take too long because of 
the effect on the dose and time pattern. The irradiation should not be considered as 
continuous pattern if the interruption exceeds 10% of the total irradiation time. If, 
exceptionally, the sources are in the patient during a visit, the visitor should be clearly 
informed of the risk and special measures like time, distance and shielding should be 
taken into consideration in order to minimize exposure. 
 
2.17   Quality Control and Quality Assurance in brachytherapy 
In brachytherapy, extreme attention should be given to quality control and quality 
assurance. The main aim of quality control and assurance is to maximize the probability 
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that each individual treatment is carried out consistently, accurately and safely. A very 
important function in brachytherapy treatment is the correct geometric localization of the 
applicator in order to treat the target volume adequately. The consistent applicator 
placement is crucially dependent on the skill of the radiation Oncologist at the facility. 
Following the application procedure, it is primarily the physicist’s responsibility to 
ensure that the treatment is delivered accurately and safely in accordance with the 
radiation Oncologist’s prescription. The treatment planning system that calculates the 
dose distributions must be provided with the correct data needed to perform these 
calculations. 
Brachytherapy source arrangements consist of individual sources, but they are often 
grouped as strings, trajectories, or applicators. One should confirm that parameter 
changes that might affect the entire group of sources are correctly made. The verification 
of dose calculations for clinical use is a very important part of the commissioning of a 
treatment planning system. A comprehensive series of test cases must be planned. These 
tests involve measurements and calculations. The results must be analyzed before any 
dose calculation is used clinically. 
 
2.18   The photon transport theory 
The transport theory is used to refer to a mathematical description of transport of particles 
or radiations through a medium. The foundation of the transport theory lies in the kinetic 
theory of gases which was developed by Ludwig Boltzmann who was a renowned 
Austrian Physicist. There are three famous equations named after Boltzmann, these are (i) 
the equation of entropy, (ii) an equation concerning particles in a gravitational field, and 
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(iii) an equation for particle or radiation transport. The latter is sometimes called the 
Boltzmann transport equation.   
The mechanism of photon transport in media (tissues) has been the subject of description 
during the last decade. If one considers photon propagation as a particle (eg. neutral 
particle) then the BTE provides an exact description of the photon transport. The photon 
transport theory may be considered as the main foundation of radiotherapy treatment. The 
photon transport equation is considered to be an expression governing the photon 
distribution in water equivalent tissue such as patient undergoing radiotherapy treatment. 
However, the complexity of the integro-differential equation makes the photon transport 
equation exceedingly difficult to solve analytically. Typically, simplifying 
approximations are made to reduce its complexity. 
‘ ’
The term transport   can be explained as the processes in which particles distribute 
themselves in a host medium, by streaming off  in various directions, interacting with 
other particle of the medium and finally being  absorbed or leaked out of the system. 
The transport theory has been applied to the study of many areas; some of them are as 
follows 
(a) neutron transport in nuclear reactors 
(b) penetration of light through the atmosphere 
(c) shielding of radioactive  sources 
(d) diffusion of holes and electrons in semiconductors 
(e) photon transport through biological tissues 
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Despite the fact that different kinds of particles (eg neutrons, gas molecules, electrons and 
photons) may be involved in the transport processes, all these phenomenon can be studied 
and described by using the same basic transport equation. 
One of the principal equations used in radiotherapy dose analysis is the transport 
equation, which is a linearized derivative of the equation developed by Ludwick 
Boltzmann for the kinetic theory of gases (Bell et al, 1970). This equation may be used to 
determine the photon distribution during brachytherapy treatment as a function of energy, 
position, direction and time. Actually, a solution for the photon source distribution is 
always sought in a given problem and the coupling between the two is treated externally. 
This study will focus on the problem of photon transport distribution in a biological 
media such as patients undergoing intracavitary brachytherapy treatment and the time-
dependent form of the linear transport equation will be used. The photon transport 
equation may be presented in different forms, and simplifications are often applied to 
tailor the equation to the requirements for a specific application. In radiotherapy physics 
applications, the transport equation is often written in terms of the angular photon flux as 
the dependent variable. Generally, the transport equation is difficult to apply and solve 
analytically hence, simplifications and numerical solutions are often necessarily sought in 
order to apply the equation in radiotherapy applications. The photon transport equation is 
an integro-differential equation that cannot often be solved simply in analytical form and 
for this reason as an alternative; researchers have reported various computational methods 
such as stochastic and deterministic method which renders the PTE solvable. 
The propagation of photons in matter might be seen as stream of particles, each with a 
localized quantum of energy. Photon transport through a turbid media can be 
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mathematically expressed by the Photon Transport Equation (PTE). The PTE is derived 
by considering the principle of conservation of energy balance of incoming, outgoing, 
absorbed and emitted photons of an infinitesimal volume element in the medium. The 
PTE is valid for an isotropic and quasi-homogeneous medium; hence it is assumed that, 
the inhomogeneities are small and uniformly distributed throughout the material. The 
tissue is assumed to be a homogeneous matrix containing absorbing and scattering 
centers. In this case, a tissue is represented by these parameters namely, absorption 
coefficient ( a ), and scattering coefficient (  s ).  A photon propagating through a 
medium or a patient is affected not only by the inverse square law but also scattering and 
attenuation. These parameters make the dose deposition in a medium a complicated 
process and its determination rendering it to be a complex task. 
Numerical methods are usually used to solve the photon transport equations which are 
divided into two forms: these are (i) stochastic (Monte Carlo) methods and (ii) 
deterministic techniques (Lewis et al, 1984). Even within the deterministic methods, 
further categorization breaks the approaches into discrete ordinates and integral transport 
approaches (Kavenoky, 1981). The integral transport approximations are derived from 
the integral form of the transport equation, in which the angular dependence of the 
angular flux is eliminated by integrating out all angular variables, based on an assumed 
order of scattering (Lewis et al, 1984). This technique can be applied to complicated, 
heterogeneous tumour geometries with multiple dimensions. 
The discretization method is one of the most widely used method for solving the PTE  an 
integro-differential form.  The accuracy of the solution obtained by using the method of 
discretization is governed by the number of discretization terms used. A wide range of 
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different techniques to solve the PTE are available but each with its advantages and 
drawbacks. 
The governing equation of transport theory is the transport equation, which is equivalent 
to the Boltzmann equation used to describe particle transport. In this particular case the 
particles are photons.   
 
2.18.1   Advantages of using the transport equation 
The following are some of the advantages of using the photon transport equation in the 
determination of absorbed dose in tissues: 
(i) The solutions are exactly reproducible from one run to another unlike that of 
Monte Carlo simulations 
(ii)   The method lends itself well for making parameter changes easily for a given 
application 
(iii)   When it applies, it can be more rapid than Monte Carlo simulations 
(iv)    It generates a solution for all points in a given phase space 
 
2.18.2 Disadvantages of the transport equation 
The main disadvantages in using the transport model in the determination of the absorbed 
dose in a medium are:  
(i) Very large grids are needed for solution with sufficient accuracy 
(ii) Little progress has been made on 3-D solutions. 
 
 
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2.19   Finite Element Method (FEM) 
A number of analytical tools such as the Sievert integral and Convolution methods are 
available for brachytherapy dosimetry (Williamson et al 1991, Carlson and Ahnesjo 
2000). The AAPM TG-43 also recommends a formulation that uses parameters such as 
geometry factors, anisotropy functions and radial dose functions derived through Monte 
Carlo methods and measurements. Monte Carlo methods are known to provide very 
accurate results for a wide range of problems of complex geometries. The efficiency of 
the Monte Carlo methods starts to degrade if the detailed distribution of the results, such 
as three-dimensional flux or absorbed dose distributions, is computed. Since the volume 
needs to be divided into very small meshes for detailed distributions, obtaining enough 
histories contributing to such small volumes may not be quite easy (Lewis and Miller 
1984).  
The FEM was first used by Clough in 1960 to solve continuum problems, however in 
1972 the tempo of development of FEM increased remarkably which became the most 
active field of interest in numerical solution of continuum problems and it has remained 
the dominant method till today.  
The FEM is a numerical technique for solving problems which are described by partial 
differential equations (PDE) or can be formulated as a functional minimization; it is well-
suited for approximating continuum solutions to boundary-value problems with complex 
geometries or domains and composed of inhomogenous materials.  The FEM is based on 
integral formulations and is one of the computational techniques for obtaining 
approximate solution to partial differential equations that arise in scientific and 
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engineering applications. It can provide approximate solutions to many complicated 
problems that would be intractable by other techniques 
The FEM is one of the numerical methods often used to solve the PTE. The numerical 
stability of the finite element model depends on the mesh resolution, error limit of the 
conjugate gradient method (that is, it refers to an iterative scheme for symmetric and 
positive definitive problems), optical properties of the simulated media and the 
modulation frequency (Arridge et al, 1993). When compared with Monte Carlo methods, 
FEM is found to be  twenty (20) times faster than MC in time - domain measurements 
and 550 times faster than MC in the continuous intensity measurements (Lapidus and 
Pinder 1982). 
The use of the FEM in transport problems has given geometrical flexibility to the 
deterministic methods which were lacking previously but are now comparable to the 
stochastic Monte Carlo methods.  After its appearance in the late fifties, the FEM has 
been finding its applications in almost all branches of science and engineering. The 
general procedure in the FEM is to subdivide the region of interest into a finite number of 
sub-regions called the finite elements. Apart from its geometrical flexibility, the FEM is 
particularly advantageous because it gives the angular flux everywhere in the system. 
The finite element method is sometimes referred to as finite element analysis. The 
solution approach of PDE is based either on eliminating the differential equation 
completely as in the case of steady state problem or rendering the PDE into an 
approximating system of ordinary differential equations which are solved using standard 
techniques such as Euler or Runge-Kutta method. It is a good choice for solving PDE 
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over complex domains. It is an approximate method for solving PDE by replacing 
continuous function by piecewise approximations defined on polygons. 
In FEM, the domains of the phase space are partitioned or discretized into sub-regions 
which are called elements or cells. In 3D, typical elements are bricks, triangular prisms 
and tetrahedrons. In 2D, the elements are usually triangular or quadrilateral in shape. A 
one dimensional (1D) interval is divided into sub-intervals. Typically, the boundaries of 
the spatial elements are located near the surfaces of different materials and one can 
assume the material properties to be the same within an element (Chai et al, 2004). The 
FEM discretization is applied to a variational form (weak form) of the original equation 
(Johnson, 1987). The variational form is achieved by multiplying the original equation by 
a test function and integrating over the whole phase space domain. When applying the 
Green's formula, the boundary conditions can be appended in the variational form. When 
using any numerical approach in finding the solution, one has to ensure that the problem 
has a solution and it is unique.  The boundedness and coercivity conditions also result in 
an estimate for the error of the FEM which is sometimes known as the Cea's estimate 
(Dautray et al, 1993). 
The principal advantage of FEM is its speed and flexibility. In addition it can produce 
photon density everywhere and as well as photon flux on the boundary. The only 
disadvantage is that there is no means of deriving individual photon histories. 
The spatial configuration of any system is described by its degrees of freedom (DOF). 
The DOF is also called generalized coordinates. If the DOF is finite, the model is called 
discrete and continuous otherwise. Since FEM is a discretisation method, the number of 
DOF is necessarily finite. The basic steps of FEM are discussed in more generality in 
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Figure 2.5. Although this diagram oversimplifies the way FEM is actually used, it serves 
to illustrate the terminology. The three key simulation steps are idealization, 
discretisation and solution; each step has its source of error. Considering the reverse 
processes, contnuification and realization will not be considered under this study now, 
however it can be considered for another study area in solving a physical system under 
modelling aspects. 
 
                   Idealization                            Discretization                     Solution                   
 Physical Mathematical Discrete  
System Model Model Solution 
 
REALIZATION 
 AND CONTINUIFICATION 
IDENTIFICATION Solution Error 
 
Discretization + Solution Error 
 
Modelling + Discretization + Solution Error 
  
 
Figure 2.5: A simplified schematic diagram of the physical simulation process 
showing FEM modeling terminologies 
 
Many researchers have preferred the use of FEM as it offers advantages in speed and 
flexibility in solving complex geometries and heterogeneous tissues (Arridge et al, 1997). 
The FEM is a numerical analysis technique used for obtaining solution to a wide variety 
of continuum problems. The basic premise of FEM is that a solution region can be 
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analytically modeled or approximated by replacing it with an assemblage of discrete 
elements. The finite discretization procedures reduce the problem to a finite number of 
unknown field variables in terms of assumed approximation functions within each 
element. The approximation functions sometimes called interpolation functions are 
defined in terms of the field variables at specified points called nodes or nodal points. 
The nodes usually lie on the boundaries where adjacent elements are connected. 
In the FEM, the domain of interest is replaced by a finite number of unknown specified 
variables in terms of so called finite elements, such that their assemblage represents the 
same domain with the same properties in terms of finite number of unknowns. Within 
each element, nodal field variables are used to determine the approximating functions. In 
other words, a continuous physical problem with an infinite number of degrees of 
freedom (DOF) is transformed into a discretized finite element problem with a finite 
number of unknown nodal parameters. For a linear problem the unknowns correspond to 
a system of linear algebraic equations that are straightforward to solve and to determine 
the nodal values (Jin, 2002). Finally, the unknown fields throughout the finite elements 
are determined from these nodal values. 
 
2.20 Photon dose deposition mechanisms 
The dose deposition in tissue from photon beams is a two step process (AAPM Report 
85, 2004).  
(1)  The TERMA step: the photons interact in the medium and impart their energy 
to charged particles 
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(2)  The dose deposition step: charged particles, launched by the interacting photons, 
deposit their kinetic energy along their tracks 
 
The TERMA is an acronym for Total Energy Released to Matter and it sometimes 
referred to as the initial photon interactions step. In the therapeutic range of photon beam 
energies and tissues constituting the human body the prevailing types of photon 
interactions, resulting in deposition of their energy in the medium, are the following: 
Compton effect, pair production and photoelectric absorption. The probability of 
Compton interaction dominates (>80%) in water-like media, but decreases for high-Z 
materials. Pair production becomes relevant for higher beam energies. The contribution 
from photoelectric absorption is negligible in water-like media, although it becomes 
greater in the high-Z materials (eg: prostheses or teeth implants). The probabilities of 
these interactions per unit distance are characterized with linear attenuation coefficients 
and their sum μ represents the probability of the photon removal from the beam. The 
attenuation coefficients are dependent both on the photon properties (energy) and tissue 
characteristics (density and atomic composition) which is given by the relation as 
t (E,, A, Z)  c (E,, A, Z)   pp (E,, A, Z)   pe(E,, A, Z)  
where: 
μt - total linear attenuation coefficient, E - photon energy, ρ – density, A - atomic number 
Z - atomic number of element, μc, μpp, μpe - linear attenuation coefficient for Compton 
effect, pair production and photoelectric absorption respectively 
 
 
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2.20.1  The dose deposition step: 
 Due to the finite range of the charged particles released in the photon interactions, the 
energy absorption occurs at some distance around the interaction point. The recoil 
charged particles are launched with a spectrum of kinetic energies and initial directions. 
The multiple scatter interactions result in energy loss and deflection from the initial 
direction.  
They dissipate their energy either locally, in multiple quasi-continuous Coulomb 
collisions with atomic nuclei or orbital electrons or in occasional radiative events, in 
which the electron is deflected from its original path and part of its energy is emitted in 
the form of a bremsstrahlung photon that carries it away from the charged particle track 
 
2.21  Cancer Staging and Grouping. 
 Since the 1930s, gynaecologic oncologists have strived for a common language to 
facilitate making diagnoses and planning treatment for their patients. The aim was, and 
still is, to reach a uniform or unified terminology that is able to provide appropriate 
prognosis to the patients and to enhance the exchange of information among health 
professionals. 
The first rules for classification and staging of female genital cancers adopted by the 
International Federation of Gynaecology and Obstetrics (FIGO) in 1958 (Creasman W. T, 
2001). 
Staging is a way of describing a cancer in terms of its location, where it has spread, 
whether it is affecting the functions of other organs in the body. There are two systems of 
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staging most types of cervical cancer. These are the International Federation of 
Gynaecology and Obstetrics (FIGO) and American Joint Committee on Cancer (AJCC).  
The two systems are very similar because they both classify cervical cancer on the basis 
of three main factors. The factors are (i) the extent of the tumour (T) (ii) whether the 
cancer has spread to the lymph nodes (N) and (iii) whether the cancer has metastasized or 
spread to other distant sites (M). There are different stage descriptions for the cancer 
types. One most important tool the oncologist use to describe the stage of cancer is the 
TNM system. The TNM is an abbreviation for tumour (T), node (N) and metastasis (M). 
This system classifies the disease in stages from 0 to IV. It is based on clinical staging 
rather than surgical staging. This means that the extent of the disease is evaluated by the 
oncologist’s physical examination and a few other tests that are done in some cases such 
as cystocospy and protoscopy. 
Using the TNM system, “T” plus a letter or number ranging from 0 to 4 is used to 
describe the size and location of the tumour. However, in some stages they are further 
divided into smaller groups that help to describe the tumour in more details. The 
following examples explain how the stages of the tumour are applied.  
Lymph node spread (N): 
N0: no spread to the nearby lymph node. 
NI: the cancer has spread to the nearby lymph nodes. 
Distant spread (M): 
M0: the cancer has not spread to distant lymph nodes, organs or tissues. 
MI: the cancer has spread to distant organs like the lungs or liver, to lymph nodes in the 
chest or neck 
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Table 2.2: Stages in cancer development 
Cancer 
Description 
stage 
the cancer cells are only found on the surface layer of the cervix that is the 
I 
layer of the cells lining the cervix without growing into the deeper tissues. 
IA there is very small amount of cancer which can be seen under a microscope. 
the cancer has grown beyond the cervix and uterus but has not spread to the 
II 
walls of the pelvis or to the lower part of the vagina. 
IIA The cancer has not spread into the tissues next to the cervix called parametria 
IIA1 The cancer can be seen but it is not larger than 4 cm . 
IIA2 The cancer can be seen and it is larger than 4 cm 
IIB The cancer has spread into the tissues next to the cervix (the parametria). 
The cancer has spread to the lower part of the vagina or to the walls of the 
III pelvis. The cancer may be blocking the ureters (tubes that carry urine from the 
kidney to the bladder). 
The cancer has spread to the lower third of the vagina but not to the walls of the 
IIIA 
pelvis. 
the cancer has grown into the walls of the pelvis and it blocking either one or 
IIIB 
both the ureters (this is called hydronephrosis) 
IV The cancer has spread to the bladder or rectum. 
 
 
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2.21.1  Stage Grouping 
Combining all the information with regard to the tumour stage (T), lymph node (N) any 
cancer spread (M) assigned to the stage of the disease is called stage grouping. 
Table 2.3: Tumour stage Grouping 
Stage Grouping Description 
of 
cancer 
I TI,N0,M0 Cancer has grown into the cervix but it is not growing outside 
the uterus. The cancer has not spread to nearby lymph nodes 
(N0) or to distant sites (M0). 
IA TIA,N0,M0 This is the earliest form of stage I. There is very small amount 
of cancer and can be seen only under microscope. The cancer 
has not spread to nearby lymph nodes (N0) or to distant sites 
(M0). 
IAI TIAI,N0,M0 The cancer is less than 3mm deep but less than 7mm wide. 
The cancer has not spread to the lymph nodes (N0) or to 
distant sites (M0). 
IB TIB,N0,M0 This stage includes stages I cancers that can be seen without a 
microscope as well as cancers that can only be seen with a 
microscope. These cancers have not spread to nearby lymph 
nodes (N0) or to distant sites (M0). 
II T2,N0,M0 In this stage the cancer has grown beyond the cervix and 
uterus but has not spread to the nearby lymph nodes (N0) nor 
to the distant sites (M0). 
IIA T2A,N0,M0 the cancer has not spread to the tissues next to the 
cervix(called parametria). The cancer may have grown into 
the upper part of the vagina. It has not spread to nearby lymph 
nodes (N0) nor to distant sites (M0). 
IIAI T2AI,N0,M0 The cancer can be seen but it is not larger than 4cm. It has not 
spread to nearby lymph nodes (N0) nor to distant sites (M0) 
 
IIB T2B,N0,M0 The cancer has spread into the tissues next to the cervix 
(parametria) It has not spread to nearby lymph nodes (N0) nor 
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to the distant sites (M0). 
III T3,N0,M0 The cancer has spread to the lower part of the vagina or to the 
walls of the pelvis. The cancer may be blocking the ureters. It 
has not spread to nearby lymph nodes (N0) nor to distant sites 
(M0). 
IIIA T3A,N0,M0 The cancer has spread to the lower third of the vagina but not 
to the walls of the pelvis. The cancer has not spread to the 
nearby lymph nodes (N0) nor to the distant sites (M0) 
IIIB T3B,N0,M0 The cancer has grown into the walls of the pelvis and has 
blocked one of the ureters but has not spread to lymph nodes 
(N0) nor to the distant sites (M0) 
 
IVA T4A,N0,M0 The cancer has spread to the bladder or rectum which are 
organs close to the cervix (T4). It has not spread to nearby 
lymph nodes (N0) nor distant sites (M0). 
IVB AnyT,anyN,MI The cancer has spread to distant organs beyond the pelvic area 
such as the lungs or liver.  
 
In addition to identifying the type and stage of the cancer of the cervix, the tumour’s 
grade should also be determined. The tumour grade is based on how the tumour cells 
appear under microscope. If they look like normal tissue, the cancer is called a low grade 
tumour. If cells do not look like normal cells the cancer is classified as a high grade 
tumour. Knowing the grade of the tumour is important in determining whether treatment 
is needed after surgery. 
 
 
 
 
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CHAPTER 3 
MATERIALS AND METHODS 
This section describes the method used in this study to model doses to the critical organs 
of cancer patients undergoing cervical cancer treatment. The theoretical model is based 
on the Boltzmann’s Transport Equation (BTE) for photons in biological tissues. The 
medium is assumed to be   heterogeneous and the finite element method is used in this 
case to solve the BTE in order to account for the complex geometries and angular 
distribution of the photons. Data from literature and results from treatment planning 
system (TPS) of the Komfo Anokye Teaching Hospital (KATH) in Kumasi, Ghana were 
used to compare results obtained by the method.   
 
3.1 Theoretical Modeling of Doses to the Critical Organs 
3.1.1 Governing Equation 
The model described in this section is based on the solution to the Boltzmann Transport 
Equation (BTE) using the finite element method. The BTE can be rewritten and 
explained in terms of the following; 
partcles at 
the net number  flux of   scattering and   
    position r being
 of  particles particles particles  absorption  
            scattered in all  entering  source  along  within the   
      directions 
 '

volume element  direction  phase element  
int o  

  
(3.1) 
 
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The balanced Boltzmann integro-differential equation for particle j can be written 
mathematically as; 
1  3j
 ( . j  j j 
' '
v t   
  dE d  S )dr dE ddt  0            (3.2) j j j j
v j 1 A j1
Thus;  
1  3j
. j  j j     j j  j dE
'd '  S j  0                                    (3.3) '
v j t 1 A j1
And 
1  3j ' '
S ' '
' j
 . j  j j     j j  j dE d        (3.4) 
v j t 1 A j1
Where, Sj indicates the source term, Φj is the photon flux distribution, Ej is the energy and 
Ω is the angular flux, υj is the particle velocity , and σj is the interaction coefficient. σj→j 
is a phase function given by: 
 2 
 j j  f (r, E, E ',, ' )  (r, E, E ',, ' )      (3.5) 
E 
 
3.1.2 Source term  
The source geometry is represented by the Manchester system using the Fletcher-Suit 
Delcos applicator as shown in Figure 3.1. In Ghana, Cs-137 low dose rate (LDR) sources 
are used for gynaecological disorders. The Cs-137 sources are made up of spherical 
pellets arranged in a stainless steel encapsulation. Secondary arrangements are made in a 
stainless steel tube similar to the Amersham “J-Type” tube. The source arrangement is 
shown in Figure 3.2.  
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In order to relate the source strength to the specific activity of the source, the spherical 
sources are assumed to be multiple isotropic point sources distributed over the surface of 
a cylinder in a heterogeneous medium. The source to the targets is assumed to be in plane 
geometry.  
 
Figure 3.1: The Manchester system showing the reference points A, B and  
        geometrical arrangement of sources in the tandem and ovoids. 
     
 
 
 
 
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 Inter Capsule    Secondary encapsulation 
Argon arc welded  stainless steel Active content      Ring marking stainless steel 
plug  
 
 
 
 
22..6655  mmmm  1 mm diameter 
 
ddiiaammeetteerr  countersunk eyelet 
  
 
Active Length 13.5mm  
   Centered in outer capsule  
 Centered in outer capsule  
    External Length 20mm 
 
 
 
 
Figure 3.2: A Cs-137 LDR afterloading unit with tube arrangements similar to an 
Amersham “J-Type” tube (Activity: Up to 1480 MBq (40 Ci) per pellet). 
     
 
 
The total source strength, S(r), representing the activity injected into a solid angle in a 
unit volume per unit time is given by: 
S (r)    S (r,, E) dEd        (3.6) 
4 E
 
Where, S(r,Ω,E,t) is the source strength for a discrete point source, which indicates the 
number of photons emitted per unit time; per unit solid angle in the direction, Ω; per unit 
energy interval at a location specified by the position vector, r. 
 
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Photon flux from the sources are formed by two components, namely the primary 
uncollided flux and the secondary scattered flux. The photons emitted by the seed form 
the uncollided flux and this does not include the characteristic radiation emission induced 
by photoelectric absorption. The flux distribution around the integrated source is as 
shown in Figure 3.3.  
 
Figure 3.3: Non-homogeneous photon flux distribution around a Cs-137 source in 
saggittal plane 
 
The uncollided flux is computed by taken the product of the total attenuation of each 
source and summing the contribution from all the sources. In this model, the source is 
taken as a collection of point sources. The angular flux induced by the point sources at a 
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point of interest, r, is subjected to attenuation along the photon path. If the angular flux is 
integrated over the solid angle, this provides the primary flux, Φp(r, Ω,E) with the source 
term, So, is given by 
r
2
          S 0 4r  p (r,, E)exp  t(r,, E)dr   
0
And the primary flux is also given as follows 
r
S
 p (r, E,) 
o exp( t (r,, E)dr      (3.7) 4r 2
0
The scattered photon flux distribution is also given by: 

 s (r, E,)   s (r, E' E, ')(r, E ', ')d 'dE '   (3.8) 
4 E
Where, µt is the total attenuation coefficient of the tissues between the source point and 
the point of interest per unit solid angle per unit energy, E; µs is the scattered photon 
attenuation coefficient; r is the spatial coordinate boundary for target boundary and 
photon source. s (r, E ' E, ') is the scattering coefficient for both coherent and 
incoherent scattering. 
 
3.1.3 Target geometry 
The targets are the rectum and the bladder, which can be represented by a slab phantom 
with specific distances from the source point depending on the patient’s topography as 
shown in Figure 3.4. The targets represent the organs at risk (OAR), where there is the 
need to limit doses to avoid complications during the treatment. The thickness of the 
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OAR and the specific distances between the source point and the targets depends on the 
patient’s size and weight.  
The target geometry can be represented by a hollow cylindrical surface matrix. In a 
cylindrical coordinate system, the flux distribution over the surface geometry is given 
by: 
H 2
S dS
            o           (3.9a) 4 r 2
0 0
or 
H 2
So Rddz             (3.9b) 4 r 2
0 0
where, dS = Rdφdz, and R is the radius of the cylinder and H is the height. 
 
Figure. 3.4: Anatomical structure of the cervix showing the target points   
 
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3.1.4 The Photon Transport Equation  
The photon transport equation is derived by considering energy conservation in a small 
volume and is given by the following 2D time dependent case for photons as 
1  E r,, t  SE r,, t. E r,, t t E r,, t
c t
2                                       (3.10) 
  s  f ,
'  E r, , , t '
0
Where, r is a position vector, c is the velocity of light and   is a unit vector in the 
direction of photon propagation. E r,,t  is the energy flux given by 
E r,,t (r,, E,t)dE . The source term SE r,,t  is given by S E r,,t= 
 S(r,, E,t)dE  and represents the activity injected into a solid angle centered on   in a 
unit volume at r. The phase function f ,'  describes the probability that during any 
scattering event, a photon with direction  ,  is scattered into the direction,  .  Where,
t  as , is the total attenuation co-efficient in tissue. 
 
3.1.5  Finite Element Model(FEM) 
The Finite Element Method (FEM) is a numerical technique for solving problems which 
are described by partial differential equations or can be formulated as a functional 
minimization. It is well-suited for approximating continuum solutions to boundary-value 
problems with complex geometries and composed of inhomogeneous materials. In the 
FEM, the domain of interest is replaced by a finite number of unknowns specified in 
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terms of so called finite elements, such that their assemblage represents the same domain 
with the same properties in terms of finite number of unknowns. Within each element 
nodal field variables are used to determine the approximating functions. In other words, a 
continuous physical problem with an infinite number of degrees of freedom (DOF) is 
transformed into a discretized finite element problem with a finite number of unknown 
nodal parameters. For a linear problem the unknowns correspond to a system of linear 
algebraic equations that are straight forward to solve and to determine the nodal values. 
Finally, the unknown fields throughout the finite elements are determined from these 
nodal values 
In the finite element method (FEM) for the Boltzmann equation, the angular flux is 
h
expressed in a finite-dimensional space Χ  
K
 (r,)h (r,) k k (r,)      (3.11) 
k1
h
Where, ( Ψk ; k=1,...,K) are a set of basis functions for X .  When applied to the first-order 
form of the Boltzmann equation with all the boundary conditions is natural and the 
resultant discretized system is again expressed as a matrix relation; 
 
BΦ = C         (3.12) 
 
where, B represents element stiffness matrix of order MN X MN and C is the element 
source vector of order MN X 1. 
The basis function can be developed separately for the spatial and angular terms, 
Ψ k (r, Ω) = ui(r)θj(Ω)        (3.13) 
This simplifies the derivation of the matrix elements Bkk’. The spatial basis is usually 
taken as a piecewise polynomial just as the angular basis.  
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3.1.5.1    Time dependent case 
Here, the finite element method is applied to solve the time dependent transport equation.  
From the general photon transport equation 
1 d
(r,, t) t (r) (r,, t)  ds (r, ' )(r,, t)  S(r,, t)  (3.14) c dt
4
        
Subject to the initial conditions 
(r,,0)  o (r,, )                                                                                                  (3.15) 
 and boundary conditions 
   (r,,t)  s (r,,t)                                                                                               (3.16) 
Defining the space, H , and choosing an arbitrary element  H  and multiplying 
equation (3.14) and integrate by parts over the phase space,  , the following equation is 
obtained; 
1 d
 ,  ,  k,  , S,  s ,                                      (3.17) 
c dt
where, the inner product ,  is defined as earlier. 
h
For the time-independent case, defining a finite dimensional subspace   H  and  
h
expanding  n (r,)  in terms of the following basis function for    
N
 n (r,)  
n
j j (r,)                (3.18) 
j1
 A similar procedure is followed for the time dependent case except that the expansion 
coefficients are time dependent: 
N
 (r,,t)  (t) nn j j (r,)                                                                                   (3.19) 
j1
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Substitution of this expansion into equation (3.14) and requiring equation(3.14) to hold 
for all   
n
 i (r,)  , i= 1,2,3,...,N  
One obtains the matrix differential equation  
1 
  P A  S                                                                                                           (3.20) 
c
  
 (t) 
1(t) 
1 S 1(t) 
      
 2 (t)   2 (t)  S 2(t) 
   .  . . 
where           ;         and S     
.  .  . 
     
.  .  . 
 (t)     N  S (t)

  N N (t)
 
Also 
n
 P  P ij ; but  Pij  j , ni  
and 
n
 A  Ai, j;  A nij  j , i  
writing the terms explicitly one obtains 
Ai, jT i, jBi, jk i, j     
where, 
T i, j  drd
n
j (r,)
n
j (r,)  

n n
B i, j
'
drdm i(r,) j (r,)  

n
k i, j  drd (r) i (r,)
n n ' '
t j
(r,)   drd i (r,) d (r,, ) (r, )  s
  4
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n
S i  drdS(r,)
n
j (r,) drdm
'o (r,) i(r,)  
 
The time dependent equations result in a system of ordinary differential equations. An 
efficient method to solve equation (3.20) is the Crank-Nicholson scheme which 
approximates the time derivative with a forward difference and other time-dependent 
terms are averaged over the present time and the incremented time: 
 (t m1) (t m) S(t m1)  S(t m)
       and S   
t m1t m 2
Using the notation, the matrix equation (3.20) becomes 
1  (m1)m m1m S (m1)S m
   P  A                                                                (3.21) 
c t 2 2
Since equation (3.21) gives o  then clearly, it can be solved for (m1)  in terms of  m  
or  
t t S (m1)S (m)
P  A  (m1) P  A  (m) t                                           (3.22)  
2 2 2
For which (m1)  can be solved for in terms of   (m)  and the known source term S. 
Thus with   (o)  known   (1) ,  (2) ,....,  (m) , can be calculated using equation (3.21). 
This is especially efficient if the LU decomposition (Cholesky method) is used to solve 
equation (3.20) 
 
 
3.1.5.2    Time-independent case in plane geometry 
In plane geometry, the general time dependent transport equation (3.8) becomes  
1
1 d d l 2l 1
   (x)(x,,t)  bl (x)Pe()dPe()(x,,t)  S(x,,t)c dt dx t l0 2 1   
(3.23) subject to the initial conditions 
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(x,,0) o (x,)
                                                                                                   (3.24) 
and the boundary conditions 
(0,,t) o (0,,t)
   
(1,,t) o (1,,t)
                                                                                                   (3.25) 
 
                                       Slab phantom (Patient) 
                                                                                                                 
 
                 
 
  Vacuum              vacuum 
 
 
 
   -1     0            1 
 
 
Figure 3.5:  Diagram of a physical domain (patient) represented by a slab 
phantom   in plane geometry 
 
Applying the finite element approximation to the equivalent integral law results in the 
following system of ordinary differential equations (ODE’s) 
1 
P A(t)  S(t)c
                                                                                                  (3.26) 
The specific matrix elements are 
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Ai, j T i, jk i, jBi, j
                                                                                                       (3.27) 
1 1
n n
P i, j dxd i (x,) j (x,)
0 1                                                                               (3.28) 
The source vector is  
1 1 1 0
n n
S i(t)  d dS(x,,t) (x,) 
n
i d0 (0,,t) i (0,)   d0 (1,,t) i (1,)                            
0 1 0 1
       
          
(3.29) 
 
3.1.5.3   Boundary Conditions for the Photon Flux 
In intracavitary brachytherapy, the high energy particles are directed to planning target 
volume (PTV) from the source inserted into the cervix which is the domain V. The 
boundary of the domain V is typically assumed to be a free surface in which particles can 
only escape from the surface V, through the surface  V   but cannot re-enter as shown in 
Figure 3.6. 
 
V  
 
             t(r, Ω) - Ω 
r 
 
V  
 
Figure 3.6: Particle trajectory from point r to the boundary V  along a straight line 
in the direction,  .  The length of the trajectory is t(r,) . 
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The assumption is approximately valid for convex domains in which a straight line 
segment connecting the two points in the region, lies entirely within the region. The free 
surface assumption is an idealization, since particles have always some probability to 
return to the volume V once they leave because of scattering process (Bell and Glasstone, 
1970; Duderstadt and Martin, 1979). However, the probability is negligible and it is 
always possible to choose the boundary to be far enough from the volume of interest so 
that the free surface assumption is valid (Bell and Glasstone,1970, Duderstadt and 
Martin, 1979, Boman, 2007).  In the following paragraph, the boundary conditions are 
defined for photon flux leaving the volume. 
In brachytherapy, photon flux is inside the domain and enters outside to the normal 
organs. This photon flux can be modeled using the transport model by setting boundary 
conditions, which states the photon outflow at the free surface. 
Let n(r)  [n1(r),n2 (r),n3(r),.] be the inward unit normal vector at a point r on the 
boundary. The particles which fulfill the condition  n(r).  0  at the point r on the 
boundary will cross the surface in an inward direction; and the particles for which 
n(r).  0  will cross the surface in the outward direction. 
The condition n(r).  0  for the outward flux gives the relation 
  n1(r)cossin n2 (r)sincos n3(r)cos  0 ,   
 which is obtained by using the relation 

      cossin ;sincos ,cos                                                           (3.30) 
u
where, (, )  are standard spherical coordinates on a unit sphere  [0,2 ]  and 
 [0, ] 
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x3 
 
 er 
 eθ 
θ 
r e  x2 
   
 
x1 
 
 
Figure 3.7: Polar coordinates (r, ψ , θ) indicating the direction of the particle in 
standard coordinate system  x = (x1, x2, x3) with r =1 
 
Once the outward flux is specified, it is possible to set a boundary condition for the 
outward flux on the boundary. The boundary condition for the outward photon angular 
flux is of the form 
0, for(r ,E ,)V PI such thatn (r )0 
o o, for(r ,E ,)PI such thatn (r )0                           (3.31) 
12  for (r, E,)VxIx  such that n(r).  0                                                    (3.32) 
where, P is the patch on the surface V  of the domain V and   0  is the photon angular 
flux per unit area with incident on P. Thus only the outward photon angular flux on P 
obtains non-zero values. Elsewhere on V the photon outward flux is zero.  The inward 
flux on the particles is not constrained. 
3.1.5.4   Error correction and Convergence 
FEM formulation is the discretization of the original problem, the error between the exact 
solution and approximation is bounded by 
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   h  Chmin(m1, p)                                                                                                (3.33) 
where ║.║  denotes the Euclidean norm, h is the maximum mesh size, p is the degree of 
basis function and m is a measure of singularity of the problem. 
In order to reduce the error, three versions of FEM  are adopted, which corresponds to 
shortening the element size by h-adaptivity,  increasing the degree of basis functions by 
p-adaptivity and swapping the edge and facet of the element by r-adaptivity respectively. 
In theory, if the maximum mesh size is sufficiently small, then the numerical solution 
could converge to the exact solution (Rao, 1999). 
 
3.1.5.5     Photon Dose Calculation 
The absorbed dose distribution can be computed from the solved particle flux and 
describes the energy absorption in the matter. For photon interaction, the spatial 
distribution of electrons can be neglected and the energy dependent photon flux can be 
used. The dose or the energy absorbed by the tissue is therefore given by the following 
equation 

    D(r, E)  en  (r, E) (E)Er
r E                                                             (3.34) 
 en (E)

where,      is the mass-energy absorption coefficient of the material or medium. 
The discretized form of equation (3.34) is given by the following: 
r N 
D(ri , E j )   (ri , E ) enj (E j ) E j

   i j            (3.35) 
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3.2      Model implementation and Simulation 
   The implementation of the FEM is based on the numerical algorithm of MATLAB and 
runs on PC/LAPTOP with a minimum of 32MB memory. For computational speed, the 
application is limited to fit into physical memory. 
 
3.2.1 Assumptions  
 a. The source term is assumed to be multiple isotropic point sources arranged in a 
line segment and with each source limited to a point r=r’ in space and time, t=t’ 
for the time dependent case. The depth, d, of the point source below the mesh is 
chosen to be equal to the mean free path before the first scattering event. i.e. 
1
d   
 s
 b. No surface reflections occur for boundaries and assume Dirichlet condition, i.e. 
(d)  0 ,  dΩ of the domain, Ω, i.e.  no surface reflections 
 c.  The boundary, dΩ, is a polygon and the general description of the FEM allows for 
many other forms of the boundary conditions.  
3.2.2 Mesh Generation 
A mesh is a collection of nodes which are points in the domain and elements are regular 
shapes- subset of the domain generally tetrahedral or hexahedral and having nodes and 
vertices. A two-dimensional slab geometry which is equivalent to a finite line source is 
described. The model comprises a two-dimensional rectangular mesh formed by 
triangular elements. For stability reasons the generated elements are regular with no 
o
internal angles greater that 90  and have approximately constant in size.  The mesh 
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resolution is characterized by the mesh parameter, h, which is defined as the longest side 
of any element , in the mesh 
                 h  max size( i)i                                                                                           (3.36) 
                   1 i  n  
Where, n is the number of elements.  Figure 3.8 shows a rectangular mesh having 
elements and nodes.  In order to keep the size of elements approximately constant we 
have a linear increase in the number of nodes per rectangular section.  A finer division of 
elements in the boundary is used in order to increase accuracy.  
 
 
 
 
 
 
 
 
 
Figure 3.8: Geometry for the generation of mesh. 
 
 
 
 
 
3.2.3 System Matrix  
 
The construction of the system matrices is performed in two steps:  first a separate set of 
e e
element matrices P , A  of order 3 x 3 is formed for each element by carrying out the 
e
integrations numerically over the element domain Ω .  Inhomogeneous properties of the 
tissues can be simulated by altering the element coefficients.  The contributions of each 
element are then assembled into the system matrices P, A of order D x D. Note that P, A 
are symmetric diagonal banded matrices with the semi band width W determined by the 
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maximal node number difference within a single element.  They can be represented by 
matrices of order W x D. As the model requires a high number of nodes it is essential to 
minimize W by reordering the node list prior to feeding it into the FEM processing 
e
algorithm. The source vector S is assembled from the element contributions S . 
 
3.2.4 Matrix solution  
In the time – independent case, a simple matrix equation of the form AФ= S where, A is 
a symmetric, positive definite, sparse, banded matrix. A variety of techniques exist for 
T
solving for Ф. First, A is reduced to the form LL , where L is a lower triangular (banded) 
T
matrix.  Then we rewrite as LL Ф=S and solve first Lc=S (forward substitution) and 
T
then L Ф=c (backward substitution).  These two procedures (i.e forward and backward 
substitutions) will make the computations simple.  It is remarked that if all that is 
required is the solution for nodes near the boundary, then if these are grouped at the lower 
end of the vector Ф the backward substitution can be stopped after these nodes are 
evaluated, leading to a slight improvement in speed. 
 
3.2.5 Time dependent case 
In the time-dependent case, sparse matrix inversion is required, but in addition, a finite 
difference scheme is used. 
The time step interval, ∆t, is chosen in accordance with the node spacing, given by the 
mesh parameter, h, and is governed by the following equation 
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h2
      t                                                                                                                (3.37) 
4kco
Where, co is the velocity of light in vacuum and υ is the refractive index of the tissue and 
c
k  o .  The choice of ∆t is governed by the requirement that the 
3[ a(r)  (1 f ) s(r)]
transport equation remains stable even for small values of t (i.e. transient behaviour).  The 
Crank-Nicholson scheme incorporates an average of the fully explicit and fully implicit 
finite difference schemes.  Both Crank-Nicholson scheme and the fully implicit scheme 
are stable for any value of time step ∆t and if only the equilibrium form is required, then 
larger time steps could be used and approach this limit quickly.  The explicit scheme is 
required to accommodate the early behaviour and this is only stable for the maximum 
value of ∆t.   
 
Application of equation (3.20) with   1  and Sj(t) = Ψj(r’)δ(t-t’), where Ψj (r’) will 2
have a nonzero value only when Nj is a node of the element containing the source, 
leading to  
 
1 1  1 1  1 ( )
    A M n1   A M n  Sn                                                       (3.38) 
2 t  2 t  t
(δ)
where, Sn  is nonzero only if t’ is contained in the interval           with the assumption    
(δ)
Sk  = 0, for all k>0, and Ф0 = 0, successive solutions of Фn is obtained, which has the 
structure 
A  B Q ( )              n1 n n  
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                A Q ( )1 0  
                Ak1 Bk ,k 1                                                                           (3.39)     
 
The matrices A and B are constant throughout the finite differencing procedure. 
Therefore the method requires one Cholesky decomposition for matrix A followed by 
another matrix multiplication applying Cholesky forward and backward decomposition 
per the time step. 
 
3.2.6   Simulation using MATLAB  
The photons traveling in a tissue(water) was simulated using the BTE. For a time 
dependent BTE, the finite element approximation in 2D is given by 
h (r, E,, t)  kk (r, E,, t)                                                   (3.40) 
At the energies around several MeVs, Compton effect is the main scattering event, which 
slows down photons and changes them to electrons. The maximum energy of the incident 
photon is approximately around 10 MeV and only Compton effect can be taken into 
account to describe the photon interactions. The total cross section for the photons was 
obtained from literature (Berger et al, 2007). 
The forward problem FEM simulations presented in this thesis were computed in two 
steps. First the smaller matrices, in which the linear system could be constructed using 
Kronecker tensor product, were generated using MATLAB in a normal PC (2.2 GHz Intel 
Core i3 with 4.0 GB memory). These matrices were then written to files in a sparse 
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matrix format. Then the resulted linear system was solved using portal extensible toolkit 
for scientific calculations (PETSc) and Krylov subspace matrix free methods. The least-
squares solution algorithm without preconditioning was used from the PETSc package for 
solving the linear system with Krylov methods. 
2
The first simulation was made in 2D spatial geometry in a (-5,5) x (0,2) cm   slab 
phantom. The incoming photon flux was located at the center of the phantom at intervals 
of (-2, 2) cm. The number of spatial nodes was 357 and the angular domain was divided 
into 164 evenly distributed elements,   (0,2 )No 164 . 
A MATLAB programme for the calculation of photon fluxes, photon source strength and 
patient doses for various tumour stages have been determined as indicated in Appendix 
III 
3.3    Optimization of Doses to the OAR 
Typically, physical or biological criteria are used in intracavitary brachytherapy treatment 
planning process. Here the physical criteria are considered. Similar theories can be used 
 
for biological criteria. The patient domain  V  R3 consists of a target volume T that is 
PTV, which includes the tumour volume and some safety margin, critical organs' region 
C and normal tissue's region N. Thus, the patient domain is a mutually disjoint union
V T CN . The dose D(r) can be computed from the particle flux  (r,E,)  using 
equation (3.34).  One can assign several physical criteria for the dose. One can simply set 
and deduce that 
 
                        DrD0    rT                                                                               (3.41) 
                        Dr 0       rV /T                                                                      (3.42)     
where D0  is a prescribed uniform dose in target T. The criterion in equation (3.41) is 
unrealistic since radiation must pass through healthy tissue to reach the tumour. That is 
why it is important to demand the equation (3.41) with conditions  
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                              D(r) Dc ;rC                                                                              (3.43) 
 
                             D(r) DN ;rN                                                                              (3.44) 
 
where,  D c and D N  are the upper bounds of the dose in critical organs C and normal 
tissue N respectively.  
Instead of condition (3.41), one can demand a feasibility condition 
 
                                      T  Dr DT              rT                                                  (3.45) 
where, T and DT  are the lower and upper bounds for the dose in PTV respectively. 
 
3.4     Experimental Validation  
A very important function in brachytherapy is to obtain the correct geometric localization 
of the applicator in order to treat the target volume adequately. The dose distributions are 
totally dependent on the inverse square law effect. As a result, proper positioning of the 
radiation sources is very important during treatment of gynaecological malignancies. This 
is study of 164 patients with cervical carcinoma treated with low dose rate brachytherapy 
source (AMRA-Curietron). In Appendix II, the patients characteristics with different 
cancer stages from stages I-IV were selected. The prescribed doses of the target volumes 
were in the range of 25-30Gy for each patient depending on the tumour stage. The dose 
reference point was placed in the target volume and normalization performed on the point 
to obtain the required dose distribution. 
In this study, source localizations were determined from the 3D co-ordinates and 
orientation of each source relative to the patient anatomy. Based on the prescribed dose to 
tumour, the dose to reference point A and critical organs are determined using a digitizer 
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as shown in Appendix II. It was accomplished by at least two images (orthogonal 
radiographs) taken from different perspectives. In order to aid dose calculations, precise 
localization of the sources in three spatial dimensions (3D) and two orthogonal 
radiographic images are used.  The dose rate distributions in relation to the tumour 
volume and critical structures were calculated using a Treatment Planning System (TPS) 
at the facility. Once the dose rate distributions and the treatment time have been 
accurately determined, the patient was ready for loading the sources for treatment. 
Depending upon the stage of the cancer, the Oncologist determines the type of treatment 
modality, source configuration for the cancer patient and positions both tandem and 
ovoids in the cervix. Depending on the tumour stage, two orthogonal projections of 
radiographic films are taken using a simulator C-arm X-ray machine. The C-arm X- ray 
machine with model number S 06U75411 and manufactured by Siemens in Germany at 
July 2006 was used for this study. 
The 3D source locations were input into the treatment planning computer, with the co-
ordinate system being orthogonal to the x, y,and z-axis. Localization began by entering 
source co-ordinates on the projection image into the computer by means of digitization 
process. A single point in space (3D) results in two co-ordinates in each projection image 
(2D). The origin of the 3D co-ordinate system was taken as reference point and was 
chosen as a point near the centre of the applicator and as a point that can easily be 
identified on both projection images. After all sources have been identified, the co-
ordinates of each end of a linear source on each projection image were sequentially 
entered into the Treatment Planning System (TPS).  
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Computerized TPS are used in radiotherapy to generate dose distributions with the intent 
to maximize tumour control and also to minimize normal tissue complications. 
Brachytherapy TPS is necessary to estimate the dose to the target volume and organs at 
risk. Patient anatomy and tumour targets are represented in 3-D models. The entire TPS 
process involves many steps and the medical physicist is responsible for the overall 
integrity of the computerized TPS to accurately and reliably produce dose distributions 
and associated dose calculations in radiotherapy. The entire treatment planning process 
involves many steps, beginning from beam data acquisition and entry into the 
computerized TPS, patient data acquisition to treatment plan generation and the final 
transfer of data to the treatment machine. The TPS used for this study was Prowess 
Panther version 4.6 which was created at the centre at October 2009. The digitizer was 
also manufactured by the Prowess systems and  a company Numonics(Accu Grid model) 
in USA.    
 
 
 
 
 
 
 
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CHAPTER 4 
RESULTS AND DISCUSSIONS 
 
The use of radiation sources in Medical Physics is very important in diagnosis and 
treatment of diseased conditions. Brachytherapy has played an important role in the 
treatment of cervical cancer patients. Analytical and computational models of the 
radiative transport equation in tissues have been applied in the interpretation of 
procedures during treatment planning stages and in dose delivery to patients.  The 
methods commonly applied include the Sievert Integral, Monte Carlo and AAPM (TG-
43). The AAPM (TG-43) uses modular dose calculation and incorporates an anisotropy 
function.  
In this chapter, the results from the study are presented and discussed. The study 
considered the solution of the Boltzmann’s Photon Transport Equation (BTE) in 
biological tissues using the Finite Element Method (FEM) and its application in cervical 
cancer treatment.  The results have been compared with published data and analytical 
reference method as well as Monte Carlo method and the treatment planning system 
(TPS) of the Komfo Anokye Teaching Hospital.  
 
4.1. General 
This section discusses the source term characteristics, angular flux distribution and dose 
profile obtained from the study.  
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4.1.1. Source Term 
Fig 4.1 shows the source term characteristics as a function of distance. The source was 
assumed to be Cs-137 spherical pellets arranged in line geometry. The source strength for 
the various treatment regimes corresponding to different cancer stages was determined. In 
the treatment of cervical cancer in brachytherapy, the same source with a specific source strength 
is used. The different stages of the cancer are treated by exposing the cancerous tissues to 
different time regimes resulting in specific photon fluxes. The photon fluxes result in specific 
dose distributions  depending on whether the treatment is delivered over a short time for a single 
large dose or over prolonged periods of time corresponding to fractionated doses. Figure 4.1 
shows that, the  source strength increases to a maximum at  1 cm and then decreases again to zero 
for the next 5 cm. From 0 to 1 cm, there is an initial build up of photon particles increeasing to 
the maximum value. From 1 cm (the maximum), there is scattering, absorption and anisotropy of 
the photon particles in the medium hence the decrease of the profile. In relation to patient 
anatomy, when the patient is big enougth(size) and with deep depth tumor, it requires more 
photon particles(higher values of Φ corresponding to higher doses) for treatment and vice 
versa. 
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2000 
Stage 1 (S1) 
1500 Stage 2 (S2) 
Stage 3(S3) 
1000 Stage 4 (S4) 
500 
0 
0 2 4 6 8 10 12 
-500 
-1000 
-1500 
-2000 
Distance (r)/cm 
 
Fig 4.1: Source strength distribution as a function of distance 
 
4.1.2. Photon Flux Distribution  
 
The photon flux distribution as a function of distance is shown in Figure 4.2. The Figure 
indicates increase in the photon fluxes over a distance of 5cm and then decreases for the 
rest of the distance. A similar trend is observed in Figure 4.3 for the corresponding doses.  
In Figure 4.3 the angular distribution of the photon flux also shows an increase from θ = 
0 to a maximum at θ = π/2 radians. This is followed by a decrease as one moves from θ = 
π/2 to θ = π radians.  This observation may be attributed to the interaction of the photon 
fluxes with the tissues, which initially results in an increase until they reach the peak, but 
slowly decreases when there is more scattering, anisotropy and attenuation effect.    
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200 Stage 4 (Φ4) 
180 Stage 3 (Φ3) 
Stage 2 (Φ2) 
160 
Stage 1 (Φ1) 
140 
120 
100 
80 
60 
40 
20 
0 
0 2 4 6 8 10 12 
Distance (r)/cm 
 
Fig 4.2: Photon Flux distribution for different cancer stages as a function of distance 
 
 
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5.00E+09 
4.50E+09 
4.00E+09 
0.5 
3.50E+09 
1 
3.00E+09 1.5 
2 
2.50E+09 
2.5 
2.00E+09 3 
5 
1.50E+09 
1.00E+09 
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 
Angle (radians) 
 
 
Fig 4.3: Variation of photon flux with angles 
 
4.1.3 Error Distribution Function 
The error distribution function from the solution of the FEM is shown in Figure 4.4. The 
error function shows a similar trend as that of the photon flux distribution in Figure 4.2. It 
is observed that the error increases to a maximum over a distance of 5cm and then 
decreases afterwards for the different values of the photon flux. This error is related to the 
mesh resolution and decreases as the mesh resolution increases until it converges to a 
certain parameter, η. For a homogeneous material, the value of η→0. In the application of 
h 2
the FEM, it is possible to overestimate the error ||Φ-Φ || ≤Ch , where C is a constant and 
h represents the mesh resolution. The error can also be related to the angular distrbution 
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of the photon flux for different mesh resolutions. Correction in the solution of the flux 
using Green’s Function is very important in order to ensure that the uses of this method 
result in high degree of accuracy in addition to its speed compared to Monte Carlo 
method. 
200 
Stage 4 (Φ'4) 
180 Stage 3 (Φ'3) 
160 Stage 2 (Φ'2) 
140 Stage 1 (Φ'1) 
120 
100 
80 
60 
40 
20 
0 
0 2 4 6 8 10 12 
Distance (r)/cm 
 
Fig 4.4: Variation of the flux error distribution function with distance in the 
solution of the photon flux using the FEM 
 
4.1.4 Dose Profile Analysis 
The dose distribution is shown in Figure 4.5. The distribution is characterized by an increase to a 
distance of 5cm followed by a decrease for the rest of the distance as it approaches the 
boundaries. The Figure 4.5  represents the corresponding dose due to the photon flux and 
therefore has similar characteristics as that of Figure 4.2.  The graph gives a general overview of 
the relationship between the dose prescriptions for the different treatment regimes corresponding 
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to the different stages of cancer disease.  The target volumes and the critical organs like the 
rectum and bladder, have specific distances from the source point depending on the 
patient’s anatomical structure. The thicknesses of the rectum and the bladder, the specific 
distances between the source point and the target volumes depend on the patient’s size 
and weight.  
140 
Stage 4 (D4) 
120 
Stage 3 (D3) 
100 Stage 2 (D2) 
Stage 1 (D1) 
80 
60 
40 
20 
0 
0 2 4 6 8 10 12 
Distance (r)/cm 
 
Fig 4.5: Variation of dose as a function of distance for different cancer stages 
 
The Figure 4.6 shows a graph of the cumulative dose with time. Depending on the type 
and stage of the cancer, the doses are normally delivered in fractions over a period of 
time. Initially, during treatment the number of photons particles begin to build up in the 
medium given rise to the various photon interactions with the medium hence the delay for 
some few seconds. Also as there is more particles streaming because of the source 
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strength of photon particles injecting from the source term, it is later overcome and shoot 
up which rise to required prescribed doses for the treatment. This shows a slow rise in the 
dose with time followed by a rapid increase after 2 seconds.  
 
900 Stage 1 (D1) 
800 Stage 2 (D2) 
Stage 3 (D3) 
700 
Stage 4 (D4) 
600 
500 
400 
300 
200 
100 
0 
0 0.5 1 1.5 2 2.5 3 3.5 
Time (t)/s 
 
      Fig 4.6 : Cummulative dose as a function of time during treatment 
 
4.2 Comparison with analytical reference method  
Solution of the transport equation for complex geometries using an analytical method 
provides an approximate solution with a number of assumptions, which limits the 
accuracy of the transport model. The FEM model is capable of dealing with very complex 
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heterogeneous geometries with certain discretization error. The analytic solution of the 
transport equation is used to calculate the discretization error of the FEM model. Exact 
solutions for the complex photon intensity in the homogeneous case are available for a 
limited number of simple geometries.  
 
In the following analytical and FEM approximations, the integrated intensity, ΦE and the 
mean time (t) have been computed for the two-dimensional circular case.  The absorption 
1
coefficient a  0.98mm  and the scattering coefficient  s 0.94mm
1
 were used as 
values representing the human tissue, with radius R=49mm.  The analytic solution for the 
integrated photon flux intensity, ΦE, in units of photon number per unit length, is given 
by  
         
1
1  I  r 
               E  cosn  n       (4.1) 2 2R n I n( R)
 
1
 2
where        a
c 
  and I n   is the modified Bessel function of the first kind of order 
 k 
n, and,  r '  R  1 ' is the radial position of the source. The value of k  is evaluated  s
c
using the expression k   obtained from the solution of the 
3[a (r)  (1 f ) s(r)]
transport equation using P1 approximation by the expansion of the radiative transport 
equation in spherical harmonics. 
The expression for the mean time (t) component is given by the expression  
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1
 r1I 1( r11 n )I n( R)  I n( r
1)I 1n ( R)   I ( r
1) 
 t   xCos(n ) n Cos(n )        
2 k I 2 n ( R)   I n( R) 
(4.2) 
when  r r1  the photon density itself is given by 
1
   I ( r )K ( R) 
            Cos(n )I ( r) 1n K n( r ) 
n n     (4.3) 
 R2   I n( R)


where, K n  is the modified Bessel function of the second kind of order n. 
Comparison of the analytic method with the result from this work is shown in Table 4.2. 
 
 
 
 
Table 4.2: Comparison of the photon flux intensity from Green’s function (GF) with 
that of the model (FEM) 
Method Photon Flux Intensity Reference 
Φ -2 -1(Nm s ) 
   
-4
     Green’s Function (GF) 7.76x10  This work 
   
Finite Element Method 9 3.27x10
(FEM)  This work 
 
 
 
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4.3 Comparison with Monte Carlo Method 
Comparison of finite element data with results obtained from the analytical Green’s 
Function shows that the FEM implementation in this study provides correct solutions of 
the transport equation. The FEM was also compared with the Monte Carlo (MC) model 
which is widely accepted as a method for simulating photon propagation in a random 
scattering media.  The three dimensional (3D) MC model assumes a collimated point 
source perpendicularly incident on the surface of an infinite cylinder.  The radiation 
emitted from the surface is given as a function of the detection angle being integrated in 
the direction of the cylinder axis to simulate line detector.  The comparison of the rectum 
and bladder doses from this study and the MC values (Miguel et al 2002) are shown in 
Table 4.3. In general, the rectal doses from this study was lower than the MC values, 
while the bladder doses from this study was slightly higher than bladder dose obtained 
using MC simulations. 
 
Table 4.3 Comparison of Monte Carlo method with this work 
Method Rectum (cGy) Bladder (cGy) Reference 
Monte Carlo 78.71±0.36 29.04±0.24 Miguel et al 
 
simulations (2002) 
FEM 46.92±0.15 48.49±0.13 This work 
 
 
 
 
 
 
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4.4 Comparison with Published Data and Treatment Planning System of Komfo   
Anokye Teaching Hospital  
 
The results from the study were compared with published data using TLD and the 
treatment planning system (TPS) of the Komfo Anokye Teaching Hospital (KATH). The 
results show variations between the method and data from published articles and the 
KATH TPS. Data from the KATH, TPS covered one hundred and sixty-four patients who 
were referred to the hospital for cancer treatment. Their ages range from 20-82 years. The 
median age was 55 years. The general characteristics of the patients are presented in the 
Appendix IV.  
Table 4.4: Comparison of the FEM with Published data and TPS of the 
Oncology centre at KATH 
 
Method Rectum (cGy) Bladder (cGy) Reference 
 
TLD 22.13 - 63.76              - Ahimedzie Cletus 
(2011) 
 
TLD - 18.99 - 46.36 Owusu-Kyere 
Reynolds (2011) 
TPS 10.13 – 85.67 21.32 – 78.81 KATH Data Sheet 
( 2012) 
  
FEM This work  
21.5 - 90.10 25.60 – 110.02 
 
 
 
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In brachytherapy, the planning target volume (PTV) is the same as the clinical target 
volume (CTV). The total dose should be as high as possible and appropriate to eradicate 
all residual macroscopic tumour (Haie-Meder et al, 2005).   
The various cervical cancer stages were grouped from stages I to IVA in accordance with 
FIGO guidelines,. Based on this, the most appropriate treatment modality was carefully 
chosen for each patient. From Figure 4.7, the most frequent stage was found to be stage 
IIIB and the least one is the stage IVA. This is due to early reporting at the facility for 
treatment or screening to assess the patient situation of the cancer before getting to the 
worst advanced stage. From Figure 4.7, stages IIB and IIIB are more pronounced than the 
rest of the stages. This agrees well with patient characteristics in Appendix IV Table 5; 
thus patients with stage IIIB (39.6%) followed by stage IIB (35.4%) and the stage with 
the least percentage is stage IV(0.6%). Of all the 164 cancer patients only one had the 
most worst advanced stage of tumour occurring in stage IV. It was observed that, the 
early cervical carcinoma for patients occurring from stage I to IIA were about 26 
patients(15.85%); and the more advanced cervical cancer from IIB-IIIB were about 137 
patients (83.53%). The most advanced stage IVA occurred only in one(1) patient which 
forms about 0.6% of the general characteristics of Table 5 in Appendix IV.  
Comparison of values from the TPS at KATH with other published values (Ahimedze 
Cletus, 2011 and Owusu-Kyere Reynolds, 2011) can be found in Table 4.4. In general the 
various dose values for the bladder and rectum were comparable and found to agree with 
each other and acceptable. 
 
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80 
70 
60 
50 
40 
30 
20 
10 
0 
I A IB IIA IIB IIIA IIIB IVA 
Stage of Cancer 
 
  
Fig 4.7:  Variation of the number of cancer patients with cancer stages                                                            
                                                        
 
 
 
 
 
 
 
 
 
 
94 
 
No. of cancer patients 
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CHAPTER 5 
CONCLUSION AND RECOMMENDATIONS 
This chapter presents the conclusion drawn from the results and the 
recommendations to stakeholder institutions to improve their treatment 
planning methods in the near future. 
5.1: Conclusion 
A number of theoretical and experimental methods have been applied in the treatment planning 
systems at various Oncology Centers. These include the AAPM (TG-43), Monte Carlo, etc. The 
method used in this study is based on the application of the finite element method (FEM) in the 
solution of the Boltzmann’s transport equation in biological tissues. Doses to the rectum and the 
bladder were determined by simulation using MATLAB and plotted with Microsoft Excel for 
clarity in order to determine the relationship between the various parameters. The results show 
doses to the rectum and the bladder to be in the range of 10.13-85.67cGy and 21.32-
78.81cGy respectively for stage I to stage IV patients. Data on dose prescriptions from the 
treatment planning system from the Radiotherapy Center of the Komfo Anokye Teaching 
Hospital in Ghana and published articles for different cancer stages were compared with the 
model results and were found to agree with standard error of ± 20% to ±34%. 
Curative radiotherapy for cancer of the cervix consists of a combination of external beam 
radiotherapy (EBRT) and brachytherapy (BT). The intracavitary brachytherapy is used to 
give a high dose to the tumour or target volume because of its characteristic steep dose 
gradient. The most important prognostic factors are tumour size, tumour extension and 
lymph node involvement. With access to modern imaging technology, it is possible to 
exactly well define the tumour in the cervix, the target volume, the organs-at-risk and the 
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applicator geometry in 3D (Gerbaulet et al 2002). Doses to the target volume can be 
optimized whilst minimizing the doses to the organs-at risk at the same time.  
 
5.2  Recommendations 
The following recommendations are addressed to stakeholder institutions: 
 
5.2.1. Oncology centres 
The Oncology Centres can adopt this method after extensive study in order to verify their 
treatment planning system (TPS) to improve treatment planning systems at the 
hospitals(Oncology centres) and also serve as a faster way of checking the dosimetric 
parameters of the patients before treatment is delivered. This method could also be 
extended to External Beam Radiotherapy (EBRT) to determine the doses to critical organs 
of patients undergoing radiotherapy treatment.  
 
5.2.2 Research Institutions 
There are many aspects for future work which can be carried out at the various research 
institutions to enhance research work. Improvements to the model that can be investigated 
are the use of higher order basis for the shape functions such as quadratics; usage of a 
basis in time as well as space, adaptive re-meshing for error reduction. In addition to these 
theoretical improvements and modifications, there are also improvements to these 
parameters that are used to set up the model. The model can be extended to a fully 3-D. 
More complex boundary conditions can be incorporated and higher order differential 
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equation descriptions of the transport model could be used. Undoubtedly, more study is 
required in future to determine the applicability of the finite element method and once 
such results have been obtained, it is possible to adopt it as the method of choice for fast 
accurate investigation of complex problems of photon transport in biological tissues. 
 
5.2.3 Regulatory Authority 
It is recommended that regulators dealing with therapy facilities can employ the FEM to 
check patients’ treatment parameters for accurate delivery of doses. 
 
 
 
 
 
 
 
 
                     
 
 
 
 
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APPENDIX I 
 
FORMULATION OF THE BOLTZMANN TRANSPORT EQUATION 
 
The Boltzmann Transport Equation (BTE) is an energy balanced equation and can be 
derived from first principles by considering particle conservation within a unit volume 
element of a phase space (Case and Zweifel, 1967). Here the BTE is derived for a 
coupled system of particles including photons, electrons and positrons assuming particle 
conservation. The particles are assumed to interact with only medium atoms but not with 
each other. The particles are assumed to travel in straight lines between interactions. 
Consider a small volume v  with surface A  about a point r . The number of particles
dN j  of type  j in this volume v  surface A  with energies E and E+dE  in the direction 
  and  + d  with time dt  is given by integrating the time derivative of the angular 
density over the volume element as 
 j (r, E,, t)
    dN   dr                                                                                    (1) j t
v
where  j (r, E,,t)  is the angular flux. 
For conservation of energy, the particles with energy between E and E+dE  in the 
direction   and  + d  and time dt   enter and leave the volume  v  suffer collisions 
or attenuated, created or born in different scattering interactions and produced by sources 
in v . Thus  
       dN j  N j ,out  N j ,att  N j ,sca  N j ,so                                                                 (2) 
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where Nj,out,  is the net number of particles flowing out of the volume v  through the 
surface A  in time dt, 
Nj,att  is the number of particles j that are attenuated in volume v  in time dt ,  Nj,sca  is 
the number of particles j  that are born in the scattering interactions with the medium 
atoms in v  in time dt, and Nj,so is the number of particles j produced by the sources 
inside the volume v  in time dt. Nj,out  is obtained by integrating the angular flux  j  
over the surface A .  The surface integral can be changed into volume integral using 
Gauss’s theorem (Griffiths 1998; Case et al 1967).  
Thus 
N j ,out   j (r, E,, t).ndAdEddt  
A
               . j (r, E,, t)dndEddt                                                                     (3) 
v
where n  is the unit outward vector normal to dA . Also equation(3) can be re-written as  
    N j ,out  . j (r, E,, t).dndEddt                                                                      (4) 
v
Nj,att    is obtained using the total cross-section  j (r, E)  which is defined as the 
probability per unit path length for particle j with energy E to attenuate.  The attenuation 
frequency depends on the particle speed v j  and is given by the number of particles 
absorbed in unit distance travelled by the particle   
        v j j (r, E)                                                                                                                (5) 
Nj,att  is also obtained by integrating the number of particles within the energy range E+dE  
in the volume element v . Thus  
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  N j ,att  v j . j (r, E) j (r, E,, t).drdEddt                                                                 (6) 
v
 
or 
N j ,att   j (r, E) j (r, E,, t).drdEddt                                                                         
v
(7) 
Nj,sca  is obtained using the differential cross-section  
 '  '  f (r, E
' , E, ' ,)  
j ' j j
 2
 = (r, E ' , E, ' ,)  which is defined as the probability per unit path length that a 
E
’
particle j  with energy E  and direction  '  will produce a secondary particle j  with 
’
energy E and direction  . Assuming the particles j ' 1,2,3  with initial photon energy E  
and direction  ' . Then the rate at which these particles are scattered to secondary 
particles j  is given by  
' ' '(v11  jf1  v 2 2  jf 2  v3 3  jf 3 )dEd  
3
'
v ' j  j  jf
' ' ' '
j '
(r, E , , E)dE d                                                                              (8) 
j '1
where f ''  f ' (r, E ,
' ,t)  
j j
Nj,sca can therefore be obtained from the following general equations as 
3
N ' ' ' ' ''j ,sca    v j  ' f j (r, E , , t)dE d dt                                                          (9) j j
'
v 1 A j 1
or 
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3
N j ,sca     ' ' ' (r, E
' , ' , t)dE 'j d
'dt                                                    (9a) 
j  j
'
v 1 A j 1
In 
 radiotherapy, there are no interactions that change electrons to positrons. Thus the 
differential cross-sections   23   is zero for all energies and angles. 
      N j ,so   S j (r, E,)drdEddt                                                                              (10) 
v
where S j (r, E,)  is the source term for particle j inside the volume v  and  j  is the 
number of particles j per unit time at t in drdEd . The balanced integro-differential 
equation is thus 
1  3j
 ( . j  j j     '  j
' '
j dE d  S j )drdEddt  0            (11) 
v t j
v j 1 A j1
Thus  
1  3j ' '
. j  j j    
' '
' j '
 j j dE d  S j  0                                  (12) 
v t
1 A j1
which is the linear Boltzmann transport equation for particle j   
 
 
 
 
 
 
 
 
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APPENDIX II 
 Patient characteristics with cancer stages ,prescribed and critical organ doses 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum Bladder 
Number Stage dose to reference dose /cGy dose  /cGy 
tumour pointA/cGy 
/cGy 
001 37 IIB 2500 86.61 45.39 37.69 
002 56 IIIA 3000 143.13 18.84 38.80 
003 50 IIIB 3000 93.71 26.28 29.14 
004 62 IIIB 3000 98.81 18.94 28.83 
005 71 IIIB 3000 79.60 45.86 36.40 
006 54 IIA 2500 86.10 76.45 50.01 
007 62 IIB 2500 105.95 52.93 56.47 
008 70 IIIB 3000 81.52 58.23 36.14 
009 46 IIB 2500 84.63 38.43 30.04 
010 59 IIA 2500 96.66 17.62 42.53 
011 40 IIB 2500 80.60 47,03 35.09 
012 38 IIA 2500 80.49 30.44 62.59 
013 68 IIB 2500 80.30 72.43 68.37 
014 69 IIIA 3000 107.68 27.08 38.75 
015 52 IIIB 3000 85.06 57.29 26.99 
016 43 IIB 2500 90.18 33.63 50.18 
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017 43 IIB 2500 122.94 47.30 70.90 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA 
/cGy /cGy 
018 56 IIA 2500 79.78 34.86 45.69 
019 65 IIA 2500 86.16 69.54 40.47 
020 45 IIIB 3000 97.21 19.26 38.97 
021 48 IIB 2500 117.20 19.32 30.13 
022 59 IIIB 3000 90.96 74.71 30.28 
023 67 IIA 2500 69.63 23.55 56.22 
024 57 IIB 2500 86.79 30.16 67.47 
025 58 IIB 2500 81.31 25.22 43.22 
026 47 IIA 2500 74.00 42.45 40.23 
027 60 IIB 2500 81.88 62.43 50.86 
028 72 IIB 2500 61.33 23.34 48.39 
029 71 IIB 2500 89.94 45.87 51.77 
030 67 IIB 2500 96.65 10.13 28.69 
031 33 IIIA 3000 88.65 62.55 43.95 
032 71 IIB 2500 94.98 22.52 61.62 
033 54 IIIB 3000 94.53 51.97 30.53 
034 44 IIA 2500 83.46 30.77 41.35 
035 65 IIA 2500 93.57 33.01 45.53 
036 48 IIIA 3000 98.39 37.33 42.44 
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037 49 IIIA 3000 87.47 54.46 67.05 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
038 54 IIIB 3000 88.92 50.32 36.25 
039 59 IIIB 3000 87.88 53.58 38.12 
040 64 IIIB 3000 90.69 75.93 59.59 
041 46 IIIB 3000 91.91 34.83 43.19 
042 53 IIIB 3000 84.92 54.43 37.20 
043 50 IIIB 3000 72.03 42.18 30.72 
044 55 IIB 2500 92.59 36.60 50.30 
045 44 IIIB 3000 85.75 56.64 25.62 
046 43 IIIB 3000 78.43 55.74 38.64 
047 40 IIIB 3000 81.68 43.45 38.19 
048 56 IIB 2500 88.28 57.10 68.66 
049 63 IIIB 3000 92.18 50.97 40.76 
050 71 IIB 2500 79.73 38.03 43.40 
051 72 IB 2500 94.29 68.47 70.61 
052 56 IIB 2500 86.77 53.30 65.24 
053 61 IIIA 3000 98.36 30.36 43.90 
054 40 IIIA 3000 94.87 77.47 65.95 
055 40 IIA 2500 109.26 85.67 69.56 
056 52 IIA 2500 92.12 57.17 44.19 
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057 35 IIA 2500 91.05 78.21 58.91 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
058 63 IIA 2500 97.62 43.54 78.81 
059 55 IIA 2500 95.94 60.68 37.01 
060 49 IIIB 3000 94.43 53.35 65.85 
061 51 IIB 2500 100.43 76.05 32.94 
062 70 IIIB 3000 70.25 28.85 58.78 
063 45 IIIB 3000 84.17 62.11 52.82 
064 53 IIA 2500 81.07 33.70 65.18 
065 53 IIIA 3000 87.55 61.78 43.50 
066 53 IIIA 3000 91.33 77.87 42.59 
067 70 IIB 2500 92.12 37.17 24.19 
068 72 IIIB 3000 86.41 42.24 68.85 
069 52 IIIB 3000 82.17 26.41 40.67 
070 35 IIB 2500 82.39 40.80 35.02 
071 58 IB 2500 87.72 55.30 48.38 
072 55 IB 2500 86.05 60.36 31.16 
073 69 IIB 2500 86,63 59.57 37.52 
074 48 IIIB 3000 98.05 67.54 58.88 
075 80 IIIB 3000 92.40 35.57 60.07 
076 61 IIIB 3000 94.87 70.79 61.46 
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077 62 IIB 2500 94.98 22.52 34.62 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
078 75 IIB 3000 83.13 69.20 49.06 
079 48 IIIB 3000 90.26 38.19 59.48 
080 50 IIIB 3000 93.83 48.37 53.43 
081 52 IIIB 3000 99.40 45.07 30.16 
082 43 IIIB 3000 88.71 53.09 37.68 
083 46 IIIB 3000 79.93 54.64 38.84 
084 72 IB 2500 112.61 75.68 38.37 
085 78 IB 2500 87.43 39.62 28.68 
086 53 IIIB 3000 109.12 34.77 63.41 
087 45 IIIB 3000 91.02 53.88 39.07 
088 35 IIA 2500 91.59 25.44 58.56 
089 65 IIIB 3000 86.74 31.39 48.40 
090 46 IIA 2500 82.10 28.05 32.00 
091 65 IA 2000 109.09 35.31 51.32 
092 44 IIIA 3000 105.46 56.74 38.24 
093 53 IIB 2500 88.12 33.11 48.64 
094 45 IIB 2500 97.94 45.44 63.91 
095 57 IIIB 3000 87.23 39.86 51.69 
096 69 IIIB 3000 91.14 39.48 21.32 
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097 65 IIIB 3000 103.57 33.01 45.53 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
098 55 IIIB 3000 94.27 51.51 39.26 
099 70 IIB 2500 90.04 33.90 40.56 
100 56 IIIB 3000 81.88 42.43 62.50 
101 48 IIIB 3000 86.24 38.46 50.75 
102 79 IIIB 3000 89.82 34.77 63.41 
103 82 IIIB 3000 71.32 27.44 58.42 
104 70 IIB 2500 128.91 28.61 35.33 
105 51 IIB 2500 84.23 57.14 47.44 
106 52 IIB 2500 98.13 78.52 50.28 
107 61 IIIB 3000 82.50 57.11 60.02 
108 70 IIIB 3000 92.31 80.40 72.99 
109 63 IIB 2500 98.75 44.51 70.38 
110 36 IIB 2500 106.96 52.28 33.69 
111 75 IIIB 3000 86.09 34.36 56.16 
112 51 IIB 2500 88.11 50.94 62.79 
113 64 IIIB 3000 88.13 50.40 54.82 
114 62 IIB 2500 89.15 58.34 47.89 
115 65 IIIB 3000 85.82 52.28 43.39 
116 45 IIIB 3000 79.86 46.06 39.87 
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117 46 IIIA 3000 78.87 34.80 46.54 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
118 60 IIIA 3000 74.01 38.60 54.34 
119 75 IIIB 3000 95.83 44.52 68.89 
120 54 IIB 2500 88.89 36.50 41.53 
121 70 IIB 2500 88.51 36.39 47.84 
122 70 IIB 2500 109.04 45.80 32.62 
123 62 IIB 2500 89.99 54.74 67.80 
124 55 IIB 2500 92.05 26.76 36.60 
125 47 IIB 2500 89.91 36.70 25.64 
126 57 IIB 2500 91.49 65.77 51.32 
127 55 IIB 2500 93.55 41.63 53.91 
128 75 IIB 2500 87.66 46.52 50.25 
129 46 IIB 2500 88.57 39.93 50.03 
130 61 IIIB 3000 91.85 39.25 50.79 
131 39 IB 2500 89.84 33.01 43.42 
132 71 IIIB 3000 91.60 58.95 44.03 
133 78 IIIB 3000 83.49 66.54 56.53 
134 50 IIIB 3000 89.87 55.03 60.60 
135 70 IIB 2500 82.22 57.81 65.00 
136 38 IIB 2500 94.01 42.68 56.68 
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137 75 IIB 2500 90.20 55.45 67.65 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
138 55 IIIB 3000 94.43 75.48 61.55 
139 49 IIB 2500 90.32 22.47 73.73 
140 77 IIB 2500 109.97 53.51 70.25 
141 39 IIIB 3000 93.29 27.67 57.87 
142 20 IIB 2500 92.93 64.72 43.09 
143 40 IIB 2500 121.77 59.51 39.54 
144 36 IIIB 3000 92.51 46.48 62.96 
145 45 IA 2500 71.53 35.03 55.67 
146 70 IIIB 3000 91.13 60.05 38.61 
147 55 IIIB 3000 94.21 33.91 44.07 
148 56 IIA 2500 81.01 27.54 51.34 
149 54 III 3000 153.67 64.33 54.28 
150 42 IIIB 3000 88.80 35.56 49.01 
151 50 IIB 2500 89.50 40.76 34.96 
152 71 IIIB 3000 92.28 49.04 56.16 
153 68 IIIB 3000 105.85 65.13 42.65 
154 48 IIIB 3000 125.85 63.37 47.66 
155 45 IVA 3000 93.75 54.49 62.29 
156 45 IIIA 3000 93.22 41.97 52.52 
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157 68 IIB 2500 90.98 55.55 34.29 
Patient ID Age/yrs Cancer Prescribed Dose to Rectum  Bladder   
Number Stage dose to reference /cGy /cGy 
tumour pointA/cGy 
/cGy 
158 56 IIB 2500 93.56 54.53 60.30 
159 47 IIB 2500 91.82 41.61 55.53 
160 75 IIIB 3000 91.66 55.68 45.88 
161 71 IIIB 3000 101.61 43.48 36.48 
162 67 IIB 2500 92.29 54.53 49.92 
163 68 IIIB 3000 91.99 49.63 77.70 
164 55 IIIB 3000 97.91 57.99 43.68 
 
 
 
 
 
 
 
 
 
 
 
 
 
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APPENDIX III 
1. Angular flux distribution with time using Matlab 
meu = 0; 
t = 0; 
x = 0; 
ind = 1; 
for t = 0:0.25:1 
for x = 0:0.25:10 
    flux = -(x + 1)*(x -11)*(5 - meu^2)*(1-t); 
    f(ind)=flux; 
    xt(ind)= x; 
    %flux(index)=flux; 
    ind = ind + 1; 
end 
f; 
xt; 
disp((f    xt)') 
plot(xt,f) 
grid on 
end 
title ('ANGULAR FLUX DISTRIBUTIONS AT TIME t') 
xlabel ('DISTANCE (mm)') 
ylabel ('ANGULAR FLUX') 
gtext('t = 0') 
gtext('t = 0.25') 
gtext('t = 0.50') 
 
2. CALCULATION OF THE SOURCE TERM 
speed = 3*10^8; 
meu = 1; 
t = 0; 
x = 0; 
bo = 0.25; 
b2 = 0.05; 
sigmat=2; 
ind = 1; 
for t = 0:0.25:1 
for x = 0:0.25:10 
    T = (1/speed)*(x - 11)*(x + 1)*(5 - meu^2)-(2*x-10)*meu; 
    G = (5 - meu^2)*(1-t)-sigmat*(x - 11)*(x +1); 
    M = (5 - meu^2)*(1-t); 
    N = (14/3*bo - (meu^2-1/3)*b2)*(x - 11)*(x +1)*(1-t); 
    source = (T*G*M) + N; 
    srce(ind)=source; 
    xt(ind)= x; 
    %flux(index)=flux; 
    ind = ind + 1; 
end 
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srce; 
xt; 
disp((srce    xt)') 
plot(xt,srce) 
grid on 
end 
title ('SOURCE TERM DISTRIBUTIONS AT TIME t') 
xlabel ('DISTANCE (mm)') 
ylabel ('SOURCE TERM') 
gtext('t = 0') 
gtext('t = 0.25') 
gtext('t = 0.50') 
 
 
3. % DOSE CALCULATION 
%{   
  Do = Initial dose rate 
  t = radiation time in secs 
  Dcum = cumulative dose 
  t2 = half - life of cesium 
  lumbda = decay constant 
  %} 
Dcum = 0.662; 
t2 = 30 
%lumbda = 0.693/30; 
disp('t         Dcum') 
disp('----------------------') 
iter = 1; 
for t = 1:0.2:3        
    Dcum = 1.44*t2*Dcum*(1-exp(-reallog(2)*t/t2));     
    disp((t Dcum)) 
    time(iter) = t; 
    Dcumm(iter)= Dcum; 
    iter = iter +1; 
end 
plot(Dcumm,time) 
grid on 
xlabel('CUMULATING DOSE (Gy)') 
ylabel('TIME (s)') 
title('CUMULATIVE DOSE PER TIME IN HUMAN TISSUE USING CESIUM') 
 
 
 
 
 
 
 
 
 
 
 
 
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APPENDIX IV 
 
Table 1. The flux distribution with distance 
 
 
 
Flux 
Distance (r) 
Φ4 Φ3 Φ2 Φ1 
0 58 40 30 15 
1 100 75 50 26 
2 138 100 70 35 
3 160 120 80 40 
4 174 130 90 44 
5 182 136 95 48 
6 175 132 90 44 
7 160 120 80 40 
8 135 100 70 36 
9 100 75 50 30 
10 55 42 30 15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Table 2. Dose distribution with distance 
 
 
Doses 
Distance (r) 
D4 D3 D2 D1 
0 37.51 25.87 19.4 9.7 
1 64.68 48.51 32.34 16.82 
2 89.26 64.68 45.28 22.64 
3 103.49 77.62 51.74 25.87 
4 112.54 84.88 58.21 26.46 
5 117.72 87.97 61.45 31.05 
6 113.19 85.38 58.21 26.46 
7 103.49 77.62 51.74 25.87 
8 67.32 64.68 45.28 25.28 
9 64.68 48.51 32.34 19.4 
10 55.57 27.16 19.4 9.7 
 
 
 
 
     
         Table 3. The flux error with distance     
Distance Flux error 
(r)/mm 
Φ'4 Φ'3 Φ'2 Φ'1 
0 0 0 0 0 
1 62.47 46.15 31.23 16.24 
2 118.56 85.92 60.13 30.07 
3 151.54 113.66 75.23 37.89 
4 170.39 127.42 88.21 43.13 
5 178.39 134.99 94.29 47.64 
6 174.51 131.63 89.75 43.88 
7 159.83 119.87 79.92 39.96 
8 134.95 99.96 69.97 35.99 
9 99.98 74.99 49.99 29.99 
10 54.97 41.99 29.99 14.99 
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126 
 
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E
Table 4  Fluxes( ) and Error fluxes(  ) with distance 
Distance   4       
h h h
2 1  4  3   2  
h E E E E
1   4  3   2  1  
3
(r) 
0 58.00 40.00 30.00 15.00 58.00 40.00 30.00 15.00 0.00 0.00 0.00 0.00 
1 100.00 75.00 50.00 26.00 37.53 28.85 18.77 9.76 62.47 46.15 31.23 16.24 
2 138.00 100.00 70.00 35.00 19.44 14.08 9.87 4.93 118.56 85.92 60.13 30.07 
3 160.00 120.00 80.00 40.00 8.46 6.34 4.77 2.11 151.54 113.66 75.23 37.89 
4 174.00 130.00 90.00 44.00 3.45 2.58 1.79 0.87 170.55 127.42 88.21 43.13 
5 182.00 136.00 95.00 48.00 1.36 1.01 0.71 0.36 180.64 134.99 94.29 47.64 
6 175.00 132.00 90.00 44.00 0.49 0.37 0.25 0.12 174.51 131.63 89.75 43.88 
7 160.00 120.00 80.00 40.00 0.17 0.13 0.08 0.04 159.83 119.87 79.92 39.96 
8 135.00 100.00 70.00 36.00 0.05 0.04 0.03 0.01 134.95 99.96 69.97 35.99 
9 100.00 75.00 50.00 30.00 0.02 0.01 0.01 0.01 99.98 74.99 49.99 29.99 
10 55.00 42.00 30.00 15.00 0.01 0.01 0.01 0.01 54.99 41.99 29.99 14.99 
 
 
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Table 5. The patient characteristics with tumour stage 
Number of patients                                      N 
                                                              164(100%) 
Age 
Median                                                       55 
Range                                                        20-82 
Stage 
IA                                                               2 (1.2%) 
IB                                                               6 (3.7%) 
IIA                                                              18 (11.0%) 
IIB                                                               58 (35.4%) 
IIIA                                                             14 (8.5%) 
IIIB                                                               65 (39.6%) 
IVA                                                              1 (0.6%) 
 
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