University of Ghana  http://ugspace.ug.edu.gh
REFINED SIEVERT INTEGRAL FOR THE CALCULATION OF DOSE 
DISTRIBUTION AROUND THE NEW BEBIG Co-60 HIGH DOSE RATE 
BRACHYTHERAPY SOURCE 
 
 
 
This thesis is submitted to the University of Ghana, Legon in partial fulfillment of the 
requirement for the award of MPHIL Medical Physics degree. 
 
 
BY 
 
 
CASTROL DUMENYA 
(10550675) 
 
 
 
 
 
JULY, 2017 
  
 
 
University of Ghana  http://ugspace.ug.edu.gh
DECLARATION 
 
“This thesis is a result of a research work undertaken by Castrol Dumenya in the 
department of medical physics, School of Nuclear and Allied Sciences, University of 
Ghana, under the supervision of Dr. Francis Hasford, Professor John H. Amuasi and 
Mr. Samuel Nii Adu Tagoe.” 
I hereby affirm that except for references which have been cited, this work is a 
product of my own research and it has not been presented in part or whole for any 
other degree in this University or elsewhere.  
 
 
Sign………………………     Sign…………………… 
Castrol Dumenya      Dr Francis Hasford 
(Student)                                                                                   (Principal Supervisor) 
Date……………………...                                                       Date………………….. 
 
        Sign…………………… 
  Prof. J. H. Amuasi 
       (Co-supervisor) 
              Date……………… 
 
Sign…………………… 
        Mr. Samuel N. A. Tagoe 
       (Co-supervisor) 
                 Date………………… 
i 
 
University of Ghana  http://ugspace.ug.edu.gh
ACKNOWLEDGEMENT 
Over and over, again and again, God has shown his love and faithfulness to me by 
taking me through another educational level and my praise to Him would never end. 
My second thanks and appreciation go to my magnificent supervisors Dr Francis 
Hasford and Prof John Humphrey Amuasi who have followed every detailed progress 
of this research work to a successful end, and have used their knowledgeable and 
scholarly experience to constructively criticise and correct my blunders. To the 
outstanding co-supervisor, Mr Samuel Nii Adu Tagoe, who birthed the idea for this 
research, helped in drawing the work plan, endowed apposite alternatives just for a 
solution to be obtained, I want to say a big thank you. 
My overwhelming appreciation goes to Mr Evans Sasu, the indefatigable, incredundus 
and auriolus clinical medical physicist at the National Centre for Radiotherapy and 
Nuclear Medicine, KBTH; for his thorough on-site guide in obtaining results, making 
corrections where necessary in order to ensure accuracy. 
My gratitude also goes to the brainy lecturers of the department of medical physics, 
School of Nuclear and Allied Sciences (SNAS), most especially Prof A. W. K. Kyere 
and Prof Cyril Schandorf for the concern shown in the progress of this research work. 
I am also very thankful to Dr Raymond Edziah, Mr Michael Vowotor, Mr Charles 
Tetteh and Mrs Elizabeth Tetteh for their unquantifiable support at every step of the 
way. 
To my good friends and colleagues, Kwesi Aane Koomson, Daniel Ackom, Nii 
Korley Corquaye, Belinda Buermle Asamanyuah, Eric Acomeah, Emmanuel K. 
Nyogbe, Philip Odonkor, Blessing Ayivi, Ms Shiela Victoria Gbormittah, I am 
grateful for the love shown during the period of this programme. To the formosus and 
exquisite Ms Mavis Yorm Aleawobu, I am deeply grateful for the love and support 
shown me throughout the period of the program. 
To my relatives Mr Forgive Torwudzo and wife, Mrs Bless Nyamuame-Glover, Ms 
Felicia Nyamuame, Michael Nuwormegbe and my siblings, Believe, Blessing and 
Godsway Dumenya, your support has been immeasurable. Thanks for everything. 
God bless you all 
ii 
 
University of Ghana  http://ugspace.ug.edu.gh
DEDICATION 
This research work is dedicated to my parents and heroes, Mr Ricky Seth Dumenya 
and Mrs Rebecca Dumenya, and my siblings for their lifetime support, wise counsels 
and prayers done on my behalf throughout my educational career. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
iii 
 
University of Ghana  http://ugspace.ug.edu.gh
TABLE OF CONTENTS 
DECLARATION ...................................................................................................................... i 
ACKNOWLEDGEMENT ....................................................................................................... ii 
DEDICATION ........................................................................................................................ iii 
LIST OF FIGURES ................................................................................................................ vi 
LIST OF TABLES ................................................................................................................ viii 
APPENDICES ......................................................................................................................... ix 
LIST OF ABBREVIATIONS AND SYMBOLS ................................................................... x 
ABSTRACT ............................................................................................................................ xii 
CHAPTER ONE: INTRODUCTION .................................................................................... 1 
1.1 BACKGROUND ....................................................................................................... 1 
1.2 STATEMENT OF RESEARCH PROBLEM ............................................................ 5 
1.3 OBJECTIVE OF THE STUDY ................................................................................. 6 
1.4 SPECIFIC OBJECTIVES .......................................................................................... 6 
1.5 SCOPE OF THE STUDY .......................................................................................... 6 
1.6 JUSTIFICATION AND RELEVANCE OF THE STUDY ....................................... 7 
1.7 ORGANISATION OF THESIS ................................................................................. 7 
CHAPTER TWO: LITERATURE REVIEW ....................................................................... 9 
2.1 INTRODUCTION ..................................................................................................... 9 
2.2 DOSE CALCULATION ............................................................................................ 9 
2.3 EVOLUTION OF DOSE CALCULATION ALGORITHMS AND DOSIMETRIC 
SYSTEMS ........................................................................................................................... 11 
2.4 DOSE-RATE CALCULATION AROUND A POINT AND LINE   SOURCE ..... 14 
2.5 TWO-DIMENSIONAL DOSE CALCULATION FORMALISM .......................... 16 
2.6 FAST FOURIER TRANSFORM ............................................................................ 23 
2.6.1 Implementation of FFT Convolution ............................................................... 26 
2.7 MONTE CARLO (MC) BASED DOSIMETRY ........................................................... 28 
2.8 SIEVERT INTEGRAL ............................................................................................ 30 
CHAPTER THREE: MATERIALS AND METHODOLOGY ......................................... 33 
3.1 SIEVERT INTEGRAL TECHNIQUE .......................................................................... 33 
3.2 DOSE CALCULATION .......................................................................................... 35 
3.3 MATLAB ....................................................................................................................... 36 
3.4 IMPLEMENTATION OF THE SIEVERT INTEGRAL ............................................... 39 
60
3.5 NEW BEBIG Co SOURCE ........................................................................................ 41 
3.6 EQUIPMENT ................................................................................................................ 43 
3.7 AAPM TG 43 DOSE CALCULATION FORMALISM ............................................... 48 
iv 
 
University of Ghana  http://ugspace.ug.edu.gh
3.8 COMPARISON OF SIEVERT INTEGRAL VALUES AND HDRplus RESULTS .... 48 
CHAPTER FOUR: RESULTS AND DISCUSSION .......................................................... 49 
4.1 HDRplus Dose Computations .................................................................................. 49 
4.2 SIEVERT INTEGRAL RESULTS ................................................................................ 52 
4.3 ANALYSIS OF RESULTS ........................................................................................... 52 
4.3 DISCUSSION .......................................................................................................... 58 
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION ................................... 61 
5.1 CONCLUSION .............................................................................................................. 61 
5.2 CLINICAL INTERPRETATION ............................................................................ 61 
5.3 RECOMMENDATION ........................................................................................... 62 
5.3.1 FOR THE RESEARCH COMMUNITY ......................................................... 62 
5.3.2 FOR THE CLINICAL COMMUNITY ........................................................... 62 
REFERENCES ....................................................................................................................... 63 
APPENDICES ........................................................................................................................ 69 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
v 
 
University of Ghana  http://ugspace.ug.edu.gh
LIST OF FIGURES 
 
Figure 2. 1: The Manchester system used in the calculation of doses at two points   .....        
A and B 12 
Figure 2. 2: Paris System prescription for breast implant ............................................ 12 
Figure 2.3: Geometry used in the calculation of dose distribution near a linear   ...........        
source based 18 
Figure 2.4: Schematic diagram illustrating the FFT convolution method for 
calculating the dose distribution for one dimensional case of three sources with 
different lengths ........................................................................................................... 27 
Figure 2.5: A pictorial representation of MC method of sampling primary and 
scattered radiation photons emitted by a radioactive source. ...................................... 30 
Figure 3. 1: Diagram displaying the geometric relations used in the evaluation of 
exposure at point P, from an elementary source. (Khan, 2007) 34 
Figure 3. 2: Display of the dose calculation in the command window of MATLAB .. 38 
Figure 3. 3: Schematic diagram of the new BEBIG 60Co HDR brachytherapy source
...................................................................................................................................... 42 
Figure 3. 4: Monitor displaying the interface of the HDRplus .................................... 44 
Figure 3.5 a: Source dwell position at y = 0 cm………………………………………….. . 45 
Figure 3.5 b: Source dwell position at y = 1 cm ...................................................................... 46 
Figure 3.5 c: Source dwell position at y = 2 cm ...................................................................... 46 
Figure 3.5 d: Source dwell position at y = -1 cm ..................................................................... 47 
Figure 3.5 e: Source dwell position at y = -2 cm ..................................................................... 47 
Figure 4. 1: Isodose lines for source at y = 1 cm      51 
Figure 4. 2: Isodose lines for source at y = -1 cm ........................................................ 51 
vi 
 
University of Ghana  http://ugspace.ug.edu.gh
Figure 4. 3: Isodose lines for source at y = 2 cm ......................................................... 51 
Figure 4. 4: Isodose lines for source at y = -2 cm ........................................................ 51 
Figure 4. 5: Isodose lines for source at y=0 cm ........................................................... 51 
Figure 4. 6: Bar charts comparing dose outputs for source at y = 1cm ....................... 54 
Figure 4. 7: Curves to compare dose outputs for source at y = 1cm............................ 54 
Figure 4. 8: Bar charts comparing dose outputs for source at y = 2cm ....................... 55 
Figure 4. 9: Curves to compare dose outputs for source at y = 2 ................................ 55 
Figure 4. 10: Bar charts comparing dose outputs for source at y =- 1cm .................... 56 
Figure 4. 11: Curves to compare ose outputs for source at y = -1cm .......................... 56 
Figure 4. 12: Bar Charts for comparing dose outputs at y = -2cm............................... 57 
Figure 4. 13: Curves to compare dose outputs at y = -2cm ......................................... 57 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
vii 
 
University of Ghana  http://ugspace.ug.edu.gh
LIST OF TABLES 
Table 3 1: Actual and fitted values of linear attenuation co-efficient for source and 
filtration materials μ_s and μ_f, respectively for the new BEBIG source. .................. 40 
Table 3 2: Details of the BEBIG Co-60 source including some initial parameters and 
their corresponding values by 18th March, 2017 ........................................................ 43 
Table 4 1:  CP parameters for Source at y = 0 cm…………………………………49 
Table 4 2: Dose Control Point Report for Source at y = 1 cm ..................................... 49 
Table 4 3: Dose Control Point Report for Source at y = 2 cm ..................................... 49 
Table 4 4:  Dose Control Point Report for Source at y = -1 cm .................................. 50 
Table 4 5: Dose Control Point Report for Source at y = -2 cm ................................... 50 
Table 4 10: Comparison of dose values for both methods for source at y = 1cm ....... 52 
Table 4 11: Comparison of dose values for both methods for source at y = 2 cm ...... 53 
Table 4 12: Comparison of dose values for both methods for source at y = -1 cm ..... 53 
Table 4 13: Comparison of dose values for both methods for source at y = -2 cm ..... 53 
 
 
 
 
 
 
 
 
 
 
 
viii 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDICES 
APPENDIX A:  Dose Control Point Report for source at y = 0 cm ............................ 69 
APPENDIX B: Dose Control Point Report for source at y = 1 cm ............................. 70 
APPENDIX C: Isodose report for source at y = 1cm .................................................. 71 
APPENDIX D: Dose Control Point Report for source at y = 2 cm ............................. 72 
APPENDIX E: Isodose Report for source at y = 2 cm ................................................ 73 
APPENDIX F: Dose Control point Report for source at y = -1 cm ............................ 74 
APPENDIX G: Isodose Report for source at y = -1 cm .............................................. 75 
APPENDIX H: Dose control point Report for source at y = -2 cm ............................. 76 
APPENDIX I: Isodose Report for source at y = -2 cm ................................................ 77 
APPENDIX J: The New BEBIG Co-60 HDR Source Details .................................... 78 
APPENDIX K: MATLAB SYNTAX .......................................................................... 79 
APPENDIX L: General Sievert Integral Table ............................................................ 80 
APPENDIX M: Sievert Integral Table for angles of CPs............................................ 81 
APPENDIX N: Dose Control Point Report as computed using the Sievert integral for 
source at y = 1cm ......................................................................................................... 82 
APPENDIX O: Dose Control Point Report as computed using the Sievert integral for 
source at y = 2 cm ........................................................................................................ 82 
APPENDIX P: Dose Control Point Report as computed using the Sievert integral for 
source at y = -1 cm ....................................................................................................... 83 
APPENDIX Q: Dose Control Point Report as computed using the Sievert integral for 
source at y = -2 cm ....................................................................................................... 83 
 
ix 
 
University of Ghana  http://ugspace.ug.edu.gh
LIST OF ABBREVIATIONS AND SYMBOLS 
AAPM  - American association of physicists in medicine 
DVH  - Dose volume histograms 
EBRT  - External beam radiotherapy treatment 
FFT  - Fast Fourier transform 
HDR  - High dose rate 
ICWG  - Interstitial Collaborative Work 
KERMA - Kinetic energy released in matter 
LDR  - Low dose rate 
MDR  - Medium dose rate 
MC  - Monte Carlo 
OAR  - Organs at risk 
PTV  - Planning target volume 
GTV  - Gross tumour volume 
CTV  - Clinical target volume 
RTP  - Radiotherapy practice 
RTPS  - Radiotherapy treatment planning system 
TPS  - Treatment planning system 
TERMA - Total energy released in matter 
CP  - Control point 
x 
 
University of Ghana  http://ugspace.ug.edu.gh
TG43  - Task group 43 
1D  - One dimension 
2D  - Two dimension 
3D  - Three dimension 
60
Co/Co-60 - Cobalt 60 
192
Ir  - Iridium 192 
ICRU  - International Commission on Radiation Units and Measurement 
ROI  - Region of interest 
POI  - Points of interest 
KBTH  - Korle Bu Teaching Hospital 
NCRNM - National Center for Radiotherapy and Nuclear Medicine 
QA  - Quality Assurance 
QC  - Quality control 
TLD  - Thermoluminiscent dosimeter 
IFT  - Inverse Fourier tansform 
ISL  - Inverse square law 
CT  - Computerised tommography 
MRI  -  Magnetic Resonance Imaging 
 
 
xi 
 
University of Ghana  http://ugspace.ug.edu.gh
ABSTRACT 
A very good reason why calculation of dose distribution is important is that it is 
essential to plan and replicate the treatment prior to the actual delivery of the radiation 
dose to the tumour. In modern radiation therapy, computer software is used for 
performing treatment planning. Different algorithms are employed at every stage of 
treatment including dose calculation algorithms. The dose calculation used for the 
HDRplus TPS is the TG43 formalism and just like every other TPS, the HDRplus, 
version 3.0.5, may produce erroneous results, (Daskalov, 1998) and it is essential to 
eliminate or reduce these errors through a quality control procedure of manual 
calculations. This will further improve treatment outcomes since TPS approximations 
are used in executing treatment. The Sievert integral algorithm is commonly used in 
radiotherapy treatment planning systems (RTPS) for evaluation of dose around 
125 169 137 192
brachytherapy sources and it is an established tool for I, Yb, Cs and Ir 
(Bhola et al, 2012). The source under consideration is the new model (Co0.A86) of 
60
the BEBIG Co HDR source, hence the need to establish the Sievert as a valid tool 
for quality control for the output of the TG43 based HDRplus TPS using this source. 
From the results obtained, the percentage deviation range of the dose output from the 
Sievert integral in comparison to that of the output from the TPS is 0.03 – 10.51% 
 o o
with a mean of 3.13% for angle range of 0  < θ < 70 . The Sievert integral breaks 
down at respectively large angles and therefore neglecting the breaking point oblique 
direction, i.e.      , the range becomes 0.03 – 5.63% with a mean value of 2.55% 
 o o
for the angle range of 0  < θ < 48 . Following the ICRU report 24 that percentage 
deviation of dose values calculated should not exceed ±5% of the true value, the 
Sievert integral has proven to be a valid tool for quality control of the HDRplus using 
the range with less error. 
xii 
 
University of Ghana  http://ugspace.ug.edu.gh
 CHAPTER ONE 
 
 
INTRODUCTION 
 
 
1.1 BACKGROUND 
In Ghana, cancer forms 16% of the top 50 causes of death and two-thirds of cancer 
deaths occur in developing countries (Health Profile: Ghana, 
http://www.worldlifeexpectancy.com/country-health-profile/ghana.html). This has 
resulted in a large number of cancer related research. Different modes of cancer 
treatment have been adopted in health facilities in Ghana and these include surgery, 
chemotherapy and radiotherapy. 
Radiotherapy, radiation oncology or therapeutic radiology, is one of the key 
modalities used in the management of malignant diseases (cancer). In dissimilarity to 
other medical areas that rely mainly on the clinical expertise and experience of 
medical officers, radiotherapy, with its use of ionizing radiation in the treatment of 
cancer, relies heavily on modern technology and the collective efforts of numerous 
professionals, including medical physicists. The coordinated team approach for 
treating cancers greatly impacts the treatment outcome (Podgorsak, 2005). 
Radiotherapy involves the use of high energy ionizing radiation in the treatment of 
tumour cells. It is classified into two categories; external beam radiation therapy 
(EBRT) and brachytherapy. For the purpose of this research, much emphasis will be 
laid on brachytherapy. 
1 
 
University of Ghana  http://ugspace.ug.edu.gh
Brachytherapy, sometimes referred to as curietherapy, is used to describe the short 
distance treatment of cancer with radiation from small, encapsulated radionuclide 
sources (Podgorsak, 2005). This method of treatment is used to deliver radiation to 
localised tumors by interstitial, intracavitary, or surface application (Kyeremeh et al, 
2012).  
Intracavitary treatments are at all times short-term, whereas interstitial treatments may 
either be short-term or permanent. Intracavitary brachytherapy involves placing 
sources into cavities proximate to the tumour volume, while interstitial brachytherapy 
deals with implanting sources surgically within the tumour volume. In temporary 
applications, dose is delivered over a short time interval compared with the half-life 
value of the radionuclide and the encapsulated radioactive source is taken off from the 
patient once the expected dosage is attained. In the case of permanent applications, 
the delivered dose lasts over the lifespan of the encapsulated source till it is 
completely decayed. Depending on the treatment duration, brachytherapy could be 
termed as low dose-rate, LDR (between 0.4 and 2 Gy/h), medium dose-rate, MDR 
(between 2 Gy/h and 12 Gy/h) and high dose-rate, HDR (greater than 12 Gy/h). 
(Acquah, 2011) 
Between 10 and 20% of all radiotherapy cases are treated using brachytherapy in 
typical radiotherapy departments. Its great value is attributed in part to the delivery of 
lethal dose to target organs with minimal dose reaching the nearby healthy tissues  
(Khan, 1994). The main advantage of this technique is the high conformal radiation 
dose delivered to the malignant tumour volume and sparing of the healthy organs or 
tissues at risk due to the inverse square law on the dose distribution around the 
radiation source thereby producing improved localized dose to the target region of 
interest (ROI) as compared with external beam radiotherapy. The setback is that 
2 
 
University of Ghana  http://ugspace.ug.edu.gh
brachytherapy can only be used in cases in which the tumour is well localized and 
relatively small (IAEA, 2007; Acquah, 2011). 
Dose calculation algorithms are the most essential software components in a 
computerized treatment planning system (TPS). These modules are accountable for 
the correct illustration of dose in the patient, and could be connected to treatment 
time, dwell times of source and localization of brachytherapy sources.  
The feats attained in the provision of radiotherapy modality of cancer treatment may 
decline if not harmonized using arduous dose calculation models and algorithms 
(Papanikolaou & Stathakis, 2009). Basically, it’s because the efficiency and 
superiority of any TPS largely depends on the sort of algorithms employed at each 
diverse phase of the process of planning. Based on manufacturer preferences, several 
types of dose calculation algorithms are used in modern TPS, but the best dose 
computational algorithm should be easy to use with its speed and accuracy well put 
together (Kyeremeh et al, 2012). The quality of the dose distribution representation of 
a good algorithm depends strongly on the data or the parameters used by the 
algorithm (Oguchi et al 2005). The concurrence between the calculated and delivered 
dose is of great importance in radiotherapy since the accuracy of the absorbed dose 
based on prescription determines the clinical outcome (Oelkfe & Scholz, 2006). 
There is a dual function of dose calculation algorithms in the practice of radiotherapy:  
i. for plan enhancement in the process of treatment planning  
ii. for nostalgic comparison of the relationship between the available treatment 
parameters and clinical output. 
It outlines two conjointly incompatible objectives of the particular dose calculation 
algorithm under consideration. To start with, the dose calculation algorithm has to be 
3 
 
University of Ghana  http://ugspace.ug.edu.gh
expeditious so as to facilitate the treatment planning process in medically satisfactory 
time frame and secondly the outcome of the dose calculation has to be adequately 
accurate so that the establishment of the relationship between delivered dose and the 
clinical outcome remains consistent and significant. This has called for more review 
into the subject of dose calculation (Oelkfe & Scholz, 2006). 
There are different dose calculation algorithms that are used in brachytherapy. These 
are Monte Carlo, Superposition-Convolution model, Sievert Integral model and the 
latest being the AAPM TG43 formalism. 
The Sievert integral is a substitute to experimental or other dose calculation 
algorithms. This method basically involves dividing a line source into small 
elementary sources and applying the inverse square law and filtration corrections to 
each. It is an indispensable tool for dose rate calculations around brachytherapy 
sources, combining simplicity with equitable computational times, requires limited 
input data and is computationally efficient. This gives the Sievert Integral an 
advantage over the Monte Carlo (MC) technique which requires large volume of input 
data and high computational time. The limitation of the Sievert integral not being 
accurate in predicting dose rate distributions around low energy isotopes will not be a 
60
problem in this research study because the source in this case is Co (Nani et al, 
2009). 
Various forms of the Sievert Integral algorithm have been used for over the past nine 
decades. The Sievert Integral could be modified to correct for photon attenuation in 
tissue. The classical Sievert Integral model lumps the primary beam and scattered 
beam together and this takes care of the effect of the inverse square law.  
4 
 
University of Ghana  http://ugspace.ug.edu.gh
Modifications in the basic model of the Sievert integrals increase the accuracy 
significantly, nevertheless there are still discrepancies between the output from the 
Sievert integral and other dose calculation algorithms (Nani et al, 2009). This study 
60
seeks to obtain the dose distributions around the new BEBIG Co source based on the 
version of the Sievert integral where the source strength is specified in air kerma 
(Khan, 2010). The results will then be compared to the output of the TG43 based 
HDRplus TPS which is used at National Center for Radiotherapy and Nuclear 
Medicine, Korle Bu Teaching Hospital (NCRNM, KBTH). 
 
1.2 STATEMENT OF RESEARCH PROBLEM 
As part of quality assurance (QA) procedures for HDR brachytherapy, it is required to 
perform some manual calculations to estimate doses at specific points called control 
points (CPs) around the brachytherapy source. This is used to validate the output of 
the TPS since dosimetry using a TPS may sometimes produce erroneous results 
192
(Daskalov et al, 1998) especially at proximal distances as has been recorded for Ir 
sources (Cohen et al, 2000). Sievert integral as stated earlier, is one of the methods 
which can be used to calculate the dose distributions around a brachytherapy source.  
There is therefore a need for validation of this method for the HDRplus TPS in use for 
60
dose distribution determination for the new BEBIG Co HDR Multisource 
afterloader brachytherapy machine (Eckert & Ziegler BEBIG GmbH, Germany) at the 
NCRNM, KBTH. The HDRplus TPS runs on TG43 algorithm and the afterloader 
60
machine uses Co with code name Co0.A86. 
 
5 
 
University of Ghana  http://ugspace.ug.edu.gh
1.3 OBJECTIVE OF THE STUDY 
The goal of this study is to identify a robust way of calculating the dose distribution 
around the BEBIG Multisource Afterloading brachytherapy source (Co-60) and 
establish the efficacy of the method (Sievert integral) by comparison with the output 
of the HDRplus. 
 
1.4 SPECIFIC OBJECTIVES 
a) To obtain a treatment plan for the study using the HDRplus. 
b) To digitally obtain doses at different locations in a x-y plane, called (CPs) and 
60
the corresponding dwell-times of the Co source.  
c) To obtain the Sievert integral values for the various angles of the CPs with 
respect to the source positions. 
d) To write a MATLAB program to compute the dose value for the various input 
data to be fed into the program in order to obtain values for each CP. 
e) To make a statistical analysis, through graphical representation of results 
obtained from calculation and the TPS and find the percentage deviation of 
Sievert integral doses from the HDRplus doses. 
 
1.5 SCOPE OF THE STUDY 
The study involves obtaining a treatment plan and positioning digitized sources in a 
universal applicator at different locations on the vertical axis of a 2D-plane. A 
prescribed dose of 2 Gy will be delivered, and dose measurements taken at twelve 
CPs. For each source dwell positions, the dose will be normalised to a distance of  
1 cm. The resultant dwell-times, which will automatically be calculated by the 
6 
 
University of Ghana  http://ugspace.ug.edu.gh
HDRplus, will equally be used in obtaining the dose values from the Sievert integral 
procedure. Other required parameters will be inputted into the dose calculation 
formalism under study and the results will then be analysed. 
 
1.6 JUSTIFICATION AND RELEVANCE OF THE STUDY 
192
The Sievert integral has been used for estimating the dose distribution around Ir 
brachytherapy source, and the outcome was compared with the Monte Carlo 
technique and uncertainties in a range of 3.5 – 5.8% were observed along the axis of 
the brachytherapy source (Nani et al, 2009). It is therefore prudent for this study to be 
60
performed on the new BEBIG Co source and to compare the results with the TG43 
formalism which is the most widely adopted dose calculation formalism in most 
radiotherapy centres, including the NCRNM. 
The HDRplus, version 3.0.5, just like every other TPS may produce erroneous results, 
as established by other research work, (Daskalov, 1998) and it is essential to eliminate 
or reduce these errors through a quality control procedure of manual calculations. 
This will further improve treatment outcomes since TPS approximations are used in 
executing treatment. 
 
1.7 ORGANISATION OF THESIS 
Chapter one deals with the background of the study, the problem statement, 
objectives, justification and relevance of the work, scope and limitation. Chapter two 
contains literature review relevant to the research work. Chapter three describes the 
research materials and methods used to conduct the study. Chapter four outlines the 
7 
 
University of Ghana  http://ugspace.ug.edu.gh
results obtained and subsequent discussions on the findings while chapter five gives 
the conclusion, recommendations and suggestions for further studies. 
  
8 
 
University of Ghana  http://ugspace.ug.edu.gh
CHAPTER TWO 
 
LITERATURE REVIEW 
 
2.1 INTRODUCTION 
This chapter discusses in-depth literature on the various dose calculation models with 
much emphasis on the TG43 formalism which is the universally accepted dose 
calculation model, the fast Fourier transform (FFT) or convolution formalism and MC 
formalism. 
Treatment of cancer with radiation (radiotherapy) is carried out with the ultimate goal 
of delivering the prescribed dose to a tumour precisely and simultaneously 
minimising the dose reaching critical organs. The key element of a treatment planning 
system (TPS) is the calculation of dose and the accuracy of the calculations which 
directly has bearing on the quality of a treatment while its speed greatly affects the 
clinical flow. 
 
2.2 DOSE CALCULATION 
A very good reason why calculation of dose distribution is important is that it is 
essential to plan and replicate the treatment prior to the actual delivery of the radiation 
dose to the tumour. Since the goal is to kill the tumour with radiation from a sealed 
source placed in a patient (brachytherapy) or from a distance (teletherapy), a precise 
dose needs to be absorbed by the tumour and this can only be guaranteed by 
performing dose distribution calculation through the management of radiation beams 
9 
 
University of Ghana  http://ugspace.ug.edu.gh
which are characterised by various parameters in the treatment machine used in the 
delivery of radiation. This procedure is referred to as treatment planning. 
st
Computer software is used for performing treatment planning in 21  century 
radiotherapy. This is done by making use of patient’s images to recognize and locate 
the internal body structures together with the parameters of the machine to simulate 
the actual treatment. The simulation results produces the calculated doses for the 
target as well as those for the organs at risk (OAR) and other regions of interest 
(ROI). The accuracy of the calculation of dose distribution and the high level quality 
assurance programme carried out on the TPS is essential to make sure that dose 
delivery to the tumour is equal to or close to 100% of the calculated dose (Lu, 2013). 
The total accumulated energy of ionising radiation absorbed by a unit mass of body 
tissues defines the radiation dose. Therefore, dose calculation is the computation of 
the energy absorbed by the media at any points that radiation beam particles pass or 
may not pass through, where various physical processes take place due to interactions, 
such as Compton’s scattering, between the beam particles and the media (Lu, 2013). 
At specific points of interest (POI), the accumulated dose is a contribution from the 
interaction of incident beam with points in the patient, and scattered radiation beam. 
A very good dose calculation algorithm compromises between large amounts of data 
storage and accuracy, and lengthy calculations. This compromise will only be 
meaningful if calculations are carried out with speed and accuracy (Lu 2013; 
Kyeremeh, 2011) 
 
10 
 
University of Ghana  http://ugspace.ug.edu.gh
2.3 EVOLUTION OF DOSE CALCULATION ALGORITHMS AND 
DOSIMETRIC SYSTEMS 
Developments in the field of nuclear and particle physics and computer science have 
seen the rapid evolution of dose calculation algorithm since the middle part of the 
twentieth century. This has led to an enhanced understanding of the physical 
processes involved in the interaction of radiation with a medium and has enabled 
simulation and calculation of doses for complex system within a short of period of 
time (Lu, 2013) 
Dosimetric systems and techniques have gone through phases of change for a few 
decades. A typical illustration is the idea or assumption that was made that a uniform 
distribution of sources on a surface applicator or for a single planar interstitial implant 
would result in a uniform dose distribution at the treating distance of usually 0.5 cm 
or 1.0 cm (Wickham and Degrais, 1910, Baltas et al,2007). This opinion was valid in 
clinical practice until the latter years of 1920s. As at the 1930s it became 
unambiguously proven that this was untrue and that a non-uniform placement of 
brachytherapy seeds resulted in an even distribution of dose. This laid the foundation 
for the formation of the Manchester System of Paterson and Parker, which came into 
being in the 1930s (Meredith, 1947; Baltas et al, 2007). The outcome from theoretical 
findings on the exposure rate distribution quantified in roentgens around 
brachytherapy seeds of less complex geometrical feature, such as line, annulus, sphere 
and cylinder, aided the formulation of rules for the system. More research had been 
done about line sources at an earlier period and the Sievert integral being popular by 
then (Sievert, 1921; Baltas et al, 2007). The Manchester System was commonly used 
till the 1970s before the inception of computerised treatment planning, in addition to a 
system known as the Quimby system, birthed in the early 1920s and became the 
11 
 
University of Ghana  http://ugspace.ug.edu.gh
system adopted and used in the United States. These two methods were then taken to 
be the standard of brachytherapy dosimetry for subsequent years (Baltas et al, 2007). 
 
Figure 2. 1: The Manchester system used in the calculation of doses at two points 
          A and B (Khan, 2010) 
 
 
Figure 2. 2: Paris System prescription for breast implant 
12 
 
University of Ghana  http://ugspace.ug.edu.gh
Planning of an interstitial implant for both systems entailed the determination of the 
area or volume of a target region and then consulting a table or graph for the required 
total source strength, measured in milligram-hours, per unit peripheral dose rate. The 
Manchester system (method clearly indicated in fig 2.1) sought to deliver ±10% of the 
prescribed dose throughout the region implant while the Quimby system sought to 
deliver a uniform distribution of source strength of equal linear activity and accepted 
the hot spots in the central region of the implant (Meredith, 1947; Baltas et al,2007). 
At a later time, the Paris system (method of source arrangement displayed in fig 2.2) 
was developed but was modelled for use with Ir-192 wires. Its dosimetry is based on 
the measurement of the dose rate at the central part of the volume being treated, which 
is called the basal dose rate. Its calculation is from the dwell position of the wires in 
the central plane and is the least dose rate between two wires while taking note that 
the correlation between the basal dose and the layout of the Ir-192 wires is very 
essential. 
International approvals (ICRU report 58, 1997) have been introduced for dose and 
volume requirement in interstitial brachytherapy which replace the earlier concepts 
and terminology such that in place of volumes and planes, we have the following; 
gross tumour volume (GTV), clinical target volume (CTV), planning target volume 
(PTV), treated volume and central plane. 
To give a detailed description of the dose distribution, the quantities used are: 
prescribed dose, minimum target dose, mean central dose, high dose volumes, and 
low dose volumes. 
Other systems have equally been introduced, for example, the dose-volume histogram 
(DVH). Theoretical evaluations pertaining to the distribution of dose around radium 
13 
 
University of Ghana  http://ugspace.ug.edu.gh
brachytherapy seeds began in 1916, about five years before the conception and 
publication of Sievert integral method for linear radium brachytherapy seeds and were 
sustained through many decades with different sources and modifications made to the 
integral to suit specific expected outcomes, alongside other modern techniques. 
Experimental measurements were also undertaken with rather rare options to imitate 
the neighbouring organs for an actual patient. Those used were rabbit and rat muscle 
and butter. The basic standard of the use of the butter phantom was that by creating 
slices in the butter, and decolourizing it using gamma rays from radium would 
efficiently produce isodose curves. 
For other forms of brachytherapy, such as the intracavitary gynaecological uses for 
either the cervix uteri or the corpus uteri, and the endometrium, dosimetry systems 
were also developed. Also, doses could be calculated from first principles; thus 
calculation of dose rate around point sources, line sources or encapsulated cylindrical 
source. Some advanced methods are two dimensional formalism, three dimensional 
dose calculation-convolution technique, Monte-Carlo technique, Sievert integral 
technique, etc. (Lu 2013; Kyeremeh, 2011). 
 
2.4 DOSE-RATE CALCULATION AROUND A POINT AND LINE 
  SOURCE   
A point source describes the least uncomplicated situation to calculate in the quest to 
create an approximate calculation algorithm. For point sources, radiations from 
charged particles are neglected and consideration is made only of photons. This is so 
because commercial sources are encapsulated and therefore the encapsulation scatters 
and slows down the electrons in that most do not leave the source capsule and for the 
14 
 
University of Ghana  http://ugspace.ug.edu.gh
few that escape, their range in tissue outside is much smaller than 5 mm and therefore 
assume not to contribute at clinical prescription distances. Considering a sample of 
radioactive material whose largest dimension is smaller than 0.01 mm, all the atoms 
are considered to be located at a single point. The resultant dose rate in a small 
volume, dv, of tissue located at a distance, r, from the source (in vacuum, such that 
there is no scatter) and that a photon is emitted at every time, t, with energy, E, will be 
given by the product of time rate of emission of the photon (activity), the probability 
of the photon hitting dv, the probability of photon of energy E, which imparts and 
interacts with the small volume and the average amount of energy dEabs that is 
absorbed in dv all divided by the mass, m, of the small volume. The dose rate is then 
expressed as energy/mass/time, explicitly expressed as (Daskalov et al, 1998) 
  
 ̇( )     (    )   (    )                                        (   )   
 
Where,  
dv = small volume of tissue 
r = distance from source centre 
E = energy of source (intrinsic) 
A = activity 
P(r,dv) = probability of photon hitting dv at a distance r from source center 
P(E,dv) = probability of photons with energy E which strikes the volume dv 
dEabs = average amount of energy absorbed in dv 
Line sources are considered as an extremely thin source of active length, L, of 
radioactive material and in such instances, the direction and distance with respect to 
15 
 
University of Ghana  http://ugspace.ug.edu.gh
the centre of the line is considered but symmetry still remains (Daskalov et al, 2000). 
For a line source, the dose rate at any point in the plane is derived by defining the 
activity per unit length of the source and integrating the point source expression. 
    ( )
                            ̇( )                                                    (   )  
Considering the entire length of the source, the final result is calculated using the 
relation 
 
 
 
       ( )          ( )
 ̇( )  ∫                                                  (   ) 
         
 
 
 
The length, L, is the active length of the source and          is the angle 
subtended by the source when viewed by the source when viewed from a point. 
For encapsulated sources used in clinical practice, the active source region in the 
cylindrical encapsulated source is divided into smaller points and contributions from 
these points are determined separately using equation (2.1) and the results are then 
added up. The encapsulation reduces the dose rate by an amount dependent on the 
path length through the encapsulation. Since the path length through the encapsulation 
from each point source is not the same, the individual contributions to dose rate varies 
in all directions and hence the solution to this encapsulated line source has been 
proposed by the Sievert integral. (Neuenschwander et al 1995; Miften et al 1999)  
 
2.5 TWO-DIMENSIONAL DOSE CALCULATION FORMALISM 
The AAPM in 1988 formed a Task Group (TG43) to address some outstanding issues 
emanating from brachytherapy source dosimetry formalisms. The resultant dose 
calculation model proposed by the Task Group 43 resulted from the publication of the 
16 
 
University of Ghana  http://ugspace.ug.edu.gh
ICWG in 1995 and has gained universal recognition and adoption (Nani et al, 2009). 
The proposed 2D dose distribution calculation formalism based on the TG43 is 
confined to quantities that are measurable and dissociates a few interdependent 
quantities and also includes new constants dedicated to the various sources (Meli et 
al, 1988). 2D dose distribution calculations around brachytherapy seeds have been 
made feasible with this protocol and have resulted in absolute dose rate change by 
17% with respect to traditionally used treatment planning data at that time (Ravinder 
et al, 1994; Kline and Earle, 2007; ICWG, 1990). The newly proposed TG43 
formalism demands input data comprising dose-rates obtained from a real 
brachytherapy seed placed in a tissue equivalent phantom and depends on the 
particular source, the source geometry and construction in addition to primary photon 
spectrum and medium unlike the conventional method which adopts exposure rate 
constant and tissue attenuation factors (Ling  et al 1985; Schell et al 1987; Weaver et 
al 1989; Nath et al 1995).  
A pure brachytherapy seed manifests significant anisotropy and therefore it is difficult 
to perfectly obtain dose distribution from photon fluence and free space. The TG43 
model lacks this basic setback through an undeviating implementation of measured or 
measurable dose distribution a source in a homogeneous medium (Nath et al, 1995). 
The TG43 model also puts into restrictive consideration cylindrical symmetric 
sources. For such sources, the dose distribution is in 2D and is best illustrated in terms 
of polar coordinates with its origin located at the centre of the active source.  
17 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Figure 2.3: Geometry used in the calculation of dose distribution near a linear 
          source based 
 
Based on the TG43, the dose rate at a specified point (r,θ) in a medium for the source 
of strength, Sk, is given by  
 (  )
 ̇(   )      (   ) ( )   (2.4)   (    )
Where  
r = distance (cm) from the origin to the point of interest, P 
θ = angle between the direction of radius vector r and the long axis of the source 
θo = source transverse plane and is equal to π/2 radians 
18 
 
University of Ghana  http://ugspace.ug.edu.gh
2 -1 
S k = air kerma strength of the source mGym h
Λ = dose rate constant in water 
G(r,θ) = geometry function 
g(r,θ) = radial dose function 
F(r,θ) = anisotropy function 
The equation (2.4) is used for both point and line source approximations to expedite 
the dose computation process. All parameters in the equation are referenced to the 
same single points and the product ΛSk is the dose rate in water at the reference 
position (ro,θo). The reference point is situated at a distance 1cm along the 
perpendicular bisector (transverse axis) of the source as shown in fig.2.  , i.e. ro = 1cm 
and θo = π/2  
Air kerma strength, Sk, as a measure of brachytherapy source strength is specified in 
terms of brachytherapy source at a point along the transverse axis of the source in free 
space is defined as the product of air kerma rate at a calibration distance  ̇( ) and the 
square of distance d. 
    ̇( ) 
     (2.5) 
   is the measure of the absolute amount of radionuclide available. Its source 
2
calibration unit U is equal to cGycm /hr by definition. (Kline and Earle, 2007; Nath et 
al 1995) 
The dose rate constant ( ) expresses the dose rate to water at a distance of 1cm on the 
transverse axis as a unit kerma strength source in a water phantom. It is given as 
 ̇(   )
                                                                 (   ) 
  
19 
 
University of Ghana  http://ugspace.ug.edu.gh
The dose rate constant is determined once for each of the manufacturer’s source using 
Monte Carlo modelling plus experimental measurements usually with TLDs (Nani et 
al 2009; ICWG 1990; Ling  et al 1985; Schell et al 1987). Errors are present in both 
procedures and hence results are averaged to yield a consensus value for Λ (ICWG, 
1990). As a result of the effect of the distribution of the activity within the source 
encapsulation, self-filtration within the source and scattering in water around the 
source, the evaluation of the dose rate presents a measurement uncertainty (Nani et al, 
2009).  
The parameter G(r,θ) represents the geometric function that gives detailed 
information for the geometric fall off of the photon fluence with respect to the 
distance from the source and, it is dependent on the distribution of radioactivity inside 
the material. The geometry factor suppresses the influence of inverse square law on 
the dose distribution around the source. There is a large dose rate gradient around 
interstitial brachytherapy sources and this makes it difficult to estimate accurate dose 
rate at distances less than 5 mm from the source. Also, the large variation in dose rate 
that emanates from the inverse square law makes accurate interpolation of 
intermediate dose rate values very difficult. Therefore, suppressing the inverse square 
law effects and extrapolating to small distances from the dose rate profile 
measurement at 5 mm and 10 mm as well as interpolating between scantily distributed 
measured values increases accuracy. From the TG43 model, there are two geometric 
functions that are considered for both point and line sources. (Nani et al, 2009; 
Weaver et al, 1989; Nath et al 1995)  
Point source is given as  (   )    ⁄     (2.7) 
 
Line source is given as   (   )        (2.8)       
20 
 
University of Ghana  http://ugspace.ug.edu.gh
The radial dose function, g(r) accounts for the effect of absorption and scatter in the 
medium along the transverse axis of the source. It is defined as 
 ̇(    ) (     )
 ( )                                                              (   ) 
 ̇(     ) (    )
The fall off of those rates in the direction along the transverse axis due to absorption 
and scattering in the medium is defined by the function, g(r). It is modelled by 
filtration of photons by source materials and encapsulation (Nath et al, 1995; Nani et 
al, 2009; Meli et al, 1988).  
Anisotropy within the TG43 formalism is described by the quantity F(r,θ). The 
introduction of the anisotropy of source distribution function F(r,θ), is to account for 
the differences in dose rate as a function of angle from the symmetry axis due to the 
specific geometry of the encapsulation of the radioactive source. The 2D anisotropy is 
described by the TG43 model by: 
 ̇(   ) (    )
 (   )                                           (    ) 
 ̇(    ) (   )
The variation of dose rate about the source at every point around the source is 
expressed by equation (2.10) and this variation is mainly due to the effects of self-
filtration, oblique filtration of primary photons through the encapsulating materials 
and scattered photons in the medium.  The information on the dose rate in all 
directions provides a means to determine the anisotropy from the TG43 formalism, 
Monte Carlo modelling and the Sievert integral (Nath et al 1990) 
If a substantial number of seeds are oriented haphazardly or the degree of dose 
anisotropy around single sources is restricted, the dose rate impact on tissue for each 
source can be estimated using the average radial dose rate. This is extended by adding 
21 
 
University of Ghana  http://ugspace.ug.edu.gh
up the single anisotropic points due the seed sources (Nath et al 1990). Although the 
anisotropic function is a function of r and θ, the quantity F(r,θ) can be averaged over 
the 4π geometry and F(r,θ) can be approximated using simple radial function ϕan(r), 
called the one dimensional (1D) anisotropy function which gives the same effect as 
the 2D anisotropy because of its averaging effects.  
  
 
 ̇( )  ∫  (   )                                            (    ) 
  
 
           ; for a cylindrically symmetric dose distribution (Ling et al, 1985). 
Substituting equation (2.3) into (2.12) and rearranging, we obtain as shown in 
equation (2.13). 
For scenarios where a 2D calculation cannot be used for cylindrical sources, the 
revised TG43 protocol which is called the TG43U1 recommends the use of  
 (    )
 ̇(   )         ( )                                           (    )  (     )
The quantity ϕan(r) describes the anisotropic factor which is a ratio of the dose rate at 
a distance r averaged with respect to a solid angle, to the dose rate on the transverse 
axis at the same distance (Nani et al, 2009) expressed as  
 
∫  ̇(   )      
   ( )   
                                        (    ) 
  ̇(    )
192 125 103
For sources such as Ir, I and Pb, ϕ(r) is less than one with values extending 
from 0.91 to 0.97.  
Equation (2.13) displays more accuracy when dealing with cylindrical sources at 
distances less than 1 cm as compared to the rather oversimplified equation (2.1) with 
22 
 
University of Ghana  http://ugspace.ug.edu.gh
the simple point source approximation. This is because for clinical line sources (thin 
cylinders) both ends of the source must have their orientation within the patient 
determined (Meli et al, 1988). 
The universal recognition of the TG43 brachytherapy dose calculation model 
nonetheless has clinical limitations. Dose calculation formalisms based on the TG43 
fail to account for dose distribution in finite tissues and heterogeneities in the patient, 
and the proper handling of the effects of shielding material in the calculation. 
 
2.6 FAST FOURIER TRANSFORM 
Fourier analysis of a periodic function involves extraction of the series of sine and 
cosine signals which when superimposed will reproduce the parent function. This 
analysis can be expressed as a Fourier series. The fast Fourier transform (FFT) is a 
mathematical method for transforming a function of time into a function of frequency 
(Kyeremeh, 2011).  It is often considered as a translation from the time domain to the 
frequency domain and has proven to be very useful for analysis of time-dependent 
phenomena such as dose distribution. 
Convolution which is a mathematical procedure that combines two functions in such a 
way to produce a third function. One of the input functions is the convolution kernel, 
or spread function (the energy deposition kernel in this case), and the two input 
functions are said to be convolved (Papanikolaou, 2004). Convolving two functions 
could be represented mathematically as: 
 ( )   ( ) ( )                                                   (    ) 
23 
 
University of Ghana  http://ugspace.ug.edu.gh
But 
 
   ( ) ( )   ∫  ( ) (   )   ∫ ( ) (   )                        (    )  
Where f(x)g(x) denotes convolution of the two functions f and g. 
Convolution procedure at a given point x is calculated through correctly placing each 
element of the kernel g(x) and centring it at that point. The resultant elements of the 
primary functions (TERMA at x, and the kernel element at the same point) are then 
multiplied. 
The goal of dose calculations is to find the reaction (TERMA) to a deposition of dose 
(KERMA – Kinetic Energy Released per unit Mass) in the tissue. The distribution of 
dose in 3D in the tissue is estimated by superimposing the point values of the 
absorbed dose from the knowledge of TERMA with reference to the dose deposition 
kernel. The dose distribution from the incident photon interaction point is illustrated 
by the kernel through the volume of the TERMA (Daskalov, 1998). Although the 
patient’s main source of dose emanates from photons, the dose impact from beta 
particles is substantial to the dose close to the superficial part of the volume of the 
TERMA and this is usually justified for in the algorithm by modelling a photon-only 
dose and comparing it to the dose that has already been measured at shallow depths 
(Miften, 1999).  
The method computes doses in a 3D configuration and fundamentally takes care of 
the influence of tissue heterogeneities on both incident and scattered photons 
(Papanikolaou, 2004). Apart from the corrections made for distribution of measured 
dose, the convolution calculates the distribution of dose from fundamental principles. 
The calculation is a four staged model which are: 
24 
 
University of Ghana  http://ugspace.ug.edu.gh
1. Modelling the incident energy fluence as it exits the source (accelerator), 
2. Projection of the energy fluence through the density representation of a patient 
to compute a TERMA volume, 
3. Three-dimensional superposition of an energy deposition kernel and 
4. Electron contamination model. (Papanikolaou, 2004) 
The fast Fourier convolution technique follows from two functions f(t) and g(t) with 
their respective Fourier transforms as F[f] and F[g]. The convolution theorem states 
that, the Fourier transform of the convolution is the complex product of the individual 
Fourier transforms. 
Thus 
  [   ]   [ ]   [ ]   (2.16)  
Hence convolving the two functions f(t) and g(t), the result is given by:  
  
 ( )   ( )  ∫  ( ) (   )                           (    ) 
  
The dose distribution at the point x by the conventional model is expressed as  
  
 ( )  ∫  (  ) (    )                                                         (    ) 
  
Where,  
  
 (  )  ∑   (   
 )                                                                          (    ) 
   
  
25 
 
University of Ghana  http://ugspace.ug.edu.gh
Equation (2.19) defines the geometric arrangements of the seeds and the 
corresponding strengths spatially described by the Dirac function  ( ). Equation 
(2.18) defines a convolution operation on the kernel f(x) with the a function g(x). A 
solution to the equation (2.18) could be expressed in the form of an inverse Fourier 
transform (IFT), shown by equation 2.20: 
 
 ( )     [    [ ( )]  [ ( )]]                                      (    ) 
 
    denotes the IFT and N is the sampling size. The extension of equation (2.20) in 
3D gives the three dimensional dose distributions as (Kemmerer et al, 2000)  
 
 ( )     [    [ (     )]  [ (     )]]                     (    ) 
      
 
2.6.1 Implementation of FFT Convolution 
The discrete convolution of two discrete functions           with period N, defined 
by their samples            , is given by  
 ⁄   ⁄ 
(   )   ∑         ∑                                            (    ) 
  
          
  
The discrete nature of the convolution is based on the assumption that the two 
functions are periodic and of equal length. Their periodic nature leads to a 
wraparound effect which is prevented through the extension of the functions with zero 
padding. Since the FFT requires only positive indices, the putting into effect of 
equation (2.22) demands a shift of the function, and this is obtained by a wraparound 
ordering of one of the functions as shown in fig. 2.4.a. for three sources in a one 
26 
 
University of Ghana  http://ugspace.ug.edu.gh
dimensional representation. The fig. 2.4.c. illustrates the kernel in a wraparound order 
necessary for FFT. The IFT of the function g(x) and the kernel f(x) is shown in fig. 
2.4.g 
 
Figure 2.4: Schematic diagram illustrating the FFT convolution method for 
calculating the dose distribution for one dimensional case of three sources with 
different lengths 
27 
 
University of Ghana  http://ugspace.ug.edu.gh
2.7 MONTE CARLO (MC) BASED DOSIMETRY 
MC based dosimetry has been generally recognised as one very useful tool in 
curietherapy forming one of the dosimetric requirements for regular clinical use of 
modern low energy photon interstitial radiation sources (Williamson et al 1998; 
Weaver, 1998). 
The MC procedure operates by performing a statistical simulation of every process 
related to photon transport and emission through the use of arbitrary numbers and 
suitable probability distribution functions. Its key feature is its probabilistic feature, 
implying that in contrast to some analytical procedures, each evaluation of a problem 
will, generally, yield varying results. The accuracy of the outputs is dependent, among 
other factors, on the number of statistical simulation processes. This makes MC 
calculations more rigorous and time-consuming than other analytical methods but the 
accessibility of computational resources and the major improvement in computer 
st
processor speed (although it requires a lot of resources) in the 21  century has greatly 
decreased calculation times. Nonetheless, MC simulation is still too CPU intensive to 
103
support commercial TPS. MC dosimetry has been carried out on sources such as Pd 
125 241 169 192 137
and I (low energy), Am and Yb (medium energy), and Ir and Cs nuclides 
60
(higher energy), but excludes rarely used Co, and lots of discussions have been 
made for energies ranging from 20 to 700 keV such that electronic equilibrium may 
be safely assumed. This is to ensure that collision kerma is used to estimate absorbed 
dose and secondary electron tracking is omitted. Electronic non-equilibrium or 
imbalance occurs only at points in close proximity, such as those less than 1 mm, to 
192
high energy Ir sources. However, dose rate improvement effect due to the emission 
192
of beta particles by Ir has been observed in this distance range (Wang et al, 2000; 
Karaiskos et al, 2001; Baltas et al, 2001) 
28 
 
University of Ghana  http://ugspace.ug.edu.gh
The foundation of any MC simulation code is a random number generator. It is 
important that this source of random numbers (RN) results in an even distribution 
within the unitary interval [0, 1], and this means that all numbers that fall within this 
interval has equal probability.  
In brachytherapy, MC based dose calculation in a given medium is a three-step 
procedure. 
1. The energy fluence of photons emitted by the active source core must be 
simulated.  
2. These photons must be transported through the source core and source 
encapsulation materials. 
3. Photons must be transported in the medium near the source in order to 
evaluate the dose delivered in predefined volume elements: i.e., scoring 
voxels. 
29 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Figure 2.5: A pictorial representation of MC method of sampling primary and 
scattered radiation photons emitted by a radioactive source. 
MC results from the above process could be analysed through comparison with non-
stochastic quantities applicable to the simulation (Baltas et al, 2007).  
 
2.8 SIEVERT INTEGRAL 
The Sievert Integral was first introduced and named after its founder, Professor Rolf 
Maximilian Sievert; a Swedish professor of medical physics in 1921 and it has a 
general form (Sievert, 1921): 
 
 (   )  ∫                                                        (    ) 
 
30 
 
University of Ghana  http://ugspace.ug.edu.gh
It is a unique function usually encountered radiation transport related calculations.  
th
(Sievert Integral, http://en.wikipedia.org/wiki/Sievert_integral.html, 18  March, 
2017). Since the introduction of the integral almost about ten decades ago, different 
models of the Sievert integral have been developed for the calculation of dose 
distribution in order to obtain much more accurate results. The most popular of all 
models is that used by Williamson et al, 1996, and it has been cited in other published 
works. Karaiskos et al, 2000, also implemented a different model, shown in equation 
2.24, of the integral which was used by Pantelis et al in 2002. 
  
(   ∙     ∙     𝑤   )
𝑤    
 ̇( ,  ) 1   [ +  
(  𝑤 ∙ ) ∙    ( ) ∙  ( , )] 
   ( )
=   ∑ 𝑐 2 (
(22.3.204) ) 
   𝜌         =1
  
 
 ̇(   ) = Dose rate 
   = Source air kerma strength 
N = number of small segments in which active source core is divided into 
      
(   ) = water-to-air ratio of mass-energy absorption coefficient 
    
   = linear attenuation coefficient of source 
   = linear attenuation coefficient of filter 
   = linear attenuation coefficient of water (medium of measurement) 
   ( ) = Scatter-to-primary ratio of dose in water 
 (   ) = Empiric correction describing the deviation of scatter radiation from 
isotropy 
31 
 
University of Ghana  http://ugspace.ug.edu.gh
 (  ) = Normalization factor correcting that corrects for capsule filtration and is given 
by: 
 
 [ (   ∙      ∙   )]
 (  )  ∑[  ]             (    )    
    
 
Various corrections are employed in implementing the Sievert integral for the 
calculation of dose to attain a targeted result 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
32 
 
University of Ghana  http://ugspace.ug.edu.gh
CHAPTER THREE 
 
METHODOLOGY 
 
3.1 SIEVERT INTEGRAL TECHNIQUE 
The Sievert integral, equation (3.1), introduced in 1921 by Professor Sievert, is a 
method that can be used to calculate the exposure rate around a linear brachytherapy 
source. The method is implemented by dividing the line source into small elementary 
sources and putting into consideration inverse square law (ISL) and filtration 
corrections to each of these sources (Khan 2010) 
 
 (   )  ∫                                                                       (   )        
 
For a source with the following parameters: 
L = active length 
t = filtration 
A = activity 
Γ = exposure rate constant  
  = effective attenuation coefficient for the filter 
33 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Figure 3. 1: Diagram displaying the geometric relations used in the evaluation of 
exposure at point P, from an elementary source. (Khan, 2007) 
 
The exposure rate dI at a point P(x,y) contributed by the source element of length dx, 
as shown in fig 3.1 is given by: 
  
  (   )  ∙  ∙   ∙ ∙    
 ∙ ∙                                                    (   ) 
   
Making substitution with the following equations (3.3), (3.4) and (3.5): 
                                        (   ) 
                                         (   ) 
                                   (   ) 
and integrating equation (3.2), the exposure rate I(x,y) obtained for the whole source: 
    
 (   )  ∫    
 ∙ ∙                                               (   ) 
    
34 
 
University of Ghana  http://ugspace.ug.edu.gh
Equation (3.6) takes after (3.1) which is the general form of the Sievert integral. The 
above Sievert integral may be evaluated by numerical methods but this would not be 
done in the case of this research. 
For a source with intensity specified in terms of exposure rate   ̇ at a specified 
distance, s, far from the source (i.e.    ), then the  Sievert integral can be written 
as: 
  ̇ ∙  
   
  
 (   )  ∙    ∫    ∙ ∙                                 (   ) 
    
The source strength used for this research is specified in air kerma strength, hence the 
Sievert integral model that will be implemented is: 
      
 (   )  ∙    ∫    ∙ ∙                              (   ) 
 ̅
  ( )    
3.2 DOSE CALCULATION  
Equation 3.8 calculates the exposure rate from the source, hence the need for 
conversion to dose. This was done by multiplying the emergent exposure rate 
expression by a quantity, fmed or simply the f-factor which is sometimes called the 
roentgen-to-rad conversion factor. This factor is a constant dependent on the medium 
in which measurement is done (in this case, water) and the photon energy from the 
radioactive source. Tables containing the values of the fmed were referred to and the 
60
value for a Co photon energy, 1.25 MeV, in water is 37.6 Gy.kg/C. Aside the value 
of the constant being available in tables, the fmed can be calculated using equation 3.9 
below. 
( ̅   𝜌)      
                                             (   ) ( ̅   𝜌)   
35 
 
University of Ghana  http://ugspace.ug.edu.gh
For a dimensional balance, the average work done per unit charge,  ̅̅̅  was converted 
 
from  
0.876 cGy/R to cGykg/C using the factor: 
                                           (    ) 
After the introduction of the f-factor, the resulting quantity is the dose rate. The dose 
is then calculated by multiplying the dose rate by the source dwell time (tdwell) or 
treatment time which will be produced by HDR 3.0 Plus treatment planning system. 
The equation (3.8) then becomes modified into equation (3.11) for calculation of the 
60
dose around the new BEBIG Co source. 
 ∙  ∙               
  ∙    ∫    ∙ ∙                             (    ) 
 ̅
 ∙  ∙ ( )    
The variable x in equation 3.11 has been used to replace y in equation 3.8 because the 
CP (point of measurement), unlike what is illustrated in figure 3.2 is on the x or 
horizontal axis while the source is on the y or vertical axis. 
 
3.3 MATLAB 
MATLAB is a powerful programming tool for handling the calculations (simple and 
complex) involved in scientific and engineering problems. The name MATLAB is an 
acronym which stands for MATrix LABoratory, because the system was designed to 
make computations of matrices particularly easy (Hahn and Valentine, 2007). 
This programming tool distinguishes itself from others such as C++ and Java in the 
sense that it is user friendly and very interactive. This implies that you input some 
commands at the unique MATLAB prompt, known as the command window, and 
36 
 
University of Ghana  http://ugspace.ug.edu.gh
obtain results immediately. The tool solves problems ranging from simple ones such 
as, finding a square root, or complex ones, like deriving the solution to a set of 
differential equations or solving an integral such as the Sievert integral. For some 
quite technical problems it would be required to enter just a few commands, and the 
answers are produced. (Hahn and Valentine, 2007) 
The derived equations and the values for each parameter were used to write a 
MATLAB program and the calculations of the dose for each control point were 
obtained. One of such is displayed in figure 3.2. 
 
37 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Figure 3. 2: Display of the dose calculation in the command window of MATLAB 
38 
 
University of Ghana  http://ugspace.ug.edu.gh
3.4 IMPLEMENTATION OF THE SIEVERT INTEGRAL  
An analytical solution of the Sievert integral does not exist; hence it will require either 
a numerical procedure or the use of a Sievert integral table of values such as that in 
appendix L. 
The Sievert integral in equation (3.11) has to be modified due to the difference in the 
definite interval between that in 3.11 [θ1, θ2] and that used in the Sievert integral table 
[0, θ] which is identical to equation 3.1 and represented in equations 3.12 and 3.13 
      ∙   ∙        
   ∙      
 
∫  ∙ ∙                                    (    ) 
 ̅
 ∙  ∙ (  )
 
    ∙   ∙  
  
       
   ∙    ∫    ∙ ∙                                    (    ) 
 ̅
 ∙  ∙ ( )   
The variation in the angles under consideration in this research and those used in 
obtaining the integral values in the table, and the corresponding x-values were 
obtained through extrapolation and interpolation calculations. The x used in equation 
3.11 differs from that in equation 3.1. In comparison to that in 3.1, the product of the 
effective attenuation coefficient,   , calculated using an equation (3.14) given by 
Williamson et al , and the wall thickness, t, which is 0.05 cm, yielded the value of x. 
     
 ∑   (      𝜌 )  
    ∙  
 
  ( )   ( )                               (    ) 
      ∑    
[  
    ( 𝜌 )
 ]
Where   = denotes the number of photons with energy    emitted per disintegration  
    
 (   ) = mass energy absorption coefficient in air for photon of energy      
39 
 
University of Ghana  http://ugspace.ug.edu.gh
 d = filter thickness 
 Ei = photon energy 
The value of the attenuation     substituted into equation 3.14 to calculate for the 
effective attenuation co-efficient of the filter material for the source energy being 
 -1considered, is the fitted value for the attenuation of the filter,   = 0.25cm . This was 
to ensure accuracy in calculated values and minimise the deviations between the 
Sievert model and any other model it is compared to which in this case is the TG43 
(Nath et al, 1995). The other values for the attenuation are shown in table 3.1. 
Table 3 1: Actual and fitted values of linear attenuation co-efficient for source 
and filtration materials μ_s and μ_f, respectively for the new BEBIG source. 
(Bhola et al, 2012) 
Linear attenuation co-efficient  (     )   (  
  ) 
Actual value 0.47 0.43 
Fitted values 0.25 0.25 
 
The Sievert integral value for each angle, θ1, and θ2, were obtained individually and 
the expression for their doses and a correction factor, calculated using the Meisbeger 
polynomial (equation 3.15), were multiplied. The difference in the doses obtained for 
each angle yielded the dose at the specified control point. All calculations were done 
by a MATLAB program as displayed in figure 3. 
                 
                                     (    ) 
  √                                                          (    ) 
Where      = Meisberger polynomial/function  
40 
 
University of Ghana  http://ugspace.ug.edu.gh
 r = distance from the source to control point calculated using x and y values  
A, B, C and D are constants or polynomial coefficients whose values are source 
60
dependent and for Co: 
A = 0.99423   (3.17) 
B = -0.005318   (3.18) 
C = -0.002610  (3.19) 
D = 0.0001327  (3.20) 
 
60
3.5 NEW BEBIG Co SOURCE 
60
The radioisotope used for this research is Co and its details are given in table 3.2. 
60
Co as a source is more economical for use in a brachytherapy treatment unit due to 
192
its longer half-life (5.26 years) in comparison with Ir. It is a well-established and 
clinically proven isotope for all HDR brachytherapy treatments. Co-60 sources are 
available for SagiNova, MultiSource and GyneSource afterloaders and since their 
introduction, lots of tremendous successes have been achieved. It has shown to be a 
good choice for treating gynaecological, rectal, prostate, dermatological and other 
body sites. 
Treatment with Co-60 guarantees lower dose to OAR as compared to Ir-192 and this 
is due to its higher average energy of 1.25 MeV, hence less scatter is produced in its 
photon interaction with tissues. The Eckert & Ziegler BEBIG Co-60 source is loaded 
with an activity of about 81.4GBq. The capsule design of the source fulfils the 
demanding regulatory and governmental standards of the EU and other nations. The 
capsule and source wire are connected by high quality laser welding method and 
41 
 
University of Ghana  http://ugspace.ug.edu.gh
crafted to withstand 100,000 source transfers. This has been verified by stress tests in 
straight and curved applicators exceeding this figure to ensure safety and long-term 
precision. With a value of 100, 000 cycles in a period of five years, the brachytherapy 
unit of a clinic could have an estimate of up to seven patients a day receiving a ten 
channel interstitial treatment or twenty-four patients a day for the treatment with a 
three-channel applicator. 
60
 The new model of the BEBIG Co HDR (model Co0.A86) brachytherapy source 
design has a small active core with a diameter of 0.5mm and a more rounded capsule 
tip as compared to the old BEBIG source design. Its central cylindrical active core 
60
length is 3.5mm which is made of metallic Co. This active core is covered by a 
cylindrical stainless steel capsule with an external diameter of 1.0 mm (more details 
are given in figure3.3) 
 
Figure 3. 3: Schematic diagram of the new BEBIG 60Co HDR brachytherapy 
source 
 
 
42 
 
University of Ghana  http://ugspace.ug.edu.gh
Table 3 2: Details of the BEBIG Co-60 source including some initial parameters 
and their corresponding values by 18th March, 2017 which is the date of 
measurement. 
Parameter Value 
Source Name Multisource Co-60 HDR 
Radionuclide Co-60 
Model Number Co0.A86 
Serial Number BB-AC 547 
th
Initial values [18  Jul, 2014]  
2
Air KERMA Strength 23210 cGy.cm /h 
Activity in GigaBecquerel 78.85 GBq 
Activity in Curie 2.05 Ci 
th
Activity for planning [18  Mar, 2017]  
2
Air KERMA Strength 16343.6  cGy.cm /h 
Activity in GigaBecquerel 53.41 GBq 
Activity in Curie 1.4435 Ci 
 
3.6 EQUIPMENT 
The TPS used in the simulation of the sources is the HDRplus manufactured by 
BEBIG. It provides treatment comfort due to its user friendly interface (displayed in 
figure 3.4), it’s fast flexible, precise and delivers an optimised treatment plan. The 
HDRplus supports all HDR applications, including intracavitary, interstitial and 
intraoperative treatments. 
43 
 
University of Ghana  http://ugspace.ug.edu.gh
Its user interface features an ultramodern and easy-to-use structure with the many 
options to customise the layout and parameter settings which further allows the user 
to decide a personal planning approach. Another unique feature of the HDRplus is the 
complete applicator database which is used to select applicators and place them 
directly on an X-ray film or CT/MR images which eliminates the procedure of 
lengthy reconstruction of the applicator. The automated nature of the applicator 
reconstruction makes the HDRplus to identify implanted breast needles and visualise 
them in 3D for simplification of the treatment planning. Automatic image fusion 
feature of this treatment planning system enables its users to match CT and MR 
images without any user interference within a time frame of less than 30 seconds.  
  
Figure 3. 4: Monitor displaying the interface of the HDRplus 
The version of the HDRplus TPS used for the procedure is version 3.0.5. The 
software was launched from the desktop of the system and a patient profile with name 
“Sievert integral” was created. The universal applicator with a dimension of 12cm and 
name LLA1200-20 was selected and a (0,0) dose plane was defined on the orthogonal 
view. The origin was also defined, i.e. (0,0) point and the midpoint, 5.5 cm, of the 
source location in the applicator was placed on the (0,0) point. 
44 
 
University of Ghana  http://ugspace.ug.edu.gh
CPs were generated with a spacing of 1 cm apart on the x-axis to the left and right of 
the  
y-axis. A prescribed dose of 2 Gy was normalised to the x = 1 cm CP which is the first 
point from the origin. The TPS generated a dose distribution and the doses to the 
various CPs were recorded. The same procedure was repeated for the (0,1), (0,2), (0,-
1) and (0,-2) cm points as clearly illustrated in figures 3.5a to 3.5e. 
      Source 
      Control Point 
 
Figure 3.5 a: Source dwell position at y = 0 cm 
45 
 
University of Ghana  http://ugspace.ug.edu.gh
      Source 
      Control Point 
 
Figure 3.5 b: Source dwell position at y = 1 cm 
      Source 
      Control Point 
 
Figure 3.5 c: Source dwell position at y = 2 cm 
46 
 
University of Ghana  http://ugspace.ug.edu.gh
      Source 
      Control Point 
 
Figure 3.5 d: Source dwell position at y = -1 cm 
      Source 
      Control Point 
 
Figure 3.5 e: Source dwell position at y = -2 cm 
47 
 
University of Ghana  http://ugspace.ug.edu.gh
 
3.7 AAPM TG 43 DOSE CALCULATION FORMALISM 
The TG43 report (Nath et al, 1995, Rivard J et al, 2004) recommended the dose 
calculation algorithm for establishing the two-dimensional dose rate distribution in a 
water medium around cylindrically symmetric photon-emitting brachytherapy 
sources. The TG 43 formalism, which is well expatiated in chapter two, is a more 
advantageous method because it contains quantities or parameters measured solely in 
the medium. The data for a specific source can be compiled as a function of position 
and the effects of several physical factors on the value of the dose rate distribution are 
considered separately. (Kyeremeh, 2011) 
3.8 COMPARISON OF SIEVERT INTEGRAL VALUES AND HDRplus 
RESULTS 
The dose obtained from the two methods were analysed by producing a bar chart  of 
dose against control point for each dwell position. In addition to the bar chart, two 
curves on the same plane for both methods, per dwell position were also plotted. 
These graphical representations were done for better understanding of the deviation of 
the outcomes of the two procedures. 
Percentage deviations of the Sievert integral dose values from that of the HDRplus 
which was designed based on the TG 43 algorithm, were calculated using the relation 
|            |
                 
        
Where     = Dose from Sievert integral 
          = Dose from HDRplus  
 
48 
 
University of Ghana  http://ugspace.ug.edu.gh
CHAPTER FOUR 
 
RESULTS AND DISCUSSION 
 
4.1 HDRplus Dose Computations 
From the procedures spelt out in section 3.6, the acquired results for the individual 
control points are shown in the tables 4.1 to 4.5 and the corresponding isodose lines in 
figures 4.1 to 4.5. 
Table 4 1:  CP parameters for Source at y = 0 cm  
Prescribed Dose (Gy) 2.00  
Total Dwell Time (s) 40.25 
 Dwell Position (cm) (0,0) 
2
Total Reference Air Kerma (cGy.m ) 0.01827 
Minimum Dose (Gy) 0.05  
Average Dose (Gy) 0.49 
Maximum Dose (Gy) 2.00 
 
Table 4 2: Dose Control Point Report for Source at y = 1 cm 
 
Prescribed Dose (Gy) 2.00 
Total Dwell Time (s) 78.68 
 
Dwell Position (cm) (0,1) 
2
Total Reference Air Kerma (cGy.m ) 0.03572 
 Minimum Dose (Gy) 0.1 
 Average Dose (Gy) 0.6 
Maximum Dose (Gy) 2.00 
 
  
    
 
 
49 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Table 4 3: Dose Control Point Report for Source at y = 2 cm 
 
Prescribed Dose (Gy) 2.00 
 
Total Dwell Time (s) 202.72 
 Dwell Position (cm) (0,2) 
2
Total Reference Air Kerma (cGy.m ) 0.09203 
 
Minimum Dose (Gy) 0.23 
 Average Dose (Gy) 0.84 
Maximum Dose (Gy) 2.00 
 
 
Table 4 4:  Dose Control Point Report for Source at y = -1 cm 
Prescribed Dose (Gy) 2.00 
Total Dwell Time (s) 81.81 
Dwell Position (cm) (0,-1) 
2
Total Reference Air Kerma (cGy.m ) 0.03714 
Minimum Dose (Gy) 0.1 
Average Dose (Gy) 0.61 
Maximum Dose (Gy) 2.01  
 
Table 4 5: Dose Control Point Report for Source at y = -2 cm 
 Prescribed Dose (Gy) 2.00 
 Total Dwell Time (s) 208.64 
Dwell Position (cm) (0,-2) 
 
Total Reference Air Kerma (cGy.m^2) 0.09472 
 Minimum Dose (Gy) 0.24 
 Average Dose (Gy) 0.85 
Maximum Dose (Gy) 2.01 
 
50 
 
University of Ghana  http://ugspace.ug.edu.gh
  
Figure 4. 1: Isodose lines for source Figure 4. 2: Isodose lines for source 
at y = 1 cm at y = -1 cm
 
  
Figure 4. 3: Isodose lines for source Figure 4. 4: Isodose lines for source 
at y = 2 cm at y = -2 cm
 
Figure 4. 5: Isodose lines for source at y=0 cm
51 
 
University of Ghana  http://ugspace.ug.edu.gh
4.2 SIEVERT INTEGRAL RESULTS 
The computations from the Sievert integral yielded results recorded in tables for each 
control point. The dose outcome to control points equidistant and at equal subtended 
angles from the source, which were to the left and right of the source were equal. 
Therefore, x values represented in the following tables, for this section, are the 
absolute values of x. 
The Sievert integral could not be used in calculating the dose delivered to the various 
control points when the source was positioned at the perpendicular bisector, i.e. at (0, 
0), and this was because the angle or value of θ is equal to zero. 
 
4.3 ANALYSIS OF RESULTS 
The means of the doses recorded at each control with equal distance and oblique 
directions from the source were calculated to carry out comparison of the obtained 
dose values from both methods under consideration. Tables of values showing both 
sets of dose values for each source dwell positions and the corresponding deviations. 
Table 4 6: Comparison of dose values for both methods for source at y = 1cm 
Sievert_Dose(Gy) HDR_Plus_Dose(Gy) % Deviation 
1.9036 1.9950 4.58 
0.7603 0.7800 2.53 
0.3752 0.3800 1.26 
0.2170 0.2200 1.36 
0.1386 0.1400 1.00 
0.0953 0.1000 4.70 
 
52 
 
University of Ghana  http://ugspace.ug.edu.gh
 
Table 4 7: Comparison of dose values for both methods for source at y = 2 cm 
Sievert_Dose(Gy) HDR_Plus_Dose(Gy) % Deviation 
1.7854 1.9950 10.51 
1.1702 1.2400 5.63 
0.7225 0.7500 3.67 
0.4638 0.4800 3.38 
0.3137 0.3200 1.97 
0.2230 0.2300 3.04 
 
Table 4 8: Comparison of dose values for both methods for source at y = -1 cm 
Sievert_Dose(Gy) HDR_Plus_Dose(Gy) % Deviation 
1.9793 2.0050 1.28 
0.7905 0.7900 0.06 
0.3901 0.3900 0.03 
0.2256 0.2300 1.91 
0.1442 0.1500 3.87 
0.0990 0.1000 1.00 
 
Table 4 9: Comparison of dose values for both methods for source at y = -2 cm 
Sievert_Dose(Gy) HDR_Plus_Dose(Gy) % Deviation 
1.8375 2.0050 8.35 
1.2043 1.2500 3.66 
0.7436 0.7600 2.16 
0.4774 0.4900 2.57 
0.3229 0.3300 2.15 
0.2295 0.2400 4.37 
 
Subsequently, bar charts and curves were plotted to illustrate how clearly the two 
methods’dose outcome deviate from each other with respect to the control points. 
 
53 
 
University of Ghana  http://ugspace.ug.edu.gh
A graph of control points against doses for source at y = 1cm 
2.5000
Sievert_Dose(Gy)
HDR_Plus_Dose(Gy)
2.0000
1.5000
1.0000
0.5000
0.0000
1 2 3 4 5 6
Control point, x (cm) 
 
Figure 4. 6: Bar charts comparing dose outputs for source at y = 1cm 
A graph of control points against doses for source at y = 1cm 
2.5000
2.0000 Sievert_Dose(Gy)
1.5000
1.0000
0.5000
0.0000
0 2 4 6 8
Control point (cm) 
 
Figure 4. 7: Curves to compare dose outputs for source at y = 1cm 
54 
 
Dose (Gy) Dose (Gy) 
University of Ghana  http://ugspace.ug.edu.gh
A graph of control points against doses for source at y = 2 
cm 
 
2.5
2
1.5
Sievert_Dose(Gy)
1
HDR_Plus_Dose(Gy)
0.5
0
1 2 3 4 5 6
Control point, x (cm) 
 
Figure 4. 8: Bar charts comparing dose outputs for source at y = 2cm 
A graph of control points against doses for source at y = 2 
cm 
2.5
2
1.5
Sievert_Dose(Gy)
1
HDR_Plus_Dose(Gy)
0.5
0
0 2 4 6 8
Control point, x (cm) 
 
Figure 4. 9: Curves to compare dose outputs for source at y = 2 
55 
 
Dose (Gy) Dose (Gy) 
University of Ghana  http://ugspace.ug.edu.gh
A graph of control points against doses for source at y = -1 
cm 
2.5000
2.0000
1.5000
Sievert_Dose(Gy)
1.0000
HDR_Plus_Dose(Gy)
0.5000
0.0000
1 2 3 4 5 6
Control Points, x (cm) 
 
Figure 4. 10: Bar charts comparing dose outputs for source at y =- 1cm 
A graph of control points against doses for source at y = -1 
cm 
2.5000
2.0000
1.5000
Sievert_Dose(Gy)
1.0000
HDR_Plus_Dose(Gy)
0.5000
0.0000
0 2 4 6 8
Control Points, x (cm) 
 
Figure 4. 11: Curves to compare ose outputs for source at y = -1cm 
 
56 
 
Dose, D (Gy)  Dose, D (Gy) 
University of Ghana  http://ugspace.ug.edu.gh
A graph of control points against doses for source at y = -2 cm 
2.5000
2.0000
1.5000
Sievert_Dose(Gy)
1.0000
HDR_Plus_Dose(Gy)
0.5000
0.0000
1 2 3 4 5 6
Control point, x (cm) 
 
Figure 4. 12: Bar Charts for comparing dose outputs at y = -2cm 
A graph of control points against doses for source at y = -2 cm 
2.5000
2.0000
1.5000
Sievert_Dose(Gy)
1.0000
HDR_Plus_Dose(Gy)
0.5000
0.0000
0 2 4 6 8
Control point, x (cm) 
 
Figure 4. 13: Curves to compare dose outputs at y = -2cm 
 
57 
 
Dose (Gy) Dose (Gy) 
University of Ghana  http://ugspace.ug.edu.gh
4.3 DISCUSSION 
The observations made from the results obtained from the TPS clearly showed a 
general equality in dose values for points with the same distance and oblique 
directions from the source. However, this was not the case for the x = 1 cm control 
point (CP) readings for the source at the (0,-2), (0,-1), (0,1) and the (0,2) cm x-y co-
ordinate location. The 100% mark to which the dose normalisation to x = 1 cm was 
done was only achieved for the scenario where the source was positioned at the 
perpendicular bisector or at the (0, 0) location, while a dose percentage of 99.5% and 
99.6% of the prescribed 2 Gy was recorded for the (0,1) and (0,2) cm source  
dwell-positions respectively. When the source was positioned at the (0,-1) and (0,-2) 
cm  
dwell-positions, the dose percentage, with respect to the 2 Gy prescribed, recorded 
was 100.5% and 100.4% respectively. These deviations from the expected outcome 
(100%) could be attributed to the influence of the encapsulation design as shown in 
figure 3.3. The non-uniformity in the distance from the core of the sealed source to 
the external part of the capsule could have resulted in the slight increase in the 
distance of the Co-60 source from the CP. Equality in the deviations for equidistant 
and equiangular points was expected but the greater thickness of the head of the 
capsule could have contributed to this variation and this resulted in the increment in 
the source dwell-time when the source was positioned below the perpendicular 
bisector. This is because the shielding of source was greater, hence a greater treatment 
time, 81.81 seconds for (0,-1) cm and 208.64 seconds for (0,-2) cm, was required to 
attain the prescribed dose as compared to the 78.68 seconds and 202.72 seconds used 
for the (0,1) cm and (0,2) cm respectively.  
 
58 
 
University of Ghana  http://ugspace.ug.edu.gh
The Sievert integral technique, as stated earlier, was incapable of calculating the dose 
for the source positioned at (0,0) cm because the angle subtended with reference to 
the CPs was 0 degrees. However, it calculated the dose for all other source dwell 
positions, producing dose values that were in acceptable ranges in comparison to what 
was expected. Early approximations and rounding off of figures resulted in an 
increment in the error produced; hence the decision to work with figures with at least 
four places of decimal to increase the accuracy of produced results, and further 
reduces the percentage deviation. The simplicity and speed of the MATLAB program 
utilised in carrying out the calculations enabled the dose computations per control 
point to take an average period of 30 seconds. Also, the magnitude of x and y values 
were used in the computations to produce positive dose values (as a negative dose 
value would be meaningless); hence the dose value for CPs equidistant and 
equiangular from the source dwell-position were equal. 
The general equality in the dose output at CPs equidistant and equiangular for both 
methods resulted in the use of the mean of the doses from each pair of points to 
compare and analyse the results. The dose outputs from both methods were generally 
close with less percentage deviation. It was observed that for extreme oblique 
directions, the percentage deviation of the Sievert integral dose output from the TG43 
based HDRplus TPS was larger as compared to points of lesser angular value. This 
verifies what Williamson et al (1983) established, that the Sievert approach introduces 
significant errors and practically breaks down in the extreme oblique directions. 
This is clearly shown from the variation in the heights of the bar charts in figures 4.6, 
4.8, 4.10 and 4.12. The slightly large height difference between the bar for the x = 1 
cm CP for the (0,2) cm and (0,-2) cm which represents a percentage deviation of 
10.51% and 8.35% respectively.  
59 
 
University of Ghana  http://ugspace.ug.edu.gh
In exception of very high deviations obtained from the extreme oblique directions, the 
Sievert integral dose output for the Co-60 has shown to be a reliable method to 
perform a quality control on the HDRplus TPS. This is evident in the overlap of the 
curves plotted for dose against distance for both methods, most especially for the (0,-
1) cm source dwell-position.  
The deviations recorded for the (0,-1) cm dwell-position had a range of 0.03 – 3.87% 
as compared to its counterpart, i.e. (0,1) cm dwell position with a range of 1.00 – 
4.70% could be used to assume that the Sievert integral produces much accurate dose 
values with a higher treatment time. This could be confirmed with the range for the 
(0,2) cm and (0,-2) cm dwell position. 
 
 
 
 
 
 
 
 
 
60 
 
University of Ghana  http://ugspace.ug.edu.gh
CHAPTER FIVE 
 
CONCLUSION AND RECOMMENDATION 
 
5.1 CONCLUSION 
The model of the Sievert integral used in this research alongside the approximations 
applied, i.e. the Meisberger polynomial, has shown to be a valid model to calculate 
the dose distribution around the new BEBIG Co-60 except for the fact that extreme 
oblique directions need to be exempted in case it will be utilised for clinical QC 
procedures.  
The percentage deviation range of the dose output from the Sievert integral in 
comparison to that of the output from the TPS is 0.03 – 10.51% with a mean of 3.13% 
 o o
for angle range of 0  < θ < 70 . The Sievert integral broke down at respectively large 
o
angles, which in this case was from above 60 . Therefore, neglecting the breaking 
point, i.e.      , the range becomes 0.03 – 5.63% with a mean value of 2.55% for 
 o o
the angle range of 0  < θ < 48 . 
 
5.2 CLINICAL INTERPRETATION 
Following the ICRU report 24 that the percentage deviation of dose values calculated 
should not exceed ±5% of the true value, the Sievert integral has proven to be a valid 
tool for quality control of the HDRplus using the range with less error. Also, for 
purposes of radiation protection and patient safety, the TG43 formalism has shown 
superiority because the Sievert integral required more exposure or treatment time to 
yield the expected dose in most cases. 
61 
 
University of Ghana  http://ugspace.ug.edu.gh
5.3 RECOMMENDATION 
5.3.1 FOR THE RESEARCH COMMUNITY 
For further research, more CPs should be considered at points away from the 
horizontal axis, so as to observe for possible consistency in the correlation of the dose 
outputs between the two methods. This also has to be with source dwell positions 
equally positioned at points away from the vertical axis, for single and multiple 
sources as well. 
The consistency that might be obtained could yield a factor that will be incorporated 
in the Sievert integral model implemented, for possible elimination of errors. 
In addition to the variation in position of sources digitally, measurements should also 
be carried out in a medium of choice so that three set of outputs would be obtained for 
a much more comprehensive analysis of the outcomes. 
5.3.2 FOR THE CLINICAL COMMUNITY 
It should be ensured that points located equidistant and equiangular from the source 
should receive equal dose during treatment through ensuring that the source 
encapsulation is uniform radially. 
The Sievert integral has proven to be quite an accurate tool for calculation of dose 
around a Co-60 source just as established for Ir-192, it will therefore be recommended 
that a facility using the new Bebig Co-60 source would consider implementing the 
model of the Sievert integral used for this research to carryout quality control 
procedures on their HDRplus TPS. 
 
 
62 
 
University of Ghana  http://ugspace.ug.edu.gh
REFERENCES 
 
Acquah G. F., (2011). Fitting And Benchmarking Of Monte Carlo Output Parameters 
 For Ir-192 High Dose Rate Brachytherapy Source (MPhil Thesis).  University 
 of Ghana, Accra, Ghana. 
Baltas, D., Karaiskos, P., Papagiannis, P., Sakelliou, L., Loeffler E., and Zamboglou 
 N. (2001) Beta versus gamma dosimetry close to Ir-192 brachytherapy 
 sources, Med. Phys., 28:1875,. 
Baltas D., Zamboglou N., Sakelliou L., (2007) The Physics of Modern Brachytherapy 
 For Oncology; Series in Medical Physics and Biomedical Engineering; CRC 
 Press Taylor & Francis Group 6000 Broken Sound Parkway NW, pp 393-396. 
Bhola S.  et al (2012) An analytic approach to the dosimetry of a new BEBIG Co-60 
 HDR Brachytherapy Source, Journal of Medical Physics, doi: 10.4103/0971-
 620399228, India 
Cohen G. N., Amols H. I., Zaider M. (2000) An independent dose-to point calculation 
 program for the verification of high-dose-rate brachytherapy treatment 
 planning. Int J Radiation Oncology Biology & Physics; 48:1251–1258. 
Daskalov G. M., Loffler E., Williamson J. F. (1998), Monte Carlo–aided dosimetry of 
 a new high dose-rate brachytherapy source. Journal of Medical Physics; 1998; 
 25:2200–2208. 
Hahn B. D., Valentine D. T.,(2007) Essential MATLAB for Engineers and Scientists 
rd
 (3  Edition), Elsevier Ltd, Oxford, U. K. 
International Atomic Energy Agency (IAEA) textbook (2007) “Radiation oncology 
 physics: a handbook for teachers and students”, chapter 13, Brachytherapy: 
 physical and clinical aspects. 373-375. 
63 
 
University of Ghana  http://ugspace.ug.edu.gh
Interstitial Collaborative Working Group (ICWG) (1990); Interstitial Brachytherapy 
 Physical, Biological and Clinical Considerations. Edited by Anderson L. L., 
 Nath R. and Weaver K. A. Raven – New York,  
International Commission on Radiation Units and Measurements, (1997) Dose and 
 volume specification for reporting interstitial therapy, ICRU Report 58.ICRU, 
 Bethesda. 
Karaiskos P., Papagiannis P., Sakelliou L., Anagnostopoulos A., and Baltas D., 
125
 (2001) Monte Carlo dosimetry of the select Seed I interstitial brachytherapy 
 seed, Med. Phys., 28,  pp 1753 
Kemmerer T., Lahanas M., Baltas D., Zamboglou N., (2000) DVH Computation 
 Comparisons Using Conventional Methods And Optimized Fast Fourier 
 Transform Algorithm For Brachytherapy; Journal of Medical Physics 
Khan, F. M., (1994) Brachytherapy In: “The physics in radiation therapy “(William 
 M. P. ed), 2nd edition, pp 418-473. Williams & Wilkins, Baltimore, U.S.A.  
th
Khan F. M., (2010), The Physics of Radiation Therapy, Philadelphia (4  Edition), 
 Lippincott Williams & Wilkins, Philadelphia, U.S.A. 
Kline R. W., Earle J. D. (2007) Implementation of TG43 for dose prescription and 
192
 calculation of Ir Eye Plaques; Mayo Clinic, Rochester, MN update 9 
Kyeremeh P. O., (2011). Three Dimensional Implementation of Anisotropy Corrected 
 Fast Fourier Transform Dose Calculation Around Brachytherapy Seeds 
 (MPhil Thesis). University of Ghana, Accra, Ghana. 
Kyeremeh P.O., Nani E.K., Addison E.K.T., Hasford F., (2012) Implementation of 
 3DAnisotropy Corrected Fast Fourier Transform Dose Calculation around 
 Brachytherapy Seeds, International Journal of Science and Technology 
 Volume 2 No.3, ISSN 2224-3577 
64 
 
University of Ghana  http://ugspace.ug.edu.gh
125
Ling C. C.  et al: (1985) Two-Dimensional dose distribution of I, Journal of 
 Medical Physics, 1985: 12:652-655 
Lu L., (2013) Dose Calculation Algorithms in External Beam Photon Radiation 
 Therapy, International Journal of Cancer Therapy & Oncology; 1(2):01025.  
 DOI: 10.14319/ijcto.0102.5 
Meli J. A., Meigooni A. S., Narth R., (1988) On The Choice of phantom materials for 
 the dosimetry of Ir-192 sources. International Journal for radiation oncology, 
 biology and physics; 14:586-595. 
Meredith W.J, ed. (1947), The Manchester System, Livingstone, Edinburgh. 
Miften M., Wiesmeyer M., Monthofer S., et al, (1999) Implementation of FFT 
 Convolution and Multigrid Superposition Models In The FOCUS RTP 
 System, Physics, Medicine & Biology Journal; 45:817-833 
Nani E.K., Akaho E.H.K., Kyere A.W.K., Amuasi J.H. et al (2009); Approximating 
 Sievert Integrals to Monte Carlo methods to calculate dose rate distributions 
 around Ir-192 brachytherapy source, Journal of Applied Science & 
 Technology, Vol. 14, page 27-31. 
Nath R, Anderson L, Luxton G, Weaver A, Williamson F, Meigooni S. (1995) 
 Dosimetry of interstitial brachytherapy sources Recommendations of the 
 AAPM Radiation Therapy Committee Task Group 43. Med Phys.; 22:209-34 
 [PubMed] 
Nath R, Park C H, King C R and Muench P (1990) A dose computation model for 
 241Am vaginal applicators including the source-to-source shielding effects 
 Med. Phys. 17 833–42 
65 
 
University of Ghana  http://ugspace.ug.edu.gh
Neuenschwander H., Machie T. R., Reckwerdt P. J. (1995); MMC-A High 
 Performance  Monte Carlo Code For Electron Beam Treatment Planning.  
 Physics, Medicine & Biology Journal; 40:543-574 
Oelke U., Scholz C., (2006). New Technologies In Radiation Oncology, Medical 
 Radiology, Part 3, pp 187-196, DOI: 10:1007/3-540-29999-8_15 
Oguchi H. et al (2005). Advantage of Multiple Algorithm In Treatment Planning 
 System For External Beam Dose Calculation   
Pantelis E., Baltas D, Dardoufas K, et al, (2002) On the accuracy of the Sievert 
 Integral Model in the proximity of Ir-192 HDR sources, Int J. Radiation 
 oncology Biol. Phys, Vol 53, No 4, pp 1071-1084 
Papanikolaou N., Battista J. J., Boyer A. L., Kappas C., Klein E., et al (2004) Tissue 
 Inhomogeneity Correction For Megavoltage Photon Beams, An AAPM Report 
 No. 85, Report of task group No. 65 of the radiation therapy committee of 
 AAPM, pp 47-64 
Papanikolaou N., Stathakis S., (2009), Dose Calculation Algorithms In Context of 
 Inhomogenous Correction For High Energy Photon Beams, Department of 
 Radiation Oncology , Cancer Therapy  and Research Center , University of 
 Texas Health Sciences Center, San Antonnio, Texas 78229, 
 USA,36(10):4765-75 
Podgorsak E.B., (2005). Radiation Oncology Physics: A Handbook For Teachers And 
 Students, International Atomic Energy Agency, Vienna, Austria 
Quimby, E.H. (1922) The effect of the size of radium applicators on skin doses, 
 Am. J. Roentgenol., 9, 671 – 683. 
66 
 
University of Ghana  http://ugspace.ug.edu.gh
Ravinder N., et al, (1994) dosimetry of interstial brachytherapy sources: 
 recommendations of the AAPM radiation therapy Committee Task Group 
 No.43, Journal of Medical Physics; 22:; 210 
Rivard J, Coursey M, Dewerd A, Hanson F, Huq S, Ibbott S, et al. (2004) Update of 
 AAPM Task Grooup 43 Report: A revised AAPM protocol for brachytherapy 
 dose calculations. Med Phys.; 31:633-744. [PubMed] 
Schell M. C., Ling C. C., Gromadzki Z. C., Working K. R. (1987); Dose distribution 
125
 of model 6702 I seeds in water; International Journal of Radiation 
 Oncology, Biology and Physics:13; 795-799 
Sievert, R. (1921) Die Intensita¨ tsverteilung der primaeren Gammastrahlung in der 
 Naehe medizinscher Radiumpra¨parate, Acta. Radiol., 1, 89 – 128. 
th
Sievert Integral, http://en.wikipedia.org/wiki/Sievert_integral.html [18  March, 2017] 
Wang, R. and Li, X.A. (2000) A Monte Carlo Calculation Of Dosimetric Parameters 
 Of 90Sr/90Y And 192Ir SS Sources For Intravascular Brachytherapy, Med. 
 Phys., 27, 2528 
125 103
Weaver K. (1998) Anisotropy functions for I and Pd sources, Med. Phys., 25, 
 2271. 
125 192
Weaver K. A., et al, (1989) Dose Parameters of I and Ir seed sources, Journal of 
 Medical Physics, 16:636-643 
Wickham, L. and Degrais, P. (1910) Radiumtherapy, English ed., Cassell, London,. 
Williamson, J.F., Coursey, B.M., DeWerd, L.A., Hanson, W.F., and Meigooni, A.S. 
 (1998) Dosimetric Prerequisites For Routine Clinical Use Of New Low 
 Energy Photon Interstitial Brachytherapy Sources, Med. Phys., 25, 2269,. 
Williamson J. F., Morin R.L., Khan F. M. (1983) Monte Carlo evaluation of Sievert 
 integral for brachytherapy dosimetry. Phys Med Biol. 28:1021 
67 
 
University of Ghana  http://ugspace.ug.edu.gh
Williamson J. F., (1996) The Sievert integral revisited: Evaluation and extension to I-
125, Yb-169, and Ir-192 brachytherapy sources, Int. Journal of radiation oncology 
Biol Phys 36:1239-1250 
World Health Rankings; Health Profile: Ghana, http://www.worldlifeexpectancy. 
th
 com/country-health-profile/ghana.html, [12  January, 2017] 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
68 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDICES 
APPENDIX A:  Dose Control Point Report for source at y = 0 cm 
 
 
 
 
 
 
 
 
 
 
69 
 
University of Ghana  http://ugspace.ug.edu.gh
 
APPENDIX B: Dose Control Point Report for source at y = 1 cm 
 
 
70 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX C: Isodose report for source at y = 1cm 
 
 
71 
 
University of Ghana  http://ugspace.ug.edu.gh
 
APPENDIX D: Dose Control Point Report for source at y = 2 cm 
 
 
 
 
72 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX E: Isodose Report for source at y = 2 cm 
 
 
 
 
73 
 
University of Ghana  http://ugspace.ug.edu.gh
 
APPENDIX F: Dose Control point Report for source at y = -1 cm 
 
 
 
 
 
74 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX G: Isodose Report for source at y = -1 cm 
 
 
75 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX H: Dose control point Report for source at y = -2 cm 
 
 
 
 
 
 
 
 
 
76 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX I: Isodose Report for source at y = -2 cm 
 
 
77 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX J: The New BEBIG Co-60 HDR Source Details 
  
 
78 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX K: MATLAB SYNTAX 
% This program calculates the dose distribution around 
the High Dose Rate  
% (HDR) Multisource Afterloader of the New Bebig Machine 
% Source: Cobalt-6o 
% Number of control points (points of measurement): 12 
% Number of source positions: 5 
% y = (-2, -1, 0, +1, +2) 
% Parameters used and their corresponding units are: 
% D = Dose (Gy) 
% Sk = Air Kerma Strength (cGy.cm^2.h^-1) 
% L = active source length (m) 
% y = source posoition (cm) 
% mu = effective attenuation coefficient (cm^-1) 
% t = wall thickness (cm) 
% x = control point (cm) 
% t_dwell = dwell time (seconds) 
% W/e = average enrgy per unit charge (cGy.kg/C) 
% fmed = roentgen-to-rad conversion factor or f factor 
(Gy.kg/C)  
% SIV = Sievert Integral value for each angle 
% SPR = scatter-to-primary ratio 
clear all 
close all 
clc 
x = input('Enter the value of x: ') 
  
y1 = input('Enter the value of y1: ') 
y2 = input('Enter the value of y2: ') 
  
r1 = sqrt((x^2)+(y1^2)); 
r2 = sqrt((x^2)+(y2^2)); 
  
SIV1 = input('Enter the Sievert Integral Value for y1: ') 
SIV2 = input('Enter the Sievert Integral Value for y2: ') 
t_dwell = input ('Enter the source dwell time: ') 
time = t_dwell/3600; 
  
A = 0.99423; B = -0.005318; 
C = -0.002610; D = 0.0001327;  
j = exp(mu*t); 
num = fmed*Sk*time*j; 
den = L*x*We;  
D1 = (num/den)*SIV1*SPR1 
D2 = (num/den)*SIV2*SPR2  
Doses = [D1 D2] 
diff = D2-D1 
 
 
79 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX L: General Sievert Integral Table 
 
 
 
 
 
80 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX M: Sievert Integral Table for angles of CPs 
Sievert Integral Values for θ 
x/θ 9.46 11.31 14.04 18.43 21.8 26.57 33.69 45 63.43 
0.125 0.16314 0.19503 0.24209 0.31777 0.37539 0.458 0.58083 0.77579 1.08986 
 
 
Sievert Integral Values for θ1 
x/θ1 7.83 9.37 11.65 15.38 16.92 20.05 22.42 24.52 31.31 39.52 42.38 61.28 
0.125 0.13504 0.16159 0.20089 0.26519 0.29174 0.34569 0.38652 0.42269 0.53968 0.68228 0.73125 1.05345 
 
 
Sievert Integral Values for θ2 
x/θ2 11.08 13.22 16.37 19.93 21.39 23.51 28.54 30.43 35.94 47.4 49.6 65.31 
0.125 0.19106 0.22796 0.28226 0.34362 0.36877 0.40529 0.49193 0.5245 0.61953 0.81719 0.854 1.12108 
81 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX N: Dose Control Point Report as computed using the Sievert integral for source at y = 1cm 
θ1 θ2 
x (cm) y (cm) y1 (cm) y2 (cm) θ (deg) (deg) (deg) SIV1 SIV2 D1(Gy) D2(Gy) Sievert_Dose(Gy) 
1 1 0.825 1.175 45.00 39.52 49.60 0.682276 0.853998 7.6771 9.5807 1.9036 
2 1 0.825 1.175 26.57 22.42 30.43 0.386516 0.524497 2.1494 2.909700 0.7603 
3 1 0.825 1.175 18.43 15.38 21.39 0.265190 0.368774 0.9675 1.3427 0.3752 
4 1 0.825 1.175 14.04 11.65 16.37 0.200890 0.282256 0.5391 0.7561 0.2170 
5 1 0.825 1.175 11.31 9.37 13.22 0.161587 0.227955 0.3393 0.4779 0.1386 
6 1 0.825 1.175 9.46 7.83 11.08 0.135040 0.191064 0.2307 0.326 0.0953 
 
APPENDIX O: Dose Control Point Report as computed using the Sievert integral for source at y = 2 cm 
x (cm) y (cm) y1 (cm) y2 (cm) θ (deg) θ1 (deg) θ2 (deg) SIV1 SIV2 D1(Gy) D2(Gy) Sievert_Dose(Gy) 
1 2 1.825 2.175 63.43 61.28 65.31 1.053450 1.121084 30.2248 32.0102 1.7854 
2 2 1.825 2.175 45.00 42.38 47.40 0.731250 0.817187 10.3854 11.5556 1.1702 
3 2 1.825 2.175 33.69 31.31 35.94 0.539675 0.619531 5.0344 5.7569 0.7225 
4 2 1.825 2.175 26.57 24.52 28.54 0.422688 0.491933 2.9029 3.3667 0.4638 
5 2 1.825 2.175 21.80 20.05 23.51 0.345692 0.405291 1.8592 2.1729 0.3137 
6 2 1.825 2.175 18.43 16.92 19.93 0.291737 0.343624 1.2773 1.5003 0.223 
82 
 
University of Ghana  http://ugspace.ug.edu.gh
APPENDIX P: Dose Control Point Report as computed using the Sievert integral for source at y = -1 cm 
θ1 θ2 
x (cm) y (cm) y1 (cm) y2 (cm) θ (deg) (deg) (deg) SIV1 SIV2 D1(Gy) D2(Gy) Sievert_Dose(Gy) 
1 -1 0.825 1.175 45.00 39.52 49.60 0.682276 0.853998 7.9825 9.9618 1.9793 
2 -1 0.825 1.175 26.57 22.42 30.43 0.386516 0.524497 2.2349 3.0254 0.7905 
3 -1 0.825 1.175 18.43 15.38 21.39 0.265190 0.368774 1.0060 1.3961 0.3901 
4 -1 0.825 1.175 14.04 11.65 16.37 0.200890 0.282256 0.5606 0.7862 0.2256 
5 -1 0.825 1.175 11.31 9.37 13.22 0.161587 0.227955 0.3528 0.497 0.1442 
6 -1 0.825 1.175 9.46 7.83 11.08 0.135040 0.191064 0.2399 0.3389 0.0990 
 
APPENDIX Q: Dose Control Point Report as computed using the Sievert integral for source at y = -2 cm 
x (cm) y (cm) y1 (cm) y2 (cm) θ (deg) θ1 (deg) θ2 (deg) SIV1 SIV2 D1(Gy) D2(Gy) Sievert_Dose(Gy) 
1 -2 1.825 2.175 63.43 61.28 65.31 1.053450 1.121084 31.1075 32.945 1.8375 
2 -2 1.825 2.175 45.00 42.38 47.40 0.731250 0.817187 10.6887 11.893 1.2043 
3 -2 1.825 2.175 33.69 31.31 35.94 0.539675 0.619531 5.1814 5.925 0.7436 
4 -2 1.825 2.175 26.57 24.52 28.54 0.422688 0.491933 2.9877 3.4651 0.4774 
5 -2 1.825 2.175 21.80 20.05 23.51 0.345692 0.405291 1.9135 2.2364 0.3229 
6 -2 1.825 2.175 18.43 16.92 19.93 0.291737 0.343624 1.3147 1.5442 0.2295 
 
83