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Fourier transform Raman spectroscopy of polyacrylamide gels for use in radiation dosimetry Jirasek, Andrew 2002

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F O U R I E R T R A N S F O R M R A M A N S P E C T R O S C O P Y OF P O L Y A C R Y L A M I D E GELS FOR USE IN R A D I A T I O N D O S I M E T R Y by Andrew Jirasek B . S c , University of Calgary, 1994 M . S c , University of Guelph, 1996 A DISSERTATION S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Philosophy in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Physics and Astronomy We accept this dissertation as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A May 2002 © Andrew Jirasek, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date K ffej ^hol DE-6 (2/88) Abstract Polyacrylamide gel dosimeters are three dimensional, tissue equivalent, dose inte-grating, high spatial resolution dosimeters that show promise for use as dose ver-ification tools in radiotherapy. To date, however, implementation of this class of dosimeters has been limited, partly due to the lack of complete understanding of the gel dose response mechanisms. The aim of this thesis is to further the under-standing of chemical changes occurring in irradiated polyacrylamide gels. Fourier transform Raman spectroscopy is used to probe these chemical changes under a range of experimental conditions. Monomer consumption and polymer formation curves are constructed by cross-correlating relative Raman peak intensities for spec-tra acquired on individual gel samples which have each been irradiated to known doses. The results of the thesis are divided into three main categories. The first component involves preliminary work pertaining to establishing adequate experimental parameters. Sample housing, data acquisition parameters, data analysis, and spectral reproducibility are all examined. Reproducibility in spectral peak area is established to be better than 1% for spectra acquired on an individual gel sample, and ~1.5% or ~3.5% for spectra acquired on irradiated gels manufactured intra or inter batch, respectively. Secondly, monomer consumption and polymer formation is studied for gels ir-ii radiated with 6 MV x-rays. The monomer consumption data are extended to include gels of varying initial composition. Results indicate that monomer consumption is, in general, highly non-linear as a function of absorbed radiation dose. A qualitative model, based on the structure of formed polymer, is used to explain the differences between the consumption curves for the different gels. It is also shown that, within any given gel, the polymer structure varies as a function of absorbed dose. Finally, the dependence of the gel response on ionizing density is studied. Polymer gels are irradiated in two different regions of a 74 MeV spread out Bragg peak proton beam (i.e. two regions of differing incident particle linear energy transfer (LET)). Monomer consumption curves are compared with 6 MV x-ray irradiated gel curves, thus arriving at a gel "relative effectiveness" (RE) as a function of LET. The theory of track structure is used to model the system and predict the gel RE in these same two regions. Track structure calculations confirm the LET dependence of the gels' response, indicating that the dependence is primarily due to the saturation of gel sensitive elements close (~ 10~6 cm) to the proton track. Track calculations are extended to different experimental situations and to gels of varying initial composition. iii Contents A b s t r a c t i i L i s t of Tables x L i s t of F i g u r e s x i i A c k n o w l e d g e m e n t s x x i C h a p t e r 1 I n t r o d u c t i o n 1 1.1 Radiation Therapy Overview 2 1.1.1 Historical Background 2 1.1.2 Tumour and Healthy Tissue Response 4 1.1.3 Conformal Radiation Therapy 6 1.1.4 Modalities of Radiation Therapy 6 1.2 Basic Concepts 9 1.2.1 Absorbed Dose and Kerma 9 1.2.2 Photon Interactions with Matter 10 1.2.3 Charged Particle Interactions with Matter 15 1.3 Dosimetry Techniques 18 1.3.1 Ionization Chamber 18 iv 1.3.2 Diode Detector 20 1.3.3 Thermoluminescent Detectors 21 1.3.4 Film Dosimetry 22 1.3.5 Chemical Dosimetry 23 1.3.6 Gel Dosimetry 24 1.4 Dosimeter Requirements 27 1.4.1 Dose Resolution, Sensitivity and Accuracy 27 1.4.2 Spatial Resolution 28 1.4.3 Dose Integration 28 1.4.4 Tissue Equivalence 29 1.4.5 Anatomical Equivalence 30 1.5 Objectives of this Thesis 30 Chapter 2 Raman Spectroscopy 33 2.1 Historical Background 33 2.2 Theoretical Considerations 35 2.2.1 Origin of Raman Spectra 35 2.2.2 Origin of IR spectra 44 2.2.3 Raman Versus IR Spectra 45 2.2.4 Energy Units and Regions of Molecular Spectra 47 2.3 Experimental Considerations 47 2.3.1 Fourier Transform Spectrometer 49 2.3.2 Grating Spectrometer 55 2.3.3 Advantages and Disadvantages of Fourier Transform over Grat-ing Spectrometers 56 2.4 Applications of Raman Spectroscopy 58 v Chapter 3 Polyacrylamide Gel Dosimeters 60 3.1 Mechanisms of Polymer Formation 61 3.1.1 Water Radiolysis 61 3.1.2 Polymerization of Monomer 63 3.2 Polyacrylamide Gels 0 66 3.3 Factors Affecting Gel Polymerization and Stability 69 3.3.1 Oxygen 69 3.3.2 Temperature Effects 69 3.3.3 Post Irradiation Gel Stability 71 3.4 Literature Review 72 3.4.1 Studies on the Properties of Polyacrylamide Gels 72 3.4.2 Studies on the Applications of Radiation Dosimetry PAGs . . 76 3.5 Other 3-D Chemical Dosimeters 76 3.5.1 Fricke Dosimeters 76 3.5.2 Other Polymer Dosimeters 77 Chapter 4 Materials and Methods 79 4.1 Gel Manufacture 79 4.1.1 Glove Box 79 4.1.2 Gel Preparation 81 4.2 Gel Irradiation: Photons 82 4.2.1 Linear Accelerator 82 4.2.2 Experimental Method 83 4.3 Gel Irradiation: Protons 85 4.3.1 Cyclotron 85 4.3.2 Experimental Method 86 vi 4.4 Raman Spectroscopy of Polyacrylamide Gels 87 4.4.1 Raman Spectrometer 87 4.4.2 Raman Spectra Acquisition 88 4.5 Data Analysis 89 4.5.1 Background Subtraction 90 4.5.2 Characterization of Peak Intensity 91 Chapter 5 Results and Discussion I: Gel Response to X-Ray Irradi-ation 95 5.1 Initial Investigations 96 5.1.1 Housing of Gel Samples 96 5.1.2 Signal Intensity 96 5.1.3 Signal to Noise Ratio in Gel Spectra 97 5.2 Spectral Identification 99 5.2.1 Glass 100 5.2.2 Water 102 5.2.3 Gelatin 102 5.2.4 Acrylamide/Bis 102 5.3 Reproducibility Studies 103 5.3.1 Same Sample Reproducibility 104 5.3.2 Intra-batch Reproducibility 104 5.3.3 Inter-batch Reproducibility 106 5.4 Peak Fitting 107 5.5 Consumption of Monomer 110 5.6 Formation of Polymer 113 5.7 Compositional Studies 119 vii 5.7.1 Monomer Consumption Curves 120 5.7.2 Effects of Initial Bis Fraction on Gel Sensitivity 122 5.7.3 Effect of Initial Bis Fraction on Polymer Structure I: Overall Consumption Rate 124 5.7.4 Effect of Initial Bis Fraction on Polymer Structure II: Math-ematical Course of Consumption 128 5.8 Irradiation History 130 Chapter 6 Results and Discussion II: Gel Response to Proton Irra-diation 132 6.1 Introduction 133 6.2 Experimental Determination of Polymer Gel Relative Effectiveness . 134 6.2.1 Proton LET Distribution at Points of Measurement 134 6.2.2 Consumption Curves and Relative Effectiveness 136 6.3 Track Structure Determination of Polymer Gel Relative Effectiveness 138 6.3.1 Basic Theory of Track Structure 138 6.3.2 Radial Point Dose Distribution 141 6.3.3 Average Dose in Sensitive Element 144 6.3.4 Calculated RE 145 6.3.5 Extension of Calculation to Different Element Radii and Sen-sitivities 147 6.3.6 Extension of Calculation to Raw Bragg Peak 147 6.4 Discussion 149 Chapter 7 Conclusions 152 7.1 Conclusions 152 viii 7.2 Future Directions 156 Appendix A Data Handling Software 157 Bibliography 159 ix List of Tables 5.1 Raman frequencies for polymer gel constituents. Vibrations are: va = symmetric stretching, va = anti-symmetric stretching , 5 — bending, UJ — wagging, p = rocking. Molecules are: AA= acrylamide, BA= bis-acrylamide, PA= poly-acrylamide, GN= gelatin, W= water, GL= glass 101 5.2 Variance-covariance matrix for fit parameters of equation (5.1), fit to an unirradiated sample spectrum 109 5.3 Gel composition with the maximum consumption rate in the given dose range 123 6.1 Experimental and calculated relative effectiveness of polymer gel to proton irradiation using a 74 MeV SOBP. An a0 = 0.25 nm and D0 = 8 Gy were used in the track calculations 146 6.2 Calculated relative effectiveness of polymer gel to proton irradiation for different sensitive element radii (o0) and sensitivity parameters {D0) 148 x Calculated relative effectiveness of polymer gel to proton irradiation using a 74 MeV raw Bragg peak (BP). RE calculated at surface (i.e. just beyond glass wall) and end of raw Bragg peak. D0 = 8 Gy, a0 = 0.25 nm. Point of calculation is given as water equivalent depth. 149 xi List of Figures 1.1 Normal tissue and tumour response to radiation as a function of ab-sorbed dose, (a) The 'classical' diagram illustrating the higher sensi-tivity to radiation of tumour cells relative to normal tissue. In this scheme, judicious choice of treatment dose leads to high levels of tu-mour response with minimal damage to normal tissue. Also shown is the therapeutic ratio (D(,/D0, see text), the ratio of doses for a given response (horizontal line), (b) An alternative schematic of tumour and normal tissue response. Adapted from [9] 5 1.2 Schematic diagram illustrating a 'beams eye view' of different ways of conforming radiation to a target (shown in black), (a) collimator jaws (light grey) are closed close to the target and custom designed lead blocks (dark grey) are positioned below the collimators (blocks are shown in full for clarity), (b) multileaf collimators conform radiation more closely to the shape of the target, (c) moving leaves can create conformed intensity maps, shown by the grayscale image of the target. 7 1.3 Examples of the dose absorbed as a function of depth in a medium (water) for a (a) 6 MV photon beam, (b) 9 MeV electron beam, and (c) 74 MeV proton beam 8 Xll 1.4 (a) Schematic example of the number of electrons ejected from atoms as a function of depth in the medium. Decreasing numbers are caused by photon attenuation, (b) Kinetic energy released in the medium (kerma, dashed line) and dose absorbed in the medium (solid line). Surface contamination by electron backscatter has not been shown. Adapted from [11] 10 1.5 The photoelectric effect. A photon interacts with an inner bound atomic electron. Also shown are characteristic x-rays, caused when an outer electron moves to fill the inner orbital vacancy. 12 1.6 In a Compton interaction, an incident photon interacts with an loosely bound (assumed free) electron. Both the photon and electron are scattered 13 1.7 Schematic illustrating pair production. An incident photon interacts with the field of the nucleus, creating an electron and positron pair. 14 1.8 Relative importance of the three main photon interactions as a func-tion of atomic number and incident photon energy. The lines indicate regions of equal probability for each adjacent effect. Adapted from [13]. 15 1.9 Schematic of a typical thimble ionization chamber 19 1.10 Schematic of a typical diode detector. The power supply provides a 'reverse bias' voltage <~300 mV. Current generated by photon bombardment is measured with the meter 20 1.11 Basic electronic structure of a thermoluminescent detector, (a) Upon irradiation, electrons are excited and trapped within the electronic structure of an impurity, (b) By heating the device, the electrons can be liberated. They subsequently emit radiation which is then detected. 22 xiii 1.12 Overview of the gel dosimetry technique, (a) Gel preparation, (b) gel irradiation, (c) gel imaging, and (d) data analysis and image processing. 25 1.13 Example of (a) axial, (b) sagittal and (c) coronal x-ray CT images of a gel housed in a spherical flask and irradiated with a 4 arc stereotactic treatment. Reproduced from [24] 26 2.1 Schematic diagram of the Raman effect. Incident radiation of energy hu excites a molecule from the ground electronic state (\I/0) to a virtual state. Rayleigh, Stokes and anti-Stokes Raman scattering are shown 36 2.2 Normal Modes of vibration for CO2. a) symmetric stretch, b) sym-metric bend c) anti-symmetric stretch. "+" indicates vibration out of page, "-" indicates vibration into page 40 2.3 Polarizability ellipsoid (1/a) for the three normal modes of vibra-tion of CO2. a) symmetric stretch, b) symmetric bend and c) anti-symmetric stretch. The central configuration in each row corresponds to the molecule's equilibrium position 42 2.4 Schematic diagram showing the polarizability (a) as a function of atomic displacement q for the a) symmetric stretch and b) symmetric bend and anti-symmetric stretch of CO2. Equilibrium position is at q = 0 42 2.5 Change in dipole moment for the three normal modes of vibration of C02- a) symmetric stretch (no dipole moment), b) symmetric bend and c) anti-symmetric stretch. The central configuration in each row corresponds to the molecules equilibrium position. There is no net dipole moment in the equilibrium position 46 xiv 2.6 Relationship for three different sets of units used in Raman and/or IR spectroscopy. Shown are the ranges for the various physical processes. 48 2.7 Schematic diagram of a basic Michelson Interferometer 50 2.8 A sine function (equation (2.25)) whose central peak has a maximum width of A - 1 53 2.9 Illustration of aliasing. In the time domain, the waveform I(t) is sampled at intervals T by a "comb" function (7(f)). In the frequency domain, the Fourier transform of the initial waveform is the ideal spectrum (B(i>)) which is convolved with the FT of the sampling comb function (r(i/)) which has spacings of 1/T. Shown are the effects (aliasing) of an insufficient sampling interval (T), i.e. when 1/T < 2umax 54 3.1 Chemical structure of (a) acrylamide, (b) bis-acrylamide and (c) poly-acrylamide 68 4.1 Schematic diagram of glove box used in polymer gel manufacture, (a) left side, (b) right side and (c) front 80 4.2 Schematic diagram of components of a standard radiotherapy linear accelerator operating in photon mode. Diagram not to scale 83 4.3 Schematic diagram of sample tube phantom used for gel irradiations. Phantom material is acrylic. All dimensions in mm 84 4.4 Schematic diagram showing the medical beamline at TRIUMF, Van-couver, BC, Canada 86 xv 4.5 Experimental set-up for proton beam irradiations. Sufficient acrylic build-up is positioned in front of the acrylic phantom such that gels are irradiated with the (a) central portion of the SOBP and (b) end SOBP 87 4.6 Optical layout of Raman spectrometer 88 4.7 Sample holder designed to house 10 mm NMR sample tubes and mount to the commercially available 1-D translational stage. Sample is positioned through 10 mm bore. All dimensions in mm 89 5.1 Raman spectra of polymer gel housed in two different sample vials. Both are reported to be manufactured of boro-silicate glass. The gel signal is virtually completely masked by the fluorescence of sample vial #1 97 5.2 Unirradiated gel Raman signal intensity as a function of incident laser power 98 5.3 Signal to Noise Ratio (SNR) as a function of the number of interfer-ometer scans and laser power for spectra acquired on an unirradiated polymer gel 99 5.4 Raman spectrum of an unirradiated polacrylamide gel 100 5.5 Raman spectra of individual gel constituents and glass vial 101 5.6 Integrated peak intensity as a function of trial for a single vial of unirradiated gel 104 5.7 Reproducibility of acrylamide and bis peak area for gels manufactured from the same batch of gel (transferred to several vials) and irradiated to 5 Gy. 105 xvi 5.8 Reproducibility of acrylamide and bis peak area for gels manufactured from different gel batches and either left unirradiated, or irradiated to 5 Gy. 106 5.9 Raw acryalmide and bis Raman peak intensity as a function of ab-sorbed dose 107 5.10 Raw acrylamide and bis peaks fit to equation (5.1) 108 5.11 Fi t parameter errors (S) as a function of dose (i.e. decreasing SNR). u>2, A<i and k refer to parameters in equation (5.1) 109 5.12 F i t background subtracted from acrylamide and bis peak data. . . . 110 5.13 Acrylamide and bis monomer consumption curves for a 6%T, 50%C gel irradiated up to 50 Gy. Lines connecting experimental data points are for visual guidance I l l 5.14 Polymer formation observed in FT-Raman spectra of irradiated poly-mer gel. (a) 27 c m - 1 mode, origin unknown (see text), (b) 1126 c m - 1 C - C stretch, (c) 1450 c m " 1 C H 2 bend and (d) 2936 c m " 1 C H 2 stretching mode of polyacrylamide. Note the different y-axis intensi-ties in the plots 115 5.15 Polymer formation curve for a 6%T, 50%C gel irradiated up to 50 Gy. Lines connecting experimental data points are for visual guidance. 118 5.16 Consumption of acrylamide (acr) and bis in irradiated polymer gel dosimeters with different initial bis fractions: (a) 0%C, (b) 30%C, (c) 50%C, (d) 70%C and (e) 100%C. Figure (c) reproduced from figure (5.13) for ease of comparison. Lines connecting experimental data points are for visual guidance 121 xvi i 5.17 Consumption rates for (a) acrylamide and (b) bis monomers for the compositions and doses cited. Note the differing y-axis intensities. . 123 5.18 (a) Links created by combination of acrylamide monomer and bis crosslinker: (i) singlet, (ii) free radical linear chain, (iii) loop and (iv) doublet. Open circles indicate reactive sites. Closed circles denote ends, (b) Progression in polymer structure as a function of initial crosslinker concentration, (i) A "gel" solely composed of monomer (acrylamide). Long, linear chains are formed with no crosslinks, (ii) Gel composed of low initial bis fraction. Predominant gel formation is an ordered, crosslinked network, (iii) Gel composed of high initial bis fraction. Gels begin to form a larger number of knots, (iv) A gel composed solely of crosslinker (bis). Predominant structures are knots, loops and doublets which together form beads 126 5.19 The average number of acrylamide molecules per bis molecule avail-able for reaction as a function of absorbed dose and initial bis fraction (%C) 129 xviii 5.20 (a) Acrylamide and (b) bis consumption rate comparison for two gels: i) solid markers show consumption rate for an initial 6%T, 50%C gel (figure (5.16)c). ii) open markers, dashed line show consumption rate for an initial 2.3%T 38%C gel, corresponding to a concentration of unreacted monomer in 6%T, 50%C gel at 7 Gy. Thus, comparison is made for consumption rates in gels with (i) pre-existing polymer network present (radiation history) and (ii) no pre-existing polymer network present (no radiation history). Solid x-axis pertains to 6%T, 50%C gel. Dashed x-axis pertains to 2.3%T, 38%C gel. Lines con-necting experimental data points are for visual guidance 131 6.1 Depth dose curve of a 74 MeV proton beam in water, spread out over 23 mm using an acrylic modulator wheel. Shown are the raw Bragg peak curves (gray lines) obtained for protons traveling through each step in the modulator wheel. Relative heights of curves indicate the proton fluence weight of each step [165]. The spread out Bragg peak curve (dark line) is obtained from the fluence weighted sum of all the individual curves 135 6.2 The distribution and relative fluence contribution (weight) of protons with given LET present at the point of measurement for the mid SOBP (dark bars) and end SOBP (light bars) regions 135 6.3 Consumption of acrylamide in a polymer gel as a function of dose. Shown are gel consumption curves for x-ray and proton (mid and end SOBP region) irradiation. The x-ray irradiated gel consumption curve is reproduced from figure (5.13). Lines connecting experimental data points are for visual guidance 136 xix 6.4 Schematic representation of the sensitive detector element used in the track structure calculations. The point dose function is integrated over the volume of the sensitive element 144 6.5 The average dose deposited by electrons in an acrylamide molecule as a function of radial distance from a proton track. Sensitive element radius a0 = 0.25 nm in this calculation 146 6.6 Probability of activating a sensitive element as a function of radial distance from the proton track. Plot is for the thinnest step of the modulator wheel. a0 = 0.25 nm, D0 = 8 Gy in this calculation. . . . 147 A.l Software used to display and manipulate up to four spectra in parallel. 158 xx Acknowledgements I would like to thank Dr. Cheryl Duzenli for her supervision during the course of this thesis. She has provided me with an invaluable blend of support, encouragement, feedback and freedom alongside a truly enjoyable and creative research atmosphere. My supervisory committee of Dr. John Eldridge, Dr. Ellen Grein and Dr. Alex Mackay have contributed useful critical discussions during the course of the thesis. Dr. John Eldridge has provided the use of the Raman instrument and lab. Family and friends have provided continual friendship and support. I would like to thank: family Jirasek, family Hilts, Lauren MacArthur, Adam Monahan, James Pond, Karl Otto, James Robar and Jen Solmes. I thank my wife, Michelle, for her unwavering commitment to expressing the positive, as well as for her patience in listening to all the minute details of this work. Funding for this work has been provided by both the University of British Columbia and the BC Cancer Agency. A N D R E W J I R A S E K The University of British Columbia May 2002 xxi Chapter 1 Introduction The aim of cancer therapy by radiation treatment is to deliver a highly localized lethal dose of radiation to a tumour while sparing surrounding healthy tissue. Ra-diation dose distributions used in such treatments are increasing in complexity, in step with the technology used to deliver them. Three dimensional (3D) experi-mental verification of radiation therapy dose distributions remains an issue under investigation. To this end, a class of dosimeters have been introduced [1, 2] based on radiation induced chemical changes occurring in gel materials. In conjunction with an appropriate imaging modality, these materials can be used to perform 3D dose verification of arbitrarily complex radiotherapy treatments. It is the goal of this thesis to gain a further understanding of the dose response mechanisms of one of these types of dosimeters (the polyacrylamide gel (PAG) system). Fourier Transform Raman (FT) spectroscopy is used to characterize the chemical changes occurring in irradiated polyacrylamide gel. Chemical changes occurring in response to ionizing radiation are characterized for a range of initial gel compositions, using both photon and proton irradiation. By comparing the gel dose response to protons 1 of different energies with that of x-rays, a further understanding of the dependence of gel response on ionizing density is gained. 1.1 R a d i a t i o n T h e r a p y O v e r v i e w Cancer, the group of diseases related to the growth and spread of tumours, is the most significant health care problem in the western world and exceeds heart disease as the leading cause of potential years of life lost [3]. In Canada, 130 000 people are diagnosed with cancer annually [3]. Approximately one in three people will develop some form of cancer in their lives, and one in four will die from their cancer. Of the people diagnosed with the disease, approximately one half will receive some form of radiation treatment. Surgery and chemotherapy are also used to treat cancer. The clinical process of using radiation for the treatment of cancer is termed radiation therapy. The aim of radiation therapy is to deliver a very accurate lethal dose of radiation to a defined target (tumour), while minimizing the dose imparted to surrounding regions. The results of radiation therapy can be either curative (i.e. eradication of the tumour) or palliative (i.e. relief of disease symptoms and pain management). 1.1.1 Historical Background The use of radiation to treat various malignant diseases began just one year after the discovery of x-rays in 1895 by Wilhelm Conrad Roentgen. The first patient cured with radiation was reported in 1899 [4]. Advances in radiation therapy in the early 1900's were centred around several aspects. Brachytherapy, the treatment of disease by sealed radioactive sources that deliver radiation by interstitial, inter-cavity, or surface application of the radiation source was first performed in 1910 2 4 and the use of this modality grew substantially in the years that followed [5]. Ad-vances also occurred in the area of x-ray tube design, and by 1922 tubes with peak energies in the 200 kV ranges were developed for the treatments of deeper seated tumours. Understanding the radiobiological basis of cancer radiotherapy was also pursued at this time, and experiments were performed demonstrating the radiobi-ological advantages of delivering multiple daily irradiations to the same treatment site (termed fractionation) [6]. Further significant advances occurred in the 1940's when particle accelerators were introduced and the betatron became available for megavoltage x-ray treatments. Linear accelerators (LINACs) specifically dedicated to the treatment of disease achieved routine clinical use by the 1960's and remain, in developed countries, the most popular method of delivering radiation treatments to patients. The greatest advances in the calculations of delivered doses occurred after computers were introduced for such purposes in the 1960's. This area developed from universal atlases of isodose distributions available in the 1960's to individual patient treatment plans of isodose distributions, such as are available today. The most significant advances in the area of accurate tumour localization came in the early 1970's when x-ray computed tomography (CT) was introduced as an imaging tool for radiotherapy treatment planning. Other significant imaging advances occurred in the 1980's with the introduction of magnetic resonance imaging (MRI). Recently, three dimensional (3D) conformal therapy has been developed which utilizes a full array of the most sophisticated linear accelerator control sys-tems, treatment planning software, and patient imaging technology in order to de-liver a high dose of radiation to a tumour (visualized in 3D) whilst sparing sur-3 rounding healthy tissue. Multi-leaf collimation, intensity modulation of the radi-ation beam, and radiation arcs are all tools available for the 3D conformation of radiation to tumour sites [7, 8]. 1.1.2 Tumour and Healthy Tissue Response Even with the sophisticated modern techniques of treatment planning and radiation dose delivery, inevitably the radiation beams must traverse healthy tissue in order to reach the tumour site. Hence, normal tissue irradiation occurs. In general, the re-sponse of healthy tissue to dose differs from that of tumour and may be different for different tissues. Figure (1.1) shows schematic diagrams of (a) 'classical' (ideal) tu-mour and normal tissue dose response curves and (b) alternative [9] representations of the two response curves. Figure (1.1a) indicates that beyond a certain threshold dose tumour cell kill becomes possible and that the tumour cell response beyond this threshold is a steep function of dose. As the dose is further increased, the slope of the tumour response curve lessens and the response (cell kill) approaches 100%. A similar curve exists for the normal tissue response and, in the classical picture, the normal tissue response curve is drawn to the right of the tumour response curve. Thus, by judicious choice of dose, reasonable tumour response can be obtained while minimizing the damage of the normal tissue cells. The therapeutic ratio (defined as the ratio of doses at a specified level of response (e.g. 50%) for normal tissue and tumour [10, 11]- see [9] for alternate definitions) can be used to determine an ideal treatment dose Figure (1.1a) illustrates the therapeutic ratio (D0/Da, D& = dose at point b, D a = dose at point a) defined at 50% response (shown by horizontal line). Although many dose response curves have been generated using animal experimen-tation, few full response curves have been determined clinically for human tumours 4 Dose (Gy) Dose (Gy) (a) (b) Figure 1.1: Normal tissue and tumour response to radiation as a function of absorbed dose, (a) The 'classical' diagram illustrating the higher sensitivity to radiation of tumour cells relative to normal tissue. In this scheme, judicious choice of treatment dose leads to high levels of tumour response with minimal damage to normal tissue. Also shown is the therapeutic ratio (D;,/Da, see text), the ratio of doses for a given response (horizontal line), (b) An alternative schematic of tumour and normal tissue response. Adapted from [9]. in vivo. Normal tissue and tumour response may vary with individual tissue type (e.g. skin versus liver versus spinal cord), physiological condition (e.g. degree of oxygenation, nutritional status), dose fraction size as well as the biological endpoint being measured. Figure (1.1b) represents a possible scenario where the normal tissue response would be the limiting factor in delivering a lethal dose to tumour tissue. In such a case it would be essential to avoid the normal tissue at risk in order to use radiation to treat the tumour effectively. Further understanding of the biological response in a clinical environment can only be gained by improving the accuracy of tumour delineation and radiation dose delivery. 5 1.1.3 Conformal Radiation Therapy As mentioned above, several methods are currently available for conforming a radi-ation beam to a target. Figure (1.2) illustrates examples of conformal radiotherapy through (a) the use of collimators in conjunction with shielding blocks (figure (1.2a)), (b) multi-leaf collimation (figure (1.2b)), and (c) multi-leaf collimation with inten-sity modulation of radiation beams (figure (1.2c)). Although significant advances in the delivery technology and treatment planning of conformal therapies have been made in recent years [7], it has been suggested that clinical implementation of these therapies has been limited by difficulties with dose verification [12]. Two aspects of conformal therapy complicate experimental dose verification. The first pertains to the fact that conformal therapies often produce dose distributions with high gradients, and so a dosimeter with high spatial resolution in 3D is required. The second pertains to the fact that since many conformal therapies are delivered with temporal and spatial modulation of beam intensity, a dosimeter is required which has the ability to integrate dose over the duration of the treatment. The common dosimeters in use (see section (1.3)) do not meet both requirements simultaneously. Gel dosimeters (chapter (3)) in conjunction with an imaging modality such as MRI, x-ray CT or optical CT have the capability of meeting these requirements. 1.1.4 Modalities of Radiation Therapy Several modalities may be used in treating cancer through radiation and may be subdivided into either ionizing (e.g. x-rays, particles) or non-ionizing (e.g. hyper-thermia (heat), cryogenic, or photodynamic). The ionizing radiation can be further subdivided as being either directly or indirectly ionizing. Charged particles such as electrons and protons form the directly ionizing radiations, as they produce ioniza-6 (a) (b) (c) Figure 1.2: Schematic diagram illustrating a 'beams eye view' of different ways of conforming radiation to a target (shown in black), (a) collimator jaws (light grey) are closed close to the target and custom designed lead blocks (dark grey) are positioned below the collimators (blocks are shown in full for clarity), (b) multileaf collimators conform radiation more closely to the shape of the target, (c) moving leaves can create conformed intensity maps, shown by the greyscale image of the target. tion by collision (Coulombic interactions) as they pass through matter. Uncharged particles such as photons and neutrons form the indirectly ionizing radiations as they liberate charged particles as they interact in matter. The deposition of dose (section (1.2.1)) in matter varies greatly as a function of depth in the matter for the different radiations. Figure (1.3) illustrates typical 'depth dose' curves for 6 M V photons, 9 M e V electrons and 74 MeV protons in water. As can be seen from the figure, different radiation modalities can be chosen for a particular treatment, depending on the requirements of that treatment. This study uses x-ray and proton beams, and so these are discussed in more detail in sections (1.2.2) and (1.2.3). 7 T • 1 1 1 • 1 I 1 1 1 1 1 r 100 Depth (mm) Depth (mm) w 0>) 1 ' ' ' ' ' ' ' ' • 1 o1—•—'—•—•—•—•—•—•—1 0 10 20 30 40 50 Depth (mm) (c) Figure 1.3: Examples of the dose absorbed as a function of depth in a medium (water) for a (a) 6 MV photon beam, (b) 9 MeV electron beam, and (c) 74 MeV proton beam. 8 1.2 B a s i c C o n c e p t s 1.2.1 Absorbed Dose and Kerma The transfer of energy from a high energy photon beam to a medium takes place in two stages. In the first stage, the photon interacts with an atom and causes an electron to be ejected. The second stage involves the transfer of energy from the electron that has been set in motion to the medium through ionization or excitation. The initial transfer of energy from the photon to the medium is termed kerma (kinetic energy released in the medium) and is defined as where Etr is the energy transferred to the medium in a mass m of material. The absorption of energy from the electron to the medium is termed absorbed dose and is defined as where is the energy absorbed by the medium in a mass m of material. The electron can travel an appreciable distance from the point at which ki-netic energy was imparted to the point at which the electron energy is fully absorbed in the medium. Hence, kerma and dose do not necessarily occur at the same point in the medium. This is illustrated in figure (1.4). Initially, as the photon beam enters the medium, electrons are set in motion and travel predominantly 'downstream' from the source of the photon beam. This is shown in figure (1.4a) and as the dashed line in figure (1.4b). Significant amounts of dose are not deposited in the surface region, since few electrons are absorbed here, as shown by the solid line in figure (1.4b). In this initial 'build-up' region, electronic equilibrium has not been established (i.e. (1.1) (1.2) 9 100 95 a) 90 86 absorbed dose b) depth Figure 1.4: (a) Schematic example of the number of electrons ejected from atoms as a function of depth in the medium. Decreasing numbers are caused by photon at-tenuation, (b) Kinetic energy released in the medium (kerma, dashed line) and dose absorbed in the medium (solid line). Surface contamination by electron backscatter has not been shown. Adapted from [11]. more electrons are set in motion than stop in a given region). As the electrons travel along their path they deposit their energy in the medium. Simultaneously, photons that have traveled further into the medium continue to set new electrons into motion. Were photon attenuation neglected, the region downstream from the 'build-up' region would achieve electronic equilibrium. However, since the photon beam is attenuated as it travels through the medium, true electronic equilibrium is never achieved. The number of electrons stopping at a given point in the medium is always higher than the number set into motion at the same point, since the electrons stopping in the medium were set in motion upstream where more photons from the beam were present. 1.2.2 Photon Interactions with Matter There are six primary photon interactions with matter: coherent (Rayleigh) scat-tering, photo-electric effect, Compton effect, pair production, triplet production, 10 and photodisintegration. While the latter two do occur for clinical energies, their probability is low and so are not discussed. The remaining four are described below. Rayleigh Scattering In Rayleigh, or coherent, scattering, incident photons interact with atomic electrons. The electrons are set into oscillation by the incident photon. However, the electrons re-radiate this energy at the same frequency as the incident photon and hence no net energy is transferred to the electrons in the process. As a result, no dose is absorbed in the medium due to this process. Rayleigh scattering occurs at low incident photon energies (eV range). See chapter (2) for a further description of Rayleigh scattering. Photoelectric Effect The photoelectric effect is a phenomenon which involves the interaction of an inci-dent photon with a bound atomic electron (figure (1-5)). The entire energy of the photon is absorbed by the bound electron which is then ejected from the atom. The kinetic energy of the emitted electron, Ee (called a photoelectron) will be Ee = hv-Eb (1.3) where Eb is the electron binding energy, and hu is the energy of the incident photon (h = Plank's constant, v = photon frequency). A vacancy in one of the inner orbitals of the atom exists once the initially bound electron has been ejected. This can be filled by an electron from an outer orbital moving in to fill the vacancy. In this process, the electron moving to fill the vacancy releases energy in the form of 'characteristic' x-rays. These characteristic x-rays have an energy equal to the difference in energy between the initial and final 11 characteristic t x-ray incident photon photo-electron Figure 1.5: The photoelectric effect. A photon interacts with an inner bound atomic electron. Also shown are characteristic x-rays, caused when an outer electron moves to fill the inner orbital vacancy. electron orbitals. Alternatively, electrons moving in to fill an inner shell vacancy can cause the emission of atomic electrons, called Auger electrons. The probability of photoelectric interaction occurring varies approximately as the inverse cube of the incident photon energy. Local probability maxima occur when the incident photon energy is close to that of the electron binding energy. Furthermore, the probability of photoelectric interaction increases approximately with the third power of the atomic number of the target medium. Compton Scattering The Compton process (figure (1.6)) involves the interaction between a photon and a 'free' electron. That is, the electron binding energy is much less then that of the incident photon. The photon is scattered by the electron and some of the incident photon energy is transferred to kinetic energy of the electron. The scattered photon and electron energies depend on both the incident photon energy and photon scattering angle [5]. The probability of Compton scattering is inversely proportional to the incident photon energy. Furthermore, since it involves the interaction of a 12 incident photon Compton electron scattered photon Figure 1.6: In a Compton interaction, an incident photon interacts with an loosely bound (assumed free) electron. Both the photon and electron are scattered. photon with a free electron, the probability of Compton scattering is essentially independent of the atomic number. Rather, it depends only on the electron density of the scattering material. Pair Production If the energy of an incident photon is greater than 1.02 MeV, it may interact with matter through the process of pair production (figure (1.7)). In this process, the photon interacts with the electromagnetic field of the nucleus of an atom. The photon is completely absorbed and causes the production of an electron and positron. Both the electron and positron have a rest mass energy of 0.511 MeV/c 2 , hence the photon requires a threshold energy of 1.02 MeV. Any additional energy that the incident photon possesses is transferred to the electron/positron pair as kinetic energy. Hence, the total kinetic energy available to the electron and positron is where hv is the incident photon energy. The most probable distribution of this energy is for each particle (electron and positron) to receive half the kinetic energy, although any distribution satisfying energy and momentum conservation is possible. EKE — (hf — 1.02) MeV (1.4) 13 electron incident photon (>1.02MeV) positron Figure 1.7: Schematic illustrating pair production. A n incident photon interacts with the field of the nucleus, creating an electron and positron pair. The positron created in this process loses energy through ionizations, excitations, and bremsstrahlung as it traverses matter. Near the end of the range of the positron, it combines with an unbound electron to give rise to two 'annihilation photons' each of energy 0.511 MeV. By the conservation of momentum, these photons are ejected in opposite directions. The probability of pair production varies as the square of the atomic number and approximately exponentially with incident photon energy. Total Attenuation The total probability of photon interaction by any of the above processes is related to the total attenuation coefficient (u), where fx is the sum of the individual attenuation coefficients Here r , oCoh, Oinc and K are the photoelectric, Rayleigh, Compton, and pair produc-tion attenuation coefficients, respectively. The energy absorption coefficient (fxao) is more closely related to dose and can be expressed as fi = T + Ocoh + Vine + « (1.5) f*>ab — T M — Tab *+" &abj,nc ~r Kab (1.6) 14 100 80 N $ 60 S 3 o 40 o < 20 0 1E-2 1E-1 1 1E1 1E2 Photon Energy (MeV) Figure 1.8: Relative importance of the three main photon interactions as a function of atomic number and incident photon energy. The lines indicate regions of equal probability for each adjacent effect. Adapted from [13]. where E a i is the average energy absorbed by the medium and hv is the incident pho-ton energy. Note that the Rayleigh scattering process does not cause any absorption of energy and hence is omitted from equation (1.6). The relative contribution of each process as a function of both incident pho-ton energy and the atomic number of the medium is shown in figure (1.8). The curves in figure (1.8) indicate where each adjacent process has an equal probability of contributing to the overall attenuation coefficient. 1.2.3 Charged Particle Interactions with Matter As described above, the relevant photon interactions with matter are photoelectric, Compton, and pair production processes. Charged particles (protons, electrons, al-pha particles etc) interact primarily through Coulombic interactions between the electric field of the particle and the electric field of the electrons and nuclei of atoms in the medium. Collisions between incident particles and electrons present Photoelectric Effect Pair Production Compton Effect 15 iii the medium result in ionizations or excitations of the atoms. Interactions be-tween incident particles and atomic nuclei can result in radiative loss of energy (bremsstrahlung) or nuclear reactions. Furthermore, if the incident particle velocity in the medium exceeds that of the speed of light in the same medium, Cherenkov radiation can be emitted. For protons in the energy ranges of interest (<100 MeV) bremsstraMung, nuclear reactions and Cherenkov radiation are all extremely weak (or non-existent). As a note, distinction must be made between the amount of bremsstrahlung produced by a heavy charged particle (e.g. proton) and an electron or positron. Electrons will produce on the order of 106 times more bremsstrahlung radiation than protons, due to their lighter mass (mass of a proton ~1800 times the mass of an electron). The rate of incident particle energy loss (due to ionizations) per unit length of material traversed can be described by the ionizational stopping power dE/dx, where E is the energy of the particle and x is the length traversed. In general, dE/dx is proportional to the square of the particle atomic number and inversely proportional to the square of the particle velocity. Thus, as the incident particle slows down, it's rate of energy loss increases and therefore the ionization or absorbed dose in the medium also increases. The dose absorbed in the medium increases sharply at the end of the incident particle range (for protons and heavier ions). This sharp increase is known as the Bragg peak (see figure (1.3c)). The ionizational stopping power of the incident particle can be written as (see [14]): 'ion — In 2m0c2/32 7(1 -0*) P Z (1.7) 16 where z = incident particle atomic number p = v/c = incident particle relative velocity m0 = electron mass c = speed of light ra = classical electron radius = number of electrons per gram I = mean atomic ionization potential a z = shell correction The mass stopping power is defined by dividing the quantity dE/dx by the density of the material. This gives the energy loss per unit thickness of material (in units of g/cm2). In the above equation the semi-empirical quantity J (the mean atomic ion-ization potential) has been used. It is a geometric-mean value of all ionization and excitation potentials of an atom of an absorbing medium [14]. This takes into account the fact that the electron may be bound to an atom with some energy. At some large values of impact parameter (i.e. large distances of closest particle approach to the electron), the energy exchange between the incident particle and electron may be insufficient to overcome the electron binding energy. The mean ionization potential (I) has been incorporated to account for this fact. The quantity ^- in equation (1.7) is called a shell correction and accounts for the errors introduced in equation (1.7) when the incident particle velocity ceases to be much greater than that of the atomic electrons in the stopping medium [14]. As the incident particle velocity drops below that of the atomic electrons (occurring for 17 K shell electrons first, followed by L shell etc.) these electrons cease participating in the stopping power process, resulting in an over-estimate of I. The shell correction accounts for this over-estimate. The terms in square brackets in equation (1.7) are collectively a slowly in-creasing function of energy. For example, for protons between 0.5 MeV and 100 MeV the term in square brackets changes approximately by a factor of two. Clearly, the major factors contributing to the stopping power are the particle atomic number (z) and its velocity (B), as noted above. 1.3 Dosimetry Techniques The following section describes several techniques currently available for measuring the dose deposited in a medium by incident photons or charged particles. The more 'traditional' dosimeters are described first, followed by a brief introduction to gel dosimetry measuring devices. The basic principle of operation of each dosimeter is given, along with a typical use in clinical radiotherapy. This serves as a brief introduction to the dosimeters before describing the requirements of the dosimeters for general, as well as 3D specific, radiation dosimetry. Detailed descriptions of each traditional dosimeter are given in standard texts [5, 11, 14]. 1.3.1 Ionization Chamber A host of ionization chambers have been developed, each with their distinct ad-vantages for measuring dose. A basic thimble ionization chamber (very common in radiotherapy clinics) is described below. A typical thimble chamber (figure (1.9)) is a cylindrical (or spherical) cham-ber of < 6 mm diameter and ~3 cm length encompassing a volume of gas of ap-18 Thin electrode (inner wall) -central electrode thimble wall 7 7 air cavity insulating material Figure 1.9: Schematic of a typical thimble ionization chamber. proximately 0.1 to 1 cm3. The chamber wall material can be typically either air or water equivalent. The chamber is typically unsealed and hence the gas inside the chamber is air. A collecting electrode is placed inside the middle of the chamber. Ions created by incoming radiation interacting with the chamber gas are collected by the anode and the resulting current is measured by an electrometer. The different types of ion chambers vary in the details of their operation, however, their general operating principles are similar. The ionization chamber is a point detector. Placed in a phantom (e.g., water, plastic etc), ionization chambers can be used to make point dose measurements. Rel-ative dose measurements are made which are then related back to absolute absorbed dose through the use of a cross calibration method. Ionization chamber measure-ments can be extended to take 2D, and even 3D measurements of static radiation fields by moving the detector through the phantom in a raster type pattern. The dose distribution of the radiation beam can then be reconstructed at the resolution of the raster pattern. 19 incident Figure 1.10: Schematic of a typical diode detector. The power supply provides a 'reverse bias' voltage <~300 mV. Current generated by photon bombardment is measured with the meter. 1.3.2 Diode Detector Diode detectors (figure (1.10)) made of semiconductor material can be used to mea-sure the ionization produced by radiation. A typical diode detector contains a wafer of silicon into which impurities are introduced to make 'p' and 'n' type silicon. The 'p' type silicon is an electron acceptor (i.e. it contains an excess of holes) and the 'n' type silicon is an electron donor (i.e. it contains an excess of electrons) [11, 15]. A reverse bias potential (V^ j, — Vp ~ 300 mV) may be placed across the p-n junction and creates an enhanced 'depletion' region (~2 /im thick) at the p-n junction. The incident radiation ionizes the silicon in the depletion layer, producing electron-hole pairs in this region. Due to the reverse bias, electrons and holes are swept to the 'n' and 'p' sides of the junction, respectively. The resulting current can be detected and ultimately related to absorbed dose. As with ion chambers, diode detectors are essentially point detectors, with extension to 2D and 3D being possible in the same ways as described for the ion chamber. They can typically be made very small, offering good spatial resolution, and are relatively sensitive. However, diodes suffer from energy and temperature 20 dependence and are damaged over time. Hence, they can only be used as relative dosimeters (i.e. absolute dosimetry is not possible). 1.3.3 Thermoluminescent Detectors The phenomenon of thermoluminescence involves the emission of light from a crys-talline material that has been heated. When certain crystalline materials (e.g. LiF) are irradiated and then heated, they emit light in proportion to the amount of radiation damage inflicted on them. When the crystalline material is doped with impurities, imperfections in the crystal lattice occur that can be used as energy traps [5, 11]. When the crystal is irradiated, it absorbs energy from the beam and electrons in the material are excited and trapped in the energy levels of the impurities (see figure (1.11)). Upon sub-sequent heating, trapped electrons are excited from their trapped state and return to their ground state, emitting light in the process. The amount of light emitted in this process is proportional to the number of electrons trapped. This in turn is proportional to the amount of energy absorbed in the detector from the radiation beam and can be calibrated in a known radiation field. Thermoluminescent dosimeters (TLDs) can be used in radiotherapy as point detectors. Since TLDs can be compartmentalized into small units, they can be placed on the body of a patient, or into a solid phantom, to make point measurements in 3D. Furthermore, since the excited electrons can be held in their traps for long periods of time (days, months), TLDs can be used as radiation dosimeter badges which are worn by personnel working with radiation. Every few months the badges can be tested for the amount of radiation they received and, in this way, can be used to monitor the radiation dose received by the personnel wearing the badge. 21 conduction band emitted photon valence ^ band incident photon Figure 1.11: Basic electronic structure of a thermoluminescent detector, (a) Upon irradiation, electrons are excited and trapped within the electronic structure of an impurity, (b) By heating the device, the electrons can be liberated. They subse-quently emit radiation which is then detected. 1.3.4 Film Dosimetry Several types of film are available for dosimetry purposes, the most common being radiographic film and the newer radiochromic film. Radiographic film consists of small crystals of silver bromide. Upon irradiation, ions of silver (Ag+) and bromide (Br -) are produced. Some of the silver ions are converted to atomic silver (Ag). In the film development process the bromine is removed and only opaque clusters of silver atoms remain. The variation in cluster density throughout the film produces variations in optical density which can be measured and correlated to dose. Radiographic film is a relative dosimeter. That is, a calibration curve re-lating optical density to dose must be measured in order to accurately calculate radiation dose delivered to a film. Film is a common 2D dosimeter used in routine quality assurance testing of linear accelerators in situations where highly accurate dose information is not required. Its use can be extended to some degree to the mea-surement of 3D dose distributions of small fields by stacking the film between sheets 22 of tissue equivalent material. However, since the film itself is not tissue equivalent, this imposes a limit on the thickness of the plastic spacers between the sheets of film [16]. It has been shown [16] that film exhibits a differential response for large field sizes (>~ 10 cm) and depths in a phantom (~10 - 20 cm). Furthermore, film response has also been shown to vary depending on the film's orientation relative to the incident radiation beam. Radiochromic film is a relatively new type of film and the changes in optical density are based on a number of polymer reactions, depending on the exact model of the film [17, 18]. Radiochromic film does not need to be 'developed' by a chemical process. It's uses are similar to that of radiographic film, although it is relatively insensitive to radiation and hence high doses are required (~ 40 Gy) in order for radiochromic film to register a response. One advantage of this relative insensitivity is that this type of film can be exposed to limited amounts of ambient light, making working conditions easier. The main disadvantage of radiochromic film is that it is difficult to produce a sheet of the film that has a highly uniform response to dose. This distorts the relative dose information. At present radiochromic film is expensive and available only in relatively limited sheet size. 1.3.5 Chemical Dosimetry By far the most popular chemical dosimeter is the Fricke dosimeter. The Fricke dosimeter is a solution containing ferrous ions (Fe2+) which, when irradiated, are oxidized into ferric ions (Fe3+). The ferric ion concentration can be determined through spectrophotometry. This concentration can be related to the dose absorbed in the solution. For this, a knowledge of the number of ferric ions produced per unit energy absorbed (known as the G value) must be known. The G value for the Fricke 23 dosimeter is available in a number of references [19-21]. The chemical dosimeter can be used as an absolute dosimeter. However, no information about the spatial characteristics of the radiation beam can be deter-mined with this chemical dosimeter in liquid form. 1 . 3 . 6 Gel Dosimetry It is currently difficult to measure with high spatial resolution, high dose accuracy, and in a reasonable amount of time, complex 3D dose distributions of arbitrary shape and size. The recent development of gel dosimeters, in conjunction with an imaging modality such as magnetic resonance imaging (MRI), x-ray CT, or optical CT, shows potential for such measurements. A brief overview of the gel dosimetry process is given here, details and literature references are given in section (3.4). The overall gel dosimetry process is shown schematically in figure (1.12). Gels are prepared from their individual constituents and poured into the desired phantom (figure (1.12a)). The gel phantom is then irradiated with a desired treatment (figure (1.12b)). The irradiated gel is imaged with one of the above imaging modalities (figure (1.12c)). Finally, necessary data analysis and processing is performed to produce a dose map (figure (1.12d)). Although there are several different types of gel dosimeters (see section (3.5)), the most common are the Fricke and polyacrylamide polymer gels. The Fricke gel is based on the same principles as the Fricke liquid dosimeter described in section (1.3.5), with the addition of gelatin or agarose into the system. The gelatin ma-trix inhibits the migration of the ferric ions, thus retaining the spatial information of the dose distribution with which the dosimeter was irradiated. The polyacry-lamide system is based on the polymerization of acrylamide and N, N' methylene-24 Figure 1.12: Overview of the gel dosimetry technique, (a) Gel preparation, (b) gel irradiation, (c) gel imaging, and (d) data analysis and image processing. bis-acrylamide into long crosslinked macromolecules. The concentration of macro-molecules is proportional to the dose imparted to the gel. Furthermore, gelatin or agarose is added to the system to inhibit polymer molecule migration, thus the gel retains spatial dose information. The work in this thesis pertains to the polyacry-lamide gel systems. More details of both systems are given in chapter (3). It has been determined that a polymer gel irradiated to a certain dose will exhibit a different proton nuclear magnetic spin relaxation rate (Ti and T2) relative to an unirradiated gel [2, 22]. Thus, by calibrating the T i or T2 MR signal to dose given to the polymer gel, a 'dose map' from an irradiated polymer gel can be established. A similar type of dose map can be obtained by imaging the irradiated polymer gel with an x-ray CT scanner, where now changes in CT Hounsfield units are correlated to absorbed radiation dose in the polymer gel. As an example, figure 25 Figure 1.13: Example of (a) axial, (b) sagittal and (c) coronal x-ray CT images of a gel housed in a spherical flask and irradiated with a 4 arc stereotactic treatment. Reproduced from [24] (1.13) shows an example of a dose map obtained from an x-ray CT scan of a gel irradiated with a four arc stereotactic treatment. The lighter areas indicate regions of the gel which have received a higher dose of radiation. The figure shows the truly 3D nature of gel dosimetry, as axial (figure (1.13a)), saggital (figure (1.13b)), and coronal (figure (1.13c)) slices have been extracted from the data. The physical basis of the x-ray polymer gel technique is likely based on the change in density in areas of high polymer molecule concentration, which contributes to the x-ray CT signal [23]. Overlaid on the gel image are corresponding isodose contours generated from the treatment planning software (BrainLab®). It can be seen that agreement between the treatment planning generated isodose contours and the irradiated gel volume are in general excellent. Both Fricke and polymer gels are relative dosimeters. They require a cali-bration curve to relate signal intensity to absorbed dose. 26 1.4 Dosimeter Requirements In the sections below, a brief mention is made of the general parameters needing consideration when making accurate dose measurements. When possible, extension is made to describe the impact that making 3D measurements has on the given parameter. 1.4.1 Dose Resolution, Sensitivity and Accuracy The detector sensitivity (i.e. the slope of the detector response vs. dose curve) which in part determines the dose resolution (i.e. the minimum difference in dose that can be measured by the detector) are both important factors in determining the utility of a detector for a given application. Some detectors are particularly sensitive to radiation dose, for example, some radiographic film saturates after receiving ~2 -3 Gy. Conversely, some radiochromic film is less sensitive to radiation, requiring ~40 - 50 Gy to achieve a reasonable signal. The dose resolution of a detector is related to its sensitivity in that a sensitive detector has a steep dose response curve and hence a small change in dose translates into an appreciable change in detector response. Conversely, a weakly responding detector exhibits a more shallow dose response curve and so a larger dose is required to cause an incremental change in detector response. The dosimeter precision further affects the resolution of a given detector. In terms of dose accuracy, the International Commission on Radiation Units and Measurements (ICRU) Report #24 recommends that the overall accuracy in dose delivered to a patient be within 5% of the true dose [25]. It is thus recommended that the absorbed dose to a reference point (and also to points away from the reference point) in water be accurate to within 2 - 2.5% [9, 26]. Thus, for purposes 27 of linear accelerator calibration, a detector must be accurate to better than 2 - 2.5%. 1.4.2 Spatial Resolution The spatial resolution of a detector refers to the detector's ability to measure ab-sorbed dose at two closely spaced points in a phantom. Ultimately, it is determined largely by the effective volume of the measuring element. For example, a typical thimble ion chamber has an effective measurement volume on the order of 1 cm3 and hence its ability to resolve dose spatially is worse than, for example, film, which has sensitive elements (AgBr crystals) on the order of microns in size (note: the spatial resolution of the film dosimetry technique is ultimately limited by the film scanning device). In terms of 2D or 3D measurements, the spatial resolution can be a significant factor in determining a detector's ability to accurately map a dose distribution. High spatial resolution is a particularly important dosimeter quality for verification of highly conformal treatments where high dose gradients can occur within a spatial range of 1 - 2 mm. 1.4.3 Dose Integration Dose integration refers to a detector's ability to integrate dose as a function of time. All of the detectors described above have this ability. Hence, the recorded reading from a detector for a given measurement translates to a total accumulated dose at that point for the duration of that measurement. However, the detector must re-main at the same point in space for the duration of the measurement otherwise the correlation between accumulated dose and position is lost. Only in static treatments (e.g. fixed single beam) can the detector (e.g. ion chamber, diode) be moved in a raster position in order to acquire 2D and 3D information. In treatments which 28 axe dynamic in nature (e.g. arced or intensity modulated beams) it is therefore not possible to move the detector through the course of the delivery of the irradiation, since the total integrated dose at any given point will not be obtained if the detector is moved. Thus, the only possibility of acquiring dose information for a dynamic treatment using point detectors (i.e. ion chambers, diodes) is to repeat the treat-ment for each spatial position of the detector. By placing several TLDs at different points in space, one can acquire rough dose maps of dynamic treatments, although this method proves to be time consuming and, furthermore, the accuracy and re-producibility of TLD's is typically ~5%. Film has the ability to acquire 2D maps of dynamic treatments. By stacking film vertically, 3D measurements can be made, although there are limitations to the minimum spacing between sheets of film [16]. Only gel dosimeters have the ability to make truly 3D measurements of dynamic treatments. 1.4.4 Tissue Equivalence If a detector's radiological properties (e.g. effective atomic number, see [11]) are not close to that of tissue, the absorbed dose recorded by the detector may need to be easily related to a dose in tissue. The perturbations of dose deposition by an ion chamber are accounted for by introducing a number of correction factors [14] which can introduce an error of ~1.5% [27] in the net determination of dose. Non-tissue-equivalent detectors must be minimized in dimension (e.g. a single sheet of radiographic film set between two blocks of tissue equivalent plastic) such that the perturbations to the dose measurement due to the presence of the detector can be neglected. Radiochromic film and polymer gel dosimeters are considered to be essentially tissue equivalent. Tissue equivalence of polymer gel dosimeters is 29 discussed in section (3.4.1). 1.4.5 Anatomical Equivalence Anatomical equivalence refers to the ability of the detector to take the shape of the patient site being treated (e.g. head and neck area). None of the traditional detectors have this ability. Gel detectors, however, have the ability to take the shape of the container in which they are housed. Hence, if anatomically shaped containers are constructed, gel dosimeters can approach anatomical equivalence. Furthermore, gel dosimeters are the only detectors in which the phantom is the detector. That is, the gel dosimeters themselves become the phantom for the dosimetric study. All traditional detectors require an additional phantom in which to house the given detector. As an additional note, it is possible to insert objects of varying composition (i.e. inhomogeneities) into the gel dosimeter. This may make it possible to study the dose perturbation due to the inhomogeneities. 1.5 Objectives of this Thesis Polymer gel dosimeters have the potential for useful measurements of complex 3D dose distributions created by comformal radiotherapy techniques. However, the polymer gel dosimeters have not, to date, gained widespread use in radiation dosime-try. A number of reasons contribute to this fact and they may be roughly broken into two categories. The first category pertains to imaging and gel manufacture considerations. To date the most common form of gel imaging is MRI which is still relatively inaccessible, on a regular basis, to a large number of radiotherapy clinics. Hence, alternative imaging modalities such as optical CT [28, 29] and x-ray CT [30] 30 have been investigated for potential use in gel imaging. A second consideration per-tains to gel manufacture. Currently gel manufacture requires specialized equipment and is time consuming. The second category deals with the more fundamental properties of the gels and the current lack of complete understanding of the gel dose response mechanisms. This thesis addresses this lack of fundamental data currently available pertaining to polymer gel radiation dose response mechanisms. Fourier transform Raman spec-troscopy is used to probe, on a molecular level, the chemical changes occurring in irradiated polymer gels. Specifically, the project can be thought of as having three components, as outlined below. The first component to this thesis involves establishing an adequate exper-imental framework within which further experiments are to be conducted. From the standpoint of gel sample preparation, this requires establishing a robust and reproducible gel manufacture technique. From the standpoint of spectral data ac-quisition, since at the onset of this project a scant one paper had appeared in the literature [31] pertaining to the use of FT-Raman spectroscopy to study irradi-ated polyacrylamide gels, the overall suitability of using FT-Raman spectroscopy to study irradiated polymer gels is established. Sample housing, Raman spectroscopy acquisition parameters, spectral reproducibility, and spectral data analysis are all examined. To aid in data analysis, customized software is written. This first com-ponent is described in chapter (4) and in the first sections of chapter (5). The second component involves studying the consumption of monomer and crosslinker and the formation of polymer for gels of 'standard composition' (see sec-tion (3.2)) irradiated with 6 MV photons. The analysis of monomer consumption is then extended to gels manufactured with a range of initial constituent concen-31 trations. Curves of consumption of monomer as a function of absorbed dose are constructed. A qualitative model is used to explain the differences observed in monomer consumption curves for the different gels. All 'compositional studies' are performed using 6 MV photons. These studies are described in chapter (5), following the initial studies described above. The third component of the thesis involves studying the dependence of gel response on ionizing density. To this end, the polymer gel dose response to 74 MeV proton irradiation is measured. Raman spectra are acquired at two points on the proton beam depth dose curve (i.e. at two points with different ionizing density) and resulting monomer consumption curves are compared to those obtained with gels irradiated using 6 MV photons. Hence, the effectiveness of polymer gel in recording dose as a function of ionizing density is established. The mathematical model of track structure [32] is used to predict the effectiveness of the polymer gel in measuring dose in the same two regions of the proton depth dose as in the experiments. As well, track structure aids in understanding the experimental results. The model can also be used to predict the polymer gel effectiveness in measuring absorbed proton dose for polymer gels of varying initial compositions and positioned in different regions of incident proton ionizing density. This work is described in chapter (6). 32 Chapter 2 Raman Spectroscopy This chapter presents the theoretical and experimental basis of Raman spectroscopy. Section (2.1) presents a brief history of Raman spectroscopy. Section (2.2) presents the basic theory behind Raman and infrared spectroscopy and a brief comparison is made between the techniques. Section (2.3) presents the experimental considera-tions required in performing Fourier transform Raman spectroscopy. Finally, a few applications of Raman spectroscopy are described in section (2.4). 2.1 H i s t o r i c a l B a c k g r o u n d The inelastic scattering of light by molecules was predicted by Smekal [33] shortly after the 1923 demonstration of the Compton effect. Similar predictions surfaced in 1925 by Kramers and Heisenberg. The experimental observation of inelastic light scattering did not, however, occur until 1928 when the effect was first observed by Sir Chandrasekhara Venkata Raman (1888-1970) [34, 35]. Using the sun as a light source, a telescope as the collector, and his eyes as the detector Sir Chandrasekhara Raman was, remarkably, able to observe the feeble effect. As well as having the 33 effect named after him, Raman's efforts gained him the Nobel prize in physics in a record time of two years. Early research in the field flourished in both the theoreti-cal understanding of the Raman effect and results of studies on molecular structure, and were summarized in an extensive work by Hibben in 1939 [36]. Developments in instrumentation were concentrated on providing better excitation sources. Early lamps of helium, bismuth, lead, zinc and other elements generally proved insuffi-cient due to low light intensities. Mercury sources and burners were also developed for Raman applications. The first detection systems included photographic plates. In 1942 the first photoelectric detection device used in Raman spectroscopy was reported by Rank et al. [37]. In the early 1950's several reports were published de-scribing the use of cooled photomultiplier tubes in Raman spectroscopy applications [38]. Despite these early advances, the field of Raman spectroscopy was explored mainly by a few specialists, due to experimental difficulties. Further significant advances did not occur until the 1960's when lasers be-came readily available and reliable. He-Ne lasers with 0.5 - 80 mW power were the first lasers used. In 1964 Chantry et al. demonstrated the feasibility of using near infrared excitation for Raman spectroscopy by analyzing iodine in carbon tetrachlo-ride and carbocyanide dye in methanol [39]. Also during this time, developments in diffraction gratings improved the efficiency of light collection, with double and triple monochromators appearing in Raman instruments. Later, in 1968, collection efficiency was further improved by the use of holographic gratings. Still, Raman spectroscopy suffered from one major limitation: fluorescence. The development of infrared lasers such as the NdiYAG1, along with high sensi-tivity detectors and Fourier transform collection optics has largely, although not 'Neodymium doped Yttrium Aluminium Garnet 34 completely, eliminated this problem. These recent technological advances have all come together within the last 10-15 years to make up the Fourier transform Raman spectrometer and, as such, FT-Raman spectroscopy has become a valuable tool in both academic and industrial settings. 2.2 Theoretical Considerations The following sections describe the origin of Raman scattering. For completeness, origins of infrared (IR) absorption are also mentioned. 2.2.1 Origin of Raman Spectra Classical Description of Raman Scattering Raman scattering is an inelastic light scattering process. A photon of energy hv (where h is Plank's constant and v is the frequency of the incident radiation) in-teracts with a molecule having vibrational energy levels vi, v2 ... (see figure (2.1)) corresponding to energies of hvi, hv2 respectively. Scattered radiation of energy hv' is collected. Most of the collected radiation will be of the same energy as the incident light (v = v', i.e. Rayleigh scattering), however, a small fraction of this scattered radiation will be of energy h(v—v\) (termed Stokes scattering) or h{v+v{) (termed anti-Stokes scattering). This inelastic component to the scattered radia-tion is what is known as Raman scattering. Typically, Raman spectra are recorded as plots of intensity of scattered radiation as a function of frequency shift between incident and scattered radiation. Normal Raman scattering is a weak process, with only one photon in ~105 to 106 being Raman scattered. As is seen in figure (2.1), in basic Raman experiments the incident energy is chosen such that the lowest elec-35 2 v=0 virtual state Rayleigh Scattering Raman Scattering v m V V 3 2 m Figure 2.1: Schematic diagram of the Raman effect. Incident radiation of energy hv excites a molecule from the ground electronic state (\I/0) to a virtual state. Rayleigh, Stokes and anti-Stokes Raman scattering are shown. tronic state (v&i) of the molecule is not excited. The radiation, therefore, excites the molecule from \I>0 to a virtual state from where the molecule relaxes back to the ground state. This minimizes the excitation of fluorescence within the sample which, when present, can completely mask the Raman signal. There are several special techniques (e.g. Resonant Raman scattering) in which incident energy is comparable to the energy required to excite the first electronic state of a molecule [40]. These techniques are not used in this study and are therefore not discussed. The presence of Stokes, anti-Stokes and Rayleigh lines may be predicted using classical theory. Consider an incident beam of electromagnetic radiation (e.g. a laser beam) whose electric field E fluctuates with time, Here E0 = (Eox, Eoy, Eoz) is the maximum amplitude of the radiation electric field, v is the frequency of oscillation, and t the time. If a molecule is irradiated with this E = E0 cos 2-Kvt (2.1) 36 field, a dipole P is induced such that (2.2) where a is a constant of proportionality known as the polarizability. In Cartesian coordinates equation (2.2) can be written in the general form 'zy-^y (2-3) Px = a Py = a Pz = a -- ctE, Px Py = Pz xx ^xy ^xz yz Ex Ey (2.4) Ez &zx Q-zy &zz Note that in most cases the polarizability matrix is symmetric (i.e. axy — ayx). If the molecule is vibrating with frequency vn = UQ, v\,V2— then the nuclear displacements q can be written as q = qa cos 2ixvnt (2.5) where q0 is the maximum amplitude of the displacement. These oscillations affect the polarizability. For small nuclear displacements a is a linear function of q, a = a 0 + — qt + Mi)a (2.6) where a0 is the polarizability at the equilibrium position and (Sa/Sqi)0 is the rate of change of a with respect to the ith coordinate q, evaluated at the equilibrium position. Substituting equations (2.6) and (2.1) into equation (2.2) gives P = aE0 cos 2-ixvt = a0E0cos2trvt + ( T—\ qiE0cos2irut 37 (2.7) Using equation (2.5) in equation (2.7) gives P = aoE0 COS(27TJ4) + ( T—) Qi0E0 cos(27r^mt) cos(2irut) (2.8) The second term in equation (2.8) can be rewritten using the trigonometric identity 2cos vlcos-B = cos^ + B) + cos(vl — B), hence P = a0E0 cos(27ri4) 1 (8a\ (2-9) + 2 \j£ j <lioEo {COS[2TT(I/ - um)t] + cos[2vr(z/ + um)t]} The three terms in equation (2.9) represent the three components observed in nor-mal Raman scattering. The first term corresponds to light scattered at the same frequency as the incident radiation (Rayleigh scattering, see figure 2.1). The second term corresponds to light scattered at a lower frequency than the incident radiation (Stokes scattering) and the third term corresponds to light scattered at a higher frequency than the incident radiation (anti-Stokes scattering). From equation (2.9) it is seen that for a Raman mode to be active, the change in polarizibility about the equilibrium position (6a/5q)0 ^ 0. That is, the derivative of at least one of the elements in the polarizabilty matrix (equation (2.4)) must be non-zero for Raman scattering to occur. The above classical description predicts Rayleigh scattering to be more in-tense than the Raman counterpart (because generally a0 > (8a/5q)0q). This is in fact observed. Furthermore, the theory predicts a linear dependence in Raman intensity with incident beam intensity (Ea). In normal Raman spectroscopy ex-periments this is also observed. However, the classical theory fails to predict the relative intensities of Stokes to anti-Stokes scattering. Classical theory predicts the intensity / of the scattered radiation to be proportional to the fourth power of the incident light intensity and the ratio of Stokes scattering intensity Is to anti-Stokes 38 scattering intensity Ias to be ( 4 V — V, m (2.10) m This is not typically observed. Since at room temperature the vo (ground state) is generally much more populated than the higher vibrational levels (vi, V 2 . . . ) the Stokes scattering (which originates from the ground state) is always more intense than the anti-Stokes scattering. Since the populations of the vibrational levels are described by the Boltzmann distribution, Placzek [41], for example, has shown that the ratio of Stokes to anti-Stokes scattering is more accurately given by where k is Boltzmann's constant and T is the temperature of the system. As a result, measuring the ratio of Stokes to anti-Stokes scattering can provide a rough estimate of the sample temperature. Normal Modes of Vibration Polyatomic molecules can perform a complex set of molecular vibrations, with each atom performing its own array of harmonic oscillations. It can be shown [42] that these complicated molecular vibrations can be expressed as a superposition of a number of "normal" vibrations that are completely independent of one another. These normal vibrations form a basis set of vibrations for the molecule from which all other vibrations are built upon. As an example, the normal modes of vibration of the linear molecule C O 2 are shown in figure (2.2). The symmetric in-phase stretching mode involves both oxygen atoms simultaneously oscillating along the bond direction (figure (2.2a)). Figure (2.2b) illustrates the symmetric bending mode, which involves simultaneous atomic displacements perpendicular to the length of the molecule. The Is = iy - "m)4 las iy + exp(hum/kT) (2.11) 39 o=c=o a) symmetric stretch symmetric b) bend (doubly degenerate) c) anti-symmetric stretch o - — • — -o 1 t o -f — • — 1 — • — - o + - o + Figure 2.2: Normal Modes of vibration for CO2. a) symmetric stretch, b) symmetric bend c) anti-symmetric stretch. "+" indicates vibration out of page, "-" indicates vibration into page. anti-symmetric stretching vibration involves the simultaneous shrinking of one CO bond and elongation of the other CO bond (figure (2.2c)). Note that the centre of mass is conserved in each of these motions. If the molecule is made to vibrate, it will undergo a complex set of vibrations which are superpositions of these three normal modes of vibration. In general, since each atom has three degrees of freedom in its motion (x, y, z), an iV-atom molecule will have 3N degrees of freedom. However, three of these degrees are due to purely translational motions of all atoms moving simultaneously in the same direction and, similarly, three degrees are due to pure rotations of the entire molecule. Hence, the molecule contains 3iV — 6 vibrational degrees of freedom. If the molecule has a symmetry property such that one of these six motions does not exist (e.g. rotation of a linear molecule about the axis parallel to the bonds), then the molecule contains 3iV — 5 degrees of freedom. In the above example, the CO2 molecule possesses 3(3) — 5 = 4 degrees of freedom. The one additional degree of freedom in this example originates from the doubly degenerate symmetric bending mode, which can be performed in two mutually exclusive planes (e.g. parallel and perpendicular to the plane of the page). 40 Selection Rules for Raman Spectra To predict whether any given mode of vibration will be Raman active (non-zero (8a/6q)0) is in general difficult. Equation (2.9) only demonstrates that the 6a/8q terms need be non-zero, but it does not say anything of whether the terms are non-zero for a particular mode. For simple molecules the Raman activity of a particular mode can be predicted by considering the polarizability ellipsoid (see below). For more complex molecules this becomes impractical. In these situations, by using information about the symmetry of the molecule, group theory can be used to predict whether a mode is Raman active or inactive. Group theory is covered in standard texts [42-44]. A demonstration of Raman active and inactive modes is given here for a simple molecule (CO2). As described above, at least one of the terms in the polarizability tensor (equation (2.4)) must change during a molecular vibration. As a demonstration, consider the CO2 molecule whose electron cloud has an elongated shape parallel to the length of the molecule. The polarizability of the molecule can be plotted in all directions from the centre of mass of the molecule. Plotting the reciprocal of the polarizability (1/aj, i = x,y,z) yields a "polarizability elipsoid." Figure (2.3) shows the changes of the polarizability ellipsoid for the different normal vibrations of the CO2 molecule. In terms of the polarizability ellipsoid, the mode is Raman active if the size, shape, or orientation of the ellipsoid changes during the vibration. In the example of CO2, the symmetric stretching mode is Raman active, since the size of the polarizability ellipsoid changes linearly as the molecule oscillates from its least to most elongated displacements. The symmetric bend mode (figure (2.3b)) is, however, not Raman active. In this case the polarizability of the molecule does not change size between the two extremes of its motion, and hence for small 41 » a o a o o b) o-c> o—#-o o—•—o o-*—o Figure 2.3: Polarizability ellipsoid (1/a) for the three normal modes of vibration of C O 2 . a) symmetric stretch, b) symmetric bend and c) anti-symmetric stretch. The central configuration in each row corresponds to the molecule's equilibrium position. a) a / b) / (5a/5q)o*0 a 0 (8a/5q)o q=0 +q q=0 +q Figure 2.4: Schematic diagram showing the polarizability (a) as a function of atomic displacement q for the a) symmetric stretch and b) symmetric bend and anti-symmetric stretch of C O 2 . Equilibrium position is at q = 0. displacements (Sa/Sq)0 = 0). This is further illustrated in figure (2.4) which shows a plot of the change in a as a function of atomic displacement for the vibrational modes of C O 2 . As with the symmetric bend, the anti-symmetric stretch mode of C O 2 is not Raman active since for small displacements, the polarizability does not change around the molecule's equilibrium position. For complex molecules this type of analysis becomes impractical. Quantum mechanical considerations in combination with group theory can be used to predict Raman activity in these more complicated cases. A full description is given in 42 standard texts [42]. Briefly, the selection rules for Raman activity are determined by the set of integrals where: i = x,y,z; j = x,y,z; ctij are components of the polarizability tensor (equation (2.4)); if>y and Vv are wavefunctions corresponding to the initial (v) and final (v') vibrational states respectively; and qa is now the normal coordinate for the vibration a [42]. For the vibrational mode to be Raman active, at least one of the six integrals in equation (2.12) must be non-zero. Equation (2.12) can be considered from two perspectives: a) As an example, the ground state wavefunction I/J0 of the harmonic oscilla-tor is invariant under any symmetry operation (see [40, 42] for a detailed description of symmetry operations). The wavefunction for the first excited state ipi oc qaij}0 and hence has the same symmetry properties as qa. Therefore, for the integrand in equation (2.12) to be symmetric and hence the integral non-zero, at least one component of the polarizability tensor ctij must have the same symmetry properties as qa. In general, the vibration associated with normal coordinate qa is Raman ac-tive if at least one of the components of the polarizability tensor belongs to the same symmetry species as qa [42]. As well, the rate of transition between states v and v' is proportional to the squared magnitude of the above transition matrix elements ([ajj]^). Since this is an observable, the non-zero value of this element must be the same for all indistinguishable orientations of the molecule. Thus the non-zero value of the integral must remain the same for all symmetry operations on the molecule. b) Equation (2.6) with the qi now as the normal coordinates qa can be sub-(2.12) 43 stituted in to equation (2.12): 'a + fp*{Qa)Qaipv' (ga) dqa (2.13) The first term in equation (2.13) is zero by the orthogonality of the wavefunctions Vv(ga) and ipvi(qa) (if v = v' then there is no transition). In the second term both for the matrix element to be non-zero. The requirement of equation (2.14) is the change in polarizability requirement (see [40] for details). The integral in equation (2.15) is non-zero when the product il)y(qa)Qa'*Pv'{Qa) is symmetric (for the harmonic oscillator this condition is met when Av = 1). 2.2.2 Origin of IR spectra Raman and infrared (IR) spectra are both generally concerned with vibrational ex-citations of a molecule. However, the origins of each phenomenon are quite different. Raman spectroscopy deals with light scattering and originates with a change in po-larizability of a molecule. Infrared absorption spectroscopy deals with absorption of energy by a molecule and originates with a change in the molecule's dipole mo-ment. Typical IR spectra are recorded as plots of percent transmission (T) through the sample as a function of incident radiation energy. Thus, as opposed to Raman spectra, IR spectra are acquired with polychromatic (preferably continuum) light sources. The intensity of IR absorption is governed by the Beer-Lambert law (2.14) and (2.15) I = V —ecd (2.16) 44 where I0 is the incident light intensity, I the transmitted light intensity, e the molec-ular absorption coefficient, c the concentration of absorbing molecules within the sample, and d the sample thickness. In contrast to Raman scattering, the intensity of IR absorption is not directly proportional to the concentration of the molecu-lar constituent in the sample. Therefore, it is sometimes customary to define the absorbance A — log(I0/I) = ecd, which is directly proportional to the molecular concentration within the sample. Selection Rules for IR Spectra The selection rules for IR spectroscopy are not the same as those for Raman spec-troscopy. Active IR vibrations require a change in molecular dipole moment. The CO2 molecule is used to demonstrate active and inactive IR modes. Figure (2.5) illustrates the changes in dipole moment in the four normal vibrations of CO2. Fig-ure (2.5a) shows that there is no net change in dipole moment in the symmetric stretch of CO2, therefore this mode is IR inactive. On the other hand, both the anti-symmetric stretch and symmetric bending mode of CO2 undergo changes in dipole moment as a function of atomic displacements, therefore both these modes are IR active. 2.2.3 Raman Versus IR Spectra Both Raman and IR spectroscopies provide vibrational information about a molecule. Since the origin of Raman and IR spectra are markedly different, the techniques are, in many ways, complementary. Perhaps the best example of the complementary na-ture of the two techniques is given by the "rule of mutual exclusion" which states that if a molecule has a centre of symmetry (e.g. CO2) then vibrations can not be both 45 *) Q • Q O • O O • O c> O # - 0 O • O 0 - + O Figure 2.5: Change in dipole moment for the three normal modes of vibration of CO2. a) symmetric stretch (no dipole moment), b) symmetric bend and c) anti-symmetric stretch. The central configuration in each row corresponds to the molecules equilibrium position. There is no net dipole moment in the equilibrium position. Raman and IR active. In general, vibrations that do not distort the molecule (e.g. symmetric vibrations) are strong in Raman. Vibrations which distort the molecule (e.g. bending vibrations) are strong in IR. Several differences are mentioned briefly, as follows: Generally, vibrations are strong in Raman if the bond is covalent (eg. C—C, large distortions in polarizability) and strong in IR if the bond is ionic (e.g. O—H, large distortions in dipole moment). Raman spectra are generally acquired by use of a laser as the exciting source. As a result, Raman spectra can be acquired on small samples. This is more difficult with conventional IR spectroscopy where the exciting source is a lamp. Since glass is a weak Raman scatterer, sample housing is less problematic than in IR spectroscopy where glass is a strong absorber. Similarly, water is a weak Raman scatterer and so Raman spectra of samples in solution can readily be obtained. Water exhibits a strong IR spectrum and so acquisition of IR spectra of samples in solution is problematic. Intense laser radiation may cause sample photo-degradation or heating. This 46 is less of a problem in IR spectroscopy. Since incident radiation in Raman experiments can still be in the visible or even ultra-violet, sample fluorescence may be a problem. 2.2.4 Energy Units and Regions of Molecular Spectra Several different units can be used when discussing spectra. The wavelength (A) is typically expressed in meters (or some variation thereof, for example, Angstroms A= 10~10 m). The frequency (v = c/A, c =speed of light) is expressed in inverse seconds. By far the most popular unit in vibrational spectroscopy is the wavenumber, v — 1/A, expressed in cm - 1 . The energy of a transition (AE) can be expressed in any of these forms, AE = hv = hc/X = hcv. Expressing spectra in terms of either frequency or wavenumber has the advantage that these terms are directly proportional to the energy of the transition. Vibrational transitions occur in the energy range of 102 cm - 1 to 104 cm - 1 . Figure (2.6) illustrate the relationships between the various units and the energy range for Raman and IR transitions, as well as a few other common physical processes. 2 . 3 E x p e r i m e n t a l C o n s i d e r a t i o n s The following section describes a few of the fundamental experimental concepts for performing Raman spectroscopy. The two main methods of acquiring spectra, Fourier transform spectroscopy (used in this thesis) and grating spectroscopy, are described. Section (2.3.1) describes the basic operation of a Fourier transform in-strument. For completeness, section (2.3.2) briefly mentions the operation of a grating instrument. Finally, a comparison between the two techniques is made. 47 Infrared, Raman N d : Y A G FT-Raman Stokes 1 Radio , Radio , M i c r o - Infrared, Ultra- X- ray y-ray Nuclear Electron Wave Raman violet, Magnetic Spin Resonance Resonance Vi s ib l e (NMR) (ESR) I I I I I I I I io" 4 io" 2 i o ° 10 2 10 4 1 10 6 10 8 1 0 1 0 Wavenumber (cm 1 ) i i i i i i i i 10 4 10 2 10° 10"2 IO' 4 "IO"6 10" 8 10" 1 0 Wavelength, % (cm) i i i i i i i i 3x10 6 3x10 8 3 x10 1 0 3 x 1 0 1 2 3 x10 1 4 3 x 1 0 1 6 3 x 1 0 1 8 3x10 2 Frequency, v (Hz) Figure 2.6: Relationship for three different sets of units used in Raman and/or IR spectroscopy. Shown are the ranges for the various physical processes. 48 2.3.1 Fourier Transform Spectrometer Basic Interferometer The central component of a Fourier transform (FT) instrument is the Michelson interferometer. A Michelson interferometer is shown in its most basic form in figure (2.7) and consists of two perpendicular mirrors, one of which is allowed to move, a beam splitter and a compensating plate. Light from a source is incident on the beamsplitter, which divides the light into two components and directs the light onto the mirrors. Light reflected from the mirrors interferes back at the beamsplitter and a portion of this interfered light is detected at the detector. The compensating plate is made of the same material as the beamsplitter and ensures that both components of the incident light travels through equal amounts of beamsplitter material. Thus, if the beamsplitter material absorbs or scatters any incident light, this scattering will be the same for both components of the incident light. Light reflected from each mirror will interfere back at the beamsplitter. The interference will either be constructive or destructive depending on the path length difference (termed retardation, 5) traveled by each component of the light. This in turn depends on the difference in distance between each mirror and the beamsplitter. For example, for a monochromatic light source, if the path length difference traveled by the light is an integer multiple of the wavelength of the source, then constructive interference occurs and is detected. If the path length difference is 1/2 integer multiples of the wavelength of the source (i.e. the difference in distance between each mirror and the beamsplitter is 1/4 wavelength) then destructive interference occurs at the beamsplitter. If one mirror is allowed to move, the path length difference varies and a pattern of constructive and destructive interference occurs. If light 49 Detector t Lens Stationary L^Mirror •Beamsplitter Compensating ""Plate Movable Mirror Figure 2.7: Schematic diagram of a basic Michelson Interferometer. intensity is plotted as a function of mirror displacement, or retardation, the result is an interferogram. For the case of monochromatic incident light the interferogram is a sine wave pattern. In general, the wavelength (or frequency) of the light source may be obtained by performing a Fourier transform on the interferogram. Interferogram Intensity For the simple ideal case of monochromatic incident radiation, the intensity of the interferogram I at a given retardation <5 is given by where I(v) is the intensity of the light source at wavenumber v. The first term in equation (2.17) describes the DC component of the interferogram and the second term describes the AC component, which is the component of interest. In practice, = 0.5 ! ( * ) ( ! + c o s 27ritf) (2.17) 50 the intensity of the interferogram is not only a function of the light source but also of the beamsplitter efficiency, detector response, and amplifier response. Of these factors, only the light source intensity changes from one set of measurements to another. Hence, a single wavenumber-dependent correction factor may be applied to account for the above instrumental characteristics and the interferogram intensity may be written as 1(8) = B (P) cos 2IT95 (2.18) where now B(u) is the source intensity as modified by the'instrumental character-istics and only the AC component has been written. For the more realistic case of a polychromatic light source, the interferogram can be written as the integral /+ 0 O B(v)oa&2iru8du (2.19) •oo which is one half of a Fourier transform pair. To obtain the spectrum of the incident light source, as modified by the instrument characteristics (B(p)), the other half of the Fourier transform pair is used, /+ 0 0 1(8) cos 2ixv8d8 (2.20) •oo Since 1(8) is an even function, r+oo B(v) = 2 1 1(8) cos 2-KD8 dS (2.21) Jo Equation (2.21) shows that in order to obtain a spectrum B(v) with infinitely high resolution, an infinitely high retardation would be required. In practice this is unattainable. The effect of finite retardation is to introduce a finite resolution in the wavenumber spectrum. 51 Finite Resolution The dependence of the spectrum resolution on the retardation of an instrument may be illustrated conceptually by considering a source consisting of two spectral lines, v\ and V2 both of equal intensity and separated by Av = vi — v\. The interferogram of each individual line is a sinusoid. At zero retardation both interferograms are in phase. At a retardation of \Av~l both inteferograms are out of phase and after a further \Av~x the two interferograms are back in phase. Thus, in order to resolve two lines separated by Ai/, a retardation of at least A ^ - 1 is required, i.e. 8=4- (2-22) The mathematical effect of moving the mirror a finite distance A is to mul-tiply equation (2.20) by a boxcar function, D(S) = 1 for - A < 6 < +A ~ ~ (2.23) D(S) =0 for S > | A | Hence equation (2.20) becomes /+ 0 O I{6)D{6) cos 2nP5d5 (2.24) -oo The Fourier transform of the product of two functions is equivalent to the convolu-tion of the Fourier transform of each function. This can be used to gain insight into the mathematical consequence of a finite retardation. The Fourier transform of the interferogram 1(6) is simply the ideal spectrum, B(v). The Fourier transform of the boxcar function DS is a sine function / (&) f(u) = 2 A S m „ ( 2 7 r ^ A ) = 2A sine (2TTPA) (2.25) Equation (2.25) is plotted in figure (2.8). When f(p) (i.e. equation (2.25)) is convolved with B(i7i), which is taken to consist of a single spectral line, the result is 52 Figure 2.8: A sine function (equation (2.25)) whose central peak has a maximum width of A - 1 . a line the shape of a sine function, centred around the frequency of the spectral line. That is, the recorded spectral line now has a maximum width of A (cm). Therefore, in order to completely resolve two spectral lines, they must be separated by at least A, which is the same result as derived above. Apodization The sine function described above has sub-lobes, or feet, of intensity on either side of the central peak. Steps can be taken to reduce the size of the feet. Apodization refers to the suppression of the side-lobes, or feet. By changing D(8) from a boxcar function to some other function (e.g. cosine, cosine squared etc), the magnitude of the side lobes may be reduced. Griffiths and de Haseth [45] have illustrated the effect of using several different "apodization functions" {D(8)) on the resulting spectral line shapes. Generally, reducing the magnitude of the side lobes has the negative effect of broadening the width of the central peak (line) and vice versa. Thus, care must be taken in choosing an apodization function. 53 Time Domain 7(t) Multiplication I(t)x7(t) t FT Pair t Convolution FT Pair V(max) V(max) V Frequency Domain Figure 2.9: Illustration of aliasing. In the time domain, the waveform I(t) is sampled at intervals T by a "comb" function (7(t)). In the frequency domain, the Fourier transform of the initial waveform is the ideal spectrum (B(i/)) which is convolved with the FT of the sampling comb function (T(i/)) which has spacings of 1/T. Shown are the effects (aliasing) of an insufficient sampling interval (T), i.e. when 1/T < 2 iw-Effect of Finite Sampling Since the data collected from the interferometer is digitized (for computer storage), the data is not available as one continuous set but, rather, as a discrete subset of the analog spectrum. This can have an effect on the quality of the spectra and consideration must be given to the data sampling frequency. The mathematical effect of sampling the data at discrete intervals is to multiply the interferogram by a constant impulse, or comb, function (figure (2.9)). A comb function is a series of Dirac delta functions separated by equal intervals. Again, this multiplication has the effect of convolving the Fourier transform of the comb function with the Fourier transform of the interferogram. The Fourier transform of a comb function with 54 an interval between impulses (i.e. sampling frequency) T is another comb function with an interval of 1/T separating impulses. This is convolved with the Fourier transform of the interferogram (i.e. the spectrum). The effect of this convolution is to repeat the spectrum ad infinitum. If the spectrum covers a bandwidth of frequencies between 0 and vma,xi then the intervals between impulses (samples) in the transformed comb function must be at least 1/T > 2umax, otherwise the spectra begin to overlap and useful information is obscured. This overlap is termed aliasing. Thus, in order to avoid aliasing, a sampling frequency of at least twice the highest frequency in the spectrum is necessary. 2.3.2 Grating Spectrometer The Fourier transform instrument is one of two of the most popular methods for acquiring the spectrum of a light source. Dispersive instruments can also be used to this end. In this system, light from the source is passed through a monochromator which filters all but a chosen narrow frequency range of the light source. This light is then detected in a detector (e.g. photomultiplier tube). The monochromator essentially consists of an entrance slit of variable width, a grating, and an exit slit, also of variable width. Double or even triple monochromators are available which have two or three gratings in succession. This helps eliminate stray light from reaching the detector. In typical operation the angle of the grating with respect to the entrance and exit slits is set such that the desired frequency of light is impinged on the exit slit (and hence the detector). Light is collected in the detector for a given period of time. The grating is then rotated incrementally such that a new frequency of light is detected in the detector and light is again collected for a period of time. Thus, 55 each frequency of light from the source is collected in the detector sequentially. The resolution of the instrument is determined in large part by the width of the entrance slits. 2.3.3 Advantages and Disadvantages of Fourier Transform over Grat-ing Spectrometers The Fourier transform instrument has several advantages as well as disadvantages compared to a grating instrument. Of the principle advantages of the interferome-ter, the first is what is sometimes known as the multiplex, or Fellgett's, advantage and essentially pertains to the fact that all wavelengths of light are collected simul-taneously with the interferometer whereas each wavelength is collected sequentially with the grating instrument. Thus, theoretically, given the same acquisition time, resolution, source, detector, optical throughput, and optical efficiency, the signal to noise ratio (SNR) in spectra acquired with an interferometer should be greater than the SNR in spectra acquired with a grating instrument by a factor of \fM [45] where M is the number of resolution elements. Furthermore, because all wavelengths of light are collected simultaneously with the interferometer, acquisition time of com-parable spectra are greatly reduced with the interferometer as compared to a grating instrument. The second advantage of the interferometer is termed the Jacquinot's ad-vantage and is essentially a throughput advantage. In order to avoid resolution degradation in grating instruments, the solid angle of light incident on the grating must remain small. This is accomplished by narrowing the entrance slits on the instrument. As a result, the optical throughput is decreased in the grating instru-ment. On the other hand, the interferometer is able to collect a wide solid angle of 56 light with minimal loss in resolution. The resolution of an interferometer is limited primarily by the distance traveled by the mirror (section (2.3.1)) and not by the solid angle of light collected. For Raman measurements, this is a significant advantage as the increase in signal resulting from the higher throughput in the interferometer ul-timately allows for the use of infrared lasers as the excitation sources for scattering. That is, the increase in throughput can be used to combat the u4 [y =frequency of incident radiation) dependence on the intensity of scattering. For example, a visible laser whose incident frequency is twice that of an infrared laser will produce scat-tering which is 16 times as intense as the infrared laser. The increase in throughput of the interferometer instrument can be used to help combat the decrease in signal which occurs when an infrared laser is used. The use of infrared lasers is advanta-geous as it minimizes sample fluorescence and allows for a wider range of molecules to be studied. A third advantage of the interferometer over the grating instrument pertains to the accuracy of the x-axis calibration. Since the interferometer mirror position is monitored by a laser (e.g. HeNe), the mirror position and hence line frequency is known very accurately. A grating instrument is seldom calibrated to such accuracy. A disadvantage of interferometer instruments occurs if the light source is not constant in intensity over the interferometer scan. In this case fluctuations as a function of time are transformed into noise at all frequencies in the resultant spectrum. This problem can be avoided with a grating instrument by, for example, dwelling on each frequency for a given beam current. A second disadvantage occurs if the source of light (i.e. sample) contains several extremely strong lines as well as very weak features which are desired. If the detector collecting light from the interferometer has several gain settings, then 57 the gain must be set so as not to saturate the detector. This, in turn, may make detection of the weak features difficult. This problem can be overcome with grat-ing instruments by increasing the sensitivity of the detector collecting light from the monochromator and dwelling only on frequencies where the weak features are expected, hence avoiding detector saturation. 2 . 4 A p p l i c a t i o n s o f R a m a n S p e c t r o s c o p y In the last 10 to 15 years Fourier transform Raman spectroscopy has become an extremely popular method of probing molecular structure. This dramatic rise in popularity was fuelled largely by the technological developments (e.g. infrared lasers and interferometers) of prior decades. Studies performed using Raman spectroscopy number in the thousands per year and are thus far too many to list here. Raman spectroscopy remains a useful method for analyzing a variety of samples. The ap-plication of Raman spectroscopy to biological and biochemical assemblies is a vast field. Earlier works are summarized in several books by Spiro [46], Parker [47], Carey [48], and Tu [49]. More recently, the literature of Raman spectroscopy applied to biological assemblies is reviewed by Levin and Lewis [50] and Hanlon et al. [51]. Polymers have also been extensively studied. Early work, with literature references, is described in the book by Painter et al. [52] and in review articles by Gerrard and Maddams [53] and Koenig [54]. More recently, articles by Bulkin [55], Hall-mark [56], and the book by Koenig [57] review more current studies on polymers. Many other applications of Raman spectroscopy exist, for example: environmental applications (e.g. determination of ionic species in ground water) [58]; applications in the paint industry (e.g. monitoring polymerization reactions) [59]; forensics (e.g. identification of illicit drugs) [60, 61]; and petroleum applications (e.g. probing the 58 composition of gasoline) [62]. The above list is certainly not exhaustive. The Inter-national Conference on Raman Spectroscopy (ICORS) is held biannually and most papers from the conference have been collected and published in volumes since the early 1970's. Further series of published works are contained in Advances in Infrared and Raman Spectroscopy [63] and Vibrational Spectra and Structure [64] and offer reviews of many applications of Raman spectroscopy. 59 Chapter 3 Polyacrylamide Gel Dosimeters The first parts of this chapter describe the basic chemical reactions involved in poly-merization of monomers present in the polyacrylamide gel. Section (3.1.1) describes the basic reactions between radiation and water which lead to the production of water radicals. Section (3.1.2) describes the specific reactions involved in polymer-ization of monomers of the type present in the polyacrylamide gels. The latter sections of the chapter describe the specific polymer gels used in this thesis (section (3.2)) and known environmental factors affecting the polymerization within the gel (section(3.3)). A literature review of some of the work done in the area of polyacry-lamide radiotherapy gel dosimetry is given in section (3.4). Other gel dosimeters used in radiotherapy are briefly mentioned in section (3.5). 60 3.1 Mechanisms of Polymer Formation 3.1.1 Water Radiolysis Upon irradiation, most of the initial interaction of the incident photons or particles will be with water molecules, since the polyacrylamide gel is composed of 89% water. The interaction between radiation and water, termed radiolysis, has been extensively studied [65-67]. A brief review of only the most important reactions is given below. The initial stage of radiolysis involves the transfer of energy between the incident radiation and the medium. The incident radiation produces ionizations and excitations in the water, H20 ~» e~+H20+ (3.1) H20 ~» H20* (3.2) A molecule in an excited state is denoted with an asterix (*). The duration of this first stage is on the order of 10 - 1 5 s. The second stage of water radiolysis usually takes place within 1 0 - u s or less and involves the hydration of the electron, the creation and hydration of the hydronium ion (H^O+), as well as the dissociation of excited water. These reactions are, e~ + nH20 —• e" (3.3) H20+ + H20 —• H30++OH (3.4) H20* —*• -H+OH (3.5) Radicals are denoted with a dot (•) next to the species, which indicates an unpaired electron. e~q denotes a hydogenated electron (an electron surrounded by water molecules). It is postulated [66] that the contribution of excited water molecules to 61 the production of primary radicals is of secondary importance in comparison with that of the ionization process. The third stage of radiolysis consists mainly of the reaction of primary species and the establishment of chemical equilibrium. These processes begin approximately 10 - 1 1 s after the passage of the radiation. The more important reactions are, 2e-g + 2H20 —> H2 + 2OH- (3.6) e~g + -H + H20 —> H2 + OH- (3.7) e-0 + -OH —• OH- (3.8) e-g + H30+ —• H + H20 (3.9) e-g + H202 —> -OH + OH- (3.10) e-g+H20 —• -H + OH- (3.11) H30+ + OH- —• 2H20 (3.12) •H+H —> H2 (3.13) •OH + -OH —» ifsOa (3.14) •H+OH —»• i? 20 (3.15) •OH + H202 H20+H02 (3.16) A number of other radical reactions are known to take place involving the primary radicals and the radiolysis products [65, 66]. Only the most important reactions have been shown here. The polymerization of monomers (3.1.2) present in the gel is primarily initiated with the above reactive species. 62 3.1.2 Polymerization of Monomer For the types of monomers present in the gel, the most predominant form of polymer-ization is referred to as vinyl polymerization and involves the breaking of the carbon double bond on the vinyl (CH2 = CH) group of the monomer. Vinyl polymerization is chain-like in nature, as described below. In general, vinyl polymerization can be initiated by ionic as well as free radical species [68, 69]. However, for radiation-chemical initiated polymerization in aqueous media, water molecules dominate the reaction system, as described above, and ionic species are scavenged before they can initiate polymerization [68]. Hence, the predominant form of polymerization in the gel systems under study is free-radical chain polymerization. Detailed descriptions of chain polymerization are given in standard texts [70, 71]. A brief description is given here. Radical Chain Polymerization Chain polymerization requires an initiator species. In this case, radiation acts to form reactive radicals of water (section (3.1.1)). Polymerization occurs by the initial reaction of the radical with a monomer, and the subsequent propagation of this reactive centre through the successive additions of monomer molecules. For example, the vinyl monomer CH 2 = CHY can be polymerized by interaction with a radical •R, •R C H 2 = C H Y > R - CH 2 - YHC- C H 2 = C H Y ) R - CH 2 - YHC - CH 2 - YHC- C H z = C H Y ) . . . (3.I7) As seen, polymerization only occurs through the reaction of monomer with the prop-agating reactive centre. In radical chain polymerization it is possible to produce a 63 high molecular weight polymer at all percents of monomer conversion (i.e. in early and late stages of polymerization). The molecular weight of the polymer does not grow in a step wise fashion that is dependent on the percent conversion of monomer to polymer. Instead, high molecular weight polymer is produced immediately. At low percent monomer conversion (i.e. early stages of polymerization), there is less high molecular weight polymer than at high percent conversions. Hence, in radical chain polymerization the polymer size is generally independent of percent conver-sion, although the amount of polymer does depend on this factor. Initiation, Propagation and Termination of Polymer The radical chain polymerization reaction proceeds in three steps, initiation, propa-gation, and termination. The initiation stage is considered to involve two reactions [71]. The first reaction involves the production of free-radicals (R-) from some ini-tiator (I), I —• -R (3.18) In this case the initiator is radiation and the free radicals are obtained from radi-ation interaction with water, as described above. The second part of the initiation involves the interaction of this radical with a monomer molecule to produce the chain initiating species (Mi-), •R + M A -Mi (3.19) Here M represents the monomer molecule, its subscript indicates the number of monomers in the chain, and k{ is the rate constant for the initiation step. The propagation step of polymerization involves the growth of monomer 64 radical Mi by chain reaction with other monomer molecules, •Mi + M •M 2 (3.20) •M 2 + M •M 3 (3.21) •M 3 + M •M 4 (3.22) or, in general •M„+M - M n + 1 (3.23) Here kp is the rate constant for the propagation stage. In general, propagation of the polymer to high molecular weights occurs very rapidly. The situation is complicated somewhat when two monomers (Ma and M6) are present in the system. This is the case for the polyacrylamide system, where two separate monomers are initially present (see section (3.2)). In this case the following propagation reactions may occur, •M" + M° •MS (3.24) •M" + M 6 kpab •Mf (3.25) M Q + -Mj •Mb2a (3.26) •M* + M 6 fcp&6 •Mb2 (3.27) The final step in the polymerization process, termination, occurs by the annihilation of the reactive centre. In general, termination can occur via some of 65 the following reactions •M a + -M a •t,im+n (3.28) •M a +-M 6 (3.29) (3.30) ivJ-m ~ • l v in (3.31) M° +M 6 (3.32) •M 6 + -Mb M 6 + M a (3.33) Here fcj is the rate constant for the termination process. In all of the above termi-nation processes the result is an unreactive, sometimes termed dead, polymer. Equations (3.18) - (3.33) describe an idealized system. The presence of other constituents within the initial mixture can affect the resulting polymerization. A common example of this is inhibition, where the polymerization is prematurely ter-minated by the presence of another substance (inhibitor). In the case of polyacry-lamide gels, oxygen acts as a strong inhibitor of polymerization. A typical reaction involving the inhibition of polymer by oxygen is •M n + 0 2 — • M„ -OO (3.34) The peroxy radical then reacts with another radical to form inactive products. Thus, oxygen must not be present in the gel system in order for polymerization to proceed under normal circumstances. 3.2 Polyacrylamide Gels Radical chain polymerization is the basis for the reaction mechanism in the poly-acrylamide gel (PAG). PAG consists of acrylamide (molecular weight = 71.04 g/mol, 66 figure (3.1a)) and N,N' methylene-bis-acrylamide (bis, 154 g/mol, figure (3.1b)) monomers. Radical chain polymerization proceeds through the breaking of the car-bon double bonds on both monomers. Acrylamide contains only one double bond and hence, in general, forms linear polymer structures. The bis monomer, on the other hand, contains two carbon double bonds, both of which are able to react, and hence a 'crosslinked' polymer can be formed upon the polymerization of bis. Figure (3.1c) shows the general form of a crosslinked polymer structure. A detailed discus-sion of the types of polymer formed in an irradiated polyacrylamide gel is given in section (5.7.3). The initial composition of polyacrylamide gel is characterized by the total weight fraction (%T) of both monomers and the relative weight fraction (%C) of bis to total monomer. That is, %T = 100^±^% (3.35) msoi %C = 100^-% (3.36) ™*a+b where ma+{, is the total mass of the acrylamide and bis monomers, mb is the mass of bis, and msoi is the total mass of the solution. A typical PAG will consist of 3% acrylamide, 3% bis, 5% gelatin, and 89% water. Hence, this gel would be designated as a 6%T, 50%C gel (since the total monomer comprises 6% of the solution and half of the total monomer is bis). Changing the composition of the gel has a significant effect on the resulting dose response (see section (5.7)) and polymer structure (see section (3.4) and section (5.7)). However, the changes in composition are limited due to the solubility of bis. Bis can only be dissolved to a maximum concentration of approximately 3% by weight. Some workers report on 67 C H , = C H C = 0 I N H , (a) C H , = C H I c = o I N H C H 2 I N H I C = 0 C H , = C H (b) • C H C H 2 - C H -I c = o I N H , C H 2 - C H -I c = o I N H I C H 2 I N H C H 2 - C H -C = 0 C H , N H , C = 0 I - C H 2 - C H -I c = o I N H C H 2 - C H -I c = o I N H , C H 2 - C H C H 2 - C H -I c = o I N H , C H 2 -C H 2 I N H I C = 0 I • C H 2 - C H -(c) Figvire 3.1: Chemical structure of (a) acrylamide, (b) bis-acrylamide and (c) poly-acrylamide. 68 manufacturing gels with higher bis concentrations (4-5%) [72-74] although this is difficult to achieve in practice. 3.3 Factors Affecting Gel Polymerization and Stability The following sections describe the major factors affecting the polymerization pro-cess as well as the post-irradiation gel stability. 3.3.1 Oxygen The single most significant inhibitor of monomer polymerization in polyacrylamide gels is oxygen. Oxygen is highly reactive and hence acts as a radical scavenger. Atmospheric concentrations of oxygen are sufficient to completely inhibit polymer-ization, hence polyacrylamide gels must be manufactured under anoxic conditions. Hepworth et al. quantified the mean diffusion time of oxygen through the gel and report a mean diffusion coefficient of (8 ± 2)-10-6 cm 2s - 1 [75]. Although they made an estimate on the concentration of oxygen necessary to inhibit polymerization, the margins of error on their measurements deem these estimates of limited usefulness. It is generally believed that the concentration of oxygen must be kept on the order of 0.01 parts per million (ppm) in order to ensure reproducible gel polymerization [76]. 3.3.2 Temperature Effects Gel Manufacture During gel manufacture (section (4.1)) gels are heated so as to allow gelatin and monomer to dissolve in the solution, as well as to expel as much oxygen from the 69 system as possible. The maximum temperature to which the gel is heated appears to have some effect on the resulting R 2 = 1/T2 (T2 =MRI relaxation rate) dose response [76]. A slight inverse variation between R 2 and maximum temperature of manufacture is reported [76], for temperatures between 40°C and 90° C. However, keeping the temperature of manufacture to within 2-3° renders this effect negligible. Heating the bis to too high a temperature (above ~45°C) can cause the monomer to spontaneously polymerize. Hence, if the gel is to be heated to a temperature above 45° C then bis is added only once the gel has cooled to below this temperature. Gel Irradiation The temperature at which the gel is irradiated appears to have little effect on the overall polymerization and subsequent stability of the gel. There is no change in dose response characteristics between a gel irradiated at 2°C or at 20°C, as measured with nuclear magnetic resonance (NMR) [22]. Gel Imaging/Readout Depending on the measurement technique, the temperature of the gel at the time of measurement can affect the signal measured from that gel. NMR measurements on polymer gels appear to be the most sensitive to this parameter. NMR studies performed by Audet and Maryanski [22, 77] indicate an inverse dependence between gel sensitivity and measurement temperature. The effect is significant enough (on the order of 10-20% variation in response over 20° C) that gels must be kept to within 1-2° C in order to obtain reproducible results. Typically, gels are measured at room temperature. On the other hand, gel temperature during x-ray CT measurements appears 70 to have little effect on the resulting measurements. Hilts et al. report negligible difference in CT measurements between gels measured 4°C and 20° C [30]. All studies performed in this thesis utilize Raman spectroscopy measurements made at room temperature (22°C). 3.3.3 Post Irradiation Gel Stability Two main types of post-irradiation gel instabilities have been identified. Each is described in turn. The first instability pertains to post-irradiation polymerization. It has been shown that the polymerization of monomers in unsaturated gels can proceed for up to ~12 hours post-irradiation [78, 79]. This is further supported by the NMR studies of DeDeene et al. [76]. This instability can effect all types of measurements (NMR, x-ray CT, Raman) and hence needs to be considered in order to obtain reproducible results. The second type of instability pertains to the long-term gelation of gelatin gels. Gelatin is essentially composed of polypeptide chains (—CNH=0 is a peptide bond), with each one wound into a left-handed helix. These chains are grouped in threes and each group is wound into a tight right-handed helix [80]. A growing network of these chains constitutes a gelatin gel. The network is established by cooling the gel below 35°C, allowing the junctions to be joined through hydrogen bonding. A number of studies have been performed on gelatin gels [81-84]. All of these studies demonstrate that the gelation is extremely rapid during the first minutes after the gel has been cooled below 35°C. However, although the evolution of the gelatin does slow considerably, true equilibrium may not occur for up to 30 days. This has the effect of changing the y-intercept of the R 2 vs dose response curve 71 in NMR measurements (~10-20% over 60 days) [76]. This phenomenon appears to have little effect on the Raman measurements, since the gelatin features in the Raman spectra are weak and independent of the monomer and polymer features. 3.4 Literature Review The literature pertaining to polyacrylamide gels is vast. A summary of the main studies present in the literature is given for each of the main areas of research pertinent to this thesis. 3.4.1 Studies on the Properties of Polyacrylamide Gels Gel Electrophoresis and Chromatography Polyacrylamide gels used in electrophoresis or chromatography differ from those used in radiation dosimetry in that electrophoresis gels do not contain gelatin. The monomer solution is polymerized using a chemical catalyst and used to separate mixtures of macromolecules into components of different sizes. Early work in the area of polyacrylamide gels for use in electrophoresis established that the formed polymer structure consists of a heterogenous network containing the water phase and the polymer phase [85-87]. The structure and kinetics of the heterogenous gel was further established in a number of studies [88-93]. Riichel et al. [87, 93] used transmission-electron microscope images to deduce the structure of formed polymer for a variety of %C and %T monomer solutions. Briefly, they found that the structure of the formed polymer varies from leaflets of polymer at low %C to random collections of bead-like structures at high %C. This is discussed in more detail in section (5.7). The pores containing the water phase also vary as the composition is 72 changed. As the amount of crosslinker increases (increase %C), the pores initially decrease in size, achieve a minimum at ~5%C and then begin to increase in size. Increasing the overall amount of monomer (%T) decreases the overall size of the pores. Radiation Dosimetry Polyacrylamide gels The effects of radiation on several different polymers were documented as early as the 1950's and early 1960's [94-98]. Most of these dosimeters consisted of high molecular weight polymer which degraded upon irradiation. The absorbed dose was determined by monitoring the amount of degraded polymer. However, these dosime-ters did not gain widespread use in 3D radiation dosimetry due to the fact that the absorbed dose was not spatially correlated in these dosimeters. The popularization of polymer gel irradiation for use in radiotherapy occurred in 1993 when Maryanski et al. [2] proposed the use of polyacrylamide gel imaged with magnetic resonance imaging (MM) as a potential tool for 3-D mapping of complex dose distributions. The early work on the fundamental properties of the gels largely involved establish-ing the dose response characteristics of the gel using NMR [22, 77]. These early studies reported the NMR dose response characteristics for gels of differing initial %C and %T. In all compositions, a linear dose response was reported, with the 50%C gel being the most sensitive to absorbed dose. Since this initial study, the linearity of these response curves has been put into question. For example, DeDeene et al. report on non-linear NMR dose response characteristics for a 50%C and 6%T gel [76]. Other early works involved studying the optical properties of the polyacry-lamide gels [28, 29]. These studies reported a negligible change in refractive index 73 for gels of varying initial composition, irradiated to a range of doses. Also reported was the changes in gel optical density. These changes were observed as a function of wavelength of incident light (restricted to visible wavelengths). Results for all compositions and all dose ranges indicate a sharp decrease in changes in optical density as longer wavelengths of light were used. Ultimately, these studies were used in aiding in the design of an optical scanner (see section (3.4.2)). Several of the more recent studies have been mentioned in section (3.3). Other recent studies into the fundamental properties of the polyacrylamide gels included establishing the radiological water equivalence of the gels. Using Monte Carlo simulations Keall and Baldock established that, when used with megavoltage photon beams, gels are water equivalent to within 2% [99]. That is, the linear attenuation coefficient for the gels and for water are equal, within 2%. They report a deviation of the linear attenuation coefficient from that of water for gels irradiated with photons of 100 keV or less. The water equivalence of the gels at megavoltage photon energies discards the need for correction factors in determining absorbed dose to water and, in turn, tissue. Also, an extensive study was recently performed which established a firm understanding of observed artefacts in MRI images of polyacrylamide gels [100, 101]. Several different imaging modalities have been investigated as possible re-placements for MRI imaging. X-ray and optical CT have been investigated as gel imaging modalities by several groups [28, 30, 102]. Also, ultrasound has been re-cently proposed for imaging of polymer gel dosimeters [103]. 74 Fourier Transform Raman Spectroscopy of Polyacrylamide Gels Fourier transform Raman spectroscopy was first used to probe radiation dosimetry PAGs by Baldock et al. [31]. This initial study was, largely, a "proof of principle" study which established the feasibility of using Raman spectroscopy to probe the gels. Prior to the commencement of this thesis, the above study was the only published work which utilized Raman spectroscopy to study PAGs used in radiation dosimetry. Since this initial study, and apart from the work in this thesis, several reports have appeared in the literature pertaining to Raman spectroscopy of PAG [104, 105]. These studies establish the non-linearity of the polymer gel response, and correlate this non-linearity to observed NMR dose response characteristics of the gels. Furthermore, several different initial gel compositions (i.e. variations in %T and gelatin concentration) are tested for possible new gel formulations. The rates of monomer consumption are shown to vary by a factor of ~ 1.5 - 2 for gels of initial monomer concentration varying between 2%T and 4%T and initial gelatin concentrations varying between 3% and 7%. Although the number of studies using Raman spectroscopy to study radiation dosimetry polyacrylamide gels is relatively few, there exists a moderate amount of literature pertaining to utilizing Raman spectroscopy to study polyacrylamide gels used in electrophoresis [88, 106-114]. Recall, electrophoresis gels do not contain gelatin and use chemical initiators as opposed to radiation to initiate polymerization. Most of these studies were concerned with understanding the reaction kinetics of the electrophoresis gels as well as identifying the features observed in the Raman spectra. With respect to this thesis, the above studies have provided "line lists" which allow for the identification of spectral features. 75 3.4.2 Studies on the Applications of Radiation Dosimetry PAGs Applications of polyacrylamide gels to problems in radiation dosimetry occurred almost immediately following the first reports on the system. Maryanski et al. used the polymer gel to characterize radiotherapy photon and electron beams as well as 1 9 2Ir and 1 3 7 Cs sources used in brachytherapy applications [2, 115, 116]. Also shown was the potential of using polymer gels in the area of stereotactic radiosurgery [115, 117]. The initial enthusiasm for polymer gel dosimetry has not waned, with re-ports appearing each year on the applications of polymer gel to radiation dosimetry. For example, Oldham et al. used the gel to investigate the dosimetry of a static nine-field intensity modulated irradiation [118]. Other reports have appeared us-ing polymer gel to verify intensity modulated radiation beams [119], radiation from brachytherapy sources (137Cs) [72, 120], stereotactic radiosurgery [24] and as a po-tential dosimeter in boron-neutron capture therapy [121]. The "Is* International Workshop on Radiation Therapy Gel Dosimetry" was held in Lexington, Kentucky in 1999. A large number of articles pertaining to applications of polymer gels to radiation dosimetry appear in the proceedings to the conference [122]. A second conference was held in Brisbane, Australia in 2001 [123]. 3 . 5 O t h e r 3 - D C h e m i c a l D o s i m e t e r s 3.5.1 Fricke Dosimeters By far the most popular gel dosimeter aside from the polyacrylamide gel is the Fricke dosimeter, as described in section (1.3.6). Although these dosimeters have been applied to problems in radiation dosimetry (e.g. [122, 124, 125]), the dosime-76 ter suffers from one major drawback. Since the ferric ions (Fe3+) ions are small, they slowly migrate through the gel and hence spatial resolution is degraded with time [126, 127]. This process begins immediately post-irradiation and is significant enough that a few hours post-irradiation the spatial information is essentially com-pletely degraded. Hence, Fricke gel imaging must occur as soon after irradiation as possible. An advantage of the Fricke system over the polyacrylamide gel is that the Fricke gels do not require anoxic conditions for manufacture. Furthermore, Fricke gels are not toxic. This simplifies the preparation technique. Recently a novel gel matrix has been proposed which reduces the diffusion coefficient of the ferric ions by up to a factor of 5 at room temperature [128-130]. The matrix is based on a polyvinyl alcohol polymer which is water soluble and can be crosslinked by freezing. 3.5.2 Other Polymer Dosimeters In recent years a number of other polymer gel dosimeters have been proposed for use in 3D radiation dosimetry. A polymer gel based on the radical polymerization of 1-vinyl-2-pyrrolidinone was proposed in 1999 [131]. Although this gel, termed VIPAR (1-Vinyl-2-Pyrrolidinone ARgon), has a lower dose sensitivity from the PAG system, its advantage is the non-toxicity of the monomer, making gel preparation easier. It has since proven useful in measuring stereotactic beam profiles [132]. Other monomers which have been considered include: acrylic acid [115, 133, 134], methacrylic acid [133, 135], sodium methacrylate [136], 2-hydroxyethyl acry-late [133] and 2-hydroxyethyl methacrylate [133]. All the above monomers exhibit slightly different dose response characteristics and many are currently under further study. Recently, Park and Schreiner have proposed an epoxy gel for use in radiation 77 dosimetry [137]. Finally, Fong et al. have recently proposed a methacrylate based gel which can be manufactured under normal room atmosphere [138]. This gel shows promise in utility as a 3D dosimeter and is currently still under development. 78 Chapter 4 Materials and Methods This chapter describes the experimental materials and methods and data analysis tools used for the studies in this thesis. Section (4.1) describes the gel manufac-ture materials and techniques. Sections (4.2) and (4.3) describe photon and proton irradiations of polymer gels. Section (4.4) describes the details of the Raman spec-trometer, and the methods used in performing spectroscopy on the polymer gels. Finally, section (4.5) describes the data analysis technique of correlation. 4 . 1 G e l M a n u f a c t u r e 4.1.1 Glove Box Polyacrylamide gels must contain minimal amounts of oxygen in order to allow for uninhibited polymerization. To aid in the anoxic manufacture of the gels, a glove box was designed and constructed (figure (4.1)). The glove box was manufactured from 0.8 cm thick clear Perspex with outer dimensions of 75 cm length, 54 cm width at base, 60 cm height, and a 10° front pane tilt. One side of the box has 79 (a) -Thumbscrew -Crossbar support -Crossbar / door latch -21--Door 29 (b) One-way outlet valve-|~Glove Holes -31--18--20-Electrical outlet ti fe AC 75 (c) Figure 4.1: Schematic diagram of glove box used in polymer gel manufacture, (a) left side, (b) right side and (c) front. 80 a 24 cm x 36 cm rectangular opening to allow entry into the box. The opening is sealed by a removable Perspex plate which is secured to the box by three aluminium braces clamped to the outside of the box. An airtight seal is maintained between the Perspex door and glove box. Nitrogen flow into the glove box is regulated by a combination of regulator valve on the nitrogen bottle as well as a flow meter (Gilmont Instruments, Barrington, IL, USA) mounted outside the box. Outflow from the box is via a custom made one-way valve. Outflow is constricted minutely so as to create a slight positive pressure inside the box, thus minimizing 0 2 contamination from the outer atmosphere. To allow for handling of materials within the box, two accordion gloves are mounted on 20 cm diameter circular openings on the front of the box. The inside of the glove box contains one hotplate/magnetic stir plate combination unit, a second magnetic stir plate unit, and a small fan to facilitate air/gas circulation. Two electrical outlets are mounted on the inside wall of the box and are powered by an external outlet. 4.1.2 Gel Preparation For the studies in this thesis, gels were manufactured in bulk (200 mL -1 L quanti-ties) and transfered to smaller sample tubes. The gels were composed of acrylamide monomer, N,N' methelyne bis-acrylamide crosslinker (6%T unless otherwise stated, %C as needed, both electrophoresis grade, Sigma Chem. Co, St. Louis, MO, USA), 5% gelatin (300 bloom, Sigma Chem. Co.) and 89% de-ionized water. The appro-priate amounts of each gel constituent were weighed on an electronic scale outside the glove box. All appropriate materials were placed in the glove box which was then sealed. To begin manufacture, the gelatin and water were stirred, using the stir-rer/hotplate unit, and deoxygenated at the same time as the glove box was allowed 81 to purge oxygen. In-flowing nitrogen was split into two flow tubes, one was allowed to fill and purge the glove box atmosphere, and a second was placed directly over the gel solution to aid in de-oxygenation of the gel. The length of time for purging was determined by calculating the length of time required to displace the volume of the box with nitrogen at the given flow rate. While it is acknowledged that the purging time may not be sufficient to completely remove all oxygen from the glove box, reproducibility studies (section (5.3)) established that constant, minimal oxy-gen levels were present in the glove box over the course of the experiments. After the given time, the gel was heated from room temperature to 30°C, at which point acry-lamide was added. Further heating and stirring was allowed, until the gel reached a temperature of ~55°C. At this point the gel was transferred to a cool stirring plate and cooled to ~44°C. Bis-acrylamide (bis) was then added and the gel was stirred until the bis dissolved. Finally, the gel was transferred to Pyrex glass NMR sample tubes (10 mm outer diam, 0.46 mm wall thickness, 178 mm length, Wilmad glass, Buena, N.J, USA). Sample tubes were capped with solid rubber stoppers (Fisher Scientific, Napean, ON, Canada). Once capped, the samples could be removed from the glove box and refrigerated to allow gels to solidify. 4.2 Gel Irradiation: Photons 4.2.1 Linear Accelerator All photon irradiations of the gels were undertaken using a Varian 21EX linear accel-erator (Varian Assoc., Palo Alto, CA, USA). A schematic of a typical linear acceler-ator design is shown in figure (4.2). Briefly, electrons are emitted from the "electron gun" and accelerated through the waveguide by ~3 GHz microwaves. Electrons 82 Bending magnets-j -Accelerator waveguide Microwaves- 1 1 L 1 l _ l L _ J L _ J 1—1 1 1 l — l l — l l — P r—I r—l 1—1 I I gun •Target •Primary collimator -Flattening filter :V-==]—Ion chamber C7 \ X \ X J^vIovable collimators Patient / phantom position Figure 4.2: Schematic diagram of components of a standard radiotherapy linear accelerator operating in photon mode. Diagram not to scale. exiting the waveguide are bent through 270° and incident on a removable tungsten target which is thick enough to stop the electrons. Forward peaked Bremsstrahlung photons are emitted from the slowing down of the electrons in the target. The beam then passes through the primary collimators (made of a heavy metal). In order to even out the lateral distribution, photons pass through a "flattening filter" typically made of tungsten. The flattened x-ray profile then passes through beam monitoring devices (e.g. ion chambers) and through movable collimators which determine the field size of the radiation beam at the exit of the accelerator. For detailed discussions on radiotherapy linear accelerator designs see, for example, [11, 139]. 4.2.2 Experimental Method Photon irradiations were performed using 6 MV x-rays, a field size of 20x20 cm2, and a dose rate of 300 cGy/min. Samples were placed in a custom designed acrylic phantom (17 cm length, 15 cm width, 12 cm height) with a 1 cm diameter cylindrical channel along the length and at 1.5 cm depth (figure (4.3)). The phantom surface 83 lOi 10, 10) 10i 15f> 170--75- ? 77 10 mm bon 120 120 Front Side Figure 4.3: Schematic diagram of sample tube phantom used for gel irradiations. Phantom material is acrylic. All dimensions in mm. to accelerator source distance was 100 cm. Sufficient acrylic to produce full scatter conditions was placed around the phantom. One sample was left unirradiated, as a control, and the remaining samples were irradiated to doses between 2 and 90 Gy, as needed. Gel samples were irradiated ~3 hours post manufacture. Samples were allowed to polymerize for ~24 hours post-irradiation. After this time the gels were exposed to oxygen. This is a conservative estimate on the time required to complete polymerization, as discussed in section (3.1.2). Oxygen exposure ensured inhibition of any further, accidental, polymerization. Samples were stored in a light tight environment and oxygen was allowed to diffuse through the samples for 7 days by replacing the rubber stoppers with oxygen permeable plastic caps (ensuring minimal gel dehydration). This amount of time ensured complete oxygen diffusion through the sample [75]. Raman spectra were acquired on the gel samples after they were rendered inactive. 84 4.3 Gel Irradiation: Protons 4.3.1 Cyclotron The TRIUMF cyclotron facility (Vancouver, BC, Canada) was utilized for the proton irradiations of gel samples. The cyclotron is capable of producing monoenergetic proton beams between 65 MeV and 500 MeV. Negative hydrogen ions (H~) are accelerated up to energies of 500 MeV and extracted by intercepting the beam with a carbon wire or foil which strips off the two electrons. The resulting proton is bent away from the cyclotron into the appropriate beamline. Details of cyclotron operation can be found elsewhere [11]. The medical beamline is shown schematically in figure (4.4). Briefly, acceler-ated protons pass through a beam profile monitor, scatterer, collimator (collimator #1 in figure (4.4)), and neutron absorber (part of collimator #1). An acrylic range shifter is positioned after the first collimator which, when in place, allows for the range of the protons to be modified. A rotating modulator wheel can be posi-tioned next to the range shifter. The modulator wheel is a variable thickness piece of acrylic which, when spun, creates a range of proton energies which produce a spread out Bragg peak (SOBP) downstream from the wheel. Beyond the modulator wheel a secondary collimator (collimator #2 in figure (4.4)), neutron absorber (part of collimator #2) and an ion chamber are positioned. Finally, an adjustable brass collimator is positioned at the end of the beamline. This collimator ultimately sets the field size of the beam specific for a particular treatment. 85 Collimator #1-|—Range shifter Proton beam Beam profile monitor scatterei Figure 4.4: Schematic diagram showing the medical beamline at TRIUMF, Vancou-ver, BC, Canada. 4.3.2 Experimental Method The experimental set-up for the proton irradiations of the polymer gels is shown in figure (4.5). These experiments utilized a 74 MeV proton beam operating at ~5 nA and which has been commissioned for clinical use. The proton beam Bragg peak was spread over 23 mm using a rotating 20 step acrylic modulator wheel. A SOBP was chosen as opposed to a raw Bragg peak for these measurements in order to achieve a uniform physical dose over the sample volume and therefore minimize the dose error associated with uncertainty in determining the actual point of measurement in the sample with the Raman instrument (see following section). The beam was collimated with a 2.5 cm diameter circular collimator. Gels were placed in an acrylic phantom (figure (4.5)) and positioned at either 26 mm depth in acrylic (31 mm water equivalent depth, including 0.46 mm glass wall thickness), allowing for samples to be irradiated with the end of the SOBP, or at 22 mm (26.4 mm water equivalent depth), allowing for gels to be irradiated with the central portion of the SOBP. Gels were irradiated to physical doses between 2 and 50 Gy, leaving one sample unirradiated for a control. Proton dose on this beamline was calibrated using an air-equivalent ion chamber according to the International Commission on Radiation 86 a) 74 MeV Protons Modulator Wheel! gel_ sample SOBP PDD V b) 74MeV Protons Modulator Wheel"! acrylic ^ phantom gel sample SOBP PDD acrylic phantom Figure 4.5: Experimental set-up for proton beam irradiations. Sufficient acrylic build-up is positioned in front of the acrylic phantom such that gels are irradiated with the (a) central portion of the SOBP and (b) end SOBP Units and Measurement (ICRU) report 59 recommendations [140]. As in the case with x-ray irradiations, gels were exposed to oxygen ~24 hours post-irradiation. Raman spectra were acquired on gel samples 7 days post oxygen exposure. 4.4 Raman Spectroscopy of Polyacrylamide Gels 4.4.1 Raman Spectrometer Raman spectra of polymer gels were acquired on a Bruker FTS Raman spectrometer (Bruker Spectrospin, Milton, ON, Canada). Figure (4.6) illustrates schematically the Raman instrument. The spectrometer is equipped with an Nd:YAG laser oper-ating at 1064 nm and a liquid nitrogen cooled germanium detector. Sample com-partment optics are designed such that light scattered at either 90° or 180° to the incident light can be collected. Laser light was deflected upon a small mirror placed 87 Sample holder r-Sample y-16 mm lens A //-Mirror Filter module— Interferometei -Detector }•—Laser '—Micro-adjust Figure 4.6: Optical layout of Raman spectrometer. on a 45° angle at the centre of a 16 mm focal length collimating lens. This mirror deflects the incident laser light on to the sample while allowing for scattered light to be collected at 180° to the incident light. Backscattered light from the sample was collimated by the lens, deflected by a mirror, and passed through an interferometer and a notch filter which filtered out the Rayleigh component of the scattered light. The remaining light intensity was collected by the detector. Bruker software was used to collect and Fourier transform the data. 4.4.2 Raman Spectra Acquisition Since the commercially available sample holders provided with the Bruker instru-ment could not house the 10 mm NMR sample tubes used in these studies, a custom designed sample holder was built (figure(4.7)). This holder was constructed such that it could be mounted on the commercial 1-D translational stage. This allowed for accurate and reproducible sample positioning. Spectra were acquired on portions of the gel closest to the inner wall of the glass tube. For the proton irradiated sam-ples, tube orientation was marked prior to irradiation and samples were positioned accordingly in the spectrometer. Knowledge of the laser sampling point in the gel 88 Front View 1-10-1 6 mm boi 12 18 10 mm sample bqre 5 mm bore -Screw holes 32 1 l? 1 3.54H -20— J -50 1 12 Side View 50— 10 mm sample-bore Set •-r-3.5 -20 J -37 Screw holes mm bore mm bore r-10-i 32 N r—Set screw 25 -Screw holes -20 J -50 —j—5 Screw 1 jf 1-75 i10'65 -10 mm bore -37-i mm bore F R F S - i Back View Top View Figure 4.7: Sample holder designed to house 10 mm NMR sample tubes and mount to the commercially available 1-D translational stage. Sample is positioned through 10 mm bore. All dimensions in mm. is estimated to be accurate to ~0.5 mm. The precision of sample re-positioning in the Raman instrument is 1/100 mm. A laser power of the maximum allowable 200 mW and 1000 interferometer scans (~30 min.) were used for each acquisition (see section (5.1.3) for details). Instrument resolution was kept at the maximum allowable 4 cm - 1 for all acquisitions. 4 . 5 Data Analysis The resulting spectra were transferred from the dedicated Bruker PC to an indi-vidual PC. Data analysis software was written in-house (Visual C++, Microsoft Foundation Classes, Microsoft, WA, USA) which allowed for multiple spectra to be displayed at user defined magnifications, arithmetic operations to be performed on the spectra, peaks within a spectrum to be integrated, or peaks within a series of 89 spectra to be auto or cross-correlated together (see section (4.5.2) and appendix (A)). The results of each calculation could be output to a file. Functional fitting of single peaks within a spectrum was implemented into the code (Levenberg-Mardquart) but was not used for the actual fitting of the data. Peak fitting and background sub-traction were performed using FORTRAN code and MINUIT (CERN, Switzerland), running under LINUX. 4.5.1 Background Subtraction As will be seen in section (5.4), the background in each spectrum can vary minutely from one dataset to another as more polymer is formed in the gel. This is most likely due to the increase in purely translational and long chain bending modes in the polymer, creating a "wash" of background frequencies. Since the monomer and polymer features lie on top of this background, subtraction of the background is required. A host of techniques are available for background subtractions, some of the more common being: a) simple linear interpolation between band endpoints, b) functional fitting of the entire baseline, and c) functional fitting of the immediate region containing the bands under study. The first two methods are not used in this work, for the reasons described below. The simplest method of subtracting a background from a spectrum, or dataset, is to choose a linear baseline which is projected between the endpoints of the band in question. The baseline contribution to each data point is subtracted from each point in the dataset [141]. This method is inadequate, as results may be distorted if endpoints are chosen where noise is excessive or if endpoints lie upon some varying substructure. In effect, two datapoints are accounting for an entire baseline and are being projected onto all datapoints in the dataset. 90 Functional fitting of the entire baseline, and subsequent baseline subtraction using the fit, is not adequate in this study primarily because a priori knowledge of the functional form of the entire baseline is not known. Empirical determination of the functional form is possible, but does not yield the most certain results. Fitting a restricted dataset to a functional form which includes a separate background and peak function results in better estimates of the background than either of the two aforementioned techniques and is hence used here. Regions en-compassing the spectral features of interest were fit to a combination of linear back-ground and either Lorentzian or Gaussian peaks (best fit, depending on the spectral features). Results of these fits are given in section (5.4). The fit background from each spectrum was subtracted from the dataset and the resulting peak intensities were quantified, as described below. 4.5.2 Characterization, of Peak Intensity A host of techniques are available to characterize the peak intensities. Several of these techniques are peak height, peak area (integration), and correlation. Corre-lation offers a slight advantage over integration, as discussed below, and hence is chosen as the technique for characterization of acrylamide and bis bands. The poly-acrylamide data could not be characterized using correlation and hence integration is used. Peak Height Although measurements of peak height for a particular Raman band are the simplest type of measurements that can be made, this type of analysis is plausible only if the signal to noise ratio of the Raman band is high. In the high dose spectra acquired 91 for these studies this is not the case and peak height measurements become overly sensitive to high frequency noise. This effect is amplified by the fact that the entire measurement consists of a single data point. As a result of these considerations, peak height is not used. Peak Area Integration calculations are generally more robust estimates of the intensity of a particular spectral band. This technique utilizes all points in the region of the peak to be integrated, thus lessening the effect of high frequency noise. Peak area is easily calculated via simple summing algorithms which are fast and convenient. This technique is viable and is used to characterize the polyacrylamide peak intensity variation as a function of dose. It must be noted that this technique gives equal weight to each point in a dataset, that is, both low and high signal to noise portions of a spectrum are weighed equally. As seen below, this is a slight disadvantage when compared with the technique of correlation. Correlation Correlation of two functions can be used to quantify the similarity between two functions and can be used to study spectra acquired in series type measurements. Correlation between two functions r(r) and s(r) is defined by equation (4.1), /oo r(T)s(t + T)dT (4.1) -00 where z(t) is the correlation function. Correlation can be visualized as successive multiplications of two functions, r and s, separated by a distance t. The entire cor-relation function, z(t), can be used to detect periodicity in noisy signals. Important in this analysis is the fact that at t — 0 the correlation function gives a quantitative 92 degree of similarity between two peaks. This is the case because the value of z(0) is equivalent to overlaying the two datasets r and s, multiplying the sets together and integrating. Auto-correlation refers to the correlation of a dataset with itself, whereas cross-correlation refers to the correlation of two different data sets. In this application, r(r) can be chosen as a reference spectrum, or dataset, and S(T) as a sample spectrum, or dataset. If the reference and sample datasets are very similar (i.e. they contain similar quantities of the analyte being studied), then the value of the correlation function at t = 0 will be greater than if the two datasets are dissim-ilar (i.e. the sample set contains little analyte present in the reference). Ideally, if the sample dataset contains none of the analyte which was present in the reference dataset, then the correlation value at t = 0 should be zero. The successful use of correlation is not restricted solely to it's application to spectra containing isolated peaks with no superimposed background. First, it must be noted, correlation can be applied to a particular region of the spectra being studied, individual peaks can even be singled out for analysis. If the peaks of interest overlap with other peaks in the spectrum, the correlation value at t = 0 will attain a minimum when the analyte is absent from the mixture, as opposed to the ideal value of zero under this condition. Problems using this technique arise when the spectral band being studied heavily overlaps with other bands whose intensity also varies from one measurement to another. In this case additional pre-treatment may be required prior to correlation. Fortunately, for the features studied this was not required. Correlation has been successfully applied to problems in spectroscopy. One of the first applications was by Horlick, who identified components of a mixture by cross-correlating a reference analyte spectrum with a sample mixture spectrum [142]. 93 Tyson et al. and Mann et al. studied the properties of the correlation function using simulated data as well as applying their results to the determination of o-xylene in m-xylene and p-anisaldehyde in m-xylene using infrared spectroscopy [143, 144]. Further applications of correlation in spectroscopy are cited in references [145-147]. Acrylamide and bis peaks in this study were correlated using equation (4.1). The peaks were correlated separately for each dataset. The reference spectrum was chosen as the unirradiated PAG spectrum, which is the highest signal to noise dataset in the series. This reference spectrum was correlated with itself to produce the autocorrelation value, and then cross-correlated with the remaining spectra in the series to produce correlation values which could be compared relative to the autocorrelation value. It is noted that correlation produces a weighted measure of the dataset being studied. Since in this case the reference spectrum exhibits the highest signal to noise, the remaining sample spectra are weighed by this high signal to noise. That is, greater weight is given to the portions of the sample spectra that are meant to contain signal (i.e. the peak) as opposed to the tail ends of the peaks, which contain noise. This is an advantage as compared to integration which weighs each point equally. 94 Chapter 5 Results and Discussion I: Gel Response to X-Ray Irradiation This chapter presents results of the preliminary investigations undertaken for this thesis, as well as results and discussion of chemical changes occurring in photon ir-radiated polymer gels. Initial investigations include: determination of an appropri-ate sample holder (section (5.1.1)); determining appropriate acquisition parameters (section (5.1.2) and (5.1.3)); spectral identification (section (5.2)); reproducibility studies (section (5.3)) and results of peak fitting (section (5.4)). Monomer consump-tion and polymer formation curves for a "standard" composition gel (6%T, 50%C) are presented and discussed in sections (5.5) and (5.6). Monomer consumption curves for a range of gel compositions are presented and discussed in the context of a qualitative model in section (5.7). Finally, an example of the effects of pre-existing polymer in the gel on subsequent monomer consumption is given in section (5.8). Portions of the work presented in this chapter have resulted in two paper publications, first authored by the author of this thesis. Specifically, parts of sections 95 (5.4), (5.5) and (5.6) appear in [148]. Portions of sections (5.7) and (5.8) appear in [149]. 5.1 I n i t i a l I n v e s t i g a t i o n s 5.1.1 Housing of Gel Samples The material that is chosen to house polymer gel must be compatible with the following requirements: gel housing (minimal oxygen permeation); gel irradiation (minimization of dose perturbation); and acquisition of adequate Raman spectra (container should be a weak scatterer and thin). Most plastics which are currently used to house gel samples (e.g. Perspex) do not meet at least one of the above requirements. Hence, glass walled containers were investigated as an alternative. Glass has very low oxygen permeability [150] as well as typically exhibiting a weak Raman signal. Even so, care must be taken. Figure (5.1) illustrates Raman spectra acquired on unirradiated polymer gel samples housed in two different sample tubes, both specified to be constructed of borosilicate glass. The thin-walled Pyrex sample tubes (vial #2 in figure (5.1), Wilmad Glass) were chosen since they most adequately meet the above conditions. 5.1.2 Signal Intensity Each 10 mm sample tube was placed in the custom made sample holder and the distance between the sample and the collecting lens was adjusted with the thumb-wheel adjustment on the mount assembly. Spectra were acquired of a single sample for a range of thumbwheel positions, until spectra of optimized signal to noise ratio were acquired. This thumbwheel position was then recorded and held constant for 96 Wavenumber (cm"') Figure 5.1: Raman spectra of polymer gel housed in two different sample vials. Both are reported to be manufactured of boro-silicate glass. The gel signal is virtually completely masked by the fluorescence of sample vial #1. all experiments performed thereafter. With the sample optimally positioned, spectra of a single sample were ac-quired over a range of laser powers. Figure (5.2) shows signal intensity of the 1256 cm - 1 bis peak for three different laser powers, ranging from 50 mW to 200 mW. As expected, signal intensity increases with an increase in laser power. This result is discussed in the next section, in conjunction with the results therein. 5.1.3 Signal to Noise Ratio in Gel Spectra The laser power and the number of interferometer scans (AT) were chosen keeping in mind the following two factors: a) the signal to noise ratio (SNR) of the spectra increases with both an increase in incident laser power and with increasing N and, b) the potential photo-degradation of the sample at high laser powers and long scan times. As shown above, a more intense incident laser power has the effect of increasing the intensity of scattered light, hence increasing the signal detected. Furthermore, increasing N increases the resultant SNR by a factor of VN, in theory. 97 5 JE •: J i I i I 50 100 200 Laser Power (mW) Figure 5.2: Unirradiated gel Raman signal intensity as a function of incident laser power. To establish the SNR of the system, a single sample was scanned with a range of interferometer scans (N) and laser powers. The results are shown in figure (5.3). It can be seen from figure (5.3) that the SNR increases as the number of scans is increased, for all three laser powers. The SNR increase generally follows the expected \/N increase. Given the moderately intense laser power (200 mW) and potentially long acquisition time per spectra (30 min per 1000 scans), several test were applied to assess the potential photo-degradation of the sample. Both visual inspection and inspection of multiple spectra acquired on a single sample held in the same position for each acquisition revealed no evidence of photo-degradation. Also, Schrader et al. noted that by comparing the ratio of stokes to anti-stokes intensities for a given peak, the thermal properties of the sample can be studied [151]. The Raman spectra acquired here do not lend themselves to such an analysis easily, as anti-stokes data can be collected only to -606 cm - 1 . There is one weak acrylamide band at 304 cm - 1 . The anti-stokes intensity of this band is extremely weak, however, there 98 Number of Scans Figure 5.3: Signal to Noise Ratio (SNR) as a function of the number of interferometer scans and laser power for spectra acquired on an unirradiated polymer gel. is no significant change in the Stokes/anti-Stokes ratio of this band as the laser power is increased from 50 mW to 200 mW, or for spectra acquired with longer scan times. As a final note, Baldock et al. have used a laser power of 400 mW to study polyacrylamide gel and do not report any sample photo-degradation [31]. From the above considerations it is concluded that photodegradation should not be an important factor in determining the laser power and number of scans in acquiring spectra. As a result, 200 mW laser power and 1000 interferometer scans were chosen as the data acquisition parameters for all studies in this thesis. It was determined that these parameters were sufficient to achieve a reasonable SNR while not making scan times prohibitively long. 5 . 2 S p e c t r a l I d e n t i f i c a t i o n A full Raman spectrum of an unirradiated polymer gel is shown in figure (5.4). Vibrational band assignments were made from a combination of two methods. First, spectra were acquired on pure water as well as aqueous samples of monomer, 99 15 2000 5 L 0 b~ 2000 2500 l , 3000 3500 4000 Wavenumber (cm1) Figure 5.4: Raman spectrum of an unirradiated polacryiamide gel. crosslinker, and gelatin. A spectrum of an empty glass vial was also taken. The spectra of these constituents are shown in figure (5.5). Second, Raman frequencies of polyacrylamide gels used in electrophoresis have been studied by a number of workers [88, 106-114]. From these studies, a line list was compiled which enables labelling of observed peaks with particular vibrational assignments. Raman frequen-cies, and the corresponding vibration (if available) are listed in table (5.1). Each individual component of the unirradiated gel is briefly described below. Polymer features are discussed in section (5.6). 5 .2 .1 Glass Few features appear in the Raman spectrum of glass. A broad feature is present at 430 cm - 1 . Although a peak at 1070 cm - 1 is reported in the literature [31], it is not observed in figure (5.5). 100 0 500 1000 1500 3000 3500 Wavenumber fcm"'-) Figure 5.5: Raman spectra of individual gel constituents and glass vial. Table 5.1: Raman frequencies for polymer gel constituents. Vibrations are: va = symmetric stretching, va = anti-symmetric stretching , 8 = bending, u> = wagging, p = rocking. Molecules are: AA= acrylamide, BA= bis-acrylamide, PA= poly-acrylamide, GN= gelatin, W= water, GL= glass. Freq. (cm x) Vibration Molecule 308 8C-C-C AA 430 v Si-0 GL 495 8O-C-N AA 623 w C - 0 AA 818 PCR2 AA 833 v C - C AA 913 BA 979 u CH 2 AA 1052 AA 1114 AA 1126 v C - C PA 1256 8 CH BA 1285 8 CH AA 1414 8 CH 2 BA 1439 6 CH 2 AA Freq. (cm x) Vibration Molecule 1450 6 CH 2 PA/GN 1592 <5NH2 AA 1629 v C=C BA 1630 8 OH W 1634 v C=C AA 1648 PA/GN 1654 v C - 0 BA 1670 v C - 0 AA 2880 v CH PA/GN 2936 vs CH 2 PA/GN 3042 vs CH 2 BA 3050 vs CH 2 AA 3119 Va CH 2 AA 3256 vs OH W 3360 va OH W 101 5.2.2 Water Water is a comparatively weak Raman scatterer. A weak, broad band appears at 1630 cm - 1 . Stronger symmetric and anti-symmetric O—H stretching modes appear at 3256 cm - 1 and 3360 cm - 1 , respectively. The broadness of these features is due primarily to the large amount of hydrogen bonding occurring in water. 5.2.3 Gelatin The Raman spectrum of gelatin exhibits several moderately weak features. A CH2 bending mode is present at 1450 cm - 1 . An un-assigned band appears at 1648 cm - 1 . C—H stretching modes attributed to gelatin occur at 2880 cm - 1 and 2936 cm - 1 . 5.2.4 Acrylamide/Bis A number of features appear in the spectra of acrylamide and bis. A number of very weak and not completely identified features for both monomers appear be-tween 304 cm - 1 and ~1150 cm - 1 . A relatively strong, narrow and unobscured peak attributable to the vinyl C—H bending mode of each monomer occurs at 1256 cm - 1 (bis) and 1285 cm - 1 (acrylamide). A similar pair of peaks attributable to the CH2 bending modes appear at 1414 cm - 1 (bis) and 1436 cm - 1 (acrylamide). The lower frequency shift of the bis band for a given mode relative to the acrylamide band for the same mode is not completely understood. A number of features for both monomers appear between 1500 cm - 1 and 1700 cm - 1 . C=C stretching modes appear at 1629 cm - 1 (bis) and 1634 cm - 1 (acrylamide). C=0 stretching modes appear at 1654 cm - 1 (bis) and 1670 cm"-1 (acrylamide). Finally in this region, an NH2 bending mode appears for acrylamide at ~1592 cm - 1 . In the region encom-passing 3000 cm - 1 the strongest monomer features are the C-H stretching modes at 102 3042 cm - 1 (bis) and 3050 cm - 1 (acrylamide). A weak feature due to the C H 2 anti-symmetric stretch in acrylamide is observed at 3119 cm - 1 . Several other features in this region which are listed in table (5.1) are not observed, possibly due to the fact that they may be obscured by the broad water peaks. The acrylamide and bis features used for analysis in this study are the 1256 cm - 1 (bis) and 1285 cm - 1 (acrylamide) vinyl CH bending modes. These features are chosen as they are well resolved, relatively intense, and not obscured by other bands in the same region or by background. The pair of features occurring at 1414 cm - 1 (bis) and 1436 cm - 1 (acrylamide) are not used since this region is obscured by a polyacrylamide band at 1450 cm - 1 (see section (5.6)). The number of features occurring in the neighbourhood of 1550 cm - 1 to 1700 cm - 1 are not used as these features are not well resolved from one another. This makes quantitative charac-terization more difficult with these bands than with the 1256 cm - 1 and 1285 cm - 1 bands. 5.3 R e p r o d u c i b i l i t y S t u d i e s Reproducibility of the Raman spectrum of polyacrylamide gel had to be established before further experiments could be conducted. Furthermore, the gel dose response reproducibility was also established at this time. The reproducibility studies pre-sented here are divided into three sub-sections. First a single, unirradiated polymer gel was sampled along the length of the vial to establish the sensitivity of the sys-tem to variations in sample tube thickness, and local variations (if any) in polymer gel along the vial. Next, an "intra-batch" study was conducted to establish repro-ducibility of the spectra within a single batch of gel. Finally, an "inter-batch" study established the reproducibility of the dose response of the gel, for different batches 103 0.4 B D >, 0.3 < I02 B O B 0.1 0.0 • Bis, Ave: 0.266 +/- 0.002 o Acrylamide, Ave: 0.169+/-0.001 1 2 3 4 Trial Figure 5.6: Integrated peak intensity as a function of trial for a single vial of unir-radiated gel. of gel manufactured under identical conditions. 5.3.1 Same Sample Reproducibility A single sample tube was filled with 6%T, 50%C gel prepared under normal atmo-spheric conditions. Raman spectra were acquired at four separate locations along the sample. Since the background in each spectrum is constant (no polymer formed), the background what not subtracted and peak areas were calculated directly for the 1256 c m - 1 bis and 1285 c m - 1 acrylamide peaks. Results are shown in Figure (5.6). The standard deviation about the mean for each peak is <1% of the respective peak area, for both peaks. This result established the errors expected on each scan for the "mtra-batch" reproducibility study. 5.3.2 Intra-batch Reproducibility To test intra-batch reproducibility, a single batch of 6%T, 50%C gel was made and transferred to 4 sample tubes. Each tube was then irradiated uniformly to 5 Gy 104 • Bis, Ave: 0.220 + 0.003 o Acrylamide, Ave: 0.138 + 0.002 _ i i i i i i i 1 2 3 4 5 Sample Number Figure 5.7: Reproducibility of acrylamide and bis peak area for gels manufactured from the same batch of gel (transferred to several vials) and irradiated to 5 Gy. using 6 M V photons and the set-up described in section (4.2). A Raman spectrum was acquired on each sample. Since each tube was irradiated to the same dose, the background in each spectrum remained nearly constant and hence the background was not subtracted and peak areas were calculated directly for the acrylamide and bis peaks, as above. Results are shown in figure (5.7). Errors for each data point were taken as the standard deviation about the mean (expressed as a percentage) for each of the peaks in the "same sample" study described above. The standard deviation of the average peak intensity for the bis-acrylamide peak at 1256 c m - 1 is 1.3% of the average, and for the acrylamide peak at 1285 c m - 1 the standard deviation is 1.4% of the average. Hence the bis and acrylamide peaks respond reproducibly under these conditions. This allows for the comparison of samples from the same batch of gel, irradiated to different doses. Furthermore, it establishes that the Raman spectroscopy technique is sufficiently stable and repro-ducible to study the polymer gels used in radiotherapy. As an additional note, the errors obtained above are encouraging in that they are comparable to established e >> b 0.2 fi 0.1 OS n n 105 . . . ! . . JL 1 1 1 1 1 1 at I I — • 1 ' 1 — i n XL X L * * » * * S * • OGy, Bis, Ave: 0.243 +/- 0.002 • 5Gy, Bis, Ave: 0.197 +/-0.007 • OGy, Acrylamide Ave: 0.169+/-0.002 • 1 I o 5Gy, Acrylamide Ave: 0.146+/-i . i i . i . i 0.005 1 2 3 4 5 Batch 6 7 8 Figure 5.8: Reproducibility of acrylamide and bis peak area for gels manufactured from different gel batches and either left unirradiated, or irradiated to 5 Gy. dosimetry techniques such as TLD dosimetry (2% error at best, 5% typical). 5.3.3 Inter-batch Reproducibility Eight separate batches of gel were made and transferred to separate sample vials. Two sample vials were filled with each gel. One sample from each batch was left unirradiated while the remaining sample was irradiated to 5 Gy in the same manner as above. Raman spectra were acquired on all 16 sample vials and peak areas for the acrylamide and bis peaks were calculated. Results are shown in figure (5.8). Errors for each data point were taken as the standard deviation about the mean (expressed as a percentage) for each of the peaks in the "intra-batch" study described above. Results indicate that the reproducibility of the peak areas in different gel batches is 3.4% for acrylamide and 3.6% for bis peaks, calculated as the standard deviation on the mean (expressed as a percentage). The larger errors observed in the inter-batch gels as compared to intra-batch gels are likely due to slight variations in O2 concentration during manufacture. 106 1150 1200 1250 1300 1350 1400 Wavenumber (cm') Figure 5.9: Raw acryalmide and bis Raman peak intensity as a function of absorbed dose. 5.4 Peak Fitting Figure (5.9) illustrates the decrease in acrylamide and bis peak intensities as each gel sample is irradiated to a higher dose. As well, it is observed that the background of each spectrum can vary minutely when spectra from gels irradiated to different doses are compared, as discussed in section (4.5). To account for this, the acrylamide and bis datasets in this case were fit to a combination of linear baseline with a superposition of three Lorentzian peaks, as described by equation (5.1). f(x) = k + mx + , 4 * + 2 2 + (5.1) The third Lorentzian was included to account for the asymmetry in the bis band (evident upon inspection of spectra acquired on individual constituents) at the high frequency side of the feature. Apart from creating background, the asymmetry does not appear to significantly influence the acrylamide peak. Data were fit to these functional forms using MINUIT (Cern, Switzerland). Figure (5.10) illustrates the results of the fitting routine to the bis and acrylamide bands in a spectrum of an 107 0 Gy Raw data -. 0 Gy Fit 1150 1200 1250 1300 1350 1400 Wavenumber (cm') Figure 5.10: Haw acrylamide and bis peaks fit to equation (5.1). unirradiated gel. The choice of functions to fit to data was made primarily on goodness of fit considerations. Peak fitting parameters were not used directly in characterizing peak inten-sity since, as samples are irradiated to higher doses and monomer is consumed, spectra exhibit progressively lower signal to noise. Hence, uncertainties in peak fitting parameters increase as a function of dose, reaching values as high as 10% relative error for typical high dose spectra (figure (5.11)). Errors in background fit parameters, however, remain constant at ~1.5% for all doses. Therefore, subtracting the background fit from the raw dataset and subsequently characterizing the peak intensity minimizes the errors resulting from fitting peaks of low signal to noise. For this technique to be rigorous, it must be noted that the background and peak fit parameters can not be interdependent. That is, the correlation between the fit parameters must be low. To establish if this is indeed the case, a variance-covariance matrix was calculated in MINUIT for each dataset fit to equation (5.1). Table (5.2) shows an example of a variance-covariance matrix for the unirradiated gel spectrum shown above. Upon inspection of the variance-covariance matrix for 108 Table 5.2: Variance-covariance matrix for fit parameters of equation (5.1), fit to an unirradiated sample spectrum. k m Ai A2 w2 A3 w3 k 1.000 -0.997 0.082 0.136 -0.012 0.086 0.274 0.517 m -0.997 1.000 -0.101 -0.094 0.018 -0.102 -0.295 -0.551 Ai 0.082 -0.101 1.000 Wl 0.136 -0.094 1.000 A2 -0.012 0.018 1.000 W2 0.086 -0.102 1.000 As 0.274 -0.295 1.000 W3 0.517 -0.551 — 1.000 109 i < 1 1 r 0 I • • . i i i i i—=< I 1150 1200 1250 1300 1350 1400 Wavenumber (cm') Figure 5.12: Fit background subtracted from acrylamide and bis peak data. any given fit it is noted that, although correlation between the elements of the back-ground (i.e. rn and k in equation (5.1)) are high, correlation between background parameters and peak parameters remains low. Restrictions can be placed on fit-ting parameters to ensure that this remains the case for all datasets. For example, peak widths (w) are restricted such that peaks do not become so wide so as to be rendered indistinguishable from the baseline. As a result, the fit background was subtracted from the raw dataset spectra, see figure (5.12) for results. The resulting peak intensities were characterized using the technique of correlation (section (4.5)). 5.5 Consumption of Monomer Figure (5.13) illustrates the results of the correlation calculation as applied to the 1256 cm - 1 bis and 1285 cm - 1 acrylamide bands for a 6%T, 50%C gel irradiated between 0-50 Gy. Errors were calculated by propagating a combination of the error in reproducibility and the error in the baseline estimate through the correlation 110 1.2 — Q — bis-acrylarnide, 1256 cm —•—acrylamide, 1285 cm"1 -i j _ ——1 0 10 20 30 40 50 Dose (Gy) Figure 5.13: Acrylamide and bis monomer consumption curves for a 6%T, 50%C gel irradiated up to 50 Gy. Lines connecting experimental data points are for visual guidance. calculation. The 0 Gy auto-correlation value corresponds to initial concentrations of 3% acrylamide and 3% bis. The cross-correlation minimum at doses > 20 Gy corresponds to 0% acrylamide and bis concentration. Data in figure (5.13) were normalized by subtracting residual background intensity produced by the correlation function, and normalizing auto-correlation values to 1. Consumption of acrylamide can be well modeled as mono-exponential over the dose range 0-50 Gy. Acrylamide data was fit to a single exponential, where Da is the "sensitivity parameter" for acrylamide. Equation (5.2) is in analogy to the "single hit" model of radiobiology, where a mono-exponential expression is the result of assuming a given target (e.g. a cell or molecule) requires one hit for inactivation [152]. In rough analogy, the acrylamide molecule requires one hit to break the carbon double bond and render the monomer reactive. Fitting equation (5.2) to the acrylamide data yields a sensitivity parameter D0(acr) — 8.1 ± 0.4 Gy. y = e-D'D° (5.2) 111 Consumption of bis can also be reasonably well fit to a mono-exponential, particularly in the low dose range between 0-13 Gy. Fitting the data to equation (5.2) yields D0(bis) = 5.5 ±0.5 Gy. However, while the bis data does fit to a mono-exponential reasonably well (K2 = 0.992), a slightly better fit can be achieved by considering other functions. For example, one can use the radiobiological "linear quadratic" model, which attributes a given effect to either a single event (hit) or a combination of two events, y = e-aD-(,D> ( 5 3 ) Here a and /3 are "inactivation constants" for single or double events occurring within the molecule [152]. Using equation (5.3) produces a fit which better represents the bis data {R2 = 0.998, Z>i = 1/a = 7.3 ± 0.5 Gy, D2 = 1/0 = 12 ± 1 Gy2). Another model which can be used is the "multi-hit" model, which postulates that a target (e.g. bis molecule) must be hit m times for a given effect. Considering the probability of an event (i.e. hit) occurring to be Poisson distributed (a common assumption made in radiobiology), and also that two hits are required to break both carbon double bonds, yields [11, 152, 153] y = (1 + D/D0)e-DID° (5.4) Using equation (5.4) yields a fit similar to the linear quadratic model (R? = 0.995, D0(bis) = 2.7 ± 0.1 Gy). It must be noted that, although the above equations do produce reasonable fits to the data, there appears to be little physical basis for using these equations at this time. Baldock et al. reported on experiments using Raman spectroscopy to char-acterize monomer and crosslinker consumption [31]. The results presented in figure (5.13) are qualitatively similar to those presented by Baldock et al. , however scatter 112 in data points is minimized in this study, due to several factors. A customized sam-ple holder provided reproducible sample positioning in the spectrometer. Precision-manufactured, thin-walled glass NMR sample tubes were used to ensure constant laser scatter and absorption for each sample. A robust background subtraction technique was employed to account for the variability in background between spec-tra. Also, correlation offers the advantage of weighting the dataset by the high SNR reference spectrum, in this case the 0 Gy spectrum. Quantitative comparison between the results in this thesis and those of Baldock et al. is difficult, as the functional parameters of the fit by Baldock et al. were not reported. However, in a later study, Lepage et al. performed similar experiments and, using a mono-exponential fit, quote D0(acr) = 12 ± 1 Gy and D0(bis) = 6.5 ± 0.7 Gy for the acrylamide and bis sensitivity parameters [104]. Although the consumption rates obtained by Lepage et al. for bis agree, within experimental error, with the results presented here, the acrylamide consumption rates for the two studies do not agree within experimental error. This may be a result of different preparation conditions (different amounts of oxygen, different manufacturing temperatures) and chemicals from different manufacturing batches. 5.6 Formation of Polymer Two methods have been used to identify regions in the gel Raman spectra where polymer formation may be observed. First, a solution of water, acrylamide and bis was manufactured and irradiated to doses between 0 and 50 Gy, as described in section (4.1). Second, bands which were observed to increase in intensity at higher doses were cross-referenced with the results of Gupta and Bansil, Mohan, and Murugan who acquired Raman spectra on pure polyacrylamide samples [106, 113 107, 110]. The water, acrylamide and bis solution was manufactured keeping the ratios of acrylamide to water, bis to water, and acrylamide to bis the same as in stan-dard gel samples. It is known that, upon irradiation, this solution exhibits a rapid monomer consumption rate [2, 154]. Hence, if polymer is to be detected in the Raman spectra at all, this dosimeter should provide the simplest system in which to detect the formation, since full monomer consumption occurs after relatively little dose, and possible complications due to gelatin do not occur. Spectral features in four separate wavelength regions are observed to form once the water, acrylamide and bis solution is fully polymerized. These features appear at 27, 1126, 1450 and 2936 cm - 1 . Polyacrylamide vibrations have been characterized, in detail, in both solid and aqueous phases of polyacrylamide [106, 107, 110]. The 1126, 1450 and 2936 cm - 1 are observed and identified by Gupta and Bansil, Mohan, and Murugan [106, 107, 110]. The 27 cm - 1 feature is not reported by the above workers, probably due to limitations in their instrumentation (they do not report on features below ~180 cm - 1). Knowledge of the spectral location of polymer formation in the Raman spec-tra was then applied to a complete polymer gel (i.e. one with gelatin). All four spectral features observed in the "no-gelatin" dosimeter were also observed in the spectra of irradiated polymer gel, see figure (5.14). The addition of gelatin does not appear to shift the frequency of vibration of these bands significantly. In addition, a simple aqueous gelatin sample was irradiated to doses as high as 50 Gy with no noticeable distortions in the resulting Raman spectra, indicating that gelatin acts as a passive component in the system. Although not all four features are fully characterized, each is discussed briefly 114 Figure 5.14: Polymer formation observed in FT-Raman spectra of irradiated poly-mer gel. (a) 27 cm - 1 mode, origin unknown (see text), (b) 1126 cm - 1 C—C stretch, (c) 1450 cm - 1 CH2 bend and (d) 2936 cm - 1 CH2 stretching mode of polyacrylamide. Note the different y-axis intensities in the plots. 115 below. The CH2 stretching mode at 2936 cm 1 is analyzed in more detail, as this band offers the simplest analysis. 27 cm - 1 possible libration A reproducible band of weak observed intensity is detected at a very low frequency of 27 cm - 1 (see figure (5.14a)). The intensity of the band is likely affected by the Rayleigh filter. The band is most probably attributable to a slow polymer libration, or gentle twisting of larger polymer arms. Its' over-all intensity increases with dose similarly to the CH2 stretching mode at 2936 cm - 1 (see below) up to 20 Gy. A turnaround is observed at 20 Gy and the band decreases in intensity, reaching a steady state intensity at 30 Gy. Due to the inconclusive assignment, no further exploration of this band was undertaken. 1126 cm"1 C-C stretch A C—C stretching mode for polyacrylamide occurs at 1126 cm - 1 . This feature is also weak, and furthermore, is obscured by an acrylamide band of moderate intensity and unknown origin (see figure (5.14b)). As the gel is irradiated to high doses, the intensity of this combined band decreases, but not at the same rate as that observed for 1285 cm - 1 the acrylamide band. Due to the relatively weak intensity of the polyacrylamide band, and the fact that it is obscured by an overlapping acrylamide peak, this band is not investigated further. 1450 cm- 1 C H 2 bend A CH2 bending mode of both gelatin and polacrylamide occurs at 1450 cm - 1 . This band is partially obscured by the CH2 stretch in acrylamide at 1436 cm - 1 (figure 116 (5.14c)). At 0 Gy, the band appears as a moderately intense feature. At 30 Gy, the intensity of the overall feature has fallen to approximately half it's original intensity, but the peak centre has shifted in frequency from 1437 cm - 1 at 0 Gy to 1450 cm - 1 at 50 Gy. The original acrylamide band at 1437 cm - 1 has decreased to minimal intensity between 0 and 50 Gy, while the polyacrylamide/gelatin band has increased from zero intensity up to it's maximum at 50 Gy. This polymer band is not analyzed in detail since it is obscured by the acrylamide band and, also, the 2936 cm - 1 polymer band offers simpler analysis. 2936 c m - 1 C H 2 stretch The 2936 cm - 1 CH 2 symmetric stretching mode of polyacrylamide is unobscured by other bands in close proximity and is relatively intense. This band is therefore analyzed in more detail, in order to characterize polymer formation as a function of dose. Figure (5.14d) shows a detailed view of the 2936 cm - 1 CH 2 bending mode of polyacrylamide observed in the Raman spectra. Note the increase in intensity of this band as a function of dose, while the 3050 cm - 1 acrylamide CH 2 stretch decreases to minimal intensity over this same dose range. Both features are superimposed on the tail of the broad band due to both the symmetric and anti-symmetric stretch in water. The background intensity is subtracted from the polyacrylamide peak by fitting a combination of a Gaussian peak superimposed upon a linear baseline in the proximal region of the peak (MINUIT). Correlation could not be used to quantify the variation in peak intensity as a function of dose due to the absence of a high signal to noise reference spectrum (the 0 Gy spectrum in this case has zero spectral 117 — polyacrylamide, 2936 cm"' U i I i I i I i I i L 0 10 20 30 40 50 Dose (Gy) Figure 5.15: Polymer formation curve for a 6%T, 50%C gel irradiated up to 50 Gy. Lines connecting experimental data points are for visual guidance. peak intensity). Hence, integration was used and the results of the calculation are shown in figure (5.15). Polymer formation can be reasonably fit to a mono-exponential, particularly in the low dose region (0 - 13 Gy). When data is fit to y = 1 - e~DlD° (5.5) a D0(poly) = 9.0±0.5 Gy is obtained. The D0(poly) is greater than D0(bis) and equal, within experimental error, to D0(acr). This indicates that polymer formation occurs at a rate similar to the rate of consumption of acrylamide. Alongside the publication of the data presented in this thesis, Lepage et al. have also characterized the formation of polymer in irradiated polymer gel [104]. Lepage et al. report a polymer formation rate of D0 — 9 ± 3 Gy which is equal to the Da(poly) obtained above. The large uncertainty on the D0 of Lepage et al. appears to stem from the relatively low signal to noise ratio of their spectra. Both the monomer consumption and polymer formation results of this study, and the results of Lepage et al. indicate that the formation of polymer is a non-118 linear function of dose. These results are significant in the fact that, traditionally, experimenters measuring gel response with MRI (specifically R2 = l/Tfe, spin-spin relaxation rate) have assumed plots of R2 vs absorbed dose to be linear in nature by assuming i) a direct link between polymer formation and R2 response, and ii) that polymer formation is a linear function of dose [22, 77]. The second assumption has been put into question by these, and other, studies. For example, exponential R2 vs. dose gel response curves have recently been obtained by DeDeene et al. by using careful gel manufacture and MRI scanning techniques [76]. As well, Lepage et al. and Zhang et al. have both used mathematical modelling to predict the gel dose response and both results (preliminary in the case of Zhang et al. ) indicate non-linear R2 dose response curves [104, 155]. The results of the modelling by Lepage et al. agree with their experimentally determined R2 dose response curves. However, it must be noted that in the low dose regions (i.e. the first few Grays of irradiation) the non-linearity of the dose response is weak, and hence experiments performed under less stringent manufacture and scanning conditions could estimate a linear response in this region, due to the typically higher scatter in the data points of such experiments. Also, for studies in low dose regions, linear response curves are approximated as this permits simple relative dosimetry to be performed and dose calibration curves are not necessary. 5.7 C o m p o s i t i o n a l S t u d i e s The dose response characteristics of the polymer gel may be drastically altered by tuning the ratio of acrylamide to bis in the initial gel composition. Results of experiments performed to measure the monomer consumption in irradiated gels of varying initial composition are described below. 119 5.7.1 Monomer Consumption Curves Monomer consumption data was generated by applying the correlation calculation (equation (4.1)) to experimantal data obtained for irradiated gels of the following compositions: 0 and 100%C (3%T), 30, 50 and 70%C (6%T). These curves are shown in figure (5.16). Although acrylamide is soluble to high concentrations, a 3% acrylamide gel was manufactured so as to be able to compare it's rate of consumption upon irradiation with that of bis. All curves were normalized and error bars were calculated as in section (5.5). During the course of these studies, difficulty was experienced in making stable gels at low bis concentrations (e.g. 10%C). Gels at these low bis concentrations that were irradiated to moderate doses (5 - 13 Gy) appear to form minute white "beads" ~24 hours post irradiation. These beads are absent in samples irradiated to high doses. The cause of such phenomena is unknown. Raman spectra acquired at various points throughout the sample indicate that the beads have the structure of poly-acrylamide, with no monomer or crosslinker present. This indicates that the monomer and crosslinker are reacting to form tightly knotted polymer, well after irradiation. The fact that samples irradiated to high doses do not form the observed beads may be due to the fact that at these high doses all monomer/crosslinker is consumed during irradiation, leaving none to participate in the post-irradiation phenomenon. Unirradiated samples also do not form the beads, indicating that the radiation acts as a form of long term catalyst for this polymer formation. The long term nature of this instability is interesting. Baldock et al. have shown, using FT-Raman spectroscopy, that polymer formation occurs for as long as 12 hours post irradiation for a gel composed of 6%T and 50%C [79]. De Deene et al. have also observed instabilities related to monomer/polymer reactions for up to 12 hours post 120 1.0 0.8 | 0.6 H •3 0.2 oi 0.0 1 1 1 ' 1 • 1 1 1 1 1 • 1 • r B —o— acr 0%C • i i i i i i » 1.0 0.8 | 0.6 S 0.4 3 0.2 oi 0.0 10 20 30 40 50 60 70 Dose (Gy) (a) 1 • 1 •— —I • 1 1 r —•—bis —o—acr 50%C -T»-i-Jr5—— Zjl— 5 H i i — i . i . t 1.0 0.8 | 0.6 a g 0.4 'S 3 0.2 oi 0.0 — i • 1 . 1 1 j-—•—bis —o— acr : j i i i 40 50 0 10 20 30 40 50 60 70 80 90 Dose (Gy) (e) Figure 5.16: Consumption of acrylamide (acr) and bis in irradiated polymer gel dosimeters with different initial bis fractions: (a) 0%C, (b) 30%C, (c) 50%C, (d) 70%C and (e) 100%C. Figure (c) reproduced from figure (5.13) for ease of compar-ison. Lines connecting experimental data points are for visual guidance. 121 irradiation, for a 6%T, 50%C gel [76]. As stated in section (5.5), acrylamide and bis consumption curves for a 50%C, 6%T gel can be reasonably well modeled as mono-exponential, particularly in the low dose region. Furthermore, for bis consumption, several alternate functional forms were suggested which improved the overall fit to the data. However, for the different %C polymer gel consumption curves shown in figure (5.16) a range of shapes (mathematical course) of the curves are observed and may be fit by a variety of mathematical functions. For example, the 100%C bis consumption curve may be fit to either a single exponential with a large D0 — 361 Gy, or as a straight line. Fit errors are, in fact, much lower in the straight line fit to these data. From the standpoint of polymerization kinetics the observation that the con-sumption curves exhibit a variety of shapes is not necessarily unexpected, as will be discussed in section (5.7.3). From the standpoint of gel dosimeter sensitivity (monomer consumption rate) this points to a differential sensitivity based not only on initial bis fraction (%C) but also to dose range, as discussed in the following section. 5.7.2 Effects of Initial Bis Fraction on Gel Sensitivity As shown in figure (5.16), monomer/crosslinker consumption does not follow lin-ear functional forms. This points to maximum monomer consumption rate being dependent not only on %C, but also on dose range. To demonstrate this effect, the derivatives of the consumption curves given in figure (5.16) have been calcu-lated. Consumption rates for acrylamide and bis were compared at several doses for the different %C gels. Results are illustrated in figure (5.17), which summarizes the behaviour of acrylamide and bis consumption rates in low, mid and high dose 122 Dose (Gy) Dose (Gy) (a) (b) Figure 5.17: Consumption rates for (a) acrylamide and (b) bis monomers for the compositions and doses cited. Note the differing y-axis intensities. Table 5.3: Gel composition with the maximum consumption rate in the given dose range. Dose Range (Gy) % C 0-5 30 5-15 50 >15 70 ranges. It is found that in low dose ranges (0-5 Gy) maximum consumption rate for both monomers occurs for 30%C gels while in mid dose ranges (5-15 Gy) maximum consumption rate occurs for 50%C gels. Above 15 Gy maximum consumption rate occurs for higher % C gels (eg 70%C). At ~5 Gy the 30%C and 50%C gels have the same consumption rate, similarly at ~15 Gy 50%C and 70%C gels have approxi-mately the same consumption rate. Table (5.3) summarizes these results. As shown above, the formation of polymer has been directly correlated with the consumption of monomer (see also [104]). Hence, polymer formation should follow a pattern similar to that of monomer consumption. A few comments are made to put into perspective the extent of this differential sensitivity. For the low dose region (0-5 Gy) the 30%C gel is potentially 1.8 times more sensitive than any other gel studied. Similarly, in the mid dose ranges (5-15 Gy) a 50%C gel is potentially 1.3 times more 123 sensitive than all other gels studied. Finally, in the higher dose region (>15 Gy) the 70%C gel is potentially 1.5 times more sensitive than any other gel studied. Maryanski et al. have used NMR transverse relaxation rate measurements to study the effects of varying relative bis fraction on the dose sensitivity (i.e. slope of i?2 vs dose response curve) of polymer gel dosimeters [77]. It must be noted that the two measurements (R2 vs dose and monomer consumption) do not measure the same physical phenomenon, however, both can be used as a measure of the gel "dose response" to radiation. Hence, only qualitative comparisons are made between this work and the work of Maryanski et al. . They report that peak polymer gel dosimeter sensitivity (slope of R2 vs dose plot) occurs for a 50%C gel. The present studies indicate that the 50%C gel exhibits a monomer consumption rate which is greater than any other gel studied here only in the mid-dose region (5-15 Gy). In the low (0-5 Gy) and high (>15 Gy) dose regions maximum monomer consumption does not occur for a 50%C gel (see above). The qualitative discrepancy between the present results and those of Maryanski et al. are not surprising given that linear fits were assumed for the response curves of Maryanski et al. . Overall, the present results indicate that a simple change in initial bis fraction of a polymer gel can act as a useful optimization tool in radiotherapy dosimetry applications. 5.7.3 Effect of Initial Bis Fraction on Polymer Structure I: Overall Consumption Rate As demonstrated above, initial bis fraction affects the overall monomer consumption rate. It also has an effect on the variations in consumption rate (mathematical course, or shape of the dose response curve) over a range of doses for an initial 124 polymer gel composition (section (5.7.4)). Both variations are discussed, in turn, in terms of the effect the initial bis fraction has on the resulting polymer formed in the gel. Much has been said on the structure of polyacrylamide gel used in elec-trophoresis (see for example [89, 90, 93]). Although electrophoresis gels do not contain gelatin, which has a moderating effect on the reaction kinetics of polymer gel [22], some of the literature is nonetheless useful in understanding, qualitatively, the rates of monomer consumption observed in this study. Primarily, the model of Righetti is followed [90]. A gel composed of a linear monomer such as acrylamide and a bi-functional co-monomer (crosslinking agent, e.g. bis) is capable of forming several types of links. Acrylamide chains may form whose ends are connected to two co-monomer units (singlets, in the terminology of electrophoresis literature, see figure (5.18a-i)). Linear acrylamide chains may form whose one end remains reactive (termed free radical linear chains, figure (5.18a-ii)). Linear or crosslinked chains may form where the "heads" have reacted with the "tails" of a given chain (loops) (figure (5.18ar iii)). Finally, bis units where the ends have reacted with each other are possible (doublets) (figure (5.18a-iv)). Termination of reactive sites is also possible through several mechanisms. Singlets are terminated by reaction of the chain with monomer or crosslinker radicals or reactive chains to form a dead polymer. Similarly, free radical linear chains are terminated by additional monomer or crosslinker radicals, or longer radical chains. Loops and doublets are terminated within themselves, since they react with their own ends (loops) or with another identical molecule (doublet). Large conglomerates can form into small beads, or knots. Different overall types of polymer will be formed as the gel is irradiated, depending on the initial fraction of 125 ^ " ' <? CZ3 (i) (ii) (iii) (iv) (a) (i) (ii) (iii) (iv) (b) Figure 5.18: (a) Links created by combination of acrylamide monomer and bis crosslinker: (i) singlet, (ii) free radical linear chain, (iii) loop and (iv) doublet. Open circles indicate reactive sites. Closed circles denote ends, (b) Progression in polymer structure as a function of initial crosslinker concentration, (i) A "gel" solely composed of monomer (acrylamide). Long, linear chains are formed with no crosslinks, (ii) Gel composed of low initial bis fraction. Predominant gel formation is an ordered, crosslinked network, (iii) Gel composed of high initial bis fraction. Gels begin to form a larger number of knots, (iv) A gel composed solely of crosslinker (bis). Predominant structures are knots, loops and doublets which together form beads. 126 bis present. This is demonstrated by considering several cases. a) 0%C A gel absent in crosslinker (see figure (5.18b-i)) has a comparatively slow reaction rate with dose. This is most likely due to the fact that a linear acry-lamide chain is highly viscous [89, 93]. This strongly hinders monomer migration to reactive linear chain sites. As a result, monomer diffusion rates are low and re-active monomers experience difficulty migrating to other reactive sites. The further decrease in acrylamide consumption at very high doses (>50 Gy, see figure (5.16a)) is also most likely due to the ever increasing viscosity of the system. b) 100%C A gel composed solely of bi-functional crosslinking agent (figure (5.18b-iv)) is also slow to react with dose, albeit for a different reason than the linear monomer system. The crosslinking agent does not form highly viscous chains, rather, bis reacting with itself primarily forms tight loops, doublets and knots which combine to form "beads". Ruchel et al. obtained transmission electron microscope images of a 100%C gel illustrating the tight form of these beads [93]. It has been postulated that these regions become so tightly bound that unreacted crosslinkers have difficulty in penetrating into these "beads" [89]. Hence, consumption of bis remains small as a function of dose. The lower viscosity of the system allows for "beads" to be formed even at very high doses, thus a further slowing of the consumption of bis is not observed (figure (5.16e)). c) Progression from 0 - 100%C Several interesting effects occur as the initial crosslinker fraction is increased from 0 to 100%C (figure (5.18b-ii,b-iii)). By intro-ducing crosslinks between the linear monomers, the high viscosity of the long linear acrylamide chains is decreased. This allows for much improved monomer/crosslinker migration to reactive sites with the net effect of a markedly increased rate of con-sumption of unreacted monomer/crosslinker. As the initial fraction of bis is in-127 creased, the average number of bis crosslinkers per acrylamide monomer continues to increase and the relative number of loops and knots increases. As the loops and knots become more prominent in the overall structure, this in turn has the negative effect of impeding monomer/crosslinker migration into these "beads", hence a slower overall reaction rate is observed at high %C. Hence, an overall increase in monomer consumption rate is observed as the initial bis fraction is increased from 0%, followed by a decrease in the overall monomer consumption rate as the bis fraction is further increased to sufficiently high concentrations. 5.7.4 Effect of Initial Bis Fraction on Polymer Structure II: Math-ematical Course of Consumption The variation in the reaction rates of monomers for different %C gels can, at least in part, be understood by considering the average number of acrylamide units per bis unit available for reaction in each case. Clearly, the initial %C of a gel dictates the initial ratio of number of acrylamide molecules per bis molecule available for reaction. Due to the differential rates of consumption of monomer and crosslinker within a given gel (see figure 5.16), the average number of acrylamide molecules per bis molecule available for reaction does not remain constant as the gel is irradiated to higher doses. This has implications for the structural forms of polymer formed. To demonstrate the effect of differential monomer/crosslinker consumption on the acrylamide to bis ratio, the average number of acrylamide molecules per bis molecule available for reaction are calculated for each gel as it is irradiated to higher doses. This ratio (R{D), D = dose) is readily obtained from a knowledge of the molar mass of acrylamide (Ma = 71.08g/mol) and bis (M0 = lb4g/rnol) and the 128 Dose (Gy) Figure 5.19: The average number of acrylamide molecules per bis molecule available for reaction as a function of absorbed dose and initial bis fraction (%C). consumption curves of figure (5.16), fa(D)maNA , R(m = M. = fa(D)maMb [ ' h(D)mbNA fb(D)mbMa Ml , (5.6) Here ma and mb are the mass of acrylamide and bis initially present in each sample (known from the initial composition of the gel), fa(D) and fb(D) are the fractions of acrylamide and bis left in each sample after a dose D (obtained from figure (5.16)), and NA is Avagadro's number. Figure (5.19) shows the results for 30 -70%C gels. Figure (5.19) illustrates that one can expect very different types of polymers to be formed in different dose ranges for different %C gels. The greatest variation in polymer structure over a dose range 0 - 50 Gy occurs for a 50%C gel. Initially a 50%C gel, on average, contains ~2 acrylamide molecules per bis molecule. This would tend to create a significant proportion of loops and knots, which is primarily responsible for the marked opacity change of the gel [89]. Due to higher reactivity of bis, however, this ratio increases up to values of ~20 acrylamide/bis molecules at doses around 15 Gy. Thus a much more linear polymer is formed in this dose range. Similarly a 30%C gel contains ~5 acrylamide/bis molecules initially, but this value 129 increases up to ~15 acrylamide/bis molecules at ~15 Gy. The fact that this ratio changes by a factor of 10 for 50%C gels over 0 - 50 Gy but only by a factor of 3 for 30%C gels may be one of the factors in causing the differential rates of reaction between the two gels. A 70%C gel, on the other hand, maintains a relatively constant acrylamide to bis molecule ratio over 40 Gy, varying from ~1 acrylamide to bis molecule to ~2 acrylamide to bis molecules at 40 Gy. This relatively constant ratio (changing by only a factor of 2 over 40 Gy) indicates a polymer similar in structure over this entire dose range. The fact that the ratio is low results in a large fraction of loops, doublets and knots, creating an overall slow rate of reaction. 5.8 Irradiation History The fact that a gel has been irradiated with a certain dose, and hence polymer is present in the system, has an effect on the subsequent monomer consumption rate. This point is illustrated using a 6%T, 50%C gel. After 7 Gy of irradiation, accord-ing to figure (5.16c), the 6%T, 50%C gel contains ~2.3%T and ~38%C unreacted monomer and crosslinker. A gel was manufactured with the above initial concen-trations (i.e ~2.3%T, ~38%C) and a dose response study was undertaken on this gel. The rate of monomer consumption for the ~2.3%T, ~38%C gel was compared with that of the 6%T, 50%C gel at 7 Gy. Results are shown in figure (5.20). Note that the initial monomer peak intensities of the ~2.3%T, ~38%C gel have been normalized to their respective concentrations, if they were present in the 6%T, 50% gel. These curves have then been overlaid on the 6%T, 50%C consumption curves. Results indicate that the monomer consumption rate of the ~2.3%T, ~38%C gel is lower than the monomer consumption rate of the 6%T, 50%C gel beginning at a 130 0 7 10 20 30 40 50 0 7 10 20 30 40 50 Dose (Gy) Dose (Gy) (a) (b) Figure 5.20: (a) Acrylamide and (b) bis consumption rate comparison for two gels: i) solid markers show consumption rate for an initial 6%T, 50%C gel (figure (5.16)c). ii) open markers, dashed line show consumption rate for an initial 2.3%T 38%C gel, corresponding to a concentration of unreacted monomer in 6%T, 50%C gel at 7 Gy. Thus, comparison is made for consumption rates in gels with (i) pre-existing polymer network present (radiation history) and (ii) no pre-existing polymer network present (no radiation history). Solid x-axis pertains to 6%T, 50%C gel. Dashed x-axis pertains to 2.3%T, 38%C gel. Lines connecting experimental data points are for visual guidance. dose of 7 Gy. This indicates that the polymer network present in the 6%T, 50%C gel at 7 Gy has a positive effect on the subsequent monomer consumption rates. It is postulated that, in this particular case, the increase in viscosity due to the newly formed polymer is outweighed by the host of reactive sites available to the unreacted monomers in the polymer network. It must be noted that this result is valid only for this particular gel. Due to the different types of polymer formed in the different gel compositions, this result may not be general. More research in this area would be required to fully develop a complete understanding of this effect. However, the result does indicate the general importance of the pre-existing polymer network on subsequent monomer consumption rates. 131 Chapter 6 Results and Discussion II: Gel Response to Proton Irradiation This chapter presents results of studying the dependence of polymer gel response on ionizing density (LET). To this end, a range of LET gel responses are explored by combining experimental results with track structure modelling of gels irradiated with 74 MeV protons. After a brief introduction (section (6.1)), experimental results are presented of polymer gels irradiated with protons (section (6.2)). Monomer consumption curves are used to establish an experimental polymer gel effectiveness in measuring proton dose for gels exposed to two different regions of a SOBP (i.e. two regions with differing ionizing density). The theory of track structure is then used to predict the effectiveness of polymer gel in measuring proton dose (section (6.3)). A brief discussion follows (section (6.4)). A large portion of the work presented in this chapter has resulted in the publication of a paper, first authored by the author of this thesis [156]. 132 6.1 Introduction Heavy charged particles such as protons exhibit dose deposition distributions which are clinically very attractive. The advantage of proton beams over x-ray or high energy electron beams for use in a clinical setting is the well defined particle range and minimal lateral scatter for protons in tissue. This gives rise to very sharp dose fall-off at the distal and lateral margins of the treated volume. A monoenergetic proton beam will deposit a very high density of energy (i.e. dose) in a small region at the end of the proton range (i.e. Bragg peak). Clinically relevant dose distributions may be obtained by modulating the energy, and therefore also the range, of the incident protons, hence creating a "spread out Bragg peak" (SOBP) (see section (6.2)). Proton and heavy ion therapy is a growing field. Currently 30 centres worldwide perform some form of proton or heavy ion therapy, and a minimum of 12 additional centres are proposed to be in operation by the end of 2003 [157, 158]. The nature of the dose distributions obtained with protons or heavy ions, as well as several technological advances in the field (e.g. scanned beams) require a careful characterization of these beams [159]. Current beam characterization methods using ion chambers can be exceedingly time consuming and tedious. Clearly, a 2D or 3D detector would be valuable in this situation. Currently most 2D methods (e.g. film) are less than ideal in characterizing such beams, primarily due to the inability of these systems to accurately characterize the Bragg peak region [160]. A 3D system based on the Fricke gel has also been tested, and a linear energy transfer (LET) dependence of the gel response was confirmed [161]. Due to the high spatial resolution and 3D nature of polymer gel dosimetry, this type of dosimeter could be very useful for particle (proton) therapy. However, there has been some evidence that the gel response is dependent on the LET of the 133 incident particle [162-164]. Ramm et al. demonstrated the inability of polymer gel in adequately characterizing a beam of carbon ions [163, 164]. Hilts et al. showed MRI and x-ray CT data of polymer gel irradiated with 70 MeV protons, again, with the Bragg peak not adequately represented in the MRI or x-ray CT data [162]. However, neither of these studies undertook a thorough investigation of the dose response characteristics of polymer gel irradiated with protons, nor did these prior studies adequately explain the physical processes involved in causing these phenomena. 6 . 2 E x p e r i m e n t a l D e t e r m i n a t i o n o f P o l y m e r G e l R e l a -t i v e E f f e c t i v e n e s s 6.2.1 Proton L E T Distribution at Points of Measurement As stated in section (4.3), the raw proton Bragg peak was spread over 23 mm using a 20 step acrylic modulator wheel. The SOBP is shown in figure (6.1). This has the effect of creating several proton energies and LETs at each point of measurement (central or end region of the SOBP, see section (4.3) and figure(4.5)). Figure (6.2) illustrates the proton LET contributions for both the central and end points of measurement. Each LET contribution is weighed by the relative proton fluence through each step in the modulator wheel. The weighted average LET in the two regions differs by ~23%. As a comparison, the range in stopping power for protons traversing water varies between ~10 MeV/cm for protons at the entrance of raw Bragg peak to ~105 MeV/cm for protons at the distal end of a raw Bragg peak. For further comparison, the stopping power of electrons set in motion in water by 6 MV x-rays is ~2 MeV/cm. 134 Depth (mm) Figure 6.1: Depth dose curve of a 74 MeV proton beam in water, spread out over 23 mm using an acrylic modulator wheel. Shown are the raw Bragg peak curves (gray lines) obtained for protons traveling through each step in the modulator wheel. Relative heights of curves indicate the proton fluence weight of each step [165]. The spread out Bragg peak curve (dark line) is obtained from the fluence weighted sum of all the individual curves. J2 S O 0.5 0.4 0.3 0.2 0.1 0.0 20 LET mid SOBP | LET end SOBP 30 40 50 60 70 LET (MeV/cm) Figure 6.2: The distribution and relative fluence contribution (weight) of protons with given L E T present at the point of measurement for the mid SOBP (dark bars) and end S O B P (light bars) regions. 135 Pi 10 20 30 40 5 0 0 Dose (Gy) Figure 6.3: Consumption of acrylamide in a polymer gel as a function of dose. Shown are gel consumption curves for x-ray and proton (mid and end SOBP region) irradiation. The x-ray irradiated gel consumption curve is reproduced from figure (5.13). Lines connecting experimental data points are for visual guidance. 6.2.2 Consumption Curves and Relative Effectiveness Figure (6.3) illustrates the results of the correlation calculation as applied to the 1285 cm - 1 acrylamide band for a 6%T, 50%C gel irradiated to between 0 and 50 Gy with 6 MV x-rays and the central and end portions of a 74 MeV proton SOBP. All curves were normalized by subtracting residual background intensity generated by the correlation calculation and normalizing resulting data to the respective auto-correlation values. Error bars were calculated by propagating the intra-batch gel reproducibility and the fit background error throughout the correlation calculation, as described previously. The sensitivity parameter for acrylamide was quoted in section (5.5) as D0 = 8.1 ±0.4 Gy. Similarily, the acrylamide consumption curves for the proton irradiated gels yield sensitivity parameters of Z?§+'e = 24±2 Gy (end position of proton SOBP) and Dl+'m = 17 ± 1 Gy (mid position of proton SOBP). The ratio of 6 MV x-ray to 74 MeV proton dose required to give a desired 136 effect (e.g. to reduce the number of unreacted monomers to, say, 50% of their initial concentration) can be used to quantify the "relative effectiveness" (RE) of the gel in measuring proton dose. This relative effectiveness is in analogy to relative biological effectiveness (RBE) used in radiobiology [11]. The relative effectiveness can be calculated from figure (6.3). All consumption curves in figure (6.3) are assumed to be mono-exponential in nature. Hence, for a given decrease in the number of unreacted monomer units (i.e. a given y in equation (5.2)) in the end of the SOBP, exp(-Dx/D0) = exp(-Dp+'e/Dp+'e) (6.1) Dx/D0 = Dp+'e/Df'e (6.2) where Dx is the x-ray dose required to give the effect and Dp+>e is the proton dose at the end of the SOBP required to give the same effect. Hence the relative effectiveness RE is RE = Dx/Dp+'e (6.3) = D0/Dp+'e (QA) Similarly for the gel exposed to the mid portion of the SOBP, RE = D0/Dp+'m (6.5) The experimental effectiveness was calculated from the above data, giving an REend = 0.33 ± 0.03 and REmid = 0.47 ± 0.04. The data in figure (6.3), along with the above measured RE's, illustrate i) the large difference in response between proton and x-ray irradiated gels and ii) the less dramatic yet significant difference in gel response between the mid and end regions of the proton SOBP. Overall, the data illustrate that the polymer gel is sensitive to the LET of the incident proton. The RE of the gel in the end and mid 137 regions of the SOBP can also be calculated through the use of the theory of track structure (section (6.3)). This theory also illuminates some of the physical processes occurring in the gels which cause the differential response of the gels exposed to the different regions of the SOBP. 6.3 Track Structure Determination of Polymer Gel Rel-ative Effectiveness 6.3 .1 Basic Theory of Track Structure The (5-ray theory of track structure developed by Katz et al. allows for the prediction of the change in dose response characteristics of a physical (i.e. non-biological) detector when the detector is exposed to radiations of different incident heavy ions [32, 166, 167]. The detector is assumed to consist of sensitive elements (in this case acrylamide molecules) which have a given response to dose. This response may be obtained by exposing the detector to sparsely ionizing (e.g. x-ray or high energy electron) radiation. Typically, physical detectors exhibit a lower response per unit physical dose in response to heavy ions (high LET) compared with low LET radiation. The <5-ray theory of track structure attributes this fact to the highly inhomogeneous dose distribution in close proximity to the path of the heavy ion. That is, detector saturation effects take place close to the path of the heavy ion. With knowledge of the radial dose distribution of secondary electrons around the path of the incident particle, one can make predictions of the relative amount of dose "wasted" close to the ion path. The "effectiveness" of the detector response to the incident particles can then be established. Track structure theory requires knowledge of only two detector parameters. 138 First, the "sensitivity parameter" of the detector must be known (D0 from above). This is established by exposing the detector to sparsely ionizing radiation. Second, an estimate of the size of the sensitive element is required. As shown previously, the fraction (/) of acrylamide molecules not activated after a dose D is given by / = exp(—D/D0). D0 is the dose at which e _ 1 (37%) of the elements are not activated. The probability of sensitive element activation (P) is therefore written as P = 1 — exp(—D/D0). From track structure theory, the response of the detector (sensitive elements) to ejected tf-rays from heavy ion (proton in this case) bombardment follows the same functional form. It is assumed that the sensitivity of the detector molecules is the same for the proton and x-ray irradiations (i.e. the molecules posses the same D0 for the two cases). However, the protons traveling through the polymer gel eject S—rays (electrons) whose dose deposition around the track of the proton is highly inhomogeneous. Hence, the probability of sensitive element activation depends on the radial distance of the element from the proton track, Here D(t) is the average dose deposited in the sensitive element as a function of radial distance t from the proton track. More completely, where a0 is the radius of the sensitive element, z* is the effective charge of the incident particle, and /3 = v/c it's velocity relative to the speed of light. Similarly, the probability of activation P(t) = 1 - exp(-D(t)/D0) (6.6) D(t)=D(t,a0,z*,P) (6.7) P(t) = P(t,a0,z*,fi) (6.8) 139 The total effect produced by the ion, or the beam, can be obtained by integrating equation (6.6) over all radial distances from the proton track, out to the maximum distance traveled by the ejected J-rays (tmax). This yields the total activation cross-section (a), ptmax a = a(a0,z*,8,Do) = 2ir P(t,a0,z* ,B)tdt (6.9) J o The relative effectiveness (RE) of the detector to the given radiation is de-fined as the ratio of proton to x-ray radiosensitivities of the detector. The radiosen-sitivity of the detector to x-ray radiation is simply given by the sensitivity parameter 1/D0. The radiosensitivity of the detector to the particle radiation can be defined as a/LET, or the ratio of total activation cross-section to the total energy deposited at that point (LET) [168]. Hence, REcaic = o-Do/LET (6.10) Track structure therefore predicts, by use of the "relative effectiveness", the detector efficiency to heavy ion radiation, given the low LET (x-ray) detector response. Track structure has been applied to both solid detectors (e.g. alanine, ra-diochromic film)[32, 169, 170] as well as liquid detectors (e.g. Fricke) [32, 171]. In the case of solid detectors, the sensitive element is taken as the size of the individ-ual target. In the cases of liquid detectors, it is assumed that the water radiolysis mechanisms are the same for the proton and x-ray irradiated samples. Hence, the detailed radiolysis mechanisms are not of significant importance in this theory, as the response produced by a given x-ray dose deposited in a detector element (which implicitly takes into account the effects of water radiolysis) is transferred to an equivalent response for a corresponding 6 ray dose (which is assumed to have simi-lar water radiolysis mechanisms). Alternatively, the sensitive element can be taken 140 to be a volume around the target such that dose deposited within this volume can initiate a series of water radiolysis events which diffuse to the target and interact with it to produce an effect (e.g. sensitive element activation). In the study of the Fricke dosimeter, the sensitive element volume was taken to be a sphere of radius 6 nm around a Fe 2 + ion [171]. For the present work, it is shown that for targets between 0.25 nm in radius (i.e. the size of an acrylamide molecule) and 6 nm, the calculated gel relative effectiveness varies by only ~3% (see section (6.3.5)). Note that track structure does not consider dose rate (fluence) effects. If the average separation between proton tracks is smaller than twice the average maximal 6 ray travel, overlapping dose and probability distributions result. This in turn degrades the quality of the track structure results. However, the TRIUMF proton beamline was operated at ~5 nA (~107 protons/cm2/s), which converts to an average separation between proton tracks of ~2-10-5 cm for gel irradiations taking up to a few minutes in duration. This indicates that dose rate effects should not play a significant role in the track calculations (see below). 6.3.2 Radial Point Dose Distribution The starting point for the track calculations is the dose function describing the point dose deposited by delta rays at increasing radial distances away from the heavy ion path. Over the years of development of track structure theory different workers have used several forms of radial dose distributions [32, 170, 172]. In this study the dose distribution proposed by Waligorski et al. and Chunxiang et al. is used [173, 174]. It is a semi-empirical formula derived from a combination of theoretical considerations, fitting to experimental data, and by comparison to results from Monte Carlo simulations of particle tracks in liquid water. It is reported that the 141 dose equation appears to be able to reproduce the experimental data to within 15% for a variety of particles and particle energies in liquid water. The complete point dose equation can be written as: Dp(t) = DiiW + K(t)) (6.11) where Di(t) is the uncorrected point dose function, K(t) is a correction term used to account for the underestimates in dose of Di(t) at small radial distances, and t is the radial distance from the ion track. Each term is described in turn. The uncorrected point dose distribution function is given as: 2TT aP2t _ t+e \ l / a t + e (6.12) for a heavy ion with effective charge z* and moving with relative velocity The derivation of equation (6.12) assumes ejection of electrons normal (i.e. perpendicu-lar) to the incident particle track, an electron ionizing potential J and a power law range-energy relationship, r = kwa (6.13) where r is the range of the ejected electron, w its energy and a and k are empirical parameters obtained by fitting to experimental data: k = 6 • 10 - 6 g/cm2/keVa; a = 1.667 for io > 1 keV, a = 1.079 for w < 1 keV. The maximum electron range (tmax) is obtained by substitution of the maxi-mum delta-ray energy (wmax) into equation (6.13). The maximum delta ray energy is determined by kinematic considerations, 2mc2f32 Wmax 2£ ^~ (6.14) where m is the electron rest mass and c the speed of light. 142 e is the range of an electron having an ionizing potential I = 10 eV [175]. The effective charge for protons is given by the equation of Ziegler [176], z* = 1 - exp[-(.2 [E/Mfl2 + 0.0012 (E/M) + 1.443 • 10-*(E/M)2)} (6.15) where E is the proton energy in keV and M the proton mass in amu. Over the proton energy ranges of interest in this study, the effective charge is essentially unity. C is a constant given by: mc1 where N is the electron density of the medium, e is the elementary charge, and k = l/47re 0 = 8.99 • 109 Nm 2 /C 2 . For water, C = 1.37 • 10~14 J/cm. The second term in equation (6.11), K(t), is a correction term accounting for the underestimate of equation (6.12) at distances close to the path of the ion (typi-cally between 1-10 nm). It is a semi-empirical expression obtained by Waligorski et al. [173], K(t)=A(±=£)exp(-*-=£) t>0.1nm (6.17) K(t) = 0 t<0.1nm (6.18) with A = 8 • Bllz 8 < 0.03 (6.19) A = 19 • Bll* 8 > 0.03 (6.20) B = 0.1nrn (6.21) C = 1.5 nm + 5 nm -8 (6.22) 143 element Figure 6.4: Schematic representation of the sensitive detector element used in the track structure calculations. The point dose function is integrated over the volume of the sensitive element. Notice that at distances far from the ion track equation (6.11) reduces to Dp(i) — 6.3.3 Average Dose in Sensitive Element The average dose deposited in a sensitive element is calculated by integrating the point dose function over the volume of the element. The average dose function _ 0 (D(t,a0,z*,fi)) is then calculated by considering sensitive elements at all radial distances traveled by the ejected electrons. To do this it is assumed that the sensitive element is a cylinder of radius a0 and length I and that its axis is aligned parallel to the track of the ion. The average dose deposited in the sensitive element is hence given by (see figure (6.4) for details), 2 fn fa° D(t,a0) = ^ / Dp(r,0,t)rdrdO (6.23) ™ o Jo Jo The length of the sensitive element (assumed small) cancels in the calculation. Note that the point dose function depends on the initial particle relative velocity and effective charge. Since the points of interest (i.e. the central and end SOBP) have dose contributions from several different incident particle energies, D(t,a0) must be calculated for each step in the modulator wheel. 144 The probability of activation for each average dose distribution (each step in the modulator wheel) is calculated using equation (6.6). The total activation cross section for each probability distribution is calculated using equation (6.9). The relative effectiveness is then calculated using equation (6.10) and by weighing each result by the relative fluence contribution of each step in the modulator wheel, thus arriving at a "cumulative" relative effectiveness for the gel positioned at either the middle or end of the SOBP. All track calculations were performed using Mathematica (Wolfram Research, IL, USA). 6 . 3 . 4 Calculated RE Protons traveling through the thicker portions of the modulator wheel do not reach the gels. In fact, only 9 steps in the modulator wheel need to be considered for gels positioned in the central SOBP region and 5 steps for gels positioned at the end of the SOBP. This has the effect of reducing the total number of calculations necessary and also changing the relative weights of the steps. The average radius of the acrylamide molecule (a0) is taken to be 0.25 nm [89], although the calculation is performed for a range of radii in order to i) ac-count for water radiolysis effects and ii) demonstrate the lack of sensitivity to this parameter (see below). Figure (6.5) illustrates the average dose in the acrylamide molecule (calculated from equation (6.23)) due to electrons ejected by protons trav-eling through the thinnest step in the modulator wheel. As illustrated, close to the path of the proton, doses are much larger than the average D0 of acrylamide (8.1±0.4 Gy). Hence close to the proton track the sensitive elements can be ex-pected to be saturated. The average dose drops off rapidly as a function of distance 145 1E5 7 r i i n 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 Radial Distance, t (cm) Figure 6.5: The average dose deposited by electrons in an acrylamide molecule as a function of radial distance from a proton track. Sensitive element radius a0 = 0.25 nm in this calculation. Table 6.1: Experimental and calculated relative effectiveness of polymer gel to pro-ton irradiation using a 74 MeV SOBP. An a0 = 0.25 nm and D0 = 8 Gy were used in the track calculations. RE Exp RE Calc % Diff end SOBP mid SOBP 0.33±0.03 0.47±0.04 0.32 0.34 3 32 from the track and by ~1.5-10~5 cm the average dose << D0. Since the doses are high close to the proton track, the probability of activation is, essentially, unity in this same region (see figure (6.6)). Again, because the average dose drops off rapidly as a function of radial distance, the probability of activation becomes negligible at large distances from the track, as may be expected. Results of the calculation for relative effectiveness, and a comparison with experimentally determined values, are given in table (6.1). 146 o 0.6 U 0.0 I— OE-6 2E-6 4E-6 6E-6 8E-6 10E-6 Radial Distance, t (cm) Figure 6.6: Probability of activating a sensitive element as a function of radial distance from the proton track. Plot is for the thinnest step of the modulator wheel. a0 = 0.25 nm, D0 = 8 Gy in this calculation. 6.3.5 Extension of Calculation to Different Element Radii and Sen-To demonstrate the sensitivity (or lack thereof) of the track structure calculations to the x-ray sensitivity parameter (D0), the calculations were performed for a range of D0 values. Furthermore, for each D0 the calculation was performed for a range of sensitive element radii. This was performed in order to establish the overall sensitivity of the calculation to the detector element size. Results of the calculations are shown in table (6.2). 6.3.6 Extension of Calculation to Raw Bragg Peak The track structure calculations can be extended to situations which are difficult to measure experimentally. For example, to demonstrate a "worst case scenario", a track structure calculation was made of the relative effectiveness of the gel at the end and at the surface (just behind glass wall) region of a 74 MeV raw Bragg peak. The sitivities 147 Table 6.2: Calculated relative effectiveness of polymer gel to proton irradiation for different sensitive element radii (a0) and sensitivity parameters (D0). a0 (nm) D0 (Gy) RE Calc (end) RE Calc (mid) 0.10 6 0.314 0.328 0.25 6 0.315 0.328 0.50 6 0.315 0.329 1.0 6 0.316 0.330 6.0 6 0.322 0.337 0.10 8 0.321 0.335 0.25 8 0.322 0.335 0.50 8 0.322 0.336 1.0 8 0.323 0.337 6.0 8 0.331 0.347 0.10 10 0.327 0.341 0.25 10 0.327 0.341 0.50 10 0.328 0.342 1.0 10 0.329 0.343 6.0 10 0.339 0.356 0.25 12 0.332 0.347 0.25 14 0.337 0.351 points of calculation were taken to be at 3.26 cm depth in acrylic (3.87 cm water equivalent depth) for the end and 0.46 mm glass (0.8 mm water) for the surface Bragg peak regions (full range = 3.88 cm water equivalent depth). Experimental difficulties make physical measurement of such a setup difficult. Specifically, sample positioning uncertainty as well as uncertainty in the effective Raman sampling point in each sample introduce errors in dose determination at the point of measurement for the end region of the Bragg peak, due to the high dose gradients in this region. Note that this problem was minimized in the above experiments by using a SOBP, which creates a uniform physical dose over the point of measurement. The track calculation was undertaken in the same manner as described above, with a0 = 0.25 nm and D0 = 8 Gy. Table (6.3) summarizes the results of the calculation. 148 Table 6.3: Calculated relative effectiveness of polymer gel to proton irradiation using a 74 MeV raw Bragg peak (BP). RE calculated at surface (i.e. just beyond glass wall) and end of raw Bragg peak. D0 = 8 Gy, a0 = 0.25 nm. Point of calculation is given as water equivalent depth. Point of calc (cm) Energy (MeV) LET (MeV/cm) RE Calc end BP 3.87 3.37 107.71 0.205 entrance BP 0.08 64.81 10.15 0.391 6 . 4 D i s c u s s i o n As seen in sections (6.2), (6.3.4), (6.3.5) and (6.3.6), both experimental results and track calculations indicate a decrease in gel effectiveness as incident proton LET increases. Reasonable agreement between the experimental and calculated relative effectiveness is observed for both the end SOBP region (3% difference) and mid SOBP region (32% difference). The larger difference between experiment and calculation in the middle SOBP region is most likely attributable to the 'first order' nature of the calculation. Specifically, the point dose function is known to be a first order approximation to the 'true' radial dose distribution, both in the initial energy distribution of the S rays, as well as their maximum radial extent [173]. Furthermore, the middle SOBP region has a greater number of contributions from different incident proton energies, possibly degrading the calculation. Overall, the stated % difference between experiment and calculation are comparable with the results of other workers using track structure [169-171]. As mentioned above, the high doses deposited at radial distances close to the proton track (see figure (6.5)) saturate the sensitive elements in the detector. This result illuminates the difficulty in extracting a quantitative dose map from gels irradiated with protons, and heavy ions in general. Both experimental and track calculation results indicate a measurable dependence of gel response to proton LET. 149 The track calculations, being the more conservative in the estimate of the difference in response between the two regions, predict a 6% difference in gel response between the middle and end SOBP regions, with a decrease in gel effectiveness as proton LET increases. A greater difference in calculated RE is observed in the case of a raw Bragg peak (table (6.3)). In this situation, track structure predicts a large (49%) difference in gel effectiveness between the two regions, with a lower gel effectiveness in the end region of the raw Bragg peak. This is consistent with the results described above, in that a lower gel effectiveness is observed at higher proton LET. Given the comparatively low LET of protons relative to heavier ions (e.g. carbon), these results indicate that this saturation effect would only be amplified by irradiating with higher LET particles (e.g. see [163, 164]). Table (6.2) reports the variation in REcai^mid) a n d REcalcrend) for different a0 and D0 in the SOBP case. The results in table (6.2) illustrate that, in this application of track structure, there is only a weak dependence in RE^i,.^^ and -R-^ 'coZc(eud) t° th e s * z e °f the sensitive element. For sensitive element radii in the 0.1 to 1 nm range, both i? C^a/c(mtd) -R-^ caJcCewZ) v a r v by less than 1%. At larger radii the relative effectiveness tends to increase slightly (~4%) due to the fact that the average dose in the sensitive elements decreases as the size of the element increases. The variation of REcai^^ and RE^i,.^^ with D0 is understood by con-sidering that a higher D0 indicates a less sensitive detector element. Hence less saturation in the high dose regions is expected. This in turn increases the relative effectiveness of the dosimeter, as indicated in table (6.2). Two possible strategies to overcome the problem of saturation in high LET regions are considered, although both appear limiting in their scope. The first involves establishing an LET calibration response curve to the heavy ion [163, 164]. 150 Construction of this curve is a formidable task, given the high dose gradients in the proton depth dose curve. Furthermore, it is unclear how this curve would be applied in more complex settings (e.g. multiple beams, inhomogeneities etc). The second possibility involves testing less sensitive gel compositions in the hope that saturation effects would be minimized. As shown in section (5.7), polyacrylamide gel sensitivity can be drastically varied by tuning the ratio of acrylamide to bis in the initial composition. Results of section (5.7) indicate that gels extremely high or low in crosslinker fraction (eg 0% or 100% crosslinker fraction) exhibit low sensitivities to x-ray irradiation. One of these dosimeters has been tested experimentally (100% crosslinker fraction). Saturation effects are still apparent, and upon inspection of the x-ray dose response curve it is clear that the "sensitivity parameter" for this gel is still orders of magnitude lower (i.e. more sensitive) than the average doses deposited close to the proton track (figure (6.5)). Track calculations for this dosimeter predict a 50% difference in response between the entrance and end region of the raw BP. A 4% difference is calculated for the difference in RE for protons in the same end and mid regions of a SOBP as in the above cases (i.e. same regions as described in section (4.3)). Lepage et al. have further obtained sensitivity parameters for polymer gels of different %T and gelatin concentrations [105]. The sensitivity parameters obtained in their experiments range between ~4 and 12 Gy. This indicates that the D0 parameters for these gels are small as compared to the doses deposited by delta rays close to the track of the proton. Again, saturation effects are expected and LET dependencies in gel response would be observed in these gels. Hence varying initial polyacrylamide gel composition does not appear to be sufficient in this respect. 151 Chapter 7 Conclusions 7.1 Conclusions This thesis has been an investigation into the dose response mechanisms in irradiated polymer gel dosimeters. Fourier transform Raman spectroscopy has been used to probe the chemical changes occurring in irradiated polymer gels. Roughly, the results of the thesis are sub-divided into three sections. The first component involved establishing adequate experimental parameters from which further experiments were conducted. A robust and reproducible polymer gel sample preparation technique was established. Adequate vials to house the sam-ples were found which were compatible with both the requirements of the Raman spectrometer and the polymer gels. Adequate Raman scanning parameters (laser power and number of interferometer scans) were established. The reproducibility of spectral peak areas was established for Raman spectra acquired on a single sample, samples manufactured from the same batch of gel and irradiated to the same known dose (5 Gy), and samples manufactured from different batches of gel and either left 152 unirradiated, or irradiated to the same known dose (5 Gy). Results indicate that acrylamide and bis peak reproducibility is better than 1% for spectra acquired on a single unirradiated sample, ~1.5% for spectra acquired on gel samples manufac-tured from the same batch of bulk gel and irradiated to 5 Gy, and ~3.5 % for spectra acquired on gel samples manufactured from different batches of gel and irradiated to 5 Gy. The results indicate that the overall system was sufficiently robust and reproducible to allow further studies to be conducted. Since, in spectra acquired on gels irradiated to different doses, the spectral background may vary minutely, a robust background subtraction technique was developed. This involved fitting the relevant spectral peaks to a combination of linear background and either Lorentzian or Gaussian line-shapes and subsequently subtracting the fit background from the raw data. This minimized errors associated with fitting the spectral peak to the given functions. The peak intensities for the resulting spectral datasets were quan-tified using either the technique of correlation (for the monomer peak data) or, in the absence of high signal to noise reference spectra, integration (for the polyacry-lamide peak data). The overall data analysis technique was robust and minimized errors associated with background variation, peak fitting, and low signal to noise portions of the spectra. Customized software was written to aid in the data analysis of the spectra. The second component of the thesis involved examining the consumption of monomer and formation of polymer for a 6%T, 50%C gel irradiated with 6 MV photons. Monomer consumption and polymer formation curves were constructed. Results indicate that both the monomer consumption and polymer formation curves are non-linear. In fact, monomer consumption and polymer formation data can be modelled reasonably well as mono-exponential. This non-linearity is significant 153 when comparing these response curves with traditional (MRI) polymer gel response curves which were, until recently, thought to be linear. Recent data by other workers confirm the non-linearity of the gel dose response observed in this thesis. The rate of polymer formation is shown to be equal to the rate of acrylamide consumption. The characterization of monomer consumption is extended to polymer gels of dif-fering initial composition. The initial bis fraction (%C) is varied while keeping the total amount of monomer (%T) as constant as possible (limited by the solubility of bis). In general highly non-linear monomer consumption is observed for the different composition gels. This non-linearity is put into context using a qualitative model, which essentially attributes different rates of consumption as being intimately linked to the types of polymer formed in each gel. Furthermore, it is demonstrated that different types of polymer are formed within each gel as it is irradiated to higher doses. This is due to the differential rates of acrylamide and bis consumption within each gel. Thus, the average number of acrylamide molecules per bis molecule avail-able for reaction in each gel is not constant as a function of dose, creating different polymer forms in the different dose regions. Finally, it has been demonstrated that the pre-existence of a polymer network in the gel has an effect on the subsequent monomer consumption rate. The final component of the thesis involved studying the dependence of poly-mer gel response on ionizing density. Raman spectra were acquired on gels irradiated in two separate regions of a 74 MeV SOBP. Monomer consumption curves were con-structed from the Raman data. By comparing these consumpion curves with the 6 MV x-ray irradiated gel consumption curves, a gel "relative effectiveness" in mea-suring unit proton dose was established. It was found that, in general, the gel has a weaker response per unit proton dose as compared with unit x-ray dose. The gel rela-154 tive effectiveness was found to be RE(end) = 0.33±0.02 and RE{mid) = 0.49±0.02. This illustrated the differential response (i.e. LET dependence) of the gel in measur-ing proton dose. The theory of track structure was used to predict the gel relative effectiveness in the same two regions of the proton depth dose curve. The agreement between experimental and track structure results was ~30%, which is on the order of accuracy achieved by other workers using track structure. The track structure results also illuminated some of the physical processes occurring in the gels which lead to the general gel under-response (as compared to x-ray irradiated response curves) when irradiated with protons. Ultimately, the highly inhomogeneous dose distributions around the track of a proton lead to saturation (overkill) effects at distances very close (~ 10 - 6 cm) to the track of the proton. In sum, this thesis has extended the base of knowledge in the fundamental dose response characteristics of the polymer gels. In particular, the non-linear na-ture of the response of a 6% T, 50% C (i.e. standard composition) gel has been established. Furthermore, the generally complex, non-linear nature of the response of gels of varying initial composition has been established and this new knowledge is an advancement over pre-existing notions of gel response as a function of initial bis fraction (%C). This should aid in future applications of polymer gels in dose veri-fication studies. Finally, significant advances have been made in the understanding of gel response as a function of ionizing density. The dependence of gel response on LET has exposed the difficulty of characterizing high LET beams using polymer gels. 155 7.2 Future Directions There are several areas in which further research could be conducted. Recently, mathematical modelling of the polymerization reactions occurring in irradiated poly-mer gel has been attempted [155]. It may become possible to correlate the results of those studies with the experimental monomer consumption data and qualitative model of polymer formation presented in chapter (5). This would further the un-derstanding of the different types of polymer being formed in the gel as a function of dose. Ultimately, the correlation of the results of the mathematical model with the above experimental results may allow for the shapes of the response curves to be understood on a more quantitative level. Recent developments in polymer gel dosimetry have included the introduc-tion of a number of new dosimeters (section (3.5.2)). One of the most promising of these is the dosimeter of Fong et al. , which has the practical advantage of gel man-ufacture under normal room atmosphere (i.e. anoxic conditions are not required) [138]. It is likely possible to undertake Raman spectroscopy studies to determine the chemical changes occurring in a number of these gels. Correlation of the chemical changes occurring as a function of dose in these recently developed gels with those occurring in the traditional polyacrylamide system would be of practical interest to the gel dosimetry community. 156 Appendix A Data Handling Software Data analysis software was written in Visual C++ (Microsoft Foundation Classes). The software is able to display up to 4 individual spectra simultaneously. User op-erated control sliders are used to manipulate the x-axis display (magnification) of the spectra (i.e. "x-axis zooming"). User operated push buttons are used to define the y-axis magnification. The user can perform simple mathematical operations (addition, subtraction, multiplication, division) on any one or two of the displayed spectra. It is further possible to correlate user defined portions of any two dis-played spectra together. Limited peak fitting (single peak, Lorentzian of Gaussian functions) on a user defined portion of the spectrum is possible. 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