{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","Extent":"http:\/\/purl.org\/dc\/terms\/extent","FileFormat":"http:\/\/purl.org\/dc\/elements\/1.1\/format","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Physics and Astronomy, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Kwa, William","@language":"en"}],"DateAvailable":[{"@value":"2009-05-28T23:27:47Z","@language":"en"}],"DateIssued":[{"@value":"1998","@language":"en"}],"Degree":[{"@value":"Doctor of Philosophy - PhD","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"Asymmetric collimation of photon beams produces non-trivial alterations in absolute\r\noutput, depth dose and beam profile. The full potential of asymmetric collimation can only be\r\nrealized with a proper treatment planning algorithm specific for asymmetric collimation. In\r\nthis thesis the dosimetric characteristics of asymmetric fields are investigated and a new\r\ncomputation method for the dosimetry of asymmetric fields is described and implemented into\r\nan existing treatment planning algorithm. Based on this asymmetric field treatment planning\r\nalgorithm, the clinical use of asymmetric fields in cancer treatment is investigated, and new\r\ntreatment techniques for conformal therapy are developed. Dose calculation is verified with\r\nthermoluminescent dosimeters in a body phantom.\r\nAn asymmetric field is referred to as an off-set radiation field whereby the central axis\r\nof the radiation field does not coincide with the collimator axis as the opposite pair of\r\ncollimators no longer are equidistant from the collimator axis. Here, the corresponding\r\nsymmetric field is a radiation field centered at the collimator axis with the opposite pair of\r\ncollimators set equidistant from the collimator axis and to the largest asymmetric collimator\r\nsetting. Usually the dose distribution in an asymmetric field is represented by some form of\r\nbeam modeling. In this thesis, an analytical approach is proposed to account for the dose\r\nreduction when a corresponding symmetric field is collimated asymmetrically to a smaller\r\nasymmetric field. This is represented by a correction factor that uses the ratio of the\r\nequivalent field dose contributions between the asymmetric and symmetric fields. The same\r\nequation used in the expression of the correction factor can be used for a wide range of asymmetric field sizes, photon energies and linear accelerators. This correction factor will\r\naccount for the reduction in scatter contributions within an asymmetric field, resulting in the\r\ndose profile of an asymmetric field resembling that of a wedged field.\r\nThe output factors of some linear accelerators are dependent on the collimator settings\r\nand whether the upper or lower collimators are used to set the narrower dimension of a\r\nradiation field. In addition to this collimator exchange effect for symmetric fields, asymmetric\r\nfields are also found to exhibit some asymmetric collimator backscatter effect. The proposed\r\ncorrection factor is extended to account for these effects.\r\nA set of correction factors determined semi-empirically to account for the dose\r\nreduction in the penumbral region and outside the radiated field is established. Since these\r\ncorrection factors rely only on the output factors and the tissue maximum ratios, they can\r\neasily be implemented into an existing treatment planning system. There is no need to store\r\neither additional sets of asymmetric field profiles or databases for the implementation of these\r\ncorrection factors into an existing in-house treatment planning system. With this asymmetric\r\nfield algorithm, the computation time is found to be 20 times faster than a commercial system.\r\nThis computation method can also be generalized to the dose representation of a two-fold\r\nasymmetric field whereby both the field width and length are set asymmetrically, and the\r\ncalculations are not limited to points lying on one of the principal planes.\r\nThe dosimetric consequences of asymmetric fields on the dose delivery in clinical\r\nsituations are investigated. Examples of the clinical use of asymmetric fields are given and the\r\npotential use of asymmetric fields in conformal therapy is demonstrated. An alternative head\r\nand neck conformal therapy is described, and the treatment plan is compared to the conventional technique. The dose distributions calculated for the standard and alternative\r\ntechniques are confirmed with thermoluminescent dosimeters in a body phantom at selected\r\ndose points.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/8435?expand=metadata","@language":"en"}],"Extent":[{"@value":"6736857 bytes","@language":"en"}],"FileFormat":[{"@value":"application\/pdf","@language":"en"}],"FullText":[{"@value":"ASYMMETRIC COLLIMATION: DOSIMETRIC CHARACTERISTICS, TREATMENT PLANNING ALGORITHM, AND CLINICAL APPLICATIONS. by WILLIAM KW A B.Sc, The University of British Columbia, 1978 M.Sc.A., McGill University, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT S FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1998 \u00a9 William Kwa, 1998 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 The University of British Columbia Vancouver, Canada DE-6 (2\/88) ABSTRACT Asymmetric collimation of photon beams produces non-trivial alterations in absolute output, depth dose and beam profile. The full potential of asymmetric collimation can only be realized with a proper treatment planning algorithm specific for asymmetric collimation. In this thesis the dosimetric characteristics of asymmetric fields are investigated and a new computation method for the dosimetry of asymmetric fields is described and implemented into an existing treatment planning algorithm. Based on this asymmetric field treatment planning algorithm, the clinical use of asymmetric fields in cancer treatment is investigated, and new treatment techniques for conformal therapy are developed. Dose calculation is verified with thermoluminescent dosimeters in a body phantom. An asymmetric field is referred to as an off-set radiation field whereby the central axis of the radiation field does not coincide with the collimator axis as the opposite pair of collimators no longer are equidistant from the collimator axis. Here, the corresponding symmetric field is a radiation field centered at the collimator axis with the opposite pair of collimators set equidistant from the collimator axis and to the largest asymmetric collimator setting. Usually the dose distribution in an asymmetric field is represented by some form of beam modeling. In this thesis, an analytical approach is proposed to account for the dose reduction when a corresponding symmetric field is collimated asymmetrically to a smaller asymmetric field. This is represented by a correction factor that uses the ratio of the equivalent field dose contributions between the asymmetric and symmetric fields. The same equation used in the expression of the correction factor can be used for a wide range of ii asymmetric field sizes, photon energies and linear accelerators. This correction factor will account for the reduction in scatter contributions within an asymmetric field, resulting in the dose profile of an asymmetric field resembling that of a wedged field. The output factors of some linear accelerators are dependent on the collimator settings and whether the upper or lower collimators are used to set the narrower dimension of a radiation field. In addition to this collimator exchange effect for symmetric fields, asymmetric fields are also found to exhibit some asymmetric collimator backscatter effect. The proposed correction factor is extended to account for these effects. A set of correction factors determined semi-empirically to account for the dose reduction in the penumbral region and outside the radiated field is established. Since these correction factors rely only on the output factors and the tissue maximum ratios, they can easily be implemented into an existing treatment planning system. There is no need to store either additional sets of asymmetric field profiles or databases for the implementation of these correction factors into an existing in-house treatment planning system. With this asymmetric field algorithm, the computation time is found to be 20 times faster than a commercial system. This computation method can also be generalized to the dose representation of a two-fold asymmetric field whereby both the field width and length are set asymmetrically, and the calculations are not limited to points lying on one of the principal planes. The dosimetric consequences of asymmetric fields on the dose delivery in clinical situations are investigated. Examples of the clinical use of asymmetric fields are given and the potential use of asymmetric fields in conformal therapy is demonstrated. A n alternative head and neck conformal therapy is described, and the treatment plan is compared to the iii conventional technique. The dose distributions calculated for the standard and alternative techniques are confirmed with thermoluminescent dosimeters in a body phantom at selected dose points. iv TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS v LIST OF TABLES viii LIST OF FIGURES ix ACKNOWLEDGMENTS xiv 1. INTRODUCTION 1 1.1. Objective and Contents of this Thesis 7 2. BACKGROUND 9 2.1. Radiation Absorbed Dose Calculations in External Beam Radiotherapy 9 2.1.1. The depth of maximum dose (dmax) 9 2.1.2. The monitor unit (MU) and output factor (OPF) 10 2.1.3. The percentage depth dose (PDD) 13 2.1.4. The tissue maximum ratio (TMR) and tissue air ratio (TAR) 15 2.1.5. The backscatter factor (BSF) or the peak scatter factor (PSF) 15 2.1.6. The scatter air ratio (SAR) 18 2.1.7. The off axis ratio (OAR) and the depth dose profile 20 2.1.8. The isodose curve, and isodose chart or distribution 20 2.1.9. The collimator scatter factor (CSF) or head scatter factor 22 3. MATERIALS AND METHODS 23 3.1. Linear Accelerator (linac) 23 v 3.2. Dosimetry Phantoms 23 3.2.1. Water tank system 23 3.2.2. Body phantom 24 3.3. Dosimetric Measuring Instruments 25 3.3.1. A thimble type ionization chamber 25 3.3.2. Thermoluminescent dosimeter (TLD) 27 3.4. Treatment Planning 28 3.4.1. Treatment planning algorithm 28 3.4.1.1. An in-house treatment planning algorithm 29 3.4.1.2. A commercial treatment planning system 29 3.4.2. Absorbed dose distribution in a patient 30 4. DOSIMETRIC CHARACTERISTICS OF ASYMMETRIC BEAMS 35 4.1. Overview\/Historical Development 35 4.2. The Output Factor (OPF) of an Asymmetric Field (AF) 40 4.2.1. Direct implementation of the Day's equivalent field method 40 4.2.2. Derivation of a correction factor (CF) 47 4.2.3. Influence of the collimator settings on the output factor (OPF) 50 4.2.4. Influence of the collimator settings on the asymmetric field output factor (AF OPF) 56 4.3. Asymmetric Field Dose Calculation 72 4.3.1. The asymmetric field dose representation at depth 72 4.3.2. The wedge effect of asymmetric collimation 77 vi 4.4.. An Asymmetric Field Treatment Planning Algorithm 86 4.4.1. Generation of a set of correction factors 86 4.4.2. Implementation of the correction factors as a FORTRAN subroutine 93 4.4.3. Summary 98 4.5. Asymmetric Collimation in Both Field Dimensions 100 5. CLINICAL USE OF ASYMMETRIC COLLIMATORS 112 5.1. An Overview 112 5.2. Half-collimated Asymmetric Breast Treatment Technique 118 5.3. Head and Neck Conformal Therapy 124 6. CONCLUSIONS 137 BIBLIOGRAPHY 140 APPENDIX 1: AsymCorrFactor- a FORTRAN subroutine for the asymmetric field treatment planning algorithm 147 APPENDIX 2: Glossary of Terms 151 vii LIST OF TABLES Page Table 1. Ratios of calculated asymmetric field output factors to measured asymmetric field output factors for a 6 MV photon beam. 64 Table 2. Ratios of calculated asymmetric field output factors to measured asymmetric field output factors for a 10 MV photon beam. 65 Table 3. Difference in \"T sym\" readout of a Clinac 2100C linac with the Clinac dose servo disabled for different 6 MV asymmetric fields. 69 Table 4. Difference in \"T sym\" readout of a Clinac 2100C linac with the Clinac dose servo disabled for different 10 MV asymmetric fields. 70 Table 5. The measured ion chamber ratios of asymmetric fields and symmetric fields for a 6 MV photon beam at the asymmetric field center are compared to the calculated correction factors at the same point. 75 Table 6. The measured ion chamber ratios of asymmetric fields and symmetric fields for a 10 MV photon beam at the asymmetric field center are compared to the calculated correction factors at the same point. 76 Table 7. Table 8. The effective tissue air ratios at dm a x and at 10 cm depth are calculated based on the Clarkson's scatter-sector integration method. 82 The average percentage dose difference between the calculated and measured dose values for points within the radiation field and 1.5 cm away form the field edges along the asymmetric central plane and the principal plane are shown. 110 Table 9. TLD verification of doses received with the standard technique at discrete points in a prestwood head phantom is shown. 127 Table 10. TLD verification of doses calculated using the in-house treatment planning system with the alternative technique at discrete points in a prestwood head phantom is shown. 132 LIST OF FIGURES Page Figure 1. An illustration of a medical linear accelerator and its rotational and translational degrees of freedom. 2 Figure 2. A linear accelerator head assembly in photon beam mode is shown. 3 Figure 3. The upper and lower collimators defining a pyramid of useful radiation are illustrated. 4 Figure 4. The relationships between field sizes and distances in an asymmetric field and a symmetric field are illustrated. 6 Figure 5. A typical photon beam depth dose curve is shown. 11 Figure 6. The irradiation geometry used in the derivation of the output factor (OPF) is shown. 12 Figure 7. The irradiation geometry used in the derivation of the percentage depth dose (PDD) at a fixed source to surface distance is shown. 14 Figure 8. The irradiation geometry used in the derivation of the tissue maximum ratio (TMR) at a fixed source to axis distance with field size W x L is shown. 16 Figure 9. The irradiation geometry used in the derivation of the tissue air ratio (TAR) at a fixed source to axis distance with field size W x L is shown. 17 Figure 10. The irradiation geometry used in the derivation of the backscatter factor (BSF) or the peak scatter factor (PSF) at a fixed source to axis distance with field size W x L is shown. 19 Figure 11. The irradiation geometry used in the derivation of the off axis ratio (OAR) is shown. 21 Figure 12. A schematic diagram of a typical thimble ionization chamber is shown. 26 Figure 13. The storing of a matrix of dose points is shown. Dose points are represented by 13 fan lines and 5 transverse lines. 31 Figure 14. A schematic diagram illustrating the absorbed dose in a rectangular field of dimensions W x L at a point Q. 32 Figure 15. A three-dimensional representation of the patient outline by the superior (sup), central, and inferior (inf) transverse cross-sectional planes or slices. 33 Figure 16. Isodose distributions for a three field treatment are illustrated on several cross-sectional CT slices. 34 Figure 17. The equivalent field contributions used in the Day's equivalent dose calculation method are illustrated. 41 Figure 18. The off-axis ratios (OARs) for both the (A) 6 MV and (B) 10 MV photon beams at dmax and in-air are shown. 45 Figure 19. Illustration of primary and scattered radiation to points on center and at the edge of the radiation field. 46 Figure 20. The output factors (OPFs) were measured for square and rectangular fields with the upper Y collimator and the lower X collimator defining the field width for a 6 and a 10 MV photon beam. 52 Figure 21. The geometric relationships between x-ray target, flattening filter, and upper and lower secondary collimators relative to the isocenter are illustrated. 54 Figure 22. The asymmetric field output factors (AF OPFs) were measured at the asymmetric field center for a 6 MV 12 x 40 cm2 symmetric field collimated to 9, 6, and 4 x 40 cm2 asymmetric fields. 59 Figure 23. The asymmetric field output factors (AF OPFs) were measured at the asymmetric field center for a 10 MV 12 x 40 cm2 symmetric field collimated to 9, 6, and 4 x 40 cm2 asymmetric fields. 60 Figure 24. The asymmetric field output factors (AF OPFs) were measured at the asymmetric field center for a 6 MV 40 x 20 cm2 symmetric field collimated to 30, 20, 18, and 10 x 40 cm2 asymmetric fields. 61 Figure 25. The asymmetric field output factors (AF OPFs) were measured at the asymmetric field center for a 10 MV 40 x 20 cm2 symmetric field collimated to 30, 20, 18, and 10 x 40 cm2 asymmetric fields. 62 Figure 26. The collecting plate configuration of the beam monitor chamber is shown. 68 Figure 27. Typical absorbed dose profiles for a symmetric field, a half-collimated asymmetric field, and a symmetric field with the same field width as the half-collimated asymmetric field are shown. 78 Figure 28. Illustration of a 20 x 20 cm2 field with the 10\u00b0 sectors used in the calculation of effective tissue air ratio (eff TAR) and effective peak scatter factor (eff PSF) at collimator axis and 7 cm off-axis is shown. 80 Figure 29. The measured and calculated asymmetric field dose profiles are compared for a 6 MV photon beam at dmax and at 100 cm source to axis distance. 83 Figure 30. The measured and calculated asymmetric field dose profiles are compared for a 10 MV photon beam at dmax and at 100 cm source to axis distance. 84 Figure 31. The measured and calculated dose profiles are compared for a 6 MV photon beam at depths of 2, 10 and 25 cm and at 100 cm source to axis distance. 87 Figure 32. A schematic diagram illustrating the dose profiles of a symmetric field and a smaller asymmetric field is shown. The positions where the various correction factors used in the present algorithm are applied for asymmetric fields are indicated. 89 Figure 33. The measured and calculated dose profiles are compared for a 10 MV photon beam at depths of 2, 10 and 25 cm and at 100 cm source to axis distance. 92 Figure 34. The measured and calculated isodose distributions for a 10 MV photon beam at 100 cm source to surface distance are shown. 94 Figure 35. The measured and calculated isodose distributions for a 6 MV photon beam at 100 cm source to surface distance are shown. 95 Figure 36. The calculated isodose distribution for two adjacent half-collimated beams and a single symmetric field for a 6 and a 10 MV photon beam are shown. 97 Figure 37. Illustration of the equivalent field principle for a 2-fold asymmetric field is shown. 101 Figure 38. The measured and calculated dose profiles along one of the principal planes at dmax, 10 and 25 cm depths and at 100 cm source to axis distance are compared for a 6 MV photon beam. 105 Figure 39. The measured and calculated dose profiles along one of the principal planes at dmax, 10 and 25 cm depths and at 100 cm source to axis distance are compared for a 10 MV photon beam. 106 Figure 40. The measured and calculated dose profiles along the asymmetric central plane (2.5 cm away from the principal plane) at dmax, 10 and 25 cm depths and at 100 cm source to axis distance are compared for a 6 MV photon beam. 107 Figure 41. The measured and calculated dose profiles along the asymmetric central plane (2.5 cm away from the principal plane) at dm a x, 10 and 25 cm depths and at 100 cm source to axis distance are compared for a 10 MV photon beam. 108 Figure 42. The measured and calculated dose profiles of a 20 x 40 cm2 symmetric field collimated in both dimensions to a 10 x 20 cm2 asymmetric field along the asymmetric central plane (10 cm away from the principal plane) at dm a x, 10 and 25 cm depths and at 100 cm source to axis distance are compared for a 6 MV photon beam. 111 Figure 43. Comparison of treatment plans for a concavely shaped planning target volume typical of nasopharyngeal cancer with retrostyloid tumor extension using a multiple asymmetric fields technique and a conventional three-field technique. 115 Figure 44. A treatment plan for an L-shaped planning target volume typical of nasopharyngeal cancer. 117 Figure 45. Schematic illustrating the set-up of the conventional tangential breast treatment with gantry tilt. 119 Figure 46. Schematic illustrating the SSD set-up of the half-collimated asymmetric field breast treatment. 120 Figure 47. The dose distributions for breast irradiation with two tangential opposed beams using the conventional and the half-collimated asymmetric field technique. 121 Figure 48. Schematic illustrating the isocentric SAD set-up of the half-collimated asymmetric field breast treatment. 123 Figure 49. The dose distributions in the central plane of a prestwood head phantom for a standard technique and an alternative technique. 126 Figure 50. The dose distribution for a single arced asymmetric field in a circular phantom as calculated with the proposed asymmetric field algorithm and with the GE Target system. 130 Figure 51. The dose distributions for a standard technique and an alternative technique along the mid-plane of a patient's neck are shown. 133 Figure 52. The dose distributions of the same patient using the same alternative technique at a plane 5 cm superiorly is shown. Figure 53. The dose distribution of the same patient is shown at a plane 5 cm inferiorly using the alternative technique but with the large lateral pair angled at an oblique angle so as to avoid the shoulder. ACKNOWLEDGMENTS I would like to express my gratitude to the following people: Dr. E. El-Khatib, my thesis supervisor and head physicist, for her continuous support and guidance. Dr. R. O. Kornelsen, my retired head physicist, for his original suggestion towards this topic. Dr. S. Hussein, my colleague and mentor, for his encouragement and advice. My co-authors, and in particular, Mr. R. W. Harrison, whose work constitutes invaluable contributions to our joint projects. My colleagues in the Clinical Physics Department, oncologists in the Division of Radiation Oncology, and staff of the Medical Illustration Department for their patience, assistance, and support. Dr. J. Scrimger and Ms. O. L. Chow for their help. My dear wife Joyce, parents, relatives and friends for their patience, encouragement, and support. xiv 1. INTRODUCTION Radiotherapy is a major treatment modality for cancer. In external beam radiotherapy a radiation emitting device is used to aim a beam of photons or electrons at the tumor volume from some distance away. The equipment of choice for external beam therapy is the linear accelerator (linac). As illustrated in Figure 1, the gantry of the linear accelerator rotates along an axis that is centered at the isocenter as represented by GG. The isocenter is a reference point in space that is the common axis of rotation for the gantry, collimator and turntable. The couch rotates with the turntable as represented by 0T and moves vertically, longitudinally and laterally in order to align the patient's treatment center with the isocenter. The collimator consists of the field-defining jaws that also have the isocenter as the axis of rotation as represented by 9c. The collimator is part of the structure of the treatment head of the linear accelerator as shown in Figure 2. In photon mode, electrons strike a target and produce bremsstrahlung photons. These photons are first collimated by the primary collimator to form a cone of radiation. The bremsstrahlung photons produced are forward peaked in intensity and therefore to produce a uniformly intense radiation field need to pass through a flattening filter. This radiation beam is then collimated by two pairs of continuously moveable collimators or jaws to produce a pyramid of useful radiation where the radiation field size can have any square or rectangular shape typically between 0 x 0 to 40 x 40 cm2 at 100 cm from the target as shown in Figure 3. The two pairs of collimators are perpendicular to each other and are referred to as the upper and lower collimators. For the Varian Clinac linear accelerators used (Varian Associates, Palo Alto, CA), the upper pair of collimators are 1 Figure 1. An illustration of a medical linear accelerator and its rotational and translational degrees of freedom. 2 Figure 2. A linear accelerator head assembly in photon beam mode is shown. \u00abm target Figure 3. The upper and lower collimators defining a pyramid of useful radiation are illustrated. 4 referred to as the Y i and Y 2 collimators, and the lower collimators are labeled X i and X 2 . With the collimator angle 0c set at 0\u00b0, the lower pair defines the radiation field width, W, and the upper pair defines the radiation field length, L. Normally, each pair moves equidistant from the center of the field so as to define a symmetric radiation field of rectangular or square shape. By convention, the position of each collimator is defined as the projected dimension at the cross plane of the isocenter which is normally 100 cm from the radiation source. For example, a symmetric field of 10 x 20 cm2 has the collimators set at X i = 5 cm, X 2 = 5 cm, Y i = 10 cm, and Y 2 = 10 cm. Some newer linear accelerators allow the independent movement of each pair of the upper and lower collimators. This feature is called asymmetric collimation. With this feature, each pair of the upper and lower collimators do not need to be set equidistant from each other. For example, X i is set at 5 cm whereas X 2 is set at 2 cm. The result is a radiation field width of 7 cm wide whose center is now located at r a s y m which is 1.5 cm offset towards the X i collimator (Fig. 4). This off-centered field is referred to as the asymmetric field and has a width AW. The clinical use of asymmetric collimation is quite limited.1 This is because most radiotherapy centers do not have the proper treatment planning algorithm to calculate the dose distribution within an asymmetric field. In radiotherapy, accurate dose delivery is crucial since the aim is to deliver maximum absorbed dose to the tumor but minimum dose to normal tissue. In particular, the absorbed dose to critical structures such as the spinal cord must not exceed organ tolerance or else the treatment will fail because of treatment complication. 5 Figure 4. The relationships between field sizes and distances in an asymmetric field (AF) and a symmetric field (SF) are illustrated. 6 1.1. Objective and Contents of this Thesis The aim of this thesis is to develop a dose computation method that would accurately predict the radiation dose at any point in an asymmetric field and that requires minimal data acquisition and minimal computation time. Some of the dosimetric characteristics of asymmetric fields are investigated. The clinical use of asymmetric fields is discussed and new treatment techniques are developed. In Chapter 2, some of the dosimetric parameters commonly used in radiotherapy physics are explained. Materials and Methods are covered in Chapter 3. Topics include the linear accelerators and phantoms used in the work of this thesis. The dose measuring methods and the algorithms used in treatment planning are also described. In Chapter 4.1, some of the published papers on the dosimetry of asymmetric fields are described. In Chapter 4.2, the influence of the asymmetric field settings on radiation output is investigated and a correction factor is derived to account for the difference in radiation dose in symmetric and asymmetric fields. In Chapter 4.3, the correction factor is extended to represent absorbed dose at any off axis point and at any depth along the principal plane parallel to the asymmetric field dimension. The correction factor is also shown to account for the observed tilt of the radiation dose profile (wedge effect) of an asymmetric field. In Chapter 4.4, the implementation of the correction factor into an existing treatment planning program is discussed. Comparisons are made between this correction factor method and the dose calculations made with those obtained from a commercial system based on the Cunningham's IRREG program.2 In Chapter 4.5, the correction factors are extended for dose 7 representation of two-fold asymmetric fields, i.e., with both the field width and length collimated asymmetrically. In Chapter 5.1, the clinical use of asymmetric collimators and its implication in dose optimization are investigated. In particular, the wedge effect (as discussed in Chapter 4.3) on treatment optimization is illustrated. An improved tangential breast treatment technique in terms of ease of patient set-up is introduced in Chapter 5.2. Again, the importance of the use of a proper asymmetric field dose calculation algorithm for the optimal optimization and delivery of a treatment plan is illustrated. In Chapter 5.3, a head and neck treatment technique with static and rotational asymmetric fields is proposed. The calculated dose distributions are confirmed by thermoluminescent dosimeter measurements in a body phantom. 8 2. B A C K G R O U N D 2.1. Radiation Absorbed Dose Calculations in External Beam Radiotherapy The use of external beam radiotherapy involves careful treatment planning. The aim of external beam radiotherapy planning is to deliver maximum absorbed dose to a well defined planning target volume in a patient to obtain the desired biological or clinical effects while at the same time sparing the surrounding normal tissues. As a result, a precise knowledge of the absorbed dose at all points of interest in an irradiated patient is required. Unfortunately, verification of dose using direct measurement is seldom possible within the patient. Indirect methods are therefore employed. Conventionally, absorbed dose measurements are carried out in a well defined manner with some water-equivalent phantom. The dose distributions are then used to relate and predict the dose distributions within the patient. Ultimately the dose at all points in an irradiated 3 dimensional volume is related to the dose at the point where the accelerator is calibrated. Some of the terminology used in the dose representation for external beam radiotherapy is described here. 2.1.1. The depth of maximum dose (dmax) This is the depth in an irradiated medium at which maximum dose is observed. For megavoltage photon beams, the photon interaction sets in motion electrons that deposit their energies at some distance away from the site of photon interaction. Thus, the maximum 9 absorbed dose occurs at some depth beyond the irradiated surface as shown in Figure 5 where all doses are shown normalized to 100% at the maximum. The actual depth of maximum dose depends on the photon energy used. For photon energies below the megavoltage range, the dmax is just below the surface, but for megavoltage photon beams, the dm a x in water ranges from 0.5 cm for Cobalt 60 radiation to 5 cm for a 25 MV photon beam. 2.1.2. The monitor unit (MU) and output factor (OPF) The radiation beam emanating from a medical linear accelerator is pulsed and the radiation output over time may fluctuate. In order to monitor the dose delivered to the patient, most linear accelerators employ a beam monitor chamber situated in the treatment head to integrate the radiation delivered over a set period. The unit used is the monitor unit (MU). Upon the delivery of a preset MU, the linear accelerator terminates the radiation production automatically. Since the actual dose delivered to the patient is dependent on various factors such as the size of the radiation field used, the term output factor (OPF) is used to relate the dose delivered at a particular point in a water phantom with the MU given. Linear accelerators are typically calibrated for a reference field size of 10 x 10 cm2 at 100 cm from the source at a depth of maximum dose, d,^, on the central axis in a water phantom. At this point the MU is adjusted so that 1 MU = 1 centigray. The output factor for a 10 x 10 cm2 field as shown in Figure 6A is then defined as: 10 max depth (cm) Figure 5. A typical photon beam depth dose curve is shown. 11 Figure 6. The irradiation geometry used in the derivation of the output factor (OPF) is shown. The absorbed dose at depth of maximum ionization (d^x) and at 100 cm source to axis distance (SAD) along the collimator axis is obtained for (A) a 10 x 10 cm2 field and (B) a radiation field W x L. 12 OPF (10x10) = D (dmax, 10x10) \/ M U . (1) where D(dmax,10xl0) is the absorbed dose measured for the number of MU delivered. It is specified in units of centigray per monitor unit, cGy \/ MU. The output factor for any other field size W x L as shown in Figure 6B is then given as: Therefore the output factors give the relationship between radiation absorbed doses at point of maximum dose in a water phantom for different radiation field sizes. 2.1.3. The percentage depth dose (PDD) To relate the absorbed dose at depth d and at d m a x along the collimator axis for a particular field size and photon energy, the term percentage depth dose (PDD) is used. For PDD the source to phantom surface distance (SSD) remains fixed, typically at 100 cm, and the distance of the measurement point from the source is SSD + d for a point located at depth d and SSD + d m a x for a point located at depth d m a x as shown in Figures 7A and B, respectively. The percentage depth dose is defined as the ratio of the absorbed dose at depth d, D(d,WxL,SSD), to the absorbed dose at d^x, D(dmax,WxL,SSD), and is given as: The percentage depth dose (an example of which is shown in Fig. 5) varies with photon energy, field size and source to surface distance. OPF (WxL) = OPF (10x10) x D (dmax,WxL) \/ D (d^, 10x10). (2) PDD (d,WxL,SSD) = 100 % x D (d,WxL,SSD) \/ D (dmax,WxL,SSD). (3) 13 D(d,WxL,SSD) E K d m a x , W x L , S S D ) Figure 7. The irradiation geometry used in the derivation of the percentage depth dose (PDD) at a fixed source to surface distance (SSD) is shown. The absorbed dose, D(d,WxL,SSD), at depth d is shown in (A) and that at depth d m a x , D(dmaX,WxL,SSD), is shown in (B). Both doses are measured on the collimator axis and for the same field size defined at SSD. 14 2.1.4. The tissue maximum ratio (TMR) and tissue air ratio (TAR) Alternatively, the relationship between two doses at different depths can also be related by the tissue maximum ratio (TMR). For the TMR, however, the distance from the source to the measurement point (source to axis distance, SAD) is kept constant at 100 cm and depth is varied by changing the water level above the measurement point. This is illustrated in Figure 8 with the tissue maximum ratio given as: TMR (d,W*L) = D (d ,WxL) \/ D ( d m a x ,WxL) . (4) The field size W x Lis defined at SAD which is the position of the measurement point. Hence for a given MU in a radiation field W * L and at 100 cm source to axis distance, the absorbed dose at a depth d along the collimator axis can be calculated from Eqs. 1, 2 and 4 as: D (d,WxL) = MU x OPF (W*L) x TMR (d,WxL). (5) Another useful term is the tissue air ratio (TAR). It is defined as the ratio of the absorbed dose at a depth d, D(d,WxL), to the absorbed dose \"in air\", Dair(WxL), at the same source to axis distance (Fig. 9A and B, respectively) and is given as: TAR (d ,WxL) = D (d,WxL) \/ D a i r ( W x L ) . (6) The Dair(WxL) is the absorbed dose in air or in free space with field size W x L , where the dosimeter is surrounded by sufficient phantom-like material to establish electronic equilibrium but not enough to significantly attenuate the primary radiation fluence. 2.1.5. The backscatter factor (BSF) or the peak scatter factor (PSF) 15 SAD D(d,WxL) D C d ^ W x L ) Figure 8. The irradiation geometry used in the derivation of the tissue maximum ratio (TMR) at a fixed source to axis distance (SAD) with field size W x L i s shown. The absorbed dose, D(d,WxL), at depth d is shown in (A) and that at depth dm a x, D(dmax,WxL), is shown in (B). 16 Figure 9. The irradiation geometry used in the derivation of the tissue air ratio (TAR) at a fixed source to axis distance (SAD) with field size W x L is shown. The absorbed dose, D(d,WxL), at depth d is shown in (A) and the dose in air, Dair(WxL), is shown in (B). 17 The backscatter factor (BSF) or the peak scatter factor (PSF) is the special case of a TAR at d m a x and is given as: BSF ( W * L ) or PSF ( W x L ) = TAR (dm ax,WxL) = D(dmax , W x L ) \/D a i r ( W x L ) . (7) This is a measure of the phantom scatter contribution at dmax for the specified field size W x L since the absorbed dose at dmax, D(dm ax,WxL), would include phantom scatter contribution whereas the absorbed dose in free space, D a i r ( W x L ) , has minimal scatter contribution (Fig. 10A and B). The peak scatter factor generally, increases with field size as more scattered radiation reaches the collimator axis but levels off at large field sizes as the scattered radiation can no longer reach the point where absorbed dose is measured. 2.1.6. The scatter air ratio (SAR) The scatter air ratio (SAR) is a measure of the contribution from scattered radiation at depth. It is the difference between the tissue air ratio (TAR) for a radiation field of finite size W X L and for a zero area field size. SAR (d,WxL) = TAR (d,W><0). (8) The tissue air ratio for zero area field size, TAR(d,0><0), can be determined from a measured narrow beam attenuation curve or from a plot of tissue air ratio versus field size extrapolated to zero field size. It represents the attenuation of the primary photon beam in material of thickness d. ' 18 S A D D ( d m a x > W x L ) Dai\/WxL) Figure 10. The irradiation geometry used in the derivation of the backscatter factor (BSF) or the peak scatter factor (PSF) at a fixed source to axis distance (SAD) with field size W x L is shown. The absorbed dose in water at dmax, D(dmax,WxL), is shown in (A) and the dose in air, Dair(WxL), is shown in (B). 19 2.1.7. The off axis ratio (OAR) and the depth dose profile The previously defined parameters all relate doses measured on the center of the beam at the collimator axis. For any points away from the collimator axis, the off axis ratio (OAR) is used to relate the absorbed dose at the off axis point to a point along the collimator axis and at the same depth. The OAR is the ratio of absorbed dose at a point a distance r away from the collimator axis, D(r,d,W> I \/ y ' \/ , I I I 1 I 1 \\ I \/ I 1 1 , I 1 1 1 I I I \\ _ _ _ r (- n n i - i - r r T -\\ \/ \/ ' I , I I I 1 I \\ I \\ \/ \/ ' I , I I I 1 I \\ \\ I I _ l_ i J _,_ i J J _ \u00b1 J _ \\_ A _ _ \/ i 1 I. , I I I ' 1 I \\ V \/ I 1 i , I I I 1 I \\ \\ V \/ I 1 I \u2022 , I I I 1 I I \\ I \/ _ _ \/ _ ! _ ' _ 4. _ L i J _ L _ L i _\u00bb _ \/ i ' 1 i i i i i i \\ \\ \\ I I 1 I , I I I 1 I V \\ \\ \/ \/ ' \/ I I I 1 I I I ' ; ; ' > ; i i i 1 i \\ i \u2014 t \/ , ' I I I I I 1 I I I \\ \/ , ' I , I I I I > 1 I V \/ \/ ' ' I I I I 1 I I I \\ \/ \/ ' I i I I I I I \\ v \\ \/ \/ ' I I I I I I \\ \\ \\ \\ y I I I 1 \\ ' \\ \\ Figure. 13. The storing of a matrix of dose points is shown. Dose points are represented by 13 fan lines and 5 transverse lines. 31 J Primary i i i i I i u 0 2 4 6 8 radius (cm) Figure 14. A schematic diagram illustrating the absorbed dose in a rectangular field of dimensions W x L at a point Q. The absorbed dose at point Q is the average absorbed dose for sectors of radius n, r2 ... into which the field is divided. The primary contribution is constant and is independent of the radius, whereas the scatter contribution is a function of the radius, increasing with the radius of the radiation field. 32 Figure 15. A three-dimensional representation of the patient outline by the superior (sup), central, and inferior (inf) transverse cross-sectional planes or slices. 33 Figure 16. Isodose distributions for a three field treatment are illustrated on several cross-sectional CT slices. 34 4. DOSIMETRIC CHARACTERISTICS OF ASYMMETRIC BEAMS Some modern linear accelerators (linacs) are equipped with asymmetric collimators or jaws. The independent jaw movement allows closing down one side of the field without affecting the opposite side. For instance, a 10 x 10 cm2 symmetric field has collimator settings of Xi, X 2 , Yi, and Y 2 each equal to 5 cm. This is the distance from the collimator axis at 100 cm from the source. When a 7 cm wide asymmetric field is desired, one can close down one of the jaws, for example, Xi to 2 cm. Consequently, the resultant asymmetric field is smaller in dimension than the corresponding symmetric field of 10 x 10 cm2 (Fig. 4) and the absorbed dose is therefore reduced. This absorbed dose reduction is non-trivial producing alterations in output at collimator axis, depth dose and beam profile. The data that is required for dose calculations are: percentage depth doses, profiles, off axis ratios, and output factors. 4.1. Overview\/Historical Development Klemp et al.6 have measured and stored beam data for half-collimated photon fields in a commercial treatment planning system (GE Target 2 system). Such a practice is time consuming and is limited to the fixed number of asymmetric fields one can store. Several investigators have developed calculation methods to determine the output factor of asymmetric fields.7\"10 Others have developed methods to account for the dosimetry of asymmetric fields at depths along the asymmetric field axis.11'12 Palta et al} have demonstrated that the percentage depth dose along the asymmetric field axis (i.e., the center 35 of the asymmetric field), and the shape of the isodose distribution of an asymmetric field differ from that of a symmetric field having the same field size as the asymmetric field. They have attributed the observed difference to the difference in primary beam profile produced because the off-set asymmetric field and the centrally-positioned symmetric field having the same field size as the asymmetric field intercept different positions of the flattening filter. Khan et al.13 have proposed to correct for the variation in beam quality observed at positions away from the collimator axis by using narrow beam attenuation, which is determined either by transmission measurements or by scatter subtraction technique. They have incorporated their primary beam modification method into an older version of the TP 11 treatment planning system (Atomic Energy of Canada Limited, Ottawa, Canada). Woo et a\/.14 have evaluated the accuracy of a newer version of the commercially available TP 11 (Theraplan versions 4.2 and 5, Theratronics International Limited, Kanata, Canada) specifically as applied to the dosimetry of asymmetric fields, which in this system is based on Khan's methodology. A totally different approach to dose calculation in asymmetric fields has been proposed by Chui et al.15 Their method is an extension of their off-center ratios (OCRs) method for 3D dose calculation,16 and hence is only applicable to their specific treatment planning system, or similar systems that compute the OCR at a point using the product of the primary OCR (POCR) and the boundary factors (BFs). Because of the approximation used in deriving the POCR, Loshek and Keller17 have observed some discrepancy between calculations and measurements for large field size and\/or heavily collimated asymmetric field with large field offset. They have refined and minimized the errors by using a piecewise re-construction technique to model the POCR, however, this requires an additional set of dose profiles to be 36 measured and manipulated for each depth. Thomas and Thomas1\" have modeled the POCR using only a single measured beam profile in air, in addition to a number of narrow beam attenuation coefficients measured at various positions to account for the radial variation of beam quality at depth. Combining both the techniques of Palta8 and Chui16, Loshek and Parker19 have used the OCR method to determine the variation in primary dose at dmaX across the field, while employing an attenuation function to account for the scatter and primary beam attenuation at depths and at off axis distances. Recently, Cadman has extended his field edge correction (FEC) method7 to generate accurate POCRs and OCRs for asymmetric fields.20 Alternatively, Storchi and Woudstra have represented the off-axis profiles by a computed envelope profile and two boundary profiles.21 Some commercial treatment planning systems do not have specific algorithms for calculating dose in asymmetric fields, but allow blocking of a field by an externally mounted block. Simulating the collimators as blocks, the calculation would then account for the asymmetric collimation. Most blocked field calculations and, in particular the one used in the GE Target 2 system, are based on the Clarkson's scatter-sector integration method5 or the Cunningham's IRREG program.2 Such an algorithm is well suited for irregular field calculation but is rather inefficient for asymmetric rectangular or square field calculation. This is because the irregular field calculation method calculates the absorbed dose by separating the primary and the scatter contributions with the scatter contribution calculated over 360\u00b0 and is very time consuming. A more efficient approach is to use the Day's equivalent field calculation method22 as the basis for asymmetric field calculation.9'11'23 Several examples of this method include the 37 direct implementation of the Day's method by Tenhunen and Lahtinen,9 the use of the effective field for the field size dependent factors by Rosenberg et al.11 and the use of the ratio of the equivalent field contributions between the asymmetric and symmetric fields by Kwa et al2' In this thesis, an analytical approach that accounts for the dosimetric characteristics of an asymmetric field is proposed.23 It is based on the concept that for every asymmetric field, where the field center is not aligned with the collimator axis, there is a corresponding symmetric field in which the opposite pair of collimators are equidistant from the axis and this distance is equal to the distance in the asymmetric field where the collimator is furthest away from the axis (Fig. 4). This approach is unique in that it does not need to separate the dose into the primary and the scatter contributions. The method calculates the difference in absorbed dose at any point between the smaller asymmetric field and the corresponding symmetric field. Therefore, the absorbed dose in an asymmetric field can be determined from existing beam data for symmetric fields. As a result, minimal data acquisition is required. This correction method can be applied to most existing treatment planning algorithms, and computation time is not significantly longer than that for symmetric fields. Furthermore, the correction factor proposed can account for differences in collimator settings and hence output factors as observed for rectangular fields when the collimators defining field width and length are switched (collimator exchange effect). In this thesis, it is also shown that part of this collimator exchange effect with asymmetric fields is due to the asymmetric collimator backscatter effect (i.e., radiation scattered from the collimators back into the beam monitor chamber). Since the proposed correction factor is also position specific with respect to the 38 asymmetric and symmetric field settings, it is also shown to account for the scatter contributions and thereby the observed tilt of the radiation dose profile (wedge effect) of an asymmetric field. In a symmetric field the radiation dose profile would be symmetric and uniform whereas in an asymmetric field it may be tilted to one side because some scatter dose is missing on that side (see Fig. 27). 39 4.2. The Output Factor (OPF) of an Asymmetric Field (AF) The absorbed dose at a given point in an irradiated medium is conventionally represented by the product of MU, OPF, TMR and OAR as given in Eq. (10) for symmetric fields. Various methods to determine the output factor of an asymmetric field have been described. Khan et al.13 have suggested the use of the output factor of the symmetric field having the same field size as the asymmetric field. Others require extensive measurements to account for the effects of each of the four collimator positions on dose.12'15 In between, there are the straightforward and efficient approaches of using the Day's equivalent field method.9'11'23 The use of the Day's method for the dosimetry of asymmetric fields and some of its limitations are discussed below. 4.2.1. Direct implementation of the Day's equivalent field method The Day's equivalent field method has been adopted to calculate absorbed dose distributions in water irradiated by symmetric megavoltage photon beams.24 It has also been used for calculations in asymmetric photon fields. ' ' The Day's equivalent field method calculates the scattered dose at any off-axis point in a radiation field by making it equal to the average scattered dose contribution of the four quarter fields centered at the same point. For example, for a W x L field with the point R along the transverse principal plane being a distance r away from the collimator axis, the two rectangular fields providing contribution to the dose at R would be (W + 2r) x L and (W - 2r) x L as shown in Figure 17(A). This 40 Figure 17. The equivalent field contributions used in the Day's equivalent dose calculation method are illustrated. For (A), the absorbed dose to a point R in a symmetric field WxLis equivalent to the average of the two doses at the center of the two symmetric fields of (W + 2r) x L and (W - 2r) x L. For (B), the absorbed dose at a point R in an asymmetric field AW x L is equivalent to the average of the two doses at the center of the two symmetric fields of (W + 2r) x L and (2AW - W - 2r) x L. For (C), the absorbed dose at a point P along the collimator axis in an asymmetric field is equivalent to the average of the two doses at the center of the two symmetric fields of W x L and (2 AW - W) x L. 41 equivalent field method is a simplified version of the widely used Clarkson's method.5 One limitation is that unlike the Clarkson's method, the Day's method is not suited for irregular field dose calculation but can only be applied to square or rectangular fields. However, the Day's method is much more efficient than the Clarkson's method for the dose computation of rectangular and square asymmetric fields. This is because with the Day's method two equivalent field contributions need to be determined, whereas, with the Clarkson's method 72 scatter contributions need to be calculated based on a 5\u00b0 per sector integration. Most asymmetric field algorithms use some form of beam modeling to generate asymmetric field dose profiles. Usually, they separate the dose into primary and scatter components. For example, Tenhunen and Lahtinen9 use the off axis ratio in free space, OARa^r,40x40), to represent the primary component of dose at distance r from the center of a 40 x 40 cm2 field, and then use the Day's method to account for the scatter component of dose originating in the phantom. In the situation when one of the collimators is set asymmetrically to width A W (Fig. 17B), the asymmetric field output factor (AF OPF) at an off-axis point a distance r away from the collimator axis (Fig. 4) and at dmax in water would be Here the AF OPF(r,AWxL) is a function of the off axis distance r, the asymmetric field size AW x L , and the corresponding symmetric field W x L . The symmetric field width W is defined as twice the largest collimator setting of the asymmetric field. For the example given in Figure 4, the largest collimator setting of an asymmetric field width of 7 cm is 5 cm, the setting of the X ] collimator. Therefore the corresponding symmetric field width is 10 cm with both X i and X 2 collimators set at 5 cm. The asymmetric field offset is then defined as 42 (W - AW) \/ 2. This is represented by the asymmetric field (AF) axis in Figure 4, and the offset distance is defined as r a s y m and is 1.5 cm away from the collimator axis for the example given in Figure 4. The OARair(r,40x40) is the off-axis ratio in free space at the off-axis point for the maximum field opening of 40 x 40 cm2. It gives the ratio of the dose at an off axis position such as R (Fig. 17) to the dose on the center of the axis and therefore relates the doses in air at those two points. The output factors relate doses in water at dmax for different field sizes and unless specified otherwise are all defined at the collimator axis or central axis. For example, OPF((W+2r)xL) is the output for a field of length L and width (W + 2r) on the central axis of this field. Here, the off-axis distance r is positive when the off-axis point is on the asymmetrically collimated side and is negative when it is on the opposite side. Hence, rasym would always be negative (Fig. 4). Kwa and El-Khatib25 have suggested that Eq. (13) which represents a direct implementation of the Day's method for the output factor calculation of an asymmetric field would lead to some discrepancy with measured values. This is because when the asymmetric field width AW approaches the symmetric field width W (Fig. 17A), Eq. (13) leads to: AFOPF(r,AW^W* L) - OAR.JrA0 x 40) * ^ ' L ) * \u00b0 P F ^ ~ \u00bb > * L \u00bb (14) But AF OPF(r,AW->WxL) is basically the output factor for the symmetric field W x L at the off-axis point r, in which, OPF(r,W x L) = OAR(dmax,r,W x L) x OPF(W x L). (15) Note that OAR(dmax,r,WxL) is the ratio of the doses at positions r and on center in water at dmax and for field size W x L . The OARair(r,40x40) and OAR(dmax,r,WxL) are generally 43 equal as shown in Figure 18 except at points when r is in close proximity to the field edge. The difference in the off axis ratios at points close to the field edge between in-air and in-phantom (at dmax) measurements is attributed to the relative lack of side-scatter contribution at points near the field edge within the irradiated medium (Fig. 19). Hence based on Eqs. (14) and (15), AF OPF(r,AW->WxL) < OPF(r,WxL) in general.25 This is because output factors do not increase linearly with field width but level off at large field width (see Fig. 20). Therefore, OPF\u00abW + 2r) x L) + OPF((W - Ir) x L) ^ Q p f { } y ^ \u00a3 ) ( 1 6 ) An example is given using measured off axis ratios from Figure 18 and output factors from Figure 20. For an off-axis point with r = 5 cm in a 6 MV 20y x 20x cm2 photon field (superscripts x and y indicate collimator positions X and Y), the calculated output factor based on Eq. (14) would be 1.034 x (1.016 + 1.076) \/ 2 = 1.082 cGy\/MU but the actual output factor at the off axis point (based on Eq. (15)) would be 1.033 x 1.058 = 1.093 cGy\/MU. Thus, the calculated output factor for an off-axis point using this direct implementation of the Day's method would be in error (1% for the example shown above) when the asymmetric collimator setting approaches that of the symmetric field.25 However, this error will vanish when r approaches zero since the two equivalent field sizes would become the same, W x L . The inequality in Eq. (16) will increase with increasing off axis distance r but this inequality will be compensated for somewhat as OARair(r,40x40) > OAR(dmax,r,WxL) (Errors up to 1.5% are expected with r = 8.5 cm for points close to the field edge of a 6 MV 20 x 20 cm2 field). 44 A. 6 MV 1.08 1.04 + 1.00 0.96 \u2022 \u2022 . \u2022\" S. -in air jrf^ * dmax \\ \\ . *. in air 40x40 cm2 d V max \\ \u2022 20x20 cm2 1 1 \u2014 i 1 0 B. 10. MV 1.08 1.04 1.00 0.96 10 15 Off axis distance, r (cm) 20 \u2022 in air 40x40 cm2 20x20 cm2 1 1 10 15 Off axis distance, r (cm) 20 25 25 Figure 18. The off-axis ratios (OARs) for both the (A) 6 MV and (B) 10 MV photon beams at d^x and in-air are shown. 45 Figure 19. Illustration of primary and scattered radiation to points on center and at the edge of the radiation field. (A) In air the dose contribution is predominantly from primary radiation (dashed arrows) and radiation scattered from the head and collimators and therefore differences along the profile are mainly due to primary transmission through the flattening filter. (B) At d^x in phantom in addition to the primary transmission through the flattening filter and head scatter, there is phantom scatter (solid arrows) which is relatively less for points near the edge of the field than for points near the center. 46 4.2.2. Derivation of a correction factor (CF) A new approach is proposed here that can avoid the inherent limitations of using the Day's method directly in the absorbed dose representation of asymmetric fields. Instead of generating asymmetric field profiles using some beam modeling as is conventionally done, a correction factor (CF) that modifies the existing symmetric field profiles to account for the dose reduction in an asymmetric field is proposed.9 The symmetric field can either be square or rectangular in shape and the resultant asymmetric field could be a square or rectangular field of smaller dimension. The correction can then be applied to dose distributions in symmetric fields to obtain those in asymmetric fields. The method is first described as it applies to situations where only one independent jaw is closed down along the field width and with the dose calculation plane along the transverse principal plane (Fig. 17B or C). However, it can readily be extended to situations where the field length is being collimated asymmetrically or where both field dimensions are being collimated asymmetrically as discussed in Chapter 4.5. Let's consider the radiation output of an asymmetric field AW x L and the corresponding symmetric field WxL. As discussed in Chapter 2.1.1, the output factor at dmax along the collimator axis in a symmetric field of W x L is defined as OPF(WxL). The output factor along the collimator axis in an asymmetric field (where r = 0), AF OPF(AWxL), is then approximated by the arithmetic mean of OPF(WxL) and OPF((2AW-W)xL) (Fig. 17C), which is given as: AF OPF(AWxL) - {OPF(WxL) + OPF((2AW-W)xL)}\/2. (17) 47 The dose reduction in an asymmetric field AW x L as compared to the corresponding symmetric field WxL can be represented by a correction factor which consists of the ratio of the output factor in an asymmetric field to the output factor in a symmetric field. This correction factor (CF) at the collimator axis is given as: AFOPF(AWxL) CF(dmBX,AW x L) V max ' OPF(W x L) {OPF(W x L) + OPF((2AW -W)x L)} 12 (18) OPF(W x L) Since the equivalent field W x L is common between the asymmetric field and the symmetric field, the above CF (Eq. 18) is expressed as the geometric mean instead of the arithmetic mean of the output factors so as to minimize the computation steps and time, i.e., OPF(W x L) \u2022 = V OPF\u00ab2AW -W) x L)_ OPF'W x L) Then, the output factor along the collimator axis in an asymmetric field can be calculated from the correction factor as: AF OPF (A W x L) = OPF(W x L) x CF(dmax , AW x L) = OPFiW x L) x pPHJ2AW-W)\u00bbL) V OPF(W x L) = \u2014 < t \\ \u2014 0 \u2014 measured SF \u2014 B \u2014 measured AF ) U \u2014 \u2014 u c n c u u i I a v e w r i . . . x- - - based on CF ._ . - based on 0PF(AFj -6 -4 Off axis distance (cm) 20x20 SF&5X20AF -10 -8 -7 Off axis distance (cm) Figure 29. The measured and calculated asymmetric field (AF) dose profiles relative to the OPF( 10x10) are compared for a 6 MV photon beam at dmax and at 100 cm source to axis distance. The calculated AF dose profiles based on ave OPF are determined by multiplying the OARair(r,40x40) with the average OPF for the two field sizes as determined by the Day's equivalent field method (Eq. 13). The calculated AF dose profiles based on CF are determined by multiplying the dose profile of the symmetric field (SF) and the proposed CF (Eq. 25). The calculated AF dose profiles based on OPF(AF) are determined by multiplying the OARair(r,40x40) with the OPF having the same field size as the AF (as Eq. 31 with r instead of rasym). 83 (A) 20x20 SF& 15x20 AF 1.100 1.080 a> in g 1.060 .1 5 1.040 a> DC 1.020 1.000 < ) I r ' A measured S F \u2014 B \u2014 measured AF \u2014 A \u2014 based on ave OPF . . . X- - - based on CF - o - - based on OPF(AF) I I -10 -8 -4 -2 Off axis distance (cm) (B) 1.100 1.080 CD ,\u00a71.060 11.040 1.020 20x20 SF 8.10x20 A F 1.000 -10 < r~~ \u20140\u2014rr teasured SF teasured AF -i \u2014 r\\cc \u2014 B \u2014 r r a udbcu un ave wrr - - -X- - - based on CF - - o - - based on 0PF(AFj -8 -6 -4 Off axis distance (cm) (C) 20x20 SF & 5x20 A F 1.100 1.080 CD S 1.060 1 1.040 CD at 1.020 1.000 \u20220 ^ > o < \u2014 6 \u2014 measure d S F d AF l ave OPF \u2014 6 \u2014 based or . . . X- - - based on CF - - o - - oasea or l u r ^ - ^ M ^ ; x -V T s^t: X \u00a3 \u2014\u2014 4< \u2014a -10 -8 -7 Off axis distance (cm) Figure 30. The measured and calculated asymmetric field (AF) dose profiles relative to the OPF( 10x10) are compared for a 10 MV photon beam at dm a x and at 100 cm source to axis distance. The calculated AF dose profiles based on ave OPF are determined by multiplying the OARair(r,40x40) with the average OPF for the two field sizes as determined by the Day's equivalent field method (Eq. 13). The calculated AF dose profiles based on CF are determined by multiplying the dose profile of the symmetric field (SF) and the proposed CF (Eq. 25). The calculated AF dose profiles based on OPF(AF) are determined by multiplying the OARairO\",40x40) with the OPF having the same field size as the AF (as Eq. 31 with r instead of rasym). 84 calculated dose profiles based on OPF(AF) are calculated by multiplying the OARair(r,40x40) with the output factor having the same field size as the asymmetric field (as Eq. 31 with r instead of rasym as off axis distance). The second method based on the proposed CF uses correction factors that vary across the asymmetric field, and thereby accounts for the difference in scatter dose contribution across the asymmetric field. Similarly, the first method based on the average output factor (ave OPF) as suggested by Tenhunen and Lahtinen9 also uses values, ave OPF, that vary across the asymmetric field. As shown in Figures 29 and 30, both methods give asymmetric field dose profiles that are in close agreement with measurement. Unlike the first two methods, the third method applies a constant value, OPF(AF), and thereby assumes a constant scatter dose contribution across the asymmetric field. As illustrated, the calculated asymmetric field dose profiles based on the OPF(AF) resemble the off axis ratio curves (Fig. 18) and fail to account for the lower dose at both ends of the field edges. In particular with the 20 x 20 cm2 symmetric field one-quarter collimated to a 15 x 20 cm2 asymmetric field, the calculated dose value at a point 2 cm from the asymmetrically collimated field edge follows the shape of the off axis ratio curve and does not slope towards the field edge as observed experimentally. As a result, Gibbons and Khan51 have modified the off axis ratio (OAR) in Khan's original equation13 with a varying scatter component that is similar to the equation suggested by Kwa et al23 85 4.4. An Asymmetric Field Treatment Planning Algorithm The correction factor (CF) proposed in Chapter 4.3.1 (Eq. 39) is used to account for the dose reduction when a symmetric field is asymmetrically collimated to a smaller asymmetric field. The absorbed dose at any point in an asymmetric field can then be determined from existing symmetric field data. One limitation is that the correction factor can only be applied across the radiation field to within 1.5 cm from the field edges. For points beyond this cutoff point, a set of correction factors to be used in the penumbral region is derived. These correction factors can then be represented by a subroutine or a function call in an existing treatment planning program. 4.4.1. Generation of a set of correction factors The details of the correction factors used across a radiation field are discussed here. The aim is to derive a set of correction factors based on equations that use similar beam, and position specific parameters, such as AW, W, L, d, r, OPF, TMR, etc. As demonstrated, they can then be used for a wide range of photon beams and asymmetric fields. These correction factors are first derived semi-empirically by beam fitting with a typical measured dose profile, such as a 6 MV half collimated 10 x 20 cm2 asymmetric field at a depth of 10 cm (Fig. 3 IB). The dose profiles at different depths are then calculated based on these parametrized correction factors and are compared with measured dose profiles. Additional parametrization of these correction factors might be required to obtain a better fit for the beam profiles at 86 -15.0 -5.0 0.0 off axis distance (cm) 15.0 ft '\"\u2022^Sfru -5.0 0.0 5.0 off axis distance (cm) 15.0 E -15.0 0.0 5.0 off axis distance (cm) 15.0 Figure 31. The measured (solid line) and calculated (dashed line) dose profiles are compared for a 6 MV photon beam at depths of 2 (open triangle), 10 (cross) and 25 (open circle) cm and at 100 cm source to axis distance. A 20 x 20 cm2 symmetric field is (A) one-quarter collimated to a 15 x 20 cm2, (B) half collimated to a 10 x 20 cm2 field, and (C) three-quarter collimated to a 5 x 20 cm2 field. Ion chamber readings were normalized to a point at a depth of 2 cm along the collimator axis of the 20 x 20 cm2 symmetric field. 87 other depths (Fig. 3 IB). Once it is done for a particular field size, one can confirm the appropriateness of the correction factors by comparing to other asymmetric field profiles, such as a symmetric field of 20 x 20 cm2 one-quarter and three-quarter collimated to a 5 x 20 cm2 and a 15 x 20 cm2 asymmetric field (Figs. 31A and 31C, respectively). At present, these correction factors are specific to our in-house treatment planning program but can be readily applied to other treatment planning programs. Based on the original treatment planning algorithm as discussed in Chapter 3.4.1.1, the correction factors are categorized into eight groups labeled from CFA to CFH and the positions where they are applicable are shown in Figure 32 where: 1. CFA- -y\/l-(l- AWIW)2 for points with OAR < 0.3 on the symmetrically collimated side where OAR is the off-axis ratio at the specified point in the symmetric field. The magnitude of the correction is small and the correction factor is approximated empirically based on some function of the width of the asymmetric field, AW, and the symmetric field, W. 2. CFB = CFA + (CFC - CFA ){OAR - 0.3) \/ 0.2 for points with off axis ratios between 0.3 to 0.5 on the symmetrically collimated side. Basically, CFB is used to smooth the transition from CFA to CFC. 3. CFC is used for points with OAR > 0.5 and to a cutoff point that is 1.5 cm away from the geometric field edge but within the radiation field on the symmetrically collimated side (Fig. 32). For a 20 x 20 cm2 field, this cutoff point would be 8.5 cm from the collimator axis. The magnitude of the correction is still small and is approximated by the correction factor at the cutoff point. Hence, it is a function of the sizes of the asymmetric and symmetric fields and depth of interest as follows: 88 Figure 32. A schematic diagram illustrating the dose profiles of a symmetric field (continuous line) and a smaller asymmetric field (dashed line) is shown. The positions where the various correction factors used in the present algorithm are applied for asymmetric fields are indicated. 89 CFC=^0PF((2AW-3) x L)\/0PF((2W-3) x L) x ^ TMR(d,{2AW - 3) x L)\/TMR(d,(2W-3) x L) 4. CFD is the correction factor (Eq. 39) used within the main radiation field between the two cutoff points 1.5 cm within both the symmetric and asymmetric field edges, and is given by: CFD = ^OPF((2AW-W-2r) x L)\/OPF((W-2r) x L) x^TMR(d,(2AW-W-2r) x L)\/IMR(d,(W-2r) x L) where r is the off-axis distance. For points within the penumbral region of the asymmetrically collimated side and outside the radiation field, the shape of the symmetric field edge is used as the basis upon which the asymmetric field edge calculation is done. Hence, the entire field edge is shifted laterally onto the asymmetric field edge, and the off axis ratio value of the symmetric field at the field edge is used to represent the off axis ratio at the field edge of the asymmetric field (shifted off axis ratio). For example, let's say the off axis ratio at the symmetric field edge is 50%, then the shifted off axis ratio (SOAR) at the asymmetric field edge is also given a value of 50%. This is analogous to the use of boundary factors16 to describe the absorbed doses at the field edge. Additional corrections are then applied to this shifted off axis ratio (SOAR) in order to smooth the transition to the cutoff point and to account for the dose reduction by the smaller asymmetric field. The term OAR used below still refers to the off axis ratio at the point of interest in a symmetric field. 5. CFE is for points with SOAR > 0.5 and to the cutoff point of the asymmetrically collimated side and its purpose is to smooth the transition to the cutoff point. Similarly, the 90 cutoff point is 1.5 cm from the geometric field edge of the asymmetrically collimated side. CFE is empirically determined to fit the measured data. CFE = ((AW -W !2-r + (1.5 - AW + W I 2 + r) x SOAR I OAR) 11.5) xCFP2x^SOAR-05) xyfCFC where CFP is the correction factor at the cutoff point on the asymmetrically collimated side given by: CFP = JOPF(3x L)\/OPF((2W-2AfV + 3)x L) x yJTMR(d,3x L)\/TMR(d,(2W - 2 A W + 3) x L) 6. CFF = (SOAR I OAR) x JCFC x CFP2H0-5-SOAR) for points with SOARs between 0.3 to 0.5. CFF reduces to CFE and CFG for points with SOARs of 0.5 and 0.3, respectively. 7. CFG = (SOAR I OAR) x JCFP x CFC x cFA4x(0-3-SOAR) for points with SOARs between 0.05 to 0.3. CFG reduces to CFF and CFH for points with SOARs of 0.3 and 0.05, respectively. 8. CFH = (SOAR I OAR) x JCFP x CFC x CFA for points with SOAR < 0.05. Again, this is an empirically determined correction factor. The significance of this correction factor is not as critical since the actual value is only a small fraction of the primary absorbed dose at the field center. Note that all the correction factors across the radiation field reduce to unity when the asymmetric field width, AW, equals the symmetric field width, W. The same equations for the various correction factors can then be used for other photon beams of different photon energies, as confirmed by the agreement between the 10 MV calculated and measured dose profiles at all three depths (Fig. 33). As these correction factors are parametrized with beam and position specific parameters, these correction factors are universal and can be applied for 91 J3 E CO -15.0 -10.0 -5.0 0.0 5.0 off axis distance (cm) 10.0 15.0 E -15.0 -10.0 -5.0 0.0 5.0 off axis distance (cm) 10.0 15.0 to E M a 100.0 \u2022 C. J T 80.0 \u2022 '\"Xi 60.0 \u2022 T 40.0 -I J *^~ I 1 ^ 0.9- 1 1 1 -15.0 -10.0 -5.0 0.0 5.0 off axis distance (cm) 10.0 15.0 Figure 33. The measured (solid line) and calculated (dashed line) dose profiles are compared for a 10 MV photon beam at depths of 2 (open triangle), 10 (cross) and 25 (open circle) cm and at 100 cm source to axis distance. A 20 x 20 cm2 symmetric field is (A) one-quarter collimated to a 15 x 20 cm2, (B) half collimated to a 10 x 20 cm2 field, and (C) three-quarter collimated to a 5 x 20 cm2 field. Ion chamber readings were normalized to a point at a depth of 2 cm along the collimator axis of the 20 x 20 cm2 symmetric field. 92 most photon energies, asymmetric field sizes, or linacs. Furthermore, these correction factors can be applied to any symmetric profile regardless of whether it represents measured data or is analytically calculated, and thus can be implemented in any dose calculation program. Any additional asymmetric field dose profiles are then measured for quality assurance purposes only. 4.4.2. Implementation of the correction factors as a FORTRAN subroutine The various correction factors from CFA to CFH were implemented as a FORTRAN subroutine into an existing treatment planning program (Appendix 1). The calculated and measured isodose distributions for two asymmetric 10 MV photon beams are shown in Figure 34. Similar results are observed for the same asymmetric fields with a 6 MV photon beam. Good agreement is observed everywhere including the penumbral regions. Additional comparisons of measured and calculated isodose distributions are also made but are not shown here with both 6 and 10 MV photon beams for 12 x 6 cm symmetric fields collimated to 9 x 6 and 6 x 6 cm asymmetric fields, and for 20 x 20 cm symmetric fields collimated to 15 x 20, 10 x 20 and 5 x 20 cm asymmetric fields. Again, this comparison is required for quality assurance only. The measured isodose distributions for a half-collimated 6 MV photon field are compared to the calculated isodose distributions using the correction factors proposed here and the GE Target full calculation option simulating an asymmetric field with blocking and are shown in Figure 35. The calculated isodose distribution based on the proposed correction factors agrees with the measured isodose distribution slightly better than that based 93 A . 10 MV (0 \"x < -10-o Figure 34. The measured (dashed line) and calculated (solid line) isodose distributions for a 10 MV photon beam at 100 cm source to surface distance are shown. (A) A 12 x 6 cm2 rectangular field is three-quarter collimated to a 3 x 6 cm2 field, and (B) a 4 x 4 cm2 square field is quarter-collimated to a 3 x 4 cm2 field. The isodose lines were normalized to a point at the asymmetric field center at a d^ of 2.5 cm. The six isodose lines shown are 100, 90, 70, 50, 30, and 10%. 94 Figure 35. The measured and calculated isodose distributions for a 6 MV photon beam at 100 cm source to surface distance are shown. A 20 x 20 cm2 square field is half-collimated to a 10 x 20 cm2 field. The calculated isodose distributions are based on the correction factors proposed here, the full calculation option of the GE Target system and the fast calculation option of the GE Target system. The calculated isodose distributions were normalized to the 90% isodose line of the measured plots at the asymmetric field center. The five isodose lines shown are 90, 70, 50, 30, and 10%. 95 on the GE Target full calculation option, but the main difference is that the computation time required for the Target full calculation option is 5.5 minutes per asymmetric field, as compared to 15 seconds for the in-house system. This is because the Target full calculation option is based on the Cunningham's IRREG algorithm,2 and hence has to separate the primary and the scatter contributions with the scatter contribution determined over 360\u00b0. As stated the proposed approach does not require the separation of the primary and the scatter, and the dose computation is much more efficient because of the way the scatter contribution is calculated. For example, the Target full calculation option calculates the scatter contribution at every 5\u00b0 interval, hence it will have to calculate for each point the scatter contribution 72 times; whereas, the correction factor proposed here only takes two field size determinations. The only limitation is that the correction factor method works only for asymmetric rectangular and square fields, and will not handle irregular fields as the Target system can. The computation time with the Target fast calculation option is much faster and is comparable with our in-house system, however, the calculated isodose distribution fails to account for the variation of scatter contribution across the asymmetric field (Fig. 35), and is unacceptable for clinical use, especially for conformal therapy.50 Figures 36 (A) and (B) illustrate the calculated isodose distributions of two adjacent half-collimated asymmetric fields for both the 6 and 10 MV photon beams, respectively. The width of the two asymmetric fields equaled the width of the symmetric field, and the combined isodose distributions should be identical to the isodose distribution of the symmetric field. The combined isodose distributions show a slight ripple at the asymmetric field junction, reflecting the limitation of the existing algorithm. These ripples are due to the gradual drop in 96 A. 6 MV Figure 36. The calculated isodose distribution for two adjacent half-collimated beams (solid curve) and a single symmetric field (dashed curve) for (A) a 6 and (B) a 10 MV photon beam are shown. The seven isodose lines shown are 100, 95, 90, 70, 50, 30, and 10%. 97 the calculated absorbed doses at and near the cutoff point of the asymmetric field edge, and the less than perfect smoothing function between the cutoff point and the point at a shifted off axis ratio of 0.5. 4.4.3. Summary In summary, a treatment planning algorithm for asymmetric collimation has been developed and implemented into an in-house treatment planning system. Conceptually, when one of the independent collimators is closed down, the resultant asymmetric field is smaller than the original symmetric field. The difference in absorbed dose between the asymmetric field and the symmetric field is accounted for by the correction factor described here which is a function of the sizes of the asymmetric and symmetric fields, depth of interest, off-axis distance, and more importantly, the distances at the point of interest in relation to the four independent collimators within the asymmetric field and the symmetric field. This correction factor accounts for the variation in scatter across the asymmetric field which resulted in the dose profile resembling that under a wedge. The absorbed doses within the penumbral regions and outside the radiation field are accounted for by some semi-empirically parametrized correction factors. These correction factors are universal, with the equations made up of constants that are beam, field size, and position specific. The resultant dose profiles and isodose distributions have been shown to be in good agreement with measured data for both the 6 and 10 MV photon beams. Because these correction factors rely only on the output factors and the tissue maximum ratios, they can easily be implemented into any existing 98 treatment planning program. There is no need to store either additional sets of asymmetric field profiles or databases. The method also lends itself to asymmetric collimation in both directions. The present algorithm is superior to a commercial system based on the IRREG program (GE Target 2 system) in that the computation time is 20 times faster for the same degree of accuracy, and that, although comparable in time, the fast calculation option of the commercial system fails to produce an acceptable dose distribution. The only limitation of the present system is that it will not account for the dose distribution within an irregular field. 99 4.5. Asymrnetric Collimation in Both Field Dimensions Here the correction factor proposed by Kwa et al.221 is extended to 2-fold asymmetric collimation where both the upper and the lower collimators are set asymmetrically. The argument that the dose to a point is related to the sum of the dose contributions between the two fields centered at the same point can be extended to the four fields centered at the same point. Figure 37(A) illustrates the four rectangular fields centered at the point P which is a distance r and s away from the mechanical axis in a symmetric field W x L. Figure 37(B) illustrates the four rectangular fields centered at the same point in a 2-fold asymmetric field AW x AL. Using the same formalism used for an asymmetric field in one field dimension23 and discussed in Chapters 4.2 and 4.3, the CF for a 2-fold asymmetric field AW x AL at a point P, a distance r and s away from the collimator axis, would be t JTMi(d,(W+2r) x(2AL-L-2s)) x 7 J V ^ ( r f , ( 2 ^ ^ - ^ - 2 r ) x ( L + 2 5 ) ) x 7 M J ( r f , ( 2 ^ ^ - ^ - 2 r ) x ( 2 ^ L - L - 2 y ) ) | lMR(d, (W+2r)x(L- 2s)) x 1MR(d, (W- 2r) x (L+2s)) x 1Ml(d, (W- 2r)x(L- 2s)) Similar to the approach used in Chapter 4.4.1, the correction factors within the penumbral region and outside the radiation field are parametrized semi-empirically. When the calculation plane that is parallel to the field width and is at least 1.5 cm away from either geometric field edge defining the length, the correction factors are again categorized into eight groups from CFA to CFH (Fig. 32). CF = 100 Figure 37. Illustration of the equivalent field principle for a 2-fold asymmetric field is shown. The absorbed dose at a point P which is a distance r and s away from the central axis is represented by the sum of the absorbed doses at the same point P between the four rectangular fields centered at point P (A) for a symmetric field of W x L and (B) for a smaller asymmetric field of AW x AL. 101 , CFA = Jl-Q-AW,W)* \u00bb ^ \u2022 ^ - J D < M I ^ - P ^ - t - i ' ) l A\" V V 0PF(Wx(L-2s)) TMR(d,Wx(L-2s)) for points with the off-axis ratio (OAR) < 0.3 on the unblocked side. 2. CFB = CFA + (CFC- CFA)(OAR - 0.3) \/ 0.2 for points with OAR's between 0.3 to 0.5 on the unblocked side. The equation is the same as that in the one-dimensional case (Chapter 4.4.1) except that the actual value would be different since CFA and CFC are different. 3. CFC is used for points with OAR > 0.5 and to the cutoff point 1.5 cm within the field edge on the unblocked side. This is basically the correction factor at the cutoff point for the 2-fold asymmetric field, and is given as: CFC = \u2022i i OPF(3 x (2AL - L - 2s)) x OPF((2AW -3)x(L + 2s)) x OPF((2AW - 3) x (2AL - L - 2s)) OPF(3 x(L- 2s)) x OPF((2W - 3) x (L + 2s)) x OPF((2W - 3) x (L - 2s)) TMR(d,3 x (2AL - L - 2s)) x TMR(d,(2AW - 3) x (L + 2s)) x TMR(d,(2AW - 3) x (2AL - L - 2s)) TMR(d,3 x(L- 2s)) x TMR(d,(2W - 3) x (L + 2s)) x TMR(d,(2W -3)x(L- 2s)) 4. CFD is the 2-fold correction factor given earlier for points within the main radiation field between the two cutoff points 1.5 cm away from the field edges (Eq. 44). CFD= OPF((W+2r) x (2AL -L-2s))x OPF((2A W- W-2r)x(L+2s)) x OPF((2A W- W- 2r) x (2AL -L- 2s)) OPF((W+ 2r) x(L- 2s)) x OPF((W- 2r) x(L+2s)) x OPF((W- 2r) x(L- 2s)) * TMR(d, (W+ 2r) x (2AL -L-2s))x lMR(d, (2 A W- W- 2r) x (L+2s)) x 1Mi(d, (2 A W- W- 2r) x (2AL -L-2s)) TMR(d,(W+ 2r) x(L- 2s)) x TMR(d, (W- 2r)x(L+2s)) x TMR(d, (W- 2r) x(L-2s)) Similarly, for points within the penumbral region of the blocked side and outside the radiation field, the shape of the symmetric field edge is used as the basis upon which the asymmetric field edge calculation is done. The off axis ratio of the symmetric field at the field edge is shifted to the field edge of the asymmetric field (shifted off axis ratio). Additional 102 corrections are then applied to the shifted off axis ratio (SOAR) in order to smooth the transition to the cutoff point and to account for the dose reduction by the smaller asymmetric field. Again the equations are identical to the one-dimensional case with the actual values being different due to the new correction factors, namely, CFA, CFC, CFD and CFP. 5 CFE is for points with SOAR > 0.5 to the cutoff point of the asymmetrically collimated width. CFE = ((AW - W12 - r + (1.5 - AW + W12 + r) x SOAR I OAR) 11.5) xCFP2