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Methods for Producing Off-Axis Ratio Tables from Mini-Multileaf Collimator Shaped Circular Fields for Input into a Stere...

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METHODS FOR PRODUCING OFFAXIS RATIO TABLES FROM MINI-MULTILEAF COLLIMATOR SHAP ED CIRCULAR FIELDS FOR INPUT INTO A STEREOTACTIC RADIOSURGE RY TREATMENT PLANNING SYSTEM By BRANDON RICE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Brandon Rice

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To my loving family and friends, who have believed in and supported me throughout my academic career and life.

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iv ACKNOWLEDGMENTS I would like to acknowledge my research advisor, Dr. Frank Bova, for his support and insight on this project. I am also mu ch indebted to Drs. David Hintenlang, David Gilland, James Dempsey, and Wesley Bolch fo r the many hours of insightful lecture and guidance pertaining to medical physics. I would also like to thank the rest of the faculty and staff in the Nuclear and Ra diological Engineering Depart ment at the University of Florida, for giving me the opportunity to learn and expand in a collegia l environment.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... ..x CHAPTER 1 BACKGROUND..........................................................................................................1 Overview of Stereot actic Radiosurgery........................................................................1 Delivery Techniques.....................................................................................................5 Dosimetry...................................................................................................................12 2 OFF-AXIS RATIO MEASUREMENT OF MINI-MLC APPROXIMATED CIRCLES....................................................................................................................18 Introduction.................................................................................................................18 Materials and Methods...............................................................................................18 Results........................................................................................................................ .26 Discussion...................................................................................................................26 3 A ROBUST ALGORITHM TO MODEL OFF-AXIS RATIOS OF MINI-MLC APPROXIMATED CI RCULAR FIELDS.................................................................33 Introduction.................................................................................................................33 Methods and Materials...............................................................................................33 Results........................................................................................................................ .39 Discussion...................................................................................................................40 4 CONCLUSION...........................................................................................................57 APPENDIX A MATLAB PROGRAM DEVELOPE D TO MODEL THE FIELDS.........................59

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vi B MATLAB PROGRAM DEVELOPED TO PERFORM ERROR ANALYSIS ON MODELED FIELDS...........................................................................................64 C MATLAB PROGRAM DEVELOPED TO ANALYZE A CALIBRATION FILM.67 D MATLAB PROGRAM DEVELOPED TO SEPARATE AND ALIGN EXPOSURES.............................................................................................................71 E MATLAB PROGRAM DEVELOPED TO AVERAGE FIEL D QUADRANTS AND PRODUCE FINAL MAP OF OFF AXIS RATIOS.........................................76 LIST OF REFERENCES...................................................................................................81 BIOGRAPHICAL SKETCH.............................................................................................83

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vii LIST OF TABLES Table page 1-1 Hypothetical treatment pl an with three isocenters.................................................11 2-1 Physical characteristics of the Wellhofer mini-MLC............................................18 3-1 Leaf alignment check.............................................................................................42

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viii LIST OF FIGURES Figure page 1-1 Linac SRS................................................................................................................3 1-2 Linac isocenter.........................................................................................................4 1-3 BRW head ring and CT localizer mounted on a Styrofoam head............................5 1-4 A typical single isocenter plan begi ns with nine equally spaced arcs of radiation...................................................................................................................6 1-5 The University of Florida SRS treatment planning algorithm.................................8 1-6 Mini-multileaf collimator used in our study............................................................9 1-7 Using a micro-multileaf collimat or to approximate circles...................................10 1-8 Relationship between absorbed dose a nd collision kerma for a megavoltage photon beam...........................................................................................................13 1-9 Cross-plane off-axis ratio profile for a 12 mm field..............................................14 2-1 Basic setup for film measurements........................................................................19 2-2 Epson flatbed scanner and processed radiographic film........................................20 2-3 Off-axis profile film measurements of a 6 cm by 7 cm rectangular field..............21 2-4 Results of the consistency check pe rformed on the Epson flatbed scanner...........22 2-5 Output of rectangular fi elds shaped by a miniMLC..............................................23 2-6 Film calibration tool...............................................................................................24 2-7 Tool used to separate and align exposures.............................................................25 2-8 Twelve millimeter mMLC circular approximation with even leaves....................28 2-9 Twenty two millimeter mMLC circular approximation with even leaves.............29 2-10 Thirty millimeter mMLC circular approximation with even leaves......................30

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ix 2-11 Twenty millimeter mMLC circul ar approximation with odd leaves.....................31 2-12 Twenty four millimeter mMLC ci rcular approximation with odd leaves..............32 3-1 Cunningham model fit to cross-plane da ta from a rectangular field formed by the mMLC..............................................................................................................35 3-2 Plots of the fitting parameters as a function of field width....................................36 3-3 The mMLC leaf positions for a 22 mm circular approximation with even leaves......................................................................................................................42 3-4 A cross-plane profile through the r ectangular open field is determined................43 3-5 The cross-plane profile is shifted to match the position of the mMLC leaf..........44 3-6 Negative field profile.............................................................................................45 3-7 Spatial map of negative field.................................................................................46 3-8 A 10 mm mMLC circular field formed by an even number of leaves...................47 3-9 A 16 mm mMLC circular field formed by an even number of leaves...................49 3-10 A 20 mm mMLC circular field formed by an even number of leaves...................51 3-11 A 26 mm mMLC circular field formed by an even number of leaves...................53 3-12 A 30 mm mMLC circular field formed by an even number of leaves...................55

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x Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science METHODS FOR PRODUCING OFFAXIS RATIO TABLES FROM MINI-MULTILEAF COLLIMATOR SHAP ED CIRCULAR FIELDS FOR INPUT INTO A STEREOTACTIC RADIOSURGE RY TREATMENT PLANNING SYSTEM By Brandon Rice August, 2005 Chair: Frank Bova Major Department: Nuclear and Radiological Engineering Radiosurgery beam profiles must be dete rmined accurately to avoid complications in critical structures adjacent to the tumo r. A micro-multileaf collimator (mMLC) may be used in place of conventional circular coll imators to approximate the circular fields typically used in stereotactic radiosurger y. We investigated two methods for producing off-axis ratio tables from the mMLC shaped circular beams for input into the in-house treatment-planning system at th e University of Florida. The first method involves direct measurement of the fields with radiographic film. The second method uses an empirically-based al gorithm to model the off-axis ratios. This was accomplished by measuring and characteri zing several rectangular fields produced by the mMLC. So-called negative field(s) we re calculated and adde d to a model of a rectangular field to arrive at the final beam model. We were able to accurately model the off-axis ratios for mMLC shaped circular beams to within (0.5 mm) distance-to-

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xi agreement error in high gradient regions and (3%) dose-dose error elsewhere in the beam profile for at least 90% of the points in each field. Given its simplicity and accuracy, we believe this method for modeling off-axis ratios of ra diosurgical beam profiles is ideal.

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1 CHAPTER 1 BACKGROUND Overview of Stereotactic Radiosurgery Stereotactic radiosurgery (S RS) is a single dose fraction form of radiotherapy that relies on the ability to accurately deliver a high dose of radiation to a stereotactically identified target.1 It is an accepted modality for treating a variety of neurologic conditions such as benign, primary, and metastatic tumors and arteriovenous malformations.2 Although distinctly different from fractionated radiotherapy or routine neurosurgery, SRS can be considered a hybrid of both. Conventional radiotherapy seeks to elimin ate diseased tissue by delivering small doses of radiation in many fractions (norma lly 1 or 2 fractions pe r day), through several static radiation fields.2-3 Fractionated treatments increase damage to the tumor because of reoxygenation and reassortment of cells into mo re sensitive phases of the cell cycle. Inaccurate beam and patient positioning is i nherent in common radiotherapy planning and must be considered when developing a tr eatment plan. Subclinical, or microscopic disease processes that do not show up on diagnos tic images must also be accounted for. Consequently, a large amount of healthy tissue is typically encompassed in the treatment plan to account for positioning errors and microscopic disease.2 This can lead to a large amount of healthy tissue being irradiated at the target dose and possibly destroyed, leading to patient complications years down th e road. For example, if a 2 cm diameter brain lesion is planned with a 5 mm margin of error to account for positioning errors, an

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2 additional 7 cm3 of healthy tissue is encompassed in the treatment plan; this additional healthy tissue represents approximately 70% of the total target volume. Stereotactic radiosurgery wo rks to avoid these pitfalls of conventional radiation therapy by administering a large dose of ra diation in one fraction so that the dose distribution has a steep gradie nt (decreases rapidly spatially) and is conformal to the disease being treated, typically with submillim eter precision (less than 0.5 mm). Radiosurgery relies on accurate, stereotac tically targeted delivery and steep dose gradients; it does not rely on taking advantage of radiobiol ogical differences between normal and diseased tissue.2 Stereotactic radiosurgery also diffe rs from common neurosurgery. Neurosurgery is an invasive, in-patient process that seeks to physically remove the pathological process. The neurosurgeon almost immediately knows wh ether or not a complication will occur. Radiosurgery, on the other hand, is a non-invasive, out-patient process. The radiosurgeon does not instantaneously know if there will be delayed complications.2 The idea of SRS was conceived in the late 1940s by a Swedish neurosurgeon named Lars Leksell.1 His concept was to have many small, focused beams of radiation impinge on a single target localized in space. He first achieved this with orthovoltage xrays and particle beams. He then invented a machine that uses many small sources of radioactive cobalt-60 focused on a single po int in space, to provide a large dose of radiation to diseased tissue while causing little or no harm to healthy tissue only millimeters away. This method of SRS is s till performed today using a commercially available system called the Gamma Knife. Our study focused on another technique that uses x-rays from a common linear accelerator (linac) to deliver the dose of radiation.

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3 The steep dose gradient of linac radiosurge ry is typically achie ved by using a basic concept of geometric optimization: many sm all beams are applied semi-isotropically about a target and are directed to converge on the target. A large dose will be delivered where the beams overlap and will fall off rapidly outside of this area. Figure 1-1. Linac SRS. The linac delivers a high dose of radiati on with a series of precisely placed arcs to control th e lesion while sparing healthy tissue. A rigid frame of reference is needed to rela te the center of the imaged target to the linac isocenter. The isocenter is the point of intersection of rotation of the gantry, patient couch, and collimator. Three-dimensional computed tomography (CT) image reconstructions are used to stereotactically id entify the target within the patient anatomy. This is often used in conjunction with othe r imaging modalities, such as angiography (a 2-D imaging modality) and magnetic resonance imaging, to allow for better interpretation of the patient anatomy.

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4 Figure 1-2. Linac isocenter. The isocenter is the point of inters ection of the patient couch, gantry, and collimator. The stereotactic frame is the rigid frame of reference that can be attached to the patient couch or pedestal of the linea r accelerator and to the patient’s skull.1 The BrownRobert-Wells (BRW) frame is a comm only used stereotactic head frame2 that is used by the radiosurgery team at the University of Florida. The frame ha s three orthogonal axes that intersect at the cente r of the circular frame.2 The BRW CT frame is equipped with a localizer that has nine fiducia l rods which appear as dots in each axial slice of the CT image.1 The location of these fiducial rods is precisely known; theref ore it is possible to relate any point in the CT reconstruction to the reference frame coordinates. Knowing precisely where the target is located with resp ect to a frame of reference is critical to accomplishing the goals of radiosurgery.

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5 Figure 1-3. BRW head ring and CT local izer mounted on a Styrofoam head. Delivery Techniques Several methods are commonly used to sh ape the small radiation fields and to produce the treatment plans needed to deliver the conformal, high dose of radiation to the target. They include the multiple isocen ter technique, micro-multileaf collimator (mMLC) conformal beams, and intensity m odulated radiation therapy (IMRT). This section provides a brief background on the multiple isocenter technique; this technique is the most preferred and commonly used deliver y method and is the method currently used by the University of Florida SRS team.

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6 The single or multiple isocenter technique for treatment delivery uses one or more sets of noncoplanar arcs of ra diation shaped by small, circul ar collimators, usually less than 3 cm in diameter. This method is idea l for producing nearly spherical or elliptical dose distributions in high -isodose regions (70-80%).2,4-5 The next discussion expands on delivery techniques used by the University of Florida radiosurgery team. The width of the dose distribu tion (size of the radiation fi eld) is approximately the diameter of the circular collimator projected to isocenter. A typical single-isocenter A B Figure 1-4. A typical single isocenter plan begins with nine equally spaced arcs of radiation. (A-B) Coronal a nd lateral views of a nine arc set. Angles in (A) refer to patient table angles. plan consists of 5-9 equally spaced arcs of ra diation; each arc span is 100 and the arcs are spaced 20 apart.2,5 These arcs of radiation are direct ed to overlap in the target region (at isocenter), taking advantage of the basic concept of geometric optimization discussed earlier. An isocenter in this context refe rs to a spherically shaped region used to approximate target dimensions and should not be confused with the mechanical isocenter

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7 of a linac. The typical characteristic beam starting angle begins 30 vertical off the superior, and extends 100 to e nd at 50 vertical off inferior.2 In most clinical circumstances, the targ et or lesion will have an elongated, ellipsoidal, or irregular geomet rical shape rather than be pe rfectly spherical. Sometimes the target will lie adjacent to a critical structure in the brain such as the optic chiasm. To account for these irregularities in target shape and to avoid cr itical structures in the brain, several parameters can be adjusted to adju st the shape and/or orientation of the dose distribution. Individual beam start and stop angles and arc weights can be adjusted to elongate or shift the orientat ion of the dose distribution.2,5 Adjusting these parameters does not drastically alter the shape of the dose distribution. For large or irregular target shapes, more than one isocenter is used to approximate the target. If multiple isocenters are used on a single target, the number of arcs is reduced from 9 to 5 for each isocenter. This is done to save time and produces near ly the same gradient as a nine arc set.2,5 Using multiple isocenters results in substant ial dose inhomogeneity in the high isodose regions.4,6 An efficient radiosurgery team such as the one at the University of Florida will spend approximately fifteen minutes treating a single isocenter using th is technique, even though the actual beam on time for this treatme nt lasts a few minutes. A collision test must be performed after each table rotation to ensure the linac does not collide with the patient upon gantry rotation. A major drawback to this tec hnique is the large amount of time it takes to treat multiple isocenters. Many cases involve more than one isocenter; on average the treatment time is one hour, but tr eatment time can take up to several hours for the most complicated cases with many isocenters.

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8 Figure 1-5. The University of Florida SR S treatment planning algorithm. Multileaf collimators (MLCs) may be used to approximate the circular fields produced by circular collimators.7 Multileaf collimators have been used to shape radiation fields for many years. They consis t of two arrays of tungs ten leaves; each leaf is driven individually by its own motor. Recently, multileaf collimators have been developed with SRS in mind. The main fact or separating conventional MLCs and MLCs specifically designed for radiosurgery is leaf width. Multileaf collimators designed for radiosurgery have leaf widths typically on the order of a few millimeters, whereas the leaf width on conventional MLCs is over a centimeter. Hence, MLCs designed for radiosurgery are usually referred to as mi ni or micro-multileaf collimators (mMLC). St. John investigated how the finite leaf width of the mMLC affects the conformality and gradient of dose distributions He showed that mMLC leaf width can

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9 have a considerable effect on the ability to use a mMLC to approximate circular beams.7 For example, a 5 mm circle cannot be approximated accurately using an MLC with 1 cm wide leaves. The leaves on the mMLC used in our study are appr oximately 5 mm wide projected to isocenter. It has been shown that dose di stributions produced by mMLCs with 5 mm wide leaves are virtually indistinguishable from those produced by mMLCs with 1 mm leaf widths.7 Figure 1-6. Mini-multileaf collimator used in our study. The collimator consists of twenty pairs of tungsten leaves, each driven by an individual motor and controlled by a computer. Circular fields shaped by a mMLC can be approximated using either an odd or an even set of leaves. The alte rnative that provides the opt imal approximation is governed by the size of the circle being approximated and mMLC leaf width. In the ideal

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10 situation, the mMLC would be allowed to rotate in between arcs to allow for better approximation of the circle. Since this is no t the case, the center of the circle being approximated is forced onto a leaf edge or leaf center, whichever is closer. If the center of the circle is on a leaf edge, an even number of leaves will approximate the circle best. If the center of the circle is on a leaf cente r, an odd set of leaves will work best to approximate the circle (when no collimator rotation is used). A B Figure 1-7. Using a micro-multileaf collimator to approximate circles. A) The circle is centered on a leaf edge so an even set of leaves are used. B) The circle is centered on a leaf center so an odd number of leaves are used. If a mMLC is used to shape the radiati on beam, in place of conventional circular collimators, it is possible to drastically redu ce the amount of time it takes to treat multiple isocenter targets. This in turn leads to a more optimal and efficient treatment delivery and increased patient satisfaction. For exam ple, consider a treatment plan containing three isocenters (Table 1-1). Using the sta ndard technique with circular collimators, it takes approximately 50 min for the University of Florida radiosurgery team to complete the treatment. Before each separate isocen ter is treated, the cent er of the isocenter (target) is placed at the isocenter of the linac and the appropriate circular collimator is mounted onto the linac. Also, a collision test must be performed for each table angle. If a mMLC is used to deliver the treatment the geometric center of the target is placed at the linac isocenter. This elimin ates the need to pl ace the center of each

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11 individual isocenter at the isocenter of the linac. It also eliminates the need to switch out collimators for different isocenters. The mMLC will approximate the shape of the necessary circle for the iso center being treated. Also, us ing a mMLC allows all three isocenters to be treated from each table angle, so the collision test needs to be performed a total of five times, as opposed to fifteen times when using the conventional circular collimators. Table 1-1. Hypothetical treatment plan with three isocenters Center Coordinatesa Isocenter Num AP (mm) Lat (mm) Ax (mm) Arc num Collim. size (mm) Table angle Start angle End angle 1 12 20 30 130 2 12 55 30 120 3 12 340 230 330 4 12 305 240 330 1 -19.0 -15.0 -40.0 5 12 270 240 330 6 18 20 30 130 7 18 55 30 120 8 18 340 230 330 9 18 305 240 330 2 -21.3 -12.0 -36.0 10 18 270 240 330 11 10 20 30 130 12 10 55 30 120 13 10 340 230 330 14 10 305 240 330 3 -16.0 -17.5 -43.7 15 10 270 240 330 aFor the conventional technique that uses circular collimators, the center coordinates of each isocenter are placed at the linac isocenter. Assuming 5 min of initial patient set up, an average of 300 monitor units (MUs) per arc of radiation, and a dose rate of 400 MUs per min, it would take approximately 16 min to complete the treatment with the mMLC; this corresponds to a 70% reduction in the amount of time necessary to complete the treatment. The reduced treatment time becomes more drastic as more isocenters ar e introduced. A compli cated eight isocenter

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12 plan could take over 2 h to treat using standard circular collimators; the plan would take about 40 min to treat using a mMLC. It has been suggested conformal and IMRT techniques are superior to the multiple isocenter technique for treating complex targets.6,8-11 The fact is multiple isocenter treatment planning can be difficult and the resulting dose distributions highly user dependent. Inhomogeneous dose distributions and poor target conformality are the result of improperly optimized multiple isocenter plan s. Wagner investigated the three methods of radiosurgery treatment delivery by im plementing optimization algorithms for each; this way each technique c ould be compared fairly.5 The multiple isocenter technique was optimized with a sphere packing algorithm that determined the placement of variablysized spheres within the target volume.5 The result of his work showed the multiple isocenter technique was able to produce a highly conf ormal dose distribution while maintaining a steep dose gradie nt outside the target volume.5 For general and complex cases, it was determined that the optimized multiple isocenter technique outperformed the other methods. Dosimetry The absorbed dose to a small mass of mate rial is defined as the amount of energy absorbed from charged particles liberated by ionizing radiation per uni t mass of material. The kinetic energy released in the medium (K ERMA) is defined as the sum of the initial kinetic energies of all the ch arged ionizing particles liberate d by uncharged particles in a material per unit mass.1 Since charged particles trav el a certain distance before depositing their energy, the absorbed dose will in crease with depth in the material up to a point where the number of charged particle s entering a small volume equals the number of charged particles leaving th at volume. This condition is known as charged particle

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13 equilibrium. The absorbed dose then begins to decrease with depth at the same rate of the collision KERMA. Figure 1-8. Relationship between absorbed dos e and collision kerma for a megavoltage photon beam. For many of the small fields used in ra diosurgery, electronic equilibrium does not exist in the lateral direction. This is due to the fact that the size of many of these small fields is smaller than the range of elec trons being produced. The resulting loss of electronic equilibrium will occur at every point in the field, even in the center.12 There are several consequences that must be taken into consideration when dealing with these small fields: the cross-beam profile is flat over only a small fraction of the full width at half maximum, the output factor decreases dramatically with a decrease in field size, and the inverse square law may not hold for large source to point distances.12 In general, a large percentage of the beam is made up of penumbra. The SRS treatment planning system at the University of Florida requires the input of several beam parameters. These include outpu t factors, dose rate at isocenter, tissue phantom ratios (TPR) at several depths, a nd off-axis ratios (OARs) for various sized

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14 fields. These parameters are derived from empirical measurements and data. An accurate assessment of these parameters is essential for minimizing the dose to normal tissues and reducing the risk of complicati ons due to nonconformal dose distributions. Figure 1-9. Cross-plane off-ax is ratio profile for a 12 mm field. Approximately 30% of the field is composed of penumbra. For any beam centered on the linac’s primar y axis at the machine isocenter, the absorbed dose to any point is calculated by Equation 1-1. 2* * * ) ( STD ScD ROF OAR TPR MU RD p Dose (1-1) where the reference dose (RD) is the dose per monitor unit, MU is the number of moniter units delivered, TPR is the tissue-phantom ratio, OAR is the off-axis ratio, ROF is the relative output factor, ScD is the source to calibration point distance, and STD is the source to target distance.

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15 Our study focused on measuring and modeli ng off-axis ratios of mMLC shaped circular fields. The empirically driven ra diosurgery treatment planning system at the University of Florida requires the input of a two dimensional matrix for each odd or even mMLC shaped field; these matrices have nor malized values (between zero and one) and contain the necessary off-axis information for each beam. The previous scheme that used circular collimators required th e input of one-dimensional arrays of off-axis ratios. There was no need for a 2-D matrix because the circ ular fields were perfectly symmetric. The penumbra of a radiation field refers to the region at the edge (of the field) over which the dose changes rapidly.1 The penumbra consists of two components: transmission penumbra and geometric penumbra. The transmission penumbra is the region in space exposed to photons that have traveled through part of the collimator. The geometric penumbra is the region that is irradiated by a spatially finite source.13 The variation in dose at the edge of the field also depends on th e amount of scattered radiation within the material of interest. The physical penumbra is thus defined as the spatial distance between two specified points of rela tive dose, namely 80% and 20% of the dose to the central axis.13 The type of detector used to measure the small the fields in ra diosurgery should be considered carefully. Lack of electronic equilibrium in the field will cause large detectors to yield erroneous results.13 The model detector for radiosurgery dosimetry should have several unique qualities. The ideal dosimeter should have submillimeter spatial resolution.12, 14-16 It should have a dose response that is energy, field size, and dose rate independent.14 It should provide a spatial ma p of the dose distribution rather than the dose at a single point, particularly for cases where it is necessary to verify

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16 conformal distributions.15 Also, the detector should be linear, reproducible, and stable.14 There is no ideal dosimeter for radiosurgery, but there are several that approach the ideal. Radiographic film has been used extensively in radiosurgery dosimetry. It is a high resolution dosimeter that records informa tion in two dimensions. Some possible problems that are encountered with film are variations in sensitivity and dose response which results from variable processing conditions.14 This problem may arise when trying to compare the doses from two separate films; the problem can be overcome by developing films from the same film ba tch under identical processing conditions. Radiographic film also suffers from spatial non-uniformity.14 Spatial variations on the order of 2% are typically seen with radi ographic film. A flat area of the measured field may be averaged to reduce the error due to these variations. This correction is not possible when taking penumbra measurements, because the field is changing rapidly over this distance. A final problem encountered wi th radiographic film is its over-response to low energy photons due to the high atomic number components that make up the film.12,14 The problem becomes worse as the field size and depth of measurement are increased, resulting in systematic error of th e measured dose in the tail of the penumbra.12 This should not be a significant problem for the small fields typically used in radiosurgery. Radiochromic film is anot her type of film that can be used for radiation therapy dosimetry; this type of film does not require chemical developing so it avoids variations in sensitivity and dose response that result from variations in processing conditions.14 Radiochromic film suffers from disa dvantages such as poor reproducibility14 and long developing time. It takes approximately 24 h for the typical radiochromic film to

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17 develop, so it is not as convenient as conventional radiographic film. Another disadvantage of using radiochromic film is th e fact that it does not respond well to low doses of radiation. Radiochromic film also needs to be handled carefully and protected from exposure to sunlight and fluorescent lighting. Radiographic film does not pe rfectly fit all the criteria for an ideal dosimeter for SRS, but it comes close. It outperforms common dosimeters such as diodes, ionization chambers, and thermoluminescent dosimeters (TLDs) because film has high spatial resolution and the ability to make a simultane ous measurement of a spatial map of dose. To accurately measure the penumbra of small fields, the detector needs to be small compared to spatial variations in dose. Volume averaging can become a significant problem in even the smallest available ionization chambers.16 The low cost and general performance characteristics of radiographic film make it a suitable detector for performing relative dosimetric measurements on the small fields used in radiosurgery.

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18 CHAPTER 2 OFF-AXIS RATIO MEASUREMENT OF MINI-MLC APPROXIMATED CIRCLES Introduction One possible method of producing the off-axis ratio lookup tables to be used in the radiosurgery dosimetry system is to direc tly measure them. Although this method can be time consuming, the results are valuable in th at they can be used to provide a means for comparison with model calculations. This chapter will go through the methodology used to measure the circular fields formed by mini-multileaf collimators and convert the measurement to a useful off-axis ratio table. Materials and Methods A Varian CLINAC 600C 6MV Radiotherapy Accelerator was used to produce the radiations fields that were measured. A miniature multileaf collimator (Wellhofer Dosimetrie, Schwarzenbruck, Germany) was us ed to shape the radiation fields. The mMLC has 20 pairs of tungsten l eaves that project 4.95 mm wi de at the isocenter of the linac. Each leaf is stepping-motor driven along a curved path so that ends of the leaves match the beam divergence.17 Table 2-1. Physical characteris tics of the Wellhofer mini-MLC Number of leaves 20 Leaf width at isocenter 4.95 mm Leaf height 8.0 cm Maximum field size at isocenter 9 X 10 cm2 Source-to-collimator distance (mounted on th e Varian 600C) 65.0 cm Leaf travel over central axis 2.0 cm Leaf velocity 1 cm/s Mass 35 kg

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19 Kodak X-OMAT V radiographic film was us ed to measure the radiation fields. Each piece of film was placed at the isocenter of the linac on top of a four centimeter slab of solid water which was used to provide sufficient backscatter. A one and a half centimeter slab of solid water was placed on top of the film to provide buildup. The film was located at the depth of maximum absorbed dose. The film was developed in a Kodak M35A-M X-OMAT processor; the temperatur e of the processing ch emicals during film development was always 32.80.5C. Films from measurements performed on the same day were taken from the same film batch and were developed unde r identical processor conditions to avoid film variations in dose response and sensitivity. Figure 2-1. Basic setup for film measurements. Kodak radiographic film is sandwiched between two slabs of solid water.

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20 A calibration exposure was produced before a set of film measurements was made to provide a means to convert optical dens ity to dose. The calibration exposure was developed along with the other films; a new calibration exposure was made each day new measurements were taken. The calibration exposures typically c ontained five or six exposures in the range of te n to eighty monitor units (MUs). The mMLC was set to produce six centimeter square fiel ds for the calibration exposures. After the calibration film was exposed, the mMLC was set to create circular field approximations using an even number of open leaves; the circular fields produced had diameters ranging from 10 cm to 30 cm in 2 cm increments. Six exposures of each circular approximation were made on the same sheet of film. The calibration film and measurement films were developed in the K odak processor and sca nned with an Epson Expression 1600 flatbed scanner at 0.1 mm/pixel resolution. The films were scanned at a high resolution to preserve the inherent hi gh spatial resolution of the detector. Figure 2-2. Epson flatbed scanne r and processed radiographic film.

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21 Several films were scanned on a Lumisys L575 densitometer at 0.124 mm/pixel resolution; this was done to determine if the flatbed scanner would produce the same results as the laser densitometer. It was determined that the flatbed scanner produced similar results to the laser densitometer. A B Figure 2-3. Off-axis profile film measurements of a 6 cm by 7 cm r ectangular field. The profiles are used to compare the flatbed s canner with the densitometer. A) is a cross-plane (direction of l eaf travel) profile and B) is an in-plane (direction of leaf width) profile. Consistency and uniformity checks were also performed on films scanned by the Epson scanner. A consistency test was perf ormed to determine if the scanning resolution was consistent throughout the length of the scan ner. A ruler was scanned at 0.1 mm/pixel resolution and the results were analyzed with a tool developed in Matlab. The tool prompts the user to click on each centimeter marker on the image of the ruler; the program then determines the distance be tween each marker. The results of the consistency check showed the scanner was pe rforming acceptably, with an average error less than 0.1 mm, which is the size of each pixel. A uniformity check was performed on unexposed radiographic film; the standard devi ation of absolute dose was determined to be less than 2%.

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22 Figure 2-4. Results of the consistency check performed on the Epson flatbed scanner. The average absolute value of position error along the scanned ruler is 0.06 mm. Once all the films were scanned and save d on the computer, a calibration curve was created to convert optical density to dose. Output factors for rect angular fields produced by the mMLC were measured beforehand with an Exradin 14P ion chamber (parallel plate, 0.002 cm3 active volume,1 mm window thic kness and 1 mm window-collector gap); the output factor for a 6 cm square fi eld was needed to convert monitor units to dose. The linac is calibrated to have a 10 cm square field deliver 1 cGy of radiation dose to water at dmax (1.5 cm) per monitor unit. The produc t of output factor and monitor units yields the amount of dose delive red to the film. The output factor for a 6 cm square field produced by the mMLC was determined to be 0.96; this value is in good agreement with the output of a 6 cm square field (0.957) when the mMLC is not attached.

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23 Figure 2-5. Output of recta ngular fields shaped by a miniMLC. The output of the field produced by the mMLC is relative to the output of a 10 cm square field produced by the primary collimator jaws alone. A film calibration tool deve loped in Matlab by St. John was modified to read in exposures saved in tagged image format (ti ff) and to convert pixel value to optical density. Equation 2-1 was used to relate pixel value to optical density for an 8-bit image. ) 255 P ( log OD10, (2-1) where OD is optical density and P is the pi xel value (between 0 and 255 for an 8-bit grayscale image). The program prompts the us er to input the output factor, number of calibration exposures and corre sponding monitor units delivered for each exposure. The user is prompted to select each exposure file and click on the background and exposure peaks in an optical density histogram. The average background is determined and

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24 subtracted from each exposure. A second de gree polynomial is fit to a plot of dose versus optical density. Figure 2-6. Film calibration tool. The user is prompted to click on the background and exposure peaks for each separate exposure. Another program was developed in Matlab to allow the user to separate, center, and align multiple exposures on a single film. The user inputs the resolution the film was scanned at and number of exposures on the film. After clicking on the (Get Image) button, the user is prompted to select the image file. The program displays the scanned image of the film and prompts the user to box in each exposure with th e mouse. After an exposure has been selected, the program displays only the selected image. The user then zooms in on the top, right, bottom, and left sides of the image, respectively. Once a side

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25 is zoomed in on, the user is prompted to cl ick on the center of the edge of the field. Having these four points, the program calculate s the equations for the lines of the central axis and the point where the tw o lines intersect (the center of the beam). The image is then rotated slightly so the central axis of the beam is perpendicular to the sides of the image. Since the center of th e beam is known, the program is able to crop the image so the center of the beam is exactly in the center of the image. Figure 2-7. Tool used to se parate and align exposures. The scanned image contains six 26 mm mMLC circular approximations. A program was developed to convert opti cal density to dose and to normalize the dose on a scale between zero and one. Since each quadrant of th e mMLC circular field is symmetric about the central axis, the program is able to split up each intensity matrix into four separate but equal quadran ts. Since six exposures were made for each field, a total

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26 of 24 field quadrants were produced for each circular field. These quadrants were averaged together by the Matlab program to produce the final result, a two dimensional map of off-axis ratios. Since multiple expos ures were averaged together to produce the 2-D maps, it was possible to calculate a 2-D ma p of statistical uncerta inty for each field. The actual error between the measured fields and the real dose di stribution is probably less than predicted by statistical uncertainties. Th is is due to the fact that a spatial error in one quadrant will be somewhat compensated by the resulting opposite spatial error in the adjacent quadrant. Results Some typical two dimensional maps of measured off-axis ratios are shown along with their statistical uncertainties in Figures 8-12. The statistical uncertainty has no units (off-axis ratios are unitless) and is generall y less than 10% in high gradient (penumbra) regions and less than 2% in low gradient re gions. A large uncerta inty in the penumbra region is expected due to the rapid falloff of dose. The plots display one quarter of the field with the origin located at the ce ntral axis of the field. Discussion Circular approximations of radiation fields form by a mMLC were measured with radiographic film; the processed films were i nput into several Matlab programs designed to separate and center the exposures, convert op tical density to dose, and to average field quadrants to arrive at the final result. It was determined the process of separating and aligning exposures introduces the most error to the final result. Some measured off-axis ratio tables had much lower uncertainty than others. This is because the off-axis ratio tables with the lower uncertainty came from exposures that were better aligned.

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27 Exposures may be over or under-rotated by as much as a few degrees. Proper alignment and centering of the exposures is cruc ial to arriving at an accurate result. A new, less user-dependent, method of aligning th e exposures is currently being developed. The new method uses an edge-detection algorith m to find the top, bottom, left, and right edges of the field. Once these edges are determined, the exposures can be rotated accurately, with errors less than 1.

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28 A B Figure 2-8. Twelve millimeter mMLC circular approximation with even leaves. A) is a contour plot of the average measurements and B) is a map of the statistical uncertainty for each pixel in the measurement.

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29 A B Figure 2-9. Twenty two millimeter mMLC circ ular approximation with even leaves. A) is a contour plot of the average measurements and B) is a map of the statistical uncertainty for each pixel in the measurement.

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30 A B Figure 2-10. Thirty millimeter mMLC circular approximation with even leaves. A) is a contour plot of the average measurements and B) is a map of the statistical uncertainty for each pixel in the measurement.

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31 A B Figure 2-11. Twenty millimeter mMLC circul ar approximation with odd leaves. A) is a contour plot of the average measurements and B) is a map of the statistical uncertainty for each pixel in the measurement.

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32 A B Figure 2-12. Twenty four millimeter mMLC ci rcular approximation with odd leaves. A) is a contour plot of the average measurements and B) is a map of the statistical uncertainty for each pixel in the measurement.

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33 CHAPTER 3 A ROBUST ALGORITHM TO MODEL OFF-AXIS RATIOS OF MINI-MLC APPROXIMATED CIRCULAR FIELDS Introduction Our goal was to develop a tool to mode l the off-axis ratios of mMLC shaped circular fields since the direct measurement of these fields is time consuming. The model is then used to produce two-dime nsional lookup tables for input into the radiosurgery treatment planning system. One alternative method that could be used to produce the off-axis ratio tables is Monte Ca rlo simulation, but it would take more time to run Monte Carlo simulations than to perform direct measurements. Also, the validation of Monte Carlo requires the dete rmination of the parameters and then validation of results. Since Monte Carlo doe s not provide results in a timely manner, another method was considered. This chapter discusses a simple technique that employs model constructions to produce th e off-axis ratio tables. Methods and Materials The model construction is based on the ne gative field techni que using the basic Cunningham model.18 This technique involves the appli cation of negative fluences to a rectangular open field to produ ce the desired dose distributio n. The algorithm that was developed to help us achieve our final result is as follows: Measure the off-axis ratios of rectangular fields with widths varying over the range of interest (10-30 mm). Fit the Cunningham model to the measur ed off-axis ratios and acquire the necessary fitting parameters.

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34 Create a model of a rectangular field using the fitting parameters acquired in Step 2. Calculate the negative field and subtract the negative field from the rectangular model. Repeat Step 4 if there is mo re than one negative field. Rectangular fields formed by the mMLC were measured with Kodak X-OMAT V film; the width of the measured rectangular fields varied from 10 mm to 30 mm. Optical density was converted to dose and normalized using the results fr om a film calibration curve obtained by the film calibration tool. On e dimensional cross-pl ane (in the direction of leaf motion) and in-plane (in the direction of leaf widt h) profiles through the central axis were determined from the data set. The Cunningham model was fit to cross-plane and in-plane profiles using the curve fit tool in Matlab. The fitting algorithm employs a nonlinear least-squares technique. The Cunni ngham model for off-axis ratio (OAR) at a point r mm from the central axis is 2 w r for ) 2 w r ( p exp t) 5 0 ( t OAR 2 w r for ) r 2 w ( p exp 5 0 1 OAR d d 2 d d 1 (3-1) where 1 and 2 are the fit parameters, wd is the field width de fined as the distance between the fifty percent isodose points, r is the radial distance from the center of the field, p is the width of the penumbra, and t is the transmission through the mMLC when all leaves are closed. Transmission wa s measured by St. John and Wagner and was determined to be 0.7%.5,7 Meeks also measured tran smission; he determined the maximum transmission between leaf sides to be 0.5% and between leaf ends to be

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35 1.2%.17 The two transmission measurements aver aged together agree with St. John’s and Wagner’s measurements. The width of the penumbra, defined as the distance between the 80% and 20% isodose points, was measured and determined to be 0.33 mm. A typical example of the Cunningham model, fit to cross-plane data, is shown (Figure 3-1). There are several other functions alternative to the Cunningham m odel that have been used to fit penumbra profiles.12, 19 These functions are continuous and have only one fitting parameter, but they don’t seem to fit the data as well as the discontinuous Cunningham model. In reality, a continuous function offers no adva ntage over a discontinuous one because our data points are sampled discretely. Figure 3-1. Cunningham model f it to cross-plane data from a rectangular field formed by the mMLC. The rectangular field is 18 mm wide.

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36 A B Figure 3-2. Plots of the fitting parameters as a function of field width. A) A step function is fit to the 1 parameters. B) A straight line is fit to the 2 parameters. These fits are input into a Matlab program and are used to create models of rectangular fields.

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37 The fitting parameters for each rectangular field were plotted against field width (Figure 3-2). A discontinuity is noted when 1 is plotted against field width; this discontinuity was also seen by St. John. Th e discontinuity is the result of adding more open leaves to the field and th e field approaching lateral e quilibrium. A step function was fit to the calculated 1 parameters. The plot of the 2 parameters versus field width is linear, so a straight line was fit to the data. The fits of the alpha parameters were input into a Matlab program that was developed to produce the final models of th e mMLC circular fields. Once the alpha parameter fits are input into the program, th e program is able to produce a 2-D model of a rectangular field. When the user enters mMLC leaf positions into the program, the program creates a model of a rectangular field with width equal to th e widest open leaf. The length of the field is equal to the produc t of the number of open leaves in the field and the leaf width. Leaf width is a variab le within the program, so the program can be applied to mMLCs of varying leaf widths. Whether or not an odd or even number of leaves is used to approximate the circle is also variable within the program. After the rectangular field is created, any negative fields that are to be subtracted from the rectangular field are calculated. Th e size, position, and weight of the negative field had to be considered carefully. If a ny of these negative field variables were not accurate, the final model would not agree well with measurement. We also had to account for the possibility of overlapping nega tive fields for cases that required the subtraction of more than one negative field. The first step in calculating the negative field is to determine the cross-plane profile through the center of the leaf wh ere the first negative field will be placed (Figure 3-4).

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38 The cross-plane profile is derived directly from the rectangular model in the case where only one negative field is applied to the rect angular open field. The total number of negative fields to be applied is equal to th e number of open leaves in the quarter field minus one. If more than one negative field is to be applied, the ne gative field that is calculated first is the largest and is located on the open leaf furthest from the center of the field. After the cross-plane profile through the rectangular fiel d has been determined, the profile is shifted to match the actual mMLC leaf position for the circular field. For example, in the case of a 22 mm mMLC shaped circle formed with an even number of leaves, the actual positions of the two open leaves in the quarter field are 10.8 mm and 8.7 mm, respectively (Figure 3-3). After the cross-plane prof ile through the modeled rect angular field (10.8 mm by 10 mm in the quarter field) has been determined (Figure 3-4), the crossplane data is shifted back, from 10.8 mm to 8.7 mm to match th e position of the second leaf (Figure 3-5). After the cross-plane profile has been shif ted, the difference between the rectangular cross-plane profile and shifted cr oss-plane profile is determined. The result is a profile of the negative field in the cross-plane direc tion (Figure 3-6). Th e previous steps are repeated for the in-plane profile; the product of the 1-D negative field cross-plane and inplane profiles yields a spatial map of the negative field. In cases where more than one negative fi eld is applied, the negative cross-plane profiles are a weighted combin ation of cross-plane profiles derived from the rectangular field and the resultant field after the previous negative field was subtracted. This is done so regions of the rectangular field are not s ubtracted twice by overlap ping negative fields.

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39 There is no need to weight the in-plane prof iles as such because of the order the negative fields are subtracted. The in-plane profiles are derived directly from the most current version of the model. For example, if one negative field had been subtracted from the rectangular field, the next in-p lane profile would be derived from the field that resulted after the subtraction of the first negative field. Results The negative field technique wa s applied to models of rect angular fields to create models of the mMLC shaped circular fields that were measured. Several metrics were used to compare the final, modeled fields, with the measured fields. The first metric is qualitative; it consists of ove rlapping contour plots of the me asured and modeled fields. By visually inspecting this plot, one can quick ly determine if the model agrees well with measurement. The next two metrics provide a means to quantitatively compare the measured and modeled dose distributions. In low gradient regions of the field (regions where the off-axis ratio is not changing rapidly), the e rror between the modeled and measured fields is best represented by the absolute dose difference between the two fields. In high gradient regions (the penumbra), the dose difference does not provide a good means of quantifying the error. Instead the error in high gradient regions is represented by the distance between points of equa l relative dose. We chose 5% and 1 mm to be the error criteria for our dose-difference and distanceto-agreement tests, respectively. A program was developed in Matlab to perform error analysis on the modeled fields using the measured fields as the re ference. The program determines the dosedifference and distance-to-agreement error for near ly all points in the modeled field. The program then places each error point into either a dose-difference or distance-to-

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40 agreement histogram depending on how well if fit the respective error criteria. The results are two-dimensional histograms of dose-difference and distance-to-agreement error. The level of agreement between th e modeled fields and reference fields is observed by these histograms. If all point s had less than 1 mm distance-to-agreement error or less than 5% dose-difference error, then the modeled field has met the error criteria. The error analysis program also de termines the percentage of error points that have either a dose-difference error less than 3% or distance-to-agreement error less than 0.5 mm. A third histogram is created to display the dose-difference error and distance-toagreement error together for a ll points. It is obvious from inspecting the 3-d histograms that high gradient regions can have a large dose-difference error, but small distance-toagreement error. And vice versa, low grad ient regions can have a large distance-toagreement error, but small dose-difference error. One can also see from these histograms whether or not the erro r criteria are met. The contour plots and histograms show th at the modeled even fields agree well with what was measured. All of the even mode ls meet or exceed the error criteria. In fact, at least 96% of the erro r points lay within 3% dose er ror or 0.5 mm distance error for the even fields presented in our study. Discussion A tool was developed, to model in 2-D, the off-axis ratios of mMLC shaped circular fields, based on a small number of meas urements. The tool proved to be fast; it can produce a 2-D off-axis ratio table in less than 1 s. The amount of time necessary to perform measurement and analysis to determin e the parameters for input into the tool takes less than half a day. The tool proved to be accurate. All of the modeled fields

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41 shaped with an even number of leaves met the error criteria (5% dose-dose error in low gradient regions and 1 mm distance-to-agreemen t error in high gradient regions). Many of the models had the majority of its errors be much less than the error criteria. The tool is also flexible. Variables such a leaf width and output size can be readily changed within the program. Some of the modeled odd fields did not ag ree well with what was measured. The error is not inherent in the model. It wa s determined that some of the leaves on the mMLC were not properly aligned when the odd field measurements were made. This effect caused the actual leaf positions to be different from the specified leaf positions by about 0.7 mm on average. To check leaf alignment, some leaf pair s of interest were opened individually and made to project 2 cm wide at isocenter. The physical distance between the leaves were measured with a micrometer and projected to isocenter. The results of these measurements (Table 3-1) show the leaves were not properly aligne d. Leaf alignment was performed about a week before the even field measurements were made. Odd field measurements were performed over six months la ter; it is quite possi ble that leaf position accuracy was affected over this time period. Sastre-Padro et al. performed stability and reproducibility measurements on an MLC and reported a maximum leaf shift of 0.27 mm over three months.20

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42 Table 3-1. Leaf alignment check Leaf Paira Physical Separation (mm) Length Projected to Isocenterb (mm) Absolute Error (mm) 8 12.68 0.05 19.51 0.08 0.49 9 12.39 0.05 19.06 0.08 0.94 10 12.05 0.05 18.54 0.08 1.46 11 12.18 0.05 18.74 0.08 1.26 12 12.60 0.05 19.38 0.08 0.62 13 12.71 0.05 19.55 0.08 0.45 aEach leaf pair was opened individually and th e distance between them was measured with a micrometer. bThe length projected to isocenter should be 20 mm. A B Figure 3-3. The mMLC leaf positions fo r a 22 mm circular approximation with even leaves. A) All leaves in the open field are shown. B) The leaves in the quarter field are shown. Because of sy mmetry, it is necessary to model only a quarter of the field.

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43 A B Figure 3-4. A cross-plane prof ile through the rectangular open fi eld is determined. A) A contour plot of the rect angular field. The black dashed lines are the hypothetical mMLC leaf positions for the rectangular model. B) The derived cross-plane profile.

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44 A B Figure 3-5. The cross-plane profile is shif ted to match the position of the mMLC leaf. A) Contour plot of the rectangular fi eld and actual mMLC leaf positions for a 22 mm circular field. B) The cross-pl ane profile before and after the shift.

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45 A B Figure 3-6. Negative field profile. A) The di fference between the cross-plane profiles is determined. B) The resulting cross-pl ane profile of the negative field is shown.

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46 A B Figure 3-7. Spatial map of negative field. A) A contour plot of the negative field. B) A color-wash plot. The negative field is a pplied to a rectangular field to produce the final model of a 22 mm mMLC circular approximation.

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47 A B Figure 3-8. A 10 mm mMLC circular field formed by an even nu mber of leaves. A) Contour plots of measured (solid line) and modeled (dashed line) fields. B) Histograms of dose-difference er ror and distance-to-agreement error comparing the modeled field to the reference measured field. C) Three-dimensional histogram of di stance error and percent error. All points have either dose error less than 3.25% or distance error less than 0.6 mm.

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48 C Figure 3-8. Continued

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49 A B Figure 3-9. A 16 mm mMLC circular field formed by an even nu mber of leaves. A) Contour plots of measured (solid line) and modeled (dashed line) fields. B) Histograms of dose-difference er ror and distance-to-agreement error comparing the modeled field to the reference measured field. C) Three-dimensional histogram of di stance error and percent error. All points have either dose error less than 4.7% or distance e rror less than 0.9 mm.

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50 C Figure 3-9. Continued

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51 A B Figure 3-10. A 20 mm mMLC circular field formed by an even number of leaves. A) Contour plots of measured (solid line) and modeled (dashed line) fields. B) Histograms of dose-difference er ror and distance-to-agreement error comparing the modeled field to the reference measured field. C) Three-dimensional histogram of di stance error and percent error. All points have either dose error less than 3.5% or distance e rror less than 0.75 mm.

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52 C Figure 3-10. Continued

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53 A B Figure 3-11. A 26 mm mMLC circular field formed by an even number of leaves. A) Contour plots of measured (solid line) and modeled (dashed line) fields. B) Histograms of dose-difference er ror and distance-to-agreement error comparing the modeled field to the reference measured field. C) Three-dimensional histogram of di stance error and percent error. All points have either dose error less than 2.8% or distance e rror less than 0.7 mm.

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54 C Figure 3-11. Continued

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55 A B Figure 3-12. A 30 mm mMLC circular field formed by an even number of leaves. A) Contour plots of measured (solid line) and modeled (dashed line) fields. B) Histograms of dose-difference er ror and distance-to-agreement error comparing the modeled field to the reference measured field. C) Three-dimensional histogram of di stance error and percent error. All points have either dose error less than 4% or distance error less than 0.8 mm.

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56 C Figure 3-12. Continued

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57 CHAPTER 4 CONCLUSION The work presented in this report ta ckled the problem of how to produce two dimensional off-axis ratio tables for circular fields approximated by a mini-multileaf collimator. The use of a mMLC in conjunction with the multiple isocenter delivery technique will allow for more efficient treatment delivery and increased patient satisfaction while maintaining conformal dose di stributions with steep dose gradients. Two methods for producing the tables of OARs were investigated. The first method was direct measurement of the fields with radiographic film. This method proved to be rather tedious and time consuming because many films needed to be exposed, processed, and analyzed. The second method investigated was an empirical one based on the negative field technique. A tool was developed in the Matlab programming environment to create models of the mMLC circular fields based on the negative field technique where negative fluences are applied to an open field to achieve the desired dose distribution. The program requires the user to input several OAR fitting parameters that can be quickly obtained from a small number of measurements. Once the fitting parameters are input into the program, the program will output a spatial map of off-axis ratios for any specified mMLC shaped circular field. The modeled fields produced by the program were compared to the fields measured with radiographic film. A tool was develope d to perform error anal ysis on the modeled fields with respect to a reference (m easured) field. This was accomplished by determining the dose-difference error in low gr adient regions of the field and distance-to-

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58 agreement error in high gradient regions of th e field. More than 90% of points in the modeled even fields had dose-difference erro r less than 3% or distance-to-agreement error less than 0.5 mm. The modeled odd fiel ds did not agree as well with measurement because the mMLC leaves were not movi ng to the specified location when the measurements were made. It should be noted from this experience that the leaves on the mMLC must be properly aligned and their pos itions accurately known for this method to work properly. Overall, the tool proved to be fast, flexible, easy to use, and accurate. Future work will include showing that the model can be applied to fields located off-axis and at different depths. It is expected that the model will work fine in these situations because it has been shown that th e penumbra of small fields does not change much with depth or off-axis position. Future work on the tool used to create the modeled fields will include having the tool be able to model any field shaped by the mMLC, as opposed to just being able to mode l circular fields.

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59 APPENDIX A MATLAB PROGRAM DEVELOPE D TO MODEL THE FIELDS msgbox('Select command window and enter inputs'); numleaf=input('Enter the number of leaves in the quarter field (default=1):'); if isempty(numleaf) numleaf=1; end for i=1:numleaf prompt = sprintf('Enter the position of leaf %d in mm: ',i); leafpos(i)=input(prompt); end %///////////////////////////////////////////////////////////////////////// %THESE VARIABLES MAY BE CHANGED TO AC COMODATE DIFFERENT MLC, ODD OR EVEN %FIELDS, ETC %EVEN OR ODD NUMBER OF L EAVES (1 FOR EVEN, 0 FOR ODD) even = 1; %LEAF WIDTH (mm) lfwidth = 4.95 %5.0; scale = 0.1; %(mm/pixel) %SIZE OF RESULING MATRIX (mm) sizematrix = 27.5; pixels = sizematrix/scale; %////////////////////////////////////////////////////////////////////////// w1 = leafpos(1)*2; %FIELD LENGTH %FIELD WIDTH if even==1 %EVEN CASE w2 = lfwidth numleaf 2; else %ODD CASE w2 = lfwidth (numleaf *2 -1); end %/////////////////////////////////////////////////////////////////// %These are the fit parameters for the rectangular fields. %THESE PARAMETERS CAN BE MODIFIED FOR A PARTICULAR MLC/LINAC COMBINATION if w1<18 alpha1cross = 2.18;

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60 else alpha1cross = 1.65; end if w2<18 alpha1in = 2.18; else alpha1in = 1.65; end alpha2cross = -.0135*w1+2.29; alpha2in = -.0135*w2+2.29; %////////////////////////////////////////////////////////////////////// t=.0075; %TRANSMISSION CONSTANT p=2.4; %PENUMBRA CONSTANT %Produce Square Field /////////////////////////////////////////////////////// matrx=zeros(pixels,pixels); for i=1:pixels m=scale*i; if m<(w1/2) crossratio(i)=(1-. 5*exp(-alpha1cross*(w1/2-m)/p)); else crossratio(i)=(t+(.5-t)*exp(-alpha2cross*(m-(w1/2))/p)); end end for i=1:pixels m=scale*i; if m<(w2/2) inratio(i)=(1-.5*exp(-alpha1in*(w2/2-m)/p)); else inratio(i)=(t+(.5t)*exp(-alpha2in* (m-(w2/2))/p)); end end matrx=zeros(pixels,pixels); l=1; for i=1:pixels k=1; for j=1:pixels matrx(i,j)=c rossratio(k)*inratio(l); k=k+1; end l=l+1; end newmatrx=matrx; %////////////////////////////////////////////////////////////////////////// %////////////////////////////////////////////////////////////////////////// %BEGIN CALCULATING AND SU BTRACTING NEGATIVE FIELDS for q=(numleaf-1):-1:1 %NUMLEAF-1 IS THE NUMBER OF NEGATIVE FIELDS centerx = round((leafpos(q+1) + (leafpos(1)-leafpo s(q+1))/2)*10); if even==1

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61 centery = (q+.5)*lfwidth*10; %EVEN CASE else centery = q*lfwidth*10; %ODD CASE end centery=round(centery); w1= leafpos(q+1); if even==1 %EVEN CASE w2new = lfwidth*(q) 2; else %ODD CASE w2new = 2* lfwidth (q-.5); end centery=round(centery); square=newmatrx(centery,:); norm=max(square); for i=1:pixels m=scale*i; if m<(leafpos(q+1)) newcross(i)=norm*(1-.5*exp(-alpha1cross*(w1-m)/p)); else newcross(i)=norm* (t+(.5-t)*exp(-alpha2cross*(m-w1)/p)); end end square2=newmatrx(:,centerx); norm2=max(square2); for i=1:pixels m=scale*i; if m<(w2new/2) newin(i)=norm2*(1-.5*exp(-alpha1in*(w2new/2-m)/p)); else newin(i)=norm2*(t +(.5-t)*exp(-alpha2in* (m-(w2new/2))/p)); end end if q==(numleaf-1) %DONT WEIGHT FIRST NEGATIVE FIELD end1=pixels; for i=1:end1 pt1 = newmatrx(centery,i); pt2= newcross(i); if (pt1-pt2)>0 negcross(i)=pt1-pt2; else negcross(i)=0; end end else %WEIGHT OTHER NEGATIVE FIELDS end1=pixels; for i=1:end1 wt1=(2*centerx-i+1)/(2*centerx); wt2= i/(2*centerx);

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62 pta=newmatrx(centery,i); ptb=matrx(centery,i); pt1 = wt2*pta+wt1*ptb; pt2= newcross(i); if (pt1-pt2)>0 negcross(i)=pt1-pt2; else negcross(i)=0; end end end negin=zeros(1,pixels); for i=1:pixels pt1 =newmatrx(i,centerx); pt2= newin(i); if (pt1-pt2)>0 negin(i)=pt1-pt2; else negin(i)=0; end end max1=max(negin); max2=max(negcross); if max1>max2 avemax=max1; else avemax=max2; end [z,end2]=size(negin); negmatrx=zeros(end2,end1); l=1; for i=1:end2 k=1; for j=1:end1 negmatrx(i,j)=(negcross(k)*negin(l))/avemax; k=k+1; end l=l+1; end if q==2 neg1=negmatrx; else neg2=negmatrx; end for i=1:end2 for j=1:end1 newmatrx(i, j)=newmatrx(i,j) -negmatrx(i,j); end end

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63 end X=(0:(pixels-1)).*scale; figure [C,h] = contour(X,X,newmatr x,[.9 .5 .1],':');hold on; %[C,h] = contour(X,X,odd28 mm,[.9 .5 .1]); clabel(C,h); Xlabel('Crossplane Distance (mm)'); Ylabel('Inplane Distance (mm)'); legend(h,'90% isodose line','50% isodose line','10% isodose line');

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64 APPENDIX B MATLAB PROGRAM DEVELOPED TO PERFORM ERROR ANALYSIS ON MODELED FIELDS clear X X1 finaldist finaldose gammadose gammadist histomatrix; %TYPE THE NAME OF THE REFERENCE DOSE PROFILE IN THE WORKSPACE refdose = even30; fieldwidth=23; %max 23 lngthofr=(fieldwidth/2)+10; numpix=275; k=1; t=1; z=1; %CHECK EACH PIXEL VALUE FOR DOSE-DOSE ERROR for theta = 0: 0.017 : pi/2 for r = 0: 0.1 : lngthofr r2=r*10; y = round(r2 *sin(theta)) +1; x = round(r2 *cos(theta)) +1; dose_err( k)=abs(newmatrx(y,x)-refdose(y,x)); match=refdose(y,x); %REFERENCE VALUE FOR DOSE WE WANT TO FIND DISTANCE TO if r<4 start=0; else start=r2-40; end for r3=start:0.5:(r2+45) y2 = round(r3 *sin(theta)) +1; x2 = round(r3 *cos(theta)) +1; temp=newmatrx(y2,x2); if temp<=match dist(k)=sqrt((y2-y)^2+(x2-x)^2)*.1; break else dist(k)=4.5; end end gammadist=dist(k)/1;

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65 gammadose=dose_err(k)/0.05; if gammadose
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66 cnt=0; sumhisto=sum(sum(histomatrix)); for i=1:m for j=1:n if (i<6)|(j<13) cnt=cnt+histomatrix(i,j); end end end ratio=(cnt/sumhisto)*100 distaxes=(0:distbin:(distbin*(m-1))); prcntaxes=(0:prcntbin:(prcntbin*(n-1))); meshz(prcntaxes(1,1:35),dis taxes(1,1:35),lghistomatrix(1:35,1:35)); xlabel('Percent Error','Rotation',15,'Fontsize',14); ylabel('Distance Error (mm)','Rotation',-4 5,'Fontsize',14); zlabel('Log(Frequency)','Fontsize',12); set(gca,'zscale','linear');view(153,54); figure N=zeros(100,1); [N,X]=hist(finaldose,100); P=zeros(20,1); [P,X1]=hist(finaldist,20); subplot(1,2,1) bar(X*100,N); grid on; Xlabel('Dose Error (%)'); Ylabel('Frequency'); pbaspect([1 1 1]);%Title('Histogram'); subplot(1,2,2) bar(X1,P); grid on; Xlabel(' Distance Error (mm)'); Ylabel('Frequency'); pbaspect([1 1 1]);%Title('Histogram');

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67 APPENDIX C MATLAB PROGRAM DEVELOPED TO ANALYZE A CALIBRATION FILM function calib_separate %CALIB_SEPARATE % This function opens calibration tiff files, selected by operator, one at a time % and plots a histogram of optical dens ity. By clicking on the background peak % and exposure peak for each file, a dose calibration curve is calculated. % Coefficients a, b, & c are returned fo r the equation dose = a*OD^2 + b*OD + c. msgbox('Select command window and enter inputs'); num=input('Enter the number of calibration exposures (default=8):'); if isempty(num) num=8; end output=input('Enter the output factor (default=1.0):'); if isempty(output) output=1; end MU(1,1)=0; for i=1:num prompt=sprintf('Enter MUs for exposure # %d: ',i); MU(i+1,1)=input(prompt); end OD=zeros(num+1,1); dose=MU.*output %film_type=input('Enter the film type(XV or EDR),(default{XV}):'); %if isempty(film_type) % film_type='XV'; %end %film_type for j=1:num prompt=sprintf('Select file number %d',j) [scanfile,direc]=uigetfile('*.tif',prompt); if scanfile==0 return; end cd(direc); fid1=fopen(scanfile); if fid1 ~=-1 raw=imread(scanfile,'tif'); else

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68 msgbox('File not found\n'); return; end if j==1 f1=figure; set(f1,'position',[120 40 600 480]); end [A,B]=size(raw); raw2=double(raw); for i=1:A for z=1:B if raw2(i,z)==0 raw2(i,z)=1; end end end raw2=-log10(raw2/255); %Convert to OD N=zeros(500,1); % HIST(raw) creates a matrix with histog rams of each column in the matrix(raw) [N1,X]=hist(raw2,500); % Combine all histograms into a single histogram (X contai ns bin centers) for i=1:B N(:,1)=N(:,1) + N1(:,i); end M = max(max(raw2)); %Use M to set axis limits Mn=max(max(N)); % Draw the plots p1 = subplot(2,2,1);imagesc(raw),colormap gray, axis square;Title('Raw Image'); p2 = subplot(2,2,2); plot(0,0) p3 = subplot(2,1,2); %[N,X]=hist(raw1,50);his t(raw1(25,:),50);hold on bar(X,N); grid on; Xlabel('Optical Density'); Ylabel('Frequency'); h1=uicontrol('style','text','position',[ 100 230 400 20 ],'BackgroundColor','w'); set(h1,'string','****Click on BACKGROUND peak****'); [X,Y]=ginput(1); BKGpts(j)=X; set(p3,'xlim',[0 M],'ylim',[0 Mn]);pbaspect([2 1 1]); set(h1,'string','****Click on Exposure peak****'); [X,Y]=ginput(1); ODpoint(j)=X; % Leave row 1 col 1 for zero dose OD(j+1)=ODpoint(j); delete(h1); end BKGpts BKG=mean(BKGpts); %std(BKGpts(:,1))/BKG; % = standard deviation OD(1)=BKG; OD; dose;

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69 for t=1:num+1 OD(t)=OD(t)-BKG; end OD dose OD_size=size(OD) dose_size=size(dose); dose=output.*MU; Mdose=max(dose); MaxOD=max(OD); %****** facts=polyfit(OD,dose,2); a=facts(1); b=facts(2); c=facts(3); for i=1:(MaxOD)*50 dose_fit(i,1)=a*(i/50)^2 + b*(i/50) + c; Xaxis(i)=i/50; end dose_fit; %*****linear fit %facts=polyfit(OD(2:6),dose(2:6),1); %a=0; %b=facts(1); %c=facts(2); %for i=1:num % dose_fit(i,1)=b*OD(i,1) + c; %end axes(p2); cla plot(Xaxis,dose_fit,'color','r','LineWidth',2); %plot(dose_fit,OD/1000,'col or','r','LineWidth',2); hold on; plot(OD(1:num+1),dose(1:num+1),'*');grid on; YLabel('Dose (cGy)');XLabel('Optical density (OD) ');Title('Calibration Curve (BKG-corrected)'); % Write the fitting parameters at the bottom of the plot h1=uicontrol('style','tex t','position',[60 225 500 20 ],'BackgroundColor','w'); result=sprintf('The calibration curve is Dose = %7.3e*OD^2 + %7.3e*OD + %4.3f ',a,b,c); set(h1,'ForeGroundColor','r','Font Weight','Bold','string',result); % Write calibration factors fid=fopen('calib.info','w'); fprintf(fid,'Calibration film: %s\n',pwd); fprintf(fid,'%7.3e \t %7.4f \t %7.4f\n',a,b,c); fprintf(fid,'%7.4f',BKG); set(findobj('tag','cal_a'),'string',a); set(findobj('tag','cal_b'),'string',b); set(findobj('tag','cal_c'),'string',c); fclose(fid);

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70 % write dose and OD to file for Excel fid=fopen('plot3.txt','w'); for i=0:num-1 j=num-i; fprintf(fid,'%f\t%f\n',OD(j),dose(j)); end fclose(fid);

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71 APPENDIX D MATLAB PROGRAM DEVELOPED TO SE PARATE AND ALIGN EXPOSURES function varargout = separate_exp(varargin) % SEPARATE_EXP M-file for separate_exp.fig % SEPARATE_EXP, by itself, creates a new SEPARATE_EXP or raises the existing % singleton*. % % H = SEPARATE_EXP returns the handle to a new SEPARATE_EXP or the handle to % the existing singleton*. % % SEPARATE_EXP('CA LLBACK',hObject,eventData,h andles,...) calls the local % function named CALLBACK in SEPARAT E_EXP.M with the given input arguments. % % SEPARATE_EXP('Proper ty','Value',...) creates a new SEPARATE_EXP or raises the % existing singleton*. Starting from the left, property value pairs are % applied to the GUI before separate_exp_OpeningFunction gets called. An % unrecognized property name or invalid value makes property application % stop. All inputs are passed to separate_exp_OpeningFcn via varargin. % % *See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one % instance to run (singleton)". % % See also: GUIDE, GUIDATA, GUIHANDLES % Edit the above text to modify the response to help separate_exp % Last Modified by GUIDE v2.5 22-Feb-2005 05:11:31 % Begin initialization code DO NOT EDIT gui_Singleton = 1; gui_State = struct('gui_Name', mfilename, ... 'gui_Singleton', gui_Singleton, ... 'gui _OpeningFcn', @separ ate_exp_OpeningFcn, ... 'gui_O utputFcn', @separ ate_exp_OutputFcn, ... 'gui_LayoutFcn', [] ... 'gui_Callback', []); if nargin & isstr(varargin{1}) gui_State.gui_Callback = str2func(varargin{1}); end if nargout [varargout{1:nargout}] = gui _mainfcn(gui_State, varargin{:}); else gui_mainfcn(gui_State, varargin{:}); end % End initialization code DO NOT EDIT % --Executes just before separate_exp is made visible. function separate_exp_OpeningFcn(hObject, eventdata, handles, varargin)

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72 % This function has no output args, see OutputFcn. % hObject handle to figure % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % varargin command line arguments to separate_exp (see VARARGIN) % Choose default command line output for separate_exp handles.output = hObject; % Update handles structure guidata(hObject, handles); set(hObject,'MenuBar','figure'); % UIWAIT makes separate_exp wait for user response (see UIRESUME) % uiwait(handles.figure1); % --Outputs from this function are returned to the command line. function varargout = separate_exp_Ou tputFcn(hObject, eventdata, handles) % varargout cell array for return ing output args (see VARARGOUT); % hObject handle to figure % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Get default command line output from handles structure varargout{1} = handles.output; % --Executes during object creation, after setting all properties. function numexp_CreateFcn(h Object, eventdata, handles) % hObject handle to numexp (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles empty handles not created until after all CreateFcns called % Hint: popupmenu controls usually hav e a white background on Windows. % See ISPC and COMPUTER. if ispc set(hObject,'BackgroundColor','white'); else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor')); end % --Executes on selection change in numexp. function numexp_Callback(hObject, eventdata, handles) % hObject handle to numexp (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hints: contents = get(hObject,'String') returns numexp contents as cell array % --Executes on button press in getfile. function getfile_Callback(hObject, eventdata, handles) % hObject handle to getfile (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) num=get(handles.numexp,'Value'); %returns selected item from numexp scale=str2double(get(handles.s cale,'String')); %returns contents of scale as a double

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73 numpixel=floor(85/scale); %was 55 check=rem(numpixel,2); %force number of pixels to be even for ease of calculations if check==0 else numpixel=numpixel+1; end [scanfile,direc]=uigetfile('*.tif','Select image' ); cd(direc); fid1=fopen(scanfile); if fid1 ~=-1 I=imread(scanfile,'tif'); else fprintf('File not found\n'); return; end cla; uiwait(msgbox('Box in each exposure. Hold shift key for square.','modal')); for i=1:num cla; axes(handles.image) %Allows the user to separate different exposures newimage=imcrop(I); %Check to see if user wants to center and align exposures num2=get(handles.align,'Value'); if num2==1 cla; contimage=double(newimage); imagesc(contimage),colormap gray, axis equal, axis tight; uiwait(msgbox('Zoom in and click on each fiducial starting at the top and working clockwise.','modal')); for z=1:4 k=20; zoom on; k=waitforbuttonpress; if k==0 [x(z),y(z)]=(ginput(1)); end zoom out; end %Determine the equations for the lines for the central axis m1=(y(2)-y(4))/(x(2)-x(4)); b1=y(2)-m1*x(2);

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74 if x(3)==x(1) %so don't divide by zero xint=round(x(3)) m2=inf; else m2=(y(3)-y(1))/(x(3)-x(1)); b2=(y(3)-m2*x(3)); %Determine intersection (center of beam) xint=round((b2-b1)/(m1-m2)); end yint=round(m1*xint+b1); % %if odd % yint=yint-25; % %end % %Crop the image so the center of beam is center of image [A,B]=size(newimage); if ((xint<(B/2))&(yint>(A/2))) % first quadrant if xint>(A-yint) I2=imcrop(newimage,[xint-(A-yin t),2*yint-A,2*(A-yint),2*(A-yint )]); % closer to upper half else I2=imcrop(newimage,[1,yintxint,2*xint,2*xint]); % closer to left edge...xmin set to 1 instead of 0 b/c dimensions work out that way. end elseif ((xint<(B/2))&(yint<(A/2))) % third quadrant if xint>yint I2=imcr op(newimage,[xint-yin t,1,2*yint,2*yint]); else I2=imcr op(newimage,[1,yint-xi nt,2*xint,2*xint]); end elseif ((xint>(B/2 ))&(yint>(A/2))) % second quadrant if ((B-xint)<(A-yint)) I2=imcrop(newimage ,[2*xint-B,yint-(B-xint),2 *(B-xint),2*(B-xint)]); else I2=imcrop(newimage ,[xint-(A-yint),2*yint-A,2* (A-yint),2*(A-yint)]); end else % fourth quadrant if yint<(B-xint) I2=imcr op(newimage,[xint-yin t,1,2*yint,2*yint]); else I2=imcrop(newimage ,[2*xint-B,yint-(B-xint),2 *(B-xint),2*(B-xint)]); end end %Next, determine how much the image needs to be rotated [A,B]=size(I2); X=A/(2*m2)+A/2; angle=rad2deg(atan(2*X/A-1)); angle %Now rotate the image I3=imrotate(I2,-angle,'bicubic','crop'); cla;

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75 [A,B]=size(I3); A=A/2-numpixel/2; B=B/2-numpixel/2; I4=imcrop(I3,[B ,A,numpixel-1,numpixel-1]); newdirec=strcat(direc,'exposure'); num2=num2str(i); newdirec=strcat(newdirec,num2); newdirec=strcat(newdirec,'.tif'); %Write each separate exposure to a file imwrite(I4,newdirec,'tif'); else newdirec=strcat(direc,'exposure'); num2=num2str(i); newdirec=strcat(newdirec,num2); newdirec=strcat(newdirec,'.tif'); %Write each separate exposure to a file imwrite(newimage,newdirec,'tif'); end % end cla; % --Executes during object creation, after setting all properties. function scale_CreateFcn(h Object, eventdata, handles) % hObject handle to scale (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles empty handles not created until after all CreateFcns called % Hint: edit controls usually have a white background on Windows. % See ISPC and COMPUTER. if ispc set(hObject,'BackgroundColor','white'); else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor')); end function scale_Callback(hObject, eventdata, handles) % hObject handle to scale (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hints: get(hObject,'String') returns contents of scale as text % str2double(get(hObject,'String')) returns contents of scale as a double % --Executes on button press in align. function align_Callback(hObject, eventdata, handles) % hObject handle to align (see GCBO) % eventdata reserved to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hint: get(hObject,'Value') returns toggle state of align

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76 APPENDIX E MATLAB PROGRAM DEVELOPED TO AVERAGE FIELD QUADRANTS AND PRODUCE FINAL MAP OF OFF AXIS RATIOS %This program is called by separate_exp when the Combine button is pressed. % The user is prompted to select six exposures. User needs to click on the background % for each exposure. Average background is subtracted from these exposures % and exposures are stored in an arra y. Program then splits each image % matrix into four equal quadrants and stores them in a separate array. %num=get(handles.numexp,'Value'); %returns selected item from numexp %Enter scale and number of exposures here (mm/pixel) scale=0.1; numexp=2; %ENTER FIT PARAMETERS TO CONVERT OD TO DOSE a=-14.82; b=224.21; c=2.861; for t=1:numexp [scanfile,direc]=uigetfile('*.tif','Select image' ); cd(direc); fid1=fopen(scanfile); if fid1 ~=-1 I=imread(scanfile,'tif'); else fprintf('File not found\n'); return; end I3=double(I); [numpixel,B]=size(I3); %CONVERT TO OPTICAL DENSITY I3=-log10((I3+1)/256); I2(t,:,:)=I3; A=B; N=zeros(500,1); [N1,X]=hist(I3,500); % Combine all row histograms into a single histogram (X contains bin centers) for i=1:A N(:,1)=N(:,1) + N1(:,i); end

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77 bar(X,N); grid on; Xlabel('Optical Density'); Ylabel('Frequency'); pbaspect([1 1 1]);Title('Histogram'); %Select background [bkg(t),y]=ginput(1); end %Determine and subtract average background avebkg=sum(bkg(:))/numexp; I2(:,:,:)=I2(:,:,:)-avebkg; pix=numpixel/2; %Convert optical density to dose for t=1:numexp for i=1:numpixel for j=1:B I2(t,i,j) = a*(I2(t,i,j))^2 + b*I2(t,i,j) + c; if I2(t,i,j)<0 %I2(t,i,j)=0; end end end end %Declare array for the sections Combine=zeros((numexp*4),pix,pix); q=1; %Split up the four images into four quadrants and store in array for m=1:numexp P=zeros(pix,pix); Q=zeros(pix,pix); R=zeros(pix,pix); S=zeros(pix,pix); %Upper left hand corner for i=1:pix for j=1:pix P(i,j)=I2(m,i,j); end end P=fliplr(P); Combine(q,:,:)=P; q=q+1; %Upper right hand corner for i=1:pix for j=(pix+1):(numpixel) k=j-(pix); Q(i,k)=I2(m,i,j); end end Combine(q,:,:)=Q;

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78 q=q+1; %Lower left hand corner for i=(pix+1):(numpixel) for j=1:pix k=i-(pix); R(k,j)=I2(m,i,j); end end R=fliplr(R); R=flipud(R); Combine(q,:,:)=R; q=q+1; %Lower right hand corner for i=(pix+1):(numpixel) for j=(pix+1):(numpixel) k=i-(pix); l=j-(pix); S(k,l)=I2(m,i,j); end end S=flipud(S); Combine(q,:,:)=S; q=q+1; end %Determine the average sum=zeros(pix,pix); Temp=zeros(pix,pix); for i=1:(4*numexp) for j=1:pix for k=1:pix Temp(j,k)=Combine(i,j,k); end end sum=sum+Temp; end Average=zeros(pix,pix); Average=sum(:,:)/(numexp*4); UncAverage = Average; max(max(Average)); Average=flipud(Average); %SUBTRACT THE AVERAGE MINIMUM [A,B]=size(Average); k=1; for i=A-25:A l=1; for j=B-25:B sample(k,l)=Average(i,j); l=l+1; end k=k+1; end

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79 minim=mean(mean(sample)); Average = Average(:,:) minim; %Normalize the data to the maximum value maxim=max(max(Average)); Average=Average(:,:)/maxim; temp=Average; sumdev=zeros(pix,pix); %DETERMINE THE STATISTI CAL ERROR OF THE AVERAGE Temp2=zeros(pix,pix); for i=1:(numexp*4) for j=1:pix for k=1:pix Temp(j,k)=Combine(i,j,k); end end for j=1:pix for k=1:pix Temp2(j,k) =(Temp(j,k)-UncAverage(j,k))^2; end end sumdev=sumdev+Temp2; end Unc=zeros(pix,pix); Unc=sqrt(sumdev(:,:) /(numexp*4-1)); Unc=Unc./maxim; %PLOT HISTOGRAM OF UNCERTAINTY N=zeros(500,1); [N1,X]=hist(Unc,500); % Combine all row histograms into a single histogram (X contains bin centers) for i=1:275 N(:,1)=N(:,1) + N1(:,i); end N=N(:,1)/75625; bar(X,N); grid on; Xlabel('Relative Uncertainty'); Ylabel('Relative Frequency'); pbaspect([1 1 1]);Title('Histogram'); xlim([0 .15]); Xax=(0:274)./10; Xtr=(274:-1:0)./10; figure imagesc(Xax,Xtr,Unc);colorbar; Xlabel('Crossplane Distance (mm)'); Ylabel('Inplane Distance (mm)'); set(gca,'Ydir','normal'); figure [C,h] = contour(Xax,Xax,Aver age,[.9 .5 .1]); clabel(C,h); Xlabel('Crossplane Distance (mm)'); Ylabel('Inplane Distance (mm)');

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80 legend(h,'90% isodose line','50% isodose line','10% isodose line');figure % flip=flipud(temp); % C=cat(1,temp,flip); % % flip=fliplr(temp); % flip2=flipud(flip); % D=cat(1,flip,flip2); % % final=cat(2,D,C); % % % imagesc(final); % colorbar figure line=Average(1,:); plot(line,':b');

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81 LIST OF REFERENCES 1. F. M. Khan, “Stereotactic Radiosurgery,” in The Physics of Radiation Therapy, edited by J. Pine, M. Standen, L. R. Ka iris, and T. Boyce (Williams & Wilkins, Baltimore, 2003). 2. F. J. Bova, W. A. Friedman, J. M. Buatti, and W. M. Mendenhall, Linac Radiosurgery: A Practical Guide, (Springer, New York, 1998). 3. E. J. Hall, Radiobiology for the Radiobiologist, (Williams & Wilkins, Baltimore, 2000). 4. R. M. Cardinal, S. H. Benedict, Q. Wu, R. D. Zwicker, H. E. Gaballa, and R. Mohan, “A comparison of three stereota ctic radiotherapy techniques; arcs vs. noncoplanar fixed fields vs. intensity modulation,” Int. J. Radiat. Oncol., Biol., Phys. 42, 431-436 (1998). 5. T. H. Wagner, Optimal Delivery Techniques for Intracranial Stereotactic Radiosurgery Using Circular and Multileaf Collimators: PhD dissertation, Nuclear Engineering Sciences, University of Florida, Gainesville, FL (2000). 6. T. D. Solberg, K. L. Boedeker, R. Fogg, M. T. Selch, and A. A. F. DeSalles, “Dynamic arc radiosurgery field shaping: a comparison with static field conformal and noncoplanar circular arcs,” Int. J. Radiat. Oncol., Biol., Phys. 49, 1481-1491 (2001). 7. T. J. St. John, Intensity-Modulated Radiosurgery Treatment Planning By Fluence Mapping Optimized Multi-isocenter Plans: PhD dissertation, Nuclear Engineering Sciences, University of Florida, Gainesville, FL (2002). 8. P. G. Archer, J. M. Balter, D. A. Ross J. A. Hayman, and H. M. Sandler, “The treatment planning of segmental, conforma l stereotactic radiosurgery utilizing a standard multileaf collimator,” Med. Dosim. 24, 13-19 (1999). 9. J. L. Nakamura, A.Pirzkall, M. P. Caro., P. Xia, V. Smith, w. M. Wara, P. L. Petti, L. J. Verhey, and P. K. Sneed, “Compari son of intensity-modulated radiosurgery with gamma knife radiosurgery for challeng ing skull base lesions,” Int. J. Radiat. Oncol., Biol., Phys. 55, 99-109 (2003). 10. S. H. Benedict, R. M. Cardinale, Q. Wu R. D. Zwicker, W. C. Broaddus, and R. Mohan, “Intensity-modulated stereotact ic radiosurgery using dynamic micromultileaf collimation,” Int. J. Radiat. Oncol., Biol., Phys. 50, 751-758 (2001).

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82 11. A. S. Shiu, H. M. Kooy, J. R. Ewton, S. S. Tung, J. Wong, K. Antes, and M. H. Maor, “Comparison of miniature multileaf collimation (mMLC) with circular collimation for stereotactic treatment,” Int. J. Radiat. Oncol., Biol., Phys. 37, 679688 (1997). 12. J. Sun and Y. Zhu, “Study of dosimetri c penumbra due to multileaf collimation on a medical linear accelerator,” Int. J. Radiat. Oncol., Biol., Phys. 32, 1409-1417 (1995). 13. K. R. Rice, J. L. Hansen, and G. K. Svensson, “Measurement of dose distribution in small beams of 6MV x-rays,” Phys. Med. Biol. 32, 1087-1099 (1987). 14. D. M. Duggan and C. W. Coffey II, “S mall photon field dosimetry for stereotactic radiosurgery,” Med. Dosim. 23, 153-159 (1998). 15. J. L. Robar and B. G. Clark, “The use of radiographic film for linear accelerator stereotactic radiosurgical dosimetry,” Med. Phys. 26, 2144-2150 (1999). 16. J. W. Sohn, J. F. Dempsey, T. S. Suh, and D. A. Low, “Analysis of various beamlet sizes for IMRT with 6 MV photons,” Med. Phys. 30, 2432-2439 (2003). 17. S. L. Meeks, F. J. Bova, S. Kim, W. A. Tome, and J. Buatti, “Dosimetric characterization of a double-focused miniat ure multileaf collimator,” Med. Phys. 26, 729-733 (1999). 18. H. E. Johns and J. R. Cunningham, The Physics of Radiology. New York: John Wiley & Sons (1983). 19. T. Kron, A. Elliot, and P. Metcalfe, “The penumbra of 6 MV x-ray beams as measured by thermoluminescent dosimetry and evaluated using an inverse square root function,” Med. Phys. 20, 1429-1438 (1993). 20. M. Sastre-Padro, U. A. van der Heide, and H. Welleweerd, “An accurate calibration method of the multileaf collimator valid for conformal and intensity modulated radiation treatments,” Phys. Med. Biol. 49, 2631-2643 (2004).

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83 BIOGRAPHICAL SKETCH Brandon K. Rice was born in October 1981, in Duluth, MN. He attended the University of Wisconsin-River Falls from 1999-2003, earning a Bachelor of Science degree in physics (with honors) in May 2003. During his undergraduate work, he served as a research assistant in the Department of Physics; he served as president of Phi Sigma Kappa national fraternity; he was a member of Sigma Pi Sigma physics honor society, and Phi Kappa Phi national honor society; and was an active member in the local chapter of the Society of Physics Students. He entered graduate school at the Univer sity of Florida in August 2003. There he completed a Master of Science degree in medi cal physics. During his work, he served as a graduate research assistant in the Radios urgery and Radiobiol ogy Laboratory at the University of Florida Brain Institute and served as vice president of the Society of Health and Medical Physics Students. He is also a member of the American Association of Physicists in Medicine and the American Nuclear Society.


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Title: Methods for Producing Off-Axis Ratio Tables from Mini-Multileaf Collimator Shaped Circular Fields for Input into a Stereotactic Radiosurgery Treatment Planning System
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Copyright Date: 2008

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Permanent Link: http://ufdc.ufl.edu/UFE0010840/00001

Material Information

Title: Methods for Producing Off-Axis Ratio Tables from Mini-Multileaf Collimator Shaped Circular Fields for Input into a Stereotactic Radiosurgery Treatment Planning System
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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METHODS FOR PRODUCING OFF-AXIS RATIO TABLES FROM
MINI-MULTILEAF COLLIMATOR SHAPED CIRCULAR FIELDS FOR INPUT
INTO A STEREOTACTIC RADIOSURGERY TREATMENT PLANNING SYSTEM














By

BRANDON RICE


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Brandon Rice

































To my loving family and friends, who have believed in and supported me throughout my
academic career and life.















ACKNOWLEDGMENTS

I would like to acknowledge my research advisor, Dr. Frank Bova, for his support

and insight on this project. I am also much indebted to Drs. David Hintenlang, David

Gilland, James Dempsey, and Wesley Bolch for the many hours of insightful lecture and

guidance pertaining to medical physics. I would also like to thank the rest of the faculty

and staff in the Nuclear and Radiological Engineering Department at the University of

Florida, for giving me the opportunity to learn and expand in a collegial environment.
















TABLE OF CONTENTS



A C K N O W L ED G M EN T S ..................................................................... ...................... iv

LIST OF TABLES ................................................... vii

LIST OF FIGURES ................................................ viii

A B S T R A C T ........................................................................................................ ........ .. x

CHAPTER

1 BACKGROUND .............................. ............ .............................. 1

Overview of Stereotactic Radiosurgery.................. ................................................
Delivery Techniques .......... ............. .. ........... ...................................... ...5
D o sim etry ............................................................................................................... .. 12

2 OFF-AXIS RATIO MEASUREMENT OF MINI-MLC APPROXIMATED
C IR C L E S ................................................................................................................. .. 1 8

In tro d u ctio n ............................................................................................................... .. 1 8
M materials and M ethods .. ..................................................................... ............... 18
R e su lts....................................................................................................... ....... .. 2 6
D isc u ssio n ................................................................................................................ .. 2 6

3 A ROBUST ALGORITHM TO MODEL OFF-AXIS RATIOS OF MINI-MLC
APPROXIMATED CIRCULAR FIELDS .............. ................33

In tro d u ctio n ................................................................................................................ 3 3
M ethods and M materials ................ .............. ............................................ 33
R e su lts....................................................................................................... ....... .. 3 9
D isc u ssio n ................................................................................................................ .. 4 0

4 C O N C L U S IO N ...........................................................................................................5 7

APPENDIX

A MATLAB PROGRAM DEVELOPED TO MODEL THE FIELDS .........................59









B MATLAB PROGRAM DEVELOPED TO PERFORM ERROR ANALYSIS
O N M O D E L E D F IE L D S ...........................................................................................64

C MATLAB PROGRAM DEVELOPED TO ANALYZE A CALIBRATION FILM .67

D MATLAB PROGRAM DEVELOPED TO SEPARATE AND ALIGN
EXPO SURES .............. .................................................................................. 71

E MATLAB PROGRAM DEVELOPED TO AVERAGE FIELD QUADRANTS
AND PRODUCE FINAL MAP OF OFF AXIS RATIOS .................................... 76

L IST O F R E FE R E N C E S ................................................. ............................................ 81

BIO GR APH ICAL SK ETCH .................................................................... ................ 83
















LIST OF TABLES

Table page

1-1 Hypothetical treatment plan with three isocenters............................................11

2-1 Physical characteristics of the Wellhofer mini-MLC .......................................18

3-1 L eaf alignm ent check ..................................................................... ................ 42
















LIST OF FIGURES


Figure page

1-1 L inac SR S ............. .....................................................................3..........

1-2 Linac isocenter ............................................................................................ . 4

1-3 BRW head ring and CT localizer mounted on a Styrofoam head............................5

1-4 A typical single isocenter plan begins with nine equally spaced arcs of
radiation .............. ............................................................................. ...... ...... . 6

1-5 The University of Florida SRS treatment planning algorithm.............................. 8

1-6 M ini-m ultileaf collim ator used in our study. ...................................... ...............9...

1-7 Using a micro-multileaf collimator to approximate circles................................ 10

1-8 Relationship between absorbed dose and collision kerma for a megavoltage
p h o to n b e am ........................................................................................................... 13

1-9 Cross-plane off-axis ratio profile for a 12 mm field .........................................14

2-1 B asic setup for film m easurem ents .................................................. ................ 19

2-2 Epson flatbed scanner and processed radiographic film...................................20

2-3 Off-axis profile film measurements of a 6 cm by 7 cm rectangular field ...........21

2-4 Results of the consistency check performed on the Epson flatbed scanner...........22

2-5 Output of rectangular fields shaped by a miniMLC .........................................23

2 -6 F ilm calib ratio n to o l ............................................................................................... 2 4

2-7 Tool used to separate and align exposures ................ .................................... 25

2-8 Twelve millimeter mMLC circular approximation with even leaves.................28

2-9 Twenty two millimeter mMLC circular approximation with even leaves ..........29

2-10 Thirty millimeter mMLC circular approximation with even leaves...................30









2-11 Twenty millimeter mMLC circular approximation with odd leaves ..................31

2-12 Twenty four millimeter mMLC circular approximation with odd leaves ...........32

3-1 Cunningham model fit to cross-plane data from a rectangular field formed by
th e m M L C .............................................................................................................. 3 5

3-2 Plots of the fitting parameters as a function of field width................................36

3-3 The mMLC leaf positions for a 22 mm circular approximation with even
leave e s.................................................................................................... ........ .. 4 2

3-4 A cross-plane profile through the rectangular open field is determined .............43

3-5 The cross-plane profile is shifted to match the position of the mMLC leaf. .........44

3-6 N negative fi eld profi le .......................................................................... ................ 45

3-7 Spatial m ap of negative field. ........................................................ 46

3-8 A 10 mm mMLC circular field formed by an even number of leaves................47

3-9 A 16 mm mMLC circular field formed by an even number of leaves................49

3-10 A 20 mm mMLC circular field formed by an even number of leaves................51

3-11 A 26 mm mMLC circular field formed by an even number of leaves................53

3-12 A 30 mm mMLC circular field formed by an even number of leaves................ 55















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

METHODS FOR PRODUCING OFF-AXIS RATIO TABLES FROM
MINI-MULTILEAF COLLIMATOR SHAPED CIRCULAR FIELDS FOR INPUT
INTO A STEREOTACTIC RADIOSURGERY TREATMENT PLANNING SYSTEM

By

Brandon Rice

August, 2005

Chair: Frank Bova
Major Department: Nuclear and Radiological Engineering

Radiosurgery beam profiles must be determined accurately to avoid complications

in critical structures adjacent to the tumor. A micro-multileaf collimator (mMLC) may

be used in place of conventional circular collimators to approximate the circular fields

typically used in stereotactic radiosurgery. We investigated two methods for producing

off-axis ratio tables from the mMLC shaped circular beams for input into the in-house

treatment-planning system at the University of Florida.

The first method involves direct measurement of the fields with radiographic film.

The second method uses an empirically-based algorithm to model the off-axis ratios. This

was accomplished by measuring and characterizing several rectangular fields produced

by the mMLC. So-called negative fields) were calculated and added to a model of a

rectangular field to arrive at the final beam model. We were able to accurately model the

off-axis ratios for mMLC shaped circular beams to within (0.5 mm) distance-to-









agreement error in high gradient regions and (3%) dose-dose error elsewhere in the beam

profile for at least 90% of the points in each field. Given its simplicity and accuracy, we

believe this method for modeling off-axis ratios of radiosurgical beam profiles is ideal.














CHAPTER 1
BACKGROUND

Overview of Stereotactic Radiosurgery

Stereotactic radiosurgery (SRS) is a single dose fraction form of radiotherapy that

relies on the ability to accurately deliver a high dose of radiation to a stereotactically

identified target.1 It is an accepted modality for treating a variety of neurologic

conditions such as benign, primary, and metastatic tumors and arteriovenous

malformations.2 Although distinctly different from fractionated radiotherapy or routine

neurosurgery, SRS can be considered a hybrid of both.

Conventional radiotherapy seeks to eliminate diseased tissue by delivering small

doses of radiation in many fractions (normally 1 or 2 fractions per day), through several

static radiation fields.2-3 Fractionated treatments increase damage to the tumor because of

reoxygenation and reassortment of cells into more sensitive phases of the cell cycle.

Inaccurate beam and patient positioning is inherent in common radiotherapy planning and

must be considered when developing a treatment plan. Subclinical, or microscopic

disease processes that do not show up on diagnostic images must also be accounted for.

Consequently, a large amount of healthy tissue is typically encompassed in the treatment

plan to account for positioning errors and microscopic disease.2 This can lead to a large

amount of healthy tissue being irradiated at the target dose and possibly destroyed,

leading to patient complications years down the road. For example, if a 2 cm diameter

brain lesion is planned with a 5 mm margin of error to account for positioning errors, an









additional 7 cm3 of healthy tissue is encompassed in the treatment plan; this additional

healthy tissue represents approximately 70% of the total target volume.

Stereotactic radiosurgery works to avoid these pitfalls of conventional radiation

therapy by administering a large dose of radiation in one fraction so that the dose

distribution has a steep gradient (decreases rapidly spatially) and is conformal to the

disease being treated, typically with submillimeter precision (less than 0.5 mm).

Radiosurgery relies on accurate, stereotactically targeted delivery and steep dose

gradients; it does not rely on taking advantage of radiobiological differences between

normal and diseased tissue.2

Stereotactic radiosurgery also differs from common neurosurgery. Neurosurgery

is an invasive, in-patient process that seeks to physically remove the pathological process.

The neurosurgeon almost immediately knows whether or not a complication will occur.

Radiosurgery, on the other hand, is a non-invasive, out-patient process. The radiosurgeon

does not instantaneously know if there will be delayed complications.2

The idea of SRS was conceived in the late 1940s by a Swedish neurosurgeon

named Lars Leksell.1 His concept was to have many small, focused beams of radiation

impinge on a single target localized in space. He first achieved this with orthovoltage x-

rays and particle beams. He then invented a machine that uses many small sources of

radioactive cobalt-60 focused on a single point in space, to provide a large dose of

radiation to diseased tissue, while causing little or no harm to healthy tissue only

millimeters away. This method of SRS is still performed today using a commercially

available system called the Gamma Knife. Our study focused on another technique that

uses x-rays from a common linear accelerator linacc) to deliver the dose of radiation.









The steep dose gradient of linac radiosurgery is typically achieved by using a basic

concept of geometric optimization: many small beams are applied semi-isotropically

about a target and are directed to converge on the target. A large dose will be delivered

where the beams overlap and will fall off rapidly outside of this area.





















Figure 1-1. Linac SRS. The linac delivers a high dose of radiation with a series of
precisely placed arcs to control the lesion while sparing healthy tissue.

A rigid frame of reference is needed to relate the center of the imaged target to the

linac isocenter. The isocenter is the point of intersection of rotation of the gantry, patient

couch, and collimator. Three-dimensional computed tomography (CT) image

reconstructions are used to stereotactically identify the target within the patient anatomy.

This is often used in conjunction with other imaging modalities, such as angiography (a

2-D imaging modality) and magnetic resonance imaging, to allow for better interpretation

of the patient anatomy.
















YF


Machine
isocenter


XF
Ceiling view Table-to-gantry view

Figure 1-2. Linac isocenter. The isocenter is the point of intersection of the patient
couch, gantry, and collimator.

The stereotactic frame is the rigid frame of reference that can be attached to the

patient couch or pedestal of the linear accelerator and to the patient's skull.1 The Brown-

Robert-Wells (BRW) frame is a commonly used stereotactic head frame2 that is used by

the radiosurgery team at the University of Florida. The frame has three orthogonal axes

that intersect at the center of the circular frame.2 The BRW CT frame is equipped with a

localizer that has nine fiducial rods which appear as dots in each axial slice of the CT

image.1 The location of these fiducial rods is precisely known; therefore it is possible to

relate any point in the CT reconstruction to the reference frame coordinates. Knowing

precisely where the target is located with respect to a frame of reference is critical to

accomplishing the goals of radiosurgery.






































Figure 1-3. BRW head ring and CT localizer mounted on a Styrofoam head.

Delivery Techniques

Several methods are commonly used to shape the small radiation fields and to

produce the treatment plans needed to deliver the conformal, high dose of radiation to the

target. They include the multiple isocenter technique, micro-multileaf collimator

(mMLC) conformal beams, and intensity modulated radiation therapy (IMRT). This

section provides a brief background on the multiple isocenter technique; this technique is

the most preferred and commonly used delivery method and is the method currently used

by the University of Florida SRS team.









The single or multiple isocenter technique for treatment delivery uses one or more

sets of noncoplanar arcs of radiation shaped by small, circular collimators, usually less

than 3 cm in diameter. This method is ideal for producing nearly spherical or elliptical

dose distributions in high-isodose regions (70-80%).2,4-5 The next discussion expands on

delivery techniques used by the University of Florida radiosurgery team.

The width of the dose distribution (size of the radiation field) is approximately the

diameter of the circular collimator projected to isocenter. A typical single-isocenter




(290 \) (270i) 4













A B

Figure 1-4. A typical single isocenter plan begins with nine equally spaced arcs of
radiation. (A-B) Coronal and lateral views of a nine arc set. Angles in (A)
refer to patient table angles.

plan consists of 5-9 equally spaced arcs of radiation; each arc span is 1000 and the arcs

are spaced 200 apart.2'5 These arcs of radiation are directed to overlap in the target region

(at isocenter), taking advantage of the basic concept of geometric optimization discussed

earlier. An isocenter in this context refers to a spherically shaped region used to

approximate target dimensions and should not be confused with the mechanical isocenter









of a linac. The typical characteristic beam starting angle begins 300 vertical off the

superior, and extends 1000 to end at 500 vertical off inferior.2

In most clinical circumstances, the target or lesion will have an elongated,

ellipsoidal, or irregular geometrical shape rather than be perfectly spherical. Sometimes

the target will lie adjacent to a critical structure in the brain such as the optic chiasm. To

account for these irregularities in target shape and to avoid critical structures in the brain,

several parameters can be adjusted to adjust the shape and/or orientation of the dose

distribution. Individual beam start and stop angles and arc weights can be adjusted to

elongate or shift the orientation of the dose distribution.2'5 Adjusting these parameters

does not drastically alter the shape of the dose distribution. For large or irregular target

shapes, more than one isocenter is used to approximate the target. If multiple isocenters

are used on a single target, the number of arcs is reduced from 9 to 5 for each isocenter.

This is done to save time and produces nearly the same gradient as a nine arc set.2'5

Using multiple isocenters results in substantial dose inhomogeneity in the high isodose

regions.4,6

An efficient radiosurgery team such as the one at the University of Florida will

spend approximately fifteen minutes treating a single isocenter using this technique, even

though the actual beam on time for this treatment lasts a few minutes. A collision test

must be performed after each table rotation to ensure the linac does not collide with the

patient upon gantry rotation. A major drawback to this technique is the large amount of

time it takes to treat multiple isocenters. Many cases involve more than one isocenter; on

average the treatment time is one hour, but treatment time can take up to several hours for

the most complicated cases with many isocenters.











Is target adjacent to very radiosensitive neural structure?

Yes No



Maximize dose gradient: Maximize conformalt)
1. Critical structure Inferior, superior, medial or Lesion shape is:
lateral to target use arc elimination tool /
2. Critical structure anterior or posterior to target -
use arc start and stop angle


Maximize conformality
* without degrading optimized dose
gradient


9 equally spaced arcs


Non-ellpooidal tirregu!ar)
Use multiple isocenters


w/principal axis in the:
1. Coronal plane use arc
weighing (elimination tool or
differential collimator sizes)
2. Sagittal plane use arc start
and stop angle tool
3. Axial plane use multiple
isocenters .


Dose selection


Figure 1-5. The University of Florida SRS treatment planning algorithm.

Multileaf collimators (MLCs) may be used to approximate the circular fields

produced by circular collimators.7 Multileaf collimators have been used to shape

radiation fields for many years. They consist of two arrays of tungsten leaves; each leaf

is driven individually by its own motor. Recently, multileaf collimators have been

developed with SRS in mind. The main factor separating conventional MLCs and MLCs

specifically designed for radiosurgery is leaf width. Multileaf collimators designed for

radiosurgery have leaf widths typically on the order of a few millimeters, whereas the

leaf width on conventional MLCs is over a centimeter. Hence, MLCs designed for

radiosurgery are usually referred to as mini or micro-multileaf collimators (mMLC).

St. John investigated how the finite leaf width of the mMLC affects the

conformality and gradient of dose distributions. He showed that mMLC leaf width can









have a considerable effect on the ability to use a mMLC to approximate circular beams.7

For example, a 5 mm circle cannot be approximated accurately using an MLC with 1 cm

wide leaves. The leaves on the mMLC used in our study are approximately 5 mm wide

projected to isocenter. It has been shown that dose distributions produced by mMLCs

with 5 mm wide leaves are virtually indistinguishable from those produced by mMLCs

with 1 mm leaf widths.7


Figure 1-6. Mini-multileaf collimator used in our study. The collimator consists of
twenty pairs of tungsten leaves, each driven by an individual motor and
controlled by a computer.

Circular fields shaped by a mMLC can be approximated using either an odd or an

even set of leaves. The alternative that provides the optimal approximation is governed

by the size of the circle being approximated and mMLC leaf width. In the ideal









situation, the mMLC would be allowed to rotate in between arcs to allow for better

approximation of the circle. Since this is not the case, the center of the circle being

approximated is forced onto a leaf edge or leaf center, whichever is closer. If the center

of the circle is on a leaf edge, an even number of leaves will approximate the circle best.

If the center of the circle is on a leaf center, an odd set of leaves will work best to

approximate the circle (when no collimator rotation is used).



-- --- L




A B

Figure 1-7. Using a micro-multileaf collimator to approximate circles. A) The circle is
centered on a leaf edge so an even set of leaves are used. B) The circle is
centered on a leaf center so an odd number of leaves are used.

If a mMLC is used to shape the radiation beam, in place of conventional circular

collimators, it is possible to drastically reduce the amount of time it takes to treat multiple

isocenter targets. This in turn leads to a more optimal and efficient treatment delivery

and increased patient satisfaction. For example, consider a treatment plan containing

three isocenters (Table 1-1). Using the standard technique with circular collimators, it

takes approximately 50 min for the University of Florida radiosurgery team to complete

the treatment. Before each separate isocenter is treated, the center of the isocenter

(target) is placed at the isocenter of the linac, and the appropriate circular collimator is

mounted onto the linac. Also, a collision test must be performed for each table angle.

If a mMLC is used to deliver the treatment, the geometric center of the target is

placed at the linac isocenter. This eliminates the need to place the center of each









individual isocenter at the isocenter of the linac. It also eliminates the need to switch out

collimators for different isocenters. The mMLC will approximate the shape of the

necessary circle for the isocenter being treated. Also, using a mMLC allows all three

isocenters to be treated from each table angle, so the collision test needs to be performed

a total of five times, as opposed to fifteen times when using the conventional circular

collimators.

Table 1-1. Hypothetical treatment plan with three isocenters
Center Coordinatesa
Isocenter AP Lat Ax Arc Collim. Table Start End
Num (mm) (mm) (mm) num size (mm) angle angle angle
1 12 20 30 130
2 12 55 30 120
1 -19.0 -15.0 -40.0 3 12 340 230 330
4 12 305 240 330
5 12 270 240 330
6 18 20 30 130
7 18 55 30 120
2 -21.3 -12.0 -36.0 8 18 340 230 330
9 18 305 240 330
10 18 270 240 330
11 10 20 30 130
12 10 55 30 120
3 -16.0 -17.5 -43.7 13 10 340 230 330
14 10 305 240 330
15 10 270 240 330
aFor the conventional technique that uses circular collimators, the center coordinates of each
isocenter are placed at the linac isocenter.

Assuming 5 min of initial patient set up, an average of 300 monitor units (MUs) per

arc of radiation, and a dose rate of 400 MUs per min, it would take approximately 16 min

to complete the treatment with the mMLC; this corresponds to a 70% reduction in the

amount of time necessary to complete the treatment. The reduced treatment time

becomes more drastic as more isocenters are introduced. A complicated eight isocenter









plan could take over 2 h to treat using standard circular collimators; the plan would take

about 40 min to treat using a mMLC.

It has been suggested conformal and IMRT techniques are superior to the multiple

isocenter technique for treating complex targets.6'8-11 The fact is multiple isocenter

treatment planning can be difficult and the resulting dose distributions highly user

dependent. Inhomogeneous dose distributions and poor target conformality are the result

of improperly optimized multiple isocenter plans. Wagner investigated the three methods

of radiosurgery treatment delivery by implementing optimization algorithms for each;

this way each technique could be compared fairly.5 The multiple isocenter technique was

optimized with a sphere packing algorithm that determined the placement of variably-

sized spheres within the target volume.5 The result of his work showed the multiple

isocenter technique was able to produce a highly conformal dose distribution while

maintaining a steep dose gradient outside the target volume.5 For general and complex

cases, it was determined that the optimized multiple isocenter technique outperformed the

other methods.

Dosimetry

The absorbed dose to a small mass of material is defined as the amount of energy

absorbed from charged particles liberated by ionizing radiation per unit mass of material.

The kinetic energy released in the medium (KERMA) is defined as the sum of the initial

kinetic energies of all the charged ionizing particles liberated by uncharged particles in a

material per unit mass.1 Since charged particles travel a certain distance before

depositing their energy, the absorbed dose will increase with depth in the material up to a

point where the number of charged particles entering a small volume equals the number

of charged particles leaving that volume. This condition is known as charged particle









equilibrium. The absorbed dose then begins to decrease with depth at the same rate of

the collision KERMA.




S -------------------------



0
0 /KERMA

0




Depth in Material >

Figure 1-8. Relationship between absorbed dose and collision kerma for a megavoltage
photon beam.

For many of the small fields used in radiosurgery, electronic equilibrium does not

exist in the lateral direction. This is due to the fact that the size of many of these small

fields is smaller than the range of electrons being produced. The resulting loss of

electronic equilibrium will occur at every point in the field, even in the center.12 There

are several consequences that must be taken into consideration when dealing with these

small fields: the cross-beam profile is flat over only a small fraction of the full width at

half maximum, the output factor decreases dramatically with a decrease in field size, and

the inverse square law may not hold for large source to point distances.12 In general, a

large percentage of the beam is made up of penumbra.

The SRS treatment planning system at the University of Florida requires the input

of several beam parameters. These include output factors, dose rate at isocenter, tissue

phantom ratios (TPR) at several depths, and off-axis ratios (OARs) for various sized










fields. These parameters are derived from empirical measurements and data. An

accurate assessment of these parameters is essential for minimizing the dose to normal

tissues and reducing the risk of complications due to nonconformal dose distributions.





/ \
0.8 -

0
0O.6 -



0.4 -
,*



0.2 -



-16 -10 -5 0 5 10 15
Crossplane Distance (mm)

Figure 1-9. Cross-plane off-axis ratio profile for a 12 mm field. Approximately 30% of
the field is composed of penumbra.

For any beam centered on the linac's primary axis at the machine isocenter, the

absorbed dose to any point is calculated by Equation 1-1.

ScD 2
Dose(p) = RD MU TPR OAR ROF (1-1)
_STD_

where the reference dose (RD) is the dose per monitor unit, MU is the number of moniter

units delivered, TPR is the tissue-phantom ratio, OAR is the off-axis ratio, ROF is the

relative output factor, ScD is the source to calibration point distance, and STD is the

source to target distance.









Our study focused on measuring and modeling off-axis ratios of mMLC shaped

circular fields. The empirically driven radiosurgery treatment planning system at the

University of Florida requires the input of a two dimensional matrix for each odd or even

mMLC shaped field; these matrices have normalized values (between zero and one) and

contain the necessary off-axis information for each beam. The previous scheme that used

circular collimators required the input of one-dimensional arrays of off-axis ratios. There

was no need for a 2-D matrix because the circular fields were perfectly symmetric.

The penumbra of a radiation field refers to the region at the edge (of the field) over

which the dose changes rapidly.1 The penumbra consists of two components:

transmission penumbra and geometric penumbra. The transmission penumbra is the

region in space exposed to photons that have traveled through part of the collimator. The

geometric penumbra is the region that is irradiated by a spatially finite source.13 The

variation in dose at the edge of the field also depends on the amount of scattered radiation

within the material of interest. The physical penumbra is thus defined as the spatial

distance between two specified points of relative dose, namely 80% and 20% of the dose

to the central axis. 13

The type of detector used to measure the small the fields in radiosurgery should be

considered carefully. Lack of electronic equilibrium in the field will cause large

detectors to yield erroneous results.13 The model detector for radiosurgery dosimetry

should have several unique qualities. The ideal dosimeter should have submillimeter

spatial resolution.12, 14-16 It should have a dose response that is energy, field size, and

dose rate independent.14 It should provide a spatial map of the dose distribution rather

than the dose at a single point, particularly for cases where it is necessary to verify









conformal distributions.15 Also, the detector should be linear, reproducible, and stable. 14

There is no ideal dosimeter for radiosurgery, but there are several that approach the ideal.

Radiographic film has been used extensively in radiosurgery dosimetry. It is a high

resolution dosimeter that records information in two dimensions. Some possible

problems that are encountered with film are variations in sensitivity and dose response

which results from variable processing conditions.14 This problem may arise when trying

to compare the doses from two separate films; the problem can be overcome by

developing films from the same film batch under identical processing conditions.

Radiographic film also suffers from spatial non-uniformity.14 Spatial variations on

the order of 2% are typically seen with radiographic film. A flat area of the measured

field may be averaged to reduce the error due to these variations. This correction is not

possible when taking penumbra measurements, because the field is changing rapidly over

this distance. A final problem encountered with radiographic film is its over-response to

low energy photons due to the high atomic number components that make up the

film.12,14 The problem becomes worse as the field size and depth of measurement are

increased, resulting in systematic error of the measured dose in the tail of the penumbra.12

This should not be a significant problem for the small fields typically used in

radiosurgery.

Radiochromic film is another type of film that can be used for radiation therapy

dosimetry; this type of film does not require chemical developing so it avoids variations

in sensitivity and dose response that result from variations in processing conditions.14

Radiochromic film suffers from disadvantages such as poor reproducibility14 and long

developing time. It takes approximately 24 h for the typical radiochromic film to









develop, so it is not as convenient as conventional radiographic film. Another

disadvantage of using radiochromic film is the fact that it does not respond well to low

doses of radiation. Radiochromic film also needs to be handled carefully and protected

from exposure to sunlight and fluorescent lighting.

Radiographic film does not perfectly fit all the criteria for an ideal dosimeter for

SRS, but it comes close. It outperforms common dosimeters such as diodes, ionization

chambers, and thermoluminescent dosimeters (TLDs) because film has high spatial

resolution and the ability to make a simultaneous measurement of a spatial map of dose.

To accurately measure the penumbra of small fields, the detector needs to be small

compared to spatial variations in dose. Volume averaging can become a significant

problem in even the smallest available ionization chambers.16 The low cost and general

performance characteristics of radiographic film make it a suitable detector for

performing relative dosimetric measurements on the small fields used in radiosurgery.














CHAPTER 2
OFF-AXIS RATIO MEASUREMENT OF MINI-MLC APPROXIMATED CIRCLES

Introduction

One possible method of producing the off-axis ratio lookup tables to be used in the

radiosurgery dosimetry system is to directly measure them. Although this method can be

time consuming, the results are valuable in that they can be used to provide a means for

comparison with model calculations. This chapter will go through the methodology used

to measure the circular fields formed by mini-multileaf collimators and convert the

measurement to a useful off-axis ratio table.

Materials and Methods

A Varian CLINAC 600C 6MV Radiotherapy Accelerator was used to produce the

radiations fields that were measured. A miniature multileaf collimator (Wellhofer

Dosimetrie, Schwarzenbruck, Germany) was used to shape the radiation fields. The

mMLC has 20 pairs of tungsten leaves that project 4.95 mm wide at the isocenter of the

linac. Each leaf is stepping-motor driven along a curved path so that ends of the leaves

match the beam divergence.17

Table 2-1. Physical characteristics of the Wellhofer mini-MLC
Number of leaves 20
Leaf width at isocenter 4.95 mm
Leaf height 8.0 cm
Maximum field size at isocenter 9 X 10 cm2
Source-to-collimator distance (mounted on the Varian 600C) 65.0 cm
Leaf travel over central axis 2.0 cm
Leaf velocity 1 cm/s
Mass 35 kg










Kodak X-OMAT V radiographic film was used to measure the radiation fields.

Each piece of film was placed at the isocenter of the linac on top of a four centimeter slab

of solid water which was used to provide sufficient backscatter. A one and a half

centimeter slab of solid water was placed on top of the film to provide buildup. The film

was located at the depth of maximum absorbed dose. The film was developed in a Kodak

M35A-M X-OMAT processor; the temperature of the processing chemicals during film

development was always 32.8+0.50C. Films from measurements performed on the same

day were taken from the same film batch and were developed under identical processor

conditions to avoid film variations in dose response and sensitivity.


Figure 2-1. Basic setup for film measurements. Kodak radiographic film is sandwiched
between two slabs of solid water.









A calibration exposure was produced before a set of film measurements was made

to provide a means to convert optical density to dose. The calibration exposure was

developed along with the other films; a new calibration exposure was made each day new

measurements were taken. The calibration exposures typically contained five or six

exposures in the range often to eighty monitor units (MUs). The mMLC was set to

produce six centimeter square fields for the calibration exposures.

After the calibration film was exposed, the mMLC was set to create circular field

approximations using an even number of open leaves; the circular fields produced had

diameters ranging from 10 cm to 30 cm in 2 cm increments. Six exposures of each

circular approximation were made on the same sheet of film. The calibration film and

measurement films were developed in the Kodak processor and scanned with an Epson

Expression 1600 flatbed scanner at 0.1 mm/pixel resolution. The films were scanned at a

high resolution to preserve the inherent high spatial resolution of the detector.


Figure 2-2. Epson flatbed scanner and processed radiographic film.










Several films were scanned on a Lumisys L575 densitometer at 0.124 mm/pixel

resolution; this was done to determine if the flatbed scanner would produce the same

results as the laser densitometer. It was determined that the flatbed scanner produced

similar results to the laser densitometer.





Densitomneter
08 I I Flatbed Scanner 0.8

S- Flatbed Scanner
I Densitometer
0.4 0 4

02 012

-40 -30 -20 -10 0 10 20 30 40 -50 -40 -30 -20 -10 0 10 20 30 40 50
A Crossplane Distance (rnmm) B Inplane Distance (mm)

Figure 2-3. Off-axis profile film measurements of a 6 cm by 7 cm rectangular field. The
profiles are used to compare the flatbed scanner with the densitometer. A) is
a cross-plane (direction of leaf travel) profile and B) is an in-plane (direction
of leaf width) profile.

Consistency and uniformity checks were also performed on films scanned by the

Epson scanner. A consistency test was performed to determine if the scanning resolution

was consistent throughout the length of the scanner. A ruler was scanned at 0.1 mm/pixel

resolution and the results were analyzed with a tool developed in Matlab. The tool

prompts the user to click on each centimeter marker on the image of the ruler; the

program then determines the distance between each marker. The results of the

consistency check showed the scanner was performing acceptably, with an average error

less than 0.1 mm, which is the size of each pixel. A uniformity check was performed on

unexposed radiographic film; the standard deviation of absolute dose was determined to

be less than 2%.













0.5 -

0.4 -

0.3 -

0.2 -

-0.1 -


02




-0.3 -

-0.4 -

-0.5 -

0 5 10 15 20 25
Position Along Scanner (cm)

Figure 2-4. Results of the consistency check performed on the Epson flatbed scanner.
The average absolute value of position error along the scanned ruler is 0.06
mm.

Once all the films were scanned and saved on the computer, a calibration curve was

created to convert optical density to dose. Output factors for rectangular fields produced

by the mMLC were measured beforehand with an Exradin 14P ion chamber (parallel

plate, 0.002 cm3 active volume, mm window thickness and 1 mm window-collector

gap); the output factor for a 6 cm square field was needed to convert monitor units to

dose. The linac is calibrated to have a 10 cm square field deliver 1 cGy of radiation dose

to water at dmax (1.5 cm) per monitor unit. The product of output factor and monitor units

yields the amount of dose delivered to the film. The output factor for a 6 cm square field

produced by the mMLC was determined to be 0.96; this value is in good agreement with

the output of a 6 cm square field (0.957) when the mMLC is not attached.











0.9 -



0.85 -

0.8 -
S0.7
0.7 -


0.65 /

0.6 -

0.55
0.5 ----
0 1 2 3 4 5 6
Equivalent Square (cm)

Figure 2-5. Output of rectangular fields shaped by a miniMLC. The output of the field
produced by the mMLC is relative to the output of a 10 cm square field
produced by the primary collimator jaws alone.

A film calibration tool developed in Matlab by St. John was modified to read in

exposures saved in tagged image format (tiff) and to convert pixel value to optical

density. Equation 2-1 was used to relate pixel value to optical density for an 8-bit image.


OD=- loglo( ), (2-1)
255

where OD is optical density and P is the pixel value (between 0 and 255 for an 8-bit

grayscale image). The program prompts the user to input the output factor, number of

calibration exposures and corresponding monitor units delivered for each exposure. The

user is prompted to select each exposure file and click on the background and exposure

peaks in an optical density histogram. The average background is determined and









subtracted from each exposure. A second degree polynomial is fit to a plot of dose

versus optical density.

Raw Image Calibration Curve (BKG-corrected)
80

-40 ---------- 8---- ---- ---------

03
0)

--------- --!---------------L------
20- -----------



200 400 600 800 0 0.5 1 1.5
Optical density (OD)
The calibration curve is Dose = 2.218e+001"OD^2 + 3.698e+001"OD + 0.374
15 ------------------------ ---------------- --- -----

B ckground Exposure Peak
S10 ------- -------- I--------- I------- .-- -------.-- I ------ .---
10 -------- -------




0 0.2 0.4 0.6 0.8 1 1.2
Optical Density

Figure 2-6. Film calibration tool. The user is prompted to click on the background and
exposure peaks for each separate exposure.

Another program was developed in Matlab to allow the user to separate, center,

and align multiple exposures on a single film. The user inputs the resolution the film was

scanned at and number of exposures on the film. After clicking on the (Get Image)

button, the user is prompted to select the image file. The program displays the scanned

image of the film and prompts the user to box in each exposure with the mouse. After an

exposure has been selected, the program displays only the selected image. The user then

zooms in on the top, right, bottom, and left sides of the image, respectively. Once a side








is zoomed in on, the user is prompted to click on the center of the edge of the field.
Having these four points, the program calculates the equations for the lines of the central
axis and the point where the two lines intersect (the center of the beam). The image is
then rotated slightly so the central axis of the beam is perpendicular to the sides of the
image. Since the center of the beam is known, the program is able to crop the image so
the center of the beam is exactly in the center of the image.




S Enter nurnbe oft exposures












*


Figure 2-7. Tool used to separate and align exposures. The scanned image contains six
26 mm mMLC circular approximations.
A program was developed to convert optical density to dose and to normalize the
dose on a scale between zero and one. Since each quadrant of the mMLC circular field is
symmetric about the central axis, the program is able to split up each intensity matrix into
four separate but equal quadrants. Since six exposures were made for each field, a total









of 24 field quadrants were produced for each circular field. These quadrants were

averaged together by the Matlab program to produce the final result, a two dimensional

map of off-axis ratios. Since multiple exposures were averaged together to produce the

2-D maps, it was possible to calculate a 2-D map of statistical uncertainty for each field.

The actual error between the measured fields and the real dose distribution is probably

less than predicted by statistical uncertainties. This is due to the fact that a spatial error in

one quadrant will be somewhat compensated by the resulting opposite spatial error in the

adjacent quadrant.

Results

Some typical two dimensional maps of measured off-axis ratios are shown along

with their statistical uncertainties in Figures 8-12. The statistical uncertainty has no units

(off-axis ratios are unitless) and is generally less than 10% in high gradient (penumbra)

regions and less than 2% in low gradient regions. A large uncertainty in the penumbra

region is expected due to the rapid falloff of dose. The plots display one quarter of the

field with the origin located at the central axis of the field.

Discussion

Circular approximations of radiation fields form by a mMLC were measured with

radiographic film; the processed films were input into several Matlab programs designed

to separate and center the exposures, convert optical density to dose, and to average field

quadrants to arrive at the final result. It was determined the process of separating and

aligning exposures introduces the most error to the final result. Some measured off-axis

ratio tables had much lower uncertainty than others. This is because the off-axis ratio

tables with the lower uncertainty came from exposures that were better aligned.






27


Exposures may be over or under-rotated by as much as a few degrees. Proper

alignment and centering of the exposures is crucial to arriving at an accurate result. A

new, less user-dependent, method of aligning the exposures is currently being developed.

The new method uses an edge-detection algorithm to find the top, bottom, left, and right

edges of the field. Once these edges are determined, the exposures can be rotated

accurately, with errors less than 10.












W 90% isodose line
50% isodose line
W 10% isodose line
















5 10 15 20 26
Crossplane Distance (mm)


10 15 20
Crossplane Distance (mm)


25 Uncertainty


Figure 2-8. Twelve millimeter mMLC circular approximation with even leaves. A) is a
contour plot of the average measurements and B) is a map of the statistical
uncertainty for each pixel in the measurement.


0.07

0.06

0.05

0.04

0.03

0.02















20


15


10


10
Crossplane


15 20
Distance (mm)


25


-- 20
E

- 15
O1

= 10


0 5 10
Crossplane


15 20
Distance (mm)


25 Uncertainty


Figure 2-9. Twenty two millimeter mMLC circular approximation with even leaves. A)
is a contour plot of the average measurements and B) is a map of the statistical
uncertainty for each pixel in the measurement.


W 90% isodose line
50% isodose line
W 10% isodose line





---
- (5



-e-^ '^ \ '
\ .4


___ i __ i 1 i __ i __ i


0.09

0.08

0.07

0.06

0.05

0.04































5 10 15 20 25
Crossplane Distance (mm)


5 10 15 20
Crossplane Distance (mm)


0.12


0.1


0.08


0.06


0.04


0.02



25 Uncertainty


Figure 2-10. Thirty millimeter mMLC circular approximation with even leaves. A) is a
contour plot of the average measurements and B) is a map of the statistical
uncertainty for each pixel in the measurement.


W 90% isodose line
50% isodose line
W 10% isodose line


-9-
as


I I


I I












25 WI 90% isodose line
560% isodose line
WI 10% isodose line
-20



215





16








5

0.1
10






0.02





0 0
0 5 10 15 20 25 Uncertainty
B Crossplane Distance (mm)

Figure 2-11. Twenty millimeter mMLC circular approximation with odd leaves. A) is a
contour plot of the average measurements and B) is a map of the statistical
uncertainty for each pixel in the measurement.





























5 10 15 20 25
Crossplane Distance (mm)


0 5 10 15 20
Crossplane Distance (mm)


0.12


0.1


0.08


1I1 IVl6


0.04


0.02


0
25 Uncertainty


Figure 2-12. Twenty four millimeter mMLC circular approximation with odd leaves. A)
is a contour plot of the average measurements and B) is a map of the statistical
uncertainty for each pixel in the measurement.


W 90% isodose line
50% isodose line
W 10% isodose line


Th 1
U~














CHAPTER 3
A ROBUST ALGORITHM TO MODEL OFF-AXIS RATIOS OF MINI-MLC
APPROXIMATED CIRCULAR FIELDS

Introduction

Our goal was to develop a tool to model the off-axis ratios of mMLC shaped

circular fields since the direct measurement of these fields is time consuming. The

model is then used to produce two-dimensional lookup tables for input into the

radiosurgery treatment planning system. One alternative method that could be used to

produce the off-axis ratio tables is Monte Carlo simulation, but it would take more time

to run Monte Carlo simulations than to perform direct measurements. Also, the

validation of Monte Carlo requires the determination of the parameters and then

validation of results. Since Monte Carlo does not provide results in a timely manner,

another method was considered. This chapter discusses a simple technique that employs

model constructions to produce the off-axis ratio tables.

Methods and Materials

The model construction is based on the negative field technique using the basic

Cunningham model.18 This technique involves the application of negative fluences to a

rectangular open field to produce the desired dose distribution. The algorithm that was

developed to help us achieve our final result is as follows:

* Measure the off-axis ratios of rectangular fields with widths varying over the range
of interest (10-30 mm).

* Fit the Cunningham model to the measured off-axis ratios and acquire the
necessary fitting parameters.









* Create a model of a rectangular field using the fitting parameters acquired in
Step 2.

* Calculate the negative field and subtract the negative field from the rectangular
model.

* Repeat Step 4 if there is more than one negative field.

Rectangular fields formed by the mMLC were measured with Kodak X-OMAT V

film; the width of the measured rectangular fields varied from 10 mm to 30 mm. Optical

density was converted to dose and normalized using the results from a film calibration

curve obtained by the film calibration tool. One dimensional cross-plane (in the direction

of leaf motion) and in-plane (in the direction of leaf width) profiles through the central

axis were determined from the data set. The Cunningham model was fit to cross-plane

and in-plane profiles using the curve fit tool in Matlab. The fitting algorithm employs a

nonlinear least-squares technique. The Cunningham model for off-axis ratio (OAR) at a

point r mm from the central axis is


OAR=1-0.5*expJ ( r for r d
Sp 2 2
A( W (3-1)
OAR= t+(0.5-t)exp 2 r- ) for r >2
S2 2


where aL and a2 are the fit parameters, wd is the field width defined as the distance

between the fifty percent isodose points, r is the radial distance from the center of the

field, p is the width of the penumbra, and t is the transmission through the mMLC when

all leaves are closed. Transmission was measured by St. John and Wagner and was

determined to be 0.7%.5 Meeks also measured transmission; he determined the

maximum transmission between leaf sides to be 0.5% and between leaf ends to be










1.2%. 17 The two transmission measurements averaged together agree with St. John's and

Wagner's measurements.

The width of the penumbra, defined as the distance between the 80% and 20%

isodose points, was measured and determined to be 0.33 mm. A typical example of the

Cunningham model, fit to cross-plane data, is shown (Figure 3-1). There are several

other functions alternative to the Cunningham model that have been used to fit penumbra

profiles. 12, 19 These functions are continuous and have only one fitting parameter, but

they don't seem to fit the data as well as the discontinuous Cunningham model. In

reality, a continuous function offers no advantage over a discontinuous one because our

data points are sampled discretely.









0.8 -
Alphal = 1.96
2 Alpha2 = 2.10


0

0.4



0.2 -
02




0 5 10 15 20 25
Crossplane Distance (mm)

Figure 3-1. Cunningham model fit to cross-plane data from a rectangular field formed by
the mMLC. The rectangular field is 18 mm wide.
















28

2as Alphal = 2.18 : f.w.<18mm
Alphal = 1.65 : f.w.>=18mm
24 -




1 2-
<






14

12


10 12 14 16 18 20 22 24 26 28 30
Field Width (mm)
A




28 -

2G -
Alpha2 = -0.014 x (field width) + 2.3
24 -
24













12
1 I I I I






10 12 14 16 18 20 22 24 26 28 30
Field width (mm)
B


Figure 3-2. Plots of the fitting parameters as a function of field width. A) A step
function is fit to the al parameters. B) A straight line is fit to the a2
parameters. These fits are input into a Matlab program and are used to create
models of rectangular fields.









The fitting parameters for each rectangular field were plotted against field width

(Figure 3-2). A discontinuity is noted when ai is plotted against field width; this

discontinuity was also seen by St. John. The discontinuity is the result of adding more

open leaves to the field and the field approaching lateral equilibrium. A step function

was fit to the calculated at parameters. The plot of the a2 parameters versus field width is

linear, so a straight line was fit to the data.

The fits of the alpha parameters were input into a Matlab program that was

developed to produce the final models of the mMLC circular fields. Once the alpha

parameter fits are input into the program, the program is able to produce a 2-D model of a

rectangular field. When the user enters mMLC leaf positions into the program, the

program creates a model of a rectangular field with width equal to the widest open leaf.

The length of the field is equal to the product of the number of open leaves in the field

and the leaf width. Leaf width is a variable within the program, so the program can be

applied to mMLCs of varying leaf widths. Whether or not an odd or even number of

leaves is used to approximate the circle is also variable within the program.

After the rectangular field is created, any negative fields that are to be subtracted

from the rectangular field are calculated. The size, position, and weight of the negative

field had to be considered carefully. If any of these negative field variables were not

accurate, the final model would not agree well with measurement. We also had to

account for the possibility of overlapping negative fields for cases that required the

subtraction of more than one negative field.

The first step in calculating the negative field is to determine the cross-plane profile

through the center of the leaf where the first negative field will be placed (Figure 3-4).









The cross-plane profile is derived directly from the rectangular model in the case where

only one negative field is applied to the rectangular open field. The total number of

negative fields to be applied is equal to the number of open leaves in the quarter field

minus one. If more than one negative field is to be applied, the negative field that is

calculated first is the largest and is located on the open leaf furthest from the center of the

field.

After the cross-plane profile through the rectangular field has been determined, the

profile is shifted to match the actual mMLC leaf position for the circular field. For

example, in the case of a 22 mm mMLC shaped circle formed with an even number of

leaves, the actual positions of the two open leaves in the quarter field are 10.8 mm and

8.7 mm, respectively (Figure 3-3).

After the cross-plane profile through the modeled rectangular field (10.8 mm by 10

mm in the quarter field) has been determined (Figure 3-4), the cross-plane data is shifted

back, from 10.8 mm to 8.7 mm to match the position of the second leaf (Figure 3-5).

After the cross-plane profile has been shifted, the difference between the rectangular

cross-plane profile and shifted cross-plane profile is determined. The result is a profile of

the negative field in the cross-plane direction (Figure 3-6). The previous steps are

repeated for the in-plane profile; the product of the 1-D negative field cross-plane and in-

plane profiles yields a spatial map of the negative field.

In cases where more than one negative field is applied, the negative cross-plane

profiles are a weighted combination of cross-plane profiles derived from the rectangular

field and the resultant field after the previous negative field was subtracted. This is done

so regions of the rectangular field are not subtracted twice by overlapping negative fields.









There is no need to weight the in-plane profiles as such because of the order the negative

fields are subtracted. The in-plane profiles are derived directly from the most current

version of the model. For example, if one negative field had been subtracted from the

rectangular field, the next in-plane profile would be derived from the field that resulted

after the subtraction of the first negative field.

Results

The negative field technique was applied to models of rectangular fields to create

models of the mMLC shaped circular fields that were measured. Several metrics were

used to compare the final, modeled fields, with the measured fields. The first metric is

qualitative; it consists of overlapping contour plots of the measured and modeled fields.

By visually inspecting this plot, one can quickly determine if the model agrees well with

measurement.

The next two metrics provide a means to quantitatively compare the measured and

modeled dose distributions. In low gradient regions of the field (regions where the

off-axis ratio is not changing rapidly), the error between the modeled and measured fields

is best represented by the absolute dose difference between the two fields. In high

gradient regions (the penumbra), the dose difference does not provide a good means of

quantifying the error. Instead, the error in high gradient regions is represented by the

distance between points of equal relative dose. We chose 5% and 1 mm to be the error

criteria for our dose-difference and distance-to-agreement tests, respectively.

A program was developed in Matlab to perform error analysis on the modeled

fields using the measured fields as the reference. The program determines the dose-

difference and distance-to-agreement error for nearly all points in the modeled field. The

program then places each error point into either a dose-difference or distance-to-









agreement histogram depending on how well if fit the respective error criteria. The

results are two-dimensional histograms of dose-difference and distance-to-agreement

error. The level of agreement between the modeled fields and reference fields is

observed by these histograms. If all points had less than 1 mm distance-to-agreement

error or less than 5% dose-difference error, then the modeled field has met the error

criteria. The error analysis program also determines the percentage of error points that

have either a dose-difference error less than 3% or distance-to-agreement error less than

0.5 mm.

A third histogram is created to display the dose-difference error and distance-to-

agreement error together for all points. It is obvious from inspecting the 3-d histograms

that high gradient regions can have a large dose-difference error, but small distance-to-

agreement error. And vice versa, low gradient regions can have a large distance-to-

agreement error, but small dose-difference error. One can also see from these histograms

whether or not the error criteria are met.

The contour plots and histograms show that the modeled even fields agree well

with what was measured. All of the even models meet or exceed the error criteria. In

fact, at least 96% of the error points lay within 3% dose error or 0.5 mm distance error for

the even fields presented in our study.

Discussion

A tool was developed, to model in 2-D, the off-axis ratios of mMLC shaped

circular fields, based on a small number of measurements. The tool proved to be fast; it

can produce a 2-D off-axis ratio table in less than 1 s. The amount of time necessary to

perform measurement and analysis to determine the parameters for input into the tool

takes less than half a day. The tool proved to be accurate. All of the modeled fields









shaped with an even number of leaves met the error criteria (5% dose-dose error in low

gradient regions and 1 mm distance-to-agreement error in high gradient regions). Many

of the models had the majority of its errors be much less than the error criteria. The tool

is also flexible. Variables such a leaf width and output size can be readily changed

within the program.

Some of the modeled odd fields did not agree well with what was measured. The

error is not inherent in the model. It was determined that some of the leaves on the

mMLC were not properly aligned when the odd field measurements were made. This

effect caused the actual leaf positions to be different from the specified leaf positions by

about 0.7 mm on average.

To check leaf alignment, some leaf pairs of interest were opened individually and

made to project 2 cm wide at isocenter. The physical distance between the leaves were

measured with a micrometer and projected to isocenter. The results of these

measurements (Table 3-1) show the leaves were not properly aligned. Leaf alignment

was performed about a week before the even field measurements were made. Odd field

measurements were performed over six months later; it is quite possible that leaf position

accuracy was affected over this time period. Sastre-Padro et al. performed stability and

reproducibility measurements on an MLC and reported a maximum leaf shift of 0.27 mm

over three months.20









Table 3-1. Leaf alignment check
Leaf Paira Physical Separation Length Projected Absolute Error
(mm) to Isocenterb (mm) (mm)
8 12.68 + 0.05 19.51 + 0.08 0.49
9 12.39 + 0.05 19.06 + 0.08 0.94
10 12.05 + 0.05 18.54 + 0.08 1.46
11 12.18 0.05 18.74 0.08 1.26
12 12.60 + 0.05 19.38 + 0.08 0.62
13 12.71 + 0.05 19.55 + 0.08 0.45
aEach leaf pair was opened individually and the distance between them was measured with a
micrometer. bThe length projected to isocenter should be 20 mm.














ft '





A Crossplane Position (mm) B Crossplane Position (rm)

Figure 3-3. The mMLC leaf positions for a 22 mm circular approximation with even
leaves. A) All leaves in the open field are shown. B) The leaves in the
quarter field are shown. Because of symmetry, it is necessary to model only a
quarter of the field.














II 0% isodose line
50% isodose line
I 20% isodose line










-02 / Crossplane Profile
10 .--- - -



-- - - - --5- -- - --- -



oi


5 87 10 15 20
Crossplane Distance (mm)


1



0.8
o


o



0.4


5 10 15 20 25
Crossplane Distance (mm)


Figure 3-4. A cross-plane profile through the rectangular open field is determined. A) A
contour plot of the rectangular field. The black dashed lines are the
hypothetical mMLC leaf positions for the rectangular model. B) The derived
cross-plane profile.







44




25-i
W 80% isodose line
60% isodose line
W 20% isodose line






I_15- -- ----- -- --- ----














5 8 7 10 15 20 25
Cr s llane Dstance (m) profile

A













0.6 '







00


S5 8.7 10 15 20 25
Crossplane Distance (mm)



Figure 3-5. The cross-plane profile is shifted to match the position of the mMLC leaf
A) Contour plot of the rectangular field and actual mMLC leaf positions for a
22 mm circular field. B) The cross-plane profile before and after the shift.


















1 -



0.8





aL
I 02 0 6



O4
04








0
0 5 10 15 20 25
Crossplane Distance (mm)
A











08


Negative Field Profile
06 -








02 \




0 5 10 15 20 25
Crossplane Distance (mm)
B


Figure 3-6. Negative field profile. A) The difference between the cross-plane profiles is
determined. B) The resulting cross-plane profile of the negative field is
shown.




































10 15
Crossplane Distance (mm)


5 10 15 20
Crossplane Distance (mrm)


L 0.25


OAR


Figure 3-7. Spatial map of negative field. A) A contour plot of the negative field. B) A
color-wash plot. The negative field is applied to a rectangular field to produce
the final model of a 22 mm mMLC circular approximation.


W 40% isodose line
20% isodose line
W 10% isodose line












I ^


20


E
E
101







5



0
0







47



II I
I 90% isodose line
25 60% isodose line
I 10% isodose line


20 -

E
(,
2 15

5
0








0 6 10 15 20 26

A Crossplane Distance (mm)

600 1200

00 -------------------------------- 1000---- ---------------

400 --------------- ---------------- 800 ------- ------- -----------

200 ---------- -------------- 400 ------- -----------


100 --- --- --------------- 200 -- -----------
100 I' I200


06 0 -
0 0.01 0.02 0.03 0.04 0 0.2 0.4 0.6 0.8
Dose Error Distance Error (mm)
B

Figure 3-8. A 10 mm mMLC circular field formed by an even number of leaves.
A) Contour plots of measured (solid line) and modeled (dashed line) fields.
B) Histograms of dose-difference error and distance-to-agreement error
comparing the modeled field to the reference measured field.
C) Three-dimensional histogram of distance error and percent error. All
points have either dose error less than 3.25% or distance error less than 0.6
mm.







48








.....


















4 10

C

Figure 3-8. Continued















25




20




- 16













0-
U


- - _-_-_- _


200


0
0*


Dose Error (%)
B


1000


800


- i- -
- - - -


0u-u- --


Distance Error (mm)


Figure 3-9. A 16 mm mMLC circular field formed by an even number of leaves.
A) Contour plots of measured (solid line) and modeled (dashed line) fields.
B) Histograms of dose-difference error and distance-to-agreement error
comparing the modeled field to the reference measured field.
C) Three-dimensional histogram of distance error and percent error. All
points have either dose error less than 4.7% or distance error less than 0.9
mm.


5 10 15 20 25
Crossplane Distance (mm)


- ----------- -----------



































Figure 3-9. Continued







51





W 90% isodose line
25 50% isodose line
m 10% isodose line


20 -

E

10

E ,

il 15 --








0 5 10 16 20 25
Crossplane Distance (mm)
A

600 2000



,,-- ,,-- - -- :- -- 1 0 0 - - -- -

9LL_ LLI
00 --- ------------------------- 1500 ----------






0 0 0
200 ------ ---r------------- 2000 i- -- -------------








0 1 2 3 4 0 0.2 0.4 0.6 0.8
Dose Error (%) Distance Error (mrn)
B

Figure 3-10. A 20 mm mMLC circular field formed by an even number of leaves.
A) Contour plots of measured (solid line) and modeled (dashed line) fields.
B) Histograms of dose-difference error and distance-to-agreement error
comparing the modeled field to the reference measured field.
C) Three-dimensional histogram of distance error and percent error. All
points have either dose error less than 3.5% or distance error less than 0.75
mm.





52











0









Figure 3-10. Continued
.-'
U- . .
U '
"w6







53




I 90% isodose line
25 560% isodose line
10% isodose line


20 -




E 1






01 161,

0-, I 1I---- I
0 5 10 15 20 25
Crossplane Distance (mm)
A


1200 1400
1200
1000l------------------ 1-00 I---------- 100

800 ------------ 1000 -
CD 800 ---------- ----- ----
0 1000

400 --__-----_--

200 - -200 -- -


0 1 2 3 0 0.2 0.4 0.6 0.8
Dose Error (%) Distance Error (mm)
B

Figure 3-11. A 26 mm mMLC circular field formed by an even number of leaves.
A) Contour plots of measured (solid line) and modeled (dashed line) fields.
B) Histograms of dose-difference error and distance-to-agreement error
comparing the modeled field to the reference measured field.
C) Three-dimensional histogram of distance error and percent error. All
points have either dose error less than 2.8% or distance error less than 0.7
mm.







54
























3. '6 -- O f

4 10
C

Figure 3-11. Continued













I I 90% isodose line
25 50% isodose line
W 10% isodose line


20 -
E

2 15 ---------- ------ -
0 .
3000 1500



















2500 ------------- --------------

..,2000 --- -- - - ------- .., 1000 ------ I------- ------- 11-------
0 ---- -














500 -------------t------- ------
O- c O \

0 0j

0 10 15 20 26
DoseCrosslane Error (%Distance (mm)
A

3000 --- 1500

2500 '

S2000 I- 1000 I
eld formed by an even number of leaves.














A) Contour plots of measured (solid line) and modeled (dashed line) ields.
=; 1500--------.-------.---------------I I
I I I I
SThree-dimensional histogram of distance error and percent error. All
points000 have either dose error less than 4% or distance error less than 0.8 mm.
6 100 -- -




0 1 2 3 4 0 0.2 0.4 0.6 0.8







56











S .. i .: '.








0



3 \ -"- 2* :.. .



4 10

C

Figure 3-12. Continued














CHAPTER 4
CONCLUSION

The work presented in this report tackled the problem of how to produce two

dimensional off-axis ratio tables for circular fields approximated by a mini-multileaf

collimator. The use of a mMLC in conjunction with the multiple isocenter delivery

technique will allow for more efficient treatment delivery and increased patient

satisfaction while maintaining conformal dose distributions with steep dose gradients.

Two methods for producing the tables of OARs were investigated. The first

method was direct measurement of the fields with radiographic film. This method proved

to be rather tedious and time consuming because many films needed to be exposed,

processed, and analyzed. The second method investigated was an empirical one based on

the negative field technique. A tool was developed in the Matlab programming

environment to create models of the mMLC circular fields based on the negative field

technique where negative fluences are applied to an open field to achieve the desired dose

distribution. The program requires the user to input several OAR fitting parameters that

can be quickly obtained from a small number of measurements. Once the fitting

parameters are input into the program, the program will output a spatial map of off-axis

ratios for any specified mMLC shaped circular field.

The modeled fields produced by the program were compared to the fields measured

with radiographic film. A tool was developed to perform error analysis on the modeled

fields with respect to a reference (measured) field. This was accomplished by

determining the dose-difference error in low gradient regions of the field and distance-to-









agreement error in high gradient regions of the field. More than 90% of points in the

modeled even fields had dose-difference error less than 3% or distance-to-agreement

error less than 0.5 mm. The modeled odd fields did not agree as well with measurement

because the mMLC leaves were not moving to the specified location when the

measurements were made. It should be noted from this experience that the leaves on the

mMLC must be properly aligned and their positions accurately known for this method to

work properly. Overall, the tool proved to be fast, flexible, easy to use, and accurate.

Future work will include showing that the model can be applied to fields located

off-axis and at different depths. It is expected that the model will work fine in these

situations because it has been shown that the penumbra of small fields does not change

much with depth or off-axis position. Future work on the tool used to create the modeled

fields will include having the tool be able to model any field shaped by the mMLC, as

opposed to just being able to model circular fields.
















APPENDIX A
MATLAB PROGRAM DEVELOPED TO MODEL THE FIELDS

msgbox('Select command window and enter inputs');

numleaf=input('Enter the number of leaves in the quarter field (default=1):');
if isempty(numleaf)
numleaf=1;
end

for i=1:numleaf
prompt = sprintf('Enter the position of leaf %d in mm: ',i);
leafpos(i)=input(prompt);
end


%THESE VARIABLES MAY BE CHANGED TO ACCOMMODATE DIFFERENT MLC, ODD OR
EVEN
%FIELDS, ETC

%EVEN OR ODD NUMBER OF LEAVES (1 FOR EVEN, 0 FOR ODD)
even = 1;

%LEAF WIDTH (mm)
Ifwidth = 4.95 %5.0;

scale = 0.1; %(mm/pixel)

%SIZE OF RESULTING MATRIX (mm)
sizematrix = 27.5;
pixels = sizematrix/scale;


w1 = leafpos(1)*2; %FIELD LENGTH

%FIELD WIDTH
if even==1 %EVEN CASE
w2 = Ifwidth numleaf 2;
else
%ODD CASE
w2 = Ifwidth (numleaf *2 -1);
end


%These are the fit parameters for the rectangular fields.
%THESE PARAMETERS CAN BE MODIFIED FOR A PARTICULAR MLC/LINAC
COMBINATION

if wl<18
alphal cross = 2.18;










else
alphal cross = 1.65;
end
if w2<18
alphalin = 2.18;
else
alphal in = 1.65;
end
alpha2cross = -.0135*w1+2.29;
alpha2in = -.0135*w2+2.29;



t=.0075; %TRANSMISSION CONSTANT
p=2.4; %PENUMBRA CONSTANT

%Produce Square Field IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
matrx=zeros(pixels,pixels);
for i=1:pixels
m=scale*i;
if m<(wl/2)
crossratio(i)=(1-.5*exp(-alphalcross*(w 1/2-m)/p));
else
crossratio(i)= (t+(.5-t)*exp(-alpha2cross*(m-(w1 /2))/p));
end
end

for i=1:pixels
m=scale*i;
if m<(w2/2)
inratio(i)=(1-.5*exp(-alphal in*(w2/2-m)/p));
else
in ratio(i)= (t+(.5-t)*exp(-alpha2in*(m-(w2/2))/p));
end
end

matrx=zeros(pixels,pixels);
1=1;
for i=1:pixels
k=1;
for j=1:pixels
matrx(i,j)=crossratio(k)*in ratio(l);
k=k+1;
end
1=1+1;
end
newmatrx=matrx;



%BEGIN CALCULATING AND SUBTRACTING NEGATIVE FIELDS

for q=(numleaf-1):-1:1 %NUMLEAF-1 IS THE NUMBER OF NEGATIVE FIELDS

center = round((leafpos(q+1) + (leafpos(1)-leafpos(q+1))/2)*10);
if even==1










center = (q+.5)*Ifwidth*10; %EVEN CASE
else
center = q*lfwidth*10; %ODD CASE
end
centery=round(centery);

wl= leafpos(q+1);
if even==1 %EVEN CASE
w2new = Ifwidth*(q) 2;
else
%ODD CASE
w2new = 2* Ifwidth (q-.5);
end

centery=round(centery);
square=newmatrx(centery,:);
norm=max(square);
for i=1:pixels
m=scale*i;
if m<(leafpos(q+1))
newcross(i)=norm*(1 -.5*exp(-alphal cross*(wl -m)/p));
else
newcross(i)=norm*(t+(.5-t)*exp(-alpha2cross*(m-wl)/p));
end
end

square2=newmatrx(:,centerx);
norm2=max(square2);
for i=1:pixels
m=scale*i;
if m<(w2new/2)
newin(i)=norm2*(1-.5*exp(-alphal in*(w2new/2-m)/p));
else
newin(i)=norm2*(t+(.5-t)*exp(-alpha2in*(m-(w2new/2))/p));
end
end

if q==(numleaf-1) DON'TT WEIGHT FIRST NEGATIVE FIELD

endl=pixels;
for i=1:end1

ptl = newmatrx(centery,i);
pt2= newcross(i);
if (ptl-pt2)>0
negcross(i)=ptl-pt2;
else
negcross(i)=0;
end
end
else %WEIGHT OTHER NEGATIVE FIELDS
endl=pixels;
for i=1:end1

wtl =(2*centerx-i+1)/(2*centerx);
wt2= i/(2*centerx);










pta=newmatrx(centery,i);
ptb=matrx(centery,i);
ptl = wt2*pta+wtl*ptb;
pt2= newcross(i);
if (ptl-pt2)>O
negcross(i)=ptl-pt2;
else
negcross(i)=O;
end
end
end

negin=zeros(1 ,pixels);
for i=1 :pixels

ptl =newmatrx(i,centerx);
pt2= newin(i);
if (ptl-pt2)>O
negin(i)=ptl-pt2;
else
negin(i)=O;
end

end

maxl=max(negin);
max2=max(negcross);
if max1 >max2
avemax=max1;
else
avemax=max2;
end

[z,end2]=size(negin);
negmatrx=zeros(end2,end1);
1=1;
for i=1:end2
k=1;
for j=1:end1
negmatrx(i,j)=(negcross(k)*negin(l))/avemax;
k=k+1;
end
1=1+1;
end

if q==2
negl=negmatrx;
else
neg2=negmatrx;
end

for i=1:end2
for j=1:end1
newmatrx(i,j)=newmatrx(i,j)-negmatrx(i,j);
end
end







63


end

X=(0:(pixels-1)).*scale;

figure
[C,h] = contour(X,X,newmatrx,[.9 .5 .1],':');hold on;
%[C,h] = contour(X,X,odd28mm,[.9 .5 .1]); clabel(C,h);
Xlabel('Crossplane Distance (mm)');
Ylabel('lInplane Distance (mm)');
legend(h,'90% isodose line','50% isodose line','10% isodose line');
















APPENDIX B
MATLAB PROGRAM DEVELOPED TO PERFORM ERROR ANALYSIS ON
MODELED FIELDS

clear X X1 finaldist finaldose gammadose gammadist histomatrix;

%TYPE THE NAME OF THE REFERENCE DOSE PROFILE IN THE WORKSPACE
refdose = even30;

fieldwidth=23; %max 23
Ingthofr=(fieldwidth/2)+1 0;

numpix=275;

k=1;
t=1;
z=1;
%CHECK EACH PIXEL VALUE FOR DOSE-DOSE ERROR

for theta = 0:0.017 : pi/2

for r = 0: 0.1 : Ingthofr
r2=r*1 0;
y = round(r2 *sin(theta)) +1;
x = round(r2 *cos(theta)) +1;

dose_err(k)=abs(newmatrx(y,x)-refdose(y,x));

match=refdose(y,x); %REFERENCE VALUE FOR DOSE WE WANT TO FIND
DISTANCE TO

if r<4
start=0;
else
start=r2-40;
end

for r3=start:0.5:(r2+45)
y2 = round(r3 *sin(theta)) +1;
x2 = round(r3 *cos(theta)) +1;
temp=newmatrx(y2,x2);
if temp<=match
dist(k)=sqrt((y2-y)A2+(x2-x)A2)*.1;
break
else
dist(k)=4.5;
end
end

gammadist=dist(k)/1;










gammadose=dose_err(k)/0.05;

if gammadose finaldose(t)=dose_err(k);
t=t+1;
else
finaldist(z)=dist(k);
z=z+1;
end

k=k+1;

end
end

dose err=dose err.*100;
%CREATE 2-D PLOT OF BINNED ERRORS
prcntbin=0.25;
distbin=0.1;

rndprcnt=prcntbin/2;
rnddist=distbin/2;

[m,n]=size(dist);

histomatrix=zeros(50,40);
Ighistomatrix=zeros(50,40);

for i=1:n
temple =dist(i);
temp2=dose_err(i);

temple =temp1 +rnddist;
temp2=temp2+rndprcnt;

temple =round(templ/distbin);
temp2=round(temp2/prcntbin);

if temp2<40
histomatrix(templ ,temp2)=histomatrix(templ ,temp2)+1;
end
end

[m,n]=size(histomatrix);
for i=1:m
forj=1:n
if histomatrix(i,j)~=0
temp = histomatrix(i,j);
Ighistomatrix(i,j)=log(temp);

end
end
end


%DETERMINE THE RATIO OF PTS THAT HAVE LESS THAN .5mm DISTANCE ERROR
%OR 3% DOSE ERROR.










cnt=0;
su mhisto=sum(sum(histomatrix));
for i=1:m
forj=1:n
if (i<6)|(j<13)
cnt=cnt+histomatrix(i,j);

end
end
end
ratio=(cnt/sumhisto)*100

distaxes=(0:distbin:(distbin*(m-1)));
prcntaxes=(0:prcntbin:(prcntbin*(n-1)));

meshz(prcntaxes(1,1:35),distaxes(1,1:35),lghistomatrix(1:35,1:35));
xlabel('Percent Error','Rotation',15,'Fontsize',14);
ylabel('Distance Error (mm)','Rotation',-45,'Fontsize', 14);
zlabel('Log(Frequency)','Fontsize', 12);
set(gca,'zscale','linear');view(153,54);

figure

N=zeros(100,1);
[N,X]=hist(finaldose,100);

P=zeros(20,1);
[P,X1]=hist(finaldist,20);

subplot(1,2,1)
bar(X*100,N); grid on; Xlabel('Dose Error (%)'); Ylabel('Frequency');
pbaspect([1 1 1]);%Title('Histogram');
subplot(1,2,2)
bar(X1 ,P); grid on; Xlabel(' Distance Error (mm)'); Ylabel('Frequency');
pbaspect([1 1 1]);%Title('Histogram');
















APPENDIX C
MATLAB PROGRAM DEVELOPED TO ANALYZE A CALIBRATION FILM

function calib_separate

%CALIB_SEPARATE
% This function opens calibration tiff files, selected by operator, one at a time
% and plots a histogram of optical density. By clicking on the background peak
% and exposure peak for each file, a dose calibration curve is calculated.
% Coefficients a, b, & c are returned for the equation dose = a*OD^2 + b*OD + c.

msgbox('Select command window and enter inputs');

num=input('Enter the number of calibration exposures (default=8):');
if isempty(num)
num=8;
end
output=input('Enter the output factor (default=1.0):');
if isempty(output)
output=1;
end
MU(1,1)=0;
for i=1:num
prompt=sprintf('Enter MUs for exposure # %d: ',i);
MU(i+1,1)=input(prompt);
end

OD=zeros(num+1,1);

dose=MU.*output
%film_type=input('Enter the film type(XV or EDR),(default{XV}):');
%if isempty(film_type)
% film_type='XV';
%end
%film_type

forj=1:num
prompt=sprintf('Select file number %d',j)
[scanfile,direc]=uigetfile('*.tif, prompt);

if scanfile==0
return;
end

cd(direc);
fidl=fopen(scanfile);

if fid1 ~=-1
raw=imread(scanfile,'tif);
else










msgbox('File not found\n');
return;
end

if j==1
fl=figure;
set(f1 ,'position',[120 40 600 480]);
end
[A,B]=size(raw);
raw2=double(raw);

for i=1:A
for z=1:B
if raw2(i,z)==0
raw2(i,z)=1;


raw2=-log10(raw2/255); %Convert to OD

N=zeros(500,1);
% HIST(raw) creates a matrix with histograms of each column in the matrix(raw)
[N1,X]=hist(raw2,500);
% Combine all histograms into a single histogram (X contains bin centers)
for i=1:B
N(:,1)=N(:,1) + N1(:,i);
end

M = max(max(raw2)); %Use M to set axis limits
Mn=max(max(N));
% Draw the plots
pl = subplot(2,2,1);imagesc(raw),colormap gray, axis square;Title('Raw Image');
p2 = subplot(2,2,2); plot(0,0)
p3 = subplot(2,1,2);
%[N,X]=hist(raw1,50);hist(raw1 (25,:),50);hold on
bar(X,N); grid on; Xlabel('Optical Density'); Ylabel('Frequency');
hl=uicontrol('style','text','position',[100 230 400 20 ],'BackgroundColor','w');
set(h1,'string','****Click on BACKGROUND peak****');
[X,Y]=ginput(1);
BKGpts(j)=X;
set(p3,'xlim',[0 M],'ylim',[0 Mn]);pbaspect([2 1 1]);
set(h1,'string','****Click on Exposure peak****');
[X,Y]=ginput(1);
ODpoint(j)=X;
% Leave row 1 col 1 for zero dose


OD(j+1)=ODpoint(j);
delete(hl);
end
BKGpts
BKG=mean(BKGpts);
%std(BKGpts(:,1))/BKG;
OD(1)=BKG;
OD;
dose;


% = standard deviation










for t=l:num+l
OD(t)=OD(t)-BKG;
end
OD
dose
OD_size=size(OD)
dose_size=size(dose);
dose=output.*MU;
Mdose=max(dose);
MaxOD=max(OD);


facts=polyfit(OD,dose,2);
a=facts(1);
b=facts(2);
c=facts(3);
for i=1:(MaxOD)*50
dose_fit(i,1)=a*(i/50)A2 + b*(i/50) + c;
Xaxis(i)=i/50;
end
dose_fit;
%*****linear fit
%facts=polyfit(OD(2:6),dose(2:6),1);
%a=0;
%b=facts(1);
%c=facts(2);
%for i=1:num
% dose_fit(i,1)=b*OD(i,1) + c;
%end
axes(p2);
cla
plot(Xaxis,dose_fit,'color','r','LineWidth',2);

%plot(dose_fit,OD/1000,'color','r','LineWidth',2);
hold on;

plot(OD(1 :num+l),dose(1 :num+1),'*');grid on;
YLabel('Dose (cGy)');XLabel('Optical density (OD)');Title('Calibration Curve (BKG-corrected)');

% Write the fitting parameters at the bottom of the plot

hl=uicontrol('style','text','position',[60 225 500 20 ],'BackgroundColor','w');

result=sprintf('The calibration curve is Dose = %7.3e*ODA2 + %7.3e*OD + %4.3f ',a,b,c);
set(hl,'ForeGroundColor','r','FontWeight','Bold','string',result);

% Write calibration factors
fid=fopen('calib.info','w');
fprintf(fid,'Calibration film: %s\n',pwd);
fprintf(fid,'%7.3e \t %7.4f \t %7.4f\n',a,b,c);
fprintf(fid,'%7.4f',BKG);
set(findobj('tag','cal_a'),'string',a);
set(findobj('tag','cal_b'),'string',b);
set(findobj('tag','cal_c'),'string',c);
fclose(fid);







70


% write dose and OD to file for Excel
fid=fopen('plot3.txt','w');
for i=0:num-1
j=num-i;
fprintf(fid,'%f\t%f\n',OD(j),dose(j));
end

fclose(fid);
















APPENDIX D
MATLAB PROGRAM DEVELOPED TO SEPARATE AND ALIGN EXPOSURES


function varargout = separate_exp(varargin)
% SEPARATE_EXP M-file for separate_exp.fig
% SEPARATE_EXP, by itself, creates a new SEPARATE_EXP or raises the existing
% singleton*.
%
% H = SEPARATE_EXP returns the handle to a new SEPARATE_EXP or the handle
% the existing singleton*.
%
% SEPARATE_EXP('CALLBACK',hObject,eventData,handles,...) calls the local
% function named CALLBACK in SEPARATE EXP.M with the given input arguments


to


% SEPARATE_EXP('Property','Value',...) creates a new SEPARATE_EXP or raises the
% existing singleton*. Starting from the left, property value pairs are
% applied to the GUI before separate_exp_OpeningFunction gets called. An
% unrecognized property name or invalid value makes property application
% stop. All inputs are passed to separate_exp_OpeningFcn via varargin.
%
% *See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one
% instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES

% Edit the above text to modify the response to help separate_exp

% Last Modified by GUIDE v2.5 22-Feb-2005 05:11:31

% Begin initialization code DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name', mfilename, ..
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @separate_exp_OpeningFcn, ..
'gui_OutputFcn', @separate_exp_OutputFcn, ..
'gui_LayoutFcn', [] ...
'gui_Callback', []);
if nargin & isstr(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end

if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code DO NOT EDIT

% --- Executes just before separate_exp is made visible.
function separate_exp_OpeningFcn(hObject, eventdata, handles, varargin)


).










% This function has no output args, see OutputFcn.
% hObject handle to figure
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% varargin command line arguments to separate_exp (see VARARGIN)

% Choose default command line output for separate_exp
handles.output = hObject;

% Update handles structure
guidata(hObject, handles);
set(hObject,'MenuBar','figure');

% UIWAIT makes separate_exp wait for user response (see UIRESUME)
% uiwait(handles.figurel);

% --- Outputs from this function are returned to the command line.
function varargout = separate_exp_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% hObject handle to figure
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)

% Get default command line output from handles structure
varargout{1} = handles.output;

% --- Executes during object creation, after setting all properties.
function numexp_CreateFcn(hObject, eventdata, handles)
% hObject handle to numexp (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles empty handles not created until after all CreateFcns called

% Hint: popupmenu controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc
set(hObject,'BackgroundColor','white');
else
set(hObject,'Backgrou ndColor',get(0,'defaultUicontrolBackgroundColor'));
end

% --- Executes on selection change in numexp.
function numexp_Callback(hObject, eventdata, handles)
% hObject handle to numexp (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)

% Hints: contents = get(hObject,'String') returns numexp contents as cell array

% --- Executes on button press in getfile.
function getfile_Callback(hObject, eventdata, handles)
% hObject handle to getfile (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)

num=get(handles.numexp,'Value'); %returns selected item from numexp
scale=str2double(get(handles.scale,'String')); %returns contents of scale as a double











numpixel=floor(85/scale); %was 55
check=rem(numpixel,2); %force number of pixels to be even for ease of calculations
if check==0
else
numpixel=numpixel+1;
end

[scanfile,direc]=uigetfile('*.tif','Select image');
cd(direc);
fidl=fopen(scanfile);
if fid1 ~=-1
l=imread(scanfile,'tif);

else
fprintf('File not found\n');
return;
end
cla;

uiwait(msgbox('Box in each exposure. Hold shift key for square.','modal'));

for i=1:num
cla;
axes(handles.image)

%Allows the user to separate different exposures
newimage=imcrop(l);

%Check to see if user wants to center and align exposures
num2=get(handles.align,'Value');

if num2==1

cla;
contimage=double(newimage);

imagesc(contimage),colormap gray, axis equal, axis tight;

uiwait(msgbox('Zoom in and click on each fiducial starting at the top and working
clockwise.','modal'));

for z=1:4
k=20;
zoom on;
k=waitforbuttonpress;

if k==0
[x(z),y(z)]=(ginput(1));
end
zoom out;
end

%Determine the equations for the lines for the central axis
ml = (y(2)-y(4))/(x(2)-x(4));
bl =y(2)-ml *x(2);










if x(3)==x(1) %so don't divide by zero
xint=round(x(3))
m2=inf;
else
m2=(y(3)-y(1))/(x(3)-x(1));
b2=(y(3)-m2*x(3));

%Determine intersection (center of beam)
xint=round((b2-bl)/(ml-m2));
end
yint=round(ml*xint+bl);

% %if odd
% yint=yint-25;
% %end
%
%Crop the image so the center of beam is center of image
[A,B]=size(newimage);

if ((xint<(B/2))&(yint>(A/2))) % first quadrant
if xint>(A-yint)
12=imcrop(newimage,[xint-(A-yint),2*yint-A,2*(A-yint),2*(A-yint)]); % closer to upper half
else
12=imcrop(newimage,[1,yint-xint,2*xint,2*xint]); % closer to left edge...xmin set to 1
instead of 0 b/c dimensions work out that way.
end
elseif ((xint<(B/2))&(yint<(A/2))) % third quadrant
if xint>yint
12=imcrop(newimage,[xint-yint,1,2*yint,2*yint]);
else
12=imcrop(newimage,[1,yint-xint,2*xint,2*xint]);
end
elseif ((xint>(B/2))&(yint>(A/2))) % second quadrant
if ((B-xint)<(A-yint))
12=imcrop(newimage,[2*xint-B,yint-(B-xint),2*(B-xint),2*(B-xint)]);
else
12=imcrop(newimage,[xint-(A-yint),2*yint-A,2*(A-yint),2*(A-yint)]);
end
else % fourth quadrant
if yint<(B-xint)
12=imcrop(newimage,[xint-yint,1,2*yint,2*yint]);
else
12=imcrop(newimage,[2*xint-B,yint-(B-xint),2*(B-xint),2*(B-xint)]);
end
end

%Next, determine how much the image needs to be rotated

[A,B]=size(12);
X=A/(2*m2)+A/2;
angle=rad2deg(atan(2*X/A-1));
angle

%Now rotate the image
13=imrotate(12,-angle,'bicubic','crop');
cla;










[A,B]=size(13);
A=A/2-n umpixel/2;
B=B/2-numpixel/2;
14=imcrop(13,[B,A,numpixel-1,numpixel-1]);

newd irec=strcat(direc,'exposure');
num2=num2str(i);
newdirec=strcat(newdirec,n um2);
newdirec=strcat(newdirec,'.tif);

%Write each separate exposure to a file
imwrite(14,newdirec,'tif);

else
newdirec=strcat(direc,'exposure');
num2=num2str(i);
newdirec=strcat(newdirec,num2);
newdirec=strcat(newdirec,'.tif);

%Write each separate exposure to a file
imwrite(newimage,newdirec,'tif);
end
%
end
cla;

% --- Executes during object creation, after setting all properties.
function scale_CreateFcn(hObject, eventdata, handles)
% hObject handle to scale (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles empty handles not created until after all CreateFcns called

% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc
set(hObject,'BackgroundColor','white');
else
set(hObject,'Backgrou ndColor',get(0,'defaultUicontrolBackgroundColor'));
end

function scale_Callback(hObject, eventdata, handles)
% hObject handle to scale (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of scale as text
% str2double(get(hObject,'String')) returns contents of scale as a double

% --- Executes on button press in align.
function align_Callback(hObject, eventdata, handles)
% hObject handle to align (see GCBO)
% eventdata reserved to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)


% Hint: get(hObject,'Value') returns toggle state of align
















APPENDIX E
MATLAB PROGRAM DEVELOPED TO AVERAGE FIELD QUADRANTS AND
PRODUCE FINAL MAP OF OFF AXIS RATIOS

%This program is called by separate_exp when the Combine button is pressed.
% The user is prompted to select six exposures. User needs to click on the background
% for each exposure. Average background is subtracted from these exposures
% and exposures are stored in an array. Program then splits each image
% matrix into four equal quadrants and stores them in a separate array.

%num=get(handles.numexp,'Value'); %returns selected item from numexp

%Enter scale and number of exposures here (mm/pixel)
scale=0.1;
numexp=2;

%ENTER FIT PARAMETERS TO CONVERT OD TO DOSE
a=-14.82;
b=224.21;
c=2.861;
for t=1:numexp

[scanfile,direc]=uigetfile('*.tif,'Select image');
cd(direc);
fidl =fopen(scanfile);
if fid1 ~=-1
l=imread(scanfile,'tif);

else
fprintf('File not found\n');
return;
end

13=double(l);
[numpixel,B]=size(13);

%CONVERT TO OPTICAL DENSITY
13=-log 10((13+1)/256);
12(t,:,:)= 13;
A=B;

N=zeros(500,1);
[N1,X]=hist(13,500);

% Combine all row histograms into a single histogram (X contains bin centers)

for i=1:A
N(:,1)=N(:,1) + N1(:,i);
end










bar(X,N); grid on; Xlabel('Optical Density'); Ylabel('Frequency');
pbaspect([1 1 1]);Title('Histogram');

%Select background
[bkg(t),y]=ginput(1);

end

%Determine and subtract average background
avebkg=sum(bkg(:))/numexp;

12(:,:,:)=12(:,:,:)-avebkg;

pix=numpixel/2;

%Convert optical density to dose
for t=1:numexp
for i=1:numpixel
for j=1 :B
12(t,i,j) = a*(12(t,i,j))^2 + b*12(t,i,j) + c;
if 12(t,i,j)<0
%12(t,i,j)=0;
end
end
end
end

%Declare array for the sections

Combine=zeros((numexp*4),pix,pix);
q=1;
%Split up the four images into four quadrants and store in array
for m=1 :numexp

P=zeros(pix,pix);
Q=zeros(pix,pix);
R=zeros(pix,pix);
S=zeros(pix,pix);

%Upper left hand corner
for i=1:pix
for j=1:pix
P(i,j)=12(m,i,j);
end
end
P=fliplr(P);
Combine(q,:,:)=P;
q=q+1;
%Upper right hand corner
for i=1:pix
for j=(pix+1):(numpixel)
k=j-(pix);
Q(i,k)=12(m,i,j);
end
end
Combine(q,:,:)=Q;










q=q+1;
%Lower left hand corner
for i=(pix+1):(numpixel)
for j=1:pix
k=i-(pix);
R(k,j)=12(m,i,j);
end
end
R=fliplr(R);
R=flipud(R);
Combine(q,:,:)=R;
q=q+1;
%Lower right hand corner
for i=(pix+1):(numpixel)
for j=(pix+1):(numpixel)
k=i-(pix);
l=j-(pix);
S(k,I)=12(m,i,j);
end
end
S=flipud(S);
Combine(q,:,:)=S;
q=q+1;

end

%Determine the average
sum=zeros(pix,pix);
Temp=zeros(pix,pix);
for i=1:(4*numexp)
for j=1:pix
for k=1:pix
Temp(j,k)=Combine(i,j,k);
end
end
sum=sum+Temp;
end

Average=zeros(pix,pix);
Average=sum(:,:)/(numexp*4);
UncAverage = Average;
max(max(Average));

Average=flipud(Average);

%SUBTRACT THE AVERAGE MINIMUM
[A,B]=size(Average);
k=1;
for i=A-25:A
1=1;
for j=B-25:B
sample(k,l)=Average(i,j);
1=1+1;
end
k=k+1;
end










minim=mean(mean(sample));
Average = Average(:,:) minim;

%Normalize the data to the maximum value
maxim=max(max(Average));
Average=Average(:,:)/maxim;
temp=Average;

sumdev=zeros(pix,pix);
%DETERMINE THE STATISTICAL ERROR OF THE AVERAGE
Temp2=zeros(pix,pix);
for i=1:(numexp*4)
for j=1:pix
for k=1:pix
Temp(j,k)=Combine(i,j,k);
end
end

for j=1:pix
for k=1:pix
Temp2(j,k)=(Temp(j,k)-UncAverage(j,k))^2;
end
end
sumdev=sumdev+Temp2;
end

Unc=zeros(pix,pix);
Unc=sqrt(sumdev(:,:)/(numexp*4-1));
Unc=Unc./maxim;

%PLOT HISTOGRAM OF UNCERTAINTY
N=zeros(500,1);
[N1,X]=hist(Unc,500);

% Combine all row histograms into a single histogram (X contains bin centers)

for i=1:275
N(:,1)=N(:,1) + N1(:,i);
end
N=N(:,1)/75625;

bar(X,N); grid on; Xlabel('Relative Uncertainty'); Ylabel('Relative Frequency');
pbaspect([1 1 1]);Title('Histogram'); xlim([0 .15]);

Xax=(0:274)./1 0;
Xtr=(274:-1:0)./10;
figure
imagesc(Xax,Xtr,Unc);colorbar;
Xlabel('Crossplane Distance (mm)');
Ylabel('lnplane Distance (mm)');
set(gca,'Ydir','normal');

figure
[C,h] = contour(Xax,Xax,Average,[.9 .5 .1]); clabel(C,h);
Xlabel('Crossplane Distance (mm)');
Ylabel('lnplane Distance (mm)');










legend(h,'90% isodose line','50% isodose line','10% isodose line');figure


% flip=flipud(temp);
% C=cat(1,temp,flip);
%
% flip=fliplr(temp);
% flip2=flipud(flip);
% D=cat(1 ,flip,flip2);
%
% final=cat(2,D,C);
%
%
% imagesc(final);
% colorbar
figure
line=Average(1,:);
plot(line,':b');















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BIOGRAPHICAL SKETCH

Brandon K. Rice was born in October 1981, in Duluth, MN. He attended the

University of Wisconsin-River Falls from 1999-2003, earning a Bachelor of Science

degree in physics (with honors) in May 2003. During his undergraduate work, he served

as a research assistant in the Department of Physics; he served as president of Phi Sigma

Kappa national fraternity; he was a member of Sigma Pi Sigma physics honor society,

and Phi Kappa Phi national honor society; and was an active member in the local chapter

of the Society of Physics Students.

He entered graduate school at the University of Florida in August 2003. There he

completed a Master of Science degree in medical physics. During his work, he served as

a graduate research assistant in the Radiosurgery and Radiobiology Laboratory at the

University of Florida Brain Institute and served as vice president of the Society of Health

and Medical Physics Students. He is also a member of the American Association of

Physicists in Medicine and the American Nuclear Society.