Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Intensity modulation of therapeutic photon beams using a rotating multileaf collimator Otto, Karl 2003

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2003-853969.pdf [ 14.77MB ]
Metadata
JSON: 831-1.0085723.json
JSON-LD: 831-1.0085723-ld.json
RDF/XML (Pretty): 831-1.0085723-rdf.xml
RDF/JSON: 831-1.0085723-rdf.json
Turtle: 831-1.0085723-turtle.txt
N-Triples: 831-1.0085723-rdf-ntriples.txt
Original Record: 831-1.0085723-source.json
Full Text
831-1.0085723-fulltext.txt
Citation
831-1.0085723.ris

Full Text

Intensity Modulation of Therapeutic Photon Beams Using a Rotating Multileaf Collimator by Kar l Otto B . S c , M c G i l l University, 1995 M.Sc . , M c G i l l University, 1997 A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard The University of British Columbia A p r i l 2003 © Kar l Otto, 2003 In present ing t h i s thes i s i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thes i s for s c h o l a r l y purposes may be granted by the head of my department or by his or her representat ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes i s for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permiss ion. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Abstract This thesis describes the development and implementation of a novel method of delivering intensity modulated radiation therapy ( IMRT) that provides greater accuracy and spatial resolution than currently available methods. Through improvements in multileaf collimator ( M L C ) based fluence generation, a dose distribution may be generated that conforms more closely to the tumour target volume. Healthy tissue surrounding the target volume w i l l therefore receive less dose, reducing the probability of side effects and allowing the physician to increase the prescribed tumour dose (dose escalation). M L C based I M R T techniques are well established but suffer several physical limitations. Dosimetric spatial resolution is limited by the M L C leaf width, interleaf leakage and tongue-and-groove effects degrade dosimetric accuracy and the range of leaf motion limits the maximum deliverable field size. Based on observations from a linear systems model of dosimetric spatial resolution degradation it is hypothesized that, by rotating the entire M L C between each sub-field, improvements w i l l be obtained in spatial resolution, dosimetric accuracy and maximum deliverable field size. To generate arbitrary fluence maps in this way, a series of unique algorithms were developed that are capable of determining the necessary rotated M L C segments. These I M R T fields may be delivered statically (with the collimator rotating to a new position in between sub-fields) or dynamically (with the collimator rotating and leaves moving simultaneously during irradiation). A full description of the rotational leaf motion algorithms is provided. 11 A n analysis of the rotating leaf motion calculation algorithms with focus on radiation efficiency, the range of collimator rotation and number of segments is provided. The mechanical and radiation producing characteristics of standard linear accelerators under collimator rotation conditions are also investigated. The technique is evaluated by characterizing the ability of the algorithms to generate rotating leaf sequences for desired fluence maps. Comparisons are also made between our method and conventional sliding window and step-and-shoot techniques. Results show improvements in spatial resolution, reduced interleaf effects and maximum deliverable field size over conventional techniques. Clinical application of these enhancements can be realized immediately with static rotational delivery although improved control of the M L C w i l l be required for dynamic delivery. i i i Table of Contents Abstract ii Table of Contents iv List of Tables x List of Figures xi Acknowledgements xxiv Chapter 1 Introduction 1 1.1 Radiation Therapy 1 1.1.1 Historical Background 2 1.1.2 Goal of Radiation Therapy 3 1.1.3 Tumour/Healthy Tissue Response to Dose 3 1.1.4 Treatment Planning 5 1.1.4.1 C T / M R Imaging 6 1.2 Dose Deposition 7 1.2.1 Photon Interactions 7 1.2.2 Electron Energy Transfer - Stopping Power 9 1.2.3 Fluence-Dose relationship 9 1.2.4 Pencil Beam Dose Deposition 11 1.3 Basic Dose Delivery Techniques 12 1.3.1 Multiple Fields 13 1.3.2 Arcs 15 1.4 Fie ld Shaping 15 1.4.1 Mult i leaf Collimator 16 iv 1.5 Intensity Modulated Radiation Therapy 18 1.5.1 Complex Fluence Generation 18 1.5.1.1 Multiple M L C Fields 20 1.5.2 Plan Optimization 21 1.5.3 Leaf Sequencing 22 1.5.4 Delivery 22 1.6 Factors Affecting Spatial Resolution in Dose Delivery 22 1.6.1 Imaging Resolution 23 1.6.2 Patient Immobilization 23 1.6.3 Delivery Technique 24 1.7 Thesis Objectives and Summary 24 1.7.1 Spatial Resolution Degradation 25 1.7.2 I M R T Delivery With M L C Rotation '. 25 1.7.3 Leaf Mot ion Derivation 26 1.7.4 Rotating M L C Evaluation 26 Chapter 2 Spatial Resolution Degradation 27 2.1 Linear Systems Theory 28 2.1.1 Spatial Invariance 29 2.1.2 Application To Imaging Systems 29 2.1.3 Fourier Mode l O f Spatial Resolution Degradation (Spatial Frequency).. 30 2.1.3.1 Modulation Transfer Function Concept 31 2.1.4 Sampling Theory 33 2.1.4.1 Aliasing 34 2.1.4.2 Nyquist Criterion 34 2.1.5 Application to Dose Delivery Systems 36 2.2 Theory: Dose Transfer Function of Spatial Resolution Degradation 36 2.2.1 M L C Sampling 37 2.2.2 M L C Function 37 v 2.2.3 Dose Spread Kernel 38 2.2.4 Dose Transfer Function 38 2.2.5 Spatial Frequency Representation O f Dose 39 2.3 Method: Application to 2-dimensional Dose Distributions 41 2.3.1 Study of Circular P T V Shaping 45 2.3.2 Study O f P T V Shapes 47 2.4 Results: Circular P T V Shaping 48 2.4.1 M L C Effects - Analytic Representation 48 2.4.2 Dependence On Leaf Width and Circle radius 49 2.4.3 Dose Spread Kernel 51 2.5 Results: Clinical Simulation O f Resolution Degradation 52 2.5.1 Collimator Angle Dependence 54 2.5.2 M L C Leaf Width - Aperture Shaping (. 56 2.5.3 Dose Spread Kernel - Blurring 58 2.6 Discussion: Implications of Resolution Degradation 59 2.7 Discussion: Application O f The D T F To I M R T 61 Chapter 3 I M R T Delivery With M L C Rotation 63 3.1 Rotational Delivery Method 64 3.2 Enhancements To I M R T With M L C Rotation 67 3.2.1 Spatial Resolution 68 3.2.2 Interleaf Effects 69 3.2.3 Maximum Deliverable Field Size 74 3.3 Mechanical Characteristics 76 3.3.1 Mult i leaf Collimator 76 3.3.2 Collimator Rotation 78 3.3.3 Collimator Angle Reproducibility 79 3.3.4 Center of Rotation 80 3.4 Linac Control 82 vi 3.4.1 Static Delivery 83 3.4.1.1 Dose Linearity 83 3.4.2 Dynamic Delivery 85 3.4.2.1 Rotation Speed Reproducibility 85 3.4.2.2 Dose Rate Stability 86 3.4.2.3 Collimator Rotation Stability 86 3.5 Feasibility of I M R T with Collimator Rotation 92 Chapter 4 Leaf Motion Derivation 94 4.1 Conventional Techniques - Review 95 4.1.1 Step A n d Shoot (Static) 95 4.1.2 Sliding Window (Dynamic) 97 4.2 Rotational Technique Leaf Mot ion Derivation 98 4.2.1 Increased Complexity 98 4.2.2 Analytic Model 101 4.2.3 Optimization Methods 104 4.2.3.1 Gradient Based Methods 105 4.2.3.2 Stochastic Methods 107 4.3 Rotational Leaf Motion Optimization 107 4.3.1 Preprocessing 108 4.3.2 Constraints 108 4.3.3 Fixed Parameters 109 4.3.4 Initialization 110 4.3.5 Optimization I l l 4.3.6 Segment Doubling 112 4.3.7 Dynamic Error Margin Control 113 4.3.8 Rotational Leaf Motion S o ftware 113 4.4 Algori thm Considerations 113 Chapter 5 Rotating M L C Evaluation 115 v i i 5.1 Method 116 5.1.1 Algorithm Assessment 116 5.1.2 Fluence Generation Parameters 119 5.1.3 Conventional I M R T Fluence Generation 119 5.2 Dosimetric Evaluation 119 5.2.1 Delivery 121 5.3 F i l m Dosimetry 121 5.3.1 Conversion To Dose 122 5.3.2 F i l m Calibration Technique 123 5.3.2.1 Enhanced Dynamic Wedge (EDW) 123 5.3.2.2 Calibration Curve 124 5.4 Results 126 5.4.1 Algorithm Characteristics 126 5.4.2 Reproducibility 128 5.4.3 Radiation Efficiency 129 5.4.4 Rotation Range 130 5.4.5 Number of Segments 131 5.4.6 Algorithm Results 132 5.4.7 Dosimetric Characteristics 135 5.4.8 Spatial resolution 138 5.4.9 Interleaf effects 140 5.4.10 Maximum field size 143 5.5 Discussion 144 Chapter 6 Conclusion 148 6.1 Conclusion 148 6.2 Future Work 150 Bibliography 152 Appendix A Circular PTV Conformity Derivation 164 v i i i Appendix B Rotational Leaf Motion Calculation Engine 167 ix List of Tables Table 2.1: A summary of the P T V shapes used in the qualitative and quantitative investigations of the model. Each shape is referred to using the acronym describing its spatial frequency characteristics 48 x List of Figures Figure 1.1: A classical representation of healthy tissue and tumour response to radiation dose is shown in (a). Both curves have a sigmoidal form. A t low doses the amount of cell k i l l is negligible but increases dramatically at a given threshold. In the classical representation the tumour response is considered to be greater than normal tissue. B y choosing a dose midway between the two curves adequate tumour control can be achieved resulting in only a small amount of healthy tissue damage. A more realistic clinical representation is shown in (b). The normal tissue curve is now to the left of the tumour curve, indicating that it is more sensitive to dose. Also , the tumour curve is less steep and plateaus before 100%, due most likely to heterogeneity of the tumour cells and prior spread of metastatic disease 4 Figure 1.2: For photons having an incident energy in the clinical range (0 to 25 M e V ) the photoelectric effect (a), Compton effect (b) and pair production (c) are the most common types of interactions resulting in a transfer of energy to electrons in the medium 8 Figure 1.3: The pencil beam model of dose deposition. A single infinitely thin pencil beam of photons w i l l generate a distribution of dose in and around the point of interaction as shown in (a). The spread of dose resulting from the pencil beam is the Dose Spread Kernel (DSK) . (b) With multiple pencil beams the total dose is the sum of all D S K s . In the limit as the number of pencil beams approaches infinity the calculation becomes a convolution of the incident fluence by the D S K (c) 11 xi Figure 1.4: A plot of absorbed dose versus depth for a square 10cm x lOcm 6 M V photon beam. Dose is given as a percentage of the maximum located at a depth of 1.5 cm 13 Figure 1.5: The linac rotates about the isocenter as shown in (a), allowing delivery of radiation to the target volume from different angular directions. A n axial C T slice showing four fields of a conformal field prostate treatment plan is presented in (b). The maximum dose is located in the target volume where the four fields intersect. The majority of surrounding tissue receives dose from only two of the four fields 14 Figure 1.6: A collimation system located below the source shown in (a) is used to shape the radiation field to the target volume. Two sets of translatable jaws are used to define a rectangular field shape (b). A more tightly conforming field shape may be obtained by adding alloy blocks below the secondary collimator as shown in (c) 16 Figure 1.7: (a) Each leaf of the multileaf collimator ( M L C ) is translated individually in and out of the radiation field using a separate motor. B y abutting the leaf edges with the edge o f the treatment volume, field shaping conforming to the tumour is created. A photograph of an M L C assembly is shown in (b) 17 Figure 1.8: A two-dimensionally varying fluence may be generated from a photon beam of constant fluence by adding the contribution from multiple uniquely shaped sub-fields. The total fluence at any point is given by the sum of all overlapping sub-fields at that point. Complex fluence maps are generated in this way by using an adequate number of sub-fields (>10) 20 Figure 2.1: Illustration of modulation transfer function concept. The Dirac delta function in (a) is input to the imaging system. Its Fourier transform (denoted by FT) is a constant throughout the frequency domain and is shown in (b). The system output t(x,y) is shown in (c). In the frequency domain, the Fourier transform of the t(x,y) is displayed (d). A l l spatial frequencies are input to the x i i system but some higher frequency information has been reduced in the output. The amount of spatial frequency reduction represents the spatial resolution capabilities of the system : 33 Figure 2.2: The function shown in (a) is sampled at a frequency 1/Ax through multiplication by the comb function (b). The result is a discrete representation of the original function shown in (c). A multiplication in the spatial domain corresponds to a convolution in the spatial frequency domain. The frequency spectrum is therefore convolved by the Fourier transformed comb function, producing multiple shifted copies of the original spectrum. When the sampling frequency is too low, adjacent frequency spectra overlap, causing a degradation of the original frequency spectrum know as aliasing 35 Figure 2.3: Degradation of a desired dose distribution due to the M L C and D S K . A n M L C sampling function for a 10 mm M L C leaf and a Gaussian Dose Spread Kernel with a=2 mm are shown in the spatial domain (a) and the frequency domain (b). The ability of this M L C and D S K to deliver a "sawtooth" dose distribution is shown in the spatial domain (c) and the frequency domain (d). Greater detail in the spatial frequency spectrum degradation can be seen in (e). 40 Figure 2.4: Illustration of M L C sampling in the degradation of an ideal dose distribution in the spatial and frequency domains using the Linear Systems model. X denotes a multiplication and <S> denotes a convolution. In the spatial domain, the ideal dose distribution (a) is multiplied by (b), sampling due to the M L C leaf spacing. The Fourier spectrum of (a) is convolved by the Fourier transformed sampling function of (b). The sampled distribution is shown in (c). 42 Figure 2.5: The sampled dose distribution (a) from the result of Figure 2.4 is convolved by the M L C leaf function (b). In the Frequency domain the sampling spectrum is multiplied by the Fourier transformed leaf function of (b). The x i i i resulting M L C shape is shown in (c). It is apparent that aliasing has occurred from the increased high frequency component that is not present in the initial spectrum of Figure 2.4(a) 44 Figure 2.6: The resulting M L C shape (a) is convolved by the dose spread kernel (b) to give the deliverable dose distribution (c). In the frequency domain the M L C shape spectrum is multiplied by the Fourier transform of the dose spread kernel in (b). The D S K acts as a symmetric low pass filter, preferentially reducing high frequency components as shown in (c) 46 Figure 2.7: M L C conformity for circular P T V s . Dark shaded areas indicate underdosed regions of the P T V . Light shaded areas indicate where healthy tissue is receiving dose. Poor coverage is observed for the peripheral leaves, i.e. when the leaf edges are oblique to the circle edge. A full derivation of the analytic solution is given in Appendix A 49 Figure 2.8: Conformity vs leaf width for circular P T V shapes with and without a rj=2 mm dose spread kernel. The decrease in conformity is a periodic function of leaf width. With the periodicity removed the decline in conformity is a linear w function of leaf width, w, with slope equal to 50 Figure 2.9: Degradation of the ideal dose distribution in the spatial and spatial frequency domain for (a), the L F S and (c) the H F S . Resulting degraded distributions are shown in (b) and (d). A 5 mm leaf width was used for both P T V shapes. Dose distributions were calculated using Equation 2.12. A Gaussian dose spread kernel with a a = 2 mm was used in all cases 53 Figure 2.10: A n ideal I M R T prostate field dose distribution is shown in (a). Degradation of the ideal dose distribution in the spatial and spatial frequency domain for the resulting distribution was calculated using Equation 2.12 and is shown in (b). A 10 mm leaf width M L C and a Gaussian dose spread kernel with rj = 2 mm was used in the calculation 54 xiv Figure 2.11: The collimator angle providing optimal conformity is plotted versus leaf width for the L F S , M F S and H F S 56 Figure 2.12: Conformity vs leaf width for the L F S , M F S , and H F S . A gaussian dose spread kernel with a=2 mm was used in all cases 57 Figure 2.13: Conformity vs leaf width for the H F S shapes using a gaussian dose spread kernels with a=0 mm, 2 mm and 3 mm. The rj=2 mm H F S curve is also plotted in Figure 2.12 59 Figure 3.1: Simplified diagram of a linear accelerator equipped with an M L C and rotating collimator. The collimator rotates about an axis that passes through the isocenter. The orientation of that axis is always perpendicular to the direction of leaf motion 65 Figure 3.2: Multiple M L C apertures at varying collimator angles contribute to the final fluence distribution. Two different M L C apertures are used to generate uniform fluence distributions in (a) and (b). The sum of (a) and (b) are displayed as a surface map in (c). Complex fluence distributions as shown in (d) are formed using several sub-fields each having a different rotation angle and M L C configuration 66 Figure 3.3: Potential improvements in spatial resolution between conventional and rotational delivery methods are shown, (a) In conventional delivery the M L C leaves move linearly in and out of the radiation field providing a rectangular pixel size whose length is limited by the leaf width, (b) Wi th rotation the direction of leaf motion changes, allowing for a smaller fluence pixel that is not limited by the leaf width 69 Figure 3.4: M L C leaves are designed with an interlocking tongue-and-groove shape on the side of each leaf. Although interleaf leakage is reduced significantly with this design there are still more photons transmitted through the interleaf gaps, causing non-uniformity in the transmitted fluence where the M L C is closed. xv Conventional I M R T delivery methods are unable to compensate for this effect and can result in overdosing at some interleaf locations 70 Figure 3.5: The tongue-and-groove effect is illustrated by considering the transmitted photon fluence at an open leaf edge. The dotted arrows indicate the direction and amount of photon transmission. The tongue causes an intermediate step in the fluence profile shown in (1). Closing this leaf and opening the adjacent leaf results in the same effect although this time caused by the groove. The sum of both profiles should ideally result in a constant fluence across both leaves. Due to the exponential nature o f photon attenuation, the sum of both (1) and (2) results in a fluence reduction error at the tongue-and-groove interface 72 Figure 3.6: A n example of 2 rotated sub-fields is shown. Because o f rotation the leaf edges of sub-field 2 are not coincident with those of sub-field 1. Wi th several sub-fields the position of leaf edges is blurred over the entire field area. Interleaf leakage and tongue-and-groove underdosing is therefore reduced with collimator rotation. In conventional delivery the leaf edges are fixed, amplifying systematic underdosing and overdosing error at the leaf edges 74 Figure 3.7: The maximum field size for conventional and rotational delivery methods in dynamic mode are shown. The field width is limited to the length of the M L C leaves in conventional delivery. With rotation the maximum field size is larger because (1) the leaves are not required to span the entire field width and (2), because the direction of leaf travel is rotated with each sub-field. The upper limit on rotational I M R T field sizes is a circle whose diameter is equal to the length of the leaf bank. 75 Figure 3.8: M L C leaf positions are calibrated individually using an infrared L E D emitter and optical sensor. The position of the optical path described by the sensor and the infrared L E D is known. Each leaf is translated into the optical path. Once the leaf blocks the infrared light the optical sensor signals that the xv i leaf has reached the calibration position. The leaf position is stored and the leaf is retracted 78 Figure 3.9: Plotted are histograms of collimator angle reproducibility. Collimator angle was verified once per month over a period of 45 months. Measurements were performed at 0, 90 and 270 degrees for Linac 1 and Linac 2. Results show that the collimator angle is reproducible to within 0.5 degrees over a period of almost 4 years 81 Figure 3.10: Histograms of the variation in the center of rotation over a 180-degree rotation. Measurements were obtained weekly for a period o f 1 year. Results are shown for Linac 1 and Linac 2 82 Figure 3.11: Measured dose versus Monitor Unit setting is plotted at dose rates of 100, 300 and 600 M U / m i n . Also plotted is a linear fit o f the 100 M U / m i n results. Results consistently show that dose is linear with M U setting within experimental error. Shown in the bottom right hand corner is a magnified section of the dose axis intercept. The dose offset indicates a monitor chamber lag time resulting in an overdose of approximately 0.06 cGy and causing significant dosimetric error of over 2% for sub-fields of less than 4 M U 84 Figure 3.12: A procedure to test the combined effects of collimator rotation instability and dose rate variations. A small aperture is opened at radius r from the central axis and rotated through 180 degrees during irradiation. The resulting profile of dose describes an arc and is measured with radiographic fi lm. Dose along the profile should ideally be constant. Larger dose rates are evaluated by repeating the rotation with a larger radial distance to the M L C opening given by equation 3.7 88 Figure 3.13: A grayscale image of the 2-dimensional dose distribution generated in the collimator rotation and dose rate stability test. Dark lines making up the arc o f the 'rainbow' correspond to exposures at each dose rate. Gradients at the beginning and end of each path are expected and are caused by the edge o f each xvn aperture exposing the tips for a smaller period than the rest of the trajectory. Slightly higher doses are seen near the end of each trajectory and are likely due to a deceleration of the collimator in advance of dose rate termination 90 Figure 3.14: Dose vs collimator rotation angle is plotted for dose rates ranging from 100 to 600 M U / m i n . Dose was measured along each trajectory of the open M L C apertures from the radiographic film of Figure 3.13. Non-uniformity in the dose profile indicates that the collimator rotation speed and dose rates are unsynchronized. Good overall uniformity is observed although there are some collimator deceleration artifacts and an increase in the standard deviation at 400, 500 and 600 M U / m i n 91 Figure 4.1: Method of generating intensity modulated fields using the 'step-and shoot" technique. The 2-dimensional fluence map in (a) is sampled along the leaf trajectory A B and is plotted in (b). The continuous fluence is then modified to produce regions of constant fluence as shown in the shaded area of (b). The leaf trajectories for leaf A and B are derived as a function of fluence index and are plotted in (c). The difference in fluence index between each trajectory is proportional to the fluence delivered at that point. The area between the two trajectories plotted as a function of leaf position is therefore equal to the shaded area in (b) 96 Figure 4.2: The sliding window technique is similar to the Varian step-and-shoot except that the fluence is divided into a large number of fluence levels as shown in (a). Also , each leaf pair moves continuously during the delivery as seen in (b), creating an opening in the M L C that slides from one side of the field to the other. The size and position of the window is calculated as a function of the fluence index. The fluence shown in (a) is generated by modifying the window width and position as plotted in (b) 98 Figure 4.3: Fluence maps are generated dynamically by rotating the collimator while the M L C leaves are in motion. Each frame in (a) to (i) shows the progressive xvm build-up of a wedge shaped fluence with the rotated M L C aperture at that instant. 99 Figure 4.4: (a) The rotating coordinate system of the M L C . (b) A sinogram of the trajectory of points A , B and C as they move through opposing leaf pairs. The points follow a sinusoidal path through the M L C leaf pairs with each point having a different phase and amplitude. The amplitude is equal to the radial distance of the point from the isocenter and the phase is a function of its initial position along the leaf bank 100 Figure 4.5: One-dimensional representation of the left and right leaf functions Ln(x,y)and Rn(x,y) from equation 4.4 and 4.5 respectively are shown in (a). Also shown is an example of a 1-dimensional aperture function Q(x)from equation 4.6 where x L and x R are the left and right leaf positions for a single leaf pair respectively. A full 2-dimensional aperture function using multiple leaf pairs is shown in (b) 103 Figure 4.6: Example of the minimization of an objective function Obj where xL(n,0) and xR(n,&) are modified after each iteration m. Gradient based methods fail to locate the global minimum when there is a high density of local minima as shown in (b) 106 Figure 4.7: A flowchart illustrating the basic mechanism of one iteration in the optimization part of the rotational leaf motion calculation algorithm 112 Figure 5.1: A 3-dimensional view of the thyroid treatment geometry. I M R T was indicated in this case due to the close proximity of the spinal cord to the P T V . A dose distribution providing minimal dose to the spinal cord and a uniform dose to the thyroid was obtained by optimizing the fluence maps of the 5 fields that are shown 117 Figure 5.2: Five clinical fluence maps as well as a 2-dimensionally varying sinusoidal fluence were used to evaluate the rotational leaf motion algorithm, (a) to (e) are xix optimized fluence maps for a thyroid treatment at gantry angles of 0, 72, 144, 216 and 288 degrees respectively. The 2 dimensional sinusoid is shown in (f). 118 Figure 5.3: Test fluence maps of varying complexity were used to evaluate the rotational delivery, (a) a Gaussian, (b) a wedge and (c) the 2-dimensionally varying sinusoid, (d) A constant intensity C-shape was used to evaluate the ability of the technique to conform to an irregular shape 120 Figure 5.4: A diagram depicting the generation of a one-dimensional gradient using an Enhanced Dynamic Wedge (EDW) is shown in (a). The collimator jaw is continuously translated across the field of incident photons during delivery. The transmitted fluence at a point is given by the amount of radiation delivered before the jaw passes over that point. Results of water tank ionization chamber measurements shown in (b) were obtained for two E D W s with different M U settings. In order to generate an adequate calibration curve, the number of M U was chosen so that the dose from the two deliveries would span the range of doses of each I M R T field 124 Figure 5.5: A scanned image of a typical calibration film is shown in (a). Pixel values are obtained at 1 cm intervals along the gradient of each field. The dose at these points was previously measured with an ion chamber as shown in Figure 5.4(b). Plotted in (b) are the pixel values versus dose values for the measurement points o f (a). The resulting curve is used to convert each measurement film to dose. A 3 r d order polynomial is fit to the plotted values to simplify the conversion. Finally, the calibration is verified by observing the overlapping dose values from the high and low doses of Field 1 and Field 2 respectively. A n y discontinuity between the two curves w i l l indicate an error in the calibration 125 Figure 5.6: A typical dynamic rotating leaf motion optimization history. The mean difference between desired and calculated fluence maps decreases rapidly over the first few thousand iterations. Once the mean difference begins to converge xx the number of segments is increased and the margin of acceptable error is decreased. The optimization proceeds in this fashion until no further improvements are observed 127 Figure 5.7: A histogram showing the reproducibility of the resulting mean difference between desired and calculated fluence maps using the rotating leaf motion algorithm. Leaf motions were calculated for the same fluence with the same parameters 100 times. Although the algorithms are stochastic in nature the resulting accuracy is highly reproducible 128 Figure 5.8: Effect of increasing the radiation efficiency. A s the efficiency is increased the quality of the calculated fluence map degrades and the mean difference increases. There is no benefit to decreasing the efficiency to values less than 50% in this example 129 Figure 5.9: Illustration of the effect of collimator rotation range on the algorithm result. The mean difference decreases rapidly as the rotation range is increased from 90 degrees to 180 degrees. Increasing the range further provides only minor benefit 130 Figure 5.10: In static mode the algorithm result is a function of the number of segments used for delivery. Accuracy of the algorithm improves rapidly as the number of segments is increased until approximately 40 segments after which adding more segments has only minimal benefit 132 Figure 5.11: Displayed is the mean difference between calculated and desired fluence maps for the 5 field thyroid I M R T plan. Results for both the rotational and conventional methods are plotted for the static delivery mode. Results are plotted for the 5 mm leaf M L C as well as the 1 cm leaf M L C 133 Figure 5.12: Displayed is the mean difference between calculated and desired fluence maps for the 5 field thyroid I M R T plan. Results for both the rotational and conventional methods are plotted for the dynamic delivery mode. Results are plotted for the 5 mm leaf M L C as well as the 1 cm leaf M L C 134 xx i Figure 5.13: Displayed are measured dose profiles for the rotational technique plotted against the desired calculated dose. Dynamic and static delivery results are shown for the (a) Gaussian and (b) Sinusoid fluence maps. A l l fluence maps were delivered with the 5mm leaf width M L C 136 Figure 5.14: Plotted is the mean difference between measured and desired dose distributions for the 2 dimensional sinusoidal test fluence. Results for rotational and conventional delivery methods in static as well as dynamic mode with both the 5mm Mil len ium M L C and the 1cm Standard M L C are shown 137 Figure 5.15: Dose conformity for the C-shape fluence maps are shown for the 5mm leaf M L C using (a) rotational and (b) sliding window techniques. 1cm leaf M L C conformity results are displayed in (c) and (d) 138 Figure 5.16: Dose profiles showing the spatial resolution capabilities of the rotational technique for a high frequency version of the sinusoidal fluence map using the l c m leaf M L C . Significant error (10%) is observed for the conventional delivery profile obtained perpendicular to the direction of leaf motion, where the limitations of leaf width are most apparent. The rotational technique profile shows only minor discrepancies 139 Figure 5.17: Leakage patterns for (a) rotational and (b) conventional techniques. ..141 Figure 5.18: Relative dose leakage profiles across the leaves and through the center o f rotation are shown for the 5mm leaf M L C and l c m leaf M L C in (a) and (b) respectively. Leakage decreases gradually at the edges of the rotational technique profiles due to the non-uniform contribution from the corners of the square aperture defined by the collimator jaws 142 Figure 5.19: A n example of the tongue-and-groove effect observed in the conventional technique delivery of the wedge shaped fluence distribution. Tongue-and-groove effects are not present when the same fluence is delivered with the rotational technique 143 xxn gure B . l : Software was developed to derive the rotating leaf motions. A graphical user interface facilitated the calculation, analysis and verification of different fluence maps. Functions of the software include: Reading in desired fluence maps, selecting fixed parameters ( M L C type, dynamic vs. static delivery, radiation efficiency, initialization method e t c . ) , displaying calculated fluence maps, evaluation of the calculated fluence maps and supplying M L C file output for delivery 167 x x m Acknowledgements I would like to thank Dr. Brenda Clark for her encouragement, constructive criticism and above all her unwavering support that occasionally went beyond the expectation a student can have of their supervisor. Members of my committee, Dr. A lex M a c K a y , Dr. Rob Kie f l and Dr. Tom Pickles were helpful in guiding my research direction throughout the thesis. I would also like to thank the staff members of the Vancouver Cancer Centre for their comments and their availability for discussing issues relating to this thesis. Physics assistants Vince Lapointe, Vince Stragar and Ron Horwood provided me with useful Q A data. Discussions with Kurt Luchka were helpful in the initial stages of the thesis. Dr. Tom Keane was particularly supportive and has earned both my respect and my gratitude. I thank my fellow students, in particular, Andrew Jirasek and Peter Petric for their help with various aspects of this thesis including many discussions and assistance with experimental measurements. Thanks go to Jyrki Alakuijala and Corey Zankowski for suggesting some of the advantages of collimator rotation in the initial stages of this work. M y fondest appreciation goes to my family for their love and support. Completing this work without a balanced life outside of research would be have been difficult. Suzie Gagnon provided me with the love and encouragement that helped maintain my motivation at all stages in the thesis. It is hard to imagine completing this work without her at my side. xxiv Chapter 1 INTRODUCTION 1.1 Radiation Therapy Cancer is a disease defined by the uncontrolled growth and spread of mutated host cells. Although it is not the main cause of fatalities it is currently the main source o f potential years of life lost in developed countries. Currently, 130,000 and 1,200,000 people are diagnosed with cancer in Canada and the United States respectively each year [1]. Radiation Therapy is the method of treating a disease through the delivery of radiation. Radiation Oncology is the sub-specialty of radiation therapy specific to cancer treatment. Cancerous tissue is destroyed primarily through damage caused by ionizing radiation. Ce l l death results from molecular changes in D N A and other critical cell components caused by direct or indirect ionization [2]. The amount of energy per unit mass deposited through ionization at a point is defined as the radiation dose (Gray). The goal of radiation therapy is to generate an accurate distribution of dose in a specified target volume while minimizing the amount of radiation to the surrounding healthy tissue. For approximately half of cancer patients the aim of radiation therapy treatment is to completely eradicate the disease (curative). For the other half the aim is to minimize pain and suffering (palliative). Other forms of treatment include surgery or chemotherapy and are often given in combination with radiation to increase tumour control. In this chapter an introduction to various aspects of radiation therapy relevant to the thesis are presented. For a more detailed 1 introduction to radiation oncology the reader is referred to Jacob V a n D y k ' s compendium 'The Modern Technology of Radiation Oncology' (Medical Physics Publishing, 1999) [3]. 1.1.1 H i s t o r i c a l B a c k g r o u n d X-rays were discovered by Wilhelm Roentgen in 1895, signaling the birth of an entirely new branch of physics research. Shortly after his discovery it was realized that high energy photons would be useful in medical applications. Firstly, the penetrating characteristics of x-rays could be used in a non-invasive way to acquire internal images of the body. Secondly, in combination with investigations performed by Henri Becquerel and Marie Curie into radioactivity, a new method o f treating disease was invented. The non-invasive and penetrating nature of radiation was seen as an alternative to surgery and, in 1899, the first patient to be cured by radiation was reported [4]. Initial uses of radiation for cancer treatment consisted mainly o f surface malignancies where dose was delivered by radioactive sources applied directly to the lesion [5]. In 1922 x-ray tubes capable of generating a photon spectrum with maximum energy of 200 keV were introduced, allowing for the treatment of more deeply located disease. Higher energy particle accelerators were developed in the 1940s which provided x-ray beams with peak energies in the M e V range. Cobalt-60 sources, produced in a nuclear reactor, were introduced in 1951. Having a high specific activity, average photon energy of 1.25 M e V and a half life of 5.26 years Cobalt-60 quickly became the most popular method of radiation treatment and is still used in developing countries as well as some larger radiation therapy centres in the western world. The most important development in radiation production as applied to radiation therapy was the introduction of linear accelerators in the 1960's. Today, the 2 linear accelerator (linac) is the most common device used to generate megavoltage energy photons and electrons for cancer treatment. 1.1.2 Goal of Radiation Therapy The goal of radiation therapy is to destroy cancerous cells without harming the surrounding normal tissue. Unfortunately, it is inevitable that some normal tissue cells are irradiated. The amount of cancerous cell destruction is therefore balanced by the limits imposed by healthy tissue. 1.1.3 Tumour/Healthy Tissue Response to Dose The response of different cell types to dose is complex. Radiation Biology is the study of cellular response to radiation. The mechanism of cell death resulting from dose deposition has been studied in detail but is still an area requiring investigation. Tumour or tissue response may be defined as the percentage probability that the tumour or tissue is permanently destroyed through the process of irradiation. A classical model of tumour response to radiation derived from in vitro experimental data is characterized by the sigmoidal dose-response curve [2] in Figure 1.1(a). A t low doses, the response of the tumour is negligible. Only at some minimum threshold, in this case 30 Gy, is there enough dose to k i l l a significant amount of cancerous tissue. Increasing the dose further results in a dramatic increase in the number of cells that are destroyed. Finally, as the percentage of cell death approaches 100%, the slope tapers off and asymptotically approaches complete tumour control. A normal tissue dose-response curve is also displayed in Figure 1.1(a). It follows the same sigmoidal form of the tumour curve but is offset to higher dose, showing that normal tissue is less sensitive to radiation. The dose selected for treatment w i l l be at a level between the two curves so that tumour and normal tissue response are maximized and minimized respectively (see example Figure 1.1(a)). 3 100 Dose (Gy) Dose (Gy) Figure 1.1: A classical representation of healthy tissue and tumour response to radiation dose is shown in (a). Both curves have a sigmoidal form. At low doses the amount of cell kill is negligible but increases dramatically at a given threshold. In the classical representation the tumour response is considered to be greater than normal tissue. By choosing a dose midway between the two curves adequate tumour control can be achieved resulting in only a small amount of healthy tissue damage. A more realistic clinical representation is shown in (b). The normal tissue curve is now to the left of the tumour curve, indicating that it is more sensitive to dose. Also, the tumour curve is less steep and plateaus before 100%, due most likely to heterogeneity of the tumour cells and prior spread of metastatic disease. Dose response curves derived from the limited clinical data available suggest that this simplified model is misleading in some respects. A more plausible clinical representation of dose response is shown in Figure 1.1(b) [3]. The normal tissue and tumour response curves have the same form but show some important relative differences. The normal tissue curve and tumour curves have similar low dose thresholds but the normal tissue curve has a steeper slope. Normal tissue is therefore more sensitive to lower doses than the tumour. The reduced slope of the tumour curve results most likely from the heterogeneity of cancerous cells throughout the 4 tumour volume. Finally, it has been suggested that the tumour curve achieves its upper threshold at less than 100% control due to metastatic disease not located at the tumour site [3]. Although the cells in the original tumour can be completely destroyed the tumour may re-grow from the metastatic cells not located at the tumour site. The model presented in Figure 1.1(b) emphasizes the need to reduce dose to healthy tissue in radiation delivery. Due to the steep slope of the healthy tissue curve only a small increase in dose w i l l result in a significant increase in healthy tissue damage. There is a clear necessity for improved accuracy and spatial resolution in dose delivery and it has therefore been a focus of research throughout the history of radiation therapy. This thesis is devoted to the development of an improved method of dose delivery that has increased spatial resolution and accuracy over currently available techniques. It should be noted that the quantification of improvement and benefit in radiation therapy is challenging. The merits of any 'improvement' have to be measured in terms of an increase in patient life span and quality of life. The relationship between an improved dose distribution that conforms more closely to the tumour volume and the overall benefit to the patient is not wel l defined. Only after years o f post irradiation follow-up with statistically large enough groups o f patients can any technical improvement be related to a clear benefit. 1.1.4 Treatment Planning With the development of penetrating megavoltage photon beams it became necessary to estimate, at least two-dimensionally, the distribution of dose that would result from multiple beams entering the patient from different directions. The arrangement of beams used to treat a given patient must be chosen so that the dose distribution w i l l conform to the radiation oncologist's prescription. Before the advent of computers, atlases of 2-dimensional dose distributions were used to predict the contribution from 5 each radiation beam [6]. The final distribution was the weighted sum of each beam obtained from the atlas. With computers it became possible to eliminate tedious manual calculation of multiple beam geometries. Dose calculation methods were developed to predict dose deposition for arbitrary geometries (see section 1.2.4), allowing more flexibility and accuracy when planning a treatment with different radiation fields. 1.1.4.1 C T / M R Imaging One of the benefits of radiation therapy is that it is relatively non-invasive. In order to preserve this benefit it is necessary that the location of the target volume also be determined through non-invasive means. Initially, physicians were guided by simple techniques such as conventional radiography and surface palpation. In the 1970s a new 3-dimensional imaging modality known as computed tomography (CT) scanning was invented [7]. B y using radiographic projections obtained at varying angles around the patient the 3-dimensional distribution of physical density inside the patient is calculated. With this improved anatomical information physicians were able to define the location of the target volume in 3-dimensional space with significantly greater accuracy. Magnetic Resonance Imaging (MRI) was introduced in the 1980s providing additional 3-dimensional information that was complementary to C T image data [7]. Three-dimensional anatomical information combined with improvements in computer technology provided an environment for modeling different beam combinations on patient anatomy prior to treatment. B y modeling different potential treatments using the patient's image data it is possible to determine the beam configuration that more closely meets the physician's prescription. 6 1.2 Dose Deposition The mechanism of photon dose deposition involves the transfer of energy from the photon to tissue through particle interactions in the tissue. In the 0 to 25 M e V energy range produced by clinical linear accelerators there are six basic types of interactions that can occur. They are: Rayleigh scattering, photoelectric effect, Compton effect, pair production, triplet production and nuclear photodisintegration. Rayleigh scattering is elastic and therefore does not contribute to dose. The photoelectric effect, Compton effect and pair production occur with the greatest probability in this energy range. 1.2.1 Photon Interactions The photoelectric effect occurs when there is a collision between a photon and an atom resulting in a bound electron being ejected from that atom. A diagram of the photoelectric process is shown in Figure 1.2(a). The kinetic energy transferred to the electron, Etrans, is given by the difference between the incident photon energy, hv, and the binding energy of the electron, BE, The probability of a photoelectric interaction occurring is greatest at photon energies that are slightly above the binding energy of the electron. Compton interactions occur when a photon interacts with a loosely bound or free electron. In this interaction the electron absorbs some of the photon energy while the rest remains with the scattered photon as shown in Figure 1.2(b). The energy transferred to the electron is given by E, trans = hv-BE. (1.1) E, trans = hv-hv' (1.2) 7 where h V is the energy of the scattered photon. Ejected Electron (b) Compton Scattered Photon Incident Photon \y\/\/\y^ o Compton Electron (c) Pair Production Positron Electron Figure 1.2: For photons having an incident energy in the clinical range (0 to 25 MeV) the photoelectric effect (a), Compton effect (b) and pair production (c) are the most common types of interactions resulting in a transfer of energy to electrons in the medium. Pair Production occurs when an incident photon is replaced by an electron-positron pair as shown in Figure 1.2(c). The interaction occurs when a photon is stimulated by the electromagnetic field of an atom nucleus. Because the rest mass 8 energy o f the electron and positron, me, must be created in the interaction, the resulting energy transferred to the positron-electron pair is: E, 'trans = h v — 2m 'e (1.3) 1.2.2 Electron Energy Transfer - Stopping Power The mechanism of energy transfer to electrons was introduced in the previous section. Energy is transferred in the form of kinetic energy, thereby imparting a velocity to the electron that brings it to a different location in the medium [6]. Electric forces act on the electron as it passes through the medium, causing it to lose energy and slow down. The stopping power of the medium is defined as the rate of particle energy loss per unit thickness. Energy losses can be divided into two categories, ionizational and radiative. The former results in further ionization of electrons along the initial electron trajectory. Each one of these ejected electrons w i l l also undergo their own ionizational and radiative losses until they come to rest. Radiative losses occur due to Bremsstrahlung processes that result in photon production and do not contribute directly to energy deposition in the medium. 1.2.3 Fluence-Dose relationship Before discussing the relationship between photon fluence and absorbed dose it is necessary to present the quantity kerma, K, defined by the kinetic energy transferred from photon to electrons per unit mass. K = TP)E' tr (1.4) 9 where O is the input photon fluence (see equation 1.8), is the mass attenuation coefficient (or photon interaction probability cross section) and Etr is the average energy transferred to the medium from each interaction. Kerma differs from dose in that it is a measure of the energy transferred to the medium at a point and not the final energy absorbed at that point [6]. In the case where there is an equilibrium between kerma and absorbed dose (i.e. the number of electrons entering the point of interest is equal to the number that are set in motion) the relationship between absorbed dose and fluence may be written: 73 = 0 V Eab (1.5) where Eab is equal to the average energy absorbed in the medium from each interaction. In general, the equilibrium condition is not completely satisfied and there is no simple method of expressing dose in terms of fluence. Monte Carlo calculation techniques which model individual interactions for large numbers of photons and electrons are the most accurate method of calculating dose. Unfortunately, calculation times on currently available computers are not yet adequate to make this method clinically useful, although commercial systems1 are currently being developed. 1 Peregrine, Livermore, C A M D S Nordion , Kanata, O N 10 1.2.4 Pencil Beam Dose Deposition A simplified model of dose deposition has been suggested by many investigators that considers the photon beam as a series of infinitely thin pencil beams as shown in Figure 1.3(a). Due to photon scatter and electron transport, when the pencil beam interacts with tissue it w i l l deposit dose in and around the area o f interaction. The distribution of radiation resulting from one pencil beam is a symmetric function referred to as the Dose Spread Kernel (DSK) [8, 9]. A radiation field is made up of multiple pencil beams. Each pencil contributes to the total dose in a cumulative fashion. The final dose distribution is therefore the sum of each D S K for each pencil beam as described in Figure 1.3(b). Each pencil beam is infinitely small making the number of D S K s that need to be summed infinite as well . The limit of the summing operation as the number of pencil beams approaches infinity is a convolution Photon Penci l Beam Mult ip le Pencil Beams Photon Beam Dose Spread Kerne l ( D S K ) D S K Convolution Resulting Dose Profile Figure 1.3: The pencil beam model of dose deposition. A single infinitely thin pencil beam of photons will generate a distribution of dose in and around the point of interaction as shown in (a). The spread of dose resulting from the pencil beam is the Dose Spread Kernel (DSK). (b) With multiple pencil beams the total dose is the sum of all DSKs. In the limit as the number of pencil beams approaches infinity the calculation becomes a convolution of the incident fluence by the DSK (c). 11 operation. D(x,y) = ®(x,y)®DSK(x,y) = oo co j J0(a,J3pSK(x —CO—CO -a,y- 0)dadp (1.6) Therefore, by convolving the incident photon fluence by the D S K it is possible to approximate the final dose distribution [10-12] as shown in Figure 1.3(c). This simplified model assumes various characteristics of the photon beam and medium that are not completely realistic. It is assumed that the photon energy spectrum is spatially invariant and that the medium is composed of a homogenous material. Variations on the basic model have been introduced in commercial treatment planning systems to account for these limitations [13]. 1.3 Basic Dose Delivery Techniques There are several methods of delivering radiation that try to meet the goal of providing a uniform dose to the target volume and minimal dose to surrounding tissue. The focus of this section is to describe the relevant methods of dose delivery used with high-energy photon beams. A plot of dose as a function of depth for a 10cm x 10cm 6 M V (peak energy of 6 M e V ) photon beam is shown in Figure 1.4. Near the surface there is a region where electronic equilibrium has not yet been attained and dose is being deposited primarily downstream (buildup region). The location of dose maximum occurs at 1.5 cm depth after which the dose slowly decreases due to attenuation and increasing distance from the source. Due to the decrease in dose with depth it is not possible to arrive at a uniform dose in the tumour volume. Multiple fields must be used to achieve the treatment goals". 12 Percentage Depth Dose (6 MV Photons) 100 _ 80 g, a> 8 60 Q > 1 40 20 "0 5 10 15 20 25 30 Depth (cm) Figure 1.4: A plot of absorbed dose versus depth for a square 10cm xlOcm 6 MV photon beam. Dose is given as a percentage of the maximum located at a depth of 1.5 cm. 1.3.1 Multiple Fields The linac is mounted on a gantry that rotates about the patient as shown in Figure 1.5(a). Dose is delivered to the patient from multiple directions with the center of rotation located inside the tumour volume [5]. The dose inside the tumour is the cumulative sum of the contribution from all beams. The dose in the surrounding tissue is the contribution from a subset of those fields that depends on the orientation and position of each field. A typical prostate treatment dose distribution is shown in Figure 1.5(b). In this case the prostate receives the total contribution of dose from two sets of opposing fields while the surrounding tissue receives a maximum dose given by the sum o f only one of the opposing beam pairs. Choice of beam orientation is a manual procedure and often requires multiple iterations before an adequate dose distribution is obtained. Fu l l computer optimization of beam orientations has been investigated by others [14, 15] but has not yet achieved clinical implementation. 13 i ' i 1 i 1 i 1 r j i I i I i I i L Field 4 Figure 1.5: The linac rotates about the isocenter as shown in (a), allowing delivery of radiation to the target volume from different angular directions. An axial CT slice showing four fields of a conformal field prostate treatment plan is presented in (b). The maximum dose is located in the target volume where the four fields intersect. The majority of surrounding tissue receives dose from only two of the four fields. 14 1.3.2 Arcs In addition to delivering multiple fields at fixed gantry angles, radiation may also be delivered while the linac gantry is rotating [5]. In some cases the location of critical healthy tissue structures is such that by spreading the dose to the surrounding tissue throughout an arc, the dose to those structures w i l l be minimized. The work presented in this thesis involves a treatment method using fixed gantry angles, therefore arc treatment techniques w i l l not be discussed further. 1.4 Field Shaping When the linac electron beam interacts with the target (photon source), high energy photons radiate from the target preferentially in the forward direction [6 ] . A 'flattening filter' placed downstream from the target is used to modify the initial fluence so that the output fluence is relatively constant over its spatial extent. Beneath the flattening filter is a fixed 'primary' collimator that defines the maximum possible field size. Below that is a secondary collimator consisting of two pairs of attenuating metal blocks that define the field used to treat the target volume (Figure 1.6(b)). These 'jaws' may be moved in and out of the beam thereby defining an arbitrary rectangular field shape up to a maximum of 40 cm x 40cm. Because tumours grow into a variety of shapes, a rectangular field w i l l be limited in its ability to conform to the treatment volume and reduce dose to the surrounding healthy tissue. One technique used to improve the field conformity is to create custom metal alloy blocks that match the shape o f the tumour [5] as shown in Figure 1.6(c). B y using low temperature alloy materials (e.g. Cerrobend) high attenuation blocks may be built that conform to the target volume. The blocks are mounted below the secondary jaws. Because the projected shape of the tumour is different at each gantry angle, a different set of blocks must be built for each field. This process of field shaping is cumbersome, requiring the apparatus and manpower to generate custom 15 blocks for each patient. Also, because there is a separate set of blocks for each field it is necessary to enter the room and change them after each field. Figure 1.6: A collimation system located below the source shown in (a) is used to shape the radiation field to the target volume. Two sets of translatable jaws are used to define a rectangular field shape (b). A more tightly conforming field shape may be obtained by adding alloy blocks below the secondary collimator as shown in (c) 1.4.1 Multileaf Collimator In recent years a new device for two-dimensional field shaping has been introduced. The Mult i leaf Collimator ( M L C ) consists o f a series of tungsten alloy leaves that move parallel to each other in and out of the radiation field as shown in Figure 1.7(a). The M L C is typically located below the secondary jaws. A picture of a partially disassembled M L C is shown in Figure 1.7(b). Each leaf is connected to a separate computer controlled motor allowing it to be positioned independently of the other leaves [16]. A field that conforms to the desired treatment shape is generated by moving the leaves to a position where the projection of the end of each leaf abuts the edge of the treatment volume. 16 Figure 1.7: (a) Each leaf of the multileaf collimator (MLC) is translated individually in and out of the radiation field using a separate motor. By abutting the leaf edges with the edge of the treatment volume, field shaping conforming to the tumour is created. A photograph of an MLC assembly is shown in (b) Due to the point source geometry of photon production, the physical leaf dimensions are smaller than their projection at the target. Typical leaf width projections are on the order of 5 and 10 mm at the isocenter (100 cm from the source). The actual physical leaf widths are roughly half that width (2.5 mm and 5 mm). The Var ian 2 M L C leaves used in this thesis are composed of a Tungsten A l l o y and are 6 cm deep, attenuating greater than 95% of the incident photon beam. For a typical maximum field size of 40x40 cm (defined at the isocenter) a minimum of 80 leaves and motors (40 on each side) are required. Smaller leaf width M L C s are available but have a more limited field size. A more detailed discussion of M L C characteristics is presented in Chapter 3. 2 Var ian Oncology Systems, Palo A l t o , C A 17 1.5 Intensity Modulated Radiation Therapy The previous section presented methods of conforming a relatively uniform photon beam to a target volume through the use of collimators. A uniform dose may be obtained in the target volume with the appropriate choice of beam orientation. In some cases, due to the particular combination of beam orientation or irregularities in the patient surface, it is also necessary to modify the intensity of the beam across its two-dimensional extent [5, 6]. For example, a wedge shaped attenuator may be placed in the field to create a one-dimensional gradient at some orientation of the collimator. The appropriate choice of gradient w i l l improve the dose distribution and bring the plan closer to the treatment goals. Historically, two-dimensional modification of photon beam intensity has been limited to simple gradients and irregular surface compensators. In the last decade there has been a large body of research and development devoted to methods for generating complex arbitrary 2-dimensional intensity maps. The rationale behind these developments is that with 2-dimensional intensity modulation there w i l l be increased flexibility in generating dose distributions thereby improving the likelihood of reaching the treatment goals. 1.5.1 Complex Fluence Generation Intensity I is defined as the energy passing through unit area, a, per unit time, t. 1 = ==^- (1.7) dadt In the case where the photon energy spectrum remains constant throughout intensity modification the process can more simply be described as fluence modulation. The Fluence 0 is the number of photons, N, passing through unit area. 18 (1.8) Henceforth, the terminology intensity modulation and fluence modulation refers to the same basic process. Generation of arbitrary two-dimensional fluence distributions is performed using the attenuation properties of photons as they pass through matter. The one-dimensional wedge gradient example described at the beginning of this section is created using an attenuator with varying thickness in one dimension. Extending to two dimensions, a custom-machined metal block may be constructed to modify the fluence for each field. Unfortunately, the machining process is typically too cumbersome for routine clinical use on large numbers of patients and the fluence and spatial resolution are inadequate in many cases. The method most commonly used to generate complex fluence maps at fixed gantry angles involves the cumulative sum of multiple uniquely shaped sub-fields [17] as shown in Figure 1.8. Although the fluence resulting from an individual sub-field is relatively constant over its spatial extent, the size and shape of each sub-field is different. A t locations where two sub-fields overlap, the fluence w i l l be the sum of each sub-field contribution. Conversely, at locations where only one field is open to the source, the contribution to the total fluence w i l l be from that field only. In this way, by using an adequate number of sub-fields, it is possible to generate arbitrary fluence maps. 19 Fluence 1 : j 4 -1 9) Fluence 1+2 t - 4 — 1 9 Fluence 1+2+3 • Figure 1.8: A two-dimensionally varying fluence may be generated from a photon beam of constant fluence by adding the contribution from multiple uniquely shaped sub-fields. The total fluence at any point is given by the sum of all overlapping sub-fields at that point. Complex fluence maps are generated in this way by using an adequate number of sub-fields (>10). 1.5.1.1 Mu l t i p l e M L C Fields Due to the ease with which field apertures may be modified with the M L C it is the most common method for defining the sub-fields used in I M R T [18]. Each aperture is defined by the M L C leaf positions. Because the leaves are computer controlled, the entire sequence of sub-fields may be loaded into the M L C controller at once. After 20 the delivery of each sub-field the leaves are immediately moved into position for the next field, making the total delivery time short enough for clinical use. Radiation may also be delivered with the leaves moving continuously throughout delivery [19, 20]. In this case the fluence is divided into a large number of continuously varying sub-fields (>100) which provide superior overall fluence resolution. The fluence map is delivered by controlling the speed of each leaf and/or the dose rate of the linac. A detailed description of M L C based I M R T delivery including the limitations of current leaf positioning methods is presented in Chapter 3. In this thesis a new method of controlling the multileaf collimator is described that improves on conventional M L C based I M R T methods. 1.5.2 Plan Optimization Although I M R T increases the flexibility of generating dose distributions it complicates the treatment planning process. The number of combinations and permutations involved when each fluence 'p ixel ' o f each field can be modified is too large to be evaluated by a human observer. For this reason with the development of fluence modulation came the introduction of treatment planning optimization methods [21-23]. The purpose of the optimization is to calculate the optimal fluence map for each field such that the physician's prescription is satisfied. In general, the fluence map for each field is derived with the intention of providing a uniform dose to the target volume while limiting dose to surrounding structures. Although plan optimization is an inherently three-dimensional problem, the fluence map required for each field is two-dimensional. Once a satisfactory plan has been obtained, the problem becomes the generation of that two-dimensional fluence map. Derivation of the optimal fluence maps and calculation of the apertures needed to generate those fluence maps are therefore two separate processes. During plan optimization it is assumed that the linac w i l l be capable of delivering the desired fluence. In this way, the problem of deriving the optimal plan, and therefore the optimal fluences, is de-21 coupled from the problem of generating the appropriate apertures needed to generate those fluences. This thesis is focused entirely on the problem of aperture generation and the concept of plan optimization is introduced in this section for completeness only. 1.5.3 L e a f Sequenc ing Once the desired fluence maps have been calculated the M L C apertures used to generate those fluence maps must be derived. Numerous leaf-sequencing techniques have been proposed in the literature [24-26]. In general, the fluence profile between each leaf and its opposing pair is considered separately. B y modifying the position and size of the gap between the leaves for each sub-field it is possible to produce virtually any fluence profile. A more detailed description of conventional leaf sequencing methods is described in section 4.1. 1.5.4 D e l i v e r y The leaf positions for each sub-field are transferred from the planning system to the M L C control computer prior to treatment. Once the patient is in position, the gantry angle, collimator jaw positions and M L C leaf positions are set for the first sub-field segment. When the beam is activated at the control console the M L C controller computer moves the leaves to each required position. Once the sequence has finished the gantry is moved to the next field. The process is repeated until all fields have been delivered. 1.6 Factors Affecting Spatial Resolution in Dose Delivery A n ideal dose distribution is one where the cancerous tissue receives 100% of the prescribed dose and the surrounding tissue receives nothing. Currently, the most important factor affecting our ability to achieve this goal is the limited information 22 available to the oncologist when determining the location of cancerous tissue [27-29]. Advances in positron emission tomography (PET) and functional M R I show promise in providing metabolic information not currently available with standard imaging techniques [30, 31]. The radiation oncologist estimates the planning target volume (PTV) from the available diagnostic information. It is then the job of the physicist and/or dosimetrist to plan the patient treatment. In practice there are several specific factors that limit our ability to deliver an ideal dose distribution to the P T V . 1.6.1 Imaging Resolution Although not the most important source of spatial resolution degradation, the resolution of diagnostic images places an upper bound on how precisely the P T V can be defined. Pixel resolution in the transverse imaging plane (cut through the body horizontally) is typically on the order of 0.5 mm. In the superior-inferior dimension the spacing ranges from roughly 1mm to 1cm. 1.6.2 Patient Immobilization Although there is significant emphasis placed on reproducing the patient set-up for treatment, there is no guarantee that the patient w i l l be in precisely the same position as when diagnostic images were obtained. Because many organs in the body are mobile (e.g. lungs and heart) and radiation is not delivered instantaneously, there w i l l always be some additional error during each treatment due to patient/organ motion. Finally, the target volume itself w i l l change over time. Although the degree of variation is different in each case there w i l l inevitably be some error due to a mismatch between the treatment fields and the true P T V at that instant. The magnitude of these effects have been studied by others [32] and is highly site dependent [32, 33]. For example, in the head and neck region an accuracy of 1 to 23 2 mm is achievable [34] while in the lung region errors as high as 3 cm have been noted [35]. 1.6.3 Delivery Technique In addition to the spatial resolution degrading factors described in the previous sub-sections the specific technique or mechanism used in delivering dose to the P T V is important. The tools available for generating an ideal dose distribution suffer from certain limitations. The photon beam has certain physical characteristics, photon scatter and electron transport in particular, that make it impossible to conform perfectly to the P T V [36]. Also , mechanical properties of the field-shaping device (e.g. M L C ) provide additional limits on spatial resolution and dosimetric accuracy [37]. A s part of this thesis a rigorous evaluation of the spatial resolution limiting characteristics of conventional delivery techniques is presented. B y using linear systems analysis it is shown how different delivery parameters affect our ability to deliver an ideal dose distribution. From the results of this investigation a new way of delivering I M R T is proposed that has potentially higher spatial resolution than existing techniques. Finally, the bulk of the thesis is devoted to the development and testing of this novel dose delivery method. 1.7 Thesis Objectives and Summary Previous investigators have shown that by improving the spatial resolution of fluence delivery dose distributions may be generated that conform more closely to the tumour volume and reduce dose to the surrounding healthy tissue [38, 39]. The primary objective of this thesis is the development of a novel method for the delivery of I M R T that has several advantages over current techniques including higher spatial resolution fluence modulation. 24 1.7.1 Spatial Resolution Degradation It was recognized in the preliminary stages of this thesis that there was a need for a more rigorous method for investigating the spatial resolution capabilities of dose delivery techniques. For this reason a new analysis method was developed by expanding on a study by Bortfeld et al. [37] applying linear systems analysis for quantifying dosimetric spatial frequency degradation (Chapter 2). Modeling dose delivery as a linear system has two main advantages. Firstly, the spatial resolution capabilities of various components in the dose delivery chain can be assessed separately. This provides insight into which component provides the best or worse spatial resolution. Secondly, with the aid of spatial frequency analysis, it is now possible to evaluate spatial resolution changes independent of the volume that is being treated. Typically, new delivery technology is assessed using relatively arbitrary treatment scenarios (e.g. a specific treatment site and beam configuration). B y quantifying spatial resolution properties of the delivery technique independent of the treatment volume an unbiased evaluation may be achieved. 1.7.2 IMRT Delivery With M L C Rotation Through observations obtained from the linear systems analysis described in Chapter 2 a new method of I M R T delivery is proposed. The method consists of rotating the entire M L C between each I M R T sub-field. These I M R T fields may be delivered statically (with the collimator rotating to a new position in between sub-fields) or dynamically (with the collimator rotating and leaves moving simultaneously during irradiation). Wi th this fundamental increase in flexibility advantages in spatial resolution are theoretically obtainable. In addition, there are potential advantages in reduced systematic interleaf errors and generating larger maximum field sizes. The majority of Chapter 3 is devoted to investigating the mechanical and radiation producing characteristics of standard linacs under collimator rotation conditions. 25 Specific experiments are used to evaluate each component in the delivery chain that may affect dosimetric accuracy. Through this investigation any dosimetric error in rotational I M R T delivery due to basic limitations of current linac hardware are determined. 1.7.3 Leaf Motion Derivation Although there are several theoretical advantages to delivering I M R T with M L C rotation, the realization of those advantages is not trivial. Deriving the appropriate rotated M L C apertures for arbitrary fluence maps is significantly more complex than with conventional fluence generation methods. The most challenging aspect of this thesis was the development of a series of algorithms capable of deriving the leaf positions for each rotated M L C aperture. Chapter 4 is devoted to a full description of the rotational leaf motion algorithms. Included is a description of conventional leaf positioning methods as well as an analytic model of the rotational leaf motion derivation problem. 1.7.4 Rotating M L C Evaluation In Chapter 5 a series of experiments that evaluate the rotational technique in both its static and dynamic modes is presented. First, the rotational leaf motion algorithms are characterized in terms of their ability to derive leaf motions for a variety of clinical fluence maps. In particular, the dependence of the algorithms on various fixed delivery parameters is investigated. Next, the dosimetric accuracy o f the rotational technique is evaluated using a series of varying complexity fluence maps. Finally, a series o f experiments are presented that focus on the advantages of using M L C rotation in I M R T delivery. The experiments described in each section are repeated for two separate linacs having different M L C models. For comparison, fluence maps are generated using both rotational and conventional delivery techniques. 26 Chapter 2 SPATIAL RESOLUTION DEGRADATION Flexibil i ty and complexity in patient treatment due to advances in radiotherapy techniques necessitate a simple method for evaluating spatial resolution capabilities of the dose delivery device. The purpose of the following investigation is to evaluate a model that describes the ability of a linear accelerator to deliver a desired dose distribution. The model, developed as part of this thesis and published in the journal Medical Physics [40], is based on linear systems theory and is analogous to methods used to describe resolution degradation in imaging systems. A qualitative analysis of spatial resolution degradation using the model is presented in the spatial and spatial frequency domains. The ability of the model to predict the effects of geometric dose conformity to treatment volumes is evaluated by varying multileaf collimator leaf width and magnitude of dose dispersion. Dose distributions for three clinical treatment shapes, circular shapes o f varying diameter and one intensity modulated fluence map are used in the evaluation. It is shown that the model accurately predicts the dependence of dose conformity on these parameters. The spatial resolution capabilities of different radiation therapy devices can be quantified, providing a simple method for comparing different treatment machine characteristics. Also , because different treatment sites have different resolution requirements, the model may be used to tailor machine characteristics to each specific site [40-43]. Validation of the model consists of two parts. The first part is a qualitative investigation into each component focusing on effects to spatial resolution in dose delivery. Included as examples are a range of typical clinical shapes varying in size 27 and complexity. The second part consists o f a quantitative analysis of the dose conformity to the same clinical treatment shapes indicated above. Results are discussed in the context of the qualitative investigation and are used to evaluate the relevance o f the model. Through observation of various aspects of the model, specifically the benefits of collimator rotation, a method of improving the spatial resolution o f I M R T distributions is proposed. This chapter provides a rigorous theoretical investigation of the limits o f spatial resolution in dose delivery. It lays a foundation for the development of a new I M R T delivery technique described throughout the remainder of the thesis. 2.1 Linear Systems Theory When characterizing the response of a given system to some form of stimulus it is advantageous i f the entire system may be modeled as linear. For a system to be linear it is required that each component that makes up the system respond in a linear fashion [44]. Expressed mathematically, H[kJx (x, y) + k2f2 (x, y)] = k,H\fx (x, y)] + k2H[f2 (x, y)]. (2.1) Where H is an operator describing any component of the system, k\ and ki are constants and, f\(x,y) and fi{x,y) are any two-dimensional functions describing a stimulus to the system. A system that adheres to equation 2.1 satisfies both the property of additivity # Di (*, y) + f2 {x, y)] = H\ft {x, y)] + H[f2 {X, y)] (2.2) as wel l as homogeneity 28 H[kj^y)] = KH[fX^y)l (2.3) 2.1.1 Spatial Invariance For two-dimensional systems it is advantageous i f each component adheres to an additional constraint. I f the location of the stimulus affects the system output then evaluating the system for arbitrary stimulus w i l l be significantly more complex. The system must therefore also have the property of spatial invariance. Expressed mathematically, for a system output g(x,y), it is required that for any j\x,y) and any Xo, yo-2.1.2 Application To Imaging Systems Many imaging systems can be considered linear and may be approximated as spatially invariant for the purposes of evaluation. Some examples include: photography, diagnostic x-ray imaging and computed tomography [7, 45-48]. In imaging applications, the ability of the imaging device to detect varying resolution objects defines its utility. B y modeling the device as a linear system it can be compartmentalized into a chain of separate resolution degrading components. Each component has a linear effect on the previous link in the chain. The advantage of this formulation is two fold. Firstly, it allows for the characterization of the imaging system by a spatial resolution degradation function. This function is not dependent on the object being imaged. Removing this dependence allows for an evaluation of the imaging system without relying on arbitrary test images. Different imaging systems may then be H[f(x x0,y-y0 )] = g(x x0,y-y0) (2.4) 29 compared in a robust fashion, with the linear degradation function of each system defining its performance. Secondly, because each component of the system can be separated, it is possible to evaluate each component individually. Different components w i l l have different effects on spatial resolution. B y evaluating the mechanisms of spatial resolution reduction from each component, it may be possible to determine methods of improving the system as a whole. The "weakest l ink" in the chain may also be determined in this fashion, allowing investigators to focus on components that w i l l result in the greatest improvement. 2.1.3 Fourier Model Of Spatial Resolution Degradation (Spatial Frequency) A large body of work has been devoted to the study of imaging systems in both the spatial and spatial frequency domains. The Fourier transform is used to convert spatial information to the frequency domain. For a 2-dimensional function f(x,y) the corresponding function in the Fourier (or frequency) domain, F(/u, v), is given by F(p,v) = FT\f(x,y)]= Ylfix^y^^dxdy (2.5) -co—oo and the inverse Fourier transform is f(x,y) = FT-l[F(M,v)] = J "jF^vy^^dfidv. (2.6) -CO -CO Analysis o f a function in the frequency domain provides direct information about the magnitude of lower and higher frequency components. Higher frequency 30 components are formed by high resolution aspects of the spatial function [44, 49]. The ability of an imaging system to preserve the high spatial frequency characteristics of an object relates directly to the spatial resolution capabilities of the system. The Fourier transform of a function w i l l contain real and imaginary parts. When converting a spatial function to its Fourier representation, absolute position information is contained in the phase component of the transformed function. The Fourier function may be written: F{^v)^\F{^v}eiM. (2.7) The phase angle, 0[ju, v), is equal to </>{ju,v)-= tan where R and I are the real and imaginary parts respectively. When describing spatial resolution characteristics of a spatially invariant system it is only necessary to evaluate the frequency spectrum, \F(JU, V)| . \F(M,v] = [R2{^,v) + I2{M,v)j (2.9) The spatial frequency spectrum |F(//,V)| contains only real components and is used for all spatial frequency domain plots displayed in this chapter. 2.1.3.1 Modula t ion Transfer Funct ion Concept A method of quantifying spatial resolution degradation is to supply an input to the system that contains all spatial frequencies and measure the resulting output spatial frequency spectrum. If all spatial frequencies are present in the input, the amplitude R{fi,v) (2.8) 31 of spatial frequencies in the output w i l l represent the ability of the system to transfer any spatial frequency. A function that contains all spatial frequencies is the Dirac delta function, as shown in Figure 2.1(a). The Fourier transform of the delta function is a function equal to unity throughout the frequency domain (i.e. it contains an equal contribution of all spatial frequencies) as shown in Figure 2.1(b). When the delta function is passed through the system it becomes a function often referred to as the point spread function, which has different spatial and spatial frequency characteristics (see Figure 2.1(c)). B y obtaining the Fourier transform of the point spread function, it is then possible to evaluate how well the spatial frequencies have been preserved by the system. The modulation transfer function (MTF) is defined as the absolute value of the output spatial frequency spectrum shown in Figure 2.1(d) scaled to its value at the zero frequency position [7, 49]. It is therefore written as: The M T F describes the spatial frequency transfer capabilities of a system independently of the object being imaged. Also , by applying the M T F to a given input to the system it w i l l be known a priori what to expect in the output. In this thesis, the same theory w i l l be applied in the context of delivering dose distributions. The input w i l l be the ideal prescribed dose distribution and the output w i l l be what can be attained with the given dose delivery device and technique. A n analogue to the M T F used in imaging systems is developed. It describes mathematically the ability o f the dose delivery system to generate the desired dose distribution, just as the M T F describes the spatial resolution abilities of an imaging system. 32 Spatial Domain (*)S(x,y) Spatial Frequency Domain (b) YT[dfx,y)] = 1 0 x,y 0 Imaging System (H) (c) t(x,y) (d) FT[t(x,y)]=Tfav) 0 x,y JU,V Figure 2.1: Illustration of modulation transfer function concept. The Dirac delta function in (a) is input to the imaging system. Its Fourier transform (denoted by FT) is a constant throughout the frequency domain and is shown in (b). The system output t(x,y) is shown in (c). In the frequency domain, the Fourier transform of the t(x,y) is displayed (d). All spatial frequencies are input to the system but some higher frequency information has been reduced in the output. The amount of spatial frequency reduction represents the spatial resolution capabilities of the system. 2.1.4 Sampling Theory In imaging there are many systems that produce an output of finitely spaced pixels. The image formed is not a continuous function but is sampled at discrete intervals. The actual function and its sampled representation have different properties. Spatial resolution of the system w i l l be affected by the sampling frequency at the image [7 , 49]. Mathematically, the sampling process can be modeled by the continuous function of the input multiplied by a series of shifted delta functions, also known as a 33 comb function (III), as shown in Figure 2.2(b) and (c). A multiplication operation in the spatial domain corresponds to a convolution operation in the frequency domain. The Fourier transform of a comb function is also a comb function in the frequency domain. Therefore, the frequency representation of the continuous function is convolved by a sequence of delta functions. The result is a series of adjacent repeated copies of the original frequency spectrum as shown in Figure 2.2 (c). 2.1.4.1 Al ias ing When the sampling interval is increased, the spacing of delta functions in the frequency domain decreases. A t some point, adjacent frequency spectra w i l l overlap. When they overlap the frequency components are summed, which degrades the true frequency spectrum of the continuous function as shown in Figure 2.2. This effect is known as aliasing [49]. It becomes more severe for higher resolution functions and as the sampling interval increases. 2.1.4.2 Nyquist Cr i te r ion If a function has a finite spatial frequency extent (band-limited) with maximum spatial frequency given by /jmax, then it is possible to fully describe that function through sampling i f the sampling frequency is equal to or greater than 2/jmax. Commonly referred to as the Nyquist criterion [44], it is derived from the duplication effect of frequency spectra due to sampling shown in Figure 2.2. A t the Nyquist frequency, adjacent spectra w i l l be as close as possible without overlapping, preserving all spatial frequencies throughout the sampling process. B y fully reproducing the spatial frequency spectrum, the spatial function w i l l also be preserved in its entirety, causing no degradation of spatial resolution as a result of sampling. 34 Spatial Domain (a) f(x) Spatial Frequency Domain FT\f(x)] = *W (b) m(x/Ax) Ax + (c) f ( x ) m ( x / A x ; •1/Ax 0 1/Ax F ( / / ; ® m r M x ; 1 X 0 -1/Ax 1/Ax Figure 2.2: The function shown in (a) is sampled at a frequency 1/Ax through multiplication by the comb function (b). The result is a discrete representation of the original function shown in (c). A multiplication in the spatial domain corresponds to a convolution in the spatial frequency domain. The frequency spectrum is therefore convolved by the Fourier transformed comb function, producing multiple shifted copies of the original spectrum. When the sampling frequency is too low, adjacent frequency spectra overlap, causing a degradation of the original frequency spectrum know as aliasing. In this chapter sampling effects are investigated in the context of generating dose distributions. The theory introduced here is applied to radiation delivery, which allows a better understanding of the limitations of current equipment and techniques. 35 Furthermore, by modifying sampling characteristics of the system it may be possible to improve spatial resolution. 2.1.5 Application to Dose Delivery Systems L S T is a wel l established mathematical tool used to describe the spatial resolution capabilities of imaging systems; [50, 51] the benefit being that it permits the quantification of resolution degradation at each stage in the imaging process independently of the object being imaged. Recently, there have been several investigations into quantifying the capabilities of dose delivery systems using similar models, [52, 53] some of which have employed methods that can be used when assuming a linear, spatially invariant system. [37, 54] In particular, Bortfeld et al. have described an investigation into the optimal leaf width of a multileaf collimator [37]. Using sampling theory they conclude that there is no benefit in decreasing the leaf width below a specific value given the range of photon scatter and electron transport in tissue. This implies an optimal leaf width on the order of 1.5 to 2 mm. In practice, however, M L C leaf widths typically vary from 3 to 10 mm. In the following investigation a complete model of resolution degradation is presented. B y understanding the processes involved in reducing spatial resolution it w i l l be possible to evaluate the limits imposed by current devices and techniques. Also , and perhaps more importantly, this investigation provides insight into a method to circumvent these limits and improve spatial resolution of the system. 2.2 Theory: Dose Transfer Function of Spatial Resolution Degradation A three-dimensional dose distribution is formed by the superposition of multiple beams where the shape and the intensity of each beam is defined by a stationary or dynamically moving two-dimensional aperture. For conformal radiotherapy, the 36 optimal dose distribution is one where there is 100% of the desired dose encompassing the greatest extent of the P T V and 0% everywhere else. The ideal dose distribution for any given beam is defined as a two-dimensional function perpendicular to beam central axis Dideal (x, y). This is equivalent to a binary mask with value 1 inside the P T V and 0 outside. For I M R T the ideal fluence distribution is defined by intensity modulation and can have dose levels ranging from 0% to 100%. The dose delivery system has inherent spatial resolution limitations that inevitably cause a degradation of the ideal distribution, which may or may not be clinically relevant. These limitations can be separated into different components. Effects due to a finite source size, the M L C leaf dimensions and scattering and electron transport in the patient are considered here. 2.2.1 M L C S a m p l i n g The M L C reduces the spatial resolution by forcing a sampling at width w equivalent to the spacing of M L C leaves in one dimension and a sampling at the incremental position of the leaves in the direction of motion in the orthogonal dimension. The sampling interval in the direction of leaf motion is much less than the interleaf sampling interval which allows the overall sampling to be described by the one-dimensional "comb" function IIl(y/w), where w is the distance between leaf centers. The sampled dose function is then convolved with the function describing the two-dimensional transmission of an M L C leaf MLC(x, y), further modifying the dose distribution. 2.2.2 M L C F u n c t i o n In this thesis a simplified model of the M L C is used. The M L C leaf is considered to be a one-dimensional square "rect" function [37] (see Figure 2.3(a) on page 40). The 37 actual M L C leaf function (i.e. to include tongue and groove effects and rounded leaf ends) may be substituted in future work. 2.2.3 Dose Spread Kernel Mohan et al. have shown that an infinitely thin pencil beam of high energy photons (effectively an impulse function) w i l l generate a point spread function of energy deposition (dose) when interacting with matter [55]. This impulse response to photon fluence allows calculation of dose using a point spread function, which is referred to as the dose spread kernel DSK{x,y). The previous result of spatial resolution degradation due to the M L C is therefore convolved with the D S K , providing a further degradation in spatial resolution. The D S K is approximated by a two dimensional Gaussian. This approximation has been shown to be an adequate model for dose calculation [56] and combines the effects of dose deposition at a given depth and finite source size into one kernel. Experimentally derived D S K s could be used to model a specific linac and beam energy in future work. 2.2.4 Dose Transfer Function Combining all o f these terms, the resulting dose distribution D(x,y) is calculated from: V J (2.11) D(x,y)= Dideal{x,y)lli^ ® MLC{x,y)® DSK{x,y) (2.12) 38 In order to approximate a spatially invariant system, it is assumed that the fluence and energy spectrum of the open beam is the same everywhere perpendicular to beam central axis. Linacs are designed with flattening filters that minimize variations in photon fluence across the beam. Small perturbations due to non-uniformity of the radiation beam do not affect the ability of the model to evaluate spatial resolution capabilities of the dose delivery device. These approximations facilitate an understanding of the underlying processes in applying L S T using Fourier analysis. Obtaining the Fourier transform of Equation 2.12, we define the "Dose Transfer Function" (DTF) where: D(M,v) = [Dideal{M,v)® lll^wyMLCi^i, v)DSK(ju, V) (2.13) The arguments (jU,v) signify the function is the Fourier transform of its spatial domain function from Equation 2.12. 2.2.5 Spatial Frequency Representation O f Dose Figure 2.3 illustrates component functions of the D T F for a one-dimensional system and provides a useful example to aid the conceptual step to a two-dimensional system. Dose degradation functions for a 1 cm wide square M L C leaf and a Gaussian D S K with rj = 2 mm are displayed in the spatial and spatial frequency domains in Figure 2.3(a) and (b) respectively. Figure 2.3(c) shows a desired simplified dose profile resembling a saw tooth function. M L C leaf motion is perpendicular to the y-axis. After sampling and convolution by the leaf function and D S K , it is apparent that this system is unable to match the desired dose profile. The sampling is too coarse and, as a result, misses the first valley and the third peak almost entirely. Also , limitations imposed by the leaf width and D S K cause a further blurring, degrading even the more adequately sampled portions and extending dose beyond the edge of 39 (a) Spatial Domain Frequency Domain 1 . 0 0 . 8 , 0.6 0.4 0 . 2 0 . 0 y (mm) S 0.8 L y (mm) MLC Leaf Sampling Leaf Function ~Dose Spread Kernel - 0 . 2 ideal Dose Profile MLC Sampling Multiplied by Leaf Functionl .0 |-Multiplied by Dose Kernel 0.8 -0.6 . 0 . 4 -0 . 2 (d) 0 . 0 •0.1 0.0 0.1 Spatial Frequency (mm') -0.1 0.0 0.1 Spatial Frequency (mm"1) 0.2 0.00 0.05 0.10 0.15 Spatial Frequency (mm"1) Figure 2.3: Degradation of a desired dose distribution due to the MLC and DSK. An MLC sampling function for a 10 mm MLC leaf and a Gaussian Dose Spread Kernel with cr=2 mm are shown in the spatial domain (a) and the frequency domain (b). The ability of this MLC and DSK to deliver a "sawtooth " dose distribution is shown in the spatial domain (c) and the frequency domain (d). Greater detail in the spatial frequency spectrum degradation can be seen in (e). 40 the desired profile. The frequency spectrum of the same process is shown in Figure 2.3(d) with an expanded scale in Figure 2.3(e). Aliasing occurs when the intensity profile is sampled at less than the optimal frequency required to completely describe it. Due to the finite sampling interval, the frequency spectrum of the profile is repeated in the Fourier domain at integer multiples of the sampling frequency (Figure 2.3(d)). Therefore, i f the sampling frequency is less than twice the maximum spatial frequency of the object (Nyquist criterion), adjacent spectra w i l l overlap and degrade the optimal profile. Aliasing of the desired dose profile due to insufficient sampling is shown in Figure 2.3(e). Multiplication by the leaf function dampens the higher spatial frequencies, removing most o f the aliased peak. Finally, multiplication by the D S K acts as a low pass filter, preferentially dampening higher frequency components. 2.3 Method: Application to 2-dimensional Dose Distributions Equations 2.12 and 2.13 are separated into a series of two-dimensional convolution and multiplication operations that are applied to an ideal distribution. The degradation in dose conformity resulting from each operation can then be qualitatively interpreted in the spatial and spatial frequency domains using two-dimensional intensity maps. For visualization of the higher frequencies the logarithm of the frequency spectrum F{JU, v ) . n t e n s i t y is assigned to each intensity value. FM***,=M>& + F{Mlv)\ (2.14) 41 Figure 2.4: Illustration of MLC sampling in the degradation of an ideal dose distribution in the spatial and frequency domains using the Linear Systems model. X denotes a multiplication and 0 denotes a convolution. In the spatial domain, the ideal dose distribution (a) is multiplied by (b), sampling due to the MLC leaf spacing. The Fourier spectrum of (a) is convolved by the Fourier transformed sampling function of (b). The sampled distribution is shown in (c). 4 2 A is a normalization factor that exploits the full range of grayscale values and 1 is added to F(/J,V) SO that v ) / n t e n s i t y does not go below 0. Figure 2.4, Figure 2.5 and Figure 2.6 show the effect of applying each term of Equations 2.12 and 2.13 to a head and neck P T V shape in the spatial domain and in the frequency domain respectively. Observing modifications to the spatial frequency spectrum allows a direct visualization of the degradation in spatial resolution resulting from each term o f the D T F . The first step is a multiplication of the ideal dose Dideal(x, y) shown in Figure 2.4(a) by the one-dimensional comb function, Figure 2.4(b). In the frequency domain this corresponds to a convolution by a series of delta functions positioned along the y frequency axis. The result of these operations is shown in Figure 2.4(c). It is apparent that at this leaf spacing there is significant aliasing causing an addition of erroneous high frequency components to the spectrum. The next step is to convolve (or multiply in the frequency domain) the result by the M L C leaf function shown in Figure 2.5(b). This causes a further modification of the frequency spectrum and dampens the higher frequency components from the previous step preferentially in the y direction (Figure 2.5(b)). The M L C shaped distribution is shown in Figure 2.5(c). Aliasing effects are apparent from the increase in higher spatial frequencies when the result is compared to the initial spectrum of Figure 2.4(a). The final step is to convolve (multiply in the frequency domain) this result by the D S K (Figure 2.6(b)). This produces a blurring of the M L C leaf fluence as shown in Figure 2.6(c), which is similar to a low pass filter, corresponding to a further symmetric dampening of higher frequency components. 43 S P A T I A L D O M A I N F R E Q U E N C Y D O M A I N Figure 2.5: The sampled dose distribution (a) from the result of Figure 2.4 is convolved by the MLC leaf function (b). In the Frequency domain the sampling spectrum is multiplied by the Fourier transformed leaf function of (b). The resulting MLC shape is shown in (c). It is apparent that aliasing has occurred from the increased high frequency component that is not present in the initial spectrum of Figure 2.4(a) 44 In conformal radiation therapy, the goal of the M L C is to block all healthy tissue around the greatest extent of the projection of the P T V . The beams-eye-view is used to derive the leaf positions that provide the maximum conformity. The degree of conformity is related to the shape of the P T V and the characteristics of the M L C . Conformity error consists of two elements, irradiation of healthy tissue and under-dosing o f the treatment volume. For the purposes of evaluating the model, geometric conformity is defined as: , .„ Area of PTV-(Area of PTV underdose + Area of healthy tissue dose) (J 1 "Aconformity = 100 x - i - - '-Area of PTV The area boundaries are defined to be the 50% isodose line. M L C leaf positions are chosen so that the center of the leaf edge intersects the edge of the P T V . This method is used because it does not bias the results toward under-dosing of the P T V or toward greater healthy tissue dose. Also , the definition of conformity in Equation 2.15 encompasses both effects and is therefore not sensitive to the leaf positioning method. 2.3.1 Study of Circular P T V Shaping Before embarking on a study using clinical P T V shapes, a baseline was needed to interpret results effectively. For that purpose a study was undertaken to evaluate the conformity o f circular P T V shapes of varying radius. A circle is an ideal shape because it is independent of collimator rotation and has a wide range of spatial resolution requirements [57]. For example, the central M L C leaf edges are almost parallel to the circle edge and provide a very close fit (see Figure 2.7). The conformity degrades for leaves farther away from the center until eventually the leaf edges and circle edges are perpendicular. The final advantage of using a circle is that it is possible to find an analytic solution to the conformity as a function of the circle radius r and the leaf width w [57]. 45 Figure 2.6: The resulting MLC shape (a) is convolved by the dose spread kernel (b) to give the deliverable dose distribution (c). In the frequency domain the MLC shape spectrum is multiplied by the Fourier transform of the dose spread kernel in (b). The DSK acts as a symmetric low pass filter, preferentially reducing high frequency components as shown in (c). 46 2.3.2 S t u d y Of P T V Shapes A P T V that has a shape with many protrusions of small diameter w i l l have more high frequency component than that of a uniform shape of similar dimensions [58]. It is therefore important to also evaluate the resolution degradation of varying shaped P T V s . For a qualitative investigation, typical clinical shapes were chosen for their diversity in shape, size and planning complexity [59]. These shapes were: a prostate, a head and neck lesion (nasopharyngeal carcinoma), and a central nervous system (CNS) lesion (an arteriovenous malformation). Table 2.1 lists their largest diameter and average spatial frequency at one tenth of the zero frequency peak. The prostate, head and neck, and C N S shapes show an increasingly larger component of high spatial frequencies. Therefore, for conciseness, these shapes w i l l henceforth be referred to as the L o w Frequency Shape (LFS), Moderate Frequency Shape (MFS) and High Frequency Shape (HFS) respectively. Two-dimensional dose distributions were calculated using Equation 2.12 with leaf width equal to 5 mm. A D S K with rj= 2 mm (comparable to a 6 M V beam at moderate depths of approx. 10 cm) was used in all cases [37]. It should be noted that the D S K is a function of depth. The D S K s from a commercial treatment planning system (CadPlan, Varian, Palo Al to , C A ) at depths ranging from 5 to 20 cm for a 6 M V photon beam have been evaluated. The a used when fitting Gaussians to these D S K s were found to increase slightly with depth. The D T F is also applicable to I M R T . The intensity levels required over the two-dimensional shape complicate resolution requirements. In order to provide a comparison with the resolution degradation of conformal treatments, an example of an I M R T prostate distribution was also calculated using a 10 mm M L C leaf width and a = 2 mm D S K . Finally, conformity was calculated as a function of M L C leaf width and D S K size for the 3 shapes listed in Table 2.1. Due to the irregularity of the shapes, results are 47 dependent on the collimator rotation angle. To remove this dependence, only the optimal collimator angles were used. The optimal angles were derived by calculating the conformity (equation 2.15) at 5 degree collimator angle increments and selecting the one with the greatest conformity. Table 2.1: A summary of the PTV shapes used in the qualitative and quantitative investigations of the model. Each shape is referred to using the acronym describing its spatial frequency characteristics. Site Largest Diameter (cm) Mean Spatial Frequency at 10% of Zero Frequency Peak (cm"1) Descriptor * Prostate 8.3 0.14 L o w Frequency Shape (LFS) I Head and Neck 6.3 0.38 Medium Frequency Shape (MFS) C N S 3.0 0.62 High Frequency Shape (HFS) 2.4 Results: Circular PTV Shaping Derivation of the analytic solution of conformity resulting from M L C shaping of circular P T V s is given in Appendix A . Presented here are specific results of that derivation given in the context of the L S T model. 2.4.1 M L C Effects - Analytic Representation The analytic solution to the conformity of circular P T V shapes involves the sum of a series of area integrals with limits of integration defined by the intersection points of the M L C with the circle, as indicated in Figure 2.7. The simplified equation is: 48 f N 7TT — 4 %conformity = 100 x V (Area A k + Area B k ) + Remainder Area Vk=0 nr (2.16) M L C 1 i K 1 r W C I R C L E Area B 2 ' Area A2 Area Br Area A i Figure 2.7: MLC conformity for circular PTVs. Dark shaded areas indicate underdosed regions of the PTV. Light shaded areas indicate where healthy tissue is receiving dose. Poor coverage is observed for the peripheral leaves, i.e. when the leaf edges are oblique to the circle edge. A full derivation of the analytic solution is given in Appendix A. 2.4.2 Dependence On Leaf Width and Circle radius The derived circular shape conformity function is plotted in Figure 2.8 as a function of leaf width for 2.5 cm and 10 cm diameter circles. The conformity is periodic and, in general, decreases with increasing leaf size. The periodic nature o f the curves is primarily due to the peripheral leaves. When the circle diameter is an integer multiple of the leaf width, the leaves furthest from the center coincide exactly with the edge o f the circle. A t this point the conformity is at a maximum. If the leaf width is increased or decreased the condition is no longer satisfied and the conformity 49 deteriorates until it becomes a minimum when the leaf width is half way to being an integer multiple of the circle diameter. Ignoring the periodic characteristics of Equation 2.16, the overall decline in conformity with leaf width can be written as: (2.17) 7 0 H — i — i — i — i — i — i — i — i — i — \ — i — i — i — i — i — i — i — i — i — I 0 1 2 3 4 5 6 7 8 9 1 0 Leaf Width (mm) Figure 2.8: Conformity vs leaf width for circular PTV shapes with and without a a=2 mm dose spread kernel. The decrease in conformity is a periodic function of leaf width. With the periodicity removed the decline in conformity is a linear function of w leaf width, w, with slope equal to . nr 5 0 Equation 2.17 is a linearly decreasing function with respect to the leaf width w and linearly increasing with respect to the circle radius r. It should be noted that the amount of high spatial frequency component contained in a two-dimensional shape is inversely proportional to its size. Therefore, a given resolution-degrading component w i l l have a more severe effect on a smaller target. This property is preserved in equation 2.17. A s the circle gets smaller, spatial resolution requirements are increased proportionately, which results in poorer conformity for a given M L C leaf width. The same argument holds for the spatial filtering effects of larger M L C leaves. A s the leaves get wider they allow less high frequency component to pass through the system, resulting in poorer conformity for a circle of given radius. 2.4.3 Dose Spread Kernel Also plotted in Figure 2.8 are results of conformity versus leaf width for the same circular P T V shapes obtained with the complete degradation function (see equation 2.12). The difference in the curves is caused by convolution with the D S K , which is not taken into consideration in the previous analytic result. When the D S K is applied, the area boundaries are defined to be the 50% isodose line. Although the periodic nature o f the curves is preserved, the inclusion of the D S K provides a higher degree of conformity at larger leaf widths. This effect seems counter intuitive at first but is explained using the L S T formulation. A t larger leaf widths the sampling interval increases. Aliasing effects due to finite sampling become more severe with increased leaf spacing. Recall that aliasing introduces erroneous high frequency component to the spatial frequency spectrum (see section 2.1.4.1 and Figure 2.2). The D S K filters out high frequency components and is therefore competing against the effects of aliasing. When the leaf spacing gets large enough, the D S K filters out more incorrect high frequency than true frequency information. Therefore, the D S K w i l l actually improve conformity at larger leaf widths. A t smaller leaf widths, the D S K becomes the dominant factor degrading the dose distribution and results in lower conformity. 51 2.5 Results: Clinical Simulation Of Resolution Degradation The L F S and H F S ideal dose distributions are displayed with their spatial frequency spectra in Figure 2.9(a) and (c) respectively. The H F S exhibits a greater proportion of high frequency components than the L F S . Also , the H F S frequency spectrum is not symmetric, showing higher spatial frequency along the axis perpendicular to the location of the thinnest part of the P T V in the spatial domain. Figure 2.9(b) and (d) show the L F S and H F S results when using a 5 mm leaf width to generate dose distributions. The L F S result shows a superior level of conformity when compared to the H F S . This is evidenced by the degree of degradation of spatial frequencies in each case. Because the L F S has a smaller high frequency component than the H F S it w i l l be less sensitive to spatial frequency filtering caused by the M L C and D S K . Detailed results for the M F S were displayed in Figure 2.4 to Figure 2.6. 52 Figure 2.9: Degradation of the ideal dose distribution in the spatial and spatial frequency domain for (a), the LFS and (c) the HFS. Resulting degraded distributions are shown in (b) and (d). A 5 mm leaf width was used for both PTV shapes. Dose distributions were calculated using Equation 2.12. A Gaussian dose spread kernel with a a = 2 mm was used in all cases. 53 The same effect is observed when using a 1 cm leaf width to generate the I M R T dose distribution as shown in Figure 2.10(b). Higher spatial frequencies have been modified, causing an overall blurring and "pixelation" of the ideal distribution. The aliasing phenomenon can be seen at the top and bottom of the resulting frequency intensity map. It is more severe than the conformal cases due to a larger M L C leaf width. SPATIAL DOMAIN FREQUENCY DOMAIN Figure 2.10: An ideal IMRT prostate field dose distribution is shown in (a). Degradation of the ideal dose distribution in the spatial and spatial frequency domain for the resulting distribution was calculated using Equation 2.12 and is shown in (b). A 10 mm leaf width MLC and a Gaussian dose spread kernel with a = 2 mm was used in the calculation. 2.5.1 Collimator Angle Dependence The collimator angle that provides optimal conformity was used for all results in the quantitative analysis o f clinical P T V shapes. Aliasing effects due to the M L C leaf spacing and multiplication by the leaf function w i l l cause a more severe deterioration 5 4 in the frequency spectrum for certain collimator angles. A plot of optimal collimator angle vs. leaf width for the clinical shapes is displayed in Figure 2.11. The L F S is fairly circular and therefore does not exhibit a preferred angle. The M F S has a preferential collimator angle at approximately 90 degrees. This same collimator angle was used in Figure 2.4. The frequency spectrum of the M F S is asymmetric (Figure 2.4(a)). The optimal collimator angle places the shape with the higher frequency components in the direction of leaf motion, which is the direction suffering the least amount of high spatial frequency degradation. The H F S shows a wide range of optimal collimator angles. Over certain ranges of leaf widths there is a preference but, in general, there is no distinct orientation that is obtained consistently. 5 5 180 135 90 45 CD 0 D) 180 c < 135 L . o 4 - » 90 05 E 45 o o 0 180 ro E 135 ^—» Q . O 90 45 0 L F S * • A A AA I A A A A A A A A A A A A A A I 1 1 1 1 ^ 1 I I MFSJ • • • • • " a • - • . " • • . m u i i i i i i i i i HFS# • •• • • • • # •••• • * •••••• i ) 1 I I I I 1 1 1 1 1 2 3 4 5 6 7 8 9 10 MLC Leaf Width (mm) Figure 2.11: The collimator angle providing optimal conformity is plotted versus leaf width for the LFS, MFS and HFS. 2.5.2 M L C Leaf Width - Aperture Shaping Results of conformity versus leaf width for the three clinical shapes are shown in Figure 2.12. The L F S conformity is a slowly decreasing function o f M L C leaf width. The M F S is slightly offset, shows a steeper slope and becomes more chaotic after approximately 7 mm leaf width. The H F S has a much more significant offset on the conformity axis than the previous 2 examples. Also , the rate of decrease and amount of variation increases more significantly for leaf widths greater than 4 to 5 mm. It is intuitive that it should be more difficult to conform to the smaller, more complex 56 volumes. It was shown in Figure 2.9 that the spatial frequency spectrum of the H F S contains a greater proportion of higher frequency information than the L F S and M F S . A s a result, the degradation in the frequency spectrum for a given M L C leaf width is more severe for the H F S . The offset toward lower conformity at very small leaf sizes for the M F S and H F S is caused by spatial resolution degradation due to the D S K . Degradation in spatial frequency resulting from the M L C leaves is significantly less and the dominant effect becomes the filtering of high spatial frequencies caused by the degradation due to the D S K . Again, this effect is more severe for the H F S than the other two shapes. 100-1 95 -\ A A A * A ^ A A 90 H ^ 85 >* 'E o 80 4— c o O 75 H 70 H A LFS m - MFS i • HFS f • • • 65- ~ i — 1 — i — • — i — 1 — i — 1 — i — 1 — i — 1 — i — 1 — i — < — i — 1 — i — 1 1 2 3 4 5 6 7 8 9 10 MLC Leaf Width (mm) Figure 2.12: Conformity vs leaf width for the LFS, MFS, and HFS. A gaussian dose spread kernel with <j=2 mm was used in all cases. 5 7 2.5.3 Dose Spread Kernel - Blurring The dependence o f conformity on the D S K for varying leaf width is shown for the H F S in Figure 2.13. A s expected, the conformity with no D S K approaches 100% for small leaf widths. Increasing the width of the D S K increases the offset of the curves on the conformity axis. For larger leaf widths the same effect observed in the circular P T V shapes in section 2.4.3 is observed here. Instead of the D S K consistently reducing the conformity at greater leaf widths there is actually an increase, as shown by the crossover in the range 3 to 5 mm. This can also be explained using the L S T formulation in Equation 2.12 and 2.13 and was also observed in the circular P T V shapes. The aliasing effect due to the sampling interval of the M L C leaves causes an addition of incorrect high frequency information that degrades the frequency spectrum. The effect is more severe for greater leaf spacing. The D S K counteracts this effect by acting as a low pass filter. The aliased high frequency information is therefore suppressed, providing better conformity in the final result. The leaf width at which these two effects are in equilibrium is located at the intersection of the curves with and without the D S K . This lies at approximately 3.5 mm and 5 mm for the G=2 mm and CT=3 mm D S K s respectively. 58 100 - i I " ' I ' ' I " I ' I "I I I I I I' I ' " I' • pill—•III- I I I l> I III.I 0 1 2 3 4 5 6 7 8 9 10 MLC Leaf Width (mm) Figure 2.13: Conformity vs leaf width for the HFS shapes using a gaussian dose spread kernels with a=0 mm, 2 mm and 3 mm. The (7=2 mm HFS curve is also plotted in Figure 2.12. 2.6 Discussion: Implications of Resolution Degradation The application of L S T to dose delivery systems provides a model for quantifying inherent limitations in obtaining desired dose distributions. The same theory has been used on imaging systems to quantify spatial resolution degradation. In imaging, the modulation transfer function (MTF) is used to describe the degradation o f spatial resolution [44]. The M T F quantifies the transfer of spatial frequencies from the object to the image and is the most commonly used comparator when evaluating different devices. This concept can be borrowed from imaging to describe an 59 effective M T F [54] which we refer to as the D T F , for a dose delivery system. In a practical case, this would involve the experimental determination of the transfer of spatial frequencies for each component that affects spatial resolution. Again, methods similar to those used in imaging could be employed. It would then be possible to compare various linacs (x-ray source), energies and M L C s (leaf width, tongue and groove, rounded leaf ends) in a simple, standardized way by comparing their DTFs . Finally, by including a depth varying D S K it w i l l be possible to extend the model to the third dimension and allow the evaluation of spatial resolution degradation resulting from a multiple beam geometry. The principle advantage in modeling dose delivery as a linear system is that it allows each spatially degrading component to be evaluated separately. The Fourier transform of each component function in the system describes the spatial frequency transfer for that component. It is therefore possible to investigate the spatial resolution degradation effects from each component independently of the volume being treated. This provides a significant benefit in that improvements in dose delivery devices no longer have to be evaluated on their ability to treat an arbitrarily chosen treatment volume. Another benefit o f using this technique for evaluation is that the desired characteristics of the linac can be tailored to the treatment site. Frequency analysis of the P T V shapes showed that smaller, more complex structures have a greater high frequency component, making them more sensitive to spatial filtering caused by the D T F . This prediction was validated in the quantitative investigation of conformity where it was shown that there is only a small benefit in using a 5 mm leaf width when treating shapes with lower spatial frequencies while shapes with higher spatial frequencies benefit significantly from the smaller leaf width. Information on the characteristics of the delivery system that are required to provide adequate conformity 60 can be obtained by comparing the spatial frequency spectrum of the P T V and the D T F defined by the delivery system. Several authors have proposed methods of improving existing dose delivery systems [60-64]. For example, Bortfeld et al. have used sampling theory to investigate the advantage of shifting the patient half a leaf width during treatment [37]. The advantages of these methods can be evaluated by calculating the effective D T F o f the modified system. The L S T model presented here is focused entirely on the dose delivery system and static patient. In a real life scenario, some notable limitations have to be considered. These include inaccuracies due to patient motion and P T V delineation using images that have their own specific resolution. It is possible to incorporate these effects into the model. For example, patient motion causes a blurring of the dose distribution. I f the motion is well defined it could be described by a low pass filter, adding another term to the D T F . To account for limited imaging resolution, the imaging M T F could be incorporated directly into the D T F , again by adding an appropriate term. 2.7 Discussion: Application Of The DTF To IMRT In the previous discussion the benefits of using the L S T formulation to describe spatial resolution degradation in dose delivery were presented. The focus has been, for the most part, on aperture shaping of the M L C and scatter and electron transport when photons interact with the patient. In I M R T , complex fluence maps are generated by superimposing multiple M L C defined apertures. The cumulative sum of these apertures is a 2-dimensionally varying fluence. The additivity property (equation 2.2) of linear systems permits an extension of the model beyond simple shaping of conformal fields. The result of applying the L S T formulation to I M R T was shown in Figure 2.10. 61 Revisiting the D T F , there are three basic terms in the spatial domain mathematical expression. Sampling of the ideal dose distribution, convolution by the M L C and convolution by the D S K . The M L C function is given by the design of M L C leaves of the given linac. The D S K is a function of the photon energy and the size of the x-ray source. Both of these functions are therefore fixed parameters for any given linac. The only remaining parameter is the M L C sampling. Although the sampling interval is fixed for any single aperture, rotation of the collimator allows flexibility in choosing the sampling direction for the M L C leaves. The benefits of selecting the optimal collimator angle were observed in the results of collimator angle versus conformity shown in Figure 2.11. These results indicate that some collimator angles provide better sampling directions than others. Extending this effect to I M R T , it should be possible to generate higher resolution fields by modifying the collimator angle for each sub-field aperture. This observation is the key to improving spatial resolution for intensity modulated fields. The remainder of this thesis is devoted to developing, testing and evaluating a new method of generating intensity modulated fields. The novelty of the method is based on rotating the collimator by an angle increment between each sub-field. Deriving the apertures necessary to generate arbitrary I M R T fields in a rotating geometry is significantly more complex due to the additional degree of freedom. A new method of calculating rotated M L C apertures has been developed as part of this thesis work. The following chapters w i l l describe the apparatus, leaf motion algorithms and experimental evaluation of this new technique. 62 Chapter 3 I M R T DELIVERY WITH M L C ROTATION M L C based I M R T techniques are well established but suffer several physical limitations. Dosimetric spatial resolution is limited by the M L C leaf width; interleaf leakage and tongue-and-groove effects degrade dosimetric accuracy and the range of leaf motion limits the maximum deliverable field size. Collimator rotation is used in standard radiation therapy to improve the conformity of the M L C shape to the target volume. Except for opposed orthogonal fields, collimator rotation has not been exploited in I M R T due to the complexity of deriving the M L C leaf configurations for rotated sub-fields. Here, a new way for M L C based I M R T delivery is proposed which incorporates collimator rotation, providing an extra degree of freedom in deriving leaf sequences for a desired fluence map. Specifically, a series of unique algorithms were developed that are capable of determining rotated M L C segments. These I M R T fields may be delivered statically (with the collimator rotating to a new position in between sub-fields) or dynamically (with the collimator rotating and leaves moving simultaneously during irradiation). In this chapter a detailed description of potential improvements in spatial resolution, reduction of systematic dosimetric error and improved field size capability with collimator rotation is provided [65, 66]. Linear accelerators were not designed to produce fluence maps with combined M L C motions and collimator rotations. For that reason the mechanical characteristics of the linac that control M L C motion, collimator rotation and radiation production are described as they apply to rotational I M R T delivery. A series of experiments were performed to evaluate the magnitude of 63 errors that are inherent in rotating the collimator during radiation production. Mechanical properties include accuracy of the center of collimator rotation and collimator angle reproducibility. Radiation production characteristics were also investigated. L o w dose sub-fields were investigated for static delivery. In dynamic delivery the uniformity of collimator rotation speed and constancy of the dose rate were evaluated. Finally, the feasibility of delivering I M R T with collimator rotation is discussed in the context of results provided by the mechanical and dosimetric experiments. 3.1 Rotational Delivery Method A field of non-uniform intensity can be divided into a series of uniform sub-fields each having a different multileaf collimator configuration, as described in section 1.5.1.1. B y delivering each one of these constant intensity sub-fields it is possible to build up a field of non-uniform intensity [17]. The technique for intensity modulation developed in this thesis is different from conventional methods in that the entire collimator, including the M L C , is rotated between each sub-field. A full description of conventional I M R T delivery methods is provided in Section 4.1. Displayed in Figure 3.1 is an example of a linear accelerator equipped with an M L C and rotating collimator. The patient is placed on a moveable table used to position the treatment site with respect to the linac isocenter. The linac gantry is rotated so that the beam enters the patient from a desired direction. The collimator rotates about an axis that passes through the isocenter and the M L C leaves move perpendicular to that axis. 64 Figure 3.1: Simplified diagram of a linear accelerator equipped with an MLC and rotating collimator. The collimator rotates about an axis that passes through the isocenter. The orientation of that axis is always perpendicular to the direction of leaf motion. Shown in Figure 3.2 is a diagram of the collimator at several distinct rotation angles. The fluence contribution from the M L C aperture at each angle as wel l as the progressive cumulative fluence distribution from each one of those apertures is also shown. The M L C leaves are used to form a uniquely shaped aperture as shown in Figure 3.2(a). Each leaf is adjacent to its neighbour and may only move linearly in and out o f the radiation field. A predetermined quantity of radiation is delivered with the first sub-field configuration before rotating the collimator and changing the leaves to the configuration of Figure 3.2(b). Again, a predetermined quantity of radiation is delivered. This aperture defines a different shape than the previous aperture. Part of the areas overlap as shown in Figure 3.2(c). The resulting fluence distribution in those 65 MLC Fluence Cumulative Fluence Figure 3.2: Multiple MLC apertures at varying collimator angles contribute to the final fluence distribution. Two different MLC apertures are used to generate uniform fluence distributions in (a) and (b). The sum of (a) and (b) are displayed as a surface map in (c). Complex fluence distributions as shown in (d) are formed using several sub-fields each having a different rotation angle and MLC configuration. 66 areas is the sum of both sub-field contributions. Conversely, where they do not overlap the contribution is only from the sub-field that is open at that point (Ignoring leakage and scattered radiation). In this fashion, multiple apertures at varying rotation angles are generated and radiation is delivered. The fluence resulting from each sub-field is added to the previous sub-field contributions. The final distribution of fluence is therefore the cumulative sum of the contributions from each individual sub-field as shown in Figure 3.2(d). 3.2 Enhancements To IMRT With M L C Rotation The M L C has certain characteristics that limit its ability to conform to a target and reduce dose to healthy tissue. Because each leaf has a finite width, the spatial resolution of fluence maps perpendicular to the direction of leaf travel is limited to that width [37, 57]. The M L C is constructed with a tongue-and-groove shape on the side of each leaf in order to minimize interleaf leakage. However, transmission through the leaves is still non-uniform and cannot be compensated for entirely [67, 68]. A l so , the tongue-and-groove creates unwanted under-dosing effects for some intensity modulated fields [69]. M L C s are constructed with enough leaves to cover a given length (e.g. 40 cm). Due to mechanical and physical limitations described in section 3.2.3, the range of leaf motion is restricted to a portion of that length [70]. The maximum intensity modulated field size is therefore limited to a rectangle whose width is given by the range of leaf motion. Several authors have reported variations on standard M L C based techniques that attempt to improve the spatial resolution of fluence maps [61, 71, 72], reduce interleaf leakage and tongue-and-groove effects [52, 73, 74], or increase the maximum deliverable field size [75]. Collimator rotation adds a degree of freedom to the sub-fields used to build up an intensity modulated field. A s a result, there are several potential advantages over conventional techniques. For example, the spatial resolution of the desired field is 67 not limited to the leaf width in the direction perpendicular to leaf travel because the leaves are rotated to a different location with each sub-field. The leaf edges are also in a different location with each rotation, reducing the effects of unwanted interleaf leakage and tongue-and-groove under-dosage. Finally, because the rotational method produces sub-fields at multiple collimator angles, the maximum possible field size is given by the superposition of all fields which removes restrictions imposed by the range o f leaf motion. 3.2.1 Spatial Resolution In Chapter 2 it was determined that by rotating the sampling geometry of the M L C leaves it should be possible to increase the spatial resolution of fluence maps. Another way of conceptualizing this improvement is to look at the minimum size of a fluence "pixel" that can be generated in a sequence of M L C defined sub-fields. Referring to Figure 3.3(a), in conventional delivery each leaf moves linearly in and out of the radiation field. The leaves can have displacements that are less than 1 mm, providing high resolution in the direction of leaf motion [37, 76]. In the orthogonal direction there is no leaf movement. The minimum size of a fluence pixel in conventional delivery is therefore a rectangle with width given by the minimum leaf displacement and length given by the width of the M L C leaf. In rotational delivery the direction of leaf motion changes at each collimator angle as seen in Figure 3.3(b). The minimum size of intensity pixel is therefore no longer limited to a rectangle but is more closely approximated by a circularly symmetric point consisting of a plateau region in the center with diameter given by the minimum leaf displacement which then gradually decreases radialy outwards. 68 (a) Conventional t Minimum size of fluence "pixel" (b) Rotational Min imum size of fluence "pixel" Figure 3.3: Potential improvements in spatial resolution between conventional and rotational delivery methods are shown, (a) In conventional delivery the MLC leaves move linearly in and out of the radiation field providing a rectangular pixel size whose length is limited by the leaf width, (b) With rotation the direction of leaf motion changes, allowing for a smaller fluence pixel that is not limited by the leaf width. 3.2.2 Interleaf Effects The M L C leaves used in this thesis are composed of a Tungsten A l l o y , are 6 cm deep and are designed to block greater than 95% of incident photons [77, 78]. For the M L C leaves to move independently, adjacent leaves are not physically connected. Also , a small gap is provided between adjacent leaves so that they may move smoothly in between sub-field irradiations. The interleaf gap allows more photons to leak through the M L C than at intraleaf locations [79]. The effect is reduced by using 69 a modified leaf edge design as described in Figure 3.4. Instead of a flat edge a tongue-and-groove shape is machined into the side of each leaf. With this design there is no location that w i l l let photons pass through un-hindered because either the tongue or the groove w i l l block incident photons. Sti l l , the thickness of metal that the photons have to pass through is less than the full thickness of the leaf, providing greater photon fluence in interleaf locations. Interleaf leakage causes non-uniformity Incident Photons L _ J L _ l L _ J L J L _ l L _ J Interleaf Leakage Figure 3.4: MLC leaves are designed with an interlocking tongue-and-groove shape on the side of each leaf. Although interleaf leakage is reduced significantly with this design there are still more photons transmitted through the interleaf gaps, causing non-uniformity in the transmitted fluence where the MLC is closed. Conventional IMRT delivery methods are unable to compensate for this effect and can result in overdosing at some interleaf locations. in delivered dose distributions that cannot be compensated for in conventional I M R T delivery [80, 81]. A consequence of introducing the tongue-and-groove shape is that for some intensity modulated fields there are underdosing effects along the leaf edges [82]. The fluence profile generated at an open leaf position is shown in Figure 3.5. Because the tongue or the groove protrudes into the field, the fluence at the leaf edge does not increase abruptly. Instead, there is a step of lower fluence before reaching the maximum. Now, consider the case where the leaf that was initially closed is now open and the adjacent leaf is closed. This situation arises frequently during I M R T delivery due to the different sub-field shapes that are required. The desired fluence for such a situation is a constant fluence profile between both leaves. Unfortunately this is not the case and there is a reduction in fluence under the tongue and the groove as shown in Figure 3.5. The physical mechanism for this discrepancy results from the exponential attenuation of photons as they pass through matter. It has not been characterized in the literature and w i l l therefore be described here. For incident intensity I0 the transmitted intensity through an M L C leaf of thickness Ueaf is given by 4„/=V~"w (3-1) where X is the linear attenuation coefficient of the leaf material. For simplicity, consider a tongue and groove design where the tongue and the groove are both equal to Vi the total leaf thickness. 71 1 2 1+2 Fluence Ileaf I t * T T M o n g u e x0 Total Fluence U ^ t Io+Ileaf ^tongue ^ g r o o v e Io+Ile Figure 3.5: The tongue-and-groove effect is illustrated by considering the transmitted photon fluence at an open leaf edge. The dotted arrows indicate the direction and amount of photon transmission. The tongue causes an intermediate step in the fluence profile shown in (1). Closing this leaf and opening the adjacent leaf results in the same effect although this time caused by the groove. The sum of both profdes should ideally result in a constant fluence across both leaves. Due to the exponential nature of photon attenuation, the sum of both (1) and (2) results in a fluence reduction error at the tongue-and-groove interface. heaf tongue groove 2 V - " - " / The total transmitted intensity through the center of the leaf in the open and then the closed position is / 0 + / w = / o ( l + (3-3) 72 and through the tongue and then the groove it is I tongue ^  I groove = ^^0e 2 • (3-4) The magnitude of tongue-and-groove error is a function of the attenuation coefficient and leaf thickness. A s the leaf transmission is decreased the effect becomes more pronounced. The transmission for a typical M L C is on the order of 2%, which corresponds to a tongue-and-groove fluence reduction of approximately 70%. In conventional I M R T delivery the location of leaf edges is fixed. Dosimetric error resulting from interleaf leakage and the tongue-and-groove effect are concentrated along the lines described by these edges. With collimator rotation the location of leaf edges change throughout delivery. Each sub-field is rotated with respect to the previous sub-field as seen in Figure 3.6. With each rotation the leaf edges fall along a different line so that any leakage or tongue-and-groove effects are spread over the 2-dimensional field area. Therefore, the maximum dosimetric error due to these effects w i l l be a fraction of that observed in conventional I M R T delivery. 73 Figure 3.6: An example of 2 rotated sub-fields is shown. Because of rotation the leaf edges of sub-field 2 are not coincident with those of sub-field 1. With several sub-fields the position of leaf edges is blurred over the entire field area. Interleaf leakage and tongue-and-groove underdosing is therefore reduced with collimator rotation. In conventional delivery the leaf edges are fixed, amplifying systematic underdosing and overdosing error at the leaf edges. 3.2.3 Maximum Deliverable Field Size The conventional dynamic method of delivering intensity modulated fields uses a 'sliding window' technique. Treatment begins with the M L C leaves forming an aperture at one side of the field. Once the beam is activated each leaf pair starts moving towards the other side of the field. Varying the speed of each leaf throughout delivery generates the desired intensity modulated field. Wi th this method each leaf pair must travel from one field boundary to the other. The maximum field width that can be generated is therefore limited to the length of the M L C leaf. Otherwise, the far edge of the leaf would pass beyond the edge of secondary jaws. Commercially available models have leaf lengths on the 74 order of 15 cm projected at isocenter. The maximum field length however is only limited to the length of the leaf bank (number of leaves multiplied by their width) and is typically 40 cm. In order to deliver wider field sizes it is necessary to deliver multiple adjacent I M R T fields [75]. C o n v e n t i o n a l R o t a t i o n a l Figure 3.7: The maximum field size for conventional and rotational delivery methods in dynamic mode are shown. The field width is limited to the length of the MLC leaves in conventional delivery. With rotation the maximum field size is larger because (1) the leaves are not required to span the entire field width and (2), because the direction of leaf travel is rotated with each sub-field. The upper limit on rotational IMRT field sizes is a circle whose diameter is equal to the length of the leaf bank. The maximum field size of intensity modulated fields that are generated with collimator rotation w i l l not necessarily be subject to the same restrictions (see Figure 3.7). First, because delivery is not performed using a sliding window the leaves are not required to move from one field edge to the other. Therefore, the maximum field 75 width may be extended to twice the leaf length. Secondly, the orientation of the leaf bank, which has a maximum length of 40 cm, is constantly rotating with respect to the desired field. The upper limit on field size is therefore a 40 cm diameter circle. 3.3 Mechanical Characteristics Linear accelerators that use an M L C to generate arbitrary field shapes are designed with collimator rotation capabilities. In standard radiation therapy it is sometimes desirable to rotate the M L C leaves so that they may fit more conformally to the target. Although this capability exists on the linacs used in this thesis, it was never intended for intensity modulation. The accuracy of M L C leaf positioning combined with collimator rotation must be assessed before embarking on the actual delivery o f rotational I M R T fields. The following subsections describe the mechanisms of the M L C to accurately position leaves in dynamic as well as static modes. The collimator rotation mechanism is also described. Finally, the results of a series of experiments used to evaluate the accuracy and stability of the collimation system are presented. 3.3.1 Multileaf Collimator Individual D C motors drive each M L C leaf. The M L C leaves consist of two opposing sets of banks that are mounted to a carriage. The carriage can be translated in the direction o f leaf motion in order to move all leaves simultaneously. The M L C head assembly includes transceiver cards that convert leaf position data for the controller computer [16]. It also includes the motor driver cards that power the leaf and carriage motors. Finally, it includes secondary feedback circuitry from the leaves and carriages that are used to independently verify leaf position. Primary feedback of leaf position is derived from each leaf motor through the leaf position encoder. The encoder divides each millimeter of leaf travel into 600 76 parts. Each encoder signals a relative position for its leaf. When the M L C is powered up a calibration procedure is used to initialize each encoder at a known leaf position. Secondary feedback for each leaf is provided from two mechanical brushes mounted on and along each leaf in the direction of travel. A s leaves move in and out of the radiation field, the brushes complete a circuit across contact points. This provides a signal that translates to leaf position. These feedback mechanisms are independent from each other. I f either mechanism indicates an incorrect leaf position or i f they become unsynchronized the delivery w i l l terminate. This provides for accurate and reliable leaf positioning throughout dynamic and static delivery to within 1 mm. The leaf calibration procedure is performed using an optical receiver and an infrared L E D [16] as shown in Figure 3.8. First, all leaves are completely retracted so that the optical beam may pass between the emitter and receiver. The position of the straight line described by the sensor and the infrared L E D is known. A given leaf is selected and slowly translated into the optical path. Once the leaf blocks the infrared light the optical sensor signals that the leaf has reached the calibration position. The leaf position is stored and the leaf is retracted. The next leaf is selected and the process continues until all leaves are calibrated on both banks. 77 M L C Leaves I v> ^ — I j r j j T 7 T \ Calibration Line ~ .. . c Infrared L E D Optical Sensor Figure 3.8: MLC leaf positions are calibrated individually using an infrared LED emitter and optical sensor. The position of the optical path described by the sensor and the infrared LED is known. Each leaf is translated into the optical path. Once the leaf blocks the infrared light the optical sensor signals that the leaf has reached the calibration position. The leaf position is stored and the leaf is retracted. 3.3.2 Collimator Rotation The collimator motor assembly is capable of driving the collimator through a rotation range of 330 degrees. The collimator is rotated with a D C motor attached to a mechanical chain that circles the outer perimeter of the collimator. Applying a signal to the motor translates the chain, which rotates the collimator system. The rotation angle is verified by a potentiometer attached to one o f the gears that guide the chain around the collimator. A s the chain translates, the signal from the potentiometer changes by an amount proportional to the angle of rotation. The linac control computer receives this signal and uses it to indicate when the collimator has reached a desired angle. 78 3.3.3 Collimator Angle Reproducibility Varian models C12100EX and C12100C/D (Linac 1 and Linac 2 respectively) were used throughout this thesis. The main difference between these two linacs is their M L C leaf designs. A full description of the M L C design differences is presented in section 5.2.1 and is not relevant to the following study. The feedback and control mechanisms of the collimation system are virtually identical. In order to evaluate the feasibility of using collimator rotation in I M R T , measurement data from the quality assurance program already in place was obtained to evaluate the stability and accuracy of the collimator for both linacs. These measurements are obtained with a high level of precision because their quality w i l l affect the quality of actual patient treatments. The reproducibility of collimator rotation was evaluated monthly over a period of 45 months. A n established quality assurance procedure was used to verify collimator angle position. First, the gantry was rotated to 90 degrees so that the collimator rotation axis was oriented perpendicular to the direction of the force o f gravity. Because part of the collimator housing is shaped like a box with four flat sides, the top edge of each one of those sides should be oriented perpendicular to gravity at increments of 90 degrees. The collimator angle was then verified at each increment using an electronic spirit level placed on the upper edge of the collimator housing. Angles of 0, 90 and 270 degrees were evaluated in turn by selecting them from the control computer and allowing the collimator to rotate freely to the desired position. Displayed in Figure 3.9 are histograms of the measured collimator angle for Linac 1 and 2 at 0, 90 and 270 degrees. The majority of the results are distributed close to the desired angle. These results show that the collimator rotation angle is accurate to within a maximum variation of 0.5 degrees. Considering that measurements were performed over a period of 45 months, the reproducibility of 79 collimator rotation is not an impeding factor to implementing rotational I M R T in a clinical setting. 3.3.4 Center of Rotation In addition to collimator angle the accuracy and reproducibility of the collimator rotation axis was also assessed. A light field produced in the head of the linac and mimicking the radiation field is used to position patients for treatment. A cross hair suspended inside the collimator is used to indicate the central axis of the radiation field, which is also the center of rotation of the collimator. The light field projects through the cross hair and can be seen on the couch surface. When the collimator rotates the cross-hair projection also rotates. The maximum displacement of the cross hair center as the collimator rotates through 180-degree was recorded weekly over a period of 1 year. Measurements were obtained using gantry angle of 0, 90 and 270 degrees. The results for Linac 1 and 2 are shown in Figure 3.10. The collimator rotation axis remained stable to within 0.5 mm on every occasion, displaying a high level of reproducibility. The error tolerances o f linear accelerators are typically on the order o f 1 to 2 mm. The collimator rotation axis exhibits a superior level of accuracy (< 1mm) than that required and w i l l therefore not contribute significantly to dosimetric error when delivering intensity modulated fields. 80 25 20 03 -»—• 03 £ 15 *#— o a5 10 E z 5 25 20 co £ 15 a3 10 _a E i 5 0 25 20 0} * - « co £ 15 a> 10 E ^ 5 Linac 1 -0.8 -0.4 0.0 0.4 0.8 Angle (deg) 89.2 89.6 90.0 90.4 90.8 Angle (deg) 269.2 269.6 270.0 270.4 270.8 Angle (deg) 25 20 03 03 £ 15H M — I 10H E l 5^ 0 25 20 15 Linac 2 Da -0.8 -0.4 0.0 0.4 0.8 Angle (deg) 03 -*-» 03 03 O 5 10 _o E ^ 5 - P -89.2 89.6 90.0 90.4 90.8 Angle (deg) 25 20 oo CO £ 15 a3 10H E i 5H 269.2 269.6 270.0 270.4 270.8 Angle (deg) Figure 3.9: Plotted are histograms of collimator angle reproducibility. Collimator angle was verified once per month over a period of 45 months. Measurements were performed at 0, 90 and 270 degrees for Linac 1 and Linac 2. Results show that the collimator angle is reproducible to within 0.5 degrees over a period of almost 4 years. 81 Linac 1 Linac 2 5 0 i 5 0 1 4 0 ^ 40 ^  30^ 3 0 J 2 0 H 2 0 ^ 10H 10H JH 1 1 1 1 1 = | , 1 . , I H ' ' ' 1 1 1 ' 1 ' 1 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 Center of Rotation Alignment Error (mm) Figure 3.10: Histograms of the variation in the center of rotation over a 180-degree rotation. Measurements were obtained weekly for a period of 1 year. Results are shown for Linac 1 and Linac 2. 3.4 Linac Control In addition to the mechanical characteristics of the collimation systems it is necessary to consider the x-ray production properties of the accelerator. Static and Dynamic delivery modes require that various components of the linac hardware function accurately and dependably. The base fluence generated by the source must be consistent and symmetric about the rotation axis. The monitor ionization chamber located in the head of the linac is divided into 4 quadrants. Each quadrant is independently verified throughout delivery. I f any one of the quadrants shows a variation outside an accepted tolerance, the electron beam steering mechanism is adjusted to compensate. In this fashion the 'flatness' as well as the beam symmetry is assured to remain constant throughout delivery. 82 3.4.1 Static Delivery Intensity modulation by multiple static segments requires that some sub-fields be delivered with relatively lower dose when compared to conventional radiation therapy. When the monitor chamber signals that the programmed delivery is complete the production of radiation is terminated. Radiation is produced in discrete pulses. Because the beam cannot be terminated within an individual pulse there is always an error o f up to 1 pulse for each sub-field. For very short irradiation periods the percentage error resulting from an additional pulse may become significant [83, 84]. Also , there is a time lag between the signaling of the monitor chamber and the complete termination of x-ray production. 3.4.1.1 Dose Linear i ty A n experiment was devised to evaluate the dosimetric error introduced by lower monitor unit segments. A monitor unit is defined as the amount of charge collected in the monitor ionization chamber located in the head of the linac that w i l l provide a specific dose under a specific set of delivery conditions. For the linacs used in this thesis 1 monitor unit w i l l provide 1 cGy of dose to water for a 10 cm x 10 cm field size at the depth of dose maximum with the point of measurement located at 100 cm from the source. A n ionization chamber was placed in a water equivalent phantom at this reference depth and field size. M U settings ranging from 1 to 10 were programmed and delivered using a dose rate of 300 M U / m i n . The results were then evaluated in terms of the linearity of radiation production with M U setting. I f the delivery system were ideal, the dose delivered with 10 M U would be exactly 10 times the dose delivered with a setting of 1 M U . Measurements were also obtained at dose rates of 100 and 600 MU/minute to evaluate whether the lag time of the monitor chamber was a function of dose rate. 83 Figure 3.11: Measured dose versus Monitor Unit setting is plotted at dose rates of 100, 300 and 600 MU/min. Also plotted is a linear fit of the 100 MU/min results. Results consistently show that dose is linear with MU setting within experimental error. Shown in the bottom right hand corner is a magnified section of the dose axis intercept. The dose offset indicates a monitor chamber lag time resulting in an overdose of approximately 0.06 cGy and causing significant dosimetric error of over 2% for sub-fields of less than 4 MU. Plotted in Figure 3.11 is measured dose versus M U setting for 100, 300 and 600 M U / m i n dose rates. Results show that dose delivered is linear with Monitor Unit setting for all dose rates. Also plotted in Figure 3.11 is a linear fit for the 100 M U / m i n results with an extrapolation of the straight line through the dose axis. A 84 blow-up o f the plot near the origin shows that the intercept is located at an offset of 0.06 cGy. If the monitor chamber feedback process were perfect the amount of dose delivered at 0 M U would equal zero. The fact that the intercept is greater than zero indicates a lag in monitor chamber feedback. The lag time between the monitor chamber and termination of the radiation beam results in an overdose of 0.06 c G y at reference depth and field size. Lag times for the other dose rates tested were identical to within experimental error. It can be concluded from these results that individual sub-field contributions w i l l not be effected by dose non-linearity. The absolute number of monitor units for each sub-field can therefore be scaled according to the prescribed dose. Due to the monitor chamber lag time, there w i l l be a large percentage error for very low dose sub-fields. Therefore, in order to reduce dosimetric error to less than 2% it is required that a minimum of 4 M U is delivered per sub-field. 3.4.2 D y n a m i c D e l i v e r y The ability of the accelerator to accurately control dose rate and collimator rotation speed in real time was evaluated. The collimator rotation speed must be constant throughout the delivery. Also , the speed of rotation must be the same for each individual delivery to avoid constant recalibration. Finally, the dose rate must also remain constant throughout each delivery. Modification of either the dose rate or the collimator rotation speed during delivery w i l l cause a phase shift between the desired location of each field and the field that is actually delivered. Following are a series of experiments that were designed to evaluate each of these potential sources of error. 3.4.2.1 Rotat ion Speed Reproducibi l i ty The time for a 180-degree rotation was measured over 14 months for Linac 1 and Linac 2. A timer was used to measure the interval between when rotation was started 85 at the control workstation and when the collimator came to a full stop. Linac 1 and 2 showed consistent rotation times of 48 +/- 0.2 s and 45 +/- 0.2 s respectively. Over the 14-month period no significant deviation from these times was noted, proving that the period of collimator rotation is highly reproducible. 3.4.2.2 Dose Rate Stability In addition to cumulative dose, the monitor chambers also verify that the accelerator maintains the desired dose rate. I f the dose rate is lower than desired the feedback mechanism of the monitor chamber w i l l allow more dose pulses to be delivered. Conversely, i f the dose rate is too high some dose pulses w i l l be removed. In the event that the dose rate changes by greater than 1% the beam is terminated. The delivery must then be restarted with M L C and collimator set to their positions when the beam was terminated. These mechanisms insure that no treatment field is delivered with an incorrect dose rate, making the chance of rotational delivery phase shift error due to dose rate variations unlikely. In order for the linac to maintain a constant dose rate it is manually 'tuned' at regular intervals. Sti l l , it is possible for the dose rate to become unstable which would cause errors in the delivery of intensity modulated fields. 3.4.2.3 Collimator Rotation Stability The previous sections described the method of dose rate control and collimator rotation monitoring that is present with the current linac hardware. The period of rotation was found to be consistent and dose rate control is robust. However, to ensure I M R T with dynamic collimator rotation is realizable within a small enough margin of error further testing is required. A testing procedure was developed to evaluate the combined effect o f collimator rotation instability and dose rate fluctuations. Although it has been found 86 that the period of rotation of the collimator is consistent, there is no reason to believe that the speed remains constant throughout the motion. This could result in an unacceptable level of error when combined with dose rate fluctuations that exist during the beginning and end of the delivery. Displayed in Figure 3.12 are a series of four sections in the dynamic irradiation of a small opening in the M L C located at radial distance r from the isocenter. Radiographic film ( X - O M A T V2) was placed in a water equivalent phantom at an arbitrary depth of 5 cm. The film was placed perpendicular to the beam central axis and irradiated with a 6 M V beam. Dose was deposited onto the fi lm during the delivery through the small M L C aperture using Linac 1. A small opening in the M L C rotated by 180 degrees w i l l describe an arc of dose on the fi lm. If the rotation speed and dose rate is constant the dose measured along the arc should also be constant. A n y variations indicate that the dose rate or collimator rotation speed was fluctuating during delivery. In order to expose the film within its dynamic range the approximate radial distance r and width of each M L C aperture w for each dose rate was determined. The aperture width was fixed at 2 cm. The fraction of dose delivered at each point along the arc throughout rotation is given by the dose that would be delivered without rotation scaled by the ratio of the aperture width w to the total path length TJT. W D r o , = D - (3.5) The total dose delivered, D, is a function of the period of rotation co (fixed) and the dose rate D. For a rotation through 180 degrees (one half period) we have. 87 Figure 3.12: A procedure to test the combined effects of collimator rotation instability and dose rate variations. A small aperture is opened at radius r from the central axis and rotated through 180 degrees during irradiation. The resulting profile of dose describes an arc and is measured with radiographic film. Dose along the profile should ideally be constant. Larger dose rates are evaluated by repeating the rotation with a larger radial distance to the MLC opening given by equation 3.7 88 (3.6) Solving for r, the radial distance from the center of rotation of each M L C aperture can be calculated with Dcow For a desired exposure Drot o f 15 cGy, which is within the dynamic range o f the fi lm, the radii r were calculated to be 3,6,9,12,15 and 18 cm for the 100, 200, 300, 400, 500 and 600 M U / m i n dose rate settings respectively. The fi lm pixel values were converted to dose using the calibration procedure described in section 5.3.1 Displayed in Figure 3.13 is a grayscale image of a radiographic fi lm exposed at all 6 dose rates. The dark lines of the 'rainbow' arcs show the dose delivered by dynamic rotation. Radius is increased with increasing dose rate. Visually, the arcs look uniform except for the beginning and the end of each trajectory. A decrease in dose is expected at those positions due to part of the M L C aperture exposing the arc ends for a smaller fraction of the delivery. A sharp increase in dose can also be seen at the ends of some trajectories. These increases indicate a deceleration o f the collimator that does not coincide with the dose rate. 89 90 600 500 400 300 200 100 MU/min Figure 3.13: A grayscale image of the 2-dimensional dose distribution generated in the collimator rotation and dose rate stability test. Dark lines making up the arc of the 'rainbow' correspond to exposures at each dose rate. Gradients at the beginning and end of each path are expected and are caused by the edge of each aperture exposing the tips for a smaller period than the rest of the trajectory. Slightly higher doses are seen near the end of each trajectory and are likely due to a deceleration of the collimator in advance of dose rate termination. Plotted in Figure 3.14 are dose profiles obtained along the aperture trajectory for all six dose rates. A l l plots have been normalized to the 90° position. Results show that overall uniformity is good although there are some dose spikes at the beginning and end of certain profiles. At the beginning of the 200 MU/min profile there is a sharp increase which is likely due to the collimator accelerating at a slower rate than the dose rate. It is only observed in this one case and is therefore most likely not a systematic error. For 400 MU/min and greater deliveries there is a dose spike at the end of the trajectory, as was visually noted in Figure 3.13. This is most likely due to the collimator decelerating at a slower rate than the dose rate. When the 90 120r 100 -CU 80 CO I o Q 60 -> ro 40 -CD CC 20 -120r 100] CU 80-CO O Q 60 -cu CC 40 -cu cc 20 -— 1 0 0 MU/min a=0.74% _ l i L_ 30 60 90 120 150 Angle (deg) 180 120 100 0J 80 CO Q 60 CD ro 40 cu CC 20 0 ^ - i < 1 > 1 • r~ —I . 1 . 1 . 1 1 r— • 300 MU/min o=0.60% _ i • i • i . l i l _ 30 60 90 120 150 Angle (deg) 180 120 100 80 60 40 20 h 0, • 500 MU/min a=1.67% _1 i I > 1— 30 60 90 120 150 Angle (deg) 180 120r 100 cu 80 CO o a 60 -> 40-cu CC 20-ol 0 • 200 MU/min o=0.80% „ J J L— 0 30 60 90 120 150 180 Angle (deg) 400 MU/min a=1.34% 0 30 60 90 120 150 180 Angle (deg) • 600 MU/min o=2.17% Angle (deg) Figure 3.14: Dose vs collimator rotation angle is plotted for dose rates ranging from 100 to 600 MU/min. Dose was measured along each trajectory of the open MLC apertures from the radiographic film of Figure 3.13. Non-uniformity in the dose profile indicates that the collimator rotation speed and dose rates are unsynchronized. Good overall uniformity is observed although there are some collimator deceleration artifacts and an increase in the standard deviation at 400, 500 and 600 MU/min. 91 monitor chamber records that the desired dose has been delivered the linac wil l stop producing radiation virtually instantaneously. The collimator cannot decelerate as quickly. There is therefore a period at the end of the trajectory where the collimator is moving slower while the linac is still at the maximum dose rate. Note that the 100 MU/min plot shows a slow increase over the first 20° and then a decrease after 160°. This effect is expected and was described earlier on page 89. It is more pronounced for the 100 MU/min plot because the overall path length relative to the aperture size is much shorter. The standard deviation a [85] was obtained over the central 120 degrees of each dose profile. The results are displayed under each respective plot in Figure 3.14. The standard deviation increases with increasing dose rate indicating that variations in dose rate and collimator speed increase as the dose rate setting is augmented. Furthermore, at higher dose rates it is evident that there is a lower dose in the center of the profile than at the outer edges, indicating a systematic difference between collimator rotation speed and dose rate. There is therefore an additional systematic error that must be considered when delivering dynamic rotational treatments at higher dose rates. The positional uncertainty resulting from the rotational instability wil l be greater with increasing radius and will therefore depend on the position and extent of the fluence map with respect to the center of rotation. 3.5 Feasibility of IMRT with Collimator Rotation In section 3.2 the potential advantages of using collimator rotation in IMRT delivery were presented. Benefits resulting from increased spatial resolution, decreased interleaf effects and larger maximum field size can only be realized i f the delivery apparatus is capable of controlling the collimation mechanism and x-ray production system accurately. In particular, rotation of the collimator adds a degree of complexity that has not previously been investigated in the context of IMRT delivery. 92 For this reason, a full mechanical and dosimetric study was performed to evaluate the feasibility of the proposed technique. In sections 3.3.3 and 3.3.4 it was shown that collimator angle and center of rotation are both accurate and reproducible. In static delivery the production of low dose segments must be considered. The results provided in section 3.4.1.1 showed that dose delivered was linear with M U setting. A dose offset due to the speed of the feedback mechanism was observed which can be avoided by only using M U settings greater than 4. These results indicate that for static rotational delivery there should be no significant increase in delivery error due to limitations of the linac. With dynamic delivery there is the additional complication of ensuring that the dose rate and collimator speed are synchronized throughout delivery. M L C leaf motion and dose rate is synchronized through a feedback mechanism that has already been established for conventional IMRT delivery. In section 3.4.2.3 the results of an investigation into dose rate and collimator rotation synchronization were presented. Although the overall results show uniform rotation speed and dose rate there is a distinct trend toward a greater systematic error at higher dose rates. Also, acceleration and deceleration of the collimator are not matched to the increasing and decreasing dose rate at the beginning and end of each delivery. The errors described in section 3.4.2.3 could substantially affect the delivery of rotational IMRT fields. By taking into consideration these discrepancies when delivering actual IMRT fields it wil l be possible to minimize their effect. 93 Chapter 4 LEAF MOTION DERIVATION In the previous chapter the potential advantages of using M L C rotation in IMRT delivery were described. The viability of delivering fluence maps in this way with conventional linear accelerators was also investigated. It was determined that although they were not designed for this purpose, they generally perform within an acceptable margin of error. The results of this investigation wil l provide guidance as to how the linac should be controlled to reduce any additional uncertainties associated with collimator rotation. Although rotation of the M L C provides an additional degree of flexibility when delivering a desired fluence map, it forces an increase in complexity when deriving the necessary leaf motions. To generate an arbitrary fluence map using a series of M L C defined sub-fields a derivation of the exact position of each leaf for each sub-field is required. With rotation, the collimator angle must also be calculated. It was therefore necessary to develop a series of specialized algorithms that are capable of determining rotated M L C segments as part of this thesis. In this chapter a detailed description of the algorithms is presented (also published in Physics in Medicine and Biology [65]). Included is a brief review of conventional leaf motion calculation methods as well as a geometric analysis of the increase in complexity with collimator rotation. Next, the mathematical formulation of the leaf position problem is derived. Finally, the bulk of the chapter contains a detailed description of the optimization-based method capable of obtaining a solution. 94 4.1 Conventional Techniques - Review Various techniques have been described to deliver IMRT. M L C based techniques are the most common and may be implemented on the majority of manufacturer's linacs. The M L C is used to define multiple uniquely shaped sub-fields at a fixed gantry angle, the sum of which result in a complex 2-dimensional fluence map. There are two basic variations on M L C based delivery techniques: static (step-and-shoot) and dynamic (sliding window). The former is a stationary method where each sub-field is shaped while the radiation beam is off and then a portion of radiation is delivered once the leaves are in position [17, 18, 25]. The latter involves moving the leaves while the beam is on and is similar to the first method but with a large number of time varying sub-fields [19, 86, 87]. 4.1.1 Step And Shoot (Static) Many static leaf motion calculation methods have been reported in the literature [19, 24, 25, 86-89]. The majority of these techniques are focused on reducing the total number of sub-fields required to deliver the desired fluence map. The step-and-shoot method is employed by the Varian Helios v.6.2 IMRT planning system which is used in the comparison study between conventional and rotation methods in Chapter 5. Consider the fluence map of Figure 4.1(a) where the M L C leaves are oriented to travel horizontally between A and B. A profile obtained between an arbitrary leaf pair is shown in Figure 4.1(b). Each leaf must move along a specific trajectory to produce the desired fluence. To deliver the fluence in a finite number of sub-fields, the continuous distribution is quantized into a series of discrete levels as shown by the shaded area in Figure 4.1(b). Choosing more levels in the quantization procedure will result in a more accurate reproduction of the desired fluence but will necessitate a larger number of segments for delivery. The leaf positions are then derived using this discrete representation of the fluence profile. Figure 4.1(c) shows a plot of fluence 95 2 0 2 Dis tance(cm) (c) 1-00 x ° - 7 5 d) 8 0 .50 c cu 3 0 .25 0 .00 Leaf A Leaf B I -2 0 2 Leaf Position (cm) Figure 4.1: Method of generating intensity modulated fields using the 'step-and shoot" technique. The 2-dimensional fluence map in (a) is sampled along the leaf trajectory AB and is plotted in (b). The continuous fluence is then modified to produce regions of constant fluence as shown in the shaded area of (b). The leaf trajectories for leaf A and B are derived as a function offluence index and are plotted in (c). The difference in fluence index between each trajectory is proportional to the fluence delivered at that point. The area between the two trajectories plotted as a function of leafposition is therefore equal to the shaded area in (b). 96 index versus leaf position for Leaf A and Leaf B. The fluence index represents the relative quantity of radiation that has been produced by the linac. The location of the open M L C area at any moment in the delivery is therefore given by the difference in leaf positions at that fluence index. Also, the total fluence delivered to a point in the field is given by the difference between the fluence indices of leaf A and leaf B at that point. Therefore, the fluence distribution of Figure 4.1(b) is equal to the area between the two leaf trajectories of Figure 4.1(c) plotted as a function of leaf position. The trajectories for each leaf pair may therefore be solved in turn by the method described above. At the beginning of each index the leaves are moved to the derived position (stepped). The beam is activated and the desired quantity is delivered (shoot). The process continues in this manner until the desired fluence has been delivered. 4.1.2 S l i d i n g W i n d o w ( D y n a m i c ) Dynamic delivery of IMRT fields is performed in a similar fashion to the step-and-shoot method described above. The fluence profile between a leaf pair is quantized into a large number of discrete levels as shown in Figure 4.2(a). Unlike the static method, M L C leaves move continuously throughout delivery [19, 86, 87]. The position of each leaf is a function of the fluence index as plotted in Figure 4.2(b). As the fluence index increases the position and length of the gap (window) between the two leaves is translated (sliding) across the area of desired fluence. As with the step-and-shoot method the fluence accumulation is given by the area between the two leaf trajectories from Figure 4.2(b) plotted as a function of leaf position. 97 Leaf Position (cm) Figure 4.2: The sliding window technique is similar to the Varian step-and-shoot except that the fluence is divided into a large number of fluence levels as shown in (a). Also, each leaf pair moves continuously during the delivery as seen in (b), creating an opening in the MLC that slides from one side of the field to the other. The size and position of the window is calculated as a function of the fluence index. The fluence shown in (a) is generated by modifying the window width and position as plotted in (b). 4.2 Rotational Technique Leaf Motion Derivation 4.2.1 Increased Complexity Conventional IMRT leaf motion calculations are effectively a one-dimensional problem. A single leaf and its opposing pair are the only two leaves that directly affect the fluence at any given point. Therefore, the necessary leaf motions required to deliver the correct fluence at that point are almost entirely dependent on those two leaves. The derivation of all leaf motions can therefore be separated into a series of one-dimensional equations involving each leaf and its opposing pair. With M L C rotation this approach is no longer possible. Different leaves will affect different points in the desired fluence map depending on their location and the 98 collimator rotation angle. This is shown in Figure 4.3 which illustrates the fluence and M L C apertures at nine collimator angles in the dynamic delivery of a wedge shaped intensity distribution. (g) P J (") © Figure 4.3: Fluence maps are generated dynamically by rotating the collimator while the MLC leaves are in motion. Each frame in (a) to (i) shows the progressive build-up of a wedge shaped fluence with the rotated MLC aperture at that instant. Referring to Figure 4.4(a), for a desired fluence pixel at a point (x,y) and the collimator rotated to angle 6, the M L C leaf position P and leaf pair L that intersect that pixel are given by: P- x' = xcos6-ysm.O L = y' = xsinf2 +ycosf? (4.1) 99 where the coordinate system (P,L)= (x',y') is fixed to the frame of reference of the M L C . (a> . \ y ( b ) Rotating MLC Sinogram ^ ^ ' \ B A < y y i i _ ! CO A ^ ^ - i f A*. ^ co \ : . I ^ J L / / / / / / / / / / K / A -f A ^ / v —1 . . . . . A o A B _ y _ l - c j 0 20 40 60 80 100120140160180 Collimator Angle (deg) / Figure 4.4: (a) The rotating coordinate system of the MLC. (b) A sinogram of the trajectory of points A, B and C as they move through opposing leaf pairs. The points follow a sinusoidal path through the MLC leaf pairs with each point having a different phase and amplitude. The amplitude is equal to the radial distance of the point from the isocenter and the phase is a function of its initial position along the leaf bank. As the collimator rotates, the M L C leaf pairs capable of modulating the fluence at a specific point change. A sinogram of the trajectory of individual fluence pixels through the M L C leaves as a function of collimator angle is shown in Figure 4.4(b). The level of complexity of the leaf motion calculation is appreciated by observing the trajectory of points A , B and C shown in Figure 4.4(a) through the M L C sinogram. As the collimator rotates, each point follows a sinusoidal trajectory through the leaf pairs. Points A and B are located at the same distance from the isocenter and have the same amplitude whereas Point C is closer and has a smaller amplitude. The amplitude of the sinusoid is equal to the radial distance of that point from the center of rotation. Points that are located further from the isocenter will 100 therefore intersect a larger number of leaf pairs throughout the rotation. Although Points A and B have the same amplitude in the sinogram their trajectories are out of phase. The phase of the sinusoid is a function of the initial position of the fluence point along the M L C leaf bank. The interdependence of leaf position and collimator angle is inherently complex and must be incorporated into the leaf motion calculation algorithms. 4.2.2 A n a l y t i c M o d e l To develop a method of calculating the leaf positions and collimator angles for arbitrary fluence maps, it was necessary to derive a mathematical model of the system. A 2-dimensional distribution of fluence <$>(x, y) generated by an M L C aperture Cl(x,y) with primary base fluence originating from the source Op(x,y) is given by 0(x,y) = Q(x,y)0P(x,y) (4.2) where Op(x,y) s i for a clinical linear accelerator (Op(x,y) is compensated for in the treatment planning phase before the desired fluence maps are generated and is therefore not included in the following calculation). A complex fluence generated by a series of M M L C apertures is M <D(*,;,) = XQm(x,v). (4.3) m=\ For the purposes of fluence generation an M L C with total number of leaves 7Y, for each leaf n, the aperture generated by the left leaf L and the right leaf R are defined by the step functions 101 Ln(x,y) = 0 x < 0,{n-l)w < y < nw Ln(x,y) = l x > 0,nw < y < (n-l)w (4.4) R„(x>y) = ° x>0,(n-l)w< y <nw n-\)w respectively where w is the leaf width. In one-dimension, the aperture function for any leaf pair may be written n(x) = Ln{x-xL)Rn(x-xR) (4.6) where xi and XR are the left and right leaf end positions. Equations 4.4 and 4.5 are plotted in Figure 4.5(a) with an example of a 1-dimensional aperture function. A full 2-dimensional aperture is described by tyx,y) = f[Ln{x-xL(n),y)Rn{x-xR{n\y) (4.7) « = i an example of which is plotted in Figure 4.5(b). 102 Figure 4.5: One-dimensional representation of the left and right leaf functions Ln(x,y) and Rn(x,y) from equation 4.4 and 4.5 respectively are shown in (a). Also shown is an example of a 1-dimensional aperture function Q(x)from equation 4.6 where XL and XR are the left and right leafpositions for a single leafpair respectively. A full 2-dimensional aperture function using multiple leafpairs is shown in (b). These equations define the aperture function for a non-rotating M L C . With collimator rotation, the angle of rotation 0 is introduced. Using the coordinate transformation defined in equation 4.1 and substituting x and y for x' and y' respectively, the aperture function is N Qe(x,y) = Y[Ln(xcosd - ysmO - xL(n),xsin0 + ycosd) n=l ( 4- 8) x Rn(xcos0 - ysinO - xR(n),xsin0 + ycosd). The fluence generated for a series of apertures through a rotation of 180 degrees is < D ( ^ ) ^ f i s ( V ) (4.9) 0=0 103 which, when expanded, is equal to n N 0(x,jv) = ^ rj[Z n(xcos^-^sint9-x i(«),xsin^ + >'cost9) ^ x Rn (x cos 6 - y sin 9 - xR («), x sin 6 + y cos 0). Therefore, given an arbitrary desired fluence \T?(x,y), it may be reproduced using a series of M L C defined sub-fields by solving equation 4.10 for the leaf positions xL{n,9) and xR{n,d) for all leaf pairs n and over all collimator angles 9. There is no method of solving equation 4.10 directly. Also, due to the discrete representation of fluence maps as a matrix with finite pixel sizes, a method to describe them as a continuous function is not obvious [90]. Still, the above formulation is an accurate description of the system that must be evaluated. The following sections describe alternative solutions to equation 4.10 using optimization techniques. 4.2.3 Optimization Methods Optimization techniques have been used in IMRT since its inception although not in the context of generating leaf sequences. Derivation of the desired fluence map for a sequence of beams is accomplished using gradient and stochastic based optimization methods [21, 22, 91, 92]. The problem, in simple terms, is to arrive at a series of 2-dimensionally varying fluence maps such that a uniform dose is delivered to the tumour while avoiding dose to the surrounding healthy tissue. See section 1.5.2 for a full description. A general discussion of optimization methods is provided here as an introduction to the rotational leaf motion calculation algorithms. Firstly, the optimization problem is described by an objective function. This function serves to define the goal of the optimization. It also quantifies the progress 104 of the optimization as it moves towards that goal. The objective function for the leaf motion derivation problem is defined as the absolute difference of the desired and calculated fluence maps summed over all pixels. Obj = desired (*» ^ )^'ca lcula ted (*, J>)| (4.11) As Obj approaches 0 the difference between desired and calculated fluence maps approaches 0. The goal of the optimization is to calculate the leaf positions xL(n,0) and xR(n,9) such that equation 4.11 is a minimum. Because equation 4.11 contains equation 4.10 it is not feasible to obtain an analytical solution. Instead, equation 4.11 is evaluated at different values of xL(n,0) and xR{n,6) until a solution is obtained. After each attempt (iteration) Obj is evaluated to determine how the new values have changed the result. The aim is to consistently reduce the objective over several iterations until it no longer improves as shown in Figure 4.6(a). How the values of xL{n,d) and xR(n,0) are determined after each iteration is given by the optimization method. 4.2.3.1 Gradient Based Methods A minimum of equation 4.11 is obtained when the gradient of Obj with respect to xL{n,6) and xR{n,G) is equal to 0. dObj n dObj . = ° „ / ^ = 0 (4-12) dxL{n,0) dxR{n,6) Gradient based optimization methods attempt to arrive at a minimum by evaluating the local gradient at some xL{n,G)m and xR(n,B)m. The values of xL{n,0) and 105 xR(n,6) for the next iteration m+1 are chosen by incrementing their value from the previous iteration m in the negative direction of the gradient *L *Li = *L (». 0)m ~ kVObj xR (n, 0)m+l = xR (n, 0)m - kVObj (4.13) which is the downward slope of the function at that point. The magnitude of the step k is, in general, reduced as the gradient approaches zero. Variations exist on the basic methodology described here that attempt to determine the minimum more efficiently and/or more accurately [93]. Figure 4.6: Example of the minimization of an objective function Obj where xL(n,6) and xR(n,6) are modified after each iteration m. Gradient based methods fail to locate the global minimum when there is a high density of local minima as shown in (b). To find the global minimum with gradient based methods, it is necessary that either the function have only one minimum or the first iteration be chosen at a value already in the valley of the global minimum. Otherwise, the optimization wil l become trapped in a local minimum as shown in Figure 4.6(b). Gradient based methods will therefore be unsuccessful for objective functions that have a topography with many local minima. 106 A n investigation into using gradient-based methods for rotational leaf motion derivation was performed as part of this thesis work. Preliminary results revealed that the topography of the objective function defined in equation 4.11 consists of a high density of local minima. The optimization consistently terminated after a small number of iterations producing a calculated fluence map with errors of greater than 50% over the majority of its area. Based on these results no further investigation was performed using these methods. 4.2.3.2 Stochastic Methods Certain optimization problems have been identified that benefit from a random selection of the argument values [94]. Problems with many shallow local minima can be solved more accurately because, through random processes, the values of the arguments are able to "jump" from one valley to another. The objective function is still used to evaluate the progress of the optimization but is more limited in determining the argument values for the next iteration. The degree of randomness is typically a parameter that is adjusted as the number of iterations increases. For example, the arguments may have values that vary wildly from one iteration to the next at the beginning of the calculation and then slowly become less erratic towards the end. This permits an overall evaluation of the objective function topography at the beginning with a more local search for the minimum at the end. 4.3 Rotational Leaf Motion Optimization The collimator rotations and leaf sequences used in this paper are derived using an optimization technique developed in-house [65, 66]. The basic method of the algorithm is stochastic in nature and it can, in some respects, be likened to simulated annealing [94]. In the calculation, leaf positions are randomly varied at prespecified 3 Patent Pending. 107 collimator angles. With each variation the resulting primary fluence is updated. The absolute difference between calculated and desired fluence maps is then evaluated. Whether the leaf position variation is accepted or not is based on two separate criteria. First, i f the variation brings the calculated fluence closer to the desired fluence then it is accepted. If there is no improvement an examination of those particular fluence pixels that are affected is performed. If those pixels fall within a dynamically controlled range of acceptable error then the variation is also accepted. If neither criterion is satisfied then the variation is discarded and a new variation is attempted. The following sections provide a detailed description of the algorithms. 4.3.1 Preprocessing Desired fluence maps are generated with a pixel size of 2.5 x 2.5 mm 2. A l l deliveries are planned with a predetermined collimator rotation range that is divided into equally spaced segments. The rotated position of M L C leaves with respect to pixels in the fluence map are precalculated for all possible leaf positions at each rotated segment. Only a fraction of the precalculated segment angles are used with the static delivery method while all segments are used when delivery is performed dynamically. This calculation is performed only once. The precalculated rotated positions are loaded into memory each time the optimization software is started. This reduces calculation times significantly because the location of changes to the calculated fluence from varying leaf positions are obtained directly from memory instead of calculated with each leaf position change using equation 4.1. 4.3.2 Constraints M L C leaves are constrained so that the maximum leaf span does not exceed the mechanical limits of the given multileaf collimator. Overlapping of opposing M L C leaves is also forbidden. In dynamic delivery there is the additional constraint that 108 the leaves can only move by a small amount between segments. This constraint is a direct result of the (currently) fixed collimator rotation speed, dose rate and maximum leaf velocity. The maximum intersegment leaf displacement smax is limited by the maximum leaf velocity vmax, the number of segments per collimator angle N and the collimator rotation rate co where V max ^ m „ = — . (4.14) I.e., at the maximum leaf speed there will be a maximum leaf displacement between any two segments. If a greater leaf speed was required the collimator would have to slow down or the dose rate would have to be reduced in order to compensate, which is currently not an option. Finally, there is a minimum gap required between opposing leaves in order to avoid collisions when both leaves are moving. A width of one pixel (2.5 mm) is used for this minimum gap due to the finite pixel size of calculated and desired fluence maps. 4.3.3 Fixed Parameters In addition to the rotation range and maximum number of segments, there are some additional fixed parameters that are defined prior to optimization. The desired fluence map is normalized to a value that represents the radiation efficiency. The radiation efficiency is defined as the fraction of the total fluence produced by the source that contributes to the point of maximum fluence. Equivalently, it is the percentage of the delivery that the multileaf collimator exposes the point of maximum fluence. E.g. a value of 100% would require that the point of maximum intensity in the desired fluence map be open to the source throughout delivery (i.e. no M L C leaf 109 will cover that point). A lower value will force the location of maximum intensity to be covered by the M L C for a certain portion of the delivery. Typical values range from 60% to 90% and are dependent on the complexity of the desired fluence map. Another parameter that is defined beforehand is whether the delivery will consist of multiple static rotated segments (static mode) or i f the collimator will rotate throughout delivery (dynamic mode). Finally, the segment weights are also fixed, which is required to maintain a constant dose rate in dynamic delivery, a current limitation of the linac control software when collimator rotation is used. 4.3.4 Initialization Before the optimization can begin it is necessary to define a starting position for all M L C leaves at each segment. With dynamic delivery the leaves are initialized flush with each other and closed so that the central leaf pair closing point is at the central axis. Only a fraction of the total number of possible segments are used at the beginning of the optimization procedure. Typically, 10 out of the 160 possible segments are initialized. These 10 segments span the entire rotation range and are equally spaced. The other segments are introduced successively throughout the optimization. The initialization procedure is slightly different when multiple static segments are used. Unlike the dynamic delivery initialization, the closing point of leaf pairs for each segment is chosen to coincide with the point of maximum fluence along the trajectory of those leaves. Therefore for each leaf pair and at each rotated segment, the leaves are closed over the points of maximum fluence. This technique has advantages in that when the optimization starts, the leaves open at a location of maximum fluence, which is where the M L C will be open for the greatest portion of the treatment. Initialization in this manner is limited to the multiple static segment delivery method because it is possible that the same leaf pair might be initialized to 110 dramatically different locations between consecutive sub-fields, which would cause the maximum leaf displacement restriction of equation 4.14 to be violated. 4.3.5 Optimization A flowchart of the basic optimization process is shown in Figure 4.7. A segment, M L C leaf and displacement direction are randomly selected. Displacements are made in 2.5 mm increments. The proposed change in the M L C configuration is verified with the mechanical and physical constraints imposed by the M L C . If the constraints are violated then a new randomly selected modification is attempted. Otherwise, the objective function is updated. The objective function is the absolute difference of the desired and calculated fluence maps summed over all pixels as described by equation If the objective function has decreased then the modification is accepted and another leaf motion is attempted. Otherwise, a second test is performed. If the mean difference between desired and calculated fluence of the pixels in the area where the modification has taken place (x,y) is within a margin of error s then the iteration is also accepted. If this second criteria is not satisfied then the modification is rejected and a new one is attempted. Due to the similarities between equations 4.11 and 4.15 only the left-hand side of equation 4.15 needs to be calculated during optimization. The tested. The optimization proceeds in this fashion until the objective function no longer improves over a pre-specified number of iterations. 4.11. (4.15) calculation result is used to update equation 4.11 after which both criteria can be 111 Randomly select rotation segment M L C leaf and leaf movement NO ^ * — I s t h e ^ ^ ^ ^^ rnod i f i ca t ion w i t h i n s ^ N i < ^ ^ ^ ^ ^ mechanical ^ ^ > ^ ^f*YES Calculate change in fluence resulting from the modification i Calculate objective NO ^ ^ ^ l a s ^ ^ ^ the ob jec t ive^v^ ^^aecreased or are the pixels^^ within the acceptable ^ ^ r a n g e ? ^CYES Accept the modification Figure 4.7: A flowchart illustrating the basic mechanism of one iteration in the optimization part of the rotational leaf motion calculation algorithm. 4.3.6 Segment D o u b l i n g With dynamic delivery, the optimization procedure starts with 10 possible segments from which the algorithm randomly chooses. Eventually, after several iterations, variations in the M L C configurations no longer reduce the objective function significantly. At this point the number of segments are doubled. The new segments are placed at collimator angles half way between the previous segments. The leaf 112 positions of new segments are derived by iinearly interpolating between the leaf positions of the adjacent segments. The optimization then continues as described above. The doubling of segments continues until all 160 segments are optimized. 4.3.7 Dynamic Error Margin Control Throughout the optimization, the error margin e from equation 4.15 is modified. Initially, the error margin is set to a maximum and, as the objective function decreases, the error margin is decreased. After each segment doubling, the error margin is reduced by a factor of 2. After the final doubling has occurred the error margin is reduced further until the objective function no longer decreases. 4.3.8 Rotational Leaf Motion Software Using Matlab 4 software, a series of computer programs were developed as part of this thesis to perform the leaf motion calculation, evaluate the resulting calculated fluence and output the M L C leaf sequence for delivery on the linear accelerator. A 'screen-shot' of the graphical user interface used to perform these tasks is shown in Appendix B (Figure B . l ) . The software also facilitated the manipulation and analysis of various fluence maps used in the experimental evaluation presented in Chapter 5. 4.4 Algorithm Considerations Derivation of the rotational leaf positions is accomplished through an optimization process. Several parameters are fixed prior to optimization that will affect the resulting fluence map. Range of rotation, number of segments, radiation efficiency and the initialization procedure are not included as optimized variables. Due to the stochastic nature of the algorithms a repetition of the leaf sequence calculation may 4 The Mathworks, Natick, M A 113 change the result depending on the random number generation seed and sequence used in the optimization. Any variation will have to be minimized i f the algorithms are to be considered robust. Finally, the actual fluence generated by the linac may not be identical to the predictions of the leaf motion calculator. The following chapter addresses these issues by including a section where desired and measured dose distributions are compared. Also, a series of simulations are performed to evaluate the dependence of the resulting fluence maps on each one of the fixed optimization parameters. 114 Chapter 5 ROTATING M L C EVALUATION This chapter describes a series of experiments devoted to a thorough evaluation of the rotational technique (also published in Physics in Medicine and Biology [65]). Included is a section describing the results of these experiments followed by a discussion of their significance focusing on clinical relevance and improvements from conventional techniques. The first series of experiments involve a thorough characterization of the algorithms. Included is a series of simulations focusing on the algorithm reproducibility. Dependence on radiation efficiency, the range of collimator rotation and number of delivery segments is also evaluated. Next, a series of clinically relevant fluence maps are used to verify the algorithms under a variety of conditions. The leaf motions for each clinical fluence map are generated for both static and dynamic rotational deliveries. Furthermore, the calculation is repeated using typical clinical M L C leaf widths of 5 mm and 1 cm for each fluence map. To interpret the results effectively comparisons are made between rotational and conventional IMRT methods. For this purpose, the entire series of leaf motion derivations were repeated for each fluence map and M L C design using the conventional step-and-shoot and sliding window methods. A dosimetric evaluation of the technique is also presented. Measurements of varying complexity fluence maps are performed using a high-resolution film dosimetry technique. In order to properly evaluate the resolution capabilities of 115 rotational IMRT delivery a robust film dosimetry system was developed as part of the thesis work and is described in this chapter. Measurements are performed for 5 mm and 1 cm leaf width designs for both static and dynamic rotational delivery methods. Again, for comparison purposes, the entire series of measurements are repeated using leaf motions derived from conventional step-and-shoot and sliding window methods. Finally, the remainder of the chapter is devoted to evaluating spatial resolution capabilities, interleaf effects and maximum deliverable field size. Specific experiments are designed to evaluate each characteristic. Experiments are repeated for both M L C designs and results are compared to conventional techniques. 5.1 Method 5.1.1 Algorithm Assessment The ability of the algorithms to accurately derive rotational leaf motions was evaluated using a 5 field thyroid treatment plan shown in Figure 5.1. IMRT was indicated for this patient due to the close proximity of the spinal cord to the thyroid planning target volume. The fields were centered on the target volume (thyroid) and oriented coplanar to the transverse plane with equally spaced gantry angles (0, 72, 144, 216 and 288 degrees). A uniform dose to the target volume with minimal dose to the spinal cord was selected for plan optimization. The fluence maps were then optimized using inverse planning software (Varian Helios v 6.2). 2 Desired fluence matrices were calculated on a 2.5 x 2.5 mm grid and are displayed in Figure 5.2(a) to (e). A 2 dimensionally varying sinusoid was also used in the evaluation and is shown in Figure 5.2(f). 116 Figure 5.1: A 3-dimensional view of the thyroid treatment geometry. IMRT was indicated in this case due to the close proximity of the spinal cord to the PTV. A dose distribution providing minimal dose to the spinal cord and a uniform dose to the thyroid was obtained by optimizing the fluence maps of the 5 fields that are shown. 117 Figure 5.2: Five clinical fluence maps as well as a 2-dimensionally varying sinusoidal fluence were used to evaluate the rotational leaf motion algorithm, (a) to (e) are optimized fluence maps for a thyroid treatment at gantry angles of0, 72, 144, 216 and 288 degrees respectively. The 2 dimensional sinusoid is shown in (/). 118 5.1.2 Fluence Generation Parameters Rotational leaf motions were calculated using a radiation efficiency of 80% and a rotation range of 180 degrees. Dynamic as well as static delivery methods were tested with 20 equally spaced segments used for static delivery. The effect of different M L C leaf widths was evaluated by calculating leaf motions for 5 mm as well as 1 cm width leaves. Resulting fluence maps for each combination of delivery parameters were then compared to the desired fluence maps. 5.1.3 Conventional IMRT Fluence Generation Results of the rotational leaf motion calculation were also compared to the fluence generating capabilities of a conventional IMRT planning system leaf motion calculator (Varian Helios v 6.2). Sliding window as well as step-and-shoot leaf motion calculations were performed for both 5 mm and 1 cm leaves. The same number of segments were used for static rotational and conventional step-and-shoot methods. 5.2 Dosimetric Evaluation A series of experiments were also performed to: (1) Evaluate the capability of linac and M L C hardware to generate accurate dose distributions with the rotational method and (2) Investigate the potential dosimetric advantages with regards to spatial resolution and reduced interleaf effects. Several fluence maps of varying complexity were used and are shown in Figure 5.3. They are: (a) a Gaussian, (b) a constant "wedge" gradient, (c) the complex 2-dimensional sinusoid and (d) a C-shape of constant intensity used to evaluate edge conformity. Other investigators have reported similar test fluences and shapes [61, 95]. Dose distributions were measured using Kodak EDR-2 film placed in a solid water cassette. The film cassette was oriented perpendicular to beam central axis and positioned at the isocenter in a square 119 phantom at 5cm depth. The film measurements provide a dose distribution resulting from the fluence given by the calculated leaf motions and collimator rotations. Measured dose distributions were compared with the desired dose at 5cm depth in water calculated from the desired fluence maps. Our technique was also compared to conventional step-and-shoot and sliding window methods using an IMRT planning system (Varian Helios v 6.2). The number of segments was forced to be equal between the "step-and-shoot" and static rotational techniques in order to provide an accurate comparison. Figure 5.3: Test fluence maps of varying complexity were used to evaluate the rotational delivery, (a) a Gaussian, (b) a wedge and (c) the 2-dimensionally varying sinusoid, (d) A constant intensity C-shape was used to evaluate the ability of the technique to conform to an irregular shape. 120 5.2.1 D e l i v e r y Rotating IMRT fields may be delivered statically (with the collimator not rotating and the leaves stationary during irradiation) or dynamically (with the collimator rotating and leaves moving simultaneously during irradiation) with no modifications to existing linac hardware required. The fields are delivered using linacs 1 and 2 (Varian 2100EX and 2100C/D, Palo Alto, CA) with different M L C models. The first has a 52 leaf model (Mark2) with 1.0 cm width leaves and the second has a 120 leaf model (Millennium) with 0.5 cm leaves over the central 20 cm and 1.0 cm leaves outside. M L C sub-field information is converted from the leaf motion and collimator rotation calculation algorithms into a format compatible with the M L C workstation software. During static delivery, individual fields are transferred to the M L C workstation and each collimator angle is programmed separately into the linac control interface. Once the collimator and M L C are in position the desired Monitor Units (MU) for that sub-field are delivered. The M U values for each sub-field were calculated to give a dose distribution that lied within the range of the film. During dynamic delivery all sub-fields are downloaded into the M L C workstation at once with M U values and dose rates chosen to generate a dose distribution that, again, is within the dynamic range of the film. Results were compared in terms of relative dose only. The collimator is rotated to the starting angle and the termination angle is programmed. At the same time that the collimator starts rotating, the beam is activated and the dynamic treatment starts on the M L C workstation. The collimator rotates with the M L C leaves moving until it reaches the termination angle. 5.3 Film Dosimetry In evaluating 2-dimensionally varying high-resolution dose distributions it is necessary that the measurement technique have resolution characteristics that do not increase the uncertainty of the result. An ionization chamber, for example, measures 121 only at a point. Furthermore, the sensitive volume of most point detectors is greater than 2 mm [96] causing a blurring of the true value at that point. Smaller detectors, although having adequate resolution, still require multiple measurements to acquire a 2-dimensional distribution. A 10x10 cm distribution measured at 1 mm intervals would require 10000 individual measurements. For the purposes of this study a radiation measurement procedure using radiographic film was developed. Used in diagnostic radiology as a high-resolution image receptor, film is capable of measuring dose deposition at a spatial resolution of < 1mm. Several investigators have evaluated the properties of film as a radiation dosimeter under various conditions [97-99]. Their results indicate that, above all, consistency throughout the measurement and development process is required to obtain accurate and reproducible results. Other considerations include increased sensitivity to low energy photons and variations in sensitivity between film batches [100-102]. The following is a description of the film measurement and calibration procedure developed specifically for planar IMRT dose measurements as part of this thesis. 5.3.1 C o n v e r s i o n T o Dose After the film has been exposed and processed, the amount of dose deposited at a point is proportional to the optical density at that point. Optical density is determined by measuring the transmission of a collimated light source through the film ( K O D A K EDR2). Using a V I D A R Dosimetry Pro film digitizer each film is scanned and then output as a 2-dimensional matrix of pixel values that are directly proportional to optical density. Next, the pixel values are converted to dose. In order to establish the relationship between optical density and dose a separate irradiation is performed using known doses. By obtaining the optical density values for a range of known 122 doses a calibration curve is generated. The 2-dimensional matrix of optical density values is converted to dose using this calibration curve. 5.3.2 F i l m Cal ibrat ion Technique Due to the higher atomic number of the radiographic film emulsion there is a higher number of photoelectric interactions for low energy photons in film than for tissue. Film is therefore more sensitive to lower energy photons than tissue (effectively water). If the spectrum of incident photon energies was the same over the area where dose is deposited there would be a simple scaling of the film measurement that would be accounted for in the calibration procedure. Unfortunately, due to scatter of low energy photons the incident spectrum varies slightly over the field area. Non uniformity of the energy spectrum becomes more significant at larger field sizes and is also a function of the depth of measurement [99]. A film calibration procedure was developed to minimize these effects. The calibration measurement is performed using a simple form of intensity modulation. This provides a calibration relationship that is more representative of IMRT scattering conditions resulting in a more accurate conversion of optical density to dose. 5.3.2.1 Enhanced Dynamic Wedge ( E D W ) A simple form of intensity modulation involves moving one of the secondary collimator jaws across a radiation field to produce a 1-dimensional 'wedge' gradient as shown in Figure 5.4(a). The 'Enhanced Dynamic Wedge' (EDW) is the most basic form of IMRT using multiple sub-fields to generate a field of non-uniform intensity. Photon scatter characteristics of EDW fields will be similar to those of complex IMRT fields. 123 (a) • J j ! . . . £ Collimator Jaw Transmitted Fluence (b) 350 300-250-u oT 150-1 </) o Q 100H 50-I 0 -1— 1 —I—•—I— 1 —I—'—I—'—1— 1 —I— 1 —r -EDW Field 1 EDW Field 2 —i—•—i—1—i—'—i—1—i—1—i—1—i—1—i—1—i— 10 -8 -6 -4 -2 0 2 4 6 8 10 Distance(cm) Figure 5.4: A diagram depicting the generation of a one-dimensional gradient using an Enhanced Dynamic Wedge (EDW) is shown in (a). The collimator jaw is continuously translated across the field of incident photons during delivery. The transmitted fluence at a point is given by the amount of radiation delivered before the jaw passes over that point. Results of water tank ionization chamber measurements shown in (b) were obtained for two EDWs with different MU settings. In order to generate an adequate calibration curve, the number of MU was chosen so that the dose from the two deliveries would span the range of doses of each IMRT field. 5.3.2.2 Cal ibra t ion Curve The optical density to dose calibration is measured using 2 EDW fields delivered side by side as shown in Figure 5.5(a). The two fields have different M U settings so that each one spans a different dose range in the calibration curve. The true dose delivered by these fields was determined in a water tank fitted with a small mechanically positioned ionization chamber (Welhdffer IC10). A l l measurements were performed at 5 cm depth and at 1 cm intervals along the gradient of each wedge. 124 The scatter contribution from the adjacent wedge was also included in each measurement. Results are displayed in Figure 5.4(b). (a) EDW Field 1 EDW Field 2 (b) 40 EbW'Field 1 Jaw Motion 15 10-CL 5H "to 30A o ro 1 5 -> 35 A 0 0 50 100 150 200 250 300 350 Dose(cGy) Figure 5.5: A scanned image of a typical calibration film is shown in (a). Pixel values are obtained at 1 cm intervals along the gradient of each field. The dose at these points was previously measured with an ion chamber as shown in Figure 5.4(b). Plotted in (b) are the pixel values versus dose values for the measurement points of (a). The resulting curve is used to convert each measurement film to dose. A 3rd order polynomial is fit to the plotted values to simplify the conversion. Finally, the calibration is verified by observing the overlapping dose values from the high and low doses of Field 1 and Field 2 respectively. Any discontinuity between the two curves will indicate an error in the calibration. A calibration film is generated for each set of IMRT measurements. Optical density values are obtained at the same points used in the ion chamber measurements. The optical density to dose calibration curve is generated by plotting the optical density values on the calibration film versus the dose obtained from the ion chamber. Plotted in Figure 5.5(b) is a typical calibration curve. The Monitor Unit (MU) settings of each EDW field are chosen such that dose points at the low end of the high 125 M U E D W overlap with dose values at the high end of the low M U EDW as seen in Figure 5.5(b). Any discrepancies due to an error in calibration, scattering characteristics, film emulsion non-uniformity or processing variations will be apparent i f there is a discontinuity between the two curves at the overlapping points. Finally, for convenience in the conversion of IMRT film pixel values to dose, the data is fit to a 3 r d order polynomial. Measurement uncertainties for film dosimetry using Kodak EDR2 film have been reported by several investigators [100-102]. From their results it is estimated that the uncertainty in measurement using the technique described here is less than 2%. Therefore, unless otherwise indicated, the results presented in the remainder of this thesis have a measurement error of 2%. 5.4 Results 5.4.1 Algorithm Characteristics Plotted in Figure 5.6 is a graph of the objective function versus number of iterations for a typical dynamic delivery (Fluence map of Figure 5.6(d)). Each successful modification of the M L C configuration is defined as one iteration. Roughly 200,000 iterations are required before the objective function ceases to improve. Convergence is achieved in less than one minute on a 1 gigahertz Pentium III (Intel Corporation) computer. Also plotted in Figure 5.6 are the error margin and the number of segments as a function of the number of iterations. With each segment doubling, the objective function is temporarily increased but quickly reduces to a value lower than that prior to the doubling. Segment doubling occurs after 10,000 iterations have passed with < 0.1 % decrease in the objective function. As the number of segments is increased the error margin is also reduced. After the last segment doubling there is a final reduction in the error margin which is used to force any points that are close to their desired value to the smallest possible difference. Increasing the number of 126 segments from 80 to 160 yields only a small benefit and there is no significant benefit with further increases. _ 20-8 18-1 16- — Mean Difference (Cost) — Error Margin • - Number of Segments r180 -160 0 0 -140 | -120 | CO ;100 -s -80 I E -60 i -40 -20 0 250 Number of Iterations (x103) Figure 5.6: A typical dynamic rotating leaf motion optimization history. The mean difference between desired and calculated fluence maps decreases rapidly over the first few thousand iterations. Once the mean difference begins to converge the number of segments is increased and the margin of acceptable error is decreased. The optimization proceeds in this fashion until no further improvements are observed. 127 5.4.2 Reproducibility The algorithm is stochastic in nature and will therefore potentially yield different results each time the calculation is performed. To evaluate this effect the optimization was repeated 100 times for a series of cases with each calculation having a different random number generation seed. Displayed in Figure 5.7 is a typical histogram of the resulting mean difference between desired and calculated fluence from 100 repeated calculations for the same fluence used in Figure 5.6 (shown in Figure 5.2(d)). The mean difference ranges from 2.30% to 2.77% with an average value of 2.50%. 451 40-35-w 30-I— 2 5 H | 20H 1 15-1 10-5--F 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 Mean Difference(% of max fluence) Figure 5.7: A histogram showing the reproducibility of the resulting mean difference between desired and calculated fluence maps using the rotating leaf motion algorithm. Leaf motions were calculated for the same fluence with the same parameters 100 times. Although the algorithms are stochastic in nature the resulting accuracy is highly reproducible. 128 5.4.3 Radiat ion Efficiency Plotted in Figure 5.8 is the result of the leaf motion calculation for the 2D sinusoidally varying fluence map (Figure 5.2(f)) as a function of radiation efficiency. Setting the efficiency parameter to values lower than 50% does not show a significant benefit. As the efficiency is increased beyond the 50% level the discrepancies between desired and calculated fluence maps increase. Although the magnitude of 30 40 50 60 70 80 90 Radiation Efficiency (%) 100 Figure 5.8: Effect of increasing the radiation efficiency. As the efficiency is increased the quality of the calculated fluence map degrades and the mean difference increases. There is no benefit to decreasing the efficiency to values less than 50% in this example. 129 these effects differ depending on the complexity of the desired fluence map the same general trends are observed in other cases. 5.4.4 R o t a t i o n R a n g e The extra degree of freedom introduced by rotating the collimator wil l provide varying results depending on the range of collimator rotation that is used. Plotted in Figure 5.9 are the calculation results for the fluence map used in Figure 5.8 and shown in Figure 5.2(d) as a function of the rotation range. The number of segments is CD o c co Z3 7.5 h 7.0 V £ 6.5 E ° 6.0 5.5 h <D O C CD | 5.0 h C CO CD 4.5 V 4.0 135 180 225 270 315 360 Rotation Range (deg) Figure 5.9: Illustration of the effect of collimator rotation range on the algorithm result. The mean difference decreases rapidly as the rotation range is increased from 90 degrees to 180 degrees. Increasing the range further provides only minor benefit. 1 3 0 the same in each case. The results presented in Figure 5.9 are typical for the fluence maps tested and a reasonable solution is usually not attainable below 90 degrees of rotation. As the rotation range is increased the mean difference decreases until approximately 180 degrees. Increasing the range further results in only minor improvements. Virtually no benefit is observed when the rotation range is increased beyond 270 degrees. Although spatial resolution considerations indicate that a rotation range of 90 degrees should be sufficient, due to the inability of the M L C to generate concave apertures in the direction opposite to leaf travel the capabilities of the M L C are only exploited when a minimum 180 degree rotation is used. 5.4.5 N u m b e r o f Segments When generating fluence maps for delivery by multiple static segments, the accuracy of the fluence will be a function of the total number of segments. Plotted in Figure 5.10 is the calculation result for the sinusoidal fluence shown in Figure 5.2(f) as a function of the total number of static segments. As the number of segments increases, the difference between calculated and desired fluences is reduced. The degree of improvement becomes less significant as the number of segments is increased. The mean difference asymptotically approaches a minimum after which any further increases in the number of segments provides no significant benefit. The same trend is observed in other cases. For lower complexity fluence maps a smaller number of segments (e.g. 10 segments) may generate a result that is considered clinically acceptable. 131 O c <D 4— X co E o c I c ro a) 0 10 20 30 40 50 60 70 80 90 100 Number of Segments Figure 5.10: In static mode the algorithm result is a function of the number of segments used for delivery. Accuracy of the algorithm improves rapidly as the number of segments is increased until approximately 40 segments after which adding more segments has only minimal benefit. 5 . 4 . 6 A l g o r i t h m Resul t s Plotted in Figure 5.11 is the absolute difference between desired and calculated fluence as a percentage of the maximum for each of the five thyroid fields shown in Figure 5.2. Results of leaf motion calculation for both rotational and conventional delivery methods in the static delivery mode are shown. Results of leaf motion 132 calculation for both rotational and conventional delivery methods in the dynamic delivery mode are shown in Figure 5.12. For all fields, the 5 mm leaf rotational method shows superior accuracy over the 5 mm leaf conventional method in both static and dynamic modes. For lcm leaves the rotational delivery method provides an T h y r o i d S t a t i c C a l c u l a t i o n R e s u l t s 10 9 8 7 6 5 4 3 2 1 0 • 5 mm Rotational A 5 mm Conventional o 1 cm Rotational A A 1 cm Conventional A A A 6 A • • O • l • FielcM Field2 Field3 Field4 Field5 Figure 5.11: Displayed is the mean difference between calculated and desired fluence maps for the 5 field thyroid IMRT plan. Results for both the rotational and conventional methods are plotted for the static delivery mode. Results are plotted for the 5 mm leaf MLC as well as the 1 cm leaf MLC. 133 even more significant improvement in accuracy when compared to conventional delivery methods in both static and dynamic modes. T h y r o i d D y n a m i c C a l c u l a t i o n R e s u l t s A • 5 mm Rotational A 5 mm Conventional o 1 cm Rotational A 1 cm Conventional A A A Field 1 " Fie'ld2 " Fie'ld3 " Fie'ld4 Field5 Figure 5.12: Displayed is the mean difference between calculated and desired fluence maps for the 5 field thyroid IMRT plan. Results for both the rotational and conventional methods are plotted for the dynamic delivery mode. Results are plotted for the 5 mm leaf MLC as well as the 1 cm leaf MLC. Another method of quantifying the conformity between desired and calculated fluence is to evaluate the distance to agreement between the desired fluence and the actual fluence maps [103, 104]. The thyroid fluence maps were assessed based on a 134 similar criterion. The area of the fluence maps where a difference in fluence of greater than 5% evaluated over a 5 mm diameter region centered on each fluence pixel was determined. This quantifier provides the area of agreement between desired and calculated fluence maps and offers additional information on spatial resolution. For the 5 mm M L C deliveries the area of agreement was greater than 95% in all cases for conventional and rotational delivery methods in both static and dynamic mode. With the 1 cm M L C rotational deliveries the area of agreement was greater than 98%> in dynamic mode and was 94% in static mode. The conventional method resulted in an area of agreement of only 88% in both step-and-shoot and sliding window modes with the lcm leaf width M L C . These results correlate well with the results presented in Figure 5.11 and Figure 5.12. 5.4.7 Dosimetric Characteristics Shown in Figure 5.13 are measured profiles for the rotational technique using the 5mm leaf M L C . Excellent agreement is observed for the Gaussian fluence shown in Figure 5.13(a) using both the dynamic and static delivery techniques, with all points within 3%> or 3mm. Results from the static and dynamic rotational deliveries of the sinusoid intensity map shown in Figure 5.13(b) exhibit good agreement throughout most of the profile except for the third peak where there is a discrepancy of 4% over 3 mm. Results from rotational delivery also agreed well for the wedge profile. The maximum deviations were from 2% to 3% and occurred near the isocenter. Static rotational delivery results showed good agreement but were slightly greater than 3% at the lower end and near the center of the wedge. Displayed in Figure 5.14 is a plot of the mean difference between desired and measured dose distributions for the sinusoidal test fluence map. This test fluence has a large number of "peaks" and "valleys" that translate into local minima in the leaf motion optimization. It therefore presents a particularly challenging test of the 135 CD CO O Q CD > ro a: 100 80 60 40 20 0 I 1 1 1 1 1 1 1 1 1 V \ it \ 1 1 - y \ / \ -/ Desired \ ff static \ Dynamic \ • J/ I . I , J i I i I i L -8 -6 8 -6 -4 -2 0 2 4 6 8 Distance (cm) -4 -2 0 2 4 6 8 Distance (cm) Figure 5.13: Displayed are measured dose profiles for the rotational technique plotted against the desired calculated dose. Dynamic and static delivery results are shown for the (a) Gaussian and (b) Sinusoid fluence maps. All fluence maps were delivered with the 5mm leaf width MLC. 136 rotational method. Results for rotational as well as conventional delivery methods are shown for static and dynamic delivery with both 5mm and lcm leaf width designs. The 5mm leaf static deliveries show similar results for both conventional and rotational delivery methods and the lcm leaf delivery shows superior agreement for the rotational method. For dynamic delivery the rotational method shows discrepancies that are larger than those seen with the conventional method in this case. CD CO O "D X CO E CD O c CD I c CO CD 10' 9-8-7-6-5-4-3-2 1-0-• 5 mm Rotational A 5 mm Conventional o 1 cm Rotational A 1 cm Conventional A O Static • Dynamic Figure 5.14: Plotted is the mean difference between measured and desired dose distributions for the 2 dimensional sinusoidal test fluence. Results for rotational and conventional delivery methods in static as well as dynamic mode with both the 5mm Millenium MLC and the lcm Standard MLC are shown. 137 5 . 4 . 8 S p a t i a l r e so lu t ion Improvements in dosimetric spatial resolution with the rotational technique are more significant at larger leaf widths. The constant intensity C-shape distributions generated using the 5mm leaf M L C are shown in Figure 5.15(a) and (b) with good conformity for the (a) rotational and (b) conventional techniques. Displayed in Figure 5.15(c) and Figure 5.15(d) are the resulting constant fluence C-shape distributions for the 1cm leaf M L C . Jagged field edges observed in the conventional technique are almost eliminated when dynamic rotation is used. Improvements in fluence map spatial resolution are most apparent perpendicular to the direction of leaf Figure 5.15: Dose conformity for the C-shape fluence maps are shown for the 5mm leaf MLC using (a) rotational and (b) sliding window techniques, lcm leaf MLC conformity results are displayed in (c) and (d). 138 motion. In Figure 5.16 this effect is demonstrated when delivering a high frequency version of the sinusoidal fluence map using the 1cm leaf M L C . A maximum deviation of 4% with good overall agreement is observed for the rotational technique. The sliding window technique is inherently incapable of generating the fluence accurately and differences of up to 10% can be observed throughout the profile. Figure 5.16: Dose profiles showing the spatial resolution capabilities of the rotational technique for a high frequency version of the sinusoidal fluence map using the lcm leaf MLC. Significant error (10%) is observed for the conventional delivery profile obtained perpendicular to the direction of leaf motion, where the limitations of leaf width are most apparent. The rotational technique profile shows only minor discrepancies. 139 5.4.9 Interleaf effects Leakage was determined by exposing Kodak X - O M A T V film with 800 M U with the M L C completely blocking a square field defined by the secondary collimator jaws. X - O M A T V film was used in this case due to the higher sensitivity required to accurately measure low doses (< 5 cGy). Displayed in Figure 5.17(a) and (b) are the M L C leakage patterns for the 1cm leaf M L C using the rotational (collimator rotated through 180°) and conventional (collimator stationary) techniques respectively. With the rotational technique the majority of interleaf leakage has been spread out over the field area although some circular rings of higher dose can still be observed. Conventional leaf sequencing methods that optimize delivery efficiency reduce interleaf leakage [24, 105]. Still, non-uniform leakage cannot be compensated for entirely in conventional IMRT delivery and is a significant source of error for some dynamic fields [52, 67, 68]. 140 Figure 5.17: Leakage patterns for (a) rotational and (b) conventional techniques. Improvements in base leakage uniformity with the rotational technique are apparent in relative dose profiles displayed for the 5mm and 1cm leaves in Figure 5.18(a) and Figure 5.18(b) respectively. 141 -4 -2 0 2 4 Distance (cm) 220 200 T 180 ° ^ 160 co 140 CO „ o 120 (b) k U • H II >i * •i •i ii i i 1cm Conventional 1cm Rotational -4 0 Distance (cm) Figure 5.18: Relative dose leakage profiles across the leaves and through the center of rotation are shown for the 5mm leaf MLC and lcm leaf MLC in (a) and (b) respectively. Leakage decreases gradually at the edges of the rotational technique profiles due to the non-uniform contribution from the corners of the square aperture defined by the collimator jaws. 142 Tongue-and-groove effects are also minimized by the rotational technique as observed in a magnified section of the wedge profile shown in Figure 5.19. Tongue-and-groove underdosing errors as high as 29% were observed in a previous study [106]. Distance (cm) Figure 5.19: An example of the tongue-and-groove effect observed in the conventional technique delivery of the wedge shaped fluence distribution. Tongue-and-groove effects are not present when the same fluence is delivered with the rotational technique. 5.4.10 M a x i m u m field size The lengths of the M L C leaves for the M L C models in this study are 14.5 cm. Therefore, the difference between the maximum and minimum leaf extension on each 143 bank may not exceed 14.5 cm. In the conventional sliding window technique each leaf pair must travel from one field edge to the other. The maximum field width that can be accommodated is therefore 14.5 cm. Techniques for combining multiple fields must be used to generate wider fields [75]. With collimator rotation the leaf length does not impose the same limit on maximum field size due to inherently different delivery geometry as described in section 3.2.3. The rotational method does not use a sliding window so individual leaves are not required to span the entire length of the field, allowing for gaps between individual leaves up to 29 cm wide. Furthermore, because the direction of leaf travel changes with respect to the desired fluence map at each collimator angle, only the length of the leaf bank dictates the upper limit of maximum field size. The maximum IMRT field size that may be delivered in one irradiation with the rotational method is therefore a 40 cm and a 29 cm diameter circle for the 120 leaf and 52 leaf MLCs respectively. 5.5 Discussion Results for the fluence generation of the 5 field thyroid treatment show a consistent improvement in fluence generation capabilities over conventional delivery methods and, for both dynamic and static deliveries, the 1cm leaf rotational delivery is on average as accurate as the 5 mm leaf conventional method. Advantages of the rotational method over conventional methods are more substantial for the 1 cm leaf than for the 5mm leaf M L C . These results correlate with the findings of section 2.5 where it was shown that resolution degradation due to dose dispersion (DSK) limits the improvements in spatial resolution of smaller leaf widths. The implication of these results wil l depend on the threshold considered acceptable by the clinician. Inaccuracy of the prescription over greater than 10% of the field (as seen with the 1 cm conventional but not the 1 cm rotational fluence maps) may be considered 144 unacceptable in some cases but will also depend on patient motion and set-up reproducibility as described in section 1.6.2. The histogram plotted in Figure 5.7 shows that the leaf motion calculation algorithms are highly reproducible, with any stochastic effects due to the optimization causing only minor variations in the resulting fluence maps. The plot in Figure 5.8 shows that radiation efficiency affects the resulting fluence map, with lower efficiency improving overall accuracy. The rotation range also affects the resulting fluence map, with rotation ranges of less than 180 degrees causing a significant reduction in accuracy as seen in Figure 5.9. Although spatial resolution considerations indicate that a rotation range of 90 degrees should be sufficient, the aperture forming capabilities of the M L C are only exploited when a minimum 180 degree rotation is used. Finally, it is shown in Figure 5.10 that increasing the total number of segments in static rotational delivery will improve the resulting fluence maps until approximately 80 segments, after which only minimal benefit is observed. Results of the dosimetric investigation seen in Figure 5.13 show that the rotational algorithms and linac delivery are able to accurately reproduce desired fluences. In a comparison between the rotational method and conventional method shown in Figure 5.14 a similar level of dosimetric accuracy is observed, with the rotational technique having slightly larger discrepancies for dynamic delivery. The rotational leaf motion calculation algorithms are based on a simplified model of the M L C . Various refinements to conventional dynamic leaf motion calculation have been reported by several investigators that have subsequently been incorporated into conventional leaf motion calculation algorithms. These include compensation for round leaf ends, M L C transmission [19] and extra-focal scattered radiation [52, 107, 108]. The above investigators and others [109] have observed discrepancies reduced to less than 3% once these effects have been taken into 145 consideration. Modifying these established refinement techniques to account for M L C rotation may produce a similar degree of improvement. In rotational delivery the stability of the collimator can affect the accuracy of fluence maps. The axis of rotation of the collimation system is highly reproducible to within 1 mm of the isocenter as shown in section 3.3.4 and is therefore not a significant source of error. When fields are delivered dynamically, the speed of the collimator and the dose rate must be synchronized or there will be a collimator phase shift error. In section 3.4.2.1 it was found that the collimator speed is highly reproducible. The time for a 180° rotation changed by less than 1% over a period of six months. At the beginning of delivery there is a period of approximately Is where the dose rate is increasing. During this time the collimator is accelerating. Errors due to a lack of synchronization were detected in an investigation of collimator rotation stability described in section 3.4.2.3, which would account for some of the delivery errors encountered in dynamic mode. Improvements in spatial resolution over conventional delivery methods are seen in Figure 5.15 and Figure 5.16. Techniques to improve the spatial resolution of intensity modulated fields have been reported by others. One method consists of indexing the couch position perpendicular to the direction of leaf motion between multiple intensity modulated fields [72]. Another uses 2 orthogonal (collimator rotated by 90°) fields [61, 71]. The principle of moving M L C leaves outside the direction of leaf travel is used in these techniques as well as the rotational technique and similar improvements in spatial resolution over conventional methods are observed. In the orthogonal delivery technique described by Siochi as well as Evans and Partridge, the high resolution fluence maps must go through a filtering process in order to separate them into two orthogonal deliveries. The amount of degradation that results from this filtering may, in some cases, cause an unacceptable modification of the desired fluence. Also, the junctioning of orthogonal leaves can cause effects 146 similar to interleaf leakage and tongue-and-groove effects. The main disadvantage of other techniques is that they use multiple intensity modulated fields and therefore generally require a larger total number of M U . Finally, although further investigation is required before the relative benefits of these techniques can be properly assessed, the fact that the rotational technique is not limited to two fixed collimator angles inherently offers more degrees of freedom for generating a desired fluence map. The amplitudes of the interleaf leakage spikes with the rotational technique are a small fraction of the interleaf leakage spikes observed in the conventional technique as seen in Figure 5.18. Although the actual leakage contribution in a clinical case wil l depend on the characteristics of the desired fluence map, the results show that the rotational method inherently provides a more uniform base leakage than conventional delivery methods. Also, no discrepancies due to the tongue-and-groove M L C leaf design have been observed with the rotational technique. The current dosimetric accuracy is not sensitive enough to discern any errors resulting from this effect. Finally, in conventional delivery there is no way to fully compensate for interleaf effects because the leaf edge positions are fixed and oriented parallel to the direction of leaf motion. This is not the case when rotation is used and it should therefore be possible to compensate for them in the rotational leaf motion calculation. It was shown that the rotational method allows an increase in maximum field size from a 14.5 cm wide field to a 29 cm and 40 cm diameter circle for the 52 leaf and 120 leaf M L C s respectively. Theoretically, increasing the field size to these upper limits could cause a reduction in the spatial resolution advantages of the rotational technique (field sizes greater that 29 cm diameter), particularly when high spatial resolution is required at the periphery of the field. Such effects have not yet been observed and, in any event, clinical IMRT field sizes exceeding 29 cm diameter are rare. 147 Chapter 6 CONCLUSION 6.1 Conclusion Currently available fluence generation techniques suffer from limitations in spatial resolution and dosimetric accuracy imposed by the multileaf collimator (MLC). In this thesis a novel method of controlling the M L C was developed that is capable of generating fluence maps with higher spatial resolution and less systematic error than conventional methods. In the first part of the thesis an investigation into the factors that reduce spatial resolution is presented. Using a linear systems theory model of the dose delivery process, a new technique was developed to evaluate the factors that limit a radiation therapy device from delivering an ideal dose distribution. The model allows the effect of the M L C and Dose Spread Kernel (DSK) to be evaluated separately, providing a greater understanding of the individual processes that modify the optimal dose distribution. Fourier analysis was used to provide insight into spatial resolution limitations at each step in the formulation. Effects of varying leaf width and D S K for treatment shapes of varying complexity were investigated. The model provided an accurate prediction of the dependence of geometric conformity on M L C leaf width and D S K size. Also, smaller more complex PTV shapes were shown to have a larger high spatial frequency component making them more sensitive to leaf width and DSK. The DTF can be used as a tool for comparing different dose delivery devices. Frequency analysis of the PTV can aid in deciding on resolution capabilities required to treat a site with a known complexity. By choosing the appropriate 148 apparatus and technique for a given treatment, the resources available to a clinician will be used more efficiently and greater overall patient care can be achieved in the clinic. The model was also applied to IMRT delivery and similar results were observed. Through this study a method of improving the spatial resolution of conventional IMRT delivery methods was identified. One-dimensional sampling of the M L C causes a degradation of the deliverable distribution in the direction perpendicular to leaf motion. It was hypothesized that an improvement in spatial resolution could be obtained by modifying the sampling geometry for each sub-field. In particular, by rotating the collimator between each sub-field it should be possible to obtain high resolution in the entire plane of fluence delivery. The mechanical and radiation producing capabilities of conventional linacs under collimator rotation conditions were evaluated. Results showed that accurate delivery is feasible with the current control mechanism although synchronization errors between collimator rotation speed and dose rate were identified in dynamic mode. Based on these findings, a new technique for the delivery of IMRT was developed that, by including collimator rotation exploits all degrees of freedom of the collimation system. A set of novel algorithms was developed to calculate the leaf positions at each collimator angle in the delivery. The algorithms were presented and analyzed in terms of their dependence on radiation efficiency, the range of collimator rotation and number of segments. A thyroid test case was used to evaluate the fluence generation capabilities of the algorithms. Advantages over conventional deliveries in generating high-resolution fluence maps were observed for 5mm and lcm M L C leaf widths in static and dynamic delivery modes. The potential dosimetric advantages of incorporating rotation were also investigated. Results showed that higher spatial resolution dose distributions are attainable with the rotational technique, allowing for superior target coverage and 149 healthy tissue sparing. It was also shown that interleaf leakage and tongue-and-groove effects are substantially reduced, decreasing the degree of systematic overdosing and underdosing observed in conventional IMRT delivery. Finally, the rotational technique removes restrictions on field size, allowing larger individual IMRT fields to be delivered. Dosimetric results show that the rotation technique provides satisfactory agreement between measured and desired dose distributions. Advantages over conventional methods are most apparent with 1 cm M L C leaves in the static delivery mode. Currently, the accuracy of rotational dynamic delivery is limited by the control mechanism of the linac. Improved linac control may provide improvements in overall accuracy and reproducibility, permitting advantages in spatial resolution, leaf edge effects and field size capabilities to be clinically realized in dynamic delivery. In summary, the rotational method is capable of generating fluence maps that have higher spatial resolution and less systematic error than those generated by conventional delivery methods. Dose distributions may be delivered that conform more closely to the target volume [38, 39], reducing the dose received by surrounding healthy tissue and decreasing the probability of negative side effects. Also, the rotational method provides physicians with the option of increasing the prescribed tumour dose and improving the probability of tumour control while maintaining the same level of healthy tissue damage. 6.2 Future Work Changes in dosimetric accuracy due to increasing or decreasing the efficiency parameter are currently under investigation. Preliminary results show that i f the efficiency is set too high there is a loss of dosimetric accuracy. Conversely, by setting it too low the sub-field M L C apertures become smaller, making leaf end and 150 transmission effects more important. Although further investigation is required, preliminary results are encouraging in that for more complex fluence maps the rotational technique attains a superior level of efficiency over conventional methods. During the optimization there is no modification of the individual sub-field weights. In dynamic delivery field weight modification is analogous to dose rate modification. By optimizing the field weights it may be possible to arrive at solutions that further improve the accuracy of the delivered fluences. Also, improvements in dosimetric accuracy may be attainable by refining the physical model of the M L C to include tongue-and-groove, interleaf leakage, non-uniform collimator scatter and rounded M L C leaf ends. The M L C and linac control software is not designed for combined automatic collimator rotation and leaf sequencing. Currently, the delivery time is only indicative of how fast the particular operator performing the delivery is able to access various linac functions and override software interlocks. It is therefore not possible to assess the delivery time with any degree of accuracy until full automatic control is implemented. Also, the possibility of increasing the maximum angular speed of the collimator is being investigated. These enhancements will allow for a more realistic evaluation of clinical delivery times. Although Varian linacs are used to deliver rotational IMRT fields in this study, other linacs and MLCs could also be used. Modifications to the rotational leaf motion calculation that incorporate Elekta M L C leaf constraints and allow rotational delivery on Elekta linacs are currently under investigation. Finally, testing of multiple 3-dimensional dose distributions with simulated patient motion and set-up reproducibility will provide a more extensive clinical evaluation. 151 Bibliography 1. Canadian Cancer Statistics 1998, in National Cancer Institute of Canada Archives. 1998. 2. Hall, E., Radiobiology for the Radiologist. 1994, Philadelphia: J.B. Lippincott Company. 3. VanDyk, J., The Modern Technology of Radiation Oncology: A Compendium for Medical Physicists and Radiation Oncologists, ed. J. Van Dyk. 1999, Madison, W l : Medical Physics Publishing. 4. C. A . Perez, L.W.B.a. J.L.R., Principles and Practives of Radiation Oncology. 1998, Philadelphia, PE: Lippincott-Raven. 1-78. 5. Kahn, F .M. , The Physics of Radiation Therapy. 2nd ed. 1984: Williams and Wilkins. 6. Johns, H.E. and J.R. Cunningham, The Physics of Radiology. 4th ed. 1983, Springfield, Illinois: Charles C. Thomas. 7. Handbook of Medical Imaging. 1st ed, ed. H.L.K. J. Beutel, R.L. Van Metter. Vol . 1. 2000, Bellingham, WA: SPIE. 8. Storchi, P.R., L.J. van Battum, and E. Woudstra, Calculation of a pencil beam kernel from measured photon beam data. Phys Med Biol, 1999. 44(12): p. 2917-28. 152 9. Storchi, P. and E. Woudstra, Calculation of the absorbed dose distribution due to irregularly shaped photon beams using pencil beam kernels derived form basic beam data Phys Med Biol, 1996. 41(4): p. 637-56. 10. Mackie, T.R., J.W. Scrimger, and J.J. Battista, A convolution method of calculating dose for 15-MV x rays. Med Phys, 1985. 12(2): p. 188-96. 11. Boyer, A . and E. Mok, A photon dose distribution model employing convolution calculations. Med Phys, 1985. 12(2): p. 169-77. 12. Chui, C S . and R. Mohan, Extraction of pencil beam kernels by the deconvolution method. Med Phys, 1988. 15(2): p. 138-44. 13. Wong, E., Y . Zhu, and J. Van Dyk, Theoretical developments on fast Fourier transform convolution dose calculations in inhomogeneous media. Med Phys, 1996. 23(9): p. 1511-21. 14. Pugachev, A . and L . Xing, Computer-assisted selection of coplanar beam orientations in intensity- modulated radiation therapy. Phys Med Biol, 2001. 46(9): p. 2467-76. 15. Pugachev, A .B . , A . L . Boyer, and L . Xing, Beam orientation optimization in intensity-modulated radiation treatment planning. Med Phys, 2000. 27(6): p. 1238-45. 16. MLC Implementation Guide. Vol . 1. 1999: Varian Oncology Systems. 17. Galvin, J .M., X . G . Chen, and R . M . Smith, Combining multileaf fields to modulate fluence distributions. Int J Radiat Oncol Biol Phys, 1993. 27(3): p. 697-705. 18. Bortfeld, T.R., et al, X-ray field compensation with multileaf collimators. Int J Radiat Oncol Biol Phys, 1994. 28(3): p. 723-30. 153 19. Spirou, S.V. and C S . Chui, Generation of arbitrary intensity profiles by dynamic jaws or multileaf collimators. Med Phys, 1994. 21(7): p. 1031-41. 20. Y u , C.X., et al., A method for implementing dynamic photon beam intensity modulation using independent jaws and a multileaf collimator. Phys Med Biol, 1995. 40(5): p. 769-87. 21. Bortfeld, T., Optimized planning using physical objectives and constraints. Semin Radiat Oncol, 1999. 9(1): p. 20-34. 22. Chen, Y . , A . L . Boyer, and L . Xing, A dose-volume histogram based optimization algorithm for ultrasound guided prostate implants. Med Phys, 2000. 27(10): p. 2286-92. 23. Wu, Q. and R. Mohan, Algorithms and functionality of an intensity modulated radiotherapy optimization system. Med Phys, 2000. 27(4): p. 701-11. 24. Ma, L., et al, An optimized leaf-setting algorithm for beam intensity modulation using dynamic multileaf collimators. Phys Med Biol, 1998. 43(6): p. 1629-43. 25. Xia, P. and L.J. Verhey, Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments. Med Phys, 1998. 25(8): p. 1424-34. 26. Siochi, R.A., Minimizing static intensity modulation delivery time using an intensity solid paradigm. Int J Radiat Oncol Biol Phys, 1999. 43(3): p. 671-80. 27. Ketting, C.H., et al., Consistency of three-dimensional planning target volumes across physicians and institutions. Int J Radiat Oncol Biol Phys, 1997. 37(2): p. 445-53. 154 28. Logue, J.P., et al, Clinical variability of target volume description in conformal radiotherapy planning. Int J Radiat Oncol Biol Phys, 1998. 41(4): p. 929-31. 29. Rasch, C , et al, Irradiation of paranasal sinus tumors, a delineation and dose comparison study. Int J Radiat Oncol Biol Phys, 2002. 52(1): p. 120-7. 30. Ter-Pogossian, M . M . , Positron Emission Tomography, in Principles of Nuclear Medicine. 1995, W. B . Saunders: Philadelphia, p. 342-377. 31. Uematsu, H., et al., Coregistration of F D G PET and MRI of the head and neck using normal distribution of FDG. JNuclMed, 1998. 39(12): p. 2121-7. 32. Bentel, G.C., Patient Positioning and Immobilization in Radiation Oncology. 1999, New York: McGraw-Hill. 33. Bieri, S., et al, Reproducibility of conformal radiation therapy in localized carcinoma of the prostate without rigid immobilization. Radiother Oncol, 1996. 38(3): p. 223-30. 34. Clark, B., et al. Immobilisation in Stereotactic Radiotherapy: Comparison of Thermoplastic Mask Systems, in 44th Annual Meeting of the American Association of Physicists in Medicine. 2002. Montreal. 35. Giraud, P., et al, Conformal radiotherapy (CRT) planning for lung cancer: analysis of intrathoracic organ motion during extreme phases of breathing. Int J Radiat Oncol Biol Phys, 2001. 51(4): p. 1081-92. 36. Ahnesjo, A . and M . M . Aspradakis, Dose calculations for external photon beams in radiotherapy. Phys Med Biol, 1999. 44(11): p. R99-155. 37. Bortfeld, T., U . Oelfke, and S. Ni l l , What is the optimum leaf width of a multileaf collimator? Med Phys, 2000. 27(11): p. 2494-502. 155 38. Chui, C.S., et al, Delivery of intensity-modulated radiation therapy with a conventional multileaf collimator: comparison of dynamic and segmental methods. Med Phys, 2001. 28(12): p. 2441-9. 39. Fiveash, J., et al, Effect of multileaf collimator leaf width on physical dose distributions in the treatment of CNS and head and neck neoplasms with intensity modulated radiation therapy. Med Phys, 2002. 29(6): p. 1116-9. 40. Otto, K. , B .G. Clark, and C. Huntzinger, Exploring the limits of spatial resolution in radiation dose delivery. Med Phys, 2002. 29(8): p. 1823-31. 41. Otto, K. , B. Clark, and C. Huntzinger. Investigation of a Linear Systems Model for Evaluating Radiation Dose Delivery, in WESCAN 2001. 2001. Fraser Valley, Canada. 42. Otto, K. , B .G. Clark, and C. Huntzinger. Degradation of Dose Conformity Using Linear Systems Analysis, in ESTRO. 2001. Seville, Spain. 43. Otto, K. , B .G. Clark, and C. Huntzinger. Evaluation of Radiation Dose Delivery Using a Linear Systems Approach, in 47th Annual Meetin of COMP. 2001. Kelowna. 44. Gonzalez, R.C. and R.E. Woods, Digital Image Processing. 1992, Reading: Addison-Wesley. 45. Cohen, G. and F.A. DiBianca, The use of contrast ~ detail — dose evaluation of image quality in a computed tomographic scanner. J Comput Assist Tomogr, 1979. 3(2): p. 189-95. 46. Arnold, B.A. , H . Eisenberg, and B.E. Bjarngard, The LSF and M T F of rare-earth oxysulfide intensifying screens. Radiology, 1976. 121(2): p. 473-7. 47. Wolf, M . , W. Steinbach, and W. Angerstein, Image quality of an image-intensifier fluorographic system. Phys Med Biol, 1984. 29(5): p. 567-77. 156 48. Atari, N .A. , R.D. Zwicker, and R.K. Schmidt-Ullrich, Performance evaluation of a prototype high resolution digital radiographic/near real-time fluoroscopic computerized tomographic system for radiotherapy simulation. Int J Radiat Oncol Biol Phys, 1995. 32(2): p. 421-36. 49. Bracewell, R., The Fourier Transform and Its Applications. 3rd ed. 1999, New York: McGraw-Hill. 50. Rossmann, K. , The spatial frequency spectrum: a means for studying the quality of radiographic imaging systems. Radiology, 1968. 90(1): p. 1-13. 51. Rossmann, K . and G. Lubberts, Some characteristics of the line spread-function and modulation transfer function of medical radiographic films and screen-film systems. Radiology, 1966. 86(2): p. 235-41. 52. Mohan, R., et al, The impact of fluctuations in intensity patterns on the number of monitor units and the quality and accuracy of intensity modulated radiotherapy. Med Phys, 2000. 27(6): p. 1226-37. 53. Shepard, D.M. , et al, A simple model for examining issues in radiotherapy optimization. Med Phys, 1999. 26(7): p. 1212-21. 54. Huntzinger, C , K . Brooks, and R. Tang. The MTF of IMRT. in World Congress on Medical Physics and Biomedical Engineering. 2000. Chicago. 55. Mohan, R., C. Chui, and L . Lidofsky, Differential pencil beam dose computation model for photons. Med Phys, 1986.13(1): p. 64-73. 56. Dong, L., et al, A pencil-beam photon dose algorithm for stereotactic radiosurgery using a miniature multileaf collimator. Med Phys, 1998. 25(6): p. 841-50. 57. Zhu, X.R. , E.E. Klein, and D.A. Low, Geometric and dosimetric analysis of multileaf collimation conformity. Radiother Oncol, 1998. 47(1): p. 63-8. 157 58. Huntzinger, C , K . Otto, and B.G. Clark. FFT of PTVs. in 43rd Annual Meeting of the American Association of Physicists in Medicine. 2001. Salt Lake. 59. Pirzkall, A., et al, Comparison of intensity-modulated radiotherapy with conventional conformal radiotherapy for complex-shaped tumors. Int J Radiat Oncol Biol Phys, 2000. 48(5): p. 1371-80. 60. Sharpe, M.B. , B . M . Miller, and J.W. Wong, Compensation of x-ray beam penumbra in conformal radiotherapy. Med Phys, 2000. 27(8): p. 1739-45. 61. Evans, P .M. and M . Partridge, A method of improving the spatial resolution of treatments that involve a multileaf collimator. Phys Med Biol, 2000. 45(3): p. 609-22. 62. Williams, P.C. and P. Cooper, High-resolution field shaping utilizing a masked multileaf collimator. Phys Med Biol, 2000. 45(8): p. 2313-29. 63. Sun, J. and Y . Zhu, Study of dosimetric penumbra due to multileaf collimation on a medical linear accelerator. Int J Radiat Oncol Biol Phys, 1995. 32(5): p. 1409-17. 64. Ma, L. , C. Yu , and M . Sarfaraz, A dosimetric leaf-setting strategy for shaping radiation fields using a multileaf collimator. Med Phys, 2000. 27(5): p. 972-7. 65. Otto, K . and B. Clark, Enhancement of IMRT Delivery through M L C rotation. Phys Med Biol, 2002. 47(Nov.): p. 3997-4017. 66. Otto, K . and B. Clark. Enhancing IMRT with MLC Rotation, in 44th Annual Meeting of the American Association of Physicists in Medicine. 2002. Montreal. 67. Williams, P.O. and A.R. Hounsell, X-ray leakage considerations for IMRT. Br J Radiol, 2001. 74(877): p. 98-100. 158 68. Klein, E.E. and D.A. Low, Interleaf leakage for 5 and 10 mm dynamic multileaf collimation systems incorporating patient motion. Med Phys, 2001. 28(8): p. 1703-10. 69. Deng, J., et al, The M L C tongue-and-groove effect on IMRT dose distributions. Phys Med Biol, 2001. 46(4): p. 1039-60. 70. Xia , P. and L.J . Verhey, Delivery systems of intensity-modulated radiotherapy using conventional multileaf collimators. Med Dosim, 2001. 26(2): p. 169-77. 71. Alfredo, R. and C. Siochi, Virtual micro-intensity modulated radiation therapy. Med Phys, 2000. 27(11): p. 2480-93. 72. Galvin, J .M., D.D. Leavitt, and A . A . Smith, Field edge smoothing for multileaf collimators. Int J Radiat Oncol Biol Phys, 1996. 35(1): p. 89-94. 73. Symonds-Tayler, J.R. and S. Webb, Gap-stepped M L C leaves with filler blades can eliminate tongue-and- groove underdoses when delivering IMRT with maximum efficiency. Phys Med Biol, 1998. 43(8): p. 2393-5. 74. Ma, L., et al, Synchronizing dynamic multileaf collimators for producing two- dimensional intensity-modulated fields with minimum beam delivery time. Int J Radiat Oncol Biol Phys, 1999. 44(5): p. 1147-54. 75. Wu, Q., et al, Dynamic splitting of large intensity-modulated fields. Phys Med Biol, 2000. 45(7): p. 1731-40. 76. Fiveash, J.B., et al, Effect of multileaf collimator leaf width on physical dose distributions in the treatment of CNS and head and neck neoplasms with intensity modulated radiation therapy. Med Phys, 2002. 29(6): p. 1116-9. 77. Pasquino, M . , V . Casanova Borca, and S. Tofani, [Physical-dosimetric characterization of a multi-leaf collimator system for clinical implementation 159 in conformational radiotherapy]. Radiol Med (Torino), 2001. 101(3): p. 187-92. 78. Chen, Y . , A . L . Boyer, and C M . Ma, Calculation of x-ray transmission through a multileaf collimator. Med Phys, 2000. 27(8): p. 1717-26. 79. Kim, J.O., et al., A Monte Carlo study of radiation transport through multileaf collimators. Med Phys, 2001. 28(12): p. 2497-506. 80. Huq, M.S., et al., A dosimetric comparison of various multileaf collimators. Phys Med Biol, 2002. 47(12): p. N159-70. 81. Saw, C.B., et al, Leaf sequencing techniques for MLC-based IMRT. Med Dosim, 2001. 26(2): p. 199-204. 82. Sykes, J.R. and P . C Williams, An experimental investigation of the tongue and groove effect for the Philips multileaf collimator. Phys Med Biol, 1998. 43(10): p. 3157-65. 83. Malet, C , et al, A study of dose delivery in small segments. Int J Radiat Oncol Biol Phys, 2000. 48(2): p. 535-9. 84. Hansen, V .N. , et al, Quality assurance of the dose delivered by small radiation segments. Phys Med Biol, 1998. 43(9): p. 2665-75. 85. Bain, L.J.E., Max, Introduction to Probability and Mathematical Statistics. 2nd ed. 1992, Boston, M A : PWS-KENT. 86. Stein, J., et al, Dynamic X-ray compensation for conformal radiotherapy by means of multi- leaf collimation. Radiother Oncol, 1994. 32(2): p. 163-73. 87. Svensson, R., P. Kallman, and A. Brahme, An analytical solution for the dynamic control of multileaf collimators. Phys Med Biol, 1994. 39(1): p. 37-61. 160 88. Langer, M . , V . Thai, and L . Papiez, Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators. Med Phys, 2001. 28(12): p. 2450-8. 89. De Gersem, W., et al, Leaf position optimization for step-and-shoot IMRT. Int J Radiat Oncol Biol Phys, 2001. 51(5): p. 1371-88. 90. Markman, J., et al, Beyond bixels: generalizing the optimization parameters for intensity modulated radiation therapy. Med Phys, 2002. 29(10): p. 2298-304. 91. Alber, M . and F. Nusslin, Optimization of intensity modulated radiotherapy under constraints for static and dynamic M L C delivery. Phys Med Biol, 2001. 46(12): p. 3229-39. 92. Shepard, D.M. , et al, Direct aperture optimization: a turnkey solution for step-and-shoot IMRT. Med Phys, 2002. 29(6): p. 1007-18. 93. Edwin K . P. Chong, S.H.Z., An Introduction to Optimization. 2nd ed. 2001, New York: Wiley-Interscience. 94. R. Cairoli, R.C.D., Sequential Stochastic Optimization. 1996: Wiley-InterScience. 95. Arnfield, M.R., et al, Dosimetric validation for multileaf collimator-based intensity- modulated radiotherapy: a review. Med Dosim, 2001. 26(2): p. 179-88. 96. Haryanto F, F .M. , Laub W, Dohm O, Nusslin F., Investigation of photon beam output factors for conformal radiation therapy—Monte Carlo simulations and measurements. Phys Med Biol, 2002. 47(11): p.N133-43. 161 97. Cadman, P., et al, Dosimetric considerations for validation of a sequential IMRT process with a commercial treatment planning system. Phys Med Biol, 2002. 47(16): p. 3001-10. 98. Childress, N.L . , L. Dong, and Rosen, II, Rapid radiographic film calibration for IMRT verification using automated M L C fields. Med Phys, 2002. 29(10): p. 2384-90. 99. Robar, J.L. and B.G. Clark, The use of radiographic film for linear accelerator stereotactic radiosurgical dosimetry. Med Phys, 1999. 26(10): p. 2144-50. 100. Olch, A.J. , Dosimetric performance of an enhanced dose range radiographic film for intensity-modulated radiation therapy quality assurance. Med Phys, 2002. 29(9): p. 2159-68. 101. Zhu, X.R., et al, Evaluation of Kodak EDR2 film for dose verification of intensity modulated radiation therapy delivered by a static multileaf collimator. Med Phys, 2002. 29(8): p. 1687-92. 102. Esthappan, J., et al, Dosimetry of therapeutic photon beams using an extended dose range film. Med Phys, 2002. 29(10): p. 2438-45. 103. Low, D.A., et al, A technique for the quantitative evaluation of dose distributions. Med Phys, 1998. 25(5): p. 656-61. 104. Harms, W.B., Sr., et al, A software tool for the quantitative evaluation of 3D dose calculation algorithms. Med Phys, 1998. 25(10): p. 1830-6. 105. Dai, J. and Y . Zhu, Minimizing the number of segments in a delivery sequence for intensity- modulated radiation therapy with a multileaf collimator. Med Phys, 2001. 28(10): p. 2113-20. 162 106. Essers, M . , et al, Commissioning of a commercially available system for intensity- modulated radiotherapy dose delivery with dynamic multileaf collimation. Radiother Oncol, 2001. 60(2): p. 215-24. 107. LoSasso, T., C S . Chui, and C C . Ling, Physical and dosimetric aspects of a multileaf collimation system used in the dynamic mode for implementing intensity modulated radiotherapy. Med Phys, 1998. 25(10): p. 1919-27. 108. Arnfield, M.R., et al, A method for determining multileaf collimator transmission and scatter for dynamic intensity modulated radiotherapy. Med Phys, 2000. 27(10): p. 2231-41. 109. Wang, X. , et al, Dosimetric verification of intensity-modulated fields. Med Phys, 1996. 23(3): p. 317-27. 163 Appendix A Circular PTV Conformity Derivation N = # of M L C leaves in one quadrant = round In cartesian coordinates the leaf centers intersect the circle at: y k + — V 2y w ( f r 2 \ X - r2- k + w2 2j J k = {0,1,2 N] The edges of the leaves intersect the circle points shown by a, b, c, e on the diagram which have values x given by: 1 6 4 xb = f ( r ^ 2 A r2- k + w2 V 2, > J Xc=(r2-(k + l)2w2) x = 0 A r e a A k i s ca l cu la t ed i n the f o l l o w i n g in tegra l : Area A k = J (r2 -x2 \ -kw \dx = f(^-^2)^Ysm-' —G-2-V^+—sin"1 2 v 6 ' 2 / v N\ v r J •kw(xa -xb) A r e a B k i s ca l cu l a t ed i n the f o l l o w i n g in tegra l : Area B k = + l)w - (r2 - x 2 )z j j x = £ w ( x 6 - x e ) - ^ ( r 2 - x 6 2 ) 2 + ysin + • + —sin f^ \ The Remainder Area is calculated using the following integral: k = N = round] I' 165 i f the remainder of — < 0.5 then: w ,1 >\ Remainder Area = jjjr2-x2)2- Nw ]dx 1 2 r . -i —sin 2 Nwx„ V J i f the remainder of — > 0.5 then: w Remainder Area = J (N + i)w- (r2 - x2 )2 jdx (7V + lK-^(r2-x62)^-ysin - l V r J The % conformity is defined as: %conformity = 100 x Area ofPTV-(Area of PTV underdose + Area of healthy tissue dose) Area of PTV f N nr - 41 %conformity -100 x (Area A k + Area B k ) + Remainder Area Vk=0 nr 166 Appendix B Rotational Leaf Motion Calculation Engine RDMIX Fluence Generator mmm - .mix E:\RDMLC expefimentsVielcE Load Fluence E:\RDMLC experimentsMieldlout Save Fluence Oesited Fluence Calculated Fluence (• 120 Leaf Millenium (5mm) C 52 Leaf Mark2(1 cm) 10 Initial 8 Segments MU Efficiency [0.8 <• Dynamic Delivery Calculate Leaf Motions Iterations; 123000 Cost: 40.3013 Number of Angles: 1 GO Maximum Segments: ISO Range of Motionfdeg); 180 0.1 0.2 Difference Histogram Refresh Results Deciease Error Window Optimize Dose Rate Increase Error Window Increase Angles (x2) C Initialize to Maximum Pause <*• Automatic Termination Continue Accept Result -| 300 •g200 o 100 0 0 S3 10 Iterations f500s1 e:\dmrc Rles\field2.mM Write DMLC File Display High Res Movie | Display DMLC Movie | Simulate Film | Points Outside Range Difference Map Figure B.l: Software was developed to derive the rotating leaf motions. A graphical user interface facilitated the calculation, analysis and verification of different fluence maps. Functions of the software include: Reading in desired fluence maps, selecting fixed parameters (MLC type, dynamic vs. static delivery, radiation efficiency, initialization method etc.), displaying calculated fluence maps, evaluation of the calculated fluence maps and supplying MLC file output for delivery. 167 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085723/manifest

Comment

Related Items