Photon Beam Dose Calculation Algorithms Kent A. Gifford, Ph.D. Medical Physics III Spring 2010 Name Title of presentation These are important to understand: What your planning system uses to calculate dose Important to understand limitations and areas of applicability. Adage: Don’t necessarily believe the dose distribution from a computer.

Dose Computation Algorithms Correction-based (Ancient!) Convolution (Pinnacle,Eclipse,…) Monte Carlo (Stochastic) Deterministic (Non-stochastic) 3 types: Older: ROCS Convolution: Pinnacle, Eclipse Monte Carlo: Not sure of commercial vendors, Research

Correction-based algorithms Standard SSD Photon Source Patient SSD, Thickness Patient composition Measurements Calculations (Correction Factors) Water Correction based: Based on lots of measurements Standard conditions 100 cm SSD Water Tank Range of field sizes, depths Patient dose obtained via Corrections: non-standard SSD irregular patient surface heterogeneities

Correction-based: Semi-empirical Empirical: Standard measurements Analytical: Correction factors for: Beam modifiers: shaped blocks, wedges… Patient contours Patient heterogeneities Correction-based algorithms Semi-empirical -measurements Correction factors account for heterogeneities irregular anatomy Beam modifiers

Measurements Percent Depth Dose Lateral Dose Profiles Beam Output Measurements Wedge Factor Measurements Measured data: PDD – all energies and field sizes Dose profiles Output factors Wedge factors

Generating Functions Convert phantom dose to patient dose Examples: Tissue-Phantom Ratio - Attenuation Inverse square factor – Distance Lookup tables, e.g. off-axis factors Generating functions: Convert dose measured in a phantom to dose in the patient Examples: TPR Inverse square factor Off axis factors, etc.

Generating Functions Accurate ONLY in case of electronic equilibrium Dmax and beyond Far from heterogeneities Issues: Small tumors in presence of heterogeneities Small field sizes Correction based algorithms use data and functions that Avoid electronic disequilibrium Not very accurate near skin surface and near heterogeneities. Small tumors in the lung are a problem So are small field sizes

Beam Modifier Corrections Must correct for attenuation through beam modifiers: 1. Wedges- WF, wedged profiles 2. Compensators- attenuation measurements 3. Blocks- OF Use Correction factors for beam modifiers as well

Contour Corrections Attenuation corrections due to “missing” tissue Effective SSD Method Uses PDD. Assumes PDD independent of SSD. Scales Dmax with inverse square factor. Contour corrections correct data acquired at a standard SSD to clinical SSDs. Effective SSD Uses PDD, assumes independent of SSD, Dose at dmax scaled according to inverse square TMR Ratio Similar, but uses TMR, Independent of SSD Isodose shift Older method, manually shifting isodose curves.

Contour Corrections TMR (TAR) Ratio Method Exploits independence of TMR and SSD More accurate than Effective SSD method.

Contour Corrections Isodose Shift Method Pre-dates modern treatment planning systems Manual method; generates isodose curves for irregular patient contours Greene & Stewart. Br J Radiol 1965; Sundblom Acta Radiol 1965

Contour Corrections

Contour corrections Effective attenuation method Corrects for average attenuation along beam direction Least accurate and easiest to apply

Heterogeneity Corrections One dimensional: 1. TMR ratio: CF=TMReff /TMRphysical Corrects for primary photon attenuation Not as accurate in heterogeneity proximity Different heterogeneity correction methods One-dimensional: TMR ratio: Most accurate when only primary attenuation dominates Ratio of TMRs for the radiological depth and the physical depth. radiological depth is a depth weighted by the density of a heterogeneity. The Batho power law uses the thickness of heterogeneities and the depth beyond the heterogeneity 3D methods The Equivalent TAR method accounts for the 3D representation of heterogeneities accounts for changes in scatter as well as primary attenuation Scales both depth and field size

Heterogeneity Corrections Batho power law

Problems with correction-based algorithms Usually assume electronic equilibrium Inaccurate near heterogeneities Errors as large as 20% Require copious measurements assume electronic equilibrium inaccuracies near the boundaries of heterogeneities, such as bone and lung. Require extensive amount of measurements.

Convolution Algorithms Rely on fewer measurements Measured data: Fingerprint to characterize beam Model beam fluence Energy deposition at and around photon interaction sites is computed Second class of algorithms-convolution algorithms. These algorithms require a minimal amount of data. Minimal amount of beam data is used to characterize the beam like a fingerprint. Once the “identity” of the beam has been determined, dose is calculated from first principles of attenuation and scatter.

Convolution: Explicitly Modeled Beam Features Source size Extrafocal radiation: flattening filter, jaws,... Beam spectrum– change with lateral position (flattening filter) Collimator transmission Wedges, blocks, compensators… Explicitly model beam source Scatter radiation from the treatment head The polyenergetic beam spectrum Tranmission through modifiers, collimators

Primary and Scatter Concepts Two types of energy deposition events Primary photon interactions. Scatter photon interactions. Convolution algorithms consider dose from both primary photons and scattered radiation: photons and electrons Primary fluence: Photons that haven’t interacted, travel along red rays until they interact at points like these. After interacting, some of the energy can be scattered from one point to another.

Dose from Scatter Interactions To calculate dose at a single point: Must consider contributions of energy scattered from points over the volume of the patient. r’ To compute the dose at a given point, contributions of scattered energy from surrounding points are considered. r’ r’

Convolution: Volume segmented into voxels (volume elements) Primary fluence(dose) Interaction sites Mathematically, this is an integration. The irradiated volume is divided into voxels shown here. Contributions to the dose at a point, r, are computed by summing contributions of dose from all voxels in the irradiated volume. Dose spread array

Convolution Algorithm: Heterogeneities Radiological path length This sagittal view of a patient is a little less abstract. Shows how path lengths through heterogeneous tissues are considered.

Convolution Algorithm This is the convolution equation expressed as an integral. The integration is over a volume. I’ll go over each of the terms in the integrand.

Primary Energy Fluence - Y(r’) Product of primary photons/area and photon energy Computed at all points within the patient from a model of the beam leaving the treatment head Primary energy fluence: Represents how much energy is passing through the patient. Product of the number of photons per unit area and their energy.

Mass Attenuation Coefficient m / r (r’) Fraction of energy removed from primary photon energy fluence per unit mass Function of electron density Fraction of energy removed from the primary energy fluence is given by the mass absorption coefficient.

TERMA - T(r’) Product of Ψ(r’) and μ/ρ(r’) Total radiation Energy Released per MAss It represents the total amount of radiation energy available at r’ for deposition Thus, the TERMA— Product of these terms total energy released in the matter per unit mass is a product of the primary Energy available for dose deposition surrounding primary photon interaction sites.

Convolution Kernel Gives the fraction of the TERMA from a primary interaction point that is deposited to surrounding points Function of photon energy and direction primary Convolution kernel Gives the fraction of the TERMA deposited at points surrounding primary photon interaction sites. Curves show lines of equal energy deposition. Typically, it is not isotropic and is a function of photon direction energy Iso energy distribution lines.2’ interactions

Convolution Superposition Algorithm Convolution equation is modified for actual radiological path length to account for heterogeneities Heterogeneities are accounted for by using radiological path lengths. Distances scaled by electron density.

Pinnacle Convolutions Collapsed-cone (CC) convolution Most accurate, yet most time consuming Adaptive convolution Based on gradient of TERMA, compromise Fast convolution Useful for beam optimization and rough estimates of dose The Pinnacle treatment planning system has 3 dose calculation algorithms, all of which are convolution calculations. They are the collapsed-cone, adaptive and fast convolution.

Collapsed cone approximation All energy released from primary photons at elements on an axis of direction is rectilinearly transported and deposited on the axis. Energy that should be deposited in voxel B’ from interactions at the vertex of the lower cone is deposited in voxel B and vice versa. Approximation is less accurate at large distances from cone vertex. Errors are small due to rapid fall-off of point-spread functions approximation that is made to simplify calculations decrease computation time. Primary interactions occur at the apex of a conical volume, located along primary ray lines. Dose deposited off axis is considered to be deposited along the ray line. More accurate closer to the apex, where voxels off axis overlap with voxels on the raylines. Not a problem at larger distances because the convolution kernel falls off rapidly with distance.

Behavior of dose calculation algorithms near simple geometric heterogeneities Fogliatta A., et al. Phys Med Biol. 2007 7 algorithms compared Included Pinnacle and Eclipse Monte Carlo simulations used as benchmark 6 and 15 MV beams Various tissue densities (lung – bone) Fogliatta and others compared 7 dose calculation algorithms of treatment planning systems Included Pinnacle and Eclipse Used Monte Carlo as benchmark They compared dose calculations near simple geometric heterogeneities.

Virtual phantom/irradiation geometry dose in a virtual water phantom containing a region shown in dark gray Where density allowed to vary from light lung to cortical bone.

Types of algorithms considered Type A: Electron (energy) transport not modeled Type B: Electron transport accounted for (Pinnacle CC and Eclipse AAA). Two types of algorithms among the 7 Type A algorithms do not model electron transport. The Type B algorithms did, and included the likes of the Pinnacle and Eclipse algorithms.

Depth dose, 15 MV, 4 cm off-axis, through “light lung”, Several algorithms Problems with algorithms that do not model electron transport. Electronic equilibrium? No problem. Better agreement between Pinnacle CC and Monte Carlo than between Eclipse AAA and Monte Carlo. 15 MV Depth dose curves through the water phantom containing a bulk heterogeneity with a density equal to lung computed by various algorithms. The baseline is the black curve. Upper curves are type A algorithms. Gray-dashed line is Pinnacle. Red line is Eclipse. In this case, Pinnacle seems superior to Eclipse. Clearly shows where it’s important to model electronic disequilibrium. All algorithms do equally well upstream and far downstream, but type A algorithms have problems at points in the lung.

Conclusions Type A algorithms inadequate inside heterogeneous media, esp. for small fields type B algorithms preferable. Pressure should be put on industry to produce more accurate algorithms type A algorithms inadequate inside heterogeneous media, especially for small fields, and that no algorithm was capable of performing adequately under all conditions. Pressure should be put on industry to complete the evolution of dose calculation algorithms.

Comparison of algorithms in clinical treatment planning Knoos T, et al. Phys Med Biol 2006 5 TPS algorithms compared (A & B) CT plans for prostate, head and neck, breast and lung cases 6 MV - 18 MV photon energies used Knoos Compared algorithms with different types of CT treatment plans. Range of photon energies. Type A & B algorithms

Conclusions – Algorithm comparisons for clinical cases Prostate/Pelvis planning: A or B sufficient Thoracic/Head & Neck – type B recommended Type B generally more accurate in the absence of electronic equilibrium As long as electronic equilibrium most algorithms about same Recommend type B, especially for thoracic

Monte Carlo (Gambling) γ σ Monte Carlo (Gambling) Particle Interaction Probabilities Monte Carlo. simulates individual photon and electron interactions Uses probability and statistics, given by interaction cross sections Name is a reference to gambling

Monte Carlo Example: 100 20 MeV photons interacting with water. Interactions: τ, Photoelectric absorption (~0) σ, Compton scatterings (56) π, Pair production events (44) Here’s an example of what Monte Carlo does: A 10 MeV photon is the most powerful photon in a 10 MV spectrum Can undergo these interactions with these probabilities Traces the photon and electrons produced as they pass through the patient

Monte Carlo

Indirect Use of Monte Carlo Energy deposition kernels Monte Carlo is also used for calculating scatter kernels for convolution

Comparisons of Algorithms Monte Carlo and Convolution Often used as gold standard in research evaluating other algorithms.

Direct Monte Carlo Planning Pros Cons Can model “everything” Requires lots of histories Accuracy improved by tracing lots of particle histories Computation times, limited by computer capabilities

Fundamentals Linear Boltzmann Transport Equation (LBTE) Sources Collision Streaming ↑direction vector ↑Angular fluence rate ↑position vector ↑particle energy ↑macroscopic total cross section extrinsic source ↑ ↑scattering source Obeys conservation of particles Streaming + collisions = production

Transport Examples Methods and Materials (External beam-Prostate)

Transport Examples Methods and Materials (External beam-Prostate)

Transport Examples Results (External beam-Prostate)

Transport Examples Methods and Materials (Brachytherapy-HDR) Dimensions in cm

Results Attila (S16) vs. MCNPX Run time*: 13.7 mins, 97% points w/in 5%, 89% w/in ±3% *MCNPX: 2300 mins

References (1/2) The Physics of Radiation Therapy, 2nd Ed., 1994. Faiz M. Khan, Williams and Wilkins. Batho HF. Lung corrections in cobalt 60 beam therapy. J Can Assn Radiol 1964;15:79. Young MEJ, Gaylord JD. Experimental tests of corrections for tissue inhomogeneities in radiotherapy. Br J Radiol 1970; 43:349. Sontag MR, Cunningham JR. The equivalent tissue-air ratio method for making absorbed dose calculations in a heterogeneous medium. Radiology 1978;129:787. Sontag MR, Cunningham JR. Corrections to absorbed dose calculations for tissue inhomogeneities. Med Phys 1977;4:431. Greene D, Stewart JR. Isodose curves in non-uniform phantoms. Br J Radiol 1965;38:378 Early efforts toward more sophisticated pixel-by-pixel based dose calculation algorithms. Cunningham JR. Scatter-air ratios. Phys Med Biol 1972;17:42. Wong JW, Henkelman RM. A new approach to CT pixel-based photon dose calculation in heterogeneous media. Med Phys 1983;10:199. Krippner K, Wong JW, Harms WB, Purdy JA. The use of an array processor for the delta volume dose computation algorithm. In: Proceedings of the 9th international conference on the use of computers in radiation therapy, Scheveningen, The Netherlands. North Holland: The Netherlands, 1987:533. Kornelson RO, Young MEJ. Changes in the dose-profile of a 10 MV x-ray beam within and beyond low density material. Med Phys 1982;9:114. Van Esch A, et al., Testing of the analytical anisotropic algorithm for photon dose calculation. Med Phys 2006;33(11):4130-4148. 50

References (2/2) Fogliatta A, et al. On the dosimetric behaviour of photon dose calculation algorithms in the presence of simple geometric heterogeneities: comparison with Monte Carlo calculations. Phys Med Biol. 2007; 52:1363-1385. Knöös T, et al. Comparison of dose calculation algorithms for treatment planning in external photon beam therapy for clinical situations. Phys. Med. Biol. 2006; 51:5785-5807. CC Convolution Ahnesjö A, Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Med. Phys. 1989; 16(4):577-592. Mackie TR, Scrimger JW, Battista JJ. A convolution method of calculating dose for 15-MV x-rays. Med Phys 1985; 12:188. Mohan R, Chui C, Lidofsky L. Differential pencil beam dose computation models for photons. Med Phys 1986; 13:64. Lovelock DMJ, Chui CS, Mohan R. A Monte Carlo model of photon beams used in radiation therapy. Med Phys 1995;22:1387.