1. 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.

2. 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

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. 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.

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

11. 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

12. Contour Corrections

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

14. 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

15. Heterogeneity Corrections Batho power law

16. 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.

17. 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.

18. 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

19. 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.

20. 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’

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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

28. 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.

29. 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.

30. 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.

31. 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.

32. 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.

33. 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.

34. 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.

35. 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.

36. 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

37. 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

38. 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

39. Monte Carlo
Example:
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

40. Monte Carlo

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

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

43. 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

44. 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

45. Transport Examples Methods and Materials (External beam-Prostate)

46. Transport Examples Methods and Materials (External beam-Prostate)

47. Transport Examples Results (External beam-Prostate)

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

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

50. References (1/2)
The Physics of Radiation Therapy, 2nd Ed., 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): 50

51. 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:
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:
CC Convolution
Ahnesjö A, Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Med. Phys. 1989; 16(4):
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.