Electron Beams:
Dose calculation algorithms
Kent A. Gifford, Ph.D.
Department of Radiation Physics
UT M.D. Anderson Cancer Center
kagifford@mdanderson.org
Medical Physics III: Spring 2015
Dose calculation algorithms
• Deterministic
– Hogstrom pencil beam (Pinnacle3)
– Phase space evolution model
– FEM solutions to Boltzmann eqn (Attila)
Dose calculation algorithms
Hogstrom pencil beam
• Mass scattering power
Dose calculation algorithms
Hogstrom pencil beam
• Fermi-equation (separated)
Dose calculation algorithms
Hogstrom pencil beam
• Fermi-equation (solution)
Dose calculation algorithms
Hogstrom pencil beam
Discrete Ordinates (FEM)-Attila
Linear Boltzmann Transport Equation (LBTE)
•
Assumptions1
1. Particles are points
2. Particles travel in straight lines
3. Particles do not interact w each other
4. Collisions occur instantaneously
5. Isotropic materials
6. Mean value of particle density distribution considered
•1EE Lewis and WF Miller, Computational Methods of
Neutron Transport, ANS, 1993.
Fundamentals
Linear Boltzmann Transport Equation (LBTE)
• ↑direction
•↑Angular
•↑position
•↑particle
vector
fluence
•↑macroscopic
vector
energy
rate
•extrinsic
total cross
•↑scattering
source
section↑ source
•Streaming
•Collision
•Sources
•Obeys conservation of particles
• Streaming + collisions = production
Fundamentals
Linear Boltzmann Transport Equation (LBTE)-angular fluence
••normalized
↑angular fluence
rate harmonics•↑↑angular fluence rate
spherical
coefficients
Fundamentals
Linear Boltzmann Transport Equation (LBTE)-scattering xsection
• •↑differential
• orthogonal
scattering
cross-section
Legendre
polynomial↑
differential
scattering
moments
↑
Fundamentals
Linear Boltzmann Transport Equation (LBTE)-scattering source
• differential
• ↑scattering
scattering
source xsection
• angular
↑ fluence rate↑
Fundamentals
Linear Boltzmann Transport Equation (LBTE)-Reaction
rate↑
cross-section of type
•↑reaction•scalar
rate fluence•↑macroscopic
whatever
Fundamentals
Attila-Energy approximation
•Multi-group approximation
• Energy range divided into g, groups
• Ordered by decreasing energy
• Cross-sections constant w/in group
Fundamentals
Attila-Angular approximation
•Discrete ordinates method (DOM)
• Requires LBTE hold for discrete angles
• Angular terms integrated by quadrature set
• Mesh swept by each angular ordinate
• As # of ordinates  , sol’n converges to exact sol’n
Fundamentals
Attila-Angular approximation
•Discrete ordinates method (DOM)-ray effects
• Non-physical buildup in fluence/rate along ordinates
• May produce oscillations or negativities
• Problematic for localized sources in weakly scattering media
Fundamentals
Attila-Angular approximation
Fundamentals
Attila-Angular approximation
•
Ray effect-remedies
•
Increase # of ordinates
•This can be computationally costly
• Employ first scatter distributed source (fsds) technique
•Less costly since a lower angular order can be used
Fundamentals
•FSDS technique
Attila-fsds
• Separate angular fluence/rate into collided and
uncollided components
• Ray trace from point source to quadrature or edit
points
• 1ST collision source generated at each tet corner
• Solve collided angular fluence/rate and add to
uncollided
Fundamentals
Attila-Spatial approximation
•
Discontinuous Finite element
method (DFEM)
•
Unstructured tetrahedral
mesh
•
Variably sized elements
•
Fluence/rate allowed to be
discontinuous across tet faces
Fundamentals
Attila-Source iteration
•Source iteration
• 4  Nelements  Nordinates  Ngroups unknowns
• Iteration started with guess for fluence
• Process may proceed slowly for problems dominated by
scattering
• Acceleration technique applied- DSA
Fundamentals
Attila-Charged particles
•Continuous scattering
operator
•LBTE
•LBTE
•Continuous slowing
down operator
Fundamentals
Attila-Cross sections
•Attila can utilize x-sections from various sources
•Multi-group processing codes
• NJOY-TRANSX (LANL)
• AMPX (ORNL)
• CEPXS (SNL)
Pros & Cons of the
deterministic method
Pros & Cons
Advantages
1. Provides solution for the entire computational domain
2. Mesh based solution lends itself to CT/MRI based
geometries
3. Typically more efficient than MC
Dose calculation algorithms
Monte Carlo
•Stochastic method for evaluating integrals numerically
•Generate N random values or points in a space, xi
•Calculate the score or tally fi for the N random values, points
•Calculate the expectation value, and standard deviation, variance
•Rely on central limit theorem
•As N approaches infinity, the expectation value will approach
reality or true value
Dose calculation algorithms
Monte Carlo
• Example:
•Particle interacting with 2 possibilities
•Absorption
•Scatter
• Random value is particle history or trajectory
•Could also tally energy or charge deposition, current, pulses
Dose calculation algorithms
Monte Carlo
• Algorithm:
•Sample random distance to the subsequent interaction site
•Transport particle to next interaction factoring in geometry
•Choose interaction type based on relative probability
•Simulate interaction
•Absorption-particle is terminated
•Scatter- choose scattering angle using appropriate scattering pdf
• Repeat until N histories are simulated
Project
• Generate MU calculation program
• Any language or spreadsheet program
• 12 e-, all field sizes, cones
– Verify correct implementation
– Demonstrate accuracy on 2 cases
Project
150 cGy to 95%, 12 MeV
Project
200 cGy to 100%, 12 MeV