Electron Beams:Dose calculation algorithms

Kent A. Gifford, Ph.D.

Department of Radiation Physics

UT M.D. Anderson Cancer Centerkagifford@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 algorithmsHogstrom pencil beam

• Mass scattering power

Dose calculation algorithmsHogstrom pencil beam

• Fermi-equation (separated)

Dose calculation algorithmsHogstrom pencil beam

• Fermi-equation (solution)

Dose calculation algorithmsHogstrom pencil beam

Discrete Ordinates (FEM)-AttilaLinear 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.

FundamentalsLinear Boltzmann Transport Equation (LBTE)

• ↑direction vector• ↑position vector• ↑Angular fluence rate• ↑particle energy• ↑macroscopic total cross section• ↑scattering source• extrinsic source ↑

• Collision • Sources• Streaming

• Obeys conservation of particles

• Streaming + collisions = production

FundamentalsLinear Boltzmann Transport Equation (LBTE)-angular

fluence

• ↑angular fluence rate coefficients

• normalized spherical harmonics ↑• ↑angular fluence rate

FundamentalsLinear Boltzmann Transport Equation (LBTE)-scattering

xsection

• differential scattering moments↑• orthogonal Legendre polynomial↑• ↑differential scattering cross-section

FundamentalsLinear Boltzmann Transport Equation (LBTE)-scattering

source

• ↑scattering source• differential scattering xsection↑• angular fluence rate↑

FundamentalsLinear Boltzmann Transport Equation (LBTE)-Reaction

• ↑reaction rate • ↑macroscopic cross-section of type whatever

• scalar fluence rate↑

FundamentalsAttila-Energy approximation

• Multi-group approximation

• Energy range divided into g, groups

• Ordered by decreasing energy

• Cross-sections constant w/in group

FundamentalsAttila-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

FundamentalsAttila-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

FundamentalsAttila-Angular approximation

FundamentalsAttila-Angular approximation

• Ray effect-remedies

• Increase # of ordinates

• Employ first scatter distributed source (fsds) technique• This can be computationally costly

• Less costly since a lower angular order can be used

FundamentalsAttila-fsds

• FSDS technique

• 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

FundamentalsAttila-Spatial approximation

• Discontinuous Finite element method (DFEM)

• Unstructured tetrahedral mesh

• Variably sized elements

• Fluence/rate allowed to be discontinuous across tet faces

FundamentalsAttila-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

FundamentalsAttila-Charged particles

• LBTE

• LBTE

• Continuous scattering operator

• Continuous slowing down operator

FundamentalsAttila-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 & ConsAdvantages

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

Project150 cGy to 95%, 12 MeV

Project200 cGy to 100%, 12 MeV