Electron Beams: Dose calculation algorithms Kent A. Gifford, Ph.D. Department of Radiation Physics UT M.D. Anderson Cancer Center Medical Physics III: Spring 2015

Dose calculation algorithms Deterministic –Hogstrom pencil beam (Pinnacle 3 ) –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) Assumptions 1 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 1 EE Lewis and WF Miller, Computational Methods of Neutron Transport, ANS, 1993.

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

Fundamentals Linear Boltzmann Transport Equation (LBTE)-angular fluence ↑angular fluence rate coefficients normalized spherical harmonics ↑ ↑angular fluence rate

Fundamentals Linear Boltzmann Transport Equation (LBTE)-scattering xsection differential scattering moments ↑ orthogonal Legendre polynomial↑ ↑differential scattering cross-section

Fundamentals Linear Boltzmann Transport Equation (LBTE)-scattering source ↑ scattering source differential scattering xsection ↑ angular fluence rate ↑

Fundamentals Linear Boltzmann Transport Equation (LBTE)-Reaction ↑ reaction rate ↑macroscopic cross-section of type whatever scalar fluence rate↑

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

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

Fundamentals Attila  -fsds FSDS technique Separate angular fluence/rate into collided and uncollided components Ray trace from point source to quadrature or edit points 1 ST 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  N elements  N ordinates  N groups unknowns Iteration started with guess for fluence Process may proceed slowly for problems dominated by scattering Acceleration technique applied- DSA

Fundamentals Attila  -Charged particles LBTE Continuous scattering operator 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, x i Calculate the score or tally f i 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