Measurement Science and Standards, National Research Council Canada
Transport under magnetic fields with the EGSnrc simulation toolkit Ernesto Mainegra-Hing, Frédéric Tessier, Blake Walters
Hugo Bouchard
Université de Montréal and
National Physics Laboratory (UK)
+
MRI
External RT
=
MRI-guided Radiation Therapy
Magnetic field effect on dose distribution
electron tracks
air
water
pencil beam of 10 MeV electrons
electron tracks
air
water
pencil beam of 10 MeV electrons
B = 1 T
pencil beam of 10 MeV electrons
water
air
air
water electron
return effect
B = 1 T
B PTW30013
60Co
air
PTW chamber irradiated by parallel 60Co beam
significant dependence on magnetic field
New dosimetry? cavity
Orange Bible
Equation of motion
𝑣 = 𝑣 0 +1
𝑚0𝛾 𝐸 𝑑𝑡′ 𝑭𝒆𝒍 𝐸 𝑡′ + 𝑭𝒊𝒏 𝐸 𝑡′ + 𝑭𝒆𝒎 𝑥 𝑡′ , 𝐸 𝑡′ , 𝑢 𝑡′
𝑡
0
The equation of motion in the force formulation for transport in a medium under the effect of an EM field can be written as
stochastic deterministic
Bielajew’s implementation
Under the assumption of very small steps such that: • Field does not changes significantly • Energy loss negligible • Negligible angular deflection the equation of motion becomes to first order:
𝑣 = 𝑣 0 +𝑡
𝑚0𝛾 𝐸0
𝑭𝒆𝒍 𝐸0 + 𝑭𝒊𝒏 𝐸0 + 𝑭𝒆𝒎 𝑥 0, 𝐸0, 𝑢 0
Bielajew’s implementation
Under the assumption of very small steps such that: • Field does not changes significantly • Energy loss negligible • Negligible angular deflection the equation of motion becomes to first order:
𝑣 = 𝑣 0 + ∆𝑣 𝑀𝐶 +𝑡
𝑚0𝛾 𝐸0
𝑭𝒆𝒎 𝑥 0, 𝐸0, 𝑢 0
Interactions with medium and external field treated independently!
Bielajew’s implementation
Neglecting lateral deflection Ds/2 one gets for the position change
Expressing the time t as a function of the total path length Ds to first order gives
∆𝑥 = 𝑢 0∆𝑠 +∆𝑠
2∆𝑢
∆𝑥 = 𝑢 0∆𝑠
∆𝑢 = ∆𝑢 𝑀𝐶 + ∆𝑢 𝑒𝑚
the change in the particle’s direction is MC step
vacuum
𝑟 = 𝑚𝑐
𝑒𝐵𝛾2 − 1
vacuum
vacuum 3 kinds of errors
vacuum 1. error in position
vacuum 2. error in radius
vacuum
3. error in energy deposition (in medium)
vacuum
1
vacuum
0.84
vacuum
0.58
vacuum
0.01
forever
Fano theorem provides a rigorous test
𝐷 = 𝑁0/𝑚𝑇 ∙ 𝐸
• Uniform electron source per unit mass N0/mT
• Medium of uniform composition but varying density
where 𝐸 is the average energy emitted
Fano theorem provides a rigorous test
If the source emits electrons of energy E0:
𝐷/𝑁0 = 𝐸0/𝑚𝑇
For a MC simulation fulfilling Fano conditions, the dose per particle in any region i is expected to be:
𝐷𝑖/𝑁0 = 𝐸0/𝑚𝑇
Use this to verify the accuracy of the electron transport algorithm!
“Fano’s theorem does not hold in the presence of static and constant external EM fields. This has the unfortunate consequence of invalidating the Fano cavity test …”
1. Isotropic uniform source per unit mass
2. Magnetic field B scales with mass density
0.001 0.01 0.1 1
identical atomic properties (air)
2 cm
in water phantom
0.001 0.01 0.1 1
uniform source of electrons, per unit mass
electron tracks
in water phantom
100 keV
0.001 0.01 0.1 1 1.01
1.00
0.99
Mo
nte
Car
lo /
Th
eory
this is what we mean when we say that EGSnrc is accurate at the 0.1% level
in water phantom
0 T
water
12 regions
Fano test 1 (PTW30013)
d d
d
d
d > RCSDA(Emax)
Same material, different densities
?
water
12 regions
Fano test 1 for a PTW30013
Same material, different densities
Powerful diagnostic
tool !!!
PTW30013 cavity dose
1 MeV
PTW30013 cavity dose
ξ = 1
𝑠2 × 𝑇𝐶𝑃𝑈
Measuring efficiency
How long needed to achieve desired uncertainty?
cavity dose
cavity dose
Transport in electromagnetic field is available in EGSnrc as a first-order correction on the velocity.
Ionization chamber dose response calculations pass Fano test in a magnetic field only with significant step size restrictions.
Larger step sizes are possible as energy increases or field strength decreases (curvature radius increases)
Considering the penalty in efficiency, a more accurate algorithm allowing larger step sizes is desirable.
Fano test: powerful tool for benchmarking radiation transport algorithms and testing the correctness of MC simulation parameters.
Conclusionss
Measurement Science and Standards, National Research Council Canada
Transport under magnetic fields with the EGSnrc simulation toolkit Ernesto Mainegra-Hing, Frédéric Tessier, Blake Walters
Hugo Bouchard
Université de Montréal and
National Physics Laboratory (UK)