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)