Derivation of initial electron beam energy spectrum
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Derivation of initial electron beam energy spectrum. Janusz Harasimowicz Establishment for Nuclear Equipment. http:// Electron beams. Range of electrons depends on the energy.

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Derivation of initial electron beam energy spectrum

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Derivation of initial electron beam energy spectrum

Janusz Harasimowicz

Establishment for Nuclear Equipment

Electron beams

  • Range of electrons depends on the energy.

  • Percentage depth dose curve shapedepends oncontribution ofparticular energies of particles directed onto the water phantom surface.

Electron beam energy

  • Electron beam mean energy at phantom surface can be derived, ex. IAEA TRS 381:

    E0 [MeV] = 0.818 + 1.935 RJ50 + 0.040 (RJ50)2 [cm]

  • However, direct measurement of full energy distribution of the beam is difficult and accelerator time consuming.Cannot be performed in oncologydepartment.

  • Is it possible to get information about initial electron beam energy spectrum from the water phantom measurements?

Energy spectrum derivation

  • Proposed method:

    • PDD measurement.

    • Simulation of dose distribution for mono-energetic beams (Monte Carlo method).

    • Derivation of each simulated beam contribution to measured PDD curve shape (error backpropagation algorithm).

  • Calculated weights of particular simulated PDD curves should give information about real energy spectrum distribution.


  • Measurements were performed forColine 15 accelerator (Establishmentfor Nuclear Equipment ZdAJ, Poland)for ”12 MeV” (nominal) electron beam.

  • To minimize energy losses, scattering foils and applicator were removed and jaws were set for 40 cm x 40 cm field.

  • Water phantom RFA-300 (Scanditronix) and plane-parallel ionization chamber NACP-02 (Scanditronix) were used.

Monte Carlo

  • BEAMnrc Monte Carlo code was used:

    • Modified Coline 15 treatment head model.

    • Radiation source: parallel circular electron beam with 2D Gaussian XY distribution (2 mm FWHM) directed onto the exit vacuum window.

    • Water phantom at SSD=100 cm.

  • Depth dose curves calculated for monoenergetic beams in energy range from 1 MeV to 15 MeV with 250 keV interval.

  • Weighted sum of simulated PDDs was fitted to measured PDD curve (inverse Monte Carlo).

Rogers DWO, Faddegon BA, Ding GX, Ma C-M, Wei J, Mackie TR. BEAM: A Monte Carlo code to simulate radiotherapy treatment units. Medical Physics 1995 22:503-524

Error backpropagation algorithm

Used error backpropagation algorithm is based on minimization of difference between measured and simulated values as stated by equation:

Q = 0.5 Σi[mi – Σj(wjcij)]2

mi→ depth dose measured at the i-th point

cij→ depth dose calculated at the i-th point for the j-th energy bin

wj→ weight of the j-th energy bin

Error backpropagation algorithm

Q = 0.5 Σi[mi – Σj(wjcij)]2

δQ/δwj = Σi[–cij (mi – Σj(wjcij))]

Δwj = –η δQ/δwj

Δwj = η Σi[cij (mi – Σj(wjcij))]

Without the „momentum term”

With the „momentum term”

Error backpropagation algorithm

To speed up the fitting procedure,a „momentum term” was added:

Δwjk+1 = –η δQ/δwjk+ α Δwjk

Fitting procedure


Mean square error Q calc.

Weights change


Q < tolerance ?

Derivative δQ/δw calc.




Depth dose [%]

Depth [mm]


~1 MeV

Relative contribution to PDD

Relative difference [%]

Energy [MeV]

Depth [mm]


  • NOTICE: Difference in PDDs <1%is not equal to method uncertainty!

  • Derivation of fitted energy uncertainty for any complicated case is rather difficult.

  • However, one can try to estimate uncertainty by analysing matched simulated beams to the known results(ex. for monoenergetic spectrum).

Uncertainty analysis

Fitted spectra

Known spectrum

Uncertainty analysis

Relative contribution to PDD

Energy [MeV]

Other results

Source: Deng J, Jiang SB, Pawlicki T, Li J, Ma C-M. Derivation of electron and photon energy spectra from electron beam central axis depth dose curves. Phys. Med. Biol. 46 (2001)

Random creep algorithm used but more advanced method adapted:

  • Four-source model for beam phase-space reconstruction.

  • Separation of photons and electrons contribution to dose distribution.

    Would be interesting to check the method with error backpropagation algorithm.


  • It seems to be possible to derive energy from depth dose measurements in water phantom.But with what accuracy?

  • Derived energy spectrum of Coline 15 is rather wide. Possible explanations:

    • Large energy intervals (250 keV).

    • Energy slit filter (located in the deflection system) has not been optimized yet (and perhaps too wide energy spectrum is getting to the exit vacuum window).

    • Wrong source definition.

    • Set up and measurement errors.

  • Further studies are needed.


Scheme of Coline 15 energy slit filter (located in the deflection system)

Appendix: Build-up region

Difference in the build-up region:2-4%

Measured dose higher than it arises form MC simulations!

Appendix: Build-up region

Bruce Faddegon (UCSF Comprehensive Cancer Center, San Francisco) found similar differences of 2-4% in parallel-plate and simulated surface dose for6-21 MeV electron beams delivered witha Siemens Primus accelerator with the jaws wide open and no applicator.Agreement was much better with diode measurements.

Appendix: Build-up region

Source:Faddegon B, Schreiber E, Ding X. Monte Carlo simulation of large electron fields. Phys. Med. Biol. 50 (2005)

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