Measurement of linear accelerator spectra, reconstructed from percentage depth dose curves by neural networks

Published:March 01, 2022DOI:


      • Measurement of linear accelerator spectra from experimental PDDs processed by neural networks.
      • Neural networks are applied to solve a Fredholm integral equation.
      • Neural networks are used to solve an ill conditioned system of equations.
      • Resolution of the Fredholm integral equation applied to noisy experimental data.
      • Linear accelerator spectra obtained without requiring knowledge of the accelerator design.


      Purpose Linear accelerator (linac) spectra, used to improve the accuracy of dose calculation and to produce a complete description of beam quality, among other aspects, are relevant in radiotherapy and linear accelerator physics.
      Methods In this work we apply neural networks in solving an ill-conditioned system of linear equations, to indirectly measure the linear accelerator spectra via the percentage depth dose curves. The photon beam spectra are related to radiation doses through a Fredholm integral equation. To address our problem we measured the percentage depth dose curve in water and solved a discretized Fredholm equation using artificial neural network. After analysing the typology of our physical problem we selected a MultiLayer Perceptron Neural Network and designed the most suitable neural network architecture.
      Results The reconstructed spectra were compared with spectra from three linacs, two of them obtained by us through simulations and the third produced by another author, achieving a concordance between 92 % and 96 %.
      Conclusions Therefore, the spectrum of any accelerator can be quickly and easily reconstructed from measured percent depth dose curves, applying a trained artificial neural network.


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        • Tsiamas P.
        • Sajo E.
        • Cifter F.
        • Theodorou K.
        • Kappas C.
        • Makrigiorgos M.
        • Marcus K.
        • Zygmanski P.
        Beam quality and dose perturbation of 6 mv flattening-filter-free linac.
        Physica Med. 2014; 30 (URL: 47-56
        • López-Sánchez M.
        • Pérez-Fernández M.
        • Fandiño J.M.
        • Teijeiro A.
        • Luna-Vega V.
        • Gómez-Fernández N.
        • Gómez F.
        • González-Castaño D.M.
        An egs monte carlo model for varian truebeam treatment units: Commissioning and experimental validation of source parameters.
        Physica Med. 2019; 64 (URL: 81-88
        • Martins J.C.
        • Saxena R.
        • Neppl S.
        • Alhazmi A.
        • Reiner M.
        • Veloza S.
        • Belka C.
        • Parodi K.
        Optimization of phase space files from clinical linear accelerators.
        Physica Med. 2019; 64 (URL: 54-68
        • Choi H.J.
        • Park H.
        • Yi C.Y.
        • Kim B.-C.
        • Shin W.-G.
        • Min C.H.
        Determining the energy spectrum of clinical linear accelerator using an optimized photon beam transmission protocol.
        Med Phys. 2019; 46 (URL: 3285-3297
        • Taneja S.
        • Bartol L.J.
        • Culberson W.
        • De Werd L.A.
        • et al.
        Measurement of the energy spectrum of a 6 mv linear accelerator using compton scattering spectroscopy and monte carlo-generated corrections, International Journal of Medical Physics.
        Clin Eng Radiat Oncol. 2020; 9: 186
        • Piermattei A.
        • Arcovito G.
        • Azario L.
        • Bacci C.
        • Bianciardi L.
        • De Sapio E.
        • Giacco C.
        A study of quality of bremsstrahlung spectra reconstructed from transmission measurements.
        Med Phys. 1990; 17 (URL: 227-233
        • Jalbout W.T.
        • Spyrou N.M.
        Spectral reconstruction by scatter analysis for a linear accelerator photon beam.
        Phys Med Biol. 2006; 51: 2211-2224
        • Landry D.J.
        • Anderson D.W.
        Measurement of accelerator bremsstrahlung spectra with a high-efficiency ge detector.
        Med Phys. 1991; 18 (URL: 527-532
        • González W.
        • Lallena A.M.
        • Alfonso R.
        Monte carlo simulation of the dynamic micro-multileaf collimator of a LINAC elekta precise using PENELOPE.
        Phys Med Biol. 2011; 56: 3417-3431
        • Verhaegen F.
        • Seuntjens J.
        Monte carlo modelling of external radiotherapy photon beams.
        Phys Med Biol. 2003; 48: R107-R164
      1. B. Juste, R. Miró, A. Jambrina, J.M. Campayo, S. Díez, G. Verd?, A new methodology to determinate linac photon spectra using the epid signal, Radiation Physics and Chemistry 95 (2014) 412–416, proceedings of the 12th International Symposium on Radiation Physics (ISRP 2012). doi: 10.1016/j.radphyschem.2013.03.005.

        • Deng J.
        • Jiang S.B.
        • 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. 2001; 46: 1429-1449
        • Andreo P.
        Monte carlo techniques in medical radiation physics.
        Phys Med Biol. 1991; 36: 861-920
      2. L. Brualla Barberà, M. Rodriguez, J. Sempau, P. Andreo, Penelope/primo-calculated photon and electron spectra from clinical accelerators doi: 10.1186/s13014-018-1186-8.

        • Ming X.
        • Feng Y.
        • Liu R.
        • Yang C.
        • Zhou L.
        • Zhai H.
        • Deng J.
        A measurement-based generalized source model for monte carlo dose simulations of CT scans.
        Phys Med Biol. 2017; 62: 1759-1776
        • Shimozato T.
        • Tabushi K.
        • Kitoh S.
        • Shiota Y.
        • Hirayama C.
        • Suzuki S.
        Calculation of 10 MV x-ray spectra emitted by a medical linear accelerator using the BFGS quasi-newton method.
        Phys Med Biol. 2006; 52: 515-523
        • Huang P.-H.
        • Kase K.R.
        • Bj?rngard B.E.
        Spectral characterization of 4 mv bremsstrahlung by attenuation analysis.
        Medical Phys. 1981; 8: 368-374
        • Archer B.R.
        • Almond P.R.
        • Wagner L.K.
        Application of a laplace transform pair model for high-energy x-ray spectral reconstruction.
        Med Phys. 1985; 12: 630-633
        • Francois P.
        • Catala A.
        • Scouarnec C.
        Simulation of x-ray spectral reconstruction from transmission data by direct resolution of the numeric system af=t.
        Med Phys. 1993; 20: 1695-1703
        • Wing G.M.
        A primer on integral equations of the first kind: the problem of deconvolution and unfolding.
        SIAM, 1991
        • Chen Z.
        • Micchelli C.A.
        • Xu Y.
        Multiscale methods for Fredholm integral equations. vol. 28. Cambridge University Press, 2015
        • Yuan D.
        • Zhang X.
        An overview of numerical methods for the first kind fredholm integral equation.
        SN Appl Sci. 2019; 1: 1-12
        • Rucci A.
        • Carletti C.
        • Cravero W.
        • Strbac B.
        Use of iaea’s phase-space files for the implementation of a clinical accelerator virtual source model.
        Physica Med. 2014; 30 (URL: 242-248
        • Sheikh-Bagheri D.
        • Rogers D.W.O.
        Monte carlo calculation of nine megavoltage photon beam spectra using the beam code.
        Med Phys. 2002; 29 (URL: 391-402
        • Kandlakunta P.
        • Momin S.
        • Sloop A.
        • Zhang T.
        • Khan R.
        Characterizing a geant4 monte carlo model of a multileaf collimator for a truebeam? linear accelerator.
        Physica Med. 2019; 59 (URL: 1-12
        • Salvat F.
        • Fernández-Varea J.M.
        • Sempau J.
        Penelope-2006: A code system for monte carlo simulation of electron and photon transport. 4. 2006: 7
        • Patlan-Cardoso F.
        • Rodríguez-Romo S.
        • Ibáñez-Orozco O.
        • Rodríguez-Vázquez K.
        • Vergara-Martínez F.J.
        Estimation of the central-axis-reference percent depth dose in a water phantom using artificial intelligence.
        J Radiat Res Appl Sci. 2021; 14 (URL: 91-104
        • Vega-Carrillo H.R.
        • Hernández-Dávila V.M.
        • Manzanares-Acuña E.
        • Gallego E.
        • Lorente A.
        • Iñiguez M.P.
        Artificial neural networks technology for neutron spectrometry and dosimetry.
        Radiat Prot Dosimetry. 2007; 126: 408-412
        • Mohammadi N.
        • Miri-Hakimabad H.
        • Rafat-Motavlli L.
        • Akbari F.
        • Abdollahi S.
        Neutron spectrometry and determination of neutron contamination around the 15 mv siemens primus linac.
        J Radioanal Nucl Chem. 2015; 304: 1001-1008
        • Panahi R.
        • Feghhi S.
        • Moghadam S.R.
        • Zamzamian S.
        Simultaneous alpha and gamma discrimination with a phoswich detector using a rise time method and an artificial neural network method.
        Appl Radiat Isot. 2019; 154 (URL:
        • Javaid U.
        • Souris K.
        • Huang S.
        • Lee J.A.
        Denoising proton therapy monte carlo dose distributions in multiple tumor sites: A comparative neural networks architecture study.
        Physica Med. 2021; 89 (URL: 93-103
        • Yang H.J.
        • Kim T.H.
        • Schaarschmidt T.
        • Park D.-W.
        • Kang S.H.
        • Chung H.-T.
        • Suh T.S.
        A multivariate approach to determine electron beam parameters for a monte carlo 6 mv linac model: Statistical and machine learning methods.
        Physica Med. 2022; 93 (URL: 38-45
      3. G. Bologna, C. Pellegrini, Three medical examples in neural network rule extraction, Phys Med 13 (1997) 183–187, iD: unige:121360.

        • Hussain S.
        Artificial neural network model for spectral construction of a linear accelerator megavoltage photon beam.
        in: 2010 International Conference on Intelligent Systems, Modelling and Simulation. 2010: 86-91
      4. G. van Rossum, Python tutorial, Tech. Rep. CS-R9526, Centrum voor Wiskunde en Informatica (CWI), Amsterdam (May 1995).

      5. F. Chollet, et al., Keras, GitHub,

      6. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G.S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mane, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viegas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, X. Zheng, Tensorflow: Large-scale machine learning on heterogeneous distributed systems (2016). arXiv:1603.04467.

      7. T. Schneider, H.-M. Kramer, A new method for an improved determination of continuous photon fluence spectra for X-ray tube voltages up to 150 kV, Radiat Protect Dosimetry 121 (4) (2006) 370–375. arXiv:, doi: 10.1093/rpd/ncl058.

      8. J. Hubbell, S. Seltzer, X-ray mass attenuation coefficients: Nist standard reference database 126, National Institute of Standards and Technology, Gaithersburg, Maryland, USA.

      9. W. IAEA, et al., Technical reports series no. 398, Absorbed dose determination in external beam radiotherapy. An international code of practice for dosimetry based on standards of absorbrd dose to water. Vienna: IAEA.

        • Kullback S.
        • Leibler R.A.
        On information and sufficiency.
        Ann Math Stat. 1951; 22 (URL: 79-86