Tikhonov regularization modified as improved method ...

Tikhonov regularization modified as improved method for determining electron energy spectra from percentage depth dose curves data of broad beams Jorge Homero Wilches Visbal and Alessandro Martins Da Costa Citation: AIP Conference Proceedings 1747, 060006 (2016); doi: 10.1063/1.4954116 View online: http://dx.doi.org/10.1063/1.4954116 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1747?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis of the Tikhonov regularization to retrieve thermal conductivity depth-profiles from infrared thermography data J. Appl. Phys. 108, 064905 (2010); 10.1063/1.3475498 Study on the Effect of Energy Parameter of Electron on the Percentage Depth Dose of Electron Beam Using Monte Carlo Method AIP Conf. Proc. 1244, 168 (2010); 10.1063/1.3462756 SU‐FF‐T‐273: Improved Calculation of Energy Spectra From Electron Depth Dose Curves Med. Phys. 33, 2110 (2006); 10.1118/1.2241193 Reconstruction of electron spectra from depth doses with adaptive regularization Med. Phys. 33, 354 (2006); 10.1118/1.2161404 Determination of percentage depth-dose curves for electron beams using different types of detectors Med. Phys. 28, 298 (2001); 10.1118/1.1350436

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Tikhonov Regularization Modified as Improved Method for Determining Electron Energy Spectra from Percentage Depth Dose Curves Data of Broad Beams Jorge Homero Wilches Visbal1, a) and Alessandro Martins Da Costa2, b) 1

Departamento de Física, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo. Av.Bandeirantes 3900,Bairro Monte Alegre. Ribeirão Preto (SP), Brasil. 2 Departamento de Física, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo. Av.Bandeirantes 3900,Bairro Monte Alegre. Ribeirão Preto (SP), Brasil. a)

[email protected] b) [email protected]

Abstract. Experiments have shown an obvious difference between the percentage dose distribution of clinical electron beams and percentage dose distribution (PDD) of monoenergetic electron beams of accelerator nominal energy in water phantom surface, because along the way, from electron’s source to phantom, electron’s beam collides with headshield accelerator and air. It is evident, then, that within PDD curves depends and contain information of spectrum. Despite modelling the source by Monte Carlo method is considered the “standard gold” and that magnetic spectrometer is a method of direct measurement; inverse reconstruction is becoming the better approach to obtain electron energy spectra since it is relatively simple, quick and inexpensive when compared with the other two. In the present work, a new method of inverse reconstruction based on Tikhonov regularization is presented. The objective of this study was establish a method that be able to obtain electron energy spectrum with greater accuracy than “Classical” Tikhonov.

INTRODUCTION Percentage depth dose of electron beams represents an important item of data in radiation therapy treatment since it describes the dosimetric properties of these. Using an accurate transport theory, or the Monte Carlo method, has been shown obvious differences between the dose distribution of electron beams of a clinical accelerator in a water simulator object and the dose distribution of monoenergetic electrons of nominal energy of the clinical accelerator in water, because the electron beams collides with of internal structures of headshield of accelerator and applicators generating secondary electron and high energy photons. Secondary electrons deposit their energy near surface while high-energy photons deposit their energy in depth. Consequently, the dose distribution actually deposited on the central-axis area differs remarkably from that of monoenergetic electrons with nominal energy1. Electron energy spectra are important parameters for both electron accelerators and accurate radiotherapy. Especially, they are necessary for accurate dose calculation2. Exist three principal approaches to obtain electron energy spectra: Monte Carlo Method to construct a complex model3 to fit a special accelerator modelling source, where MC is not flexible although the most accurate method4; Direct Measurement by a magnetic spectrometer but is expensive, delayed and inappropriate for the clinical ambient and Inverse Reconstruction, a better approach to obtain the electron energy spectra because only needs develop an appropriate mathematical model, not expensive and fast 5. Algorithms to assess the dose contribution of treatment head bremsstrahlung 5 and initial angular spread6, to electron beam PDD, in practice, have been investigated. We reconstructed the electron energy spectra neglecting both the photon contamination contribution as the influence of the initial angular spread of incident electrons. In this paper, we present an effective method to determine the electron spectra by constructing modified Tikhonov regularization model and its corresponding comparison with classical model. The testing result was new method improve the adjust practically along spectra.

Medical Physics AIP Conf. Proc. 1747, 060006-1–060006-5; doi: 10.1063/1.4954116 Published by AIP Publishing. 978-0-7354-1404-4/$30.00

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MATERIALS AND METHODS It is well established that via a Fredholm integral equation of first kind is possible link the energy spectrum to measured PDD7, through the PDDs monoenergetic electrons in the medium as a kernel function in the integral equation. ‫ܦ‬ሺ‫ݖ‬ሻ ൌ  න

ா೘ೌೣ

‫ܦ‬ሺ‫ܧ‬ǡ ‫ݖ‬ሻ߮ሺ‫ܧ‬ሻ݀‫ ܧ‬ሺͳሻ

଴

‫ܦ‬ሺ‫ݖ‬ሻ is the measured central-axis PDD in a water phantom, ߮ሺ‫ܧ‬ሻ is the energy spectrum of incident electrons and ‫ܦ‬ሺ‫ܧ‬ǡ ‫ݖ‬ሻ is the PDD from monoenergetic beam with energy E. Discretizing the previous integral equation, it is, ௡

‫ܦ‬ሺ‫ݖ‬ሻ ൌ ෍ ‫ܦ‬൫‫ܧ‬௜ ǡ ‫ݖ‬௝ ൯߮ሺ‫ܧ‬௜ ሻο‫ ܧ‬ൌ ‫ ߮ܭ‬ሺʹሻ ௜

‫ܦ‬ሺ‫ݖ‬ሻ ൌ ‫߮ܭ‬ሺ͵ሻ Where, ‫ ܭ‬ൌ ሾ‫ܦ‬൫‫ܧ‬௜ ǡ ‫ݖ‬௝ ൯ο‫ܧ‬ሿ is usually called the kernel function; ݊ is the maximum number of monoenergetic beams used. ‫ܦ‬ሺ‫ݖ‬ሻ is a m-dimensional vector, ߮ሺ‫ܧ‬ሻ is a n-dimensional vector and ‫ܭ‬is a mxn matrix. The linear system equation (3) is generally ill-posed8. Common methods to solve (3), as inverse matrix method (m =n) or generalized matrix method (m ≠ n), can’t be used in general. It is well known that Tikhonov regularization method can solve this type of problems. Its basic purpose is put additional information about the desired solution in order to stabilize the problem and to single out a useful and stable solution1,9,10. Well, in order to achieve an effective solution this problem, we purposed minimize two handles function via MatLab lsqnonlin function. One of the two was exactly the Tikhonov function, and the in other we were only interchange the classical residual norm by a similar function to the rms error, that is ଶ

ଶ

‫݊݅ܯ‬ሺ݄ܶ݅݇௖௟௔௦௦௜௖௔௟ ሻ ൌ ‫ ݊݅݉݃ݎܣ‬ቄหȁ‫ ߮ܭ‬െ ‫ܦ‬ሺ‫ݖ‬ሻȁห ൅  ߣଶ หȁ‫ܮ‬ሺ߮ െ ߮଴ ሻȁห ቅሺͶሻ ‫݊݅ܯ‬൫݄ܶ݅݇௠௢ௗ௜௙௜௘ௗ ൯ ൌ ‫ ݊݅݉݃ݎܣ‬ቐඨ

σ௡௜ሾሺ‫߮ܭ‬ሻ௜ െ ‫ܦ‬ሺ‫ݖ‬ሻ௜ ሿଶ ଶ ൅  ߣଶ หȁ‫ܮ‬ሺ߮ െ ߮଴ ሻȁห ቑሺͷሻ ݊

Where ߣ is the regularization factor that is appropriately choice by user and ‫ ܮ‬is the regularization matrix is computed by get_l function that it is into Regularization Tool Package developed by Hansen. As it has already said, ‫ܦ‬ሺ‫ݖ‬ሻ was obtained via PENELOPE-MC simulation for a size field 10x10 cm2, SSD = 100 cm and a volume water phantom of 30x30x30 cm3 from a Varian Clinac 2100C spectra of 6MeV3 being discretized with Engauge 4.1. The kernel function ‫ ܭ‬is a matrix build from monoenergetic PDDs which were generated via PENELOPE, setting the source as monoenergetic, and with steps between them of 0.25 MeV i.e. K = [PDD0.25MeV, PDD0.5MeV, … PDDEmax] where Emax represents the monoenergetic cutoff energy. It can set initial guess ߮଴ to be 1.0 for the energy bin that convers the obtained most probable energy, 0.0 for the energy bin corresponding to cutoff energy, 0.75 for the adjacent energy bin and so forth until 0.25; from here the relative weighting factors were obtained halving the prior-bin until a value just greater than 0.01 and 0.01 for the remaining energy bins. The handles functions (4) and (5) were minimized by means nonlinear least-squared (lsqnonlin) function available at optimization toolbox-MatLab. We only make two algorithms (MatLab functions), one for each, adapted to this context. Finally, the PC system utilized, both for the PDDs simulations as spectra an errors calculation, was Win7 Ultimate, SP1 CPU: 2.30 GHz, RAM: 4Gb. MatLab R2015a (Version 8.5) was employed for calculations.

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RESULTS AND DISCUSSION Figure 1 shows that all Modified Tikhonov’s absolute percentage errors are lower than Classical Tikhonov. Both methods reconstructed the spectrum with absolute percentage errors (APEs) lower than 10pp (percentage points). Modified Tikhonov showed to be able to reconstruct the spectrum with an APE lower than 5pp at all points whereas Classical Tikhonov showed APEs greater than 5pp in peak region of spectrum excepting cut-off energy.

FIGURE 1. (a) Results of reconstructed spectrum of 6 MeV, via Classical Tikhonov, when compared with original. (b) Results of reconstructed spectrum of 6 MeV, via Modified Tikhonov, when compared with original. MAPE: Median Absolute Percentage Error. RMS: Relative Median Squared Error11. RPE: Relative Percentage Error3. Swmax: Peak Spectral Weighting. ESW: Peak Energy. The peak energy of original spectrum is 6.90MeV3. On the other hand, the relative percentage error between most probable energy of two methods and original was lower than 1%. This shows that both methods are successful in obtaining the energy value associated with spectral peak. Figure 2 shows that the absolute error in all PDD regions was lower than 2.5pp and it falls as the depth increases. The RMSE, for Classical and Modified method, was 0.36 and 0.32 and MAPE was 0.14 and 0.12%, respectively. Correlation coefficient was 1.000.

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FIGURE 2. Original and reconstructed PDDs comparison. It was calculated absolute error at each point in depth being the step equal 0.2 cm. Absolute error was shown staking both contributions. The absolute error is displayed as a “stacked” bar plot, where the height (error) of each bar is the sum of the elements (sum of errors) in the row.

The most remarkable difference is at points of clinical relevance like R 50, R80 or R90. As seen in the Figure 2, R50 differences between original and modified Tikhonov was 0.000 mm while for classical was 0.072 mm; for R80 was 0.21 mm, classical method, 0.06 mm, modified method and for R90 was 0.18 mm, classical method, 0.07 mm, modified method. In general, both methods reached a good fit on the three PDD regions: build-up region, fall-off region and tail region. In build-up region, “Classical Tikhonov” achieved the best fit. In fall-off region “Modified Tikhonov” achieved the best fit. It is worth mentioning that “Modified Tikhonov” fit in fall-off region was more accurate than “Classical Tikhonov” fit in build-up region. Deng et al accepted as commissioning criterion for their equipment Varian Clinical 2100 C a R 50 difference less than 1 mm12. Their creep reconstruction algorithms achieved R50 value within 0.1 mm from their target PDD R50. Carletti et al reported a R50 value within 0.01 mm. Finally, our Modified Tikhonov Method achieved R50 value lower than 0.01 mm.

CONCLUSIONS The two methods presented by us achieved satisfactory results in order into account a reconstruction acceptable when we look for R50. Differences between original PDD R 50 and reconstructed PDDs R50 were lower than 0.1 mm, the best results got for Deng et al (2001). The Modified Tikhonov Method yields even better results, whenever differences between original PDD R50 and reconstructed PDDs R50 were less than 0.01 mm. Moreover, at all points the absolute percentage error was lower than 5pp. At last, we believe that considering photons contamination and angular dispersion this results will improve at the objective to achieve a reconstruction with the highest possible accuracy.

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ACKNOWLEDGMENTS Authors thanks Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

L. Zhengming and D. Jette, Phys. Med. Biol 44, N177-N182 (1999). L. Gui, L. Lui, W. Ai-Dong, S. Gang and W. Yi-Canl, Chin.Phys.Lett 25, 2710-2713 (2008). C. Carletti, P. Meoli and W.R. Cravero, Phys.Med.Biol. 51, 3941-3952 (2006). D.W.O Rogers, B.A.Faddegon, G.X.Ding, C-M Ma, J. We and T.R.Mackie, Med.Phys 22, 503-524 (1994). L. Gui, W. Aidong, L. Hui and W.Yi-Can, IFMBE Proccedings 19, 451-454 (2008). B. Faddegon and J. Blevis, Phys.Med.Biol. 46, 514-526 (2000). A. Chvetsov and G. Sandison, Phys.Med.Phys. 29, 578-591 (2002). L. Zhengming, Nucl. Instrum.Methods A. 255, 152-155 (1987). P.C Hansen, Numerical Algorithms 6, 1-35 (1994). P.C Hansen, Inverse Problems 8, 849-872(1992). Chai T and Draxler R, Geosci.Model Dev 7, 1247-1250 (2014). J. Deng, S.B Jiang, T. Pawlicki, J. Li and C.M Ma, Phys.Med.Biol 46, 1429-1449 (2001).

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