Reconstruction of electron spectra in radiotherapy using inverse Monte Carlo methods

R. P. Hugtenburg, Ph.D.,
Department of Medical Physics,
University Hospital Birmingham,
Birmingham B15 2TH,
UNITED KINGDOM.
richard.hugtenburg@university-b.wmids.nhs.uk

Introduction

This note descibes the determination of incident beam information for use in Monte Carlo based radiotherapy treatment planning with electrons. Although the electrons eminating from the waveguide of a therapeutic linear accelerator are fairly monoenergetic by the time the electrons have reached the level of the patient they have undergone many interactions with the intervening air volume and with the various collimating components. Several workers have examined electron spectra by direct means such as with magnetic spectrometer[1,2,3,4]. However this equipment is not generally available to oncology departments and alternative methods are required.

**Figure:** Depth-dose curves for 20 MeV electron monoenergetic source (histogram) calculated using EGS4, the 20 MeV modality on the Varian 2100C linear accelerator with a 10 cm $\times$ 10 cm applicator cone (upper continuous line) and without the applicator cone (lower continuous line). The accelerator jaw setting is 14 cm by 14 cm. The measurements were performed in a water phantom using a Markus chamber.
$\includegraphics[width=10cm]{elec.eps}$

To demonstrate the need for accurate spectral information for Monte Carlo based dosimetry, figure 1 shows a depth-dose curve calculated using EGS4 for a 20 MeV monoenergetic electron beam. This is compared to depth-dose data for the 20 MeV modality on a Varian 2100C linear accelerator with a 10x10 applicator setting and depth-dose for the same collimator jaws setting (14x14) but minus applicators. The difference between the monoenergetic depth dose and the standard clinical setting is substantial, indicating the need to use carefully determined spectra in Monte Carlo based treatment calculations. The difference between the applicator and non-applicator depth-dose indicates the deposition of secondary scatter electrons at the central axis originating in the electron applicator cones. A significant difference between the Monte Carlo generated depth-dose and the applicatorless measurement suggests a broadening and softening of the incident electron beam in the collimators and other head componentry.

The inverse Monte Carlo method

The inverse Monte Carlo method was introduced by Dunn[5]. He was able to demonstrate some useful applications to the design of photon beam compensating filters[6]. The method has been used in SPECT (single photon emission CT) reconstructions[7,8,9,10]. Lind and Brahme suggest the method for inverse planning[11].

The method has similarities to a variance reduction method called importance sampling[12] in that the measurable and computable distributions, z_j and z_ij, are determined for particle characteristics, x_i, sampled from a prior probability density function (pdf), f_i^*. The actual pdf, f_i is determined from a set of weightings, f_i=W_i f_i^*, which are obtained from a numerical inversion of z_j=z_ijW_i. In the same manner that importance sampling is used to speed convergence, the prior pdf should be chosen to minimise the sampling of non-contributive, x_i's and be based on our knowledge of the physics which defines the measured distribution.

In the following example, the measureable distribution, is a central axis depth-dose, D₀(z). The actual pdf is the incident electron spectra denoted $\Phi_E$ =d $\Phi_E$ /dE (a differential fluence) and the particles are sampled from a uniform pdf over a range of energy from 0 up to a maximum energy, $E_{\mbox{max}}$ .

The central-axis depth dose is given by,

$\begin{displaymath}\mbox{D}_{0}(z) = \int_0^{E_{\mbox{max}}} \mbox{D}_{\overline{e}}(E,z)\Phi_{E} dE + \Psi\mbox{D}_{\Gamma}(z). \end{displaymath}$

(1)

where $\mbox{D}_{\overline{e}}(E,z)$ is the Monte Carlo determined kernel we would wish to invert. The contribution of the bremsstrahlung component in the incident spectrum, $\mbox{D}_{\Gamma(z)}$ , is considered as a whole and weighted by further parameter, $\Psi$ . Various authors have suggested how to determine the contribution of photons generated in the head[13,14,15,16]. Here the depth-dose distribution of the 18 MV modality on the accelerator has been used in accordance with findings by Rustgi and Rodgers that the dose maximum of the photon component suggests an energy several MeV less than the nominal energy of the electron beam.

The inverse Monte Carlo method provides us with means to establish a relationship between a known and measurable quantity such as an in phantom dose distribution and an unknown theoretical quantity namely the phase space distribution of the particles in the beam. With the case of electron therapy dosimetry we are interested in determining spectral information from phantom measurements. This technique has been used by several workers[17,18,19] but it is surprising, given the radiation properties of electrons, that the technique has not been more widely applied.

The depth-dose curves of a 20 MeV modality electron beam for a 10 cm $\times$ 10 cm applicator cone have been measured with a Markus chamber in the tissue equivalent material Solid Water[20]. Sets of monoenergetic 10 cm diameter electron beams were simulated using EGS4[21] with the PRESTA extension[22]. The data set is represented in figure 2. The electron range is finite and, to a reasonable approximation, range related parameters such as R_p and R₅₀ are a linear function of beam energy. Consequently the matrix approximates an upper-triangular or Row-Echelon form and comes into a class of problems that are easily solved using a back-substitution or stripping technique. The implication is that broad electron beams should well define the electron spectra that contribute to them.

**Figure 2:** A set of depth-dose curves for 1 through 20 MeV monoenergetic beams calculated using Monte Carlo. 10000 histories gave a peak variance of 0.5%. A line source impinges on a semi-infinite water phantom and the dose is accumulated in a 10 cm radius cylinder of thickness 1 cm.
$\includegraphics[width=10cm]{edds.eps}$

In this problem and in others like it, a solution is best not determined by matrix inversion. The difficulty arises because dose contributions are always additive whereas a matrix inversion will generate positive and negative contributions.

The inversion is carried out by determining a best fit of the monoenergetic contributions to the measured data. A combination simulated annealing and simplex minimisation technique[23,24] was used in this case to acquire a best-fit. A useful technique for problems such as this where there is a positivity constraint on both the matrix values and weightings is the maximum likelihood method[25,26].

For a given level of uncertainty in both the measurement and calculations there is a defined minimum to the goodness of fit and it is appropriate to consider a range of spectra fullfilling this criterion.

The spectrum determined using this technique is shown in figure 3. The broadening demonstrated in the main energy range of the beam and contributions at lower energies are features that cannot be ignored. The inversion was performed ten times with a small variation in the spectra resulting in each case. This range is represented by the error bars in the figure.

**Figure 3:** A spectrum determined for the 20 MeV electron modality on the Varian 2100C. An inverse-Monte Carlo method is used and described in the text. The energy ordinate can really only be regarded as an effective energy for obliquely incident contributions cannot be distinguished from low-energy contributions by this method.
$\includegraphics[width=10cm]{elspec.eps}$

This result indicates the presence of several distinct lower energy contributions in the electron fluence. As these contributions are well defined it indicates that the beam exterior to the defined beam is being incompletely attenuated in the cone. This argument has been backed up by explicit modelling by Kassaee et al[27] comparing the old style of applicator upon which our measurements were performed and recent modifications to the Varian electron applicators.

Conclusion

Inverse Monte Carlo simulation offers a method for acquiring incident beam characteristics such as beam spectra necessary for Monte Carlo based treatment planning. Spectra can be acquired from broad electron beams. The influence of applicator cones and cutouts have been examined using Monte Carlo[27,28] and scatter integration methods[29,30] but the acquisition of spectra is a first step in both cases. The technique is a practical alternative to either full-detail Monte Carlo simulations of the internal workings of an accelerator or magnetic spectrometry.

The method described in this note can be extended to a measurement based planning algorithm. The method provides a means to extrapolate from measured sets of data to in vivo dose distributions. Potential sources of variability in dose such as tissue inhomogeneity, irregular field shapes and surface angulation can be applied as perturbations to the measured data. If these perturbations are small with respect to the total dose then an algorithm should demonstrate high levels of efficiency. With a conventional, deterministic, algorithm there is no obvious way to implement this sort of short cut and vast integrations will have to be performed in circumstances that may even be identical to the original measured data. A Monte Carlo based perturbation method will fair better in many circumstances as the algorithm can be designed to terminate upon achieving an adequate variance. Finally by basing the Monte Carlo calculation on standard measurements, such as would be acquired at the time of commissioning, one is able to ensure that the calculation is absolute.

Bibliography

1: S. W. Johnsen, P. D. LaRiviere, and E. Tanabe, ``Electron depth-dose dependence on energy spectral quality,'' Phys. Med. Biol., vol. 28, no. 12, pp. 1401-1407, 1983.
2: J. O. Deasy, P. R. Almond, and M. T. McEllistrem, ``The spectral dependence of electron central-axis depth-dose curves,'' Med. Phys., vol. 21, no. 9, pp. 1369-1376, 1994.
3: J. O. Deasy, P. R. Almond, M. T. McEllistrem, and C. K. Ross, ``A simple magnetic spectrometer for radiotherapy electron beams,'' Med. Phys., vol. 21, no. 11, pp. 1703-1714, 1994.
4: J. O. Deasy, P. R. Almond, and M. T. McEllistrem, ``Measured electron energy and angular distributions from clinical accelerators,'' Med. Phys., vol. 23, no. 5, pp. 675-684, 1996.
5: W. L. Dunn, ``Inverse Monte Carlo analysis,'' J. Comp. Phys., vol. 41, pp. 154-166, 1981.
6: W. L. Dunn, V. C. Boffi, and F. O'Foghludha, ``Applications of the inverse Monte Carlo method in photon beam physics,'' Nucl. Instrum. Meth. A, vol. 255, pp. 147-151, 1987.
7: C. E. Floyd, ``Inverse Monte Carlo: A unified reconstruction algorithm for SPECT,'' IEEE Trans. Nucl. Sci, vol. NS-32, no. 1, pp. 779-785, 1985.
8: C. E. Floyd, F. J. Jaszczak, S. H. Mangloos, K. L. Greer, and R. E. Coleman, ``Improved SPECT reconstruction using inverse Monte Carlo,'' SPIE (Physics and Engineering of Computerized Multidimensional Imaging and Processing), vol. 671, pp. 206-213, 1986.
9: C. E. Floyd, F. J. Jaszczak, K. L. Greer, and R. E. Coleman, ``Inverse Monte Carlo as a unified reconstruction algorithm for ECT,'' J. Nucl. Med., vol. 27, pp. 1577-1585, 1986.
10: K. F. Koral, X. Wang, K. R. Zasadny, N. H. Clinthorne, W. L. Rogers, C. E. F. Jr, and R. J. Jaszczak, ``Testing of local gamma-ray scatter fractions determined by spectral fitting,'' Phys. Med. Biol., vol. 36, no. 2, pp. 177-190, 1991.
11: B. K. Lind and A. Brahme, ``Optimisation of radiation therapy dose distributions using scanned electron and photon beams and multileaf collimators,'' in The use of computers in radiation therapy, pp. 235-239, Amsterdam: Elsevier, 1987.
12: P. Andreo, ``Monte Carlo techniques in medical radiation physics,'' Phys. Med. Biol, vol. 36, no. 7, pp. 861-920, 1991.
13: D. Gur, A. G. Bukovitz, and C. Serago, ``Photon contamination in 18-20 MeV electron beams fropm a liear accelerator,'' Med. Phys., vol. 6, no. 2, pp. 145-146, 1979.
14: C. C. Ling, M. C. Schell, and S. N. Rustgi, ``Magnetic analysis of the radiation components of a 10 mv photon beam,'' Med. Phys., vol. 9, no. 1, pp. 20-26, 1982.
15: S. N. Rustgi and J. E. Rodgers, ``Analysis of the bremsstrahlung component in 6-18 MeV electron beams,'' Med. Phys., vol. 14, no. 5, pp. 884-888, 1987.
16: S. C. Klevenhagen, ``An algorithm to include the bremsstrahlung contamination in the determination of the absorbed dose in electron beams,'' Phys. Med. Biol., vol. 39, pp. 1103-1112, 1994.
17: M. D. Altschuler, P. Bloch, E. L. Buhle, and S. Ayyalasomayajula, ``3D dose calculations for electron and photon beams,'' Phys. Med. Biol., vol. 37, no. 2, pp. 391-411, 1992.
18: R. Martinelli, G. Ghiso, P. Boccacci, F. Foppiano, and L. Andreucci, ``Determination of the energy of the electrons incident upon a water phantom to build composite energy beam kernels for 3D dose calculation,'' in Proc. Int. Conf. Comp. Radiat. Ther. XI, (Manchester), pp. 124-125, 1994.
19: R. P. Hugtenburg, J. R. Turner, and P. H. Butler, ``Numerical reconstruction of electron therapy beams,'' Med. Phys., vol. 22, no. 6, p. 948, 1995.
20: C. Constantinou, F. H. Attix, and B. R. Palliwal, ``A solid water phantom material for radiotherapy X-ray and X-ray beam calibrations,'' Med. Phys., vol. 9, no. 436-441, 1982.
21: W. R. Nelson, H. Hirayama, and D. W. O. Rogers, The EGS4 code system, rep. SLAC-265.
Stanford Linear Accelerator Center, California, 1985.
22: A. F. Bielajew and D. W. O. Rogers, ``PRESTA: The parameter reduced electron-step transport algorithm for electron Monte Carlo transport,'' Nucl. Instrum. Meth. B, vol. 18, pp. 165-181, 1987.
23: W. H. Press, Numerical recipes: The art of scientific computing.
New York: Cambridge University Press, 1986.
24: A. G. Ferreira and J. Zerovnk, ``Bounding the probablity of success of stochastic methods for global optimization,'' Computers Math. Applic., vol. 25, pp. 1-8, 1993.
25: J. Llacer, ``Inverse radiation planning using the dynamically penalized likelihood method,'' Med. Phys., vol. 24, no. 11, pp. 1751-1764, 1997.
26: G. H. Olivera, D. M. Shepard, P. J. Reckwerdt, K. Ruchala, J. Zachman, E. E. Fitchard, and T. R. Mackie, ``Maximum likelihood as a common computational framework in tomotherapy,'' Phys. Med. Biol., vol. 43, pp. 3277-3294, 1998.
27: A. Kassaee, M. D. Altschuler, S. Ayyalsomayajula, and P. Bloch, ``Influence of cone design on the electron beam characteristics on clinical accelerators,'' Med. Phys., vol. 21, no. 11, pp. 1671-1676, 1994.
28: M. A. Ebert and P. W. Hoban, ``A Monte Carlo investigation of electron beam applicator scatter,'' Med. Phys., vol. 22, pp. 1431-1435, 1995.
29: M. A. Ebert and P. W. Hoban, ``A model for electron beam applicator scatter,'' Med. Phys., vol. 22, pp. 1419-1429, 1995.
30: M. A. Ebert and P. W. Hoban, ``The energy and angular characteristics of the applicator scattered component of an electron beam,'' Australasian Physical and Engineering Sciences in Medicine, vol. 19, no. 3, pp. 151-159, 1996.

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Reconstruction of electron spectra in radiotherapy using inverse Monte Carlo methods

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