Pencil-Beam Fluence Evaluation Based on Monte Carlo Simulations Algorithm of High Energetic Treatment Photons.

BACKGROUND: The accurate calculation of doses during external radiotherapy sessions is necessary. Recently, the models used for this purpose have been the point kernel, pencil-beam, and collapsed cone superposition/convolution combination models.
METHODS: In this study, it is aimed to determine point/pencil-beam kernels to be used in dose calculations. For this purpose, tallying pencil-beam fluence based on Monte Carlo (MC) simulations is investigated by scoring a volume of interest centered in a cubic water phantom at a depth of 10 cm and the irradiated field of 10 cm × 10 cm. The fluence is calculated for each mono-energetic photon ranging from 0.25-6 MV at increments of 250 keV.
RESULTS: The output of the four fluence kernels along the depth Z around the central axis is categorized for both the primary and secondary photons and electrons. Here, a database for pencil-beam kernels is established for each category. For validation purposes, other MC simulations are carried out for fluence calculations as produced by the predetermined poly-energetic spectra for a Varian 6 MV linear accelerator.
CONCLUSIONS: The net results show a good fit of the two convoluted fluence spectra quantities for both mono-energetic and poly-energetics-based simulations except little singular peaks.

Introduction

Nowadays, there is a growing awareness with respect cancer patient's dose accuracy in modern dose calculations or dose engines based on various algorithms for radiotherapy applications.[1234] More accurate equipment is now available for estimating absorbed doses during radiotherapy sessions.[5678] However, it is still suffered inaccuracies despite the availability of advanced treatment methods. For this purpose, in-patients dose engine is based on predetermined energy fluence on or into the patient.[9] Most of these algorithms are constructed to facilitate dose calculations and mapping in different environments.[13] Two of these algorithms are point beam kernel (PK) and pencil-beam kernels (PBK) that have been extensively used. PBK methods superposed the predetermined dose distributions from narrow “pencils” of radiation in water.[4] Some of treatments planning systems are based on PK and PBK for dose deposition calculations.[101112] One of the examples of using PBK algorithm is presented in[13] through deconvolution process from experimental data, especially in flattening filter-free (FFF) linear accelerator (LINAC) for 3D dose calculations. Uncertainties found in the results depend on particular detail for both calculation of PBK and MC. As an example, implementation of PBK method shows low statistical uncertainty in radiotherapy,[134] however, it is high for calculation of actual proton doses makes it unsuitable for reflection of dose in some cases.[14] The most interesting approach to PBK fluence has been proposed by[15] which calculated the local spectra via dividing it into high and low energies using small and large cavity theories approximations, respectively.[16]

PBK fluence is divided into four groups:[15] primary photons, secondary photons, secondary electrons, and primary electrons. The primary photons would be the photons which remain at a depth z beyond its attenuation. Primary electrons arise from primary photons interacting with LINAC head or phantom. The scattered photons come from primary photon interactions or other scattered particles which contained bremsstrahlung and annihilation photons. At last, electrons which are released from interactions of those scattered photons are converted into scattered electrons. MC codes are extensively used in radiotherapy due to its high accuracy of dose calculation and dose distribution.[171819] Excellent results are recently reported between different MC codes for obtaining a good agreement between percent depth dose curves and depth doses.[20]

PENELOPE is a code system that performs photon-electron coupled transport process in materials with a wide-ranging energy.[21222324] It is extensively used in the simulation of x rays spectra, gamma rays and electron emission.[242526] Moreover, it is implemented in complicated geometries for dose application.[27] Recently, updated versions based on PENELOPE package is used like PENEPMA for electron probe microanalysis measurements.[28] In a recent paper,[29] PENELOPE presented excellent improvements for security purposes in coded aperture and pinhole imaging systems.

A large number of impinging particles is required to get sufficient low uncertainty; however, it is time-consuming. For this purpose, high-performance computers or supercomputers nowadays solve this limitation in a more sophisticated computation.[30] Furthermore, non-analog simulations like that using variance-reduction technique (particle splitting and Russian-roulette techniques) are extensively needed for both efficiency and lesser time for calculations.[1531]

Other advantage of the pencil-beam approach is that it could easily evaluate the deviation induced as a result of compensator or wedge. Accordingly, changing pencil-beam kernel would be easily to treat the variations resulted from wedge filtration or off-axis. So that, tuning the incident fluence over time could be extremely useful toward modeling the complex treatments like as intensity modulated radiation therapy (IMRT).[32] Some methods are well-known for creating pencil beam from measurements such as differentiation of both radial relative dose along the central axis.[3334] MC is also used for PBK determinations in any materials.[1635]

Our study aimed to implement the analytical methods of dose engine such as PK and PBK algorithms for fluence spectra calculation. Therefore, MC simulations are carried out for two different cases (mono-energetic and poly-energetic spectra calculations). The essence of PBK is to increase the calculation speed and convolve fluence over the field precedent identified.

Materials and Methods

To obtain the spectrum of the fluence to the point of interest (x, y, z) with a given irradiation field, we can apply a lateral and a convolution integration on energy range [0, Emax] convolving over the field aperture, as follows:[32]

Where is the incident energy fluence spectrum with the maximum energy Emax. Both of electron fluence kernel and photon fluence kernel spectrum are defined at the position of (x, y, z) for mono-energetic beam of energy ED.

A commercial application of the PBK is the AAA (analytical anisotropic algorithm).[16333637] The literature on PBKs shows a variety of approaches.[38394041424344]

PBK is scored in cylindrical voxels with a mono-directional pencil-beam impinging on a homogeneous unlimited phantom. Fluence is differentiated using cylindrical coordinates (radius r and depth Z from the entrance point of P (x, y, z)) at which is where the pencil beam enters the phantom.

Monte Carlo simulations setup

Pencil-beam approach

The geometry of the simulation is depicted in Figure 1, which is saved in a. geo file and shows two different colors, violet and red, representing the air cavity and the water, respectively. Eighty detectors are implemented to simulate the effects of a mono-direction beam for a maximum depth of 40 cm under the water. Each detector is described by its cylindrical voxel centered on the z-axis of a volume of 26.8 cm3. The source of the photons is set at 1 m from the air-water surface, emitting mono-directional photons perpendicular to the surface. The voxels in Figure 1 are the volume of interest in the middle cylinder of 0.5 cm. High computing performance (Aziz) technology that is available at King Abdulaziz University, Jeddah, is used for the simulation of the fluence scoring cylindrical voxel regions. The variation of the energy for the mono-energetic beam kernels is from 0.25–6 MeV with an increment of 250 keV.

Figure 1

Two-dimensional geometrical view of the simulation

EABS1, EABS2 and EABS3, cutoff energies for electrons, photons, and positrons, respectively, are set at 30 keV. Wcc and WCR, minimum energy that transfer in bremsstrahlung emission and elastic collisions, respectively, are set at 30 keV. The parameters C and C2 are the average angular deflection between hard elastic events and maximum average fractional energy loss, respectively in a single step. Both values of C and C2 are set to be small enough at 0.02 for high-resolution particle transport resolution without creating a detailed simulation.[12] DSMAX is the maximum step length for electrons which should equal to one-tenth of voxel size,[12] is set at 0.05 cm. The number of simulated primary photons, NSIMSH, is set at 20 × 106 particles.

Results and Discussion

The fluence kernels of the different categories are represented for both cylindrical radius r (cm) and the depth Z (cm) in Figure 2a-d. The incident energy is set at 6 MeV. As shown in Figure 2a, the primary photon sernel is only one voxel wide. It is clear that primary photon kernel contains only those photons that have not yet been scattered.

Figure 2

(a) Fluence kernels of the primary photons of mono-directional mono-energetic fluence in water phantom with an incident energy of 6 MV. (b) Fluence kernels of the primary electrons of mono-directional mono-energetic fluence in water phantom with an incident energy of 6 MV. (c) Fluence kernels of the secondary photon of mono-directional mono-energetic fluence in water phantom with an incident energy of 6 MV. (d) Fluence kernels of the secondary electrons of mono-directional mono-energetic fluence in water phantom with an incident energy of 6 MV

In addition, note that the primary electrons [Figure 2b] are also somewhat limited in a finite volume while the scattered photons and electrons [Figure 2c and d] have a wide span. In the meantime, a problem in the scoring code has rendered this solution almost impossible. As seen in geometry shown in Figure 1, the most voxels defined would not represent the correct value of the incident fluence. For decreasing the uncertainty errors, a source file (penelope. f) in the PENELOPE package is modified to change the rules of scoring according to the interaction history. An alternative compensation is found when the enclosure of the overall system changes. The enclosure usually represents as a giant sphere with a default radius of 3 × 107 cm. It contains all the defined geometric structures and outer sides of the enclosure would be absolute vacuum such that any tracking history of the particles would stop if they reached the interface of the enclosure. When the change in the enclosure radius is implemented to 3 × 1030 cm, the uncertainty errors of scoring becomes much less visible.

Calculation of the spectra from the pencil-beam kernels

Once PBK database has been established, the net spectrum at any point of an irradiated water slab could be determined through the superposition of the fluence pencil kernels over the field with a known incident fluence. The fluence kernel resulted from the simulation of the polar coordinate system (r, z) is transformed from to the (x, y, z) three dimensional space axes. Therefore, energy fluence spectrum could be calculated using convolution in Eq. 1.

A poly-energetic pencil kernel

A poly-energetic pencil kernel is calculated by taking a beam spectrum for the 6 MV Varian LINAC of reference.[45] The spectrum has been used to simulate the actual spectra of the accelerators. This spectrum is simulated using MC codes (BEAMnrc) and refers specifically to the spectrum of a 6 MV beam on Varian CD 2100 machine. The spectrum is validated by fitting it to that of the actual machines by comparing the measured and simulated depth doses for several field sizes.

Another set of data for a square parallel beam field of 10 cm × 10 cm has been generated to validate the results of the kernel superposition. Since the size of this simulation is enormous, the parameters are modified, and the simulation is divided into batches to fasten the overall simulation. Geometry of the simulation and the modified simulation parameters[46] are summarized in Table 1.

Table 1

Simulation parameters used for fluence calculation based on poly-energetic pencil beam kernel

The geometry of the simulation in a field size of 10 cm × 10 cm is shown in Figure 3. The volumes of interest are put at a depth of 10 cm perpendicular to the axis. Thus, a profile of the fluence varying with the position of the volume is presented. Herein, the form of the voxel is defined as a volume of the cube equal to 0.125 cm3.

Figure 3

Simulation geometry of 10 cm × 10 cm field size, volumes of interest are put in a depth of 10 cm perpendicularly to the axis

The profile of the simulated fluence is shown in Figure 4. As shown in Figure 4, the photon fluence is much wider than the fluence of the electrons and positrons.

Figure 4

A profile of the fluencies of different particle categories at 10 cm depth underwater surface and the field size of 10 cm × 10 cm

The positrons lie in almost the region around central axe of incident beam of 5 cm radius, approximately of the irradiated field as their energy is relatively low. The calculated uncertainty of the fluence is relatively higher at the off-axis region as fewer photons have arrived in these areas. The number of particles simulated in this test should be larger to avoid such high uncertainties.

A comparison of the spectra in the same positions from the simulations and convolutions is made [Figure 5]. The voxel chosen for comparison is that at the position (0, 0), where we set the axis as the original point at a depth of 10 cm. Figure 5a shows the result of the directly simulated fluence spectrum for the given field size and the energy spectra of the incident photons. Figure 5b represents the spectra convolved using the poly-energetic kernel for the given field size.

Figure 5

(a) Monte Carlo simulation results of pencil-beam fluence spectra of the voxel at the center position of the field of 10 cm × 10 cm extracted from the simulation results based on convolution for each mono-energetic fluence. (b) Monte Carlo simulation results of pencil-beam fluence spectra of the voxel at the center position of the field of 10 cm × 10 cm extracted out of pre-validated spectrum based on poly-energetic kernel

One of the major drawbacks of this convolution is that the calculated kernel is based on the mono-energetic pencil kernels while the simulated spectrum is given in the form of a histogram. This explains the peaks that appear in Figure 5a because the kernel is considered as a combination of several energy peaks, which caused this difference.

Local photon spectrum is obtained by convolving PBK spectra through simulating the irradiation of monodirectional mono-energetic beam over a field of 10 cm × 10 cm in an infinite water phantom. A successful database recalculation of the pencil kernels based on mono-energetic PBK is significant. The established database of the fluence spectra are divided into four categories according to their interactions: (1) uninterrupted incident photons or primary photons; (2) primary electrons that are created during primary photon interactions; (3) scattered photons which are resulted from Compton, Rayleigh, bremsstrahlung, pair production, and ionization effects interactions; and (4) scattered electrons that are created by scattered photons interactions. Although there are some singular peaks appearing in the convoluted spectrum [Figure 5a] the quantities and the form of the fluence spectra of the same volume agree well. In addition, note that the number of simulated particles in the test field of 10 cm × 10 cm is still quite small. This small number of simulated particles is further evidenced by the uncertainty of the spectra, which is quite large in Figure 5a. The importance of this study is essential when dealing with more sensitive solid-state dosimeters. Those dosimeters are deviated from tissue-equivalent materials such as diodes and MOSFETs. As a result, it could be easy to tackle any induced spectral variation in these dosimeters or its covering cavities. Furthermore, it is important to analyze any local spectrum perturbation based on scattered and primary photons[47] or scattered and primary electrons independently. So that, some modifications were employed in cavity theories considering the aforementioned photons and electrons categories and matched with our results.[1247] Moreover, it could be easy and fast to get correction doses factor for the sensitive dosimeters that could be linked to any local spectra perturbation.

Conclusion

The pencil-beam approach is essential for modeling complex treatments like those of the highly modulated IMRT and volumetric modulated arc therapy fields because this approach considers the off-axis doses carefully by changing the entrance points. This study investigated establishing a PBK database based on MC algorithm to calculate convoluted spectra in homogeneous medium. Such database is classified into four categories of primary photons, primary electrons, secondary photons, and secondary electrons. The results of the comparisons between two pencil-beam fluence spectra of two different methods have been quite encouraging. In the first method, the fluence is calculated by convolution of mono-energetic fluence quantities over a field size of 10 cm × 10 cm. In the second method, fluence is computed by convolution of poly-energetic fluence quantities of predetermined validated spectrum over same field size. Such comparison showed a little singular peaks; however, the quantities and the form of the fluence spectra of the same volume are good fitted. Hence, it would make a considerable saving in computation time of dose calculations. This method is considered helpful for implementation of fast calculations for dose corrections into the routines of radiotherapy clinics.

Financial support and sponsorship

None.

Conflicts of interest

There are no conflicts of interest.

BIOGRAPHIES

Hossam Donya Dr. Hossam Donya works as assistant professor at the Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia. His permanent affiliation is also at the Department of Physics, Faculty of Science, Menoufia University, Shibin El-Koom, Egypt

Email: hdunia@kau.edu.sa