Evaluation by Monte Carlo Simulation of Doses Distribution in Tumors with Hypoxia

Radiotherapy is one of the most useful modalities applied for tumor treatments, which use ionization radiation to eradicate the tumor, in major cases. Cells with normal oxygenation are more sensitive to the effects of ionizing radiation than those with hypoxic conditions, because O 2 molecules react rapidly with free radicals, produced by irradiation, originating highly reactive radicals. Thus, the different concentrations of hypoxia in tumors can modulate the response of the irradiation through the radioresistance they present and consequently the success of the treatment. This chapter deals with the dose distributions in cranial tumors with different concentrations of hypoxia through a code based on Monte Carlo simulation.


Introduction
The modality of treatment with ionization radiation, specifically, for cancer can be considered routinely in several hospital centers, which must have human and technological resources capable of conforming the radiation dose into the target volumes and trying to avoid high toxicity in the adjuvant tissues. Although all these resources exist, the treatment planning is based on prior knowledge of the structures of the patient that can be obtained through any kind of medical images. Once the tumor has been diagnosed, appropriate treatment is indicated without, often, considering some factors that may influence treatment success, due the limitation of the medical images. Factors known as repair of sublethal DNA damage, cell repopulation, redistribution of cells, and reoxygenation are not considered [1].
Experimental data showed that oxygen is the most component that modified the radiation sensitivity and hypoxic cells that can be 2-3 times more resistant to ionizing radiation, which would imply administering doses higher than doses to achieve the same effect in oxygenated cells under normal conditions [2][3][4].
An important concept in clinical radiobiology is that the tumor may have subpopulations in hypoxic areas, thus leading to success of radiotherapy. Still, there are concepts related to acute or chronic hypoxic cells that may also alter treatment outcomes [5]. The low concentration of oxygen or hypoxia in the tumor tissues is a radiobiological phenomenon that has been observed since the beginning of the twentieth century [6].

PENELOPE-Monte Carlo code
PENELOPE is a code used to simulate the transport of electrons, positrons, and photons considering interactions of photons and charged particles (such as the photoelectric effect, Rayleigh scattering, Compton scattering, production and annihilation of pairs, elastic and inelastic collisions), which are simulated in complex geometries and arbitrary materials.
In the PENELOPE package, there are subroutines written in FORTRAN distributed in various (open) source codes, applications, a database with characteristics of various materials, as well as application examples. The FORTRAN subroutines are organized into four basic files: PENELOPE.f, PENGEOM.f, PENVARED.f, and TIMER.f. PENELOPE.f contains the simulated particle scattering and absorption subroutines, primary and secondary particle generation and storage, and particle transport management and simulation as a whole.
PENGEOM.f defines the structures, or geometries, to be simulated, which may consist of several homogeneous bodies, defined by a specified material and also by their limits in space. The bounding surfaces of the geometry bodies are described by quadratic functions. Through these functions surfaces such as planes, plane pairs, spheres, cylinders, cones, ellipsoids, parables, and hyperboloids can be defined.
For each body defined in the geometry file of a given simulation, a material index must be defined, corresponding to the material that will be a constituent of the body, having an agreement between the geometry file and the material file. In the material archive, the interaction data of the radiation with the material being used are shown in tables, as interaction coefficients for electrons and photons in energies from 1 eV to 1GeV. A material file is created using the subroutines of the MATERIAL.f and PENELOPE.f source codes. One of the advantages of PENELOPE is that it uses a recent database with the characteristics of various materials of interest in radiological physics [42] and current cross section libraries and other quantities required for particle transport [43].
The PENVARED.f source code contains subroutines that perform the variational reduction methods of the code, without increasing simulation time and neither the statistical uncertainty of the simulated results.
Finally, source code TIMER.f manages the simulation time.
The simulation algorithm is based on a model that combines numerical and analytical cross section data for the different types of interaction and is applied for initial energies from 1 keV to 1 GeV. Photons transport are simulated by the conventional or detailed, and for electron and positron are simulated using a mixed algorithm. Thus, for electrons and positrons, the PENELOPE code differs from other simulation codes by using a mixed algorithm that implements two simulation models: the detailed, for strong events, defined from angular deflection (scattering angle) or energy loss above a set value, and the condensate for weak interactions with angular deflection or energy loss less than the preset values. Condensed interactions are described by a multiple-scatter approximation consisting of transforming a given number of weak interactions into a single artificial event [44]. To develop a simulation with PENELOPE, the user must edit a FORTRAN file, user.f, with calls from the subroutines PENELOPE.f, PENGEOM.f, PENVARED.fe, and TIMER.f, providing overall simulation management and creating with these five FORTRAN files a user.exe file.
The simulation is started by running the user.exe file that reaches the usersupplied input information through the input.in file, geometry information through the geometry.geo file, and cross section information for the materials involved in the simulation through the material.mat file.
Another executable program comes together with PENELOPE package, the GEOVIEW.exe, which allows the visualization of the defined bodies and the materials that constitute a simulation geometry.

Simulation of the dose distribution in tumors with hypoxia through PENELOPE code
As some tumors have concentric circles of regions of low concentration of oxygenation, which are deprived of adequate blood supply, the evaluation of those regions becomes essential to guarantee the success of radiotherapy treatments [4]. The study evaluated the hypoxia effect in the dose distribution for simulated cranial tumor with different concentrations of oxygen [18,[36][37][38][39] through the PENELOPE simulation code. The code allows the "construction" of tumors through the compound's chemical composition, mass density, mean excitation energy, and energy and oscillator strength [45,46]. The tumor with different concentrations of oxygen will be compared with one under normal oxygen conditions. The parameter OER was applied to express the decreased radiosensibilty of the tumor with hypoxia. The simulation of the dose distribution in tumors with hypoxia through PENELOPE code is based on the published by Alva-Sánchez [47].

Material and methods
The Monte Carlo code, PENELOPE ® , version 2008 was used to achieve the main objective to simulate tumors with hypoxia. Since the code allows the "construction" of materials by the compound's chemical composition, the soft tissue material from the code, number 262, was modified by adding different concentrations tumors with hypoxia, from knowledge that chemical compounds of the normal tissue are approximately equal to the tumor. The geometry of simulation used was a parallelepiped of 8 × 15 × 21 cm 3 containing six identical spherical tumors of 1.2 cm radius, as shown in Figure 1 [47]. In that work 2 × 10 9 primary particles and 0.1mm 2 pixel size and photon spectra at 6 MV [48], which was applied in the input file of the code, were used. The tumors were located before the buildup region for the This geometry used to simulate the radiation conditions was 22 × 22 cm 2 radiation field, 100 cm source-tumor distance, with a 6 MV energy beam. The obtained results from the simulation were analyzed in terms of deposited energy and values of OER relative to the pressure O 2 in terms of mmHg, following the OER equation proposed for [55].
The second part of that work published [47] analyzed the oxygen effect in the dose distributions in simulated cranial tumors: one with different oxygen concentrations and the other with normal oxygenation [18,[36][37][38][39]. A cylinder with 18 cm in diameter and 20 cm in height was simulated to represent the head of an adult, containing concentric spheres of radii of 0.5-1.2 cm that can represent the dimensions of a glioblastoma tumor. Due the characteristics of localization of this kind of tumor, the simulation tumors were centralized at 10 cm height, beyond the equilibrium range to charged particle for the energy used in this study. The bean incident was simulated in the direction to the phantom, parallel to the Z axis. In Figure 2 the geometry for radiation for a simulated tumor with different concentrations of oxygen from minor (radius sphere 0.5 cm) to greater is shown (radius sphere 1.2 cm).
The simulated material of the tumors and brain structures was soft tissue due to similar chemical compounds present. This material is available in the simulation code. In the spheres (tumors) these materials were modified with different concentrations of oxygen. The radiation conditions used were 1.2 × 1.2 cm radiation field, source-tumor distance of 100 cm, and photon spectra at 6 MV.

Results
From the results obtained to the six spheres inserted in the parallelepiped (Figure 1), the deposited energy was plotted for each spherical tumor containing different pressures of oxygen as shown in Figure 3. Values of the OER were obtained through Eq. (1) and plotted for each pressure of oxygen, as shown in Figure 4.
The penmain file of the code generates an output file with generic information, such as number of simulated primary showers, secondary-particle generation probabilities, average deposited energies, and statistical error, etc. Each simulation was written in a separate file. The program computes and delivers the statistical uncertainties (3σ) of all evaluated quantities and distributions. Thus, for all obtained results, an error at least 3.58% was reported by the code used. Figure 5a shows the dose distributions for a simulated tumor of 1.2 cm radius with different pressures of O 2 , and Figure 5b shows the same tumor of Figure 5a with normal oxygenation.
These distributions were compared through a dose profile, as shown in Figure 6, along the center of the dose maps.

Discussions and conclusions
Statistical uncertainties at least 3.58% was reported by the code for all simulations. From the results shown in Figure 3, we can observe that the deposited energy have an approximately linear behavior with the increases in pressure until 50 mmHg of O 2 ; from this pressure the deposited energy shows a constant trend for higher pressures than 60 mmHg of O 2 . This behavior was also equivalent to those presented in the literature [36,50].
The obtained values of the OER relative to pressure of O 2 shown in Figure 4 have a behavior similar to an increasing logarithm function as found in literature [49,[51][52][53][54]. After the 60 mmHg of pressure of O 2 , the OER values have a trend to value constant. Figures 3 and 4 confirm the increasing of the deposited energy for tumors with a high concentration of oxygen, because of the oxygen effect that is disregarded in accuracy of the ionization radiation treatment [53,55].
The dose distributions shown in Figure 5 visually show almost the same distribution, but in the field profile of both distributions, shown in Figure 6, a difference of 7.29% was found at a radial distance of 0.6 mm of the tumor. The code allowed evaluated the influences of effect of oxygen for tumors with hypoxia that is related  with the outcome of treatment with radiotherapy. However, the hypoxic tumor cells are resistant to radiation [8]. From the comparisons of the dose distribution in the central plane of the phantom, it was observed that there are regions where the concentration of oxygen was lower (that of sphere of less radius), thus, the energy deposited was lower, unlike the spheres with higher oxygen concentrations.
Despite the fact that the code cannot simulate the physiological factor, which it can modulate a variety of normal developmental and metabolic processes that cause injury to the tumor cell, the present study of tumors with hypoxia plays an important role in dose distribution that can compromise the treatment outcomes and individual prognosis.

Considerations
There are a number of simulation and numerical methods and codes that demonstrate the importance of the low concentration oxygen effect in response to ionizing radiation treatments. This chapter shows the work Study of the distribution of doses in tumors with hypoxia through the PENELOPE code which showed physical parameters with results similar to the literature through the PENELOPE-Monte Carlo code.