### Monte Carlo and ANN

Monte Carlo is known as a mathematical technique that employs statistical sampling for numerical experiments using the computer to estimate outcomes for uncertain events in which the deterministic methods can not give reliable predictions. This method works based on random sampling and is the best way known to explore the behavior of complex systems and geometries with multiple degrees of freedom, such as the transport of the particles in a medium. By repeatedly performing Monte Carlo simulations, many probable outcomes will generate, which become more accurate as the number of inputs grows. Finally, this method offers a clear picture including the results and the corresponding uncertainties. The necessity of performing a complete and trustworthy Monte Carlo computational model is to be used for planning the experimental work and studying possible additional optimization and improvements of the facility. Despite all these advantages, Monte Carlo is time-consuming as there is a need to generate a large number of sampling to generate reliable output, particularly for large and multi-dimensional problems. This is also the case with our problem of optimizing the thermal neutron shield for the entrance door of the treatment room. This is because the problem involves a large volume of matter through which particles must travel, leading to an increased probability of particle loss and complex interactions between particles and the medium. In addition, the problem requires the optimization of multiple parameters, such as the material composition and thickness of the shield, which are coupled and can influence each other’s effectiveness.

In recent years, the ANN tool has found widespread applications in nuclear engineering to predict the behavior of the system by assigning a model to the input data. This involves training the network using a large amount of data to learn the patterns and find relationships between the input data and the corresponding output. Once trained, the ANN can be used to make predictions or classify new data based on its learned knowledge. Considering the advantage of using ANN for predicting the results on one hand and the disadvantage of performing Monte Carlo simulations on the other hand, which involves the difficult and time-consuming task of testing all possible mixtures and thicknesses to find the optimal one, using the data generated by the Monte Carlo simulations as inputs of the ANN was proposed. It is worth mentioning that ANN is a method of predicting the results from the input data in the case of multiparameter, complicated problems having certain efficiency advantages, and is inherently different from the Monte Carlo method that uses a broad class of computational algorithms to obtain numerical results. Feeding the ANN with the Monte Carlo data and validating its generated results with proper Monte Carlo calculations must be finally done.

### Simulations

In this work, the Geant4 Monte Carlo code was employed to perform the simulations. This toolkit is a general-purpose Monte Carlo radiation transport code that is capable of tracking various particle types including leptons, photons, hadrons and ions in arbitrary three-dimensional configurations of materials and geometries over wide ranges of energies. Also, the important features that make this code interesting include being easy to use, flexible structures, and an extensive collection of cross-section data. Moreover, Geant4 provides visualization drivers and interfaces, graphical user interfaces, and a flexible framework for persistency^{32}. The other interesting aspect of GEANT4 is that it has grown over the years and changes have been made to accommodate the needs of the users so that can cover a large number of experiments and projects in a variety of application areas. Details can be found in the literature^{33}. The equipment chosen for simulation was the BSA designed for BNCT based on the D-T neutron generator yielding \(5 \times 10^12\) neutrons per second, with apertures of 6 cm radius for emission of the neutron spectrum. This equipment was simulated in accordance with the BSA proposed in our previous work^{34} for the treatment of deep tumors with the output flux of \(\sim 10^9\) n.cm\(^-2\)s\(^-1\). This system was placed in the center of the simulated treatment room based on an existing room in Imam Khomeini Hospital Complex in Tehran. The simulated treatment room featured in Fig. 1 had a square geometry of 11 \(\times \) 11 m\(^2\), with a maze and entrance door, and a height of 2.5 m. The concrete was considered as the material of both the main walls (those surrounding the BSA exit and the patient bed) and the secondary walls (those behind the maze). The entrance door, with dimensions of 1.5 \(\times \) 2 m\(^2\), was initially assumed to be made of lead in the primary simulations. The thicknesses of the walls, the beam direction, and the position of the phantoms for dose evaluation have been depicted in Fig. 1.

To assess the effectiveness of the designed shield in limiting radiation exposure, the maximum permissible doses based on the widely accepted recommendations were considered. For this purpose, nine spherical simulated water phantoms with a radius of 15 cm were placed behind the entrance door, as the present study focused on the shielding design for this area. The International Commission on Radiological Protection (ICRP) and National Council on Radiation Protection and Measurements (NCRP) publish recommendations for occupational dose limits. The NCRP limits generally agree with ICRP recommendations for dose limits and there are two types of occupational dose limits in these guidelines, including limits for specific organs or tissues and acceptable risk levels for cancer induction^{35,36}. According to these standards, the weekly limits of effective dose in controlled and uncontrolled areas are 0.1 mSv/week and 0.02 mSv/week, respectively. Taking these in to account, the shielding design was planned to ensure that the maximum allowable dose rate behind it (as an uncontrolled area) is less than 0.5 \(\upmu \)Sv h\(^-1\), assuming that the clinic plans to operate for 40 h per week.

The neutron and photon doses have been calculated by scoring the ambient dose equivalent in the simulated phantoms, defined as a weighted radiation dose that takes the quality factor of the particles depositing energy in biological matter into account. To this aim, whenever a neutron or a gamma ray traverses the phantom, the fluence spectrum inside the sphere is obtained and the fluence conversion coefficients are applied. All simulations tracked 5 \(\times \) 10\(^8\) histories, and the statistical errors associated with the results were reported.

### Shielding material

The thermal neutron shield proposed in this study has been inspired by the work of Shahram et al.^{31}, who experimentally designed a polymer composite based on PMMA (polymethyl methacrylate with chemical formula of C\(_5\)H\(_8\)O\(_2\) and density of 1.1 g cm\(^-3\)) and polyethylene powder (with chemical formula of C\(_2\)H\(_4\) and density of 0.9 g cm\(^-3\)). In these materials, hydrogen capturing of thermal neutrons is through the \(^1\)H(n, \(\gamma \))\(^2\)H reaction with a cross section of 0.33 barn^{37}. An in-situ polymerization technique was employed to increase the composite’s slowing-down feature and the boric acid powder (with chemical formula of BH\(_3\)O\(_3\) and density of 1.44 g cm\(^-3\)) was added to absorb thermal neutrons through \(^10\)B(n, \(\alpha \))\(^7\)Li reaction. The produced heavy particles can easily stop in the shielding material and, therefore, have not been considered in the dose calculations. In their work, a polyethylene layer was used as a moderator, followed by a polymer composite layer as an absorber. In the second layer, boron was added at weight fractions of 1%, 5%, 7%, and 10%. In order to evaluate the effectiveness of the designed shield, various combinations of thicknesses for the two layers and the proportion of boron in the polymer composite were experimentally tested. For these limited samples, the neutron doses were measured.

Though their study was pioneering in designing thermal neutron shields, the limited number of models tested raises the question of whether there exist other combinations, thicknesses, or weight fractions that could result in even better shielding properties. To address this question and take advantage of the benefits of ANN as discussed earlier, we tested 720 different models in the present study. It is necessary because, as the number of inputs increases, the accuracy of forecasts by ANN tends to improve, making it necessary to consider a larger number of inputs. The weight fraction of polyethylene in the polymer composite, the weight fraction of boric acid in the polymer composite, the thickness of the first layer of the shield (polyethylene), and the thickness of the second layer (polymer composite) were considered as the parameters with the specific values presented in Table 1. From here, these parameters are labeled by A, B, C, and D, respectively. By using these values, 720 sets of arrangements were generated. In our simulations, each shield has been placed in front of a typical neutron source which has a neutron energy range from a few eV up to 10 MeV, and the dose and flux beyond the shield were calculated.

### Artificial neural network

We utilized ANN for predicting both the thermal neutron flux behind the designed shield as well as the dose calculations. For each network, we had a total of 720 data sets, which have been divided into two parts: training data (690 samples) and testing data (30 samples) to validate the results. Table 2 lists the combination of the four parameters of A to C for the 30 samples that have been used as the testing data. These sets of parameters are chosen so that, in a good approximation, incorporate all the values in the ranges presented in Table 1.

The neural network used in this study for thermal neutron flux consists of a single hidden layer perceptron with 40 neurons and an output layer. The multilayer perceptron (MLP) network has been trained using the Levenberg-Marquardt (LM) algorithm which is an appropriate option for solving generic curve-fitting problems. To calculate the output of a node based on its set of specific individual inputs and their weights, the activation function is needed. In this work, the sigmoid activation function was used for each layer. The input data for the network was a \(4 \times 690\) matrix which included the neutron flux extracted from the code, and the output data was in the form of a \(1 \times 690\) matrix. For the dosimetric data (a \(4 \times 690\) input matrix), a two-layer Feedforward Backpropagation neural network with a hidden layer of 30 neurons was used, and trained using the LM algorithm. Also, the sigmoid activation function was used for each layer of the network. During the training phase of both networks, the neural network was initially trained with training data, and some parameters such as weights and biases have been regularized to prevent overfitting that occurs if the model cannot generalize and fits too closely to the training dataset. The networks have been trained for 1000 epochs to ensure that they had converged to a stable solution. Figure 2 shows the architecture of the neural network used in this work.

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