DeepBeam: a machine learning framework for tuning the primary electron beam of the PRIMO Monte Carlo software

To generate training data for the machine learning framework, the simulations were run for a total of 300 tuples (E, σ_E, s, α) within the set S such that:At the first simulation stage, 10⁸ histories (a history corresponds to a single electron of the virtual primary beam) were simulated for each tuple (E, σ_E, s, α) and the phase-space file (PSF) above the secondary collimators was saved for further purposes. At this first stage, the splitting roulette variance reduction technique [20] was used with the size of the splitting region set to the largest region, i.e. to the 40 × 40 cm² field. The saved PSFs were then used to simulate radiation transport to a homogeneous cubic water phantom for three fields: 3 × 3 cm², 10 × 10 cm², and 30 × 30 cm². The size of the phantom was set to 50 × 50 × 50 cm³. The doses in the phantom were tallied within a regular grid of 0.5 × 0.5 × 0.5 cm³ voxels. The respective faces of the phantom were set parallel to the respective main axes of the coordinate frame of reference of the accelerator. The main axis of the phantom coincided with the photon beam axis. The source-to-surface distance (SSD) was set at 100 cm, the isocentre being located at the front surface of the phantom. Splitting in the water phantom was selected as the variance reduction method [20] at this simulation stage, with a splitting factor of 300. The uncertainty of the dose values tallied in the water phantom always remained within 1.5% (which corresponds to two standard deviations of MC calculated dose). The calculated 3D spatial distribution of doses within the phantom was saved to a text file, separately for each tuple (E, σ_E, s, α) and for each field. A total of 900 3D dose files were collected. Each 3D dose file contained 10⁶ dose values calculated by PRIMO at (x,y,z) coordinates given by the following coordinate ranges:where the z axis is parallel to the radiation field axis. To generate testing data for the machine learning framework, the simulations were run further for 25 tuples (E, σ_E, s, α) with primary beam parameters sampled randomly from the following sets of values:Applying the above sampling scheme, it was assured that the primary electron beam parameters (E, σ_E, s, α) in the testing set never coincided with parameters used for generating the training set, and consequently, that the electron beam parameters for the testing set were well separated from the electron beam parameter selected for training.

All simulations were run using the PlGrid infrastructure (Prometheus grid, https://​kdm.​cyfronet.​pl/​portal/​Main_​page) and required a total real time of about 2.5 months. During the simulation period 12 Prometheus nodes run the PRIMO software, each node equipped with two Intel Xeon E5-2680v3 processors, 24 cores in total, and 128 GB RAM. The simulation of a single case, i.e., of three fields for a single tuple (E, σ_E, s, α), required about 40 CPU hours. As the operating system installed on the nodes is Linux CentOS 7, while PRIMO is a Windows application, wine software (https://​www.​winehq.​org/​) was installed and configured in order to use PRIMO in graphic mode under Linux exactly as if Windows were the operating system.

Dose profiles were measured in water for the 6 MV photon beam of a clinically exploited Clinac 2300C/D medical accelerator at the Krakow Branch of the National Research Institute of Oncology. A PTW MP3 Water Phantom and PTW Markus Type 23,343 and PTW Semiflex Type 31,010 ionization chambers were used for dosimetry. PTW Mephysto software was applied for data collection. Three experimental setups of dose profile measurements were arranged, as described in more detail in the Results section.

Let Prof = {Prof₁, Prof₂,…, Prof_n} represent a user-selected subset of Prof_MAX. A user-selected spatial range Range_i is associated with each Prof_i (Range_i would typically be the user-dependent spatial range over which Prof_i is measured under clinical settings). Each subscript i corresponds to a unique field size and a unique dose profile type (either depth or lateral, at one of the five depths: D_max = 1.4 cm, 5 cm, 10 cm, 20 cm, or 30 cm).

As each dose profile Prof_i is sampled within a given spatial resolution (usually 1 mm), it may be considered a 1D vector of some dimensionality (dependent of the sampling resolution and the sampling range Range_i). The regression task which is to be solved can be formulated as follows:where Par is any element of the tuple (E, σ_E, s, α), f_Par is the regression function and ε_Par is the residual term. The components of each Prof_i are however strongly correlated as they represent dose values measured at neighbouring spatial locations. For this reason, the model given by Eq. (4) may not be very effective, as the set of explanatory variables (arguments of f_Par) contains a high contribution of redundant information.

To resolve this redundancy problem dimensionality reduction is applied. Typically, dose profiles are specified by applying some ad-hoc features, such as width at half maximum, width of penumbra regions, “wing heights” in lateral profiles, etc. Here, rather than rely on such hand-crafted features, Principal Component Analysis (PCA) is applied to the analysed profiles [21]. PCA is a method for finding uncorrelated variables from correlated ones (in our case—consecutive dose values along depth and lateral profiles). Finding such new variables reduces to solving an eigenvalue/eigenvector problem, and the new variables are linear combination of the old ones. Moreover, each PCA feature is assigned a percentage of the total variance of profile shapes it explains. As demonstrated in what follows, three most important PCA features usually explain over 98% of the variability of shapes of the training dose profiles. PCA reduces the dimensionality of the space of explanatory variables by a factor of 10²—the final set of features consists of 3n elements (explanatory variables)—three features for each profile Prof_i in Prof. The learnt PCA models were saved in respective files (a separate PCA model M_PCA,i file for each index i) and subsequently used at the stage of model testing.

Clearly, for each i—index there are 300 training profiles {Prof_i,1, Prof_i,2,…,Prof_i,300} corresponding to 300 different tuples {(E, σ_E, s, α)₁, (E, σ_E, s, α)₂,…, (E, σ_E, s, α)₃₀₀} and a single PCA model M_PCA,i which extracts three features (F_i,1, F_i,2, F_i,3)_k from Prof_i,k. Hence, the regression problem, after dimensionality reduction, becomes:where Par is any element of the tuple (E, σ_E, s, α), \({f}_{PAR}^{PCA}\) is the PCA-based regression function and ε_PCA is the residual term. To learn the regression functions, the following training set, Tr, was applied:Support Vector Regression (SVR) with radial basis function (rbf) kernel was selected as the regressor [22] though other options are also available. SVR with rbf kernel first transforms the explanatory variables (PCA features in our case) to some high dimensional space and then, in this high dimensional space, replaces nonlinear relationship of Eq. (5) with a multilinear one between transformed explanatory variables and explained variables (electron beam parameters in our case). This high dimensional multilinear relationship is then used to make predictions. The best regression models were selected using a fivefold cross validation run on Tr. After training, four \({f}_{PAR}^{PCA}\) regressors were obtained, one per E, σ_E, s, and α. The regression models were saved to files and subsequently used in testing.

The processing pipeline described in the previous section consists of two separate steps: feature extraction, and training of four regressors. In the current section an end-to-end regression model is described which, during training, learns both dose profile data representation and regression functions simultaneously for all primary beam parameters (E, σ_E, s, α). The model presented here is based on deep learning. The architecture of the deep learning (DL) model is outlined in Fig. 1.

The architecture is designed to follow the same processing steps as the approach described in the previous section. In short, each Prof_i in Prof is a separate input for the DL model and is processed by a separate encoder block. Each encoder block consists of a few convolution blocks. Each convolution block consists of two 1D convolutions (filter size equal to 3, number of filters equal to 16, 32, 64, etc. in the consecutive convolution blocks, ReLU activation) followed by a MaxPool1D layer which reduces the size of the data by a factor of two. The number L of convolution blocks in each encoder block is selected based on the length N of the input of this block, according to the formula L = int(log₂N/3), i.e., the number of features learnt by any encoder block cannot be less than 3. Each encoder block ends with 1D convolution with a single filter of unit size. The outputs of the encoder blocks are then concatenated to form a 1D vector of features (in analogy to PCA features). This feature vector is then processed by two fully connected layers of size 100 and ReLU activation. The output of the last fully connected layer is next fed into the final fully connected layer with four outputs and no activation. These outputs are expected to deliver estimates of E, σ_E, s, and α.

The training data Tr_DL for the DL model is:

The model is trained for 300 epochs using the Adam optimizer and a constant learning rate equal to 0.0001. The loss function selected for this regression problem was mean square error between the model outputs and the values of primary electron beam parameters for which the dose profiles at the model input were obtained. A 20% portion of the training set was randomly selected for model validation. The best model found during training was saved to a file and used in subsequent testing.

At the stage of model testing, the testing profiles were fed at the input of either the PCA + SVR or DL models. The PCA + SVR model first extracts the features from the testing profiles based on PCA models learnt on the training set. These test features are then processed by SVR regressors which return the predicted values of E, σ_E, s, and α. In the case of the DL model the raw testing profiles are fed at the input of the DL model which returns the predicted values of E, σ_E, s, and α. The true and predicted values of E, σ_E, s, and α are then compared using correlation analysis and linear regression.

The regression results can be further improved by minimizing the difference between the actual profiles being fed at the input of regressors and profiles reconstructed from the regression results. In particular, the parameters of the model of the primary electron beam for the training set Tr (Eq. (6)) have the form of a regular grid S, defined in Eq. (1), embedded within a 4D hypercube H. With every node Q of S associated are PCA features corresponding to the dose profiles determined for primary electron beam model parameters (E, σ_E, s, α)_Q assigned to Q. The regressions return a point P = (E, σ_E, s, α)_PRED within H (see Fig. 2 for a 2D example). Consecutively, using interpolation, PCA features corresponding to P may be determined, and next an inverse PCA transform applied to them in order to reconstruct profiles from the results of regression models (E, σ_E, s, α)_PRED. Thus, for each Prof_i in Prof a reconstructed profile RecProf_i(P) obtains, which in general differs from Prof_i. This difference can then be further minimized using one of several optimization methods, with (E, σ_E, s, α)_PRED as the starting point for such minimization. Namely, beginning with P = (E, σ_E, s, α)_PRED, P_MIN = (E, σ_E, s, α)_MIN is sought, such that:where w_i is the weight assigned to the i-th profile. In the experiments all w_i were set to unity but in general the user may set these according to her/his actual needs. The minimization problem defined in Eq. (8) was solved using the SLSQP method [23].

All models were implemented in Python 3.6.10. The scipy library (version 1.5.2) was used to implement regression models using PCA and SVR. The same library was used to run interpolation over 3D dose distributions to extract dose profiles, optimization of the regression results and profile reconstruction from regression results. The DL model was implemented using the keras (version 2.3.1) library. All codes, pretrained models, as well as training and testing data, are freely available at https://​github.​com/​taborzbislaw/​DeepBeam.