Evaluation of a neural network‐based photon beam profile deconvolution method

Abstract Purpose The authors have previously shown the feasibility of using an artificial neural network (ANN) to eliminate the volume average effect (VAE) of scanning ionization chambers (ICs). The purpose of this work was to evaluate the method when applied to beams of different energies (6 and 10 MV) and modalities [flattened (FF) vs unflattened (FFF)], measured with ICs of various sizes. Methods The three‐layer ANN extracted data from transverse photon beam profiles using a sliding window, and output deconvolved value corresponding to the location at the center of the window. Beam profiles of seven fields ranging from 2 × 2 to 10 × 10 cm2 at four depths (1.5, 5, 10 and 20 cm) were measured with three ICs (CC04, CC13, and FC65‐P) and an EDGE diode detector for 6 MV FF and FFF. Similar data for the 10 MV FF beam was also collected with CC13 and EDGE. The EDGE‐measured profiles were used as reference data to train and test the ANNs. Separate ANNs were trained by using the data of each beam energy and modality. Combined ANNs were also trained by combining data of different beam energies and/or modalities. The ANN's performance was quantified and compared by evaluating the penumbra width difference (PWD) between the deconvolved and reference profiles. Results Excellent agreement between the deconvolved and reference profiles was achieved with both separate and combined ANNs for all studied ICs, beam energies, beam modalities, and geometries. After deconvolution, the average PWD decreased from 1–3 mm to under 0.15 mm with separate ANNs and to under 0.20 mm with combined ANN. Conclusions The ANN‐based deconvolution method can be effectively applied to beams of different energies and modalities measured with ICs of various sizes. Separate ANNs yielded marginally better results than combined ANNs. An IC‐specific, combined ANN can provide clinically acceptable results as long as the training data includes data of each beam energy and modality.


| INTRODUCTION
In the commissioning of a treatment planning system (TPS) and the periodic quality assurance (QA) of a linear accelerator (linac), it is essential to accurately measure the transverse photon beam profiles produced by the linac. 1 It is well-known that the measurements, typically performed with a finite-size ionization chamber (IC), are compromised by the volume averaging effect (VAE). 2 The VAE, caused by the signal averaging over the detector's active volume, can artificially broaden the penumbra of photon beam profiles by 2-3 mm, depending on the detector's effective size. This effect has profound implications for the planning, delivery, and QA of radiotherapy using small beam segments, such as intensity-modulated radiotherapy (IMRT) and volumetric-modulated arc therapy (VMAT). [3][4][5][6][7] For example, Yan et al demonstrated that the elimination of the VAE enabled the use of stricter criteria (from 3%/3 mm to 2%/2 mm) in patientspecific IMRT QA, which in turn led to higher chances of detecting dosimetric errors arising from either treatment planning or delivery system. 4,7,8 Direct reconstruction of the "true" beam profiles from the measurements is the preferred approach to address the VAE. 9 A process called deconvolution is performed where the VAE, modeled with a detector response function, is numerically or analytically removed from the measurement based on the convolution theorem. [9][10][11] Photon beam profile deconvolution is challenging for a few reasons.
First, since the penumbra of photon beam profiles is located in the high gradient area, Fourier-based numerical deconvolution methods suffer from high-frequency measurement noise. Second, each type of IC has a unique detector response function that is hard to determine. Additionally, there is no consensus on the exact shape and extend of the detector response function. 12,13 Third, the shape of the beam profiles varies with beam geometry (depth and field size) and beam modality [flattening-filter (FF) vs flattening-filter-free (FFF)], which makes it difficult to fit the beam profiles with functions that can facilitate analytical deconvolution. 4,14,15 In a previous proof-of-principle paper, the authors investigated the feasibility of photon beam profile deconvolution using an artificial neural network (ANN). 10

2.A | Neural network model
Here, we briefly describe the ANN model, the detail of which can be found in our previous paper. 10 The ANN consists of an input layer, a hidden layer, and an output layer (Fig. 1). While the input and hidden layers have multiple nodes, the output layer has a single node. A sliding window is used to extract inputs from the measured beam profile at 1 mm resolution. The output, corresponding to the deconvolved value at the center of the window, is given by where s denotes input signal from the measured profile; w represents the weight associated with the link connecting adjacent layers; b represents the bias of the hidden or output neurons. The deconvolved beam profile is created in a point-by-point fashion with the sliding window moving across the measured beam profile. The hidden nodes and output node use the hyperbolic tangent sigmoid activation function (σ h ) and linear activation function (σ o ), respectively.
The number of input nodes (L sw , i.e., the size of the sliding window) and the number of hidden nodes (N hn ) were determined with a parameter sweeping algorithm in our previous paper. It was found that L sw ¼ 15 and N hn ¼ 5 yielded the best performance and these parameters were used in this study.  Compared to the ICs, the EDGE diode has significantly less VAE due to its small effective measuring area (0.8 × 0.8 mm 2 ). 4,9

2.C | Network training
There are practical advantages of training and using as less ANNs as possible. Therefore, it is of great interest to know whether a separate ANN is needed for each scenario (beam energy, beam modality, and IC). Theoretically, the "real" beam profile can be expressed as the convolution between the measured beam profile and a deconvolution kernel, and the ANN is trained to simulate the convolution operation. The deconvolution kernel is directly related to the detector response function. ICs of different types have different detector response functions due to their differences in radius, effective volume, and physical construction. 16  To answer these questions, we compared the detector response function of an IC under different beam energies and different beam modalities. The measured beam profile P m x ð Þ is the result of convolving the "true" beam profile P t x ð Þ with the detector response function K σ x ð Þ, which is typically approximated using a Gaussian function with a shape parameter σ. 7,11 Given a pair of P m x ð Þ and P t x ð Þ, σ can be determined through iterative optimization, where denotes the convolution operation. Practically, for P m x ð Þ and P t x ð Þ, we use the same beam profile measured with the IC and the EDGE, respectively. 11,17,18 The EDGE-measured beam profile can be regarded as the "true" beam profile due to its negligible VAE. [19][20][21] Here, we determined σ for CC13, the most commonly used scanning IC, for the 6 MV FF beam, the 6 MV FFF beam, and the 10 MV FF beam. For each beam energy/modality combination, we calculated σ for each geometry (field size and depth) separately, resulting in 28 shape parameters.
Our study revealed that there were subtle differences in the shape parameters for different beam energies and modalities.
between the predicted output O i and the desired value P i taken from the EDGE-measured profile (N is the length of the profile). The standard Levenberg-Marquardt backpropagation algorithm was used in the training. The network was initialized with random weights and biases and the training was repeated 10 times with each attempt involving 400 epochs. The network with the smallest MSE was selected for evaluation. The test dataset, unseen by the ANN, was used to test its generalization ability. In addition to MSE, another metric called PWD (penumbra width difference) was used in the evaluation. The PWD was calculated as the difference in the penumbra width (distance between 20% and 80% intensity) where W o and W r were the penumbra width of the deconvolved and the reference profile, respectively. While the MSE evaluates the overall agreement between two profiles, the PWD focuses on the penumbra area where the VAE is most prominent. Note that, in this work, the EDGE-measured profiles are used as references in PWD calculation by virtue of the negligible VAE associated with the EDGE.
Three combined ANNs were also trained for comparison. We    Fig. 4(d)). The mean PWD decreased from 1.99 ± 0.15 mm to 0.07 ± 0.05 mm after deconvolution. The reduction of PWD is detailed in Table 2.
After deconvolution, the PWD was under 0.20 mm for all studied beam geometries.  Figure 6 illustrates the difference using CC13-measured data. In Figs. 6(a)-6(b), the combined ANN was trained for two modalities combined (6 MV FF and 6 MV FFF). Compared with separate ANNs, the combined ANN slightly overestimated or underestimated the penumbra. However, the clinical impact should be insignificant given that the magnitude of difference is negligible. In Fig. 6(c), the combined ANN was trained for two energies combined (6 MV FF and 10 MV FF). In this example, the separate ANN and the combined ANN had nearly identical performance. In Fig. 6(  Artificial neural networks excel at a variety of tasks, thanks to the representation power associated with the hidden neurons in the hidden layers, which enables well-trained ANNs to closely approximate any continuous functions. We have shown that, in the deconvolution problem, the desired outcome ("true" beam profile) is essentially a function of the measured beam profile. 10 Therefore, we can expect that ANN is suitable for the deconvolution problem even the function involves a complex, not well-defined deconvolution kernel.
The successful training of an ANN relies on the abundancy of training data, especially for fully connected, large-scale ANN. 22 In this work, the three-layer ANN has 15 input nodes and 5 hidden nodes, totaling 86 parameters (80 weightings and 6 biases). We used only 12 beam profiles to train the ANN and found that it generalized well on the unseen test data. The success could be attributed to the relatively simple architecture of the ANN and the similarity in the data. For ICmeasured data (input to the ANN) and diode-measured data (desired output of the ANN), the difference only lies in the penumbra area as characterized using PWD. For the 6 MV FF beam data measured with the same IC, the mean PWDs were within 0.3 mm across all the studied field sizes (ref Table 1 To our knowledge, this is the first report studying the mitigation of VAE in FFF photon beam profiles. The unique shape of unflattened beam profiles presents challenges to analytical deconvolution methods. These methods use analytical functions such as the difference of error functions to fit the beam profiles, 4,15,30 which is feasible for flattened beam profiles but challenging for unflattened beam profiles. However, it does not pose a problem for our ANNbased method. It learns how to mitigate the difference between ICmeasured data and diode-measured data, which occurs mainly in the penumbra area even for unflattened beams. Both separate and combined ANNs achieved excellent results for FFF beams. Not only was the penumbra area restored, the unflattened shape of the beam profiles was also well preserved.

| CONCLUSION S
We evaluated the robustness of an ANN-based photon beam deconvolution method when applied to photon beams of different energies, of different modalities, and measured with ICs of various sizes.
F I G . 6. Comparison between separate artificial neural networks (ANNs) and combined ANNs. The combined ANN was trained by combining data from 6 MV FF and 6 MV FFF in (a) and (b), combining data from 6 MV FF and 10 MV FF in (c), and combining data from 6 MV FF, 6 MV FFF and 10 MV FF in (d).
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For each IC, excellent results were achieved with ANNs separately trained for each energy and modality combination. Clinically acceptable results were also achieved with ANNs trained by combining both beam energies and modalities. Therefore, for a given IC, an ICspecific, combined ANN is sufficient for clinical use as long as the training data includes beam profiles from each energy and modality.

CONF LICT OF I NTEREST
The authors have no conflicts to disclose.