Efficient Domain Decomposition for a Neural Network Learning Algorithm, Used for the Dose Evaluation in External Radiotherapy Marc Sauget1 , R´emy Laurent1 , Julien Henriet1 , Michel Salomon2 , R´egine Gschwind1 , Sylvain Contassot-Vivier3, Libor Makovicka1, and Charles Soussen4 1
Femto-ST, ENISYS/IRMA, F-25210 Montb´eliard, France
[email protected] 2 University of Franche-Comt´e, LIFC/AND, F-90000 Belfort, France 3 University of Nancy, LORIA, Campus Scientifique, BP 239 F-54506 Vandoeuvre-l`es-Nancy Cedex France 4 Facult´e des sciences et techniques, CRAN, F-54506 Vandoeuvre-l`es-Nancy Cedex France
Abstract. The purpose of this work is to further study the relevance of accelerating the Monte Carlo calculations for the gamma rays external radiotherapy through feed-forward neural networks. We have previously presented a parallel incremental algorithm that builds neural networks of reduced size, while providing high quality approximations of the dose deposit. Our parallel algorithm consists in a regular decomposition of the initial learning dataset (also called learning domain) in as much subsets as available processors. However, the initial learning set presents heterogeneous signal complexities and consequently, the learning times of regular subsets are very different. This paper presents an efficient learning domain decomposition which balances the signal complexities across the processors. As will be shown, the resulting irregular decomposition allows for important gains in learning time of the global network. Keywords: Pre-clinical studies, Doses Distributions, Neural Networks, Learning algorithms, External radiotherapy.
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Introduction
The work presented in this paper takes place in a multi-disciplinary project called Neurad [2], involving medical physicists and computer scientists whose goal is to enhance the treatment planning of cancerous tumors by external radiotherapy. In our previous works [4,9], we have proposed an original approach to solve scientific problems whose accurate modeling and/or analytical description are difficult. That method is based on the collaboration of computational codes and neural networks used as universal interpolator. Thanks to that method, the K. Diamantaras, W. Duch, L.S. Iliadis (Eds.): ICANN 2010, Part I, LNCS 6352, pp. 261–266, 2010. c Springer-Verlag Berlin Heidelberg 2010
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Neurad software provides a fast and accurate evaluation of radiation doses in any given environment (possibly inhomogeneous) for given irradiation parameters. More precisely, the Neurad project proposes a new mechanism to simulate the dose deposit during a clinical irradiation.The neural network is used to obtain the deposit in a homogeneous environment whereas the specific algorithm expresses the rules to manage any inhomogeneous environment. The feasibility of this original project has been clearly established in [3]. It was found that our approach results in an accuracy similar to the Monte Carlo one and suppresses the time constraints for the external radiotherapy evaluation. The accuracy of the neural network is a crucial issue; therefore it is the subject of many research works on neural network learning algorithms. To optimize the learning step, we have proposed to learn regular subdomains of the global learning dataset. Although satisfactory results were obtained, we could observe different level of accuracy among the subdomains. Thus, in this paper we describe a new decomposition strategy of the global learning set to solve this problem.
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Quantification of the Complexity of the Learning Domain
The goal of this work is to propose a solution allowing a decomposition of a data set in subdomains taking into account the complexity of the data to learn. Complexity influences the time and the accuracy of the learning step for a neural network. Therefore a work about the complexity management is necessary to ensure a homogeneous learning in both time and accuracy for all subdomains composing the global data set. To identify the complexity of our learning domain, we have chosen to study the variance between the data. The interest of this technique, used in many contexts, is to establish precisely the local and the global complexities of a domain. In our case, we have evaluated the learning domain complexity using the following variation function for the following function: lCi,j =
i+r
j+r
x=i−r,x=i y=j−r,x=j
|f (x, y) − f (i, j)| −−−→ −−→ ||(x, y) − (i, j)||
(1)
So, in order to evaluate the local complexity lCi,j at spatial point (i, j), we use the variations between that point and a given set of its neighbors (x, y), each of them being distance weighted. The interest of this function is to take into account all informations of complexities composing the dataset after a sampling of the dose values. The global complexity of a global learning domain is described by the sum of all the local complexities (in absolute value). The r parameter allows to determine the size of the neighborhood taken into account to evaluate the local complexity of a point. We do not use the average of local complexities, because the number of values in a subdomain has a direct impact on the final accuracy of the learning, and even more on the learning time.
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Decomposition Strategy
In our previous works [1] we have used a classical dimensional strategy to subdivide the learning domain. This first choice was only motivated by its simplicity of use and the regularity of the resulting subdomains: as it can be seen in our previous results, a regular decomposition only based on the dimension characteristic does not provide a set of subdomains with a constant degree of complexity. In spite of that, we propose to use another decomposition strategy based on a tree structure. This strategy of decomposition, denoted URB (Unbalanced Recursive Bisection) in the literature, was proposed by Jones and Plassman [7,8]. The URB strategy divides each dimension alternatively in two parts to obtain two subdomains having approximately the same complexity estimation. The two parts are then further divided by applying the same process recursively. Finally, this algorithm provides a simple geometric decomposition for a given number of processors. One of its advantages is to provide rectangular sub-parts which do not imply any sensible management overhead. In our case, dividing the dimension alternatively gives quickly subdomains which are spatially unbalanced. To further highlight this difference, we facilitate the comparison between the complexities of the subdomains by rescaling them between 0 and 100, so that 100 corresponds to the highest complexity. And again, we can see a very important difference between the two strategies. On the one hand the regular strategy exhibits a wide range of complexities: from 4.9 to 100; on the other hand the URB strategy presents a limited range: from 30.9 to 35.4. Consequently, the proposed local correlation based on irregular domain decomposition allows us to build subdomains with homogeneous complexities. The remainder of this article shows why this characteristic is important in the context of neural network learning.
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Experimental Results
In this section, both quality and performance of our algorithm are experimentally evaluated. Our algorithm has been implemented in standard C++ with OpenMPI for the parallel part, and all computations were performed at Mesocentre de calcul de Franche-Comte machine. 4.1
Data Description
In the context of external radiotherapy, an essential parameter is the width of the beam. The radiation result is directly dependant on this parameter. Indeed, if the beam has a small width, it cannot reach the electronic balance and does not present a tray as large as could be seen in Figure 1. The main constraint in the scope of our work is the limited tolerance about the final learning error. Indeed, the neural network described here represents one tool in a very complex chain of treatments for the global evaluation of the dose deposit in complex environments. The use of the neural network is central in our overall project and
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Fig. 1. Dose deposits at the depth of 60 mm in water and in TA6V4
a too large error at this stage could imply that our final solution may be out of the medical error tolerance. For these experiments, we have trained our network using a set of data provided by Monte Carlo simulations [5]. These data represent the doses deposit for different beams. We have chosen to work specifically on small width beams to study the different behaviors of the learning process for data with and without electronic stability. The dataset is built from the result of irradiation in water and TA6V4 for three widths (2, 15 and 30 mm). The water is the principal element composing the human body (similar to the human tissue composition and used for medicinal accelerator calibrations) and the TA6V4 is a common compound used in prosthesis [6]. Since the dataset is generated using a grid of 120x100 points and considering two materials with three beam widths for each one, it is composed of 120x100x3x2 elements. Each element has seven input components: the position in the three spatial dimensions; the beam width (min and max information); the environment density and the distance between it and the irradiation source. We propose to test different decomposition configurations to quantify the accuracy of the learning and to determine the limits of our solution. 4.2
Decomposition Results
First, we will study how the complexities are balanced by comparing decompositions induced by our URB based algorithm to regular ones. Therefore Table 1 describes the minimum, maximum and means complexity value obtained for the different subdomains with increasing levels of partitioning. As explained previously, the global data set is composed of irradiation results corresponding to different configurations (two materials, three beam widths). To evaluate the complexity value of a subdomain, we have chosen to use only the sum of its local complexities. The objective is to get a finer partitioning in the areas of larger complexities. The values in Table 1 show clearly the good behavior of our approach: the complexities are better balanced with our algorithm than with the regular partitioning.
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Learning Results
As shown in the previous subsection, our approach is able to produce dataset partitions with less heterogeneity across the complexity values. Similar complexity values should mean similar learning tasks and thus similar learning times. From Table 1, which presents also the training times , we can observe that these times are still not very homogeneous. This can be explained by the complexity of the learning domain which is evaluated for each curve of dose deposit independently. Indeed, it is not realistic to find any correlation parameter between the different simulation results. The choice to use the sum of the local complexities to evaluate the global complexity does take into account all situations. This explains why all the learning times gained for URB based decomposition are not equal. Fortunately, in all cases the mean learning times resulting from our decomposition clearly outperform the ones obtained with the regular decomposition. Table 1. Global evaluation of our solution with a selection and the full dataset
subdomaine 4 8 16 24
4.4
Initial test Full dataset Decomposition learning times Qualitative evaluation URB REG URB REG URB REG SD1 SD SD Means2 SD Means Means Means 1.0 87.9 270 424 1122 739 * * 3.5 73.3 291 348 2399 880 1604 11498 3.1 60.7 72 26 1225 273 1098 8632 8.7 47.6 125 23 355 100 113 5259
Qualitative Evaluation
With this test, we want to control the quality of the learning and the accuracy of interpolation induced by of our neural networks. For this, we have enlarge the training set by adding two more beam widths (9 and 23 mm) for both water and TA6V4. The learning dataset is thus composed of 120x100x5x2 points. Globally, we can say that our optimized URB based decomposition approach gives more precise results. Furthermore, and more important, we can note very shorter computing times (Table 1) with our approach. Those overall good performances (accuracy and learning times) show the relevance of our work, and are crucial for the ongoing of the whole medical project.
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Conclusion and Future Work
A strategy for the domain decomposition has been presented. Its principle is based on a domain decomposition on the input parameters of the learning data set taking into account the complexity of the dose deposit signal in order to 1 2
Standard Deviation (in %). Means of the learning time (in seconds).
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balance the learning times and to improve the global accuracy. Qualitative and quantitative evaluations of the algorithm have been performed experimentally on real data sets. They confirm the good behavior of our algorithm in terms of performance and quality. The small differences between the subdomain learning times show the improvement of our solution in a realistic context of use. In the following of the Neurad project, it should be interesting to add another important feature to our learning process which is a dynamic load balancing who could have some interests in the context of very large learning. And with this efficient parallel learning algorithm we have the capabilities to learn all the data necessary to the medicinal context. Acknowledgments. The authors thanks the LCC (Ligue Contre le Cancer) for the financial support, the Franche-Comt´e region and the CAPM (Communaut´e d’Agglom´eration du Pays de Montb´eliard).
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