This paper investigates technique for solving spectrometry inverse problem the neural network as method for reconstruction of electron beam spectrum using depth-charge curve. The inverse problem turned into multivariable optimization and the form of spectrum is based on proposed three-parameter model. Radial basis function network calculates the parameters of this model. We developed computational experiment using Monte-Carlo technique to evaluate strengths and weaknesses of proposed approach and compare neural networks with conventional data evaluation methods.
One of the main characteristic of the irradiation processes is an energy of beam. This parameter in uences on absorbed dose in target. Therefore, standards for radiation technologies [1, 2] predetermine the upper bound of beam energy to prevent ionization of the object under irradiation. Because of accelerator features, electrons in beam have di erent energy. Thus, the beam energy represented by some function, which shows relations between particles number and their energies. This function called beam spectrum. In practice at least three parameters de ne the spectrum: average (Eav) and probably (Ep) energies and full width on half maximum (Ew). In order to measure beam energy dosimetric wedge and stack are widely used in centers of radiation technologies. These devices allow to determine only average and probable energies of beam [1{6]. Of course, these two parameters does not allow to reconstruct full energy distribution. Thereby developing of new instruments and methods of dosimetric measurements is actual problem.
Mentioned devices intend to measure distributions of absorbed dose or charge [5, 6]. The measured depth-dose (depth-charge) curves relate to beam spectrum through Fredholm integral equation and nding exact spectrum is an ill-posed inverse problem [7]. This means that evaluated spectrum obtained by conventional mathematical methods can di er with true energy distribution. There are, for example, method of least squares (MLS) or method of Tikhonov regularization (MTR). Above all, important disadvantage of the MLS and MTR is impossibility to include additional solution conditions, for example, correlations between parameters, positivity and other. This lack can bring to violation of conditions, given by physical lows. It should be mention that in common case the neural networks (NN) solve approximation tasks and nd solutions based on existing precedents after supervised training [8{12]. So the one of the way of improving dosimetry e ectiveness is developing of methods for measurement results evaluation based on neural networks. In order to apply NN for dosimetric data processing it is necessary to solve next problems: select networks topology, obtaining data for NN training, developing methods for data preprocessing and interpretation, system for evaluation network e ectiveness.
So current research is about feasibility of using neural networks for developing system of measurement results evaluation for beam spectrum monitoring of industrial electron accelerators. We will discuss mathematical model of measurement process, which was built in order to compile training set for network learning procedure (Section 2). Section 3 describes methods under investigation. In section 4, we will show approach for methods evaluation, which contains computational experiment and comparison criteria. In section 5 given comparison results of neural networks and conventional methods testing. 2
Physical process and mathematical model In order to calculate radiation energy, it is a common practice in eld of radiation technologies to measure depth-dose curve by dosimetric wedge. However, the works of recent years propose new devices based on measurement of depthcharge curve that can realize on-line energy monitoring [3{6]. In this work, we will consider mathematical abstraction of these devices and will build method for beam spectrum controlling using depth-charge curve. 2.1
Device [5] consists of two plates only and intend to calculate probable energy as a value which linearly depends on charge in rst plate to sum charge ratio. Measurer in [6] contains 10 absorbers. But in order to simplify average energy calculation the plates were combined and authors use similar to [5] dependency.
Fig. 1 shows principal schema of measurer. Dosimetric stack consists of set of plates - absorbers. The absorbers material is often aluminum, because of radiation ruggedness. The electron beam falls on the sequence of plates. Electrons stop at di erent depths depending on their energy. Thus, absorbers collect some charge which can be measured by current integrators connected to corresponding plate. The set of measured values represents the depth-charge curve.
Mathematical model of the measurement process is based on a semi-empirical
model of the depth-charge distribution for monoenergetic electrons and model
of charge measurement uncertainty. Direct problem describes relation between
known beam spectrum and depth-charge curve through equation:
(
EL f (x) =
Q(x; E)y(E) dE; x 2 [0; xR]; where y(E) - describes relation between number of particles and their energy (electrons spectrum), f (x) - describes depth distribution of charge, xR measurer full width, [EL; ER] - operating energy range of accelerator, integral kernel Q(x; E) corresponds to radiation type ( , , ) and measurer internal characteristics (including absorbers material). Works [13, 14] describe appropriate relations for monoenergetic beam and depth-charge curve.
In the research we neglect charge leakage and suppose that distance between absorbers is neglectfully small. It means that each particle from initial beam can stops in absorbers and pass through current integrator or can pass through whole device with no impact in depth-charge curve.
The measurement results of charge distribution in absorbers is set f = ff1; f2; : : : ; fng (see Fig. 1), where n - number of absorbers, fi - integral of f (x) over the depth for i-th absorber: xi+ x ER
Z Z xi
EL fi =
Q(x; E)y(E) dE dx; fi = 2x Pj pjE yj [Q(xk + (i
1) x; Ej )+ +Q(xk + i x; Ej )]; where
x - absorbers width. Equation (
EE=2 j =oth0e_rwj i=sem :
It's obvious that complexity of spectrum reconstruction grows with increasing of m (dimension of vector y). In order to reduce problem the we used parameterization of y(E). As mentioned above, the general practice is denoting spectrum by parameters: Ep, Eav, Ew. Therefore, it is reasonable to make model of the beam spectrum, which use three parameters. 2.2
{ Emax { maximal particles energy in the beam, { Ep { most probable energy, { Es { energy of 10 times decreasing of the intensity compared to Ep electrons along left slope.
In the future discussion the will denotes set of spectrum parameters, i.e. = fEs; Ep; Emaxg.
Parameters of the model correspond to characteristics of beam used in practice according to:
Ep = Ep; Ew = llnn00::51 (Ep Eav = Es + ln
Es) + Emax Ep
2
Emax Ep + 0:45(Es Ep)
4 ln0:1
and mathematical expression for spectrum is:
(
h 0.5h 0.1h
Es
Ep
Emax
; k2 = hEmax
Ep ; where E 2 [0; 1], h = y(Ep) - maximum of function y(E) and was obtained with supposition of
Emax Z Es
y(E)dE = 1: Therefore, maximum of energy distribution is: h = y(Ep) = [0:9
Es Ep + 0:5(Emax ln(0:1)
Ep)] 1:
It should be mention, that in accordance to physical laws the function y(E) is positive or equal zero for all accepted E and parameters correlates as: 0 < Es < Ep
2.3
In the real experiment measured fi di er with its real value. This error grounded on weaknesses of measurer and external in uence. We will mark set of true values of f (x) as f , and use f~ to mark set of values complemented with measurement uncertainty:
f~ = (1 + " )f;
where " - value of standard deviation of measurement error, - random variable
distributed in accordance to standard normal distribution:
(
We will use similar signature to denote evaluated parameters ~ , E~s, E~p, and E~max reconstructed spectrum y~ instead their true values without tilde. 3 3.1
Methods for spectrum reconstruction
In order to apply NN for solving spectrometry inverse problem reconstruction of spectrum can be represented as multivariable function tting. Suppose that function implements measurement process of depth-charge curve, i.e. f~ = ( ). Therefore, inverse function ~ = 1(f~) realizes transformation from depthcharge curve to beam spectrum. So approximation of 1 can be used to get spectrum using depth-charge curve. In the work we used general regression neural network (GRNN) [15] to t 1. This network needs set of precedence for supervised learning. Consider algorithm of training set creation.
Implemented measurement models allow to create pairs s = (f~; ), where f~
calculates from parameters set . The collection of s is based on di erent and
represents a reference points for 1 tting:
(
Ay~ f~ ; where k k - Euclidian norm. Method of Tikhonov regularization expands MLS through additional stabilizer function: ~ MT R = arg min
Ay~
f~ +
ky~ k ;
where > 0 - regularization parameter. It should be remind that using of
mathematical model of measurement process gives true values of electrons spectrum.
So can be calculated from [7]:
= arg min ky y~ k : (
In the work we applied Nelder-Mid simplex method numerical solution of
(
Algorithm for evaluation methods preparing and testing 4.1
Implemented models of spectrum, measurement process and methods for data evaluation compose computational experiment (Fig. 3 shows sequential diagram). The experiment aim is comparison of methods for spectrum reconstruction. The approach which was used to build experiment uses Monte-Carlo technique: system generate measurement results, each methods reconstruct spectra using samples of depth-charge cure, system calculates statistical characteristics of reconstruction error. Computational experiment consists of three steps: preparation, main part (loop Common) and results interpretation.
Preparation of an experiment includes setting parameters of models and methods. Main part is a series of subexperiments with varied measurement uncertainty ". Each of them contains two steps: training of NN and selected methods comparison. Both processes include generation of pairs s = (f~; ) which is based on prede ned set of . But these sets of reference spectrum are di erent. Testing procedure (loop Data Evaluation) repeats sampling of f~, evaluates appropriate by each method and collects reconstruction error based on truth and calculated spectra based on proposed set of indicators. The results processing step aims to build relationships that show correlations between accuracy of spectrum reconstruction and varied error of measurement.
Software for experiment execution implemented in MATLAB with Neural Network Toolbox (function newgrnn as NN), Optimization Toolbox (function fminsearch as MLS and MTR). In order to speed up computational experiment, software was executed on high performance cluster [16] with Distributed Computing Toolbox. 4.2
In order to assess the e ectiveness of methods for reconstruction of beam energy
characteristics we suggested set of indicators. The set consists of the standard
statistical estimates of data evaluation error and indicator of methods reliability.
There are two indicators type: mismatch along energy axis (estimate shift of
reconstructed spectrum along horizontal axis) and common indicator. Consider
details of each indicators.
1. Mismatch along energy axis. Average M (r) and standard deviation r of
distance along intensity axis between reconstructed and true spectra are
based on:
2. Common characteristics. Probability of method failure P . We suppose that
the method failure is a case when applying mathematical methods leads to
impossible (due to physical lows) solution, i.e. the solution brakes condition
(
In order to evaluate methods e ectiveness with suggested indicators we made computational experiment with parameters shown in Table 1.
The training and testing sets include reference spectra with parameters: 1 n r = (y y~)2; (19)
As shown in Table 1 the device consists of 15 absorbers. This con guration is chosen based on previous research [18] which was aimed to nd optimal discretization step of depth-charge curve for spectrum reconstruction by NN. It should be mention that in works [17, 18] sets for methods testing and preparation based on reference spectra with xed Ew parameter and same maximum h = 1. Therefore, seeking of optimal absorbers width is open for future research. M(r) σr 0.5
As an opposite to conventional methods, the solutions obtained by NN have smaller dependency between evaluation error and input data uncertainty. Furthermore as shown on Fig. 5a, 5b the GRNN evaluates spectra more accurate than MLS and MTR for measurement uncertainty more than 5-7%. The main advantage of NN method is that GRNN reconstruct beam spectrum parameters with no failures (see Fig 5c), i.e. all obtained solutions are compliance with physical lows. 6
The work shows GRNN method e ectiveness for solving inverse dosimetry problem of electron spectrum reconstruction using depth-charge curve. The main advantages of proposed technique compared to conventional methods is allowance to apply additional solutions conditions. It lids to getting robust evaluation method. As shown in the work methods based on NN can be used for building on-line energy monitoring systems in centers of radiation technologies.
Furthermore, we proposed comparison approach based on Monte-Carlo technique and set of e ectiveness indicators. The approach allows testing di erent types of evaluation methods and can be used for methods optimization in order to select or apply technique for industrial problems solving.