=Paper= {{Paper |id=None |storemode=property |title=Use of Neural Networks for Monitoring Beam Spectrum of Industrial Electron Accelerators |pdfUrl=https://ceur-ws.org/Vol-1000/ICTERI-2013-p-118-129.pdf |volume=Vol-1000 |dblpUrl=https://dblp.org/rec/conf/icteri/BaievLD13 }} ==Use of Neural Networks for Monitoring Beam Spectrum of Industrial Electron Accelerators== https://ceur-ws.org/Vol-1000/ICTERI-2013-p-118-129.pdf

Use of Neural Networks for Monitoring Beam
Spectrum of Industrial Electron Accelerators

Oleksandr Baiev, Valentine Lazurik and Ievgen Didenko

School of Computer Science, V. N. Karazin Kharkiv National University,
4, Svobody Sqr., 61022, Kharkiv, Ukraine

oleksandr.baiev@gmail.com, lazurik@hotmail.com,
ievgen.v.didenko@gmail.com

Abstract. 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 calcu-
lates the parameters of this model. We developed computational experi-
ment using Monte-Carlo technique to evaluate strengths and weaknesses
of proposed approach and compare neural networks with conventional
data evaluation methods.

Keywords. Neural nets, Inverse problems, Monte Carlo, Radiation tech-
nologies, Depth-charge curve

Key terms. ComputerSimulation, Methodology, MachineIntelligence

1 Introduction
One of the main characteristic of the irradiation processes is an energy of beam.
This parameter influences 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 fea-
tures, electrons in beam have different energy. Thus, the beam energy repre-
sented by some function, which shows relations between particles number and
their energies. This function called beam spectrum. In practice at least three
parameters define the spectrum: average (Eav ) and probably (Ep ) energies and
full width on half maximum (Ew ). In order to measure beam energy dosimet-
ric 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 dis-
tribution. 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
Use of Neural Networks for Monitoring Beam Spectrum . . . 119

through Fredholm integral equation and finding exact spectrum is an ill-posed
inverse problem [7]. This means that evaluated spectrum obtained by conven-
tional mathematical methods can differ with true energy distribution. There
are, for example, method of least squares (MLS) or method of Tikhonov regu-
larization (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 find solutions based
on existing precedents after supervised training [8–12]. So the one of the way
of improving dosimetry effectiveness 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 effectiveness.
So current research is about feasibility of using neural networks for devel-
oping system of measurement results evaluation for beam spectrum monitoring
of industrial electron accelerators. We will discuss mathematical model of mea-
surement 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 com-
putational 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 field of radiation
technologies to measure depth-dose curve by dosimetric wedge. However, the
works of recent years propose new devices based on measurement of depth-
charge 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 Devices

Device [5] consists of two plates only and intend to calculate probable energy
as a value which linearly depends on charge in first 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 different 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.
120 O. Baiev, V. Lazurik and Ie. Didenko

f1 f2 f3 f4 f5 fn

Fig. 1. Common schema of stack for depth-charge measurement

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:

ZER
f (x) = Q(x, E)y(E) dE, x ∈ [0, xR ], (1)
EL

where y(E) - describes relation between number of particles and their energy
(electrons spectrum), f (x) - describes depth distribution of charge, xR mea-
surer full width, [EL , ER ] - operating energy range of accelerator, integral kernel
Q(x, E) corresponds to radiation type (α, β, γ) and measurer internal char-
acteristics (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 =
{f1 , f2 , . . . , fn } (see Fig. 1), where n - number of absorbers, fi - integral of
f (x) over the depth for i-th absorber:
ER
xiZ+∆xZ

fi = Q(x, E)y(E) dE dx, (2)
xi EL

where ∆x - absorbers width. Equation (2) can be approximated as:

fi = ∆x
P E
2 pj yj [Q(xk + (i − 1)∆x, Ej )+
j (3)
+Q(xk + i∆x, Ej )],
where i = 1, n, j = 0, m, m = (ER − EL )/∆E - number of steps of function y(E)
discretization over energy axis, ∆E - step of spectrum energy discretization, yj
- value of y(E) in approximation nodes, coefficient pE
j defines method and step
Use of Neural Networks for Monitoring Beam Spectrum . . . 121

of function y(E) approximation. Then the measurement process can be shown
as system of linear equations:

 Ay = f⇔    
a1,1 a1,2 · · · a1,n y0 f1
 a2,1 a2,2 · · · a2,n   y1   f2 
  , (4)
.  .  =  . 
 
 .. .. . .
 . . . ..   ..   .. 
am,1 am,2 · · · am,n ym fn
where elements of matrix A are:

ai,j = ∆x E
2 pj [Q(xk + (i − 1)∆x, Ej )+ (5)
+Q(xk + i∆x, Ej )].

In order to approximate y(E) by method of trapezoids, coefficients pE
j are:

∆E/2 j = 0 ∨ j = m
pE
j = . (6)
∆E otherwise
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 parameter-
ization 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 Model of electrons spectrum

Fig. 2 shows geometrical interpretation of electrons spectrum model considered
in the present work. The graph of spectrum consists of two part: left exponential
and right linear slopes. The parameters of this model are:

– 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 Π will denotes set of spectrum parameters, i.e.
Π = {Es , Ep , Emax }.
Parameters of the model correspond to characteristics of beam used in prac-
tice according to:

Ep = Ep ,
E −E
Ew = ln0.5 − Es ) + max2 p
ln0.1 (Ep (7)
Emax −Ep 0.45(Es −Ep )
Eav = Es + ln 4 + ln0.1

and mathematical expression for spectrum is:
122 O. Baiev, V. Lazurik and Ie. Didenko

0.5h

0.1h

Es Ep Emax

Fig. 2. Model of electron beam spectrum


 heµ(E−Ep ) , 0 < E ≤ Ep
y(E) = k1 E + k2 , Ep < E ≤ Emax , (8)
0, Emax < E


ln(0.1) h hEmax
µ= , k1 = , k2 = , (9)
Es − Ep Ep − Emax Emax − Ep
where E ∈ [0; ∞], h = y(Ep ) - maximum of function y(E) and was obtained
with supposition of
EZmax

y(E)dE = 1. (10)
Es

Therefore, maximum of energy distribution is:

Es − Ep
h = y(Ep ) = [0.9 + 0.5(Emax − Ep )]−1 . (11)
ln(0.1)
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 ≤ Emax . (12)

2.3 Model of measurement

In the real experiment measured fi differ with its real value. This error grounded
on weaknesses of measurer and external influence. 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 + εξ)f, (13)
where ε - value of standard deviation of measurement error, ξ - random variable
distributed in accordance to standard normal distribution:
Use of Neural Networks for Monitoring Beam Spectrum . . . 123

p
ξ = cos(2πr1 ) −2ln(r2 ), (14)
where r1 , r2 - random variables which ate distributed in accordance with stan-
dard uniform distribution.
We will use similar signature to denote evaluated parameters Π̃, Ẽs , Ẽp , and
Ẽmax reconstructed spectrum ỹ instead their true values without tilde.

3 Methods for spectrum reconstruction

3.1 Neural networks

In order to apply NN for solving spectrometry inverse problem reconstruction of
spectrum can be represented as multivariable function fitting. Suppose that func-
tion φ implements measurement process of depth-charge curve, i.e. f˜ = φ(Π).
Therefore, inverse function Π̃ = φ−1 (f˜) realizes transformation from depth-
charge curve to beam spectrum. So approximation of φ−1 can be used to get
spectrum using depth-charge curve. In the work we used general regression neu-
ral network (GRNN) [15] to fit φ−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˜, Π), where f˜
calculates from parameters set Π. The collection of s is based on different Π and
represents a reference points for φ−1 fitting:

f˜1 f˜2 · · · f˜N

(s1 · · · sN ) = (15)
Π1 Π2 · · · ΠN
where N - number of elements in training set. For future discussion, we will
denote each unique Π in training and testing sets as reference spectrum. Note,
that each values of parameters for all Π from training set was normalized in
accordance to [EL ; ER ] → [0; 1]. Of course, outputs of network were scaled back
during testing.

3.2 Conventional methods

Consider methods, which is traditionally used for measurement results evalua-
tion. The data which were obtained by these methods is a base level to determine
NN effectiveness for solving spectrum reconstruction problem. The method of
least squares calculates parameters Π as:

Π̃M LS = arg min AỹΠ − f˜ , (16)
Π

where k·k - Euclidian norm. Method of Tikhonov regularization expands MLS
through additional stabilizer function:

Π̃M T R = arg min AỹΠ − f˜ + α kỹΠ k , (17)
Π
124 O. Baiev, V. Lazurik and Ie. Didenko

where α > 0 - regularization parameter. It should be remind that using of math-
ematical model of measurement process gives true values of electrons spectrum.
So α can be calculated from [7]:

ky − ỹα k
α∗ = arg min . (18)
α kyk
In the work we applied Nelder-Mid simplex method numerical solution of
(16), (17) and (18).

4 Algorithm for evaluation methods preparing and
testing
4.1 Comparison approach
Implemented models of spectrum, measurement process and methods for data
evaluation compose computational experiment (Fig. 3 shows sequential dia-
gram). The experiment aim is comparison of methods for spectrum reconstruc-
tion. The approach which was used to build experiment uses Monte-Carlo tech-
nique: 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: prepa-
ration, 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 un-
certainty ε. Each of them contains two steps: training of NN and selected meth-
ods comparison. Both processes include generation of pairs s = (f˜, Π) which is
based on predefined set of Π. But these sets of reference spectrum are different.
Testing procedure (loop Data Evaluation) repeats sampling of f˜, evaluates ap-
propriate Π 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 spec-
trum 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 Com-
puting Toolbox.

4.2 Comparison indicators
In order to assess the effectiveness 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.
Use of Neural Networks for Monitoring Beam Spectrum . . . 125

Fig. 3. Sequential diagram of computer experiment

1. Mismatch along energy axis. Average M (r) and standard deviation σr of
distance along intensity axis between reconstructed and true spectra are
based on:
1
r = (y − ỹ)2 ; (19)
n
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
(12). It is obvious that value 1 − P characterize method reliability.

5 Results and discussions

5.1 Parameters of computation experiment

In order to evaluate methods effectiveness with suggested indicators we made
computational experiment with parameters shown in Table 1.
The training and testing sets include reference spectra with parameters:
126 O. Baiev, V. Lazurik and Ie. Didenko

Table 1. Common experiment parameters

Parameter Value
Characteristic for measurement depth-charge curve
Absorbers material Aluminum (Z = 13, Am = 27)
Absorber’s width (∆x) 0.4 g/cm2
Device total width (xR ) 6 g/cm2
Uncertainty (ε) Varied from 0% to 30%, step 1%
[EL ; ER ] [0M eV ; 10.2M eV ]
Ew of reference spectra Randomly from 2% to 10% of Ep
y(E) discretization step (∆E) 0.05M eV
Number of reference spectra 9000 (training) and 41000 (testing)

Ep = r1 , r1 ∼ U [EL , ER ],
Emax = Ep (1 + 2r2 ), r2 ∼ U [0.01, 0.02], (20)
ln0.1
Es = Ep − ln0.5 r3 , r3 ∼ U [0.01, 0.08].

Fig. 4 shows examples of sampled reference spectra. Number of the spectra
for training and testing sets is reduced, but proportion saved. As shown on Fig.
4 and in Table 1 the testing set is bigger than training set. It is necessary to get
appropriate assessment of method based on NN with influence of retraining.

Testing spectra Training spectra
1.8 2.5

1.6

1.4 2

1.2
y(E) − Intensity

y(E) − Intensity

1.5
1

0.8
1
0.6

0.4 0.5
0.2

0 0
5 6 7 8 9 10 5 6 7 8 9 10
E − Energy E − Energy

Fig. 4. Reference spectra for a) NN training and b) methods testing

As shown in Table 1 the device consists of 15 absorbers. This configuration
is chosen based on previous research [18] which was aimed to find optimal dis-
cretization step of depth-charge curve for spectrum reconstruction by NN. It
should be mention that in works [17, 18] sets for methods testing and prepara-
tion based on reference spectra with fixed Ew parameter and same maximum
h = 1. Therefore, seeking of optimal absorbers width is open for future research.
Use of Neural Networks for Monitoring Beam Spectrum . . . 127

M(r) σr Failures
0.25 0.5 0.5

MLS
0.2 0.4 0.4 MTR
GRNN
M(r) 0.15 0.3 0.3

σr

P
0.1 0.2 0.2

0.05 0.1 0.1

0 0 0
0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3
ε ε ε

Fig. 5. Results of methods comparison

5.2 Results and discussion

Fig. 5 contains obtained dependencies, which describe relation between methods
evaluation error and measurement uncertainty. The charts 5a and 5b based on
indicator (19). Chart 5c shows probability of method failure.
For MLS and MTR experiment proves expected results. Methods are sensi-
tive to uncertainty in input data. Fig. 5a and 5b show that error of MLS and
MTR solutions rapidly grows with increasing of ε. With respect to probability
of failure, both methods demonstrated almost equal inefficiency. It can be mean
that stabilizing additions in MTR does not affect to the method reliability. It
should be mention that the reason of MLS and MTR error for ε = 0% is dis-
cretization inaccuracy which appears when transforming integral (1) to system
(4).
As an opposite to conventional methods, the solutions obtained by NN have
smaller dependency between evaluation error and input data uncertainty. Fur-
thermore 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 parame-
ters with no failures (see Fig 5c), i.e. all obtained solutions are compliance with
physical lows.

6 Conclusion

The work shows GRNN method effectiveness for solving inverse dosimetry prob-
lem of electron spectrum reconstruction using depth-charge curve. The main ad-
vantages 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 tech-
nique and set of effectiveness indicators. The approach allows testing different
128 O. Baiev, V. Lazurik and Ie. Didenko

types of evaluation methods and can be used for methods optimization in order
to select or apply technique for industrial problems solving.

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