Boron neutron capture therapy (BNCT) is expected to treat refractory cancers, including glioblastoma [1]. Epithermal neutrons are used as a source in BNCT, and their energy range is very wide (0.5 eV–10 keV). Evaluating the neutron energy spectrum is crucial because the biological effectiveness of the neutron on human tissue depends on its energy [2].

Our research group has attempted to develop a new type of spectrometer using a ³He position-sensitive proportional counter [3]. Fig. 1 shows our position-sensitive proportional counter. The signals regarding the detected position can be obtained in this device [4]. Additionally, ³He demonstrates a high nuclear-reaction cross-section in the epithermal neutrons [5], and the cross-section decreases as neutron energy increases (Fig. 2). Lower-energy neutrons in the epithermal range react with ³He at the shallow location from the neutron entrance surface of the detector, whereas those of higher energies react at the deeper location, due to these characteristics, assuming that the detector is placed parallel to the direction of incident neutrons. Further, a specific detector response can be obtained for each neutron energy. The obtained signals related to the detector’s depth (defined as depth distribution {y}) are then converted to the energy spectrum by the spectrum unfolding method [6]. The validation experiments of this spectrometer have been conducted [3, 7]. A depth distribution was obtained using an experimental system, and the obtained depth distribution was converted to the energy spectrum. The estimated value and the one calculated by simulation were then compared. Fig. 3 shows the results of the experiment [7], and discrepancies were found between the two. The response function in our research group was expected to be related to these discrepancies because it could not be evaluated precisely (Fig. 4). A previous study has written the following Equation (1) [3]:

where R(E,r)dr is the detector response function at energy E within the interval from r to r+dr, and ∑(E) is the macroscopic cross section at E. However, neutrons may also enter the detector from the side surface. This response function could not consider such neutrons. Hence, to solve this issue, evaluating the response function by Monte Carlo simulation is indicated as follows. The spectrometer is defined as not only the detector but also the shield that surrounds the detector. Here, the availability of this method is described.

The response function of this spectrometer was assessed by the ³He(n,p)³H reaction rate for each monochromatic neutron in the epithermal range. The F4 tally in Monte Carlo N-particle (MCNP) 5 was used for calculation [8], and JENDL-4.0 [5] was utilized for the nuclear data. The neutron energy was divided into 44 equal lethargy (10^0.1) groups in energies from 0.5 eV to 10 keV. The spectrometer was separated axially into 40 groups for the length of the detector, 40 cm, by 1 cm intervals, to estimate the response function (energy dependent {y}) of the present position-sensitive proportional counter. Fig. 5 shows the experimental system of the validation experiment. A 14 MeV neutron source in the OKTAVIAN (the facility name of Osaka University 14 MeV Neutron Source Facility) was assumed as a neutron source, the intensity of which is 10¹⁰ n/s [9]. The epithermal column was introduced to moderate the neutrons to the epithermal region. It was structured according to a previous study [7]. Concrete and polyethylene bricks were introduced around the detector to reduce the neutron incidence from the side surface. Such neutrons may affect the shape of the response function, which cannot be distinguished and extracted experimentally.

We need to convert the energy spectrum from the measured detection depth distribution by solving the following matrix equation (Equation (2)). The process is called spectrum unfolding:

where y_i denotes the i^th depth value for {y} (0≤i≤40), x_j indicates the j^th value of the energy spectrum (0≤j≤44) and r_i,j represents the normalized response function value for row i and column j. This study used the Bayesian estimation proposed by Iwasaki et al. [10] to solve the Equation (3). Bayesian estimation is based on Bayes’ Theorem [11] regarding conditional probability. Here, the estimated value of the energy spectrum is obtained using the following Equation (3) [12]:

where estj(c+1) denotes the j^th value of the energy spectrum when Bayesian estimation is repeated c times, y′_i indicates one of the values of normalized {y}, and r′_i,j represents normalized r_i,j. These values were normalized from Equation (2) because the Bayesian estimation is a theory of probability. The estimation is repeated until the estimates converge to provide the estimate of the energy spectrum. A (44,1) matrix was introduced for the initial estimated energy spectrum values, with all element values normalized by 1/44. The true spectrum is obtained by correcting for normalization effects.

This study evaluated the r_i,j, assuming the epithermal neutron as the source of the epithermal column. The energy spectrum at the source was then estimated by ‘{y}.’

Several numerical experiments were conducted after evaluating the r_i,j to investigate the effectiveness of this method. These experiments converted several candidates for {y} to the posterior distribution by Bayesian estimation to confirm the congruence of these distributions with the target value. These values were compared with the calculated energy spectrum of the source (defined as x_target, shown in Fig. 6). The following five values were the input for {y}:

(1) is the ideal case, if {y} obeys the Equation (1). As for (2)–(4), uncertainty is added for the evaluation of counting uncertainty. In general, a counting error was observed in measured {y} and evaluating the effect of uncertainty is important. Therefore, the error tolerance can be evaluated in this validation experiment by adding uncertainty to case (1) and conducting the estimation. (2) indicates that {y} has a counting error of 1%. Finally, simulating {y} by calculating in MCNP and the estimation was conducted.

Figs. 7 and 8 illustrate the evaluated response function and the results of Bayesian estimation (defined as x_pred), respectively. From Fig. 8, estimated values generally agree with the target value in the range from 0.5 eV to 4 keV although the results do not agree in the higher energy region. Additionally, the discrepancies between x_pred and x_target became great as the added error was large. We introduced the mean absolute percentage error (MAPE) to evaluate the performances of the prediction in the range from 0.5 eV to 4 keV. MAPE is a function that assesses the error rate whose formula is detailed in Equation (4). MAPE is close to 0 if the prediction accuracy is:

Table 1 shows the values of MAPE in each case. The value of MAPE increases alongside the error added. This information contributes to the determination of the number of counts required for the experiment and the duration of the experiment because the added uncertainty corresponds to counting uncertainty.

Several things should be considered to improve the discrepancies.

First, the effects of uncertainty need to be evaluated accurately. The calculated r_i,j exhibited some uncertainty in MCNP. This r_i,j is used repeatedly in this estimation; thus, each iteration increases the effect of the error. This could be the cause of the discrepancy between x_target and x_pred in case (1). Therefore, we should establish a more precise response function by increasing, the number of particles simulated in MCNP.

The same applies to counting uncertainty: the larger the error, the greater the effect of repetition on the estimate. Hence, uncertainty in the counting rate should be as small as possible. In particular, irradiating neutrons for a long time to obtain many counting rates can be effective. Second, notably, the shapes of the response functions in the higher energy region are almost the same. The more the energy increases, the smaller the slope of the r_i,j. These results reveal the poor energy resolution in a higher energy range of this spectrometer and that this spectrometer cannot discriminate higher energy neutrons. We should think of other approaches to improve the response function in the higher energy region (>4 keV), particularly, using a longer detector, increasing the ³He pressure, etc.

We have investigated the response function of the low-energy neutron spectrometer for epithermal energy region by the Monte Carlo simulation. Initially, the response function was assessed even considering the spectrometer shield. Numerical experiments were then performed, considering the measuring uncertainty to confirm the availability of this response function in practical use. The energy spectrum was estimated by the Bayesian estimation. These results indicated that the feasibility of this approach was confirmed in the energy range from 0.5 eV to 4 keV. Additionally, from the MAPE values, the influence of uncertainties produced during the calculation of the response function and uncertainty due to counting was considered. This information is useful to figure out the need for the number of counts in the experiment and the duration of the experiment. The causes of the discrepancies include poor response functions and low-energy resolution in the high-energy region in addition to counting and response function calculation uncertainty. Therefore, other approaches are crucial to represent well, particularly, using a longer detector or increasing the pressure of the detector.

In the future, we will conduct an irradiation experiment to verify whether the measured energy spectrum agrees with the numerical calculation result by the Monte Carlo simulation. More accurate production of the response function and the experimental system will be produced to suppress the neutron incidents from the side surface and obtain sufficient counts based on the results of numerical experiments to conduct this experiment accurately.