### 1. INTRODUCTION

### 2. MATERIALS AND METHODS

### 2.1 Attenuation curve measurement

^{6}photons·mm

^{-2}at 1 m from the target on the beam center axis for each pulse of 6 MeV electrons. This means that direct measurement of the pulse height spectrum is almost impossible for high energy X-ray source using ordinary X-ray spectrometry.

*E*) [

_{j}*j*=

*1–M*], through an attenuation material, the thickness of which can be changed gradually as

*t*[

_{i}*i*=

*1–N*]. An experimental arrangement for the attenuation curve measurement is shown in Figure 1. An electron linear accelerator of 6 and 9 MeV electrons switchable (Linatron-M, Varian Medical Systems Inc., Palo Alto, CA), with a pulse width of 4.2

*μ*s, is utilized as an X-ray source. At 1 m from the focal spot, the dose rate is 10 Gy·min

^{-1}for 6 MeV and 30 Gy·min-1 for 9 MeV without a flattening filter in both cases. In the X-ray CT system, there are several hundred of solid state detectors arranged within the 15° fan beam. This enables us to measure multiple attenuation curves within the fan beam at the same time.

*yθ*(

*t*

_{i}) [θ＝0 to ±15°,

*i*=1 to 59 or 30], are obtained for all the detectors at the same time.

*θ*of the X-rays as shown in Fig. 1. The actual path length is used for

*t*in the calculation of energy spectra.

_{i}### 2.2 Spectrum type Bayesian estimation

*φ*(

*E*). This method is based on the Bayesian theorem and it has been applied to radiation measurements. For instance, in energy spectrum measurements of charged particles correction of the energy loss in the sample is carried out using the spectrum type Bayesian estimation method as an unfolding technique [7-9]. Furthermore, in X-ray energy spectrum measurement, we recently investigated an approach utilizing the Bayesian estimation method combined with an attenuation curve measurement using step shaped attenuation material. This method was validated by precise measurement of energy spectrum from a 1 MeV electron accelerator. The specific features of the spectrum type Bayesian estimation method are summarized as follows.

_{j}(1) Its very simple principle makes for easier application compared to other unfolding methods.

(2) It never gives unfolded results having a negative value.

(3) It does not require any constraint like an initial guess [10].

*φ*(

*E*) and measured attenuation result

_{j}*y*(

*t*) for the attenuation curve. It can be described with a response function. The response function

_{i}*consists of*

**H***h*, which includes information on the attenuation coefficient at photon energy

_{ij}*E*, penetration thickness

*t*, and other detector characteristics such as its detection efficiency. This relationship is described as Equation 1.

_{i}*l*+1 th estimation

*est*(

_{j}^{(l+1)}*i*) as,

##### (2)

$$es{t}_{j}^{(l+1)}=\sum _{i=1}^{N}\left(y\right({t}_{i})\times \frac{es{t}_{j}^{\left(l\right)}\times {h}_{ij}}{\sum _{j=1}^{M}es{t}_{j}^{\left(l\right)}\times {h}_{ij}})$$*y*(

*t*), i.e., a revised

_{i}*est*is used as the prior information in the next iteration calculation.

_{j}^{(l)}### 2.3 Response function evaluation

*in eq. 1 is rewritten as*

**H***R*•Ｆ to connect the primary X-ray energy spectrum and the measured pulse height spectrum from preamp board as shown in Fig. 1. Matrix

*R*of

*N*rows and

*M*columns consists of attenuation coefficients varying due to X-ray energy for the column direction and the penetration thickness for the row direction. Matrix

*Ｆ*of

*M*rows and

*M*columns contains photon-to-charge conversion factor for the specific X-ray energy induced the detector.

*μ*for energies

_{j}*E*

_{1},

*E*

_{2},

*E*

_{3},…,

*E*as representative energies for energy bins are given, the X-ray energy spectrum

_{M}*φ*□(

*E*) after penetrating through the attenuation material of thickness

_{j}*t*is taken from the incident X-ray energy spectrum φ(

_{i}*E*) and exp(-

_{j}*μ*·

_{j}*t*). The pulsed X-rays, which penetrated through the attenuation material, are detected and converted into a large electric charge. Generally, the conversion coefficient, that is, the detection efficiency for the X-rays has energy dependence. Now, we define the detection efficiency

_{i}*ε*for X-ray energy of

_{j}*E*. The output signal

_{j}*y*after passing through the attenuation material of thickness

_{i}*t*is proportional to the value of summarized

_{i}*ε*·

_{j}*φ*□(

*E*) with energy bin

_{j}*j*from 1 to

*M*.

*k*to the matrix equation, which depends on the individual measurement system, we express the matrix equation to derive the attenuation curve of X-rays,

*y*(

*t*), as in the next equations.

_{i}##### (3)

$$\left[\begin{array}{c}y\left({t}_{1}\right)\\ y\left({t}_{2}\right)\\ \vdots \\ y\left({t}_{N}\right)\end{array}\right]=k\xb7\left[\begin{array}{cccc}exp(-{\mu}_{1}\xb7{t}_{1})& exp(-{\mu}_{2}\xb7{t}_{1})& \cdots & exp(-{\mu}_{M}\xb7{t}_{1})\\ exp(-{\mu}_{1}\xb7{t}_{2})& exp(-{\mu}_{2}\xb7{t}_{2})& \cdots & exp(-{\mu}_{M}\xb7{t}_{2})\\ \vdots & \vdots & \ddots & \vdots \\ exp(-{\mu}_{1}\xb7{t}_{N})& exp(-{\mu}_{2}\xb7{t}_{N})& \cdots & exp(-{\mu}_{M}\xb7{t}_{N})\end{array}\right]\xb7\left[\begin{array}{cccc}{\epsilon}_{1}& 0& \cdots & 0\\ 0& {\epsilon}_{2}& \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& \cdots & {\epsilon}_{M}\end{array}\right]\xb7\left[\begin{array}{c}\varphi \left({E}_{1}\right)\\ \varphi \left({E}_{2}\right)\\ \vdots \\ \varphi \left({E}_{M}\right)\end{array}\right]\phantom{\rule{0ex}{0ex}}y=k\xb7R\xb7F\xb7\varphi \begin{array}{}\\ \\ \end{array}$$*R*•

*Ｆ*in Equation 3 in this section, focusing on a 6 MeV X-ray source and attenuation material of aluminum as an example. The maximum energy of the response function is therefore set to 6 MeV to cover X-rays emitted from the 6 MeV X-ray source. The column of the response function

*R*is then divided into 60 bins, namely they cover the X-ray spectrum from 0.0 to 6.0 MeV with an equal bin width of 0.1 MeV. This means

*j*=

*1–M*(

*M*=60) in Eq. 3. On the other hand, the row component for the attenuation material thickness in Eq. 3 is divided into 59, that is,

*i*=

*1–N*(N=59) as mentioned later in detail. The detection efficiency matrix F is evaluated by calculating the reaction rate of incident monochromatic photons for each energy bin in the actually used detector. The obtained reaction rate is converted into the deposited energy in the detector, i.e., electric charge. We use the Monte Carlo numerical simulation code EGS5 for this evaluation.

^{1)}.

*k*, because of the dependence on the individual X-ray measurement system, we decide the value experimentally for the system we use.

### 3. RESULTS AND DISCUSSION

*φ*(

*E*), estimated from the attenuation curve measurements for 6 MeV and 9 MeV X-ray sources at emission angle of 0°. The spectrum estimation was done with the spectrum type Bayesian estimation method described in chapter 2.2 from the independent measured attenuation curve in Fig. 2 and 3 as y in Eq. 3. As an initial guess a white spectrum was used. The X-ray energy spectrum simulated by the EGS5 code is also shown. In the present simulation, we assumed the tungsten target thickness was 1 mm. As shown in Fig. 4 and 5, in spite of estimated from independent measured curve, the estimated spectra look quite a similar shape with each other. Furthermore, the experimental spectra are in good agreement with the simulated spectra except for higher energy region more than 4 MeV (for 6 MeV X-ray source) or 6 MeV (for 9 MeV X-ray source). The EGS5 results show smaller values in high energy region compared to the experiment. It is difficult to discuss the reason of this discrepancy. However, it can be said that the present unfolding method based on the Bayes’ theorem may not reproduce spectral values having such a small likelihood. Nevertheless, these differences between estimated and simulated spectra at higher energy region are not so serious because those absolute intensities are quite small. Besides, in the present measurement, several hundred attenuation curves

_{j}*yθ*(

*t*) within the fan shaped beam of 15° were measured at the same time. This means that a lot of angle-dependent X-ray energy spectra can be measured in the same experiment.

_{i}*θ*. The figure also denotes that all the angular distributions of 9 MeV source show more forward-peeked distributions with respect to emission angle compared to 6 MeV source. This means that higher energy photons are more dominant in forwarder emission angles for both X-ray sources, and the case of 9 MeV shows it more clearly. However, each measured curve is slightly larger than the curve of EGS5 calculation

^{2)}due to underestimation of EGS5 in higher energy region as mentioned previously.