### Introduction

### Materials and Methods

### 1. Optimization using a modified iterative genetic algorithm

#### 1) Optimization and genetic algorithm

#### 2) Modified iterative genetic algorithm

### 2. Calculation of detector positions for the system

#### 1) Structural design

^{3}will be utilized for the RPM system. These detectors are also utilized in other RPM researches [5, 7]. If the active area is too small, the detection performance is poor, and if it is too large, it is difficult to specify the position of the detector. From this perspective, the NaI detector seems to be the appropriate choice and is the reason why it is widely utilized in RPM systems.

#### 2) Inverse transport theory to define optimization problem

D = Measured quantity with a detector

A = the active area of a detector,

*ɛ*= the intrinsic parameter of a detector,_{ip}*d*= the distance between the source and the detector,*I*= the intensity of the source,_{0}*μ*= the absorption coefficient of the medium.

##### (2)

$$({({x}_{s}-{x}_{i})}^{2}+{({y}_{s}-{y}_{i})}^{2}){e}^{\mu \sqrt{({({x}_{s}-{x}_{i})}^{2}+{({y}_{s}-{y}_{i})}^{2})}}=\frac{A{\varepsilon}_{ip}}{4\pi {D}_{i}}{I}_{0}$$*x*= the X coordinate of the position of the source,_{s}*y*= the Y coordinate of the position of the source,_{s}*x*= the X coordinate of the position of the_{i}*i*^{th}detector,*y*= the Y coordinate of the position of the_{i}*i*^{th}detector in the Cartesian coordinate system,*D*= measurement of the_{i}*i*^{th}detector.

*x*

*,*

_{s}*y*

*, and*

_{s}*I*

*. This means that the position of the source can be calculated by solving a system of equations if the number of these equations is greater than three. It is possible to solve nonlinear equations even without the intensity of the source. The detector parameters can be confirmed by checking the specifications of the detector. In the case of the absorption coefficient, it varies with the medium that a radiation particle passes through. Because inhomogeneous media are quite complicated, only a homogeneous medium will be considered to simplify the problem in this research.*

_{0}#### 3) Problem definition

^{6}gamma rays·s

^{−1}.

^{−4}cm

^{−1}, equivalent to the microscopic cross section of air. Each detector is assumed to have an active area of ~412.9 cm

^{2}and an intrinsic efficiency of 10% for simplification. These values are reasonable for NaI scintillator detectors [5].

##### (3)

$${D}_{i}=\frac{A{\varepsilon}_{ip}}{4\pi ({({x}_{s}-{x}_{i}^{*})}^{2}+{({y}_{s}-{y}_{i}^{*})}^{2})}\hspace{0.17em}{I}_{0}{e}^{-\mu \sqrt{({({x}_{s}-{x}_{i}^{*})}^{2}+{({y}_{s}-{y}_{i}^{*})}^{2})}}$$##### (4)

$${D}_{ij}=\{\begin{array}{c}{\mathrm{\Sigma}}_{i=1}^{n}{\mathrm{\Sigma}}_{j=1}^{n}\mid {D}_{i}-{D}_{j}\mid /2,\hspace{0.17em}i\ne j\\ 0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a},\hspace{0.17em}i=j\end{array}$$*m*= the number of sources,*n*= the number of detectors,*D*= measurement of the_{i}*i*^{th}detector according to equation 1,(

*D*)_{ij}*k*= total difference in the measurements between the*i*^{th}and*j*^{th}detectors for the k^{th}source.

*D*

*is divided by 2 is to countervail the overlap between two identical differences such as*

_{ij}*D*

_{12}and

*D*

_{21}. By substituting the

*D*

*and*

_{i}*D*

*terms in equation 4 with equation 3, the objective function can be represented with the unknown detector positions.*

_{j}*x*positions are fixed on the frame. The

*y*positions are adjustable but limited to the height of the frame. For top-side detectors, the

*x*positions are adjustable but limited to the width of the frame. The

*y*positions are fixed on the frame. When each position of the detectors is represented by two vectors

*X*and

*Y*, these equality and inequality constraints can be written with vectors and matrices as follows.

##### (5)

$$\mathbf{A}\xb7\mathbf{X}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ W\\ W\end{array}\right],\hspace{0.17em}\mathbf{B}\xb7\mathbf{Y}=\left[\begin{array}{c}0\\ 0\\ H\\ H\\ 0\\ 0\end{array}\right]$$##### (6)

$$\mathbf{B}\xb7\mathbf{X}\le \left[\begin{array}{c}0\\ 0\\ W\\ W\\ 0\\ 0\end{array}\right],\hspace{0.17em}\mathbf{A}\xb7\mathbf{Y}\le \left[\begin{array}{c}H\\ H\\ 0\\ 0\\ H\\ H\end{array}\right]$$*A*= the matrix for selecting the left- and right-side detectors,*B*= the matrix for selecting the top-side detectors,*W*= the width of the RPM frame,*H*= the height of the RPM frame,*x*= the X coordinate of the position of the_{i}*i*^{th}detector,*yi*= the Y coordinate of the position of the*i*^{th}detector in the Cartesian coordinate system.

### Results and Discussion

### 1. Performance evaluation of the MIGA

^{−20}. At the beginning of the GA, the fitness value decreases rapidly. This means that the GA searches the optimal solution effectively. Then, the fitness value converges. This shows how the GA finds the optimal solution.