### Introduction

### Materials and Methods

### 1. Estimation Overview

### 2. One-Dimensional Advection-Dispersion Equation with Langmuir Isotherm Adsorption

*t*denotes time,

*z*is the spatial coordinate from up to down in this study, θ is porosity, ρ

_{b}is bulk density (kg/m

^{3}), C is the contaminant concentration in liquid waste, q is the adsorbed amount in the media,

*v*is the linear average velocity of liquid waste (m/s),

*D*is the dispersion coefficient in a column (m

^{2}/s),

*λ*is the decay constant of radionuclides (s

^{−1}), and γ(z) is the production term by insertion of liquid waste into the column. Meanwhile, the adsorption kinetics equation based on Langmuir isotherm is expressed as follows [14]:

*k*

_{1}is the adsorption constant (s

^{−1}),

*k*

_{2}is the desorption constant (s

^{−1}), and q

*is the adsorption capability of the media. To collect the dataset from the above hypothesis, numerical methods are applied to Equations (1) and (2). Equation (2) is derived as follows using an explicit finite difference method:*

_{max}##### (4)

##### (5)

### 3. Estimation Model of Radiation Damage for Biological Adsorbents

*a,b*) function, which replaces negative values with 0, was used in this study. To represent this, let

*i*-th cell at the

*j*-th frame. The relationship between

*r*

_{a}is the bio-adsorbent loss rate of radiation damage per absorbed dose (#/Gy), and

*ξ*

_{i}*is the adsorbed dose by a radiation damage model (Gy). Here, the total contaminant concentration, including in the liquid and adsorbed phases, is utilized to estimate the adsorbed dose*

^{j}*ξ*

_{i}*. Therefore, by letting*

^{j}*from Equation (3) should be modified as:*

^{max}*ξ*

_{i}*is calculated by multiplying Δ*

^{j}*t*by the absorbed dose rate. The other input values for the MCNP simulation are fixed during the simulation, except for the concentration information in the column.

### 4. Artificial Neural Network for Radiation Damage Estimation

^{−3}and 1×10

^{−7}, respectively. The loss function between the results of the neural network model and MCNP simulation was determined based on the mean squared error (MSE). A total of 500,000 normalized concentration samples were randomly collected to perform the deep learning model and were then divided into three categories: 400,000 training data, 50,000 validation data, and 50,000 test data. Using the simulation dataset, the ANN model in this study was trained by 1,000 epochs within 3 hours of running time (NVIDIA Tesla V100). The deep learning model was implemented using Tensorflow 2.0 (Google LLC), supported by Python 3.6 (Python Software Foundation).

### Results and Discussion

### 1. Simulation Results and Analysis

^{60}Co, which emits higher energy than other radionuclides, was selected only from in the liquid radioactive waste. It was assumed that the radionuclides were injected into the top of the column as the initial condition to run the simulation. The simulation is terminated after all the purified liquid waste passes through the bottom of the column.

*C*, adsorption of bio-adsorbent

*q*, and survival rate of bio-adsorbent are defined under the condition of adsorption rate

*k*

_{1}. Table 2 summarizes the bio-adsorbent loss rate of radiation damage per absorbed dose

*r*

_{a}by the present adsorption kinetic model with radiation damage discussed in Section 2 and the parameters used in this study. The concentration values were evaluated with the length

*L*of 5 cm, and during the operating time

*T*=20 minutes, the input values are

*Δt*=0.01 (min),

*Δz*=0.5 (cm), C

_{i}*=*

^{0}*δ*

_{i}_{0}, (q

*)*

_{max}

_{i}^{0}=0.1 (for 1≤

*i*≤10),

*v*=0.49 (cm/min),

*D*=0.03 (cm

^{2}/min),

*θ*=0.7,

*λ*=2.48×10

^{−7}(min

^{−1}),

*k*

_{2}=1×10

^{−6}(min

^{−1}), and

*ρ*

*=1.11 (g/cm*

_{b}^{3}). Here,

*δ*

*denotes the Kronecker delta function, which is defined as 1 when*

_{i0}*i*=0 and 0 for all other values of

*i*. The radius and height of the column should be 2.5 and 50 mm, and the column is divided into 10 nodes (Fig. 4). Table 3 summarizes the necessary information of the input variables for the MCNP code (e.g., density and molecular composition). In each MCNP simulation, the number of particle histories is selected to 100,000, which shows a relative error of the dose under 1.5%.

^{60}Co. From Fig. 5B, the contaminant concentration at the top node was initialized with a normalized concentration of 1.0 because of the injection of liquid radioactive waste, then decreased due to advection, adsorption, and dispersion, and finally converged to 0. Similarly, the concentrations at the middle and bottom nodes temporarily increased due to advection and, then, decreased due to advection and adsorption. From Fig. 5C, each level of adsorption concentration converged because the Langmuir adsorption equation was assumed. Then, the survival fraction of the bio-adsorbent shown in Fig. 5A is derived by calculating the radiation damage induced by radioactive concentration from the liquid radioactive waste and bio-adsorbent of the column using the results from Fig. 5B and 5C.

*r*

_{a}=0), the number of bio-adsorbent is fixed. The variation of the adsorption rate

*k*

_{1}rarely affects the survival rate because the difference of the curves from 1 to 3 among the scenarios is small, whereas the survival fraction decreases drastically when

*r*

_{a}is increased from 4 to 6 among the scenarios. Therefore, the results demonstrate that the survival ratio for the bio-adsorbents directly depends on

*r*

_{a}, rather than on

*k*

_{1}. However, the inference time for testing a scenario case, which has 2,400 frames, was about 45 minutes. It is found that the process for evaluating radiation damage by MCNP mainly results in bottleneck phenomena.

### 2. Verification of the ANN Damage Analysis Model

*R*

^{2}) score:

*y*

*is the actual value of the test, which means the absorbed dose obtained by MCNP,*

_{i}*ȳ*is the mean of the test value of the test, and

*n*is the number of samples.

*k*

_{1}and

*r*

_{a}), four metrics are measured: MSE,

*R*

^{2}score, MAPE, and inference time. Compared with the ANN and MCNP, the results with the test data agree well (Table 4). The

*R*

^{2}score shows 99.3% accuracy and MAPE indicates 0.062% accuracy. Meanwhile, the average inference speed of the ANN is approximately 0.031 second per a frame, which is approximately 30 times faster than that of the MCNP model (i.e., 1.01 second per a frame). As confirmation of the replacement performance, the sample results of the simulation using the ANN model show that the error range with the MCNP model is up to 0.002, and the average of the errors is at the 1×10

^{−5}scale (Fig. 8).

### Conclusion

*R*

^{2}score and MAPE, respectively. The inference time can be reduced to 74 seconds, and the MCNP code evaluated 2,400 frames within 44 minutes. Note that the performance can be optimized to estimate the damage from radioactive materials, and whether bio-adsorbents can be reused should be determined accurately and quickly by using the advantages of the ANN. As a further study, the development of bio-adsorbents would be accelerated to contribute to the maintenance and utilization of decontaminant processes for NPPs.