Fig. 1 shows a bio-adsorbent removal example for radionuclides in liquid radioactive waste in a column that contains a bio-adsorbent when the liquid radioactive waste is injected into the top of the column, and the purified liquid is extracted through the bottom. During radionuclide transport into the column, specific target ions are selectively captured by the bio-adsorbent through the diffusion process, and the rest move down by advection. Additionally, some reactions occur (e.g., dispersion and radioactive decay) in liquid radioactive waste. Meanwhile, the bio-adsorbent desorbs the adsorbed target ions and is damaged by the ionizing radiation of radionuclides.

The radiation damage to the bio-adsorbent is mainly governed by the radionuclides contained in the liquid and captured in it. It causes a reduction in the adsorption performance of the bio-adsorbent. Therefore, contaminant concentrations of radionuclides considering the abovementioned two phases obtained using a transport model should be considered to evaluate the time-dependent adsorption performance of bio-adsorbents with radiation damage (Fig. 2). In this study, the ADE with Langmuir isotherm adsorption is introduced to evaluate the contaminant concentration in the liquid and adsorbed phases. Further, an ANN model is developed and applied to improve the estimation time of radiation damages during the ADE simulation.

For the simulation of radionuclides captured by the bio-adsorbent as porous adsorbent media, the following conditions are assumed: (1) the velocity, temperature, and pressure are constant when injected into liquid waste with one-dimensional (1D) flows; (2) an ideal fluid behavior is followed; (3) no mass transfer resistance emerges between the liquid and adsorbed phases; and (4) the capture in the adsorbed phase is given by Langmuir isotherm. The desired 1D ADE with adsorption can be expressed as follows [14]:

where t denotes time, z is the spatial coordinate from up to down in this study, θ is porosity, ρ_b is bulk density (kg/m³), 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 (m²/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]:

where k₁ is the adsorption constant (s⁻¹), k₂ is the desorption constant (s⁻¹), and q_max 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:

Similarly, using the implicit scheme from Equation (1), Equation (4) can be expressed as follows:

Finally, by substituting Equation (3) into Equation (4), the following equation can be obtained:

where a=vΔt2Δz, b=DΔt2Δz2, r=λΔt2, x=ρbk1Δt2θ and y=ρbk2Δt2θ from Equation (5), a dataset that explains the interaction of the radioactive amount between the liquid waste and the bio-adsorbent in the column can be gathered.

It is assumed that the adsorption performance depends on the number of surviving bio-adsorbents in the irradiated environment. Also, in a preliminary study [15], the number of damaged bio-adsorbents linearly varied with the radiation dose. With the assumption of the linear variation, the number of surviving bio-adsorbents can be negative; hence, the max(a,b) function, which replaces negative values with 0, was used in this study. To represent this, let nij denote the number of surviving bio-adsorbent in the i-th cell at the j-th frame. The relationship between nij and nij+1 can be expressed as follows:

where r_a is the bio-adsorbent loss rate of radiation damage per absorbed dose (#/Gy), and ξ_i^j 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 ξ_i^j. Therefore, by letting sij=1-nij+1nij be the survival rate, the term of q^max from Equation (3) should be modified as:

In a previous study [15], the relationship between the absorbed dose and damages of bio-adsorbents were evaluated. To calculate Equation (7), the radioactive source is assumed to be uniformly located in each column divided for the partial damage analysis; F4 tally with the Monte Carlo N-Particle code version 6 (MCNP6) code is used for estimating the absorbed dose. The dose energy function card from International Commission on Radiological Protection 74, which provides conversion coefficients for air kerma per unit fluence of monoenergetic photons, is utilized to directly convert the flux into the dose rate [16]. The normalized concentration of each column estimated using Equation (6) is applied to the source probability as the MCNP input. In Equation (6), ξ_i^j is calculated by multiplying Δt 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.

The ANN, which is developed by referring to biological neurons, is a type of machine learning method used for prediction or classification. It has many advantages, such as real-time operation, adaptive learning and, the ability to easily analyze complex mathematical problems. Fig. 3 shows an ANN architecture comprising an input layer, wherein input features should be the information of the radioactive concentration from the liquid waste and bio-adsorbent of the column; hidden layer, including a non-linear activation function; and output layer, which predicts the adsorbed dose affected by the concentration information.

In this study, as illustrated in Table 1, the neural network consists of five fully connected layers, each activation function of the hidden layer is the “rectified linear unit (ReLU)” function, and the optimizer in the neural network is Adam with weight decay (ADAMW) [17], of which the learning rate and weight decay are 1×10⁻³ 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).

In this simulation, ⁶⁰Co, 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.

Depending on the experimental environment (e.g., sequence of bio-adsorbent and type or concentration of radionuclides), the simulation scenario, including adsorption and radiation damage to bio-adsorbent such as aptamers comprising short sequences of artificial DNA or RNA molecules, would be subject to change [18]. The adsorption rate, desorption rate, and adsorption capacity of the bio-adsorbent could be determined under various adsorption experiments [19]. Meanwhile, the damage coefficient of the bio-adsorbent could be estimated through the Monte Carlo damage simulation code, which can evaluate radiation-induced DNA damage linked with MCNP.

The contaminant concentration of liquid radioactive waste C, adsorption of bio-adsorbent q, and survival rate of bio-adsorbent are defined under the condition of adsorption rate k₁. 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_i⁰=δ_i0, (q_max)_i⁰=0.1 (for 1≤i≤10), v=0.49 (cm/min), D=0.03 (cm²/min), θ=0.7, λ=2.48×10⁻⁷ (min⁻¹), k₂=1×10⁻⁶ (min⁻¹), and ρ_b=1.11 (g/cm³). Here, δ_i0 denotes the Kronecker delta function, which is defined as 1 when 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%.

Fig. 5 presents the results for the cumulative survival and adsorption concentration of the bio-adsorbent and contaminant concentration in the liquid phase at the top, middle, and bottom nodes in scenario #1 with liquid waste insertion, including ⁶⁰Co. 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.

The survival fractions of the bio-adsorbent for the whole bio-adsorbent were estimated (Fig. 6). If the bio-adsorbent has no radiation damage similar to that of scenario #4 (r_a=0), the number of bio-adsorbent is fixed. The variation of the adsorption rate k₁ 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 k₁. 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.

The prediction accuracy of ANN models, compared with the results of the MCNP simulation, was evaluated:

MSE:

Mean absolute percentage error (MAPE):

Coefficient of determination (R²) score:

where yι^ is the predicted value of the machine learning model, y_i is the actual value of the test, which means the absorbed dose obtained by MCNP, ȳ is the mean of the test value of the test, and n is the number of samples.

As the MCNP simulation is a stochastic method, the datasets include uncertainties. It is noted that the ANN trained with the uncertain datasets provides an average value [20]. Therefore, as a preliminary study, the uncertainty of the dataset is not mainly considered in this study. During the training phase of the ANN model, the loss functions for the training and validation data have converged to near zero, and overfitting has not occurred as the differences between training and validation losses were small (Fig. 7). To evaluate the performance of the presented simulations that correspond to the presented values concerning the two main parameters of the model (i.e., k₁ and r_a), four metrics are measured: MSE, R² score, MAPE, and inference time. Compared with the ANN and MCNP, the results with the test data agree well (Table 4). The R² 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).