**1. Introduction**

Nowdays, biodegradable polymers are widely used as materials for medical devices [1]. Biodegradable polymers, used for decades, include polyesters and their copolymers, such as poly(L-lactic acid) (PLA), poly(ε-caprolactone) (PCL), poly(L-lactide-co-<sup>ε</sup>caprolactone) (PLCL), and poly(L-co-D,L lactide) (PLDLA). Among the various materials, PLCL and PLDLA are very valuable materials used in medical applications as implantable devices because of their excellent flexibility and biodegradability [2–4]. Sterilization is essential for implantable devices [5], and some established sterilization methods include dry heat, ethylene oxide, steam, and radiation methods [6]. In particular, the gamma or electron beam sterilization process is performed at room temperature and has the advantage of a short sterilization time and low risk of toxic residues [7]. In addition, it has a high sterilization effect for substances that struggle to penetrate into other sterilizing agents [8]. Packaging is used to protect the bioimplantable device from moisture and ions inside the human body [9], the material is sterilized in the entire volume of the product together with the packaging. Because of these advantages, gamma irradiation is the most commonly used method for the sterilization of materials with a high transmittance [10].

However, free radicals generated by radiation energy can propagate within the polymer chain structure and cause a chain reaction, leading to crosslinking [11–13]. Therefore,

**Citation:** Shim, H.-E.; Yeon, Y.-H.; Lim, D.-H.; Nam, Y.-R.; Park, J.-H.; Lee, N.-H.; Gwon, H.-J. Preliminary Study on the Simulation of a Radiation Damage Analysis of Biodegradable Polymers. *Materials* **2021**, *14*, 6777. https://doi.org/ 10.3390/ma14226777

Academic Editor: Vladimir Krsjak

Received: 7 October 2021 Accepted: 5 November 2021 Published: 10 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

aliphatic polyesters are decomposed in the radiation sterilization process, and the decomposition temperature may change depending on the polymer composition [7,14,15]. In general, the effect of radiation on polymers can cause changes in various properties, such as chemical composition, crystallinity, molecular weight and density, depending on the radiation dose, dose rate, and temperature [6]. In particular, because biodegradable, polymer-based, implantable devices are very important for the performance and service life of materials [16,17], it is necessary to analyze the changes in the properties of polymer materials according to irradiation dose, in preparation for the possibility of material decomposition during the radiation sterilization process. In addition, radiation sterilization is the most suitable and useful material for human insertion among the currently available sterilization methods. Therefore, it is necessary to secure the optimal sterilization dose for each polymer material according to its intended use.

In contrast, irradiation is well known as a very convenient tool for sterilizing biodegradable polymers and transforming polymer materials through crosslinking, grafting, and decomposition [18]. Because crosslinking using radiation may generate radicals within the polymer chain, the functionalization process may be omitted, and the crosslinking reaction may be performed at a low temperature due to its excellent penetrating power. In particular, irradiation with a high-energy radiation such as gamma rays and electron beams was used for the processing and crosslinking of polymers [19,20].

However, radiation accompanies the chain scission reaction and crosslinking simultaneously and may affect basic properties such as reducing the glass transition, crystallization and melting temperature of the polymer. Other physical properties, such as gas permeability, may differ from those of conventional polymers. In particular, because crosslinking generally reduces the degradability of polymers, it can negatively affect the properties of materials designed for degradable devices after the insertion into the human body [16]. Radiation treatment can control the biodegradation time [21]. A compromise must be found between the mechanical properties and the biodegradation time. Therefore, it is expected that the time and cost of material development can be dramatically shortened if the optimal radiation irradiation conditions that are suitable for the required properties can be selected through a pre-simulation study.

In addition, to analyze the damage to polymers caused by radiation, it is necessary to understand radiation physics, dosimetry, chemical analysis, and instrumental analysis applied to the modeling and simulation of the radiation environment [22]. Therefore, it is necessary to investigate the interaction of the polymer of interest with gamma rays. However, because it is difficult to rely solely on repeated testing and dose selection for an accurate analysis [23], some researchers conducted simulation-based correlation studies [24–26]. Ghosal et al. performed theoretical simulations to determine the correlation between the morphological changes due to gamma irradiation and other properties such as molecular weight distribution, intrinsic viscosity, and ionic conductivity [24]. Saha et al. conducted a computer simulation study to examine the effects of gamma irradiation on properties such as molecular weight distribution and viscosity [25]. In particular, the number average molecular weight can be an indicator of the critical dose that causes changes in the physical properties of polymers due to radiation [27]. In addition, the quantitative analysis and interpretation of the number average molecular weight is important for understanding the decomposition behavior of polymer materials according to the radiation dose within the simulation.

In this study, the average molecular weight of the polymer was selected as a parameter to conduct a basic simulation of radiation damage analysis for biodegradable polymers (PLCL and PLDLA). Figure 1 shows the chemical structures of PLCL and PLDLA. In preparation for the possibility of material decomposition during radiation (gamma-ray) sterilization, changes in the characteristics of PLCL and PLDLA according to irradiation dose were observed. In addition, a Geant4 simulation was performed to predict the structural damage dose range in the gamma ray sterilization and processing by selecting

this as a damage model, and a comparative verification study was conducted to verify the change in the average molecular weight according to the irradiation dose.

**Figure 1.** Chemical structure of polymer (**a**) PLCL and (**b**) PLDLA.

#### **2. Materials and Methods**

#### *2.1. Modeling and Simulation*

2.1.1. Geant4 Simulation for Gamma Ray Fluence Calculation

To calculate the radiation dose of polymer films, Geant4 (version 10.6p02, CERN, Conseil Européenne pour la Recherche Nucléaire, Meyrin, Switzerland) was used to perform the simulation. First, the radiation dose of the high-level gamma radiation device was simulated. Using the "GeneralParticleSource (GPS)" and "G4RadioactiveDecayPhysics" modules in Geant4, the 60Co decay scheme was modeled. The energy absorbed by an alanine dosimeter (cylindrical type with a radius of 0.24 cm and height of 0.3 cm), placed 15.8 cm from the 60Co source, was calculated. The "G4EmStandardPhysics\_option4" physics model was used to calculate the absorbed dose of the alanine (material: C3H7NO2, density: 1.42 g/cm3) dosimeter when exposed to 1173 keV and 1332 keV gamma rays simultaneously released from the 60Co source. The dose amount necessary to calculate the same radiation dose as experimental conditions at the target position was converted to fluence.

#### 2.1.2. Geant4 Simulation for Absorbed Dose Calculation

Simulations were performed to calculate the absorbed dose per unit mass ( *D*, eV/g) after irradiating the PLCL and PLDLA films with the previously calculated gamma ray fluence. The polymer film was the same size as the actual film, measuring 2 × 2 × 0.1 cm3. The PLCL film was made of HO[C3H4O2]n[C6H10 O2]mCH3, (n:m = 70:30), and had a density of 1.2 g/mL. The PLDLA film was made of HO[C3H4O2]n[C3H4O2]mCH3, (n:m = 70:30), and had a density of 1.2 g/mL. The PLCL and PLDLA films were placed 15.8 cm away from the 60Co source, and the absorbed dose per unit mass was calculated as summarized in Table 1.


**Table 1.** Absorbed dose per unit mass of PLCL and PLDLA models according to gamma radiation dose.

2.1.3. Prediction of Damage to Polymer Materials Using a Radiation Damage Model

The relationship between the radiation dose and number average molecular weight can be derived as follows: The number average molecular weight ( *M*n,0, g/mole) for the polymer sample is as follows [28]:

$$M\_{\rm n,0} \text{ (g/mole)} = wN\_{\rm A}/N\_0 \tag{1}$$

where *w* denotes the weight (g) of the polymer sample, *N*A is Avogadro's number, and *N*0 is the total number of molecules (initial molecules) in *w* before irradiation. From Equation (1), *N*0 can be rearranged as follows:

$$N\_0 = w N\_\Lambda / M\_\Pi \tag{2}$$

If the dose is expressed as *D* (eV/g), we can calculate *Dw*, that is, the total absorbed dose by the sample. Therefore, the number of newly formed molecules (*N*\*) inside the polymer due to irradiation is obtained as follows:

$$N^\bullet = KDw\tag{3}$$

where *K* denotes the polymer structure constant that represents the resistance to radiation. It can be replaced by *G*, which is defined as the number of molecules or atoms produced per 100 eV of energy.

The *G*-value is generally expressed as *G*s for the number of scissions due to exposure, or *G*x for the number of crosslinking reactions. The production of new molecules in relation to the number of scissions (*N*s\*) and crosslinking (*N*x\*) can be described as follows:

$$N\_{\sf s}{}^{\sf s} = \left( G\_{\sf s} / 100 \right) Dw \tag{4}$$

$$N\_{\chi}" = \left(G\_{\chi} / 100\right) Dw \tag{5}$$

From the perspective of the number of molecules in the polymer sample, chain scissions increase the number of molecules, whereas crosslinking decreases the number of molecules. Therefore, the number average molecular weight (*M*n\*) of the polymer when the scissions and crosslinking reactions occur competitively with the absorbed dose *D* is obtained as follows:

$$M\_{\rm n}" = w N\_{\rm A} / (N\_0 + a N\_{\rm s}" - \beta N\_{\rm x}") \tag{6}$$

where the total mass of the polymer is assumed to be constant during exposure, *α* and *β* denote constants used to consider the change in the initial number of molecules (*N*0) when chain scission and crosslinking events occur; 0.5 and 2 are applied, respectively.

Based on Equations (4)–(6), the polymer damage model that considers radiationinduced chain scission and crosslinking is as follows:

$$1/M\_{\rm n}{}^{\star} = 1/M\_{\rm n,0} + \left[ (aG\_{\rm 5} - \beta G\_{\rm x})/100N\_{\rm A} \right] D \tag{7}$$

where *M*n,0 and *G*-value (*G*s and *G*x) were obtained from the experimental results. The radiation damage model of Equation (7) was used to simulate a biodegradable polymer film.

#### *2.2. Film Preparation and Radiation Measurement for Simulation Verification*
