Abstract
BACKGROUND: Poly(lactic-co-glycolic acid) (PLGA) has been widely used as medical implants and drug vehicle. Due to various factors involved in the degradation process, it is rather difficult to simulate the mass loss properties of PLGA.
OBJECTIVE: To investigate the mass loss properties of PLGA films, and observe the correlation between the stimulation and the factual measurement.
DESIGN: Repeated measuring experiment.
SETTING: School of Materials Science and Engineering, Dalian University of Technology.
MATERIALS: PLGA 50:50, 70:30 and 75:25 with weight average molecular weight (Mw) of 63 000, 115 000 and 400 000 were purchased from Shandong Key Laboratory of Medical Polymer Materials, and 1,4-dioxane (analytically pure) was purchased from Tianjin No.1 Chemical Reagent Factory and used as solvent.
METHODS: The experiments were carried out in the School of Materials Science and Engineering of Dalian University of Technology from April to July in 2006.①PLGA were dissolved in 1, 4-dioxane according to the mass ratio of 50:50, 70:30 and 75:25, and the solution were cast into 10 mm×10 mm×1 mm polymer films.②PLGA films were immersed in Hank's simulated body fluid (pH 7.4) at 37 ℃, and then taken out every week (every 5 days for PLGA 50:50). The samples were washed and dried to measure the changes of number and Mw using gel permeation chromatography. Degradation velocity was also worked out. Meanwhile the mass loss properties of PLGA films were measured using electronic balance. Furthermore, an improved Monte Carlo method was applied to carry out the correlation analysis with experiment data.
MAIN OUTCOME MEASURES: ①degradation rates of PLGA with different ratios of lactic acid and glycolic acid.②the correlation coefficient between simulation results and experiment data.
RESULTS: ①The degradation rates for PLGA at 50:50, 70:30 and 75:25 were 0.058 5, 0.016 6 and 0.010 1 /d (based on Mn), or 0.061 0, 0.017 5 and 0.008 5 /d (based on Mw), respectively.②Correlation coefficients between simulation results and experiment data for PLGA 50:50, 70:30 and 75:25 were 0.973 6, 0.987 4 and 0.990 3 correspondingly.
CONCLUSION: The numerical simulation results for the mass loss properties of PLGA fit the experiment data well.

INTRODUCTION

Poly (lactic-co-glycolic acid) (PLGA) is widely used in tissue engineering and controlled drug release[1]. But complex degradation process in-duces the poor repeatability[2-4], moreover, the degradation cycle may cost too long, usually several months. In this paper, Monte Carlo simu-lation based on tunnel mass transfer is developed and compared with experiment results, in order to shorten the experimental duration and forecast the degradation of PLGA directly.

MATERIALS AND METHODS

Materials
PLGA 50∶50, 70∶30 and 75∶25 with weight average molecular weight (Mw) of 63 000, 115 000 and 400 000 were purchased from Shandong Key Laboratory of Medical Polymer Materials, and 1,4-dioxane (analytically pure) was purchased from Tianjin No.1 Chemical Reagent Factory and used as solvent.

Methods
PLGA with three kinds of Mw were dissolved with 1,4-dioxane to form the 5% mixture solution, then cast into thin films on the polytetrafiuoro-ethylene plate and evaporated at room temperature. Afterwards the PLGA films were taken off and cut into samples of 10 mm×10 mm×1 mm. Repeated dipping and coating were necessary for meeting the reserved thickness. The rectangle samples were dried to constant mass under vacuum and then weighed.
PLGA films were immersed in Hank's simulated body fluid (pH 7.4) at 37 ℃, and renewed every week (every 5 days for PLGA 50∶50). Accordingly, the samples were washed with distilled water and dried to constant mass. The changes of Mn and Mw were determined by gel permeation chromatography using polystyrene standards for calibration and tetrahydrofuran as eluent at a flow rate of 1 mL/min at 40 ℃. Mass loss was measured using an electronic balance according to the formula[5]

W0 as initial weight of samples, Wt as the sample weight at each time point

Modeling for biodegradable matrices
For the symmetry of membrane samples in x-y coordinate system, the simulation could be reduced to one quarter of the films. Models for two-dimension mass transfer were meshed into m×n pixels (m in x-axis and n in y-axis, m=1 000, n=100, the thickness of the membrane was one-tenth of its width). Due to retained solvent or the oligomer and monomer existed in the PLGA films more or less, a fast mass loss process often
occurs at the early stage of degradation. Because the foreign substances are predisposed to dissolve in water and diffuse to external solution, a stable stage can be obtained following the early mass loss. Therefore the present simulation began from the subsequent quasi-stable stage, with mass loss measure-ment obtained in the experiment. It is an assumption that mass loss, which induces membrane depletion, should occur evenly, so the pixel with mass loss constitutes pores. Open pores and close pores are formed depending on whether the position of pores on the surface of model or not. For dense pixels, random values which fit the first order Erlang distribution[6] are used as the lifetimes of dense pixels, also calling exponential dis-tribution. In this paper, the analysis software MATLAB was used to calculate the lifetime of the pixels generated randomly in the polymer matrices, with an order of exprnd(1/λ, n, m), λ referring to the lifetime loss velocity and presently the degradation velocity k, which could be obtained by the mo-lecular weight measurement. The calculating process of mass loss amount is described as below: A matrix named TL with n×m units is generated and the value of each unit equals to the lifetime of corresponding pixel. Then a matrix named P is used to characterize the state of each pixel, and the values of 1, 0 or -1 represent dense, open or close porous pixels correspondingly. These descriptions are all clear in the original articles. After-wards matrix TL is modified to make the lifetimes of porous pixels equal to 0 or -1 according to matrix P. When a dense pixel which exceeds its lifetime and contacts with open pores in any direction, the corresponding value in TL is replaced with 0, that means the mass loss process occurs and this pixel changes to be a pore. Afterwards the pixels at the boundary are treated spe-cially in comparison with TL value at any time point (totally 100 points, one point for one day). When less than the specific time point, the certain unit of TL will degrade beyond the lifetime (For example, TL unit is 45.99 < the specific time point 46, in-dicating the exceeded lifetime). However pure degradation can-not induce the mass loss, which how to judge? In this paper we adopted the mass transfer channel in percolation theory. When one pixel degrades and any one of adjacent pixels is open pore, the mass loss will occur (No. 3 pixel in Figure 1 is open pore, so No. 5 pixel can loss the mass). Conversely once the adjacent pixels are away from the open pores (No. 6 and 7 pixels in Fig-ure 1), the mass loss is impossible. Then the amount of pores (including close pore) at each time point is calculated, and the mass loss percentage is the divisor of this obtained amount by the total amount of the pixels.

In this paper, the analysis software MATLAB was used to calculate the lifetime of the pixels generated randomly in the polymer matrices. After repeating the procedure of mass loss many times, the average value is worked out according to Monte Carlo method.

RESULTS

Calculation of degradation rates
The molecular weights of PLGA films with different initial copolymer compositions were measured at each time point and the relationships between molecular weights of PLGA films and degradation time during the degradation period are illustrated in Figure 2.

Mn and Mw decreased exponentially with degradation time and the relationship between them can be calculated by
LgM=lgM0-k·t
where M is number or weight average molecular weight, M0 is the initial molecular weight and k is degradation rate[7]. The degradation rates for PLGA 50∶50, 70∶30 and 75∶25 were 0.058 7, 0.015 9 and 0.010 3/d (based on Mn), or 0.061 0, 0.017 7 and 0.008 4/d (based on Mw), and the degradation rates based on Mn were used in Monte Carlo simulation.

Monte Carlo simulation results
The erosion model is illustrated in Figure 3.
In this schematic diagram, white, grey and black pixels represent pores (tunnels), closed and degradated domains and non-degradated domains respectively. With the com-pletion of mass loss, most of the pixels which exceeded their lifetimes could generate pores, and after many porous pixels formation and connection with each other, the tun-nels would occur. While for some pixels which exceeded their lifetimes but surrounded by the pixels which were below their lifetimes, no tunnels could support the mass transfer of these pixels according to the assumption in this paper. As a result, the closed and degradated domains oc-curred, accumulated and disappeared. The results using Monte Carlo method to simulate the mass loss of PLGA films are shown in Figure 4.

The correlation coefficients between simulation results and experiment data for PLGA 50∶50, 70∶30 and 75∶25 were 0.973 6, 0.987 4 and 0.990 3 correspondingly. Al-though the effect of disruption and breakdown of matrices was ignored during the simulation process, the simulation results fit well with the experiment data. With the decrease of molecular weight and ratio of lactic acid/glycolic acid, the quasi-stable stage was shortened and the fast mass loss stage occurred earlier. The experiment results showed the time of the second fast mass loss for PLGA 50∶50, 70∶30 and 75∶25 were about 14, 47 and 74 days which can also be predicted by Monte Carlo simulation.

DISCUSSION

Stages during mass loss process
The degradation mechanism of PLGA was widely investigated and proved experimentally in the past, and due to the autocata-lysis effect to accelerate the breakage of ester bonds, the bulk degradation and erosion mechanism was extensively accepted. However under the effect of various physical and chemical processes, the tendency of the mass loss of PLGA was compli-cated. From above simulation, four stages are exhibited in Fig-ure 5 during mass loss process. At the first stage a sharp mass loss was measured and it was mainly because of the existence of oligomer, monomer or retained solvent during synthesis and preparation. The mass loss amount during this stage is decided by the purity of materials and the drying conditions. This value can only be measured and calculated through experiment. After the release of above impurities, the real mass loss process of PLGA began. Degradation proceeded with the decrease of mo-lecular weight, and before the degradation products can dissolve in the medium, the quasi-stable stage Ⅱ occurred and last. When stage Ⅲ began, large amount of degradation products solved and released into outer medium. For the last stage, the mechanical properties of matrices generally decreased to a very low level, and the breakdown phenomenon would occur. Fur-thermore, the closed and degradated domains (Figure 3) were accumulated during stage Ⅰ and Ⅱ, and when the amounts of closed and degradated domains decreased from peaks (tp), the fast mass loss stage Ⅲ occurred. With the appearing of another knee point (tk), the mass loss processes became slow and stage Ⅳ began. It was regarded that the change of closed and degra-dated domains induced the beginning and the end of fast mass loss stage Ⅲ. Moreover, by operating the variation of closed and degradated domains, the mass loss rates of PLGA films could be further controlled to required values.

Optimization of film structure
For the drugs with high molecular weights, the diffusion process was limited. To achieve relative stable drug release kinetics, the drug vehicles were expected to design with stable mass loss rates. Monte Carlo simulation provided an effective method to predict the mass loss behaviors of PLGA films.

PLGA 50∶50 and 70∶30 bilayer and multilayer structures were applied as the subjects of investigation, and the ratios of PLGA 50∶50 decreased gradually. The same densities of PLGA 50∶50 and 70∶30 are considered, and the total thickness of the two structures and the ratios of two components are regarded as constants. The mass loss properties of bilayer and multilayer structures with equivalent ratio of two compo-nents are presented in Figure 6. With the non-uniform compo-nent distribution, the mass loss property changed obviously. For bilayer structure, two fast mass loss processes in stage Ⅲ were expressed and the amount of closed and degradated domains split into two peaks. Due to the bilayer structure, the sharply mass loss process was buffed and the mass loss rate curve ex-hibited two low peaks instead of one sharp peak in the single component. For the multilayer structure, the mass loss property of PLGA film was further improved, and the closed and degra-dated domains for multilayer exhibited a smoother variation (Figure 7) from tp to 0 than that for bilayer.

REFERENCES

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聚乳酸-乙醇酸膜在模拟体液中降解的Monte Carlo 模拟*☆

齐 民，刘洪泽，魏志勇，黄莹莹
大连理工大学材料科学与工程学院，辽宁省大连市 116024
齐 民☆，男，1960年生，吉林省怀德镇人，汉族，1994年大连理工大学毕业，博士，教授，博士生导师，主要从事非晶、纳米材料及生物材料的研究。
国家自然科学基金资助项目（30470521）*
摘要
背景：聚乳酸-乙醇酸广泛应于医用植入物与药物缓释载体，由于影响其降解过程的因素较多，使得其失重性能难以模拟。
目的：对聚乳酸-乙醇酸膜失重性能进行数值模拟，观察模拟与实际测量的相关性。
设计：重复测量。
单位：大连理工大学材料科学与工程学院。
材料：重均分子量分别为63 000, 115 000和400 000的实验用的聚乳酸-乙醇酸 50/50，70/30 和75/25，购自山东省医用高分子材料重点实验室。1,4-二氧六环为天津化学试剂一厂生产，用作溶剂。
方法：实验于2006-04/06在大连理工大学材料科学与工程学院完成。①将聚乳酸-乙醇酸分别按质量比50∶50，70∶30及75∶25溶于1,4-二氧六环，溶液浇铸成10 mm×10 mm×1 mm的聚合物膜。②聚乳酸-乙醇酸膜在37 ℃的Hank's模拟体液（pH 7.4）中浸泡，每周取出1次（质量比为50：50的聚乳酸-乙醇酸为每5天取1次），试样清洗并干燥之后使用凝胶渗透色谱仪测量其数均与重均分子量，计算降解速率。同时使用电子天平测量其失重性能，并与Monte Carlo法改进后实际测量值进行相关性分析。
主要观察指标：① 不同质量比聚乳酸-乙醇酸的降解速率。② 模拟结果与实验值之间的相关系数。
结果：①质量比50：50，70：30和75：25的聚乳酸-乙醇酸降解速率分别为0.058 5，0.016 6和0.010 1/d（数均分子量）或者0.061 0，0.017 5和0.008 5/d（重均分子量）。②聚乳酸-乙醇酸 50/50，70/30和75/25的模拟结果与实验值之间的相关系数分别为0.973 6，0.987 4 和0.990 3。
结论：数值模拟聚乳酸-乙醇酸膜失重性能与实验值符合良好。
关键词：聚乳酸-乙醇酸；Monte Carlo法；降解；溶蚀；失重性能
中图分类号: R318.08 文献标识码: A 文章编号: 1673-8225(2008)06-01153-04
齐民，刘洪泽，魏志勇，黄莹莹．聚乳酸-乙醇酸膜在模拟体液中降解的Monte Carlo 模拟[J].中国组织工程研究与临床康复，2008，12(6):1153-1156
[www.zglckf.com/zglckf/ejournal/upfiles/12-6/6k-1153(ps).pdf]
(Edited by: Nie J/Yang Y/Wang L)