J. Phys. Chem. B 2007, 111, 4411-4416
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A Different Diffusion Mechanism for Drug Molecules in Amorphous Polymers Zhi-Jian Zhao, Qi Wang,* Li Zhang, and Ying-Chun Liu Department of Chemistry, Zhejiang UniVersity, Hangzhou 310027, People’s Republic of China ReceiVed: NoVember 8, 2006; In Final Form: March 1, 2007
Polymer materials are widely used in controlled drug release, and the diffusion property of drug molecules in these materials is of great importance. In this work, the diffusion behavior of a model drug (aspirin) in different ratios of poly(lactic acid-co-ethylene glycol) (PLA-PEG) was investigated by molecular dynamics simulations. Two major factors, which influence the diffusion of aspirin in polymer matrix: the wriggling of the polymer chain and the free volume of the polymer matrix, are discussed. The wriggling of the polymer chain mainly controls the diffusion of aspirin molecules. Free volume becomes the secondary effect. For two different polymers having a similar degree of wriggling, the free volume controls the diffusion of the aspirin molecules. Comparing with the diffusion behavior of small gas molecules in polymer matrix, a different mechanism was proposed for the drug molecules. The drug molecules can only diffuse along with the wriggling of the polymer matrix.
1. Introduction In the last two decades, poly(lactic acid) (PLA) was widely used in clinical and medicinal fields as one of the biodegradation materials owing to its good biocompatibility, biodegradability, and mechanical properties. It was extensively considered to be a kind of material for controlled drug release.1 However, it is hard to satisfy the different release speeds for different drugs only via adjusting the molecular weight or the molecular-weight distribution of PLA, and the hydrophobic surface of PLA would be easily adsorbed by proteins and caught by reticular tissues in vivo. So, the modification of PLA is necessary and important. Many researchers have focused on the copolymer of lactic acid and other monomers2-5 such as ethylene glycol.6 Large numbers of new synthesized polymers bring a new problem of being hard to quickly choose a suitable polymer for a specified drug as its controlled release material. As we know, the diffusion and transport properties of drugs in polymer materials are of great importance in controlled drug release. Recently, many studies were focused on these properties of drug molecules in different polymer microparticles. Dong and Feng7 synthesized paclitaxel-loaded nanoparticles of poly(D,L-lactide)/methoxy poly(ethylene glycol)-polylactide blends and studied the release behavior for paclitaxel. Luan and Bodmeier8 investigated the influence of the type of poly(lactideco-glycolide) on the leuprolide release from in-situ forming microparticle (ISM) systems. Unfortunately, the experiments in this field can only observe the macroscopic behavior of drug release, and the information on the diffusion mechanism of drug molecules in polymer at molecular level is still very limited. Knowledge on the diffusion behavior of drugs in polymer and the intermolecular interactions of drug molecules with the polymer chains would help us to understand the diffusion of drug molecules in polymer, and finally, it would guide people to quickly find or directly design a controlled release material for a specified drug. An effective way to obtain the diffusion and transport properties of drugs in polymer materials at molecular level is * To whom correspondence should be addressed. Fax: +86-57187951895. E-mail:
[email protected].
using molecular dynamics (MD) simulation. Recently, the motion of small molecules in polymer matrixes was properly studied using molecular simulation. Bharadwaj and Boyd9 and Hofmann et al.10,11 simulated small molecules diffusing in barrier materials. van der Vegt et al.12,13 discussed the initial guessed configurations and the temperature effects on the diffusion of molecules in polymers. Pavel and Shanks14,15 studied the diffusion of oxygen and carbon dioxide in amorphous polymers. Tamai et al.16-18 studied the diffusion process of methane, water, and ethanol in poly(dimethylsiloxane) (PDMS) by MD simulations. Some other works have also been published in this field.19-27 A mechanism named “hopping diffusion mechanism” for small molecule, mostly gas molecules, diffusion in polymer was employed in some of these studies. However, a molecularlevel mechanism underlying the diffusion of drug molecules, which are probably much larger than the extensively studied gas molecules and simple organic molecules, through the polymer matrix is still poorly understood. Correspondingly, an in-depth understanding of larger molecule diffusion in polymer is very limited. In this work, the diffusion behavior of a model drug, aspirin, in different ratios of poly(lactic acid-co-ethylene glycol) (PLA-PEG) was investigated. The aim of this study is to uncover the factors that influence the diffusion of aspirin in polymer matrix and the diffusion mechanism of large molecules in polymers. 2. Computational Methods and Simulation Details 2.1. Simulation Details. Drug molecules releasing from medicament to the internal environment of human body might be divided into three steps. The first is that the drug molecules diffuse into the polymer matrix (controlled release material). The second is that the drug molecules diffuse inside the polymer matrix, and the last is that the drug molecules diffuse into the internal environment of human body. This work concerns the second process. Structures of the polymer (PEG-PLA) and the drug molecule (aspirin) simulated in this work are shown in Scheme 1. The simulation box contained three aspirin molecules and three polymer chains with a polymerization degree of 40. Properly,
10.1021/jp0673718 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/12/2007
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SCHEME 1: Structure of Aspirin (a) and PLA-PEG (b)a
a The different monomers of PLA-PEG were randomly distributed in the polymer chain, and the total number of monomers is 40.
TABLE 1: The Number of Lactic Acid and Ethylene Glycol Mononers in the System Investigated system no.
1
2
3
4
5
6
7
8
9
total number of mononer 40 40 40 40 40 40 40 40 40 number of lactic acid 40 32 24 20 16 8 5 2 0 mononer number of ethylene 0 8 16 20 24 32 35 38 40 glycol mononer
Figure 1. Calculated density of PLA with respect to simulation time for the revised and the nonrevised intermolecular potential in NpT MD simulation.
periodic boundary conditions were used. MD simulations and data analysis were carried out using a modified version of TINKER molecular modeling package28 and employing amber force field29,30 with a cutoff distance of 0.8 nm and with the cross-interaction parameters from the Lorentz-Berthelot rules.31
12 ) x1122 σ12 )
(σ11 + σ22) 2
(1) (2)
The structure of a polymer chain was first drawn by ChemOffice package. Compared with the true structure of the polymer, the structure drawn has a high potential energy and could not be used as the initial structure for the simulations. Energy minimization and annealing were performed to optimize the structure of the polymer. The simulation box was then built by adding three aspirin molecules and three polymer chains of the optimized single chain. The size of the simulation box was determined by the mass of the system and the given initial density of 0.6 g‚cm-3. Then, energy minimization and annealing were run again to optimize the conformation of the system. Nine ratios of the polymer chains were constructed, which are listed in Table 1. The MD simulations were then performed with NpT (constant number of molecules, pressure, and temperature) and NVT (constant number of molecules, volume, and temperature) ensembles at 310 K, which is similar to the temperature of the human body. For the NpT ensemble, the pressure was kept at 101.3 kPa. The simulation time of each stage was 2 ns and the time step was 1 fs. At first, NpT MD simulation was employed to compress the system to the experimental density, and in the first nanosecond of the NpT MD simulation, the intermolecular potential was factitiously reduced to avoid forming cluster of atoms that may lead to the simulation fail. The revised potential energy was Urev ) U/k, where U is the potential energy calculated by the original expression, and Urev is the revised potential that was used to calculate the intermolecular force, and k was set to be 1.3 in this work. Figure 1 shows the density difference of PLA between the revised and the nonrevised intermolecular potential after 2 ns NpT MD simulation. The density for the revised one reaches 1.15 g‚cm-3, which is close to the experimental density of PLA (1.25 g‚cm-3). The difference might be caused by three aspirin molecules added. In contrast, for the nonrevised one, the density of the system
Figure 2. Statistically averaged mean square displacement (MSD) of aspirin molecules in poly(ethylene glycol) versus time.
quickly increased to 0.8 g‚cm-3 and then fluctuated around the value. It was considered that the conformation of the system was trapped in a local minimum of potential energy. The second nanosecond of the NpT MD simulation was conducted with the original intermolecular potential for NpT equilibrium. Subsequently, NVT MD simulation was performed. The volume and the initial conformations for the NVT MD simulation were taken from the result of the NpT simulation, and the trajectories of the second nanosecond of NVT MD simulation were used to calculate the statistical properties. 2.2. Diffusion Coefficient of Penetrant Molecule. The diffusion coefficients of the penetrant molecule in polymer matrix were calculated from the slope of the penetrant mean square displacement (MSD) as follows:32
lim 〈|r(t + ∆t) - r(t)|2〉 ) 6D∆t tf∞
(3)
where D is the self-diffusion coefficient, t is the time, and r(t) is the position vector of the center-of-mass of the drug molecule at time t. The brackets denote an ensemble average. Figure 2 shows that the similar MSD components in the x-, y-, and zdimensions mean that the motion of drug molecules is chaotic and that eq 3 could be used in this case. Generally, the MSD with respect to time is linear except in the first several picoseconds. Therefore, in this work, the MSD of the first 60 ps was discarded and the results were averaged from 60 ps to 200 ps. 2.3. Free Volume. The simplest way to measure the free volume of the polymer matrix is based on the volume unoccupied by the atoms of the system. However, in this study, small free volume in polymer matrix has little effect on the diffusion of drug molecules, and the method mentioned above cannot
Different Diffusion Mechanism for Drug Molecules
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Figure 3. Free volume calculation. Sphere no. 1 overlapping with the polymer matrix, not recorded, and sphere no. 2 not overlapping with the polymer matrix, recorded.
Figure 4. Diffusion coefficients of aspirin in PLA-PEG with different ratios of monomers.
distinguish the size of the free volume. Another method was used here to get a more accurate result. One simulation box was divided into 50 × 50 × 50 small boxes. At the corner of each box, a sphere of diameter 0.2 nm was inserted and the positions where the sphere did not overlap with the polymer matrix were recorded as shown in Figure 3. By scanning the recorded points, one can find big free volumes that contained the aspirin molecule. Other recorded points represent other small free volumes, and they were ignored in the final result. The total number of points recorded for big free volume in total scanned numbers was taken as the free volume percentage, and it was used in the following analysis. 2.4. Wriggling of the Polymer Chain. The wriggling of the polymer chain might be measured by the MSD of the chain. However, the polymer chain was very long and the wriggling of the chain was relatively weak, so it was not appropriate to evaluate the wriggling of the polymer chain by the MSD of the center-of-mass of the chain. Symmetrical moving of the chain might cause little position change of the center-of-mass. It may lead to that the polymer chain moves greatly but a less effect on the MSD is observed. In this work, the MSD of atomic groups (MSD-AG) of the polymer main chain was used to describe the wriggling of the polymer chain, and the result was averaged over the all atomic groups. In addition, since the wriggling occurs in the main polymer chain and not in the side groups, the MSD of the latter was not included. 3. Results and Discussion 3.1. Diffusion Coefficients. The diffusion coefficients of aspirin in polymer matrix were successfully obtained for each monomer ratio of lactic acid and ethylene glycol from the mean square displacement according to eq 3, Figure 4. It was found that the diffusion coefficients of aspirin in polymer are similar when the number of ethylene glycol monomer is lower than 24 (the total number of monomer is 40). However, when this
Figure 5. Free volume distribution in polymer matrix. (a) PLA, the free volume percentage is 7.09%, (b) PEG, the free volume percentage is 2.91%. Three big free volumes that are numbered 1-3 can be easily observed in b. The 4’s are small free volumes that were ignored in the final statistical result. Two free volumes 2 and 3 were cut into pieces because of the periodic boundary conditions.
number is higher than 24, the diffusion coefficient of aspirin increases sharply. To reduce the statistical error, many MD simulations (parallel experiments) were conducted using different initial conformations for each system investigated. Because the diffusion coefficients are quite close to each other for lower ethylene glycol contents, 20 parallel simulations have been performed when the monomer ratios of lactic acid to ethylene glycol are between 40:0 and 16:24. While for other ratios, eight simulations have been conducted. In general, the uncertainty of calculation for the diffusion coefficients in this work is estimated to be 25∼40%, which is reasonable compared to the results for the small molecules diffusing in the polymer (30∼50%),27 because the numerical values of the diffusion coefficients are extremely small (usually, 2 orders of magnitude lower than the conventional liquids). 3.2. Free Volume. The motion of small molecules in polymer matrix was extensively investigated. It had been found that the free volume of the polymer was the most important factor that controls the diffusion of the small molecules.14,15 The scanned free volumes of PLA and PEG are shown in Figure 5. Figure 6 illustrates the free volume for polymers with different monomer ratios, and a minimum was observed when the ratio of two kinds of monomers was 1:4. If one considers that the polymer chain is linear and that the side groups of the polymer are branches of the linear chain (Figure 7a), lactic acid has two side groups: methyl and oxygen atom of carbonyl, while ethylene glycol has no side groups.
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Figure 6. Free volume for PLA-PEG with different ratios of monomers.
Figure 8. MSD-AG of each PLA-PEG with different ratios of monomers.
More lactic acids bring more side groups. These side groups refuse two polymer chains going closer and leave larger free volumes (Figure 7b). The number of side groups decreases along with the increasing number of ethylene glycol in the polymer chain, and the free volume between two neighboring side groups becomes larger. When this free volume becomes large enough to allow the side group of another chain to go into it, the space between two polymer chains is compressed to a minimum point (Figure 6 and Figure 7c). As the number of side groups continues to decrease, the free volume starts to turn larger (Figure 7d). At last, poly(ethylene glycol) has no side groups, so two polymer chains go closer than when they have side groups (Figure 7e). Therefore, as shown in Figure 6, the free volume has a drop when there are no side groups in the polymer chain (the last point is at 40 ethylene glycol monomers). The relationship between free volume and diffusion coefficient was not apparent in this work. The free volume of PEG is small, but the diffusion coefficient of aspirin in PEG is the largest one. On the other hand, PLA has the largest free volume in this study but the diffusion coefficient of aspirin in PLA is almost the smallest one. All these phenomena imply that some other factors control the diffusion of aspirin in polymer matrix. 3.3. Wriggling of the Polymer Chain. Another factor investigated here is the wriggling of the polymer chain. The wriggling of the polymer chains is caused by the thermal motion of the atoms. This motion has some effect on the diffusion of aspirin in polymer matrix. In this work, the wriggling of the polymer chains is measured by the MSD-AG. Large MSD-AG means that the wriggling of the polymer chains is strong.
From Figure 8, it is easy to see that the more ethylene glycol monomer there is, the stronger is the polymer chain wriggling. For these two kinds of monomers, lactic acid has a carbonyl group, and the oxygen of carbonyl has very strong polarity that increases the potential energy of the internal rotation of the polymer chain and that leads to the wiggling of the polymer chain weakening. As the number of lactic acid monomers decreases, the number of carbonyl groups also decreases and the potential energy of the internal rotation of the polymer chain is lowered. Therefore, the wriggling of the polymer chain changes more strongly. Comparing Figure 4 with Figure 8, it can be easily found that the change trend of the diffusion coefficient is consistent to that of the wriggling of the polymer chain. The diffusion coefficient of aspirin increases when the content of ethylene glycol is higher than half in polymer chain, and at these contents, the wriggling of the polymer chain also increases. On the other hand, both the diffusion coefficient and the wriggling of polymer chain are similar at other contents. In other words, the diffusion of aspirin in polymer matrix is mainly controlled by the wriggling of the polymer chain. 3.4. Diffusion Mechanism. Diffusion of molecules in polymer matrix might be considered as a two-step process: (1) motion within the cavities and (2) jumps between free volumes (cavities) or movement of the cavity itself originating from the wriggling of the polymer chain. For one kind of penetrant, the first step is controlled by the size of the free volume of the polymer matrix, while the second one is considered to be strongly affected by the wriggling of the polymer chain. In the previous studies, the diffusion behavior of small molecules in
Figure 7. Schematic drawing for the free volume of the polymers; gray space represents the free volume.
Different Diffusion Mechanism for Drug Molecules
Figure 9. Trajectories of an aspirin molecule in (a) PLA and (b) PEG within a simulation time of 1 ns. The trajectories are continuous, and they indicate that almost no hopping jump occurs.
J. Phys. Chem. B, Vol. 111, No. 17, 2007 4415 of the polymer matrix when the monomer ratios of lactic acid to ethylene glycol are between 40:0 and 16:24. The difference of the diffusion coefficients may be caused by the thermal motion of the penetrant molecule in the cavity. Large cavity is favorable to the motion of the penetrant, and the diffusion coefficient is also large. So, when the MSDs of the polymer chains are comparative, the free volume strongly affects the diffusion of the penetrant molecules. In summary, the wriggling of polymer chain mainly controls the diffusion of aspirin. Free volume becomes one of the side effects. When the wriggling of the polymer chain is similar, the free volume controls the diffusion of the penetrant molecule. 4. Conclusion
Figure 10. Linear correlation of the diffusion coefficients with respect to the free volumes: the monomer ratios of lactic acid to ethylene glycol are between 40:0 and 16:24.
polymer matrix is found to be controlled by the free volume of the polymer matrix.14,15 However, different diffusion behavior was observed when the penetrant molecule becomes larger. Molecules like gas or simple organic molecules are small enough to be dissolved in a cavity of the free volume. Accordingly, the diffusion coefficient is sensitively affected by the size of the cavity. For larger molecules, a cavity is not large enough to accept it, and the entrance of the larger molecules leads to the size of the cavity increasing. Usually, at the vicinity of the penetrant molecule, the polymer matrix does not have a second cavity to accept that molecule to move into. The restriction of the cavity means that the “hopping jump” of the larger molecule cannot occur. In our simulation, the motion of the aspirin molecule was fixed in a small area as shown in Figure 9a, and no hopping jump was observed. The diffusion of the larger molecule only can move along with the wriggling of the polymer matrix. It makes the diffusion of the larger molecule much slower than the hopping jump of a small molecule. The simulation results support this conclusion. Comparing PLA and PEG systems as shown in Figures 8 and 9, one can find that the wriggling of PLA is almost the weakest while the trajectory of the aspirin molecule is restricted in a very small space. This means that the weakest wriggling leads to the weakest motion of the cavity, so the cavity nearly fixed restricts the motion of the aspirin molecule. In contrast, PEG wriggles much more strongly and the molecule has a much bigger space to be accessible than in PLA. If the polymer chain could wriggle strongly, the position of the cavity can change quickly and the larger molecule is forced to move spreadingly. Therefore, the wriggling of the polymer chain becomes the main factor that controls the diffusion of a somewhat large molecule. When the wriggling of the polymer chain is similar, the diffusion coefficient of aspirin has a small difference and the effect of free volume of the polymer matrix gradually emerges. The data in Figure 10 demonstrate an approximately linear correlation between the diffusion coefficient and the free volume
The diffusion behavior of aspirin in PLA-PEG was successfully simulated in this work. The diffusion coefficients of aspirin in different monomer ratios were calculated, and the major factors that affect the diffusion of aspirin in polymer matrix were analyzed. The diffusion coefficients of aspirin in different polymer matrixes change along with the content of ethylene glycol monomer. When the content of ethylene glycol is less than 50%, the diffusion coefficients are similar. However, the diffusion of aspirin significantly increases along with the increasing of the content of ethylene glycol monomer when it is higher than 50%. The dominant factors that control the diffusion behavior of aspirin were investigated. It was found that the wriggling of polymer chain controls the diffusion of aspirin. When the polymer chain is wriggling strongly, the diffusion coefficient is also very large. Unlike small molecules diffusing in polymers, free volume becomes the side effect. When the wriggling of the polymer chain is similar, the diffusion of the penetrant molecules is controlled by the free volume. Comparing with the diffusion mechanism of small gas molecules in polymer, a different diffusion mechanism was proposed for the drug molecules. A relatively larger volume of aspirin molecule leads to almost no hopping jump occurring, and it can only diffuse along with the wriggling of the polymer matrix. Acknowledgment. This work was financially supported by the National Natural Science Foundation of China (Grant No. 20576112 and No. 60533050). References and Notes (1) Langer, R.; Tirrell, D. A. Nature 2004, 428, 487-492. (2) Lo, C.-L.; Lin, K.-M.; Hsiue, G.-H. J. Controlled Release 2005, 104, 477-488. (3) Kumar, M. N. V. R.; Bakowsky, U.; Lehr, C. M. Biomaterials 2004, 25, 1771-1777. (4) Barrera, D. A.; Zylstra, E.; Lansbury, P. T., Jr.; Langer, R. J. Am. Chem. Soc. 1993, 115, 11010-11011. (5) Cook, A. D.; Hrkach, J. S.; Gao, N. N.; Johnson, I. M.; Pajvani, U. B.; Cannizzaro, S. M.; Langer, R. J. Biomed. Mater. Res. 1997, 35, 513-523. (6) Gref, R.; Quellec, P.; Sanchez, A.; Calvo, P.; Dellacherie, E.; Alonso, M. J. Eur. J. Pharm. Biopharm. 2001, 51, 111-118. (7) Dong, Y.-C.; Feng, S.-S. J. Biomed. Mater. Res., Part A 2006, 78A, 12-19. (8) Luan, X.-S.; Bodmeier, R. J. Controlled Release 2006, 110, 266272. (9) Bharadwaj, R. K.; Boyd, R. H. Polymer 1999, 40, 4229-4236. (10) Hofmann, D.; Fritz, L.; Ulbrich, J.; Schepers, C.; Bo¨hning, M. Macromol. Theory Simul. 2000, 9, 293-327. (11) Hofmann, D.; Fritz, L.; Ulbrich, J.; Paul, D. Comput. Theor. Polym. Sci. 2000, 10, 419-436. (12) van der Vegt, N. F. A.; Briels, W. J.; Wessling, M.; Strathmann, H. J. Chem. Phys. 1996, 105, 8849-8857.
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