Molecular Dynamics Study of Cisplatin Release from Carbon

Jul 29, 2013 - The release dynamics of cisplatin from the interior of a carbon nanotube is studied using molecular dynamics simulations. The nanotube ...
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Molecular Dynamics Study of Cisplatin Release from Carbon Nanotubes Capped by Magnetic Nanoparticles Tomasz Panczyk,*,† Anna Jagusiak,‡ Giorgia Pastorin,§ Wee Han Ang,∥ and Jolanta Narkiewicz-Michalek⊥ †

Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30239 Cracow, Poland Chair of Medical Biochemistry, Jagiellonian University Medical College, ul. Kopernika 7, 31034 Cracow, Poland § Department of Pharmacy, National University of Singapore, S4 Science Drive 4, 117543 Singapore ∥ Department of Chemistry, National University of Singapore, 3 Science Drive 3, 117543 Singapore ⊥ Department of Chemistry, Maria Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 3, 20031 Lublin, Poland ‡

S Supporting Information *

ABSTRACT: The release dynamics of cisplatin from the interior of a carbon nanotube is studied using molecular dynamics simulations. The nanotube is initially capped by magnetic nanoparticles which, upon exposure to an external magnetic field, detach from the nanotube tips, and the initially encapsulated cisplatin molecules leave the nanotube interior according to the diffusion mechanism. Diffusivities of cisplatin in bulk water and inside the nanotube were determined by analyzing the mean-square displacements, and they take the values 2.1 × 10−5 and (0.6−0.9) × 10−5 cm2 s−1, respectively, at 310 K. The release of cisplatin was found to be an activated process with the activation barrier ∼25 kJ mol−1 in an ideal system. Analysis of experimental data allowed for the estimation of the diffusion barrier in the actual system which was found to be ca. 85 kJ mol−1. The difference between these two estimations is attributed to the existence of numerous surface defects in the case of experimental system. The release dynamics proceeds according to a simple 1D Fick’s mechanism, and either simulation or experimental data follow a very simple equation derived from the above assumption. That equation predicts that the release of simple molecules from carbon nanotubes should obey the second-order kinetic equation. The time scale of the release depends on the nanotube length, initial amount of drug, and diffusivity of drug molecules inside the nanotube. Simulations predict that, for the studied ideal architecture, the release completes in a few milliseconds. Experimental data show that that process is, due to surface defects, definitely slower; i.e., it needs about 3 h.

1. INTRODUCTION Stimuli-responsive systems being able to release guest molecules in controlled manner are receiving much attention because of their potential application in targeted drug delivery.1,2 Among many possible concepts investigated to date much attention is paid to systems revealing mesoporous channels being able to collect drug molecules in their interiors. Important examples of one of such systems are mesoporous silica nanoparticles. When additionally equipped in macromolecular or nanoparticle caps, plugging of the channels entrances, they may realize the scheme of the controlled release trigged by some physical or chemical stimuli. One of such triggering factors might be pH change from physiological 7.4 to acidic one ∼5.5, which happens in solid tumors due to hypoxia.3−5 Another important example of drug carriers is carbon nanotubes, CNTs. They reveal many advantages due to their high mechanical and chemical stabilities, needle-like shape facilitating transport through cell membrane, and ability to form conjugates with various functional groups.6 The scheme of drug © 2013 American Chemical Society

encapsulation, corking, and subsequent uncorking together with drug release has already been experimentally engineered by our group using the nanoextraction processes.7,8 However, in vivo release needs triggering factors which are not harmful and provide adequate control on the process. Application of the mentioned pH change is a very promising approach though the reported results of successful realizations of such processes either need definitely lower pH than 5.59 or lead only to somewhat higher release rate at acidic pH than that at the neutral one.10 Another useful method of triggering drug release from CNTs is the application of the infrared radiation. It has been shown, using molecular dynamics simulations, that the state of water inside narrow CNTs differs qualitatively from its bulk properties, and the infrared radiation may lead to rapid increase of water pressure and temperature.11,12 Moreover, the Received: June 6, 2013 Revised: July 26, 2013 Published: July 29, 2013 17327

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magnetization and magnetic anisotropy of the MNPs.18 Considering that magnetic cores need some protective shell, the effective diameter of the MNPs (100 Å) determines the minimal length of the nanotube which should be larger (or at least equal) than the sum of the MNPs diameters. The presence of silica shell reduces the range of dispersion forces originating from metallic nanoparticles, provides electrostatic stabilization in solution, and provides functional silanol groups for attachment of poly(ethylene glycol) linkers.18 The force field associated with the NC consists of several types of interactions. The interatomic interactions are set up according to the General Amber Force Field (GAFF).19 The structure of the nanotube remains rigid since the deformation of the CNT due to interaction with e.g. magnetic NPs is, at the considered conditions, unlikely.20 Very important component of the force field is the dispersion interaction between MNPs and other species. It can be adequately described within the Hamaker theory of dispersion interactions.21 Additionally, due to large size of the simulation box (500 × 500 × 500 Å), it is impossible to treat solvent (water) molecules explicitly. Therefore, the recently developed22 implicit solvent model, based on Hamaker theory, is utilized here. The magnetic interactions due to magnetic anisotropy of the MNPs, external field, and dipole−dipole interactions complete the force field associated with the NC. Their detailed description and method of coupling the processes of magnetization reversal with the standard atomistic molecular dynamics scheme are provided in ref 18. The key assumptions are that the MNPs are single domain particles revealing the uniaxial magnetic anisotropy and the magnetization reversal proceeds according to the coherent rotation mechanism.23 Both assumptions are justified according to the results of experimental studies of cobalt and a few other nanoparticles.24 A new component of the force field utilized in this study comes from the presence of cisplatin in the simulation box. Both molecular topology and interaction parameters for cisplatin are taken from the work by Lopez et al.25 Because we are dealing with the implicit solvent model, the parameters of the interatomic interactions between cisplatin and other species are calculated according the rule derived in ref 22. That is, a given Lennard-Jones εii parameter is converted into a formal Hamaker constant for a single atom, and next the effective Hamaker constant for mixed interactions across solvent is computed. Finally, that effective Hamaker constant is converted back into Lennard-Jones εij value. Thus, the mixed Lennard-Jones εij values within the implicit solvent approximation are computed according to the equation

energy transfer between a CNT heated by the infrared radiation and the drug and water molecules is very fast.13 Precise control of the onset of drug release and ability to confirm that the drug carrier has reached the target are possible by utilizing magnetic nanoparticles, MNPs, as components of drug carriers. MNPs are widely applied in medicine as contrast agents; there is also increasing interest in applying them as hyperthermia agents or just drug delivery vehicles. When MNPs are coupled with carbon nanotubes, then a new class of magnetic field sensitive materials appears. Simple attachment of MNPs to CNTs leads to systems which can be manipulated by magnetic field gradients. More sophisticated designs may lead to materials revealing extraordinary properties though their fabrication might be a challenge. Molecular modeling is a fast and reliable tool for screening of complex molecular architectures. In that way key properties of such systems might be predicted or even created by tuning and adjusting parameters of a given model. We applied that methodology for studies of systems composed of CNTs and MNPs covalently linked to nanotube tips by simple chain-like molecules. Those systems, which we called magnetically controlled molecular nanocontainers, NC, have been carefully studied in a series of our recent publications using either Monte Carlo or molecular dynamics simulations. By applying Monte Carlo methods, we have determined the most promising architectures of the NCs in terms of topology, sizes, material properties, and key components of the force field.14−17 In the very recent publication18 we focused on dynamic properties of those systems by applying the molecular dynamics modeling. Conclusions coming from those studies are very encouraging, and we stated that the NC can realize the scheme of uncorking of carbon nanotubes induced by externally applied magnetic fields. The current study focuses on the stage of drug release from the NC during the magnetically triggered uncapping. We used cisplatin as a drug model because it is one of the most frequently applied anticancer drugs, and its administration is accompanied by severe side effects. This work mainly deals with the release mechanism of cisplatin, its diffusivity and structure in the inner space of the NC, stability of the capped state of the NC containing cisplatin, and time scales of the release process. The obtained results strongly support application of the NC as a very efficient, in terms of the release rate, drug carrier.

2. METHODS The topology and the force field controlling the properties of the NC are actually the same as those utilized in our recent work18 devoted to magnetic anisotropy effects in processes of magnetically assisted uncapping of the NC. Basically, the NC is composed of a 5-walled carbon nanotube with the inner and outer diameters 39.15 and 67.34 Å, respectively. The CNT length is 200 Å, and both its terminal rings are saturated by amide groups. The MNPs are bound to CNT tips by triethylene glycol chains. The MNPs are composed of 80 Å in diameter magnetic cores which are covered by 10 Å thick silica shells, so the total diameters of the MNPs are 100 Å. The magnetic cores are assumed to be cobalt nanoparticles while the silica shells act as protective and stabilizing agents. The assumed large size of the NC is necessary in order to satisfy the required energetic balance between the capped and uncapped configurations. Multiwalled nanotube ensures stronger biding of the MNPs at the CNT tips than at the sidewalls.20 The assumed size of magnetic cores provides sufficiently large

εij = ( εii − vi A ww /144 )( εjj − vj A ww /144 )

(1)

where Aww is the Hamaker constant of solvent (water) and vi, vj are the factors which mean how many solvent molecules are replaced by every single atom belonging to the molecule i, j. These factors can be determined from simple geometric considerations, and for the two types of molecules studied here, i.e., the NC and cisplatin, they are 0.278 and 0.514, respectively. More information concerning the derivation of eq 1 can be found in ref 22. The electrostatic interactions between partial charges are explicitly accounted for using the Coulomb potential. However, due to the implicit solvent model assumed, the electrostatic interactions are screened according to the Debye screening model, i.e., by applying the exponential decay of forces with 17328

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Nose−Hoover scheme. The cisplatin molecule was not explicitly thermostated in order to keep its motion unaffected by the artificial velocity changes imposed by the thermostat. Its temperature was maintained at the prescribed level due to collisions with water molecules. The diffusivities were obtained from the slopes of the mean-square displacements versus time, which followed linear relationships as predicted by the Einstein law. Table 1 shows the determined values of cisplatin diffusivity in water at three temperatures. For comparison, reference values

separation distance. For the considered conditions, i.e., the ionic strength of physiological solution 0.145 mol L−1 NaCl, the Debye screening length is set to 8 Å while the cutoff distance for both electrostatic and Lennard-Jones interactions is 30 Å. For the dispersion and electrostatic interactions between MNPs and cisplatin the cutoff distance is 125 Å. The very large cutoffs were necessary since the Hamaker potential decays slowly with the distance and the cutoffs must be significantly larger than the MNPs sizes. The distances assumed above were obtained from several runs aimed at determining the minimal cutoff distance for which further increase does not significantly change the pair energy. All calculations were performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code26 with several extra classes written from scratch and working on magnetic torques. Normally, calculations were performed in NVT ensemble using 1.0 fs time step. Some preliminary calculations, aimed at determining the diffusivity of cisplatin in water, were carried out in NPT ensemble using explicit TIP3P water model. The number of atoms creating the nanocontainer was 65 756, while the number of cisplatin molecules was from 12 to 306. The calculations were based on the Langevin dynamics27 which, according to the fluctuation− dissipation theorem, links the dynamics of a system with friction forces, which, in turn, allows for a proper description of the real time. The friction forces acting on each atom were determined from the Einstein−Stokes equation and were suitably rescaled in order to match the diffusivity of the cisplatin in water. That diffusivity was determined from separated calculations involving explicit water molecules.

Table 1. Free Diffusion of Cisplatin in Water Determined from Simulations and Calculated Using Eq 2

kBT 6πηr

(2)

(η = dynamic viscosity of fluid, r = radius of a spherical particle) for determination of D values.28 However, cisplatin, due to its planar shape, does not satisfy the fundamental condition for the Einstein−Stokes equation to be applicable. Therefore, the determination of the accurate value of the diffusivity of cisplatin in water is important for further studies focused on its release dynamics. Determination of the diffusion coefficient from molecular dynamics simulations is straightforward and reliable. It is based on monitoring of the mean-square displacement of a particle and taking its limit for long simulation times: D=

⟨x 2⟩ 1 lim 6 t →∞ t

D, cm2 s−1 × 105

D, cm2 s−1 × 105 (Einstein−Stokes)

293 300 310

1.49 ± 0.10 1.67 ± 0.05 2.10 ± 0.11

0.673 0.812 1.034

determined from the Einstein−Stokes eq 2 are also provided. It is clearly seen that the Einstein−Stokes equation underestimates the diffusion coefficients. They are ca. 2 times smaller than the values determined from simulations, which should be considered as reliable. The reason for those discrepancies is obviously related to the nonspherical shape of the cisplatin molecule. Thus, the cisplatin molecule turns out to be very mobile in diluted water solutions; its diffusion coefficient at 310 K is comparable to water free diffusion at 298 K, i.e., 2.3 × 10−5 cm2 s−1. 3.2. Cisplatin in the Nanocontainer Interior. The properties of cisplatin in the NC interior will differ significantly from the properties in bulk solution. This is due to the presence of an extra potential field created by the nanotube walls and MNPs. Additionally, due to the confined space, the diffusion coefficient of cisplatin will be altered. All these factors are important in terms of the cisplatin release dynamics from the interior of the NC, and they were carefully studied. The previously determined free diffusion coefficient of cisplatin has been used for adjusting the dumping factors in the Langevin dynamics in order to keep its value intact in the implicit solvent dynamics. The calculations were performed for four different numbers of cisplatin molecules in the NC interior, namely, 12, 34, 170, and 306. The first case corresponds to the saturated concentration of cisplatin in 0.9% sodium chloride solution at 298 K, i.e., 5 ×10−3 mol L−1.29 The concentration of cisplatin in the inner space of the NC is supposed to be higher than in the bulk due to the adsorption phenomenon. It is, in theory, possible to track the spontaneous filling, but the simulation would take too long to be practical. Therefore, these three larger numbers of cisplatin were intentionally selected just to give us a notion about the concentration dependence of equilibrium properties of cisplatin inside the NC interior as well as its release dynamics. The starting point in each case was the capped state of the NC, so the cisplatin molecules were encapsulated in the NC interior by the MNPs stuck to the CNT tips. Equilibration of the systems was relatively fast, as found from the analysis of the total energy profiles; however, we allowed each system to relax for at least 4 ns, and the next 6 ns was for data collection. Thus, the total time spent by the NC’s in the capped states and without the external magnetic field applied was 10 ns. We have not observed either spontaneous uncapping due to thermal

3. RESULTS AND DISCUSSION 3.1. Free Diffusion of Cisplatin in Water. The release of cisplatin from the inner cavity of the NC is driven mainly by the diffusion process. Thus, a key factor which needs a careful analysis is the diffusion coefficient D of cisplatin in water or in physiological fluid. Because the diffusivities of various simple molecules at the same conditions are normally similar, it is a common practice to use the Einstein−Stokes equation D=

T, K

(3)

These preliminary calculations were performed using 3438 water TIP3P molecules and a single cisplatin molecule. The interatomic interactions were computed using standard mixing rules, and electrostatic interactions were treated according to Ewald summation technique with the cutoff 12 Å. The water molecules were thermostated and barostated according to the 17329

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Table 2. Some Properties of the Cisplatin Molecules Encapsulated in the NC Interior Determined at the Temperature 310 Ka N 12 34 170 306

D, cm2 s−1 × 105

Da, cm2 s−1 × 1010

± ± ± ±

5.91 5.87 5.76 8.79

0.918 0.844 0.683 0.560

0.070 0.060 0.042 0.030

⟨UCP‑CP⟩, kJ mol−1 −0.04 −0.07 −0.49 −1.10

± ± ± ±

0.07 0.06 0.06 0.08

⟨UCP‑CNT⟩, kJ mol−1 −24.83 −24.67 −23.71 −21.55

± ± ± ±

1.43 0.89 0.48 0.39

⟨UCP‑MNP⟩, kJ mol−1 −2.0 −2.2 −1.3 −1.4

± ± ± ±

0.7 0.4 0.1 0.1

(−23.3) (−22.6) (−25.3) (−26.1)

D is the diffusion coefficient, Da is the diffusion coefficient for activated diffusion, i.e., Da = D exp(−EA/kBT). ⟨UCP‑CP⟩ is the mean interaction energy between cisplatin molecules, ⟨UCP‑CNT⟩ is the mean interaction energy of cisplatin with the nanotube walls, and ⟨UCP‑MNP⟩ is the mean interaction energy of cisplatin with the MNPs. All energies include van der Waals and electrostatic components of the interaction and are calculated per one cisplatin molecule. The ⟨UCP‑MNP⟩ energy per one cisplatin molecule located in the first coordination layer around the MNPs (186.7 Å away from the MNPs centers) is given in parentheses. a

fluctuations or the uncapping induced by the internal forces (pressure) exerted on the MNPs by the cisplatin molecules. Table 2 shows some key results concerning the state of cisplatin molecules confined in the NC cavity. The diffusivity of the cisplatin molecules is reduced when compared to the free diffusion of a single molecule. This is due to two obvious factors; first, the intermolecular interactions at higher concentrations limit the mobility of molecules; second, the interaction with the nanotube walls and confined space motion significantly limit the occurrence of large displacements. Hence, even for small concentrations (N = 12), the diffusivity is reduced more than twice when compared to the free diffusion in water, and this might be attributed mainly to the confined space motion. For larger concentrations, the increasing role of intermolecular interactions reduces the diffusivity even stronger. This is clearly seen in the concentration dependence of ⟨UCP‑CP⟩ energy; large displacements at higher concentrations need breaking significant potential energy barriers. The decreasing ⟨UCP‑CNT⟩ at higher concentrations might be attributed to the decreasing amount of free space on the CNT walls and the increasing probability of distortion of an optimal configuration of the cisplatin molecule by its neighbors. The ⟨UCP‑MNP⟩ component decreases with the increasing concentration, and this indicates that there is a limited number of available interaction sites adjacent to the MNPs surfaces. The release dynamics is, however, controlled by the activated diffusion mechanism. The molecules are trapped in the potential well generated by the nanotube walls. Thus, a successful escape attempt must be accompanied by a thermal excitation which occurs with the probability exp(−EA/kBT). This means that the diffusivity of cisplatin in the interior of the NC must, in fact, be multiplied by that Boltzmann factor in order to get an actual measure of the release dynamics. The activation barrier, EA, can be understood as a difference between the energy of the cisplatin molecules at infinite separation from the NC and the energy when they reside inside the NC. Thus, this is simply a negative value of the mean potential energy of cisplatin molecules in the interior of the NC, i.e., ⟨UCP⟩ = ⟨UCP‑CP⟩ + ⟨UCP‑CNT⟩. The contribution coming from interaction with the MNPs, ⟨UCP‑MNP⟩ is not included in EA since we consider the release from a fully uncapped NC. Looking at the values of Da in Table 2, we can see that they are about 4 orders of magnitude smaller than the corresponding values of D. It may imply that the release dynamics of cisplatin from the interior of the NC might be slow. This problem will be discussed in detail in the next section. A general notion about the structure of the cisplatin inside the inner cavity of the NC can be deduced from Figure 1. The density profiles along the nanotube axis (Figure 1A) reveal

Figure 1. (A) Density profiles of cisplatin along the nanotube axis. (B) Density profiles of cisplatin across the nanotube. (C) Radial distribution functions for distances between the MNPs centers and platinum, nitrogen, and chlorine atoms comprising the cisplatin molecules. (D) Simulation snapshot showing the structure of cisplatin inside the NC for the case of 170 cisplatin molecules. Carbon atoms creating the multiwalled nanotube have been removed for the presentation purposes. Only the terminal ring creating the innermost nanotube and the ensemble of amide groups terminating the outermost nanotube are displayed. The calculations performed for fully capped state of the NC at T = 310 K.

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It is assumed in eq 4 that the concentration drops from N to zero at the position of the hole (tips), and the distance within which the concentration vanishes is just the average fraction of the nanotube length belonging to a single molecule, i.e., L/N. Under the above assumptions the Ficks equation becomes an ordinary differential equation which solution is simple and straightforward. The above model is obviously very crude, but it offers a simple expression for prediction of the concentration drop within the interior of the nanotube as a function of time. It reads

distinct peaks near the nanotube tips. For any number of cisplatin molecules there is a concentration excess induced by the presence of the MNPs capping the nanotube tips. Clearly, there occurs adsorption of cisplatin on the surfaces of the MNPs. However, the slabs of the MNPs surfaces accessible from the CNT interior are not covered uniformly by the cisplatin molecules. They locate only on the circular strips of the MNPs surfaces adjacent to the CNT tips. This conclusion comes from the analysis of Figure 1B where the density profiles, as functions of radial distances from the nanotube axis, are shown. The cisplatin molecules locate on the inner walls of the nanotube creating a kind of statistical monolayer. The central space of the CNT is totally free of cisplatin molecules at the studied concentrations. Thus, the enhanced concentration at the tips is a result of the superimposition of forces coming from the nanotube walls and MNPs surfaces. We can also conclude that the cisplatin molecules are oriented in such a way that NH3 groups are closer to the surfaces of the MNPs than the Cl atoms (Figure 1C,D). This is the most optimal configuration since it enhances the electrostatic component of the interaction energy. Positive charges in the cisplatin molecule are located on hydrogen atoms whereas negative ones on the Cl atoms. The MNPs are negatively charged, thus we can observe that distinct separation of the distances shown in Figure 1C. Thus, when a cisplatin molecule is not stabilized in such an orientation, then, due to thermal fluctuations, its orientation in reference to the MNP surface fluctuates which, in turn, leads to reduction of its mean interaction energy with the MNP. Therefore, we do not observe, at least at the considered concentration range, a uniform coverage of the MNPs surfaces though the components of the interaction energy with the MNPs and the nanotube walls are similar (Table 2). The cisplatin molecules prefer location close to the nanotube walls where they are spatially stabilized by the interaction with the nanotube walls. Analysis of the simulation snapshots leads to the conclusion that the cisplatin molecules lie flat on the CNT walls no matter what is the distance from the MNPs. 3.3. Release of Cisplatin from the NC Interior. Molecular dynamics simulations offer a good methodology for tracking such processes as the release of drug molecules from a carrier. However, a serious limitation is always a short observation time when compared to macroscopic times. As found in the current study, the accessible simulation time, i.e. 10 ns, is not enough for emptying the NC spontaneously. Thus, it would be convenient to compare the obtained short-time results with some reference model being able to predict a longtime behavior in an analogous but simplified system. Let us, for that purpose, analyze a process of diffusion of some species out of a carbon nanotube inner cavity. CNTs are quasi-one-dimensional systems; thus, we can consider a onedimensional flux of molecules through a hole of area S which diameter is much smaller than the length L. According to the Fick’s first law, the flux of molecules is proportional to their diffusivity and the concentration drop at the position of the hole. Thus, we can easily find how the number of molecules N inside the nanotube will change in time when both CNT ends are open −

dN N N2 = 2Da S = 2Da 2 dt V (L / N ) L

N=

N0L2 L2 + 2Da N0t

(5)

where N0 is the initial number of molecules inside the CNT. Let us note that the release kinetics from the CNTs seems to be a second-order kinetics process, as follows from eq 4. Moreover, its integral form, eq 5, does not predict a squareroot time dependence of the concentration drop even at initial times. This is due to the assumed shape of the concentration gradient, but as we will soon show, the applied assumptions seem to be supported by either the simulation or experimental results. Using eq 5, we can estimate what is an approximate time necessary for the release of some amount of cisplatin from the NC using the diffusivities collected in Table 2. Equation 5 does not represent a good accuracy; nevertheless, predictions concerning the order of magnitude of the release times are very useful. Table 3 shows what are the times necessary for the Table 3. Estimation of the Release Times Obtained from Eq 5 N

t5%, ns

12 34 170 306

1.5 × 104 5.3 × 103 1.1 × 103 324

t50%, ns 2.8 1.0 2.1 6.1

× × × ×

105 105 104 103

t95%, ns 5.4 1.9 3.9 1.2

× × × ×

106 106 105 105

release of 5%, 50%, and 95% of the initial amount of cisplatin from the nanotube having the same parameters as that being the part of the NC. As expected, the release rate depends on the initial amount of cisplatin encapsulated in the CNT interior. The fastest is the case of the highest initial concentration; however, even in this case the release of 5% of the drug needs more than 300 ns. Unfortunately, such times are unreachable in the molecular dynamics simulations of the NC. On the other hand, the times necessary to release even 95% of the drug are very short in terms of the macroscopic time scale. Clearly, almost total emptying of the nanotube should be completed within a few milliseconds. This means that carbon nanotubes are very efficient carriers in terms of cisplatin release dynamics. This conclusion is drawn using a very crude model, though. However, we believe that more advanced approaches, e.g. based on the solution of the 3D Fick’s law, would not drastically alter the results predicted by eq 5. The molecular dynamics simulations of the cisplatin release were carried out according to the following scheme. The final states of the NC obtained in the studies of diffusion coefficients and containing various amounts of cisplatin were used as starting configurations. The uncapping processes of the NCs were triggered by applying the external magnetic fields. The conditions leading to a fast transition from the capped to the

(4) 17331

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uncapped state are well-known from the previous studies.18 Thus, we assumed that the MNPs reveal the saturation magnetization 1070 kA m−1, which is representative of cobalt nanoparticles.30 The magnetic anisotropy constant of the MNPs was assumed to be 107 J m−3, which is within the limits found for cobalt NPs, as discussed in ref 18. The external field strength was set to 9.3 T, and it acted along the direction perpendicular to the nanotube axis. These conditions lead to the double uncapped state of the NC in not more than 4 ns. During calculations the number of cisplatin molecules contained in the interior of the CNT was monitored. Thus, a given molecule was considered as released when its center of mass was beyond the cylinder defined by the coordinates of carbon atoms creating the innermost nanotube. Figure 2 shows the key results concerning the release dynamics of cisplatin obtained in this study. As mentioned, the

Equation 5 assumes that the released molecule immediately escapes from the potential field of the nanotube, so its energy immediately drops to zero as in an infinite separation. In simulations, as shown in Figure 3, the molecules leaving the

Figure 3. Simulation snapshots of magnetically triggered release of cisplatin from the NC taken after 2, 6, and 10 ns timesteps. The results are for 306 cisplatin molecules at the initial time and calculations correspond to the temperature 310 K.

internal space of the NC normally wander for some time on the external surface of the NC. Thus, their energy does not drop to zero but reaches some value determined by the interactions with the MNPs, linkers, and the external walls of the nanotube. Thus, the effective activation barrier is definitely smaller than that for full separation shown in Table 2. Thus, it can be concluded that the overall release process consists of, at least, two stages. The first is related to the transition of molecules from the interior of the NC to its external walls, and the second is a kind of desorption of molecules from the external walls into the bulk. Both stages are activated processes, but the corresponding activation barriers will be smaller than the total one found for full separation, which, in turn, must be the sum of these components. We can, thus, conclude that the actual release process should be faster than that predicted by the single-stage mechanism assumed in eq 5. This is because a multistep activated process proceeds as its slowest component associated with the highest activation barrier, which, in this case, must be lower than the barrier in the single-stage mechanism. The above considerations allow us to state that the release times shown in Table 3 are the upper limits for actual processes of cisplatin release from the interior of the NC. Given the milliseconds regime for t95%, the NC should be considered as a very effective carrier being able to blow a target with large amounts of drug molecules within a very short time. More details concerning the release process is provided in Figure 4, which shows the temporal evolution of the density

Figure 2. Release dynamics of cisplatin from the NC interior upon exposition of the system to the external magnetic field 9.3 T. Each curve represents different initial numbers of cisplatin molecules as shown in the legend. Numbers on the right-hand side show the final numbers of cisplatin molecules still residing in the NC interior after 10 ns simulation time. The dotted lines show fitting of eq 5 to the simulations results. The dash-dot-dot line shows an attempt to correlate the simulation data by the t0.5 model. The calculations were carried out for the temperature 310 K.

available simulation time 10 ns is not enough for emptying the NC though significant amounts of cisplatin have been released in some cases. For small initial numbers of cisplatin the release curves are almost flat, indicating that only one or two molecules left the NC interior within 10 ns. However, the curve for 306 molecules provides quite good resolution and allows for tracking the release dynamics in more details. The dotted lines in Figure 2 are fitting results of eq 5 to the simulations data. Only the diffusivities were adjusted for each case because L and N0 are strictly defined parameters of the model. As an illustration an attempt to fit the data by the square root of time model t0.5 is also provided. So, the release dynamics of cisplatin seems to follow, at least qualitatively, the very simple mechanism coded in eq 5. The effective diffusivities found in the fitting procedure are 3 orders of magnitude higher than Da in Table 2 for each case. This is mainly due to some differences in the definition of the released state of a molecule. 17332

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Figure 4. Density profiles of cisplatin along and across the nanotube axis taken after 2, 6, and 10 ns and measured for the preceding 2 ns time window. The results concern the system containing 306 molecules at the temperature 310 K. Grayed areas show the nanotube length (A) and nanotube wall width (B). In part B, the peaks on the left show the density inside the nanotube, whereas the peaks on the right show the density beyond the nanotube.

Figure 5. Simulation snapshots showing the edge views of the system taken after 2 and 10 ns. The results are for 306 cisplatin molecules at the initial time, and calculations correspond to the temperature 310 K.

profiles of cisplatin along and across the nanotube. The profiles were taken at the same timesteps as the snapshots in Figure 3. Figure 5 shows the edge views of the system after 2 and 10 ns of the release process. The density evolution in time should be compared to the results in Figure 1, i.e., in the capped state of the NC. We already mentioned that at the vicinity of the MNPs there is some concentration excess induced by the interaction of cisplatin with the MNPs. The density peaks were located at about ±100 Å from the nanotube center. As seen in Figure 4A, these peaks disappear in 2 ns of the magnetically triggered uncapping and develop at ±105 Å. Thus, the initially nonuniform concentration within the nanotube becomes uniform just after uncapping of the nanotube tips. It means that the release proceeds mainly through migration of cisplatin molecules on the inner surface of the nanotube, and the peaks at ±105 Å correspond to their location on the nanotube tips. This is the discussed first stage of the release process associated with a small activation barrier. Moreover, the reverse process seems to be fast as well since we observed frequent migrations from the nanotube tips into its interior as well. At the very beginning of the magnetically driven uncapping, i.e., when the MNPs are detaching from the nanotube tips, some specific phenomenon is observed. This is seen in Figure 2 for each initial concentration of cisplatin though for 306 it is the most visible. Namely, at the very beginning of the process the release does not occur because the slit sizes between the MNPs and nanotube tips are too narrow. Afterward, quite intense concentration drop occurs and next some stabilization or even small increase of concentration is observed. This phenomenon is related to dragging of cisplatin molecules by the MNPs

which, in turn, are driven by the magnetic field. As a result, larger amounts of cisplatin escape from the nanotube interior when compared to the free diffusion mechanism. However, due to local increase of concentration at the tips, the reverse transport toward the nanotube interior induces. Thus, in summary that phenomenon though interesting does not play an important role in the long term release dynamics. Analysis of further steps of the release process, i.e., after 6 and 10 ns, provides another portion of useful information. In Figure 4B, the development of the second peak located at ca. 38 Å occurs. It corresponds to cisplatin molecules wandering on the external walls of the nanotube. At 10 ns time step this peak increases and reveals a longer tail which indicates that some portion of the molecules have gone into the bulk. A similar conclusion can be drawn from the analysis of Figure 4A. There appear very sharp peaks at the vicinity of the nanotube tips which correspond to transient occupation of the nanotube edges. The growing in time ranges of the nonzero density at longer distances from the nanotube indicate the increasing number of permanently released drug molecules into the bulk. It seems that the discussed second stage of the release process proceeds with a significant rate after 6 ns from the beginning. Thus, the effective activation barrier associated with this second stage is larger than that for the first stage. 3.4. Analysis of Experimental Data. Filling of carbon nanotubes can be engineered by applying the nanoextraction method. It seems that large amounts of cisplatin can be loaded into the inner space of CNTs. As shown very recently, the mass ratio of cisplatin encapsulated inside the nanotube to the total mass of drug plus nanotubes reaches 0.621.8 The nanotubes used in that study had lengths about 1 μm, and their inner and 17333

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out an effective diffusivity of cisplatin in experimental conditions from the definition of the time constant a in eq 6. It turns out that such an effective diffusion coefficient is smaller than its counterparts in Table 2 by 9 orders of magnitude; i.e., it takes the value ∼8 × 10−19 cm2 s−1. There are a few possible explanations of that discrepancy. The first and the most obvious is a limited accuracy of eq 5 being due to very simple mechanism assumed in its derivation. Thus, the diffusivity estimated from eq 6 might be only an effective value being a result of some neglected factors. These factors are mostly due to the idealized geometry of the nanotubes assumed in eq 5. It is obvious that real nanotubes of sizes about 1 μm are curly and create bundles, so this might slow down the diffusion of cisplain inside the nanotubes, and effectively we can obtain too low value of diffusion coefficient from the time constant a in eq 6. The other reliable explanation of the discrepancy between simulation and experiment is an inherent nonideality of real systems which normally reveal many local defects in atomic structure. Those defects lead to significant increase in interatomic interaction energy, and as a result, the activation barriers for surface diffusion and desorption increase adequately as well. The TEM images provided in ref 8 show that the nanotubes tips are not ideal as shown in Figure 3 or 5. They are highly distorted and probably full of atomic length defects; therefore, the interaction energy of cisplatin at the nanotubes tips is definitely stronger than in ideal systems used in simulations. Thus, the desorption of cisplatin from the tip edges or its migration toward the sidewalls might be the rate-determining step associated with the highest activation barrier. Also, we cannot exclude the existence of various bottlenecks or other steric obstacles in the inner cavities of the nanotubes. All those factors may additionally slow down the diffusion, and by comparing the effective diffusivity found from fitting of eq 6 and the simulation results, we can find out that the effective activation barrier in experiment is about 85 kJ mol−1, thus higher by about 60 kJ mol−1 than the barriers found in ideal systems using computer simulations. Another factor which may contribute to the estimated low value of the diffusivity using eq 6 is the effect of concentration. We found that higher concentrations of cisplatin reduce the diffusivity inside the nanotubes (Table 2). A similar effect was observed by Chaban et al.13 in studies of ciprofloxacin release from carbon nanotubes. We do not precisely know the initial concentration of cisplatin in experimental conditions, so it might be definitely higher than the concentrations studied in simulations. Thus, this might be another factor which contributes to the discrepancy observed in the release rates in experiment and simulations.

outer diameters were 10 nm and 30−40 nm, respectively. So, these were definitely larger objects than the NC studied here using molecular dynamics. By applying simple geometric considerations, we can estimate the number of cisplatin molecules contained in one nanotube to be ca. 8 × 106. Generally, the nanotubes reveal extremely large capacity for cisplatin as follows from the experimentally determined mass ratio 0.621, which is due to an enhanced adsorption potential in their inner cavities. The experimental measurements of the release dynamics of cisplatin from nanotubes confirm our theoretical findings. Figure 6 shows the results concerning that process for the

Figure 6. Experimental data for cisplatin release dynamics from carbon nanotubes taken from ref 8. The dotted line is the result of fitting eq 6 to these data, whereas the dashed line is an attempt to fit the data by the square root of time model t0.5. The estimated value of the time constant a = 0.0013 s−1. The release process was carried out at pH = 5.5 and temperature 310 K.

above-discussed systems. The release data are for the uncapped nanotubes at pH = 5.5 and temperature 310 K; another set of data for pH = 7.4 reveals the same release pattern meaning that at the considered pH range the release dynamics is pH independent.8 The experimental data are fitted in Figure 6 using eq 5 which was suitably transformed in order to give the percentage of the released molecules; that is, we use % = 100% ×

at , 1 + at

a=

2Da N0

(6) L2 The results in Figure 6 confirm that the release proceeds according to very simple mechanism assumed in eq 6. It is worth noting that similarly like in the case of the simulation results the square root of time model t0.5 is not applicable here. Both simulation results and experimental data obey eq 6, and we may conclude that either simulation results are supported by the experimental data or vice versa. Thus, both approaches give us an insight into different time regimes and complete each other. However, a closer analysis of time scales leads to the conclusion that there is a big mismatch between them. In experiment, the release process needs 12 000 s to reach 95% level; in simulations, as we discussed already, the same level is reached in a few milliseconds. Of course, there is a big difference in sizes of both systems; however, according to eq 5, the release rate is controlled only by the nanotube length and the initial amount of drug molecules. Both parameters of the experimental system are roughly known, so we can easily find

4. SUMMARY AND CONCLUSIONS This work provides physical insights into the mechanism of cisplatin release from carbon nanotubes which ends are capped by magnetic nanoparticles. From molecular dynamics simulations we determined the free diffusion coefficient of cisplatin in TIP3P water as well as its diffusivities inside the inner space of carbon nanotubes. We found significant reduction of the diffusion coefficient inside the nanotube: from 2-fold to 4-fold depending on the concentration. Moreover, the release of cisplatin is accompanied by significant activation barriers being due to adsorption potential inside the nanotube. The barriers are about 25 kJ mol−1, which additionally reduces the effective diffusivities for the release process to the values of the order 6 × 17334

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The Journal of Physical Chemistry C 10−10 cm2 s−1. Within the considered concentrations range the cisplatin molecules locate on the inner walls of the nanotube creating a kind of a monomolecular layer. There was also found a significant concentration excess at the vicinity of the nanotube tips being due to extra forces coming from the interaction with magnetic nanoparticles. Studies of the release process during magnetically triggered uncapping allowed us to track its molecular mechanism and draw some conclusions of a general nature. We found that the release proceeds basically in two steps. The first one corresponds to migration of the drug molecules from the inner walls of the nanotube to its tips and external walls. Thus, this stage is driven by the surface diffusion process and is accompanied by relatively small activation barriers. The next step is the desorption of drug molecules from the external walls of the NC into the bulk. Though it formally starts once the first molecules appear on the nanotube tips, its contribution becomes significant after some time (about 6 ns for the 306 molecules case) necessary for collection of larger amounts of cisplatin on the external walls. This step is also accompanied by activation barriers related to adsorption potential of the external walls of the nanotube or magnetic nanoparticles. However, those barriers must be less than 25 kJ mol−1 since this value is the sum of all barriers heights for the overall release process. We found a very simple formula, obtained from the solution of 1D Fick’s law, being able to correlate the simulation and experimental data with surprisingly good accuracy. That formula allowed us to draw some conclusions concerning the long-term behavior of the release process. Assuming a single step mechanism with the activation barrier 25 kJ mol−1, i.e., the least effective one, almost complete emptying of the nanotube occurs in a few milliseconds. The simulation data predict that the actual release rate should be definitely faster and the above estimation is just the upper limit for an actual process. Analysis of experimental data leads, however, to the conclusion that the release rate is, in fact, definitely slower. This is due to nonideality of real systems which normally reveal much more complex geometry and a lot of atomic scale defects. These defects significantly enhance the activation barriers for desorption and diffusion in actual systems. Nevertheless, carbon nanotubes and particularly the NC are very efficient agents in terms of cisplatin release rate. At the same time, carbon nanotubes alone are not able to keep cisplatin molecules in their inner cavities for a long time. Some external factor must be applied to realize the capped states of nanotubes in the stage of drugs transport. We believe that the described NC is one of the most promising constructions satisfying known requirements of drug carriers based on carbon nanotubes.

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ACKNOWLEDGMENTS

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REFERENCES

This work was supported by the Polish National Science Centre (NCN) Grant N N204 205240.

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ASSOCIATED CONTENT

S Supporting Information *

Animation showing the release process of cisplatin from the NC containg initially 306 molecules. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected], Ph +48815375620 (T.P.). Notes

The authors declare no competing financial interest. 17335

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