Cisplatin is one of the most widely prescribed chemotherapy drugs to treat different types of cancers. However, this anticancer drug has a number of side effects such as kidney and nerve damages, anaphylactic reactions, hearing loss, nausea, and vomiting that strongly restrict its function. In the present study, single-walled carbon nanotubes (SWCNTs) are used as protective drug carriers which can decrease these severe side effects to some extent. Using the hybrid discrete-continuum model in conjunction with Lennard-Jones potential, new semi-analytical expressions in terms of single integrals are given to evaluate van der Waals (vdW) potential energy and interaction force between an offset cisplatin and a SWCNT. In addition, molecular dynamics (MD) simulations are conducted to validate the results of such a hybrid approach. The preferred location and orientation of cisplatin while entering SWCNTs are determined. It is shown that the equilibrium condition of the drug may be affected by the radius of nanotube, the orientation of cisplatin, and the distance between the central molecule of the drug (Pt) and the left end of nanotube. Furthermore, the influence of equilibrium condition on the distributions of vdW interactions is investigated.

Introduction

Carbon nanotubes (CNTs) have received considerable attention in the area of nanotechnology since the publication of Iijima’s discovery paper [(1)]. These novel nanostructured materials possess many unparallel and fascinating properties such as hollow cylindrical shape, structural perfection, low weight, and flexibility. As a result, CNTs may have a wide range of technological applications including (bio)molecule separation devices [(2)], molecular sensors [(3)], and encapsulation media for molecule storage and delivery [(4,5,6)]. The transporting capabilities of CNTs combined with appropriate surface modifications and their unique physicochemical properties show great promise in the field of drug delivery systems [(7)]. SWCNTs can become amphoteric if organic molecules possessing different electron affinities and ionization energies are encapsulated inside [(8,9)]. Encapsulation of biomolecules, such as DNA and proteins is a promising procedure for applications in gene and drug delivery [(10)]. The large surface of CNTs together with their fascinating optical and electrical properties makes these nanostructured materials suitable candidates for the delivery of drugs and biomolecules. These materials can be combined either covalently or noncovalently with drugs, nanoparticles, and biomolecules. Although there exist some pending concerns about the toxicity of them in vitro and in vivo, functionalized CNTs are found to be less toxic and are not immunogenic. Therefore, these materials are identified as one of the most favorable carriers which possess a great potential for the progression of a new-generation delivery system for drugs and biomolecules [(11)].

Since the discovery of cisplatin by Rosenberg et al. [(12)] in the mid-1960s, chemotherapies have been successful in killing free cancer cells and removing undetectable tumor microfocuses in some cases [(7)]. Cisplatin is extremely important in the medical field as a drug, approved by the Food and Drug Administration (FDA) to treat bladder cancer, ovarian cancer, testicular cancer, squamous cell carcinoma of the head and neck, cervical cancer, malignant mesothelioma, and nonsmall cell lung cancer [(13)]. This anticancer drug is used in hospitals and cancer treatment centers across the world with the brand name of Platinol. It works by binding to DNA to interfere with cellular growth, stopping cancer cells [(14)]. Although cisplatin is a lifesaving drug that is indispensable in the treatment of life-threatening cancers, there are a number of adverse side effects associated with this powerful medicine [(15,16,17,18)]. Systemic toxicity may develop at the same time due to the lack of selectivity of the drugs for cancer tissues and cells, which often leads to the failure of chemotherapies [(7)]. Accordingly, the concept of drug carriers came into consideration to improve the delivery and effectiveness of cancer drugs.

Applications of CNTs in drug delivery are investigated by Bianco et al. [(6)] and it was concluded that pristine CNTs are highly toxic, mainly due to their insolubility. Therefore, it is of high importance to verify the solubility of functionalized CNTs in physiological media. CNTs can be functionalized with bioactive peptides, proteins, nucleic acids, and drugs which can be used to deliver different cargos to cells and organs [(6)]. Because functionalized CNTs display low toxicity and are not immunogenic, such systems hold great potential in the field of nanobiotechnology and nanomedicine [(6)]. Portney and Ozkan [(19)] attempted to outline most of the current nanoparticle toolset for therapeutic release by liposomes, dendrimers, smart polymers, and virus-based systems. Advantages of nanoparticle-based imaging and targeting by the use of nanoshells and quantum dots were also explored [(19)]. Ji et al. [(20)] in their review showed how CNTs have been introduced into the diagnosis and treatment of cancer. Novel SWCNT-based tumor-targeted drug delivery systems were highlighted. Furthermore, the in vitro and in vivo toxicity of CNTs reported in recent years were summarized by them. Guven et al. [(21)] described the preparation, characterization, and in vitro testing of a new ultrashort SWCNT (US-tube)-based drug delivery system for the treatment of cancer. In particular, the encapsulation of cisplatin, a widely used anticancer drug, within US-tubes has been achieved, and the resulting cisplatin@US-tube material was characterized by high-resolution transmission electron microscopy, energy-dispersive spectroscopy, X-ray photoelectron spectroscopy, and inductively coupled optical emission spectrometry [(21)]. The relationship of the biological safety of SWNTs with their physicochemical properties such as length, purity, agglomeration state, concentration, and surface functionalization was systemically discussed and evaluated by Meng et al. [(22)]. Zhang et al. [(7)] reviewed the progress in the study on the application of CNTs as target carriers in drug delivery systems for cancer therapies.

The energetic of CNTs and drug delivery is also taken into consideration in the literature [(23,24,25,26,27)]. Hilder and Hill [(25)] modeled the encapsulation of the anticancer drug cisplatin into CNTs. They presented a hybrid discrete-continuum formulation to describe the vdW interactions between concentric cisplatin and a CNT. In their investigation, three orientations of cisplatin while entering the CNT were examined. In addition, the acceptance condition and suction energy which are two noticeable issues in medical applications were studied by the aforementioned authors. On the basis of elementary mechanics along with applied mathematical modeling techniques, the acceptance and suction energies of the cisplatin molecule inside a CNT were also studied in Ref. [26]. They investigated the radius of CNT at which suction energy is maximized and the minimum radius of CNT at which the drug molecule can be accepted by the tube. Using MD simulations, Arsawang et al. [(27)] examined the molecular properties of the encapsulation of the anticancer drug gemcitabine in SWCNTs.

In the present paper, the encapsulation of an offset cisplatin into SWCNTs is modeled based on the hybrid discrete-continuum approach together with the Lennard-Jones potential function. To this end, it is assumed that cisplatin is free to take its equilibrium position while entering CNTs. Novel semi-analytical expressions are presented in terms of single integrals to evaluate the vdW potential energy and interaction force. The results are also validated by the use of MD simulations. The influences of different geometrical parameters such as nanotube radius, cisplatin orientation, and the distance between the central molecule and the drug (Pt) from the tube left end on the preferred location and orientation of the drug and its vdW distributions are investigated. Also, the obtained results are compared with those given in Ref. [25] for the case of concentric cisplatin and a CNT.

Potential Energy and Interaction Force

The Lennard-Jones and Morse potentials are the two prevalently used empirical potentials [(28,29)]. Herein, the classical Lennard-Jones potential function is utilized as follows:
φ(ρ)=-Aρ6+Bρ12
1
in which ρ indicates the distance between a pair of atoms, and A and B are the attractive and repulsive constants, respectively. Table 1 outlines numerical values of the Lennard-Jones constants for the various atoms studied in this paper.
Table 1

Approximate Lennard-Jones constants [(25)]

A (eVÅ6)B (eVÅ12)
C–Cl113.81366602
C–Pt76.66194943
C–N38.6588933
C–H14.9414544
A (eVÅ6)B (eVÅ12)
C–Cl113.81366602
C–Pt76.66194943
C–N38.6588933
C–H14.9414544
The total potential energy can be evaluated by summing the interaction energy for each atom pair
Etot=ijφ(ρij)
2
In the prior equation, φ(ρij) denotes a potential function for atoms i and j on each molecule at a distance ρij apart. Based on the continuum approximation which assumes that atoms are uniformly distributed over the surface of each molecule, the following double surface integral can be utilized instead of the double summation in Eq. 2 as follows:
Etot=η1η2φ(ρ)dS1dS2
3
where η1 and η2 indicate the mean atomic surface density of atoms on each molecule and ρ is the distance between two typical surface elements dS1 and dS2.
It should be noted that the continuum approximation can be applied for evaluating the interaction between regular shaped molecules. Since the cisplatin molecule is normally irregularly shaped, in the present study, a hybrid discrete-continuum model is adopted as [(30)]
Etot=ηiSφ(ρi)dS
4
in which the mean surface density of nanotube is assumed to be that of graphene which is equal to 0.382 Å−2.
In addition, the vdW interaction force between the two interacting molecules can be written as
FvdW=-Etot
5

Offset Cisplatin Inside a CNT

About Cisplatin.

Cisplatin, also called cis-diamminedi-chloroplatinum (II), is a molecule made up of 11 atoms. The chemical formula of cisplatin is expressed by (NH3)2N2PtCl2. The molecular shape is square planar which in the present study is supposed that the drug is located on the xz plane or on a plane which is parallel to the xz plane. The bond lengths are given in Table 2. All the bond angles around the central molecule (Pt) are assumed to be 90 deg except Cl-Pt-Cl which is presented in Table 2. Also, for NH3 it is assumed that one of the hydrogen atoms is located on the plane of cisplatin and the others are located symmetrically above and below of the plane of the drug [(25,31,32)].

Table 2

Bond length and bond angle of cisplatin [(31)]

Bonded pairBond length (Å)
Pt–Cl2.33 ± 0.01
Pt–N2.01 ± 0.04
N–H11.05
Cl–Pt–Cl91.9 ± 0.4 deg
Bonded pairBond length (Å)
Pt–Cl2.33 ± 0.01
Pt–N2.01 ± 0.04
N–H11.05
Cl–Pt–Cl91.9 ± 0.4 deg
a

Taken from Ref. [32].

Modeling an Offset Cisplatin While Entering a SWCNT.

Consider an offset cisplatin inside a semi-infinite SWCNT of radius a, as depicted in Fig. 1. The global coordinate is considered to be at the left end of nanotube and the distance between the global coordinate and Pt is assumed to be Z. In the global coordinate, the parametric equations for the nanotube is given by (acosθt,asinθt,zt) and the coordinates of atoms in cisplatin can be generally stated as (xcis,i+ɛx,ycis,i+ɛy,Z+zcis,i) where i denotes the number of atoms in cisplatin which is 11 and ɛx is the distance between Pt and the central axis of nanotube in vertical direction and ɛy is the distance between the plane of cisplatin (which is supposed to be parallel to the central xz plane) and the central xz plane as shown in Fig. 2.

Figure 1
Cisplatin entering a CNT
Figure 1
Cisplatin entering a CNT
Close modal
Figure 2
Offset cisplatin entering a CNT
Figure 2
Offset cisplatin entering a CNT
Close modal
Thus, the distance ρi takes the following form:
ρi2=(acosθt-(xcis,i+ɛx))2+(asinθt-(ycis,i+ɛy))2+(zt-(Z+zcis,i))2
6
On the basis of the hybrid discrete-continuum method, the total potential energy between the two interacting molecules can be written as
Etot=ηai=11102π0(-Aρi6+Bρi12)dztdθt
7
In order to evaluate the integral of Eq. 7 with respect to zt, the integral of the following form must be performed:
I=0dzt(μi2+(zt-(Z+zcis,i))2)n,n=3,6
8
where
μi2=(acosθt-(xcis,i+ɛx))2+(asinθt-(ycis,i+ɛy))2
9
The two integrals appearing in Eq. 8 can be evaluated analytically. Thus, the total potential energy of system is obtained in terms of a single integral as follows:
Etot=ηai=11102πPi(θt)dθt
10
in which
Pi(θt)=j=12Mjμi1-6j[N1(j)(π2+tan-1λi)+k=13j-1Nk(j)λi(1+λi2)-k]
11
In the above equation, λi is equal to Z+zcis,iμi and the constant parameters are given by
{M1=-A,M2=BN1(1)=38,N2(1)=14N1(2)=63256,N2(2)=21128,N3(2)=21160,N4(2)=980,N5(2)=110
12
Also, due to the symmetry of the problem, the axial interaction force is only considered here. Accordingly, integrating Eq. 10 with respect to Z yields
Fztot=-EtotZ=-ηai=11102πdPi(θt)dZdθt
13
where
dPi(θt)dZ=j=12Mjμi-6j×[N1(j)(1+λi2)-1+k=13j-1Nk(j)(1+(1-2k)λi2)(1+λi2)-(k+1)]
14

Results and Discussions

In most theoretical works conducted previously on the delivery of drugs into CNTs based on the continuum approximation, for simplicity, it was assumed that the drug is limited to be on the central axis of CNT or only some special orientations were considered for the drug molecule. For instance, Hilder and Hill [(25)] supposed that the drug molecule only can move in the z direction and the orientation of the drug is constant while entering the tube. In reality, however, the molecule is free to take any orientation and location. Thus, it is of high importance to investigate these effects on the problem. Since considering all degrees of freedom is computationally costly, only two different cases are examined in this study. In the first step, it is assumed that cisplatin is located on the central xz plane and can choose its preferred position and orientation, and in the second step, it is supposed that cisplatin is free to locate on each plane which is parallel to the central xz plane which gives its equilibrium condition. It is noticeable that the equilibrium position of cisplatin occurs where the energy reaches the minimum value.

Preferred Location of Cisplatin on the Central xz Plane.

In this subsection, it is supposed that cisplatin is located on the central xz plane. In other words, the drug does not have any offset in y direction. On making this assumption, the preferred position and orientation of cisplatin while entering nanotubes are determined.

Validation of the Model.

In order to verify the discrete-continuum model developed herein, the results from the present model are compared with those from MD simulations. It should be noted that in the MD simulations conducted here, the Tersoff–Brenner [(33,34)] potential function is used to give the energy of covalence bonding between the carbon atoms. A Velocity-Verlet algorithm [(35)] is also used to integrate the equations of motion and a basic time step of 1 fs is employed to guarantee good conservation of temperature. The simulations are carried out in canonical ensemble, so called NVT, at room temperature (300 K). The Nose–Hoover thermostat [(36)] is utilized in the simulations to keep the temperature constant at 300 K. Using this thermostat results in less fluctuation of the system during the temperature stabilization. In order to impose end conditions, four layers of carbon atoms are considered to be fixed to simulate clamped–clamped boundary conditions at the end of nanotube.

Note that, the minimum radius of CNT where the cisplatin molecule can overcome the energy barrier is calculated as 4.7763 Å which takes place at 90 deg. In Fig. 3, different orientations are given and it is assumed that cisplatin is free to locate on its equilibrium position. In Fig. 3a, for a (9,9) SWCNT with radius 6.1032 Å, the distribution of potential energy on the preferred position of cisplatin is illustrated for the considered orientations. It should be mentioned that, in this geometry, the drug molecule would be definitely accepted by the nanotube. Because, the minimum radius of CNT, where the cisplatin molecule can overcome the energy barrier, is calculated as 4.7763 Å, which takes place at 90 deg.

Figure 3
Influence of orientation (a) on the distribution of energy, (b) on the preferred position of cisplatin, based on the discrete-continuum approach and the MD simulations for a (9,9) CNT
Figure 3
Influence of orientation (a) on the distribution of energy, (b) on the preferred position of cisplatin, based on the discrete-continuum approach and the MD simulations for a (9,9) CNT
Close modal

In Fig. 3b, the preferred position of the drug molecule versus Z is plotted. In accordance with this figure one can deduce that the present discrete-continuum model is accurate enough to predict the MD results.

From Fig. 3b, it can be observed that when the drug is outside of nanotube and considerably far from the left end of it, the drug prefers to locate on the central axis of nanotube, while it comes closer, the offset from central axis reaches a maximum value. When it is near to enter the nanotube, this value decreases to reach a constant value inside the nanotube which depends on the orientation.

Distribution of Potential Energy Based on Offset Configuration.

The results presented in Fig. 4 are associated with a (9,9) SWCNT. In each Z, the drug has different potential energies according to its orientation and offset from the central axis. In this figure, the potential energy in preferred position of the drug is plotted while its orientations change from 0 to 360 deg. Amongst all these orientations, the drug chooses the one which has the minimum potential energy that is shown by black spline.

Figure 4
Distribution of energy on the preferred position for a (9,9) CNT and different orientations of cisplatin
Figure 4
Distribution of energy on the preferred position for a (9,9) CNT and different orientations of cisplatin
Close modal

In Figs. 5,6,7, (8,8) and (9,9) CNTs with the radii of 5.43 Å and 6.1032 Å are used, respectively. The preferred orientation of cisplatin versus Z is plotted in Fig. 5. The results indicate that when the drug is outside of the nanotube and far from its left end, it prefers Cl atoms to be near the nanotube and generally chooses zero angle. However, while the drug comes closer, it rotates and gets different orientations. After entering cisplatin into nanotube, the orientation of the drug reaches a constant value which is equal to 270 deg for both nanotubes.

Figure 5
Preferred orientation of cisplatin on the central xz plane for (8,8) and (9,9) CNTs
Figure 5
Preferred orientation of cisplatin on the central xz plane for (8,8) and (9,9) CNTs
Close modal
Figure 6
Preferred position of cisplatin on the central xz plane for (8,8) and (9,9) CNTs
Figure 6
Preferred position of cisplatin on the central xz plane for (8,8) and (9,9) CNTs
Close modal
Figure 7
Distribution of energy for (8,8) and (9,9) CNTs
Figure 7
Distribution of energy for (8,8) and (9,9) CNTs
Close modal

The preferred position of cisplatin against Z is displayed in Fig. 6. As can be seen, when the radius of nanotube is large, the offset configuration is noticeable. The position of the drug changes while entering the nanotube so that it starts from zero and reaches a maximum value outside of nanotube then reduces to a constant position inside the nanotube.

Using Figs. 5,6, the distribution of potential energy in the preferred position and orientation of cisplatin is graphically shown in Fig. 7. As discussed in Fig. 5, when the drug is inside of nanotube, its orientation remains constant and is equal to 270 deg. Therefore, here a comparison is conducted between the results of potential energy in the preferred position and orientation of the drug and those in the case that Pt is located on the central axis of nanotube and its orientation is equal to 270 deg. As can be seen, when the radius of nanotube is small, the differences are not significant. Increasing the radius of nanotube increases the differences especially when the drug is inside of nanotube. Also, the results show that the energy distribution is exactly the same as one reported in Ref. [25] for a (8,8) CNT with orientation 270 deg, confirming the validity of the present model in the case of concentric drug molecule and a CNT.

Preferred Location of Cisplatin on a Plane Parallel to Central xz Plane.

In this subsection, it is assumed that there are no limitations for the drug in y direction while entering nanotube. In other words, cisplatin is free to move in y direction while it changes its x position or its orientation but it must be located on a plane which is parallel to the xz plane. The results presented here is for (9,9) CNT. Figure 8 compares the distribution of potential energy for the case cisplatin is located on the preferred x and y positions and preferred orientation with the results of previous section and Hilder and Hill assumption [(25)]. As can be seen, the differences are noticeable when cisplatin is inside of nanotube.

Figure 8
Importance of location of cisplatin on the distribution of energy
Figure 8
Importance of location of cisplatin on the distribution of energy
Close modal

After determining the minimum potential energy of cisplatin in each distance from the left end of nanotube, now it is desirable to investigate the location of the drug when it has the lowest level of energy. The preferred position of cisplatin in x and y directions and also its preferred orientation is depicted in Fig. 9. Figure 9a indicates that it is favorable for cisplatin to locate on the central yz plane except when it is near to the left end of nanotube and outside of it. Despite the tendency of the drug not to have offset in x direction, it prefers to have offset from central axis in y direction which can be seen in Fig. 9b. The preferred position of the drug in y direction decreases when the drug moves from outside to inside of nanotube and remains constant inside of tube.

Figure 9
Location of the drug on its equilibrium condition on a plane parallel to the central xz plane for a (9,9) CNT (a) preferred position in x direction, (b) preferred position in y direction, and (c) preferred orientation
Figure 9
Location of the drug on its equilibrium condition on a plane parallel to the central xz plane for a (9,9) CNT (a) preferred position in x direction, (b) preferred position in y direction, and (c) preferred orientation
Close modal

Analyzing the results of preferred orientation of cisplatin indicates that far from the left end of nanotube, inside and outside of it, the drug prefers its bisector to have no rotation toward the axis of nanotube but near the entrance, the drug changes its orientation as can be observed in Fig. 9c.

Using the results of Fig. 9, the distribution of interaction force while cisplatin is on its equilibrium condition is presented in Fig. 10 and is compared with the interaction forces in the location of Figs. 5,6 and the case which cisplatin is located on the central axis of nanotube. Investigating the influence of three assumptions on the distribution of interaction forces shows that there is no noticeable difference between the interaction force of case two and three but by extending our assumption to case one which is near to a real situation, it is concluded that considering the equilibrium condition of the drug can have significant influence on the distribution of interaction force when the drug is near to the left end of nanotube.

Figure 10
Influence of location of cisplatin on the distribution of interaction force for a (9,9) CNT
Figure 10
Influence of location of cisplatin on the distribution of interaction force for a (9,9) CNT
Close modal

Conclusion

In this paper, using the hybrid discrete-continuum model along with the Lennard-Jones potential, new semi-analytical expressions were presented to determine the van der Waals interactions between an offset cisplatin and a carbon nanotube. Extensive studies on the variations of potential energy were also performed while cisplatin is located on its preferred position and orientation on the plane which is parallel to the central xz plane. It was indicated that the equilibrium condition of the drug is important to take into consideration for obtaining force and energy distribution when the radius of nanotube is large. For small radii, the assumption that Pt is located on the central axis of nanotube is an adaptable assumption. The results showed that it is favorable for cisplatin to have an offset in y direction (direction which is perpendicular to the plane of cisplatin). Also, when the drug is outside of nanotube the offset is more than of when it is inside of nanotube. Partly far from the left end of nanotube regardless of being inside or outside of it, cisplatin prefers its bisector not to have any offsets in x direction and any rotations but while it wants to enter the nanotube, it is favorable for the drug rotates to the orientation which NH3 enters first. In general, the drug does not choose the central axis of nanotube for its location and prefers to maintain its position when it is inside of nanotube especially far from the left end of nanotube. In accordance with the results, it can be concluded that although the assumption of considering the drug molecule on the central axis of nanotube simplifies the calculation, the obtained results noticeably differ with the case in which the constraint is eliminated. On the other hand, since considering all degrees of freedom is time-consuming, only some of them were taken into account in this study. Accordingly, the considered situation is not as complicated as the real one and also gives a good approach of results.

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