Modeling Dual Drug Delivery from Eluting Stents: The Influence of Non-Linear Binding Competition and Non-Uniform Drug Loading

Abstract

Objective

There is increasing interest in simultaneous endovascular delivery of more than one drug from a drug-loaded stent into a diseased artery. There may be an opportunity to obtain a therapeutically desirable uptake profile of the two drugs over time by appropriate design of the initial drug distribution in the stent. Due to the non-linear, coupled nature of diffusion and reversible specific/non-specific binding of both drugs as well as competition between the drugs for a fixed binding site density, a comprehensive numerical investigation of this problem is critically needed.

Methods

This paper presents numerical computation of dual drug delivery in a stent-artery system, accounting for diffusion as well as specific and non-specific reversible binding. The governing differential equations are discretized in space, followed by integration over time using a stiff numerical solver. Three different cases of initial dual drug distribution are considered.

Results

For the particular case of sirolimus and paclitaxel, results show that competition for a limited non-specific binding site density and the significant difference in the forward/backward reaction coefficients play a key role in determining the nature of drug uptake. The nature of initial distribution of the two drugs in the stent is also found to influence the binding process, which can potentially be used to engineer a desirable dual drug uptake profile.

Conclusions

These results help improve the fundamental understanding of endovascular dual drug delivery. In addition, the numerical technique and results presented here may be helpful for designing and optimizing other drug delivery problems as well.

Appendix 1: Exact Solution for Irreversible Single Drug Binding in a Two-Layer Geometry

This Appendix derives an exact solution for two special cases of the general model, for comparison with numerical computations.

Special Case of Diffusion and Irreversible Binding of a Single Drug

First, diffusion and irreversible binding of a single drug in a two-layer geometry is considered. For simplicity, superscript A is dropped from the derivation below. In this case, it is assumed that the binding site density is much larger than the concentration of bound drug, and therefore, the problem defined by equations (11), m=1,2 in non-dimensional form can be linearized as follows:

$$\begin{array}{cc}\displaystyle\frac{\partial {\overline{c} }_{1}}{\partial \tau }={\overline{D} }_{1}\frac{{\partial }^{2}{\overline{c} }_{1}}{\partial {\xi }^{2}}-{\overline{\beta }}_{1}{\overline{c} }_{1}&\qquad 0<\xi <{\gamma }_{1}\end{array}$$

(36)

$$\begin{array}{cc}\displaystyle\frac{\partial {\overline{c} }_{2}}{\partial \tau }=\frac{{\partial }^{2}{\overline{c} }_{2}}{\partial {\xi }^{2}}-{\overline{\beta }}_{2}{\overline{c} }_{2}&\qquad {\gamma }_{1}<\xi <1\end{array}$$

(37)

where ${\overline{\beta }}_{m}={\mu }_{m}^{s}{B}_{m}$ for m = 1,2, following linearization, and diffusion coefficient in the second layer is used for non-dimensionalization.

The associated boundary and interface conditions are given by equations (14)–(15).

Due to the linearization of this problem, an exact solution may be derived using the method of separation of variables. One may write the solution in the following form

$${\overline{c} }_{1}\left(\xi ,\tau \right)=\sum\limits_{n=1}^{\infty }{p}_{n}{A}_{n}\;\mathrm{cos}\left({\omega }_{1,n}\xi \right)\mathrm{exp}\left(-{\lambda }_{n}^{2}\tau \right)$$

(38)

$${\overline{c} }_{2}\left(\xi ,\tau \right)=\sum\limits_{n=1}^{\infty }{p}_{n}{B}_{n}\;\mathrm{sin}\left({\omega }_{2,n}\left(1-\xi \right)\right)\mathrm{exp}\left(-{\lambda }_{n}^{2}\tau \right)$$

(39)

Note that sine and cosine terms are not included in Eqs. (38) and (39), respectively, in order to satisfy the boundary conditions at $\xi =0$ and $\xi =1$. In addition, in order for Eqs. (38) and (39) to satisfy the governing equations (36) and (37), one must have

$${\omega }_{1,n}=\sqrt{\frac{{\lambda }_{n}^{2}-{\overline{\beta }}_{1}}{{\overline{D} }_{1}}};\quad {\omega }_{2,n}=\sqrt{{\lambda }_{n}^{2}-{\overline{\beta }}_{2}}$$

(40)

Further, in order for Eqs. (38) and (39) to satisfy the interface conditions, one must require

$${A}_{n}\;\mathrm{cos}\left({\omega }_{1,n}{\gamma }_{1}\right)={B}_{n}\;\mathrm{sin}\left({\omega }_{2,n}\left(1-{\gamma }_{1}\right)\right)$$

(41)

$$-{\overline{D} }_{1}{\omega }_{1,n}{A}_{n}\;\mathrm{sin}\left({\omega }_{1,n}{\gamma }_{1}\right)= -{\omega }_{2,n}{B}_{n}\;\mathrm{cos}\left({\omega }_{2,n}\left(1-{\gamma }_{1}\right)\right)$$

(42)

Dividing Eq. (41) by (42), and rearranging, one may derive the following eigenequation to determine the eigenvalues ${\lambda }_{n}$:

$${\omega }_{2,n}\;\mathrm{cot}\left({\omega }_{2,n}\left(1-{\gamma }_{1}\right)\right)-{\overline{D} }_{1}{\omega }_{1,n}\;\mathrm{tan}\left({\omega }_{1,n}{\gamma }_{1}\right)=0$$

(43)

Also, without loss of generality, one may assume ${A}_{n}=1$ and obtain, from Eq. (41), ${B}_{n}=\displaystyle\frac{\mathrm{cos}\left({\omega }_{1,n}{\gamma }_{1}\right)}{\mathrm{sin}\left({\omega }_{2,n}\left(1-{\gamma }_{1}\right)\right)}$.

Finally, the remaining constants ${p}_{n}$ may be obtained using the principle of orthogonality for a two-layer body as follows:

$$\begin{array}{c}p_n=\displaystyle\frac{\int\limits_0^{\gamma_1}c_{1,in}\left(\xi\right)\cos\left(\omega_{1,n}\xi\right)\,d\xi+\int\limits_{\gamma_1}^1c_{2,in}\left(\xi\right)\frac{\cos\left(\omega_{1,n}\gamma_1\right)}{\sin\left(\omega_{2,n}\left(1-\gamma_1\right)\right)}\sin\left(\omega_{2,n}\left(1-\xi\right)\right)d\xi}{\int\limits_0^{\gamma_1}\cos^2\left(\omega_{1,n}\xi\right)d\xi+\int\limits_{\gamma_1}^1\left[\frac{\cos\left(\omega_{1,n}\gamma_1\right)}{\sin\left(\omega_{2,n}\left(1-\gamma_1\right)\right)}\sin\left(\omega_{2,n}\left(1-\xi\right)\right)\right]^2d\xi}\\\end{array}$$

(44)

This completes the exact solution for the special case of irreversible drug binding in a two-layer body, with which the numerical solution may be compared.

Special Case of Pure Diffusion of a Single Drug

The exact solution for the case of pure diffusion of a single drug in a two-layer geometry can be easily obtained from the derivation above by setting ${\overline{\beta }}_{1}={\overline{\beta }}_{2}=0$ in Eq. (40), resulting in ${\omega }_{1,n}={\lambda }_{n}/\sqrt{{\overline{D} }_{1}};\: {\omega }_{2,n}={\lambda }_{n}$. The rest of the solution is identical to the one presented in the previous section.