the Role of Phase Change in Modelling Drug-Eluting Stents



Fig. 1
Cross-section of the stent mesh, the coating, and the arterial wall: geometrical configuration and reference system (figure not to scale)



Our results demonstrate a strong coupling of the release kinetics in the polymer and the drug dynamics in the wall. The results highlight the importance of tailoring drug release to drug kinetics in the arterial wall to ensure optimal DES performance.



2 A Two-Layer Model for Drug Elution


We consider a stent that is coated with a thin layer (of thickness $$l_0$$) of a porous polymer containing a drug and that is embedded into the arterial wall. As the bulk of drug transport occurs along the direction normal to the polymeric surface (radial direction) and by assuming axial symmetry, we can restrict our study to a simplified one-dimensional model (Fig. 1). In particular, we consider a radial line crossing the metallic strut, the coating and the arterial wall and pointing outwards. Because the wall thickness is very small compared to the radius of the artery, a Cartesian coordinate system $$x$$ is used along the radial line. For simplicity, we consider the situation where the stent polymer is in direct contact with the medial layer of the arterial wall (hereafter simply referred to as wall). This layer is modeled as a homogeneous porous medium of thickness $$l_1$$. Without loss of generality, we assume $$x=0$$ is the coating-wall interface (Fig. 1).

In this paper we are interested in non-local mass transfer processes in the coating, where the drug passes from a solid (polymer-encapsulated, $$c_e$$) to a free phase ($$c_0$$) by dissolution. Similarly, a part of the dissolved and transported drug in the wall ($$c_1$$) is metabolized by the cells and transformed to a bound state ($$c_b$$). Thus, the drug delivery process starts in the coating and ends at the SMC receptors, with bidirectional phase changes in a cascaded sequence, as schematically depicted in Fig. 2. A microscopic approach would require knowledge of the specific and local geometry of the individual pore structure networks, which is unfeasible. Therefore, both the polymeric matrix and the wall are treated as macroscopically homogeneous porous media with volume-averaged concentrations. Even though we are only considering the 1D case in this study, we will express concentrations in units of $$\mathrm {mol}\,{\mathrm {m}}^{-3}$$. These chosen units have no influence on the results.

A324942_1_En_5_Fig2_HTML.jpg


Fig. 2
Schematic of the cascade mechanism of drug delivery in the coating-wall coupled system. An unbinding (resp. binding) reaction occurs in the coating (resp. in the wall) (blue arrows). In both layers, reverse reactions (red arrows) are present in a dynamic equilibrium. Drug transport occurs only in the free phases $$c_0$$ and $$c_1$$


2.1 The Two-Phase Coating Model


The coating of a DES consists of a porous polymeric matrix that encapsulates a therapeutic drug in solid phase; as such, it is unable to diffuse and to be delivered into the tissue [5]. Nevertheless, when expanded and deployed into the wall, the stent coating is exposed to the surrounding biological fluids. As a consequence, such fluids fill the interstitial spaces of the polymer and form a network of liquid channels, acting as a release medium for the drug. Thus, a fraction of the drug mass is first transferred, in a finite time, to the liquid phase, and then released and diffuses into the arterial wall. We carry out a mesoscale description of the volume-averaged drug concentrations in the liquid ($$c_0$$) and solid ($$c_e$$) phases considered separately. By using a microstructure argument [17], the governing equations are:


$$\begin{aligned} {\partial c_e \over \partial t}&= - \beta _0 c_e + \delta _0 c_0&{\text {in}} \, (-l_0,0)\,\, \end{aligned}$$

(1)



$$\begin{aligned} {\partial c_0 \over \partial t}&=D_0 {\partial ^2 c_0 \over \partial x^2} + \beta _0 c_e - \delta _0 c_0&{\text {in}} \, (-l_0,0)\,\, \end{aligned}$$

(2)
where $$D_0$$ ($${\mathrm {m}}^2\,{\mathrm {s}}^{-1}$$) is the diffusion coefficient of the unbound solute, $$\beta _0 \ge 0$$ ($${\mathrm {s}}^{-1}$$) and $$\delta _0 \ge 0$$ ($${\mathrm {s}}^{-1}$$) are the unbinding (dissociation) and binding (re-association) rate constants in the coating, respectively [10]. It should be noted that $$\beta _0$$ depends on the porosity of the coating $$\varepsilon _0$$ [17]. The binding rate constant is defined as the inverse of the characteristic solid-liquid transfer time scale, $$\delta _0=t_0^{-1}$$. The ratio of the unbinding and binding rate constants is the equilibrium dissociation constant $$K_0=\displaystyle \frac{\delta _0}{\beta _0}=\frac{1-\varepsilon _0}{\varepsilon _0}$$. $$t_0$$ and $$K_0$$ are quantities that can typically be determined experimentally.

The associated initial conditions are:


$$\begin{aligned} c_e(x,0)=C_e \qquad \qquad c _0(x,0)=0 \end{aligned}$$

(3)
expressing that, at initial time, the entire drug exists in the solid phase at a maximum constant concentration, and it is subsequently released into the liquid phase. Since the metallic strut is impermeable to the drug, no mass flux passes through the boundary surface $$x=-l_0$$; hence, we impose a no-flux condition:


$$\begin{aligned} D_0 {\partial c_0 \over \partial x} =0 \qquad \qquad {\text {at}}\, x=-l_0 \end{aligned}$$

(4)


2.2 The Two-Phase Wall Model


Drug that enters the arterial wall is transported by convection and diffusion through the tortuous paths of the extracellular matrix surrounding the wall’s cells. Similarly to the coating, a phase change can occur in the wall from the free state ($$c_1$$) of the drug to the bound state ($$c_b$$) and vice versa at the surface of the cells within the wall. The free drug ($$c_1$$) binds to specific receptors on the surface of SMCs to form a bound complex ($$c_b$$). The maximum density of specific receptors available to the drug is denoted as $$c_{b}^{max}$$. The formed drug-receptor complex is not permanent and can be dissolved after a typical time scale $$t_1$$. These two processes (binding and unbinding) are modeled by a second order reversible saturating binding equations [1]:


$$\begin{aligned}&{\partial c_1 \over \partial t} =D_1 {\partial ^2 c_1 \over \partial x^2} - V_1 {\partial c_1 \over \partial x} - \beta _1 c_1 \left( c_{b}^{max}- c_{b}\right) + \delta _1 c_{b}&\text {in} \, \, (0 , l_1) \end{aligned}$$

(5)



$$\begin{aligned}&{\partial c_b \over \partial t} = \beta _1 c_1 \left( c_{b}^{max}- c_{b}\right) - \delta _1 c_{b}&{\text {in}} \, \, (0 , l_1) \end{aligned}$$

(6)
where $$D_1$$ is the effective diffusivity of unbound drug, $$V_1$$ ($$\mathrm {m}\,{\mathrm {s}}^{-1}$$) is a convection velocity, and $$\beta _1 \ge 0$$ ($${\mathrm {mol}}^{-1} \, {\mathrm {m}}^3 \, {\mathrm {s}}^{-1}$$) and $$\delta _1 \ge 0$$ ($${\mathrm {s}}^{-1}$$) are the binding and unbinding rate constants, respectively [15]. The unbinding rate constant is defined by the inverse of the characteristic unbinding time scale, $$\delta _1 = t_1^{-1}$$. It should be noted that due to the cross product of $$c_b$$ and $$c_1$$ in the binding term of Eq. (6), the units of $$\beta _1$$ are different from those of $$\beta _0$$ ($${\mathrm {s}}^{-1}$$). $$\beta _1$$ is related to $$\delta _1$$ by the equilibrium dissociation constant $$K_1=\displaystyle \frac{\delta _1}{\beta _1}$$ ($${\mathrm {mol}} \, {\mathrm {m}}^{-3}$$). The initial conditions are:


$$\begin{aligned} c_1(x,0) = 0 \qquad \qquad c_{b}(x,0) = 0 \end{aligned}$$

(7)
Finally, a perfectly absorbing boundary condition is imposed at the wall limit:


$$\begin{aligned} c_1(x,0)=0 \qquad \qquad \text {at}\, x=l_1 \end{aligned}$$

(8)
At the coating-wall interface, we impose the balance of flux


$$\begin{aligned} D_0{ \partial c_0 \over \partial x} = D_1{ \partial c_1 \over \partial x} - V_1 c_1 \qquad \qquad {\text {at}} \, \, x=0 \end{aligned}$$

(9)
and continuity of the liquid-phase concentration:


$$\begin{aligned} {c_0 \over k_0 \varepsilon _0 } = {c_{1} \over k_{1} \varepsilon _{1}} \qquad \qquad {\text {at}} \, \, x=0 \end{aligned}$$

(10)
where $$\varepsilon $$ and $$k$$ are the porosities and the partition coefficients in the respective layers.


2.3 Physiological Parameters


The two most common drugs used for DES are paclitaxel and sirolimus. These two drugs have similar molecular size and resulting transport properties (effective diffusivity and convection velocity), but drastically different binding and unbinding dynamics in the arterial wall, resulting in different binding $$\beta _1$$ and unbinding $$\delta _1$$ rate constants and maximum binding site densities $$c_{b}^{max}$$ [14]. Based on the parameters determined in [14, 15, 17], we have chosen a set of parameters that describes a generic drug that is similar in its transport properties to paclitaxel and sirolimus and lies in between paclitaxel and sirolimus for its binding/unbinding properties. The parameters of the reference model are summarized in Table 1. We consider the entire polymer and wall tissue as accessible and thus $$k_0=k_1=1$$. The values for $$\delta _0$$ and $$\delta _1$$ correspond to a characteristic solid-liquid transfer time $$t_0 = 1$$ day and an unbinding time scale $$t_1=100$$ h, respectively.
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Oct 21, 2016 | Posted by in BIOCHEMISTRY | Comments Off on the Role of Phase Change in Modelling Drug-Eluting Stents

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