Drug release from a surface erosion biodegradable viscoelastic polymeric platform: Analysis and numerical simulation
Introduction
The drug release from a biodegradable polymeric platform in contact with a fluid, where a solid drug is initially dispersed, is a cascade of phenomena:
- (i)
The fluid enters in the polymeric structure;
- (ii)
The dissolution of the solid drug takes place when the solid drug is in contact with the fluid;
- (iii)
The dissolved drug is transported through the polymer to the exterior;
- (iv)
The polymeric structure degrades.
The mathematical modelling of the drug release from biodegradable or non-biodegradable polymeric platforms has been an object of intense research during the last years. Without being exhaustive we mention the papers [1], [2], [3], [4], [5], [6], [7], [8], [9]. Viscoelastic polymeric platforms were considered in [3], [4], [5], [6] where the non-Fickian entrance of the fluid in the viscoelastic polymeric platforms is combined with the dissolution process of the solid drug and the diffusion transport of the dissolved drug.
In [5], [6], the authors consider a penetrant diffusing in a polymeric platform that swells without erosion. In this case, the polymeric chains deform and the polymeric chains and cross-links do not break. The swelling depends on the cross-links density that maintain the internal structure organized. Polymers with high cross-links density present faster water absorption and faster swelling while polymers with lower cross-links density are characterized by slower water absorption and slower swelling. However the amount of water absorbed and the swelling in the first type of polymers are smaller than the ones observed in the second type. Polymeric degradation occurs when the polymeric chains and/or the cross-links break with the solvent by hydrolysis reactions. The balance between swelling and degradation depends on the characteristics of the polymeric chains and the cross linker agent as well as on the solvent.
In what concerns the polymeric degradation, it can be of two types: surface or bulk (Fig. 1) [7], [8], [9], [10], [11]. Bulk degradation occurs when degradation is slower than water uptake. The entire system is rapidly hydrated and polymer chains are cleaved through all polymer structures. Surface degradation occurs when degradation is faster than the entrance of water in the system. The break of polymer chains occurs mainly in the outermost polymer layers. In this case, the domain changes in time that introduces new challenges in the mathematical description of the cascade of phenomena.
In [8], Rothstein et al. propose a system of partial differential equations to describe the drug release from a biodegradable polymeric matrix considering the Fickian entrance of the fluid in the structure, the evolution of the molecular weight due to the polymeric degradation that occurs due to hydrolysis reactions, the dissolution of the solid drug and the transport of the dissolved drug.
In [1], the non-Fickian drug transport through a biodegradable viscoelastic polymeric platform is studied combining the molecular weight evolution due to the degradation process [8] with the non-Fickian drug transport due to the viscoelastic nature of the polymeric matrix. Here, it is assumed that the fluid is in equilibrium, the drug is completely dissolved and the polymer presents bulk degradation (the spatial domain is fixed).
The aim of this work is to study a system of nonlinear partial differential equations defined in a two-dimensional moving boundary domain that can be used to describe the drug release from a biodegradable viscoelastic polymeric platform, where a solid drug is initially dispersed, involving the cascade of phenomena (i)-(iv): the non-Fickian fluid uptake, the dissolution of the solid drug, the transport of the dissolved drug through the relaxed polymer, the surface degradation of the polymer. We assume that the erosion is due to hydrolysis reactions that lead to the reduction in time of the polymeric structure. We also assume that the mechanical characteristics of polymer chains, the Young modulus, and the viscosity, depending on the polymeric molecular weight [12], [13]. Moreover, we follow [14] to specify the moving boundary velocity considering that it depends on the fluid mass flux. The differential system is completed with convenient initial and boundary conditions and we remark that to the best of our knowledge, this system of partial differential equations was not yet the object of any mathematical study. We analyse the stability of the moving boundary non-linear initial boundary value problem and we propose a numerical method that will be used to illustrate the qualitative behaviour of the fluid, solid and dissolved drugs concentrations as well as the moving front of the domain. The accuracy of the spatial discretizations of the non-Fickian fluid uptake and the Fickian dissolved drug transport is established.
We would like to point out that to describe the drug release from polymeric platforms, similar to those studied here, in a target site enhanced by external stimuli like ultrasound [15], heat [16], electric [17] or magnetic fields [18], the system (1)–(5) should be adapted to take into consideration the effect of the stimulus on the drug release (polymeric platform degradation and drug transport) that should be coupled with the differential system describing the drug behaviour in the target.
The paper is organized as follows: the system of partial differential equations is introduced in Section 2 considering the drug release from a biodegradable polymeric platform that is a consequence of the cascade of phenomena described above. The stability analysis of the initial boundary value problem defined in a time-varying domain, which is a consequence of the erosion process, is presented in Section 3. Following Oberkampf in [19], in Section 4, the moving boundary initial value problem is rewritten in a fixed domain that is the basis of the numerical approach followed in Section 5. In this section, an implicit–explicit numerical scheme is proposed that is considered in Section 6 to illustrate the behaviour of the mathematical model. The convergence analysis of the discretization of the fluid uptake and the transport of the dissolved drug is presented in Section 7. We observe that the uniform boundness of the fluid approximations, which is concluded using the error estimates, has an important role in the convergence analysis of the drug approximations. Finally, in Section 8 we present some conclusions.
Section snippets
Mathematical model
We consider a two-dimensional biodegradable viscoelastic polymer (Fig. 2). We assume that initially a solid drug is homogeneous distributed in the spatial domain and that the fluid enters through the boundaries and . Consequently, it is reasonable to assume that the fluid distribution, the solid and dissolved drugs distributions in the spatial domain are symmetric with respect to the origin. These assumptions allow us to replace
Stability analysis
To simplify the presentation, we assume in what follows that , , and the diffusion coefficient are constant. Moreover, to study the stability of the initial boundary value problem (1)–(5), (6), (11), an assumption needs to be considered in the moving front velocity.
Due to surface erosion, the degradation moving front is a decreasing function in time, then is acceptable to impose that and are decreasing functions in time and their derivatives are negative. As we imposed the
Tracking the degradation fronts
In what follows we rewrite the IBVP (1)–(6), (11) in a fixed domain considering a convenient coordinate transformation. Let be the new space variables. Following Oberkampf in [19] we take We observe that if then we have and We use to denote the gradient operator with respect to the new
Numerical scheme
In this section we propose a coupled Implicit–Explicit (IMEX) method to solve the initial boundary value problem (59)–(65).
In we consider the grid , with and . We fix , and we define in the grid As we are dealing with an initial boundary problem with boundary conditions on the spatial derivatives defined on the boundary, and to obtain discrete approximations with higher
Numerical results
In what follows we exhibit some numerical results for the initial–boundary value problem (59)–(65) using the method (75)–(81). The following values for the parameters have been considered:
In Fig. 3 we plot the concentration of the water as it diffuses into the polymeric
Error analysis
To justify the behaviour of the numerical method (75)–(81), in what follows we study the spatial discretization, in one-dimension, considering only the fluid concentration without the polymeric reaction for the fluid entrance and the dissolved drug concentration . In this scenario, we analyse the convergence behaviour of the solution of the differential problem
Conclusion
A system of partial nonlinear differential equations complemented with boundary and initial conditions defined in a moving boundary domain (1)–(5), (6)–(9) is analysed from an analytical and numerical point of view. We point out that the boundary moving law (10) used here is analogous to the one proposed by Crank in [14]. This system can be used to describe the drug release from a biodegradable viscoelastic polymeric platform that presents surface erosion and where a drug is initially dispersed
Acknowledgements
J.A. Ferreira was partially supported by the Center for Mathematics of the University of Coimbra (CMUC) - UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
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