Computational modeling can help develop this understanding and can be used to .... use [28,29]. It is highly desirable t
BIODEGRADABLE POLYMERIC DRUG DELIVERY: PARALLEL SIMULATION AND OPTIMAL DRUG RELEASE PROFILES
BY ASHLEE NICOLE FORD B.S., University of Oklahoma, Norman, 2005
THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Chemical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2009
Urbana, Illinois
Adviser: Professor Richard D. Braatz
To My Family
ii
Acknowledgments
I would like to thank my advisor Professor Richard Braatz, all of my labmates, particularly Nicholas Kee, Paul Arendt, and Masako Kishida, and my collaborators Professor Daniel Pack and Kalena Stovall for their guidance. I also am thankful for the assistance of Jim Davenport, Yolanda Small, J. J. Sestrich, Eugene Frankfurt, Shaun Joy, and Professor Laxmikant Kale. I am extremely grateful for the loving support of my dear friends and family who have encouraged me throughout the research and thesis writing process. This work was made possible through support of the Department of Energy Computational Science Graduate Fellowship provided under grant number DE-FG02-97ER25308.
iii
Table of Contents
Chapter 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 2 Analysis of Optimal Drug Release from Biodegradable Polymer Microsphere Arrays . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 8 9 10 19
Chapter 3 Parallel Simulation of Drug Release from Biodegradable Polymer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model and Parallel Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 20 22 28 35
Chapter 4 Conclusions and Future Work . . . . . . . . . . . . . . . . 4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 36 36
Appendix A HAP Crystal Growth in Polycarbonate Membrane Pores A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
40 40 40 42
Appendix B Molecular Dynamics Simulations of Benzene Binding to Glucose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Model and Simulation Details . . . . . . . . . . . . . . . . . . . . . . B.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . .
44 44 47 50 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
iv
Chapter 1
Introduction
Controlled-release drug delivery systems are being developed as alternatives to conventional medical drug therapy regimens for pharmaceuticals that require frequent administrations. The types of drugs of particular interest are deactivated quickly due to short in vivo half-life and eliminated from the body before the active agent can be completely utilized. Traditional drug regimens include oral, topical, and injection dosage forms. Figure 1.1 illustrates the differences between concentration profiles for controlled-release and traditional drug delivery systems. For traditional drug delivery, the concentration peaks shortly after the dose is administered. In order to prevent peaks of high toxic concentration, traditional therapies must use sufficiently low concentration dosages. The concentration diminishes with time as the drug is used by or expelled from the body. The drug must be re-administered to maintain the concentration in the therapeutic regime and to minimize the time elapsed at low, ineffective concentrations. With controlled-release drug delivery, the drug concentration can be sustained at desired levels within the therapeutic regime for extended periods of time. Controlled-release systems have the potential to provide better control of drug concentrations, reduce side effects caused by concentration extremes and repeated administrations, and improve compliance as compared to conventional regimens [1–3]. Despite these advantages, the implementation of controlled-release drug delivery devices for human patients has been gradual as the design of controlled-release devices depends heavily on trial-and-error experiments due to
1
Drug Concentration in Blood Plasma
Controlled-release Delivery System Traditional Delivery System
Toxic
Therapeutic
Ineffective
Time
Repeated dosages
Figure 1.1: Typical release profiles for controlled-release and traditional drug delivery systems.
incomplete understanding of the mechanisms that regulate the release processes. Computational modeling can help develop this understanding and can be used to predict drug release behavior for a wide range of conditions. Such predictive capability could be a useful tool for designing microspheres for obtaining desired drug release profiles. Poly(lactic-co-glycolic acid) (PLGA) microspheres have been extensively studied for controlled-release drug delivery mainly because the polymer is biodegradable and bioabsorbable and the size range provides the release rates appropriate for many drug compounds. These qualities allow for the passive degradation of the polymer in aqueous environments such as living tissues and for the harmless incorporation of degradation products into the surrounding
2
medium [4]. Many experimental studies have been conducted to characterize the PLGA microsphere degradation and erosion processes and the release of drug molecules from the microspheres [5–21]. Several processes contribute to the overall rate of drug release from PLGA microspheres including chemical degradation of the polymer by the autocatalytic ester hydrolysis reaction, polymer erosion, pore structure evolution as a result of mass erosion, and drug release by diffusion through the polymer matrix and the aqueous pore structure. In this thesis, the term degradation refers to the process during which the polymer chains are hydrolyzed to form oligomers and monomers. The term erosion refers to the loss of mass due to diffusion of small oligomers and monomers out of the polymer matrix. These definitions are the same as those given by Gopferich [4] and have been widely adopted in the literature. Figure 1.2 shows a schematic for the overall drug release process from microspheres made of bulk-eroding polymers such as PLGA. The drug molecules are initially distributed throughout the polymer matrix in a manner dependent on the microparticle fabrication technique. Before the drug release is initiated, water from bodily fluids in vivo or from the buffer medium in vitro must hydrate the polymer matrix of the microspheres. The hydration is a relatively fast process compared with the timescales for polymer degradation and erosion—on the order of a few minutes compared to weeks or months [22]. The water in the polymer matrix can hydrolyze the polymer chains to break them into smaller fragments. Small PLGA oligomers are capable of diffusing out of the bulk leaving void volumes in the polymer due to the oligomer mass loss. The void volumes can be connected as pores. The drug is transported through the polymer matrix and through pores due to a concentration gradient. The drug diffuses more readily in aqueous pores than in the polymer matrix, so the effective drug diffusivity increases as the pore network develops due to the hydrolysis of the polymer. The hydrolytic polymer degradation, the pore network development, and the drug transport are illustrated in Steps 3 and 3
Polymer Microsphere Drug Release Process H2O
H2O
H2O
H2O
1. A polymer microsphere initially contains some drug molecules.
H2O
H2O
H2O
H2O
2. Water moves into the microsphere.
3. Pores form in the microsphere as the polymer chemically reacts with the water.
4. Drug molecules travel out of the polymer through the pores.
Figure 1.2: Drug release process for bulk-eroding biodegradable polymer microspheres.
4 in Figure 1.2 and occur in concert for the duration of the drug release process. There also can be an initial burst effect where a large percentage of the drug is released within the first day. This effect has been reported for some formulations of PLGA microspheres; however, as this process can be diminished or eliminated by adjusting the fabrication technique [23], it is not considered in this work. Due to the relatively short time required for hydration, the erosion for PLGA and similiar linear polyester materials is referred to as “bulk erosion,” which is characterized by the reaction and diffusion processes occurring throughout the microspheres rather than progressing as a front from the surface inward. It has been observed that the microsphere bulk also can erode in a heterogeneous fashion–from 4
the particle center outward–depending on the particle size; larger microspheres have been shown to experience faster erosion in their centers than smaller microspheres [17–20, 24]. The cause of the heterogeneous mass loss is generally attributed to the autocatalytic hydrolysis reaction by which the polymer chemically degrades [20, 21, 24, 25]. The hydrolysis reaction is catalyzed by acids including the carboxylic acid end groups of the polymer chains and protons present in the buffer medium or in bodily fluids. The hydrolysis reaction proceeds by cleavage of the ester bonds of the PLGA polymer chains. At the onset of degradation, all particle sizes hydrolyze PLGA at similar rates while generating acidic byproducts. Hydrolysis eventually leads to erosion when sufficiently small water-soluble oligomer fragments from degraded PLGA are transported away from the reaction site. If the diffusion process controlled the drug release without influence by polymer degradation, larger microsphere sizes would be expected to have smaller relative release rates than smaller particles as the diffusion pathways would be longer and the concentration gradients would be smaller. Contrary to this intuitive diffusion-controlled behavior, polymer degradation does influence the drug release rates for different sized particles. In domains close to the external surfaces of microspheres, the diffusion lengths are sufficiently small for the acidic oligomer hydrolysis byproducts to diffuse out of the particles; in smaller microspheres, the entire volume can have short diffusion lengths. Acidic polymer fragments which remain in particles have hindered mobility in regions farther from the external surfaces where transport is limited by greater diffusion lengths. This leads to an accumulation of acidic degradation byproducts in the interior of larger microspheres, which results in a change in the microenvironment pH. The acidic endgroups further catalyze the hydrolysis reaction leading to accelerated degradation particularly in the interior of large microspheres due to the limited acid transport out of the center. Over time, the autocatalytic effect becomes more pronounced, and microspheres can form heterogeneous, hollow interiors [25]. Small 5
microspheres without long diffusion lengths are less susceptible to acidic buildup and heterogeneous degradation. Experimentalists have reported evidence of local pH drop due to accumulation of the acidic byproducts of the polymer hydrolysis and have detected degradation rates which increase with polymer particle size as a result of autocatalysis [6, 11–15, 20, 24, 26, 27]. The autocatalytic effects compete with the diffusion effects for control of the drug release rates [6]. Controlled-release drug concentration profiles can be determined through experiments for certain conditions or can be modeled to predict release behavior for a wider set of design parameters. One problem associated with controlled-release drug delivery is determining the optimal release profile desired for a particular application. Once the desired release profile is known, the microsphere formulation conditions can be modified to achieve the drug concentration profile. Chapter 2 of this thesis contains the derivation for the optimal release profile needed by living cells for tissue regeneration when the polymer microspheres are arranged in a linear array. Modeling controlled-release drug delivery is important for investigating large ranges of conditions that are not accessible experimentally and for predicting the conditions that would be profitable to explore without wasting laboratory materials. An accurate computational model could augment the understanding of the release mechanisms and serve as an aid for planning experiments, designing drug delivery systems to be tested, and manufacturing pharmaceutical products for actual patient use [28, 29]. It is highly desirable that computational models be applicable to a large number of chemical species in the system, include multiple physics simultaneously, and have fine spatial and temporal resolutions to simulate the system accurately. The development of a parallel algorithm for implementing a simple model for autocatalytic degradation and drug release from PLGA matrices is described in Chapter 3. This algorithm could be applied readily to more sophisticated models allowing faster code execution than sequential algorithms. Faster code execution 6
would be important for applying the model to explore large ranges of particle design conditions in order to obtain the optimal conditions for desired release profiles. The final chapter in this thesis gives conclusions and highlights future work for modeling controlled-release drug delivery from biodegradable polymers. The appendices include reports on other projects tangential to this work that were completed during the course of the MS thesis project.
7
Chapter 2
Analysis of Optimal Drug Release from Biodegradable Polymer Microsphere Arrays 2.1
Introduction
The mathematical analysis presented in this chapter was performed by Ashlee N. Ford in Fall 2006. The framework and equations were shared with Masako Kishida as an introduction to the physical problem of controlled-release drug delivery for biological tissue regeneration, and she expanded on these results in her own research. Controlled-release drug delivery using polymer microspheres as described in this thesis has been used to deliver different types of small molecules and proteins both in vitro and in vivo over extended periods of time [30–33]. It is hypothesized that proteins such as growth factors could be released in a prescribed manner from a three-dimensional array of polymer microspheres in order to maintain the concentration needed by living cells to regenerate human tissue. An example application is growth factor release to ameloblast cells to produce human tooth enamel crystals that could grow in a controlled manner to produce enamel if the environment could accurately simulate the natural conditions for teeth formation. The analysis presented here provides an expression for the optimal release of growth factor from a collection of polymer microspheres to provide linear uptake at the cell-medium interface.
8
2.2
Minimization Problem
In this analysis, the total concentration of growth factor released from a one-dimensional polymer microsphere array is investigated. Future work should consider the details of the arrangement of the particles needed to produce the optimal overall concentration profile in three dimensions. The objective function for the optimization of growth factor release and subsequent uptake is Z min C(0,t)
∞
[Jdes (t) − kC(1, t)]2 dt
(2.1)
0
subject to the partial differential equation for one-dimensional diffusion of growth factor between the microsphere array and the cells separated by unit length ∂C ∂ 2C =D 2 , ∂t ∂x
0 < x < 1, t > 0
(2.2)
with the initial condition C(x, 0) = 0
(2.3)
C(0, t) = C0
(2.4)
∂C kC(1, t) + D =0 ∂x x=1
(2.5)
and boundary conditions
and
where C(x, t) is the concentration of growth factor between the source polymer microsphere array and the target cells, D is the diffusivity of growth factor through the medium, k is the mass transfer coefficient at the interface between the medium and the cells, Jdes (t) is the biological rate of uptake of growth factor by the cells, and C0 is the constant growth factor concentration released from the microsphere array.
9
2.3
Results
Solution to the Partial Differential Equation Homogeneous boundary conditions simplify the separation of variables technique for solving partial differential equations. The boundary condition at x = 0, Equation 2.4, is nonhomogeneous. To homogenize the boundary conditions, let C(x, t) ≡ CI (x) + CII (x, t) where CI is the steady-state solution and CII is the time-variant solution. Because CI is the steady-state value,
∂CI ∂t
= 0. For CI the
partial differential equation reduces to the ordinary differential equation ∂ 2 CI = 0, ∂x2
0
