Mathematical models of drug release from degradable hydrogels

CHAPTER

Mathematical models of drug release from degradable hydrogels

9

Ghodsiehsadat Jahanmir, Ying Chau Department of Chemical and Biological Engineering, The Hong Kong University of Science and Technology, Hong Kong SAR, China

Chapter Outline 1 Introduction ....................................................................................................... 233 2 Degradation, swelling, and erosion ..................................................................... 234 2.1 Statistical-kinetic models .....................................................................235 2.2 Treelike theory .....................................................................................245 2.3 Zero order surface erosion ....................................................................247 3 Drug release ...................................................................................................... 248 3.1 Diffusion controlled .............................................................................249 3.2 Degradation controlled drug release .......................................................250 3.3 Diffusion/degradation controlled release from bulk degrading networks ......257 4 Disintegration time ............................................................................................. 264 4.1 Mathematical model ............................................................................264 4.2 Experimental system and validations .....................................................265 4.3 Contributions and constraints ...............................................................265 5. Conclusion ....................................................................................................... 266 References ............................................................................................................ 266

1 Introduction Hydrogels are networks of hydrophilic polymers. Due to the presence of chemical or physical cross-linkages, they can absorb a large amount of aqueous solvent within their structure without solvation. These properties make them suitable candidate for numerous applications such as superabsorbent materials (Buchholz and Peppas, 1994), contact lens (White et al., 2011), bioadhesives (Khanlari and Dub
e, 2013), drug delivery platform, and tissue-engineered implants (Vashist et al., 2014; Park et al., 2011; Huglin, 1989; Chung and Park, 2009; Kamaly et al., 2016). In drug delivery, they can be used as a depot to encapsulate and release Biomedical Applications of Nanoparticles. https://doi.org/10.1016/B978-0-12-816506-5.00002-4 # 2019 Elsevier Inc. All rights reserved.

233

234

CHAPTER 9 Mathematical models of drug release

labile proteins (Yu and Chau, 2012; Yu et al., 2015). Degradable hydrogels are attractive as they negate the need of removal after the end of application. Hydrogels with different properties can be designed through manipulating a number of parameters, including polymer type and molecular weights, polymer concentrations, degree of chemical modification and nature of functionalities, linkers, drug attachment, and encapsulation. Proper design cannot be achieved without understanding about the relationship between influencing parameters and hydrogel macroscopic properties, measurable in terms of hydrogel swelling, mechanical strength, and profile of drug release. Mathematical modeling can accelerate the rational design of hydrogels. Modeling seeks to describe quantitatively species transport, network expansion, hydrogel degradation and erosion, and drug release rate. The studies aim to relate these properties with controllable material parameters. There are comprehensive reviews on the mathematical modeling of drug release from nondegradable hydrogels in the literature (Patel et al., 2011; Masaro and Zhu, 1999; Amsden, 1998; Ganji et al., 2010). In this chapter, we will review mathematical models relevant to the properties and drug release from degradable hydrogels. We will discuss the advantages and limitations of each model, whether it captures the underlying mechanism satisfactorily, and how it can be used by experimentalists to help design a hydrogel system.

2 Degradation, swelling, and erosion Degradation refers to bond cleavage by enzyme or water. There are two extreme cases for the process depending on the relative rate of water/enzyme penetration to the network and bond cleavage rate. Bulk degradation occurs throughout the network when water penetration into the network is much more rapid than polymer chain cleavage. On the other hand, when polymeric chains are cleaved faster than water transport to the system, surface degradation is dominant (Siepmann and Siepmann, 2008). Because of the hydrophilicity of hydrogels, bulk erosion is more common than surface erosion. Surface erosion in the hydrogels occurs in more specific cases, for example, when the rate of enzyme penetration into the network is much slower than enzymatic cleavage. In contrary to “degradation,” “erosion” is a physical phenomenon that refers to depletion or transport of cleaved/detached materials from the network to the surrounding. Important phenomena, relevant in bulk-degrading hydrogel, are swelling. According to Flory’s theory (Flory, 1953), equilibrium swollen state is the result of balancing of two opposing forces: the thermodynamic force of mixing polymer chains and water and the retractive force of the polymer chains in cross-linked conditions. In bulk-degrading hydrogels, the network undergoes continuous swelling until disintegration as the cross-linking density changes with degradation. Decreasing in the moles of cross-linked chains leads to the shift of thermodynamic equilibrium and causes more water to enter the network and thus the expansion of the hydrogel. Complex process starts by locating the hydrogels into the target place.

2 Degradation, swelling, and erosion

Balancing two aforementioned forces results in the transition from relaxed to initial swelling equilibrium and drawing first water/enzyme penetration step. As time passes, individual bonds on backbone/cross-linked chains start to be hydrolyzed/ cleaved by water/enzyme molecules. It leads to some loss in elastic cross-linked chains and changing the equilibrium toward drawing more water into the network. Meanwhile, some of detached small fragments/long chains leave the network to the surrounding. These continuous changes decrease the overall network molecular mass and mechanical strength (Lao et al., 2011). Finally, times proceed to the point in which network is too weak to maintain its structure and disintegrated and turn into polymer solution. Numerous mathematical models have been developed to model all or most of the mentioned steps occurring in bulk-degrading hydrogels.

2.1 Statistical-kinetic models In these models, the general goal is to describe theoretically the degradation behavior of bulk-degrading hydrogels using physical properties of the network as model parameters. It is called statistical-cokinetic to reflect that the model considers both the effect of network structure and individual hydrolyzable bond on the overall erosion (mass loss) behavior. Model developers assumed certain kinetics for hydrolysis of labile bonds in the cross-linked nodes and obtained the probability of individual bond cleavage at any time. Then, structural effects could come into model by relating the probability of individual bonds to the number of possible configurations of intact cross-linked nodes within degrading network. In this step, statistical approach and probability theories were used. Earlier models were developed for chainpolymerized hydrogels. Later, same concepts were utilized to quantify the degradation behavior of step-growth polymerized hydrogels. The models share some similar assumptions detailed as follows. First, degradable blocks within the network are hydrolyzed according to pseudo first-order kinetics: d ½DB ¼ K 0 ½DB dt P¼1 0

½DB 0 ¼ 1  eK t ½DB0

(1) (2)

where k is the pseudo first-order kinetics constant, [DB] the concentration of degradable blocks in the network, and P the probability that any degradable unit has been hydrolyzed. Second, because of the high swelling ratio, hydrolysis of the any degradable block is assumed to occur homogeneously throughout the hydrophilic network with no crystalline regions present. Third, degradation products diffuse out of the highly swollen network faster than degradation such that there is no need to consider transport resistance in calculating the mass loss (erosion). Moreover, all chain lengths are monodisperse, and chain transfer reactions are neglected in polymerization.

235

236

CHAPTER 9 Mathematical models of drug release

2.1.1 Chain polymerized hydrogels This kind of hydrogels is usually formed through free-radical polymerization of multivalent macromers. In these hydrogels, the network is formed through chain-growth homopolymerization. The schematic structure of chain-polymerized network is given in Fig. 1. Degradable linkers are incorporated into the end groups of each macromer. Resulting network consists of nondegradable kinetic chains—formed during chaingrowth polymerization step. The cross-linkers are the macromers with degradable endcaps. Three types of species remain after complete degradation of hydrogel: the central hydrophilic molecule, monomeric or oligomeric degradable blocks, and kinetic chains formed from the radically polymerization of the active groups.

Mathematical models A generalized bulk degradation model based on statistical, mean-field approach for this type of hydrogel networks was presented by Martens et al. (2001), Martens et al. (2003), Martens et al. (2004). It was modified from previously proposed models for the same type of gel (Metters et al., 2000a,b; Metters et al., 2001a; Anseth et al., 2002; Mason et al., 2001). Modeling starts from hydrolysis of degradable bonds according to pseudo first-order kinetics (Eq. 1). From Eq. (2), one can obtain the probability that any bond is cleaved. To obtain hydrogel mass erosion, corresponding portion of each type of degradation products can be calculated according to the following equation: %mass loss ¼ ðWxl Fxl + Wkc Fkc Þ

(3)

UV

Hydrophilic core molecule (e.g. PEG) O m Degradable segment (e.g. PLA) CH3 Reactive end group for polymerization (e.g. Vinyl)

O O

n O CH2

Formed kinetic chain after gelation

FIG. 1 Schematic representation of chain-polymerized hydrogels from its macromer and illustration of one example which is PLA-PEG-PLA.

2 Degradation, swelling, and erosion

Wxl and Wkc are mass percentage of the original cross-linked network, estimated based on the chemical composition of starting macromers. Fxl and Fkc are fraction of cross-linking macromer and kinetic chain that is extractable from the gel. The procedure for obtaining these two critical parameters is given in the next section for two conditions, ideal hydrogel with no primary cycles in the structure and nonideal hydrogel containing some degree of cycling in the initial network. Without cyclization. For ideal network, each cross-linking macromer with m reactive arms within the network is attached to m number of different kinetic chains. There are three possible ways for one selected kinetic chain to be released from the network: (1) (2) (3)

The attached arms of macromers to the selected kinetic chain are degraded. All of the macromer arms except the one attached to kinetic chain of interest are degraded. The complete degradation of the macromer.

Fkc is the summation of mentioned three probabilities to the power of n, which is the average number of macromers originally attached to each kinetic chain:
n Fkc ¼ P + ð1  PÞPm1

(4)

For one macromer to be released, there are two possible pathways: All of its arms are degraded, or all arms except the one connected to a releasable kinetic chain are degraded. Fxl is the summation of these two probabilities:
 Fxl ¼ Pm + Fkc ð1  PÞPm1

(5)

The hydrogel is disintegrated at the point in which two cross-linkers per kinetic chain remain in the network. At this point, the mass loss is complete, and the hydrogel becomes soluble. xl, the average number of cross-links per kinetic chain at any point throughout the degradation process, can be predicted by Eq. (6):   xl ¼ nð1  PÞ 1  Pm1

(6)

With cyclization. For nonideal network containing cyclization, each cross-linking macromer with m active arms does not necessarily attach to m kinetic chains. It means that it can attach to 1 to m kinetic chain(s), and several configurations exist depending on the value of m. General mass loss equation for nonideal hydrogels differs from that for ideal hydrogels, since the calculation for releasable fraction of kinetic chains and macromers must account for the presence of cycles. Here, structural configuration term means the number of ways in which one cross-linking macromer with m arms is attached to a different number (from 1 to n at maximum) of kinetic chains. A new parameter Yik is defined that is the fraction of cross-linking macromer in each structural configuration and is determined from the probability of cyclization Ψ x. There, i is the number of kinetic chains the cross-linking macromers is attached to and k an array, including all possible number of active groups reacted to each

237

238

CHAPTER 9 Mathematical models of drug release

kinetic chain. For example, Y31, 1, 2 is the probability that a cross-linking macromer with four active groups connects three kinetic chains, one of those kinetic chains has two of the active groups reacted into it and each of remaining kinetic chains has reacted with one active group. Summation of all values of k is equal to the no. of macromer arms (m) (in this example, m ¼ 4). To obtain Fkc in this case, similar approach to that for the ideal network has been used. In each Yik configuration, there are two ways to the release kinetic chain of interest. Connected arm of cross-linking macromer attached to kinetic chain is cleaved, or all the connected arms to other kinetic chains except the one attached to the kinetic chain of interest are cleaved. So, for each configuration of Yik, Xij is the summation of the probability of two mentioned events, which means the probability of breakage of one cross-linking macromer that leads to the releasing of kinetic chain of interest. j is the number of active group connected to a specific kinetic chain. Like the corresponding step in an ideal network, this probability should be raised to the power of the average number of macromers of considered configuration in each kinetic chain (nYik/i) to give the fraction of releasable kinetic chain for hydrogel: Fkc ¼

 Y Xij nYik =i

(7)

One cross-linking macromer is releasable if (1) all of its connected arms are degraded or (2) all arms except the one attached to a releasable kinetic chain are degraded: Fxl ¼ Pm + Fkc

X φij Yik

(8)

φij is the probability that a cross-linking macromer in a specific configuration is connected to a releasable kinetic chain. To calculate it, the probability that a crosslinking macromer is connected only to the kinetic chain of interest is divided by the probability of all the possible events in degradation: φij ¼

ð1  Pi ÞPmj 1  ð1  Pmj Þð1  Pi Þ

(9)

Then, the number of intact cross-links, xl, is modified due to the presence of cyclization as follows: xl ¼

 X  nYik 1  Xij i

(10)

1  Xij is the probability that a cross-linking macromer is not releasable and functions as intact cross-linked chain. It should be multiplied by the probability of one type cross-linking macromer to be in a certain configuration Yik. As mentioned at the start of this section, bulk-degrading hydrogel undergoes continuous swelling before disintegration. So, to account for this phenomenon in the model, the microscopic network properties of hydrogel are correlated to experimentally measurable hydrogel properties such as volumetric swelling ratio (Q) and compressive modulus (CM). Using Flory-Rehner equation (Flory, 1953) and elasticity theory (Metters et al., 2000a,b), one can relate network swelling ratio to the average

2 Degradation, swelling, and erosion

molecular weight of cross-linked chains and the compressive modulus. Through this equation, for highly swollen network and neglecting chain ends, Eqs. (11), (12) are achievable: 3

Q∝ ½xl 5 6

CM ∝ ½xl5

(11) (12)

Experimental systems and validation As mentioned above, starting precursors for this kind of hydrogels are hydrophilic polymers that are endcapped by functional groups, for example, multiacrylated polyethylene glycol (PEG) and polyvinyl alcohol or polysaccharide. By adding degradable segments (such as blocks of polylactic acid (PLA) or polyglycolic acid (PGA)) with reactive vinyl termini during precursor synthesis, hydrolyzable bonds are incorporated in this type of hydrogel making it degradable. Polymer chains are elongated and cross-linked under UV emission by free-radical reaction between the reactive end groups (Tibbitt et al., 2013). The gelation reaction is fast, and hydrogels with various mechanical properties can be obtained by controlling the network parameters. For example, PEG was used as core molecule that gave hydrophilicity to the whole macromer, and hydrolytically cleavable polylactide segments were used to provide hydrogel biodegradability (Metters et al., 2000a,b). Vinyl end groups attached to PLA segments were used to cross-link the macromers using UV and form hydrogel network. By changing the ratio of PLA (m) to PEG (n) and also the number of ester bonds in one PLA segment, one can obtain hydrogels with different bulk degradation profile to suit specific application (Metters et al., 2000a,b). The statistical-kinetic model has been used to predict mass loss for hydrogels composed of mixture of PLA-PEG-PLA endcapped with methacrylate functionalities and polyvinyl alcohol (PVA) modified by 5-ester acrylate molecules (AcrEst-PVA). Authors introduced two different hydrolytic rate constants, k0 , into the model to account for two different degradable linkers incorporated in the macromers (Martens et al., 2003). However, the model could only predict the observed trend in the experimental profile for the mass loss and swelling ratio approximately, but the deviation of model results from experimental data were statistically significant. The reason could be due to the fact that combining two different macromers produced network containing different chemistry and more complicated microstructure relative to the one composed of pure macromer. The model was used to fit with the experimental data from hydrogels made of PVA (Martens et al., 2004). The molecular weight of PVA was 16, 14, and 31 kD with 5, 3 and 2 arms per molecule. k0 was obtained through fitting to experimental data for the mass swelling ratio and comprehensive modulus for each formulation. The other two parameters, the number of cross-linking macromer to each kinetic chain (n) and degree of cyclization (Ψ x), are difficult to be measured and were obtained through a good fitting with experimental data of disintegration time and

239

240

CHAPTER 9 Mathematical models of drug release

mass loss profile, respectively, for PVA 16 kD. However, when the fitted parameters were used to predict the mass loss profile for PVA 14 and 31 kD, the model could not provide a satisfactory prediction for experimental mass loss. This is because structural differences exist between hydrogels made of different starting macromer. These structural diversities were reflected in the parameters n and Ψ x that were assumed constant for all the networks. The model was used to obtain hydrolysis rate constant (k0 ) for hydrogel made of diacrylated PLA-b-PEG-b-PLA copolymers with two different degrees of acrylation and numbers of ester bonds in PLA segment (Shah et al., 2006). The fitted values were compared with the hydrolysis rate constant measured for solution. The effects of macromer composition and concentration, buffer pH, and ionic strength on degradation kinetics were investigated. Assuming pseudo first-order kinetic for hydrolysis, k0 for solutions was obtained through monitoring changes in concentration of lactic acid formed when two adjacent ester bonds were degraded. k0 for hydrogels was estimated through fitting model with experimentally measured swelling ratio. Hydrolysis of ester bonds in hydrogels were found to be more influenced by bulk solution condition than macromer degradation. The results demonstrated that hydrolysis kinetic constants for degradation of both hydrogel and soluble macromer were sensitive to initial polymer concentration and pH. Unlike the hydrogel, macromer degradation kinetics in solution was not dependent on the ionic strength. This difference could be attributed to the interaction between buffer ions and bound ions on the polymeric chains of degrading network. Ionization of bound ionic species (lactic acid and acrylic acid) on the gel happened during degradation. In higher ionic strength, ionized species bound to the network were masked by the free positive ions in the buffer and resulted in the depletion of repulsion forces. As a consequence, the swelling ratio and therefore the estimated k0 for hydrogel decreased. The degradation kinetics was strongly dependent on the macromer backbone chemistry for hydrogel that was not observed for soluble macromer. This revealed that slight differences between cross-linking densities (when two macromers with different degrees of acrylation were used for making hydrogels) could affect the rate at which degradable bonds within networks were hydrolyzed and released from the network.

Contributions and constraints Experimental mass loss profile for a degradable hydrogel composed of multivinyl macromers consists of three regions. In the early time points, none or little erosion occurs because of the connectivity of network. This is followed by a relatively steady-state mass loss due to the release of cross-linking macromers from the network. Finally, kinetic chains are released, and disintegration occurs (Metters et al 2000a). Model parameters incorporated into this model to capture the phenomena during mass loss are weight fraction of kinetic chain and cross-linking macromer, kinetic rate constant, number of arms in each macromer, and number of macromers attached to each kinetic chain. Number of arms and weight fractions are known for the experimental system a priori. Kinetic rate constant for hydrolysis was obtained through fitting swelling data. It has been shown that kinetic rate constant depends on many parameters such

2 Degradation, swelling, and erosion

as degradable bond chemistry, length of linkers, and local microstructure. So, it cannot be measured directly from soluble macromer solution and assigned to hydrogel network composed of same macromer. The amount of cyclization is difficult to be measured experimentally, and some researchers obtain it through fitting to experimental mass loss data (Martens et al 2004). The last parameter is the number of cross-linking macromers attached to each kinetic chain, which is influenced by kinetic chain length and the conditions of polymerization. It can be adjusted to fit the experimental data in the disintegration region. Putting aside the number of macromer arm and macromer weight fractions, remaining parameters (kinetic rate constant, number of macromers attached to one kinetic chain, and degree of cyclization) are difficult to measure experimentally. Their results show that the model is strongly dependent on them and they are specific to each system so they are fitted to experimental data in all of the applied hydrogel systems. Although all the fitted parameters have physical meaning to the system, however, this causes the model to lose its ability for prediction and become a fitting tool instead of having power to predict.

2.1.2 Step polymerized hydrogels Step-growth polymerized hydrogels include a broad range of hydrogels that are formed through conjugation reaction between complementary end groups on the macromers. Step-growth Michael addition reaction is one example in which the gelation occurs due to the reaction between macromers/polymers and either small molecules or macromers/polymers modified at chain ends involving a complementary pair of nucleophilic and electrophilic groups (Fenoli and Bowman, 2014; Tibbitt et al., 2013). Schematic network structure of step-growth polymerized hydrogels is given in Fig. 2.

Mathematical model This kind of hydrogels bears one or more degradable bonds within each cross-linked chain. Like chain-polymerized hydrogels, they undergo continuous swelling during degradation. Degradation products contain low-molecular-weight hydrophilic macromers and small cross-linker molecules. Metters and Hubbell (2005) proposed a model (later used by Shih and Lin (2012)) for predicting the swelling of degradable hydrogels of this type (Fig. 3). This model shares the basic assumptions that were mentioned in the beginning of section. The probability that any degradable unit that has not been hydrolyzed is similar to Eq. (1): Pintact ¼

½DB 0 ¼ eK t ½DB0

(13)

By considering the number of labile sites present on each cross-link (N), the fraction of broken elastic cross-link chains (Pchain) within the network is as follows: 0

Pchain ¼ 1  ðPintact ÞN ¼ 1  e2K t

(14)

241

242

CHAPTER 9 Mathematical models of drug release

FIG. 2 Three examples of hydrogel networks formed by step-growth polymerization.

FIG. 3 Ideal 100% cross-linked (A) and nonideal networks (B) due to primary cycles formed by step-growth polymerization of four-armed PEG-acrylate and dithiol.

2 Degradation, swelling, and erosion

For the example shown in Fig. 3, N is equal to 2 as there are two labile links where the cross-linker connects with the arms of the macromer on both sides. To relate the microscopic structural changes to the experimentally measurable macroscopic properties of hydrogels, one needs to obtain changes in the concentration of active crosslinked chains (i.e., cross-linking density) as a function of time within the degrading network. To do this, it is necessary to know the number of independent connections that every macromer has with its surrounding. For ideal networks that are composed of starting precursors having functionality of fA (the number of arms) and assuming Flory distribution for all cross-links, the fraction of cross-links that have i arms connected to the network at any time is Fi, fA ¼

fA ! ð1  Pchain Þi Pchain ðfA iÞ ðfA  iÞ!i!

(15)

Cross-linking density (vc) is the summation of fraction of intact functional cross-link until the remaining intact arms reach its minimum (i.e., i ¼ 3 when the cross-link can no longer be considered active). A0 is the initial concentration of multiarmed macromer in the system: vc ¼ ½A0 

fA X i 3

2

Fi, fA

(16)

Profile of the mass swelling ratio of hydrogels with different initial network properties can be predicted by plugging the theoretical cross-link density (from Eq. 16) into Flory-Rehner equation (Flory, 1953). The chain ends present within the network are assumed negligible: vc ¼

V1 lnð1  v2 Þ + v2 + X1 v2 2  ¼ 1 2v2 v2 M c ðv 2 Þ3  fA v2 ¼

1 v2 ¼ Qv ½ðQm  1Þv1 + v2 

(17)

(18)

where Qv, Qm, v1 , v2 , V1, X1, and v2 are volume and mass swelling ratio, specific volume of the water and dried macromer, molar volume of the swelling agent (water), Flory polymer-solvent interaction parameter, and polymer volume fraction in swollen hydrogel, respectively.

Experimental systems and validation Pioneer work in preparing this type of gels is performed by Hubbell group (Elbert et al., 2001). Hydrogels were made of dithiol-(linear) PEG and (branched) PEGmultiacrylate and were tested for controlled release of albumin protein. Two ester groups were present in each cross-link rendering the hydrogel degradability. Compared with chain-polymerized network, step-growth polymerized hydrogel produces more ideal and mechanically homogenous network (Tibbitt et al., 2013). The statistical-kinetic model has been employed to predict the swelling ratio of hydrogel

243

244

CHAPTER 9 Mathematical models of drug release

composed of three-, four-, and eight-armed acrylated PEG and either small dithiothreitol (DTT) molecules or linear smaller PEG modified with thiol at both ends as cross-linkers. The initial network structure (reflected as initial swelling ratio) depended on precursor functionality, molecular weight, and interaction parameter. As the number of arms and concentration of macromer increased, experimental data approached the model predictions for initial swelling ratio. With lower values (fA < 8 and conc <50%w/v), the model result deviated from measured initial swelling ratios. This occurred due to the increase in primary cycles and unreacted groups that lower the intact effective cross-linked chains and therefore higher swelling ratio relative to ideal value. This deviation was accounted for with an additional parameter called cross-linking efficiency for nonidealities. This could be adjusted by comparing the experimental and ideal initial swelling ratios. At the next step, the effects of kinetic rate constant, functionalities, polymer concentration, and molecular weight on the hydrogel swelling ratio profile in course of degradation were investigated. Model predictions showed different k0 affected the rate of degradation without changing the initial network but hydrogels with different precursor molecular weight degrade at the same rate with different initial swelling ratio. Changes in initial functionality of precursors (fA) produced the same effect as changing k0 , without affecting the initial swelling ratio for ideal networks, while experimental data showed changes in both the initial swelling ratio and degradation rate. The effect of cross-linking efficiency on swelling ratio profile was predicted satisfactorily. When cross-linking efficiencies fell, both the initial swelling ratio and the rate of degradation increased, but the slope of swelling ratio curves were identical at a given swelling ratio value. This was attributed to the homogenously introduced cycling or imperfections within the network in the same random way. Another parameter was macromer precursor concentration. Lower values led to higher initial swelling ratio and also degradation rate of swelling ratio profile due to the lower cross-linking efficiency. The model and the experimental mass swelling ratio matched well using one single k0 for gels with different macromer concentration.

Contributions and constraints Time variation of moles of cross-linked chains is modeled and related to mass swelling ratio profile of degradable step-growth polymerized hydrogels. Although it was said that cross-linking efficiency was adjusted using the initial experimentally measured swelling ratio, however, they did not explain clearly how they plugged it into the model. The effect of the number of arms and polymer concentration on the swelling ratio profile was successfully predicted by the model through adjusting crosslinking efficiency and one single kinetic rate constant. The effect of macromer molecular weight however could not be predicted. The prediction here did not match the experimental data when the same kinetic rate constant was used. Unlike experimental data, predicted curve by the model had identical slope. It showed that changing the molecular weight of macromer in microscopic level can change the environment and water concentration that may lead to change in actual k0 . Overall, the model could be used for prediction of specific type of degradable step-growth

2 Degradation, swelling, and erosion

FIG. 4 Schematic representation of network contained tethered drug.

polymerized hydrogel (Figs. 3A and 4) when its important parameters (including cross-linking efficiency and kinetic rate constant) could be estimated or measured. Nothing is mentioned about using macromer with different range of functionalities in forming network. Therefore, the model applicability is limited to the network initially composed of homogenous mixture of macromer with identical number of arms. In addition, it is not necessarily correct that each macromer in the network has the identical cross-linking efficiency so cross-linking efficiency should be extended to a range of values that are obtained based on the probabilities of individual scenario for each macromer.

2.2 Treelike theory 2.2.1 Mathematical model Hydrogel network is supposed to have a hierarchical or treelike structure. So, treelike structure theory is based on this natural characteristic and used recursive nature of branching process and some basic probability theories. Exploiting this theory, Li et al. (2011) proposed a mathematical and chemical scheme for describing and precise controlling over degradation of hydrogels made by tetra-arm low-molecular-weight polymer with three different types of end groups without changing the gelation reaction type. The overall structure of the network is similar to Fig. 3, except that only some known portion of cross-linked chains are degradable. Degradation rate is controlled by changing the ratio of nondegradable to degradable macromer (rdeg).

245

246

CHAPTER 9 Mathematical models of drug release

Regarding theory of treelike structures, the number of moles of elastic chain per unit volume (λ) for hydrogel network of tetrafunctional macromers as one example can be represented in following manner:  λ¼U

   1 1 3 1 3 1 3 1 3 +  2   2 2 p 4 2 p 4

(20)

where U is the molar concentration of the tetra-arm polymer in a unit volume of material and p is the fraction of hydrolyzed bonds (reaction conversion) (Eq. 1). Assuming pseudo first-order kinetics for bond hydrolysis, similar reactivity of reactive moieties, and homogenous distribution of cleavable bond within network, p can be expressed as follows:     p ¼ p0 1  rdeg + rdeg exp kdeg t

(21)

where p0 and kdeg are the initial fraction of connected bonds and the rate constant of hydrolysis, respectively. After substituting Eq. (21) into Eq. (20), one can find how λ changes with time: ! ! 1 1 3 1       + λ¼U  2 2 4 p0 1  rdeg + rdeg exp kdeg t ! !3 3 1 3 1     2   2 4 p0 1  rdeg + rdeg exp kdeg t

(22)

The value of λ is plugged into Flory-Rehner equation (Flory, 1953) to obtain the change in the swelling ratio over time. To reveal if there is any parameter that can be manipulated to control the degradation, Eq. (21) was rearranged to become  ln tdeg ¼ 

   pc  p0 1  rdeg rdeg kdeg

(23)

where tdeg and pc are the time at which hydrogel network disintegrated and critical conversion, respectively. Critical conversion is the fraction of intact connected bonds within the network at that time of disintegration. At values above the critical conversion, hydrogel can maintain its 3-D structure, while at this point and below that, gel network is turned into the solution.

2.2.2 Experimental systems and validation The model is applied to the degradable hydrogels made by three different tetra-PEG macromers: modified PEG with amino termini (TAPEG), modified PEG with succinimide end groups (TCPEG), and modified PEG with succinimide end groups plus an ester group in each arm (TGPEG). Hydrogels were formed by reacting of TAPEG macromer with a mixture of TCPEG and TGPEG in aqueous solution. Degradation rate was controlled by changing the ratio of TGPEG to TCPEG (rdeg). First of all, elastic moduli (G) were measured experimentally for hydrogels containing different rdeg ratios and (Eq. 24) were used to calculate the number of moles of elastic chain

2 Degradation, swelling, and erosion

per unit volume (λ) for hydrogel networks formed with varying rdeg. The goal was to investigate if there was any change in the physical structure of hydrogels by changing rdeg: G ¼ RTλ

(24)

The result showed no change in λ, which indicated the same reactivity of TCPEG and TGPEG toward TAPEG. Similar hydrogel structures were obtained for different values of rdeg. Then, kdeg was estimated through fitting model equation (Eq. 22) with experimental data of the swelling ratio by least square method. kdeg was found to be independent of rdeg. Then, time course of reaction conversion (p) (Eq. 21) could be predicted for hydrogels with different rdeg and using fitted kdeg. From the profile of conversion (p) versus time and using experimentally measured disintegration time (tdeg) of each hydrogel formulation, critical conversion can be determined at each rdeg. They found that this value is around 0.46 and is independent of rdeg. Finally, a relationship was established between disintegration time (tdeg) and rdeg ratio by using Eq. (23). Necessary parameters are kdeg and pc that must be acquired in prior experiments. Therefore, the model provides guidelines for controlling hydrogel disintegration time by manipulating rdeg.

2.2.3 Contributions and constraints There are two unknown parameters in the model, kinetic rate constant and critical conversion. The approach for obtaining the input parameter is fitting model result to experimental data. kdeg is adjusted through least square fitting of swelling ratio data over time, and so, they concluded that their model is valid because kdeg is nearly constant independent of changing rdeg. pc is determined from experimentally measured tdeg. However, the value obtained for pc have deviated significantly from the calculated value by the theory of treelike structures. This deviation was attributed to the nonideal structure of hydrogel network. Because no parameter is considered to represent deviation from ideal treelike structure in their model, their model is only suitable for hydrogel network with close to ideal structure. It requires high concentration of macromers and fast gelation reaction between complementary end groups.

2.3 Zero order surface erosion In polymeric matrix composed of poly(anhydride esters) or poly(ortho esters) (Lin and Anseth, 2013), bond cleavage is much faster than penetration of water, and this gives rise to surface erosion. However, for hydrogels that are inherently hydrophilic and contain a great amount of water, surface erosion only occurs in enzymatic degrading systems that the transport of enzyme into the gel is slower than the rate of enzymatic degradation (Lin and Metters, 2006). Under this situation, degradation is limited to the outer layer of the system, and the network maintains its structure until the final stage of degradation.

247

248

CHAPTER 9 Mathematical models of drug release

2.3.1 Mathematical model A simple model is developed to describe erosion from a very thin hydrogel that is degradable at the presence of enzyme (Aimetti et al., 2009). Mass loss curve is proportional to the surface area of gel. If gel is thin enough, one can assume a relatively constant surface area: d ð%mass lossÞ ¼R dt

(25)

where R is the slope of the mass loss profile. Under the assumption of constant density and surface area, changes in erosion front could be obtained as 

d ðhÞ ¼ Rh0 ¼ k0 dt

(26)

0

where h, h0, and k are erosion front dimension, initial thickness of hydrogel (in erosion direction), and kinetic rate constant (length/time).

2.3.2 Experimental system and validation Aimetti et al. (2009) applied the model for biodegradable PEG hydrogels formed by thiol-ene photopolymerization. Cross-linked chains contained human neutrophil elastase (HNE)-sensitive peptide and degraded at the presence of HNE. Zero-order kinetic was observed for mass loss curve. The slope or kinetic rate constant of mass loss (Eq. 26) was affected by changing the type of peptides in the cross-linked chains, concentration of HNE, and concentration of peptides within the gel. The rate constant was estimated through fitting with experimental mass loss data for each case. The constant swelling ratio in the course of degradation also confirmed that hydrogels underwent surface-eroding process.

2.3.3 Contribution and drawback Experimental mass loss profile for the hydrogels confirmed zero-order kinetic for erosion of very thin hydrogels when surface area is maintained constant. However, due to sensitivity of mass loss to the experimental conditions, a careful extensive work is needed to investigate the exact kinetic for enzyme and peptide reaction in different conditions. Also, model application is limited to very thin hydrogels. The kinetic rate constants need to be modified for other geometries and dimensions.

3 Drug release Drug can be released from a hydrogel through several mechanisms depending on the status of the drug molecule within the hydrogel network. Drug molecules are encapsulated in the hydrogel by either physical entrapment or chemical conjugation. The rate of drug release is dominated by the rate-controlling step. Therefore, one needs to compare the rate of degradation relative with the rate of drug diffusion from bulk to surface of hydrogel. In addition, diffusion of degradation catalysts like enzymes into the system and swelling of hydrogel network during the course of degradation must

3 Drug release

be considered. Releasing of drug from a degradable drug delivery system to outside environment is accomplished through some multiple steps (Tzafriri, 2000). For chemically attached drugs, liberation of free drug further requires the cleavage of drug linker that adds one more step. Depending on the surrounding environment in which hydrogel is placed (finite or infinite volume), diffusion across external boundary layer should be accounted. This will add one more mass transfer boundary condition to the mathematical model.

3.1 Diffusion controlled When bond degradation is relatively slow, drug release from hydrogel is treated similarly as nondegradable hydrogel. Diffusion-controlled drug release is the most widely applicable mechanism for describing drug release from hydrogels. It is therefore important to have an understanding of the parameters governing solute diffusion within hydrogels and the means by which they affect diffusion. There are many excellent reviews on this topic (Amsden, 1998; Masaro and Zhu, 1999). Drug diffusion within highly swollen hydrogels is best described by Fick’s law of diffusion or Stefan-Maxwell equations (Lin and Metters, 2006):  ∂Ci ∂ ∂Ci Di ðCi Þ ¼ ∂t ∂x ∂x

(27)

where Ci, Di, x, and t are solute concentration, solute diffusion coefficient, location, and time coordinate, respectively. Drug diffusivities are determined empirically or estimated theoretically (Lin and Metters, 2006). Once diffusivity is determined, Eq. (27) can be solved to obtain the drug release profile using appropriate boundary and initial conditions that are based on the geometry and type of drug delivery system (reservoir or matrix). Solute transport within hydrogels occurs primarily within the water-filled regions in the space between polymer chains with size represented by the network mesh size (E). Factors that reduce this E will affect the movement of the solute. Such factors include the size of the solute relative to the size of the space between polymer chains, polymer chain mobility, and the existence of charged groups on the polymer chains that may bind the solute molecules. A general equation to capture the relationship is shown below: Dg ¼ f ðrs , v2, s , EÞ Do

(28)

where Dg, Do, rs, v2, s, and E are solute diffusivity within hydrogel, its diffusivity in water, solute hydrodynamic radius, volume fraction occupied by polymer chains in the swollen state, and network mesh size, respectively. More factors such as environmental changes or concentration of solute can be added to this function (Lin and Metters, 2006). Several theoretical models have been developed to define function f in Eq. (28). It means to relate drug diffusion coefficient in hydrogel (Dg) to diffusion coefficient in pure solvent (D0) through mentioned parameters. Free-volume theory,

249

250

CHAPTER 9 Mathematical models of drug release

hydrodynamic theory, and obstruction-effect theory (Amsden, 1998; Huglin, 1989) are among the main theories (Huglin, 1989; Amsden, 1998). Each of these theories has one or two model parameters that are specific to polymer-solute-solvent and are obtained through fitting to the experimental data. Amsden (1998) examined the applicability of each model based on the quality of the fit for a variety of experimental release data available in the literature. He concluded that Cukier (1984) hydrodynamic model and Amsden obstruction (Amsden, 1999) model showed a good fit for homogenous and heterogeneous hydrogels, respectively. Combinatory models did not show a good fit to small-sized solute, so their application is limited to diffu˚ ) in the hydrogel with stiff sion of large molecules (rs greater than approximately 20 A polymer chain and high volume fraction.

3.2 Degradation controlled drug release Degradation-controlled drug release occurs in two systems: hydrogels with pendant drugs and those exhibiting surface erosion. In the first type, drug is liberated when the labile linker that connects the drug to the network chain is degraded. In the second type, erosion is limited to the outer layer of hydrogel structure. A moving front is formed that marks the reducing boundary of the hydrogel. Drug release accompanies the gel erosion.

3.2.1 Pendant drugs Drug is attached to the polymer via a hydrolytically or enzymatically labile bond. That’s the rate of cleavage of bonds that can control the drug release rate (Peppas et al., 2000). This situation can exist in several ways. First, hydrogel is nondegradable, and drug release rate is purely controlled by the degradation of drug linkers. Second, the cross-linker of the hydrogel and drug linker have the same type of labile bond that degrades at same rate (DuBose et al., 2005). Third, the hydrogel consists two hydrolytically labile bonds, one for drug linker and one for cross-linker (Reid et al., 2015). Most of these linkages have been designed to be hydrolytically degradable allowing degradation and release rates to be characterized by fairly simple firstorder kinetic relationships (Lin and Metters, 2006).

Statistical-kinetic models Mathematical model. Metters and Hubbell (2005) expanded the model (Metters and Hubbell, 2005) to include the release of tethered drug from the network. One new term called “cross-linking efficiency φi” was introduced to account for network nonideality (Fig. 4), which is necessary for modeling drug release. φi represented the fraction of active group chain ends that participated in the cross-linking reaction. Eq. (15) was modified as below for nonideal hydrogel: Fi, fA , initial ¼

fA ! ð1  φi ÞfA i φi i ðfA  iÞ!i!

(29)

3 Drug release

where fA is the number of arms in the selected macromers. Fi, fA, initial is the fraction of intact macromer arms. For ideal network, Fi, fA, initial is 1. General functionality distribution Fi, fA is given in Eq. (30): Fi, fA ¼ Fi, fA , degrade  Fi, fA , initial

(30)

Fi, fA, degrade is the original fraction before modification (Eq. 15). Here, it is assumed that the linker hydrolysis rate constant is the same as that for the cross-linker of the hydrogel. Diffusion of any released drug is assumed to be fast relative to the cleavage rate, so they reach to the outside of hydrogel as soon as they are cleaved. Two forms of drug are described: (i) Drug is released through the cleavage of its linker, and (ii) drug is released with one single fA-armed macromer molecule attached to it. Higherorder attachment of drug (i.e., to more than one macromer) is not considered due to its insignificant effect on the overall release profile. P1, 0 is the probability that drug linker to the network is cleaved, and P2, 0 is the probability that all except one crosslinks to which drug is attached are cleaved: P1, 0 ¼ 1  Pintact

(31)

P2, 0 ¼ ð1  Pchain ÞðPchain ÞfA 1

(32)

The total fraction of released drug is summation over these two probabilities: PT ¼ P2, 0 + P1, 0

(33)

The fraction of intact multiarmed macromer (Eq. 34) was used to predict the disintegration time of hydrogel at which point the average number of intact cross-linked chains per macromer (Fi, fA Þ is equal or <2: Fi, fA ¼

fA X

iFi, fA  2

(34)

i¼0

Due to hydrogel disintegration, a burst release of drug is present at this point in the drug release profile. Experimental system and validation. Metters’ group (DuBose et al., 2005) expanded the previous model presented for predicting degradation behavior of hydrogel synthesized through Michael addition reaction (Metters and Hubbell, 2005) to predict the release rate of bioactive molecules tethered to the hydrogel network. A fluorescent probe was attached to multiarmed PEG-acrylate through thiol groups. The product solution was reacted with DTT to make hydrogels. The same type of labile linkers was used to tether the drug to the network and to be incorporated in the cross-linker, so it could be assumed that the hydrolysis rate for both cleavage of linker and cross-linker chains were the same. Overall, effects of four parameters on the mass swelling ratio and drug release profile were examined in the model. They were cross-linking efficiency,φi; number of arms of macromer, fA; kinetic rate con0 stant, k ; and molecular weight of single arm, Mr. fA and Mr were intrinsic properties

251

252

CHAPTER 9 Mathematical models of drug release

0

of the selected macromer. φi and k were acquired through fitting model results to experimental data of the swelling ratio, respectively. Thereafter, they were used to predict drug release from the hydrogel. To account for the effect of k0 , experimental measurement of the swelling ratio has been performed in three different temperatures. The contribution of primary (P1, 0) and secondary (P2, 0) drug release was included. It was found that primary fraction had higher contribution in total drug release profile. The prediction was validated with experimental data obtained by high-performance liquid chromatography (HPLC). The product of these two release mechanisms could be distinguished by gel permeation chromatography (GPC) analysis through their different retention time in the column. The model results were in a good match to experimentally measured fractional drug release. Contributions and drawbacks. The model satisfactorily can predict drug release from hydrogels with different cross-linking efficiencies and kinetic rate constants. To investigate the effect of functionality alone, instead of using macromers that the only difference is the number of arms, four- and eight-armed macromers with different single-arm molecular weight and initial macromer concentration were used. So, no conclusion can be drawn that the predicted result for these two macromers is due to the changes only in the functionality. Obtaining different fitted k0 for these macromers confirmed this statement that they differ in other properties in addition to macromer functionality. The shape of predicted secondary probe release profile was a satisfactory qualitative match to the GPC experimental data with a peak in the middle. However, it has some slight deviation at some time point. The claimed larger contribution of primary versus secondary drug release was observed in the experimental data as well.

Diffusion-reaction model Mathematical model. A diffusion-reaction model was developed by Kim research group (Cheng et al., 2011) to describe drug release from an enzymatically degradable hydrogel microsphere with different initial drug distribution. It was assumed that the diffusion coefficients of free drug (D) and liberated drug are the same. D is also assumed to be nearly invariant with time that is valid if most of drug is released before significant changes in hydrogel microstructure occurred by degradation. Drug distribution in the hydrogel can be obtained by solving Fick’s law with source term as  ∂Cf 1 ∂ ∂Cf ∂Cr ¼ 2 + D:r 2 ∂t r ∂r ∂r ∂r

(35)

Cf and Cr are the concentration of free and of liberated (from chain but still physically trapped inside) drug within the microsphere; r and t are radial and time coordinate, respectively. Perfect sink condition and symmetry condition were used as two boundary conditions to solve this equation. These two boundary conditions should be defined based on the system application. Liberation of immobilized drug from

3 Drug release

the chains was related to enzymatic degradation by using the Michaelis-Menten model for polymer degradation. k1 , k1 E + S , ESk2k2 E + P

(36)

∂½S ½S Vmax ½S ¼ k2 ½E0 ¼ ∂t ½S + KM ½S + KM

(37)

KM ¼

k2 + k1 k1

(38)

Vmax and KM are the constants for a given enzyme-substrate pair. S, E, and k1, k1, k2 are substrate (polymer), enzyme, and rate constants for different stages of enzymatic reactions, respectively. According to the experimental condition of enzyme and polymer concentration in the system, [S] ≪ KM condition is met. Thus, Eq. (37) is simplified to become ∂½S Vmax ¼ μ½S μ ¼ ∂t KM

(39)

Degradation rate depends on the polymer concentration, and the rate constant (μ) is dependent on enzyme concentration, which is nearly constant in experiments. To relate polymer degradation and drug release, it is assumed that drug liberation is proportional to the polymer degradation rate. Also, immobilizing capacity, σ, is treated as constant and is related to immobilized drug concentration as Cim ¼ σS. So, using Eq. (39), Cim ðtÞ ¼ σ ½S0 eμt ¼ Cim0 eμt

(40)

Cim0 is the initial distribution of immobilized drug concentration. Liberated drug and immobilized drug concentration are related through Cr ¼ Cim0  Cim, and so, ∂Cr ¼ μCim0 eμt ∂t

(41)

Experimental system and validation. Kim et al. (Cheng et al., 2011) described drug release from an enzymatically degradable charged gelatin hydrogel microsphere by a diffusion-reaction model by altering processing parameters such as enzyme and cross-linking agent concentration. Cross-linked hydrogel microspheres are placed in the drug solution after synthesis. Drug molecules were diffused and entrapped within microspheres by forming polyionic complex with the cross-linked gel or remaining free when gelatin binding capacity was full. Once microspheres were immersed in drug solution, they became hydrated. Hydration was fast, and because gelatin degradation was bulk degradation, authors assumed an invariant geometry for microsphere hydrogel. Another assumption was that heterogeneity of the crosslinking reaction did not affect drug diffusion coefficient. This assumption was made

253

254

CHAPTER 9 Mathematical models of drug release

due to insignificant observed increase in diffusion coefficient when large changes were made in the cross-linking agent concentration. To use Michaelis-Menten equation for describing enzymatic degradation, enzyme kinetics must be estimated. According to the reported values for k2/KM for the degradation of collagen with collagenase and higher degradation rate of gelatin with same enzyme in comparison with collagen and also polymer concentration in microsphere, [S] ≪ KM condition is met. D and μ were unknown model parameters. Diffusion coefficient was fitted based on free drug release of one model drug, trypan blue from microspheres in the absence of enzyme without the last source term in Eq. (35). Initial drug distribution was estimated from the analysis of confocal microscopy images. For highly cross-linked hydrogels, drug amount was lower in the center of microsphere relative to the lower cross-linked ones due to greater diffusion barrier. Fitting operation showed that diffusion coefficient was not a monotonous function of crosslinking agent concentration. Therefore, relationship between cross-linking agent concentration and enzymatic degradation rate was correlated by fitting the current model to in vitro release data for three different cross-linking agent concentrations, and μ was fitted to the equation below as a function of cross-linking agent concentration:   μ ¼ 0:36 1 + 0:1eð14:29∗CGA Þ

(42)

CGA was the concentration of cross-linking agent. Finally, Eq. (42) was used in reaction-diffusion equation to predict drug release from microsphere with two different cross-linking agent concentrations. Contributions and drawbacks. Results showed a good agreement between experimental release data and model prediction for two different cross-linking agent concentrations with a small deviation. Observed small discrepancies can be attributed to the assumption of constant diffusivity for all immobilized and free drugs in all the stages of degradation, while the structure is changing. Also, according to the initial distribution of drug that is radius-dependent, cross-linking density is also radiusdependent. Although degradation rate constant is a function of cross-linking density, however, its location dependency is not accounted in the model. This function for polymer degradation rate constant is empirically obtained, dependent on the type of enzyme and cross-linking agent.

3.2.2 Surface erosion In surface-eroding systems, drug release is mediated by the rate of surface erosion. As mentioned, in these systems, matrix maintains its structure until later stage of degradation. However, due to the inherently high water content of hydrogels, surface erosion only occurs in enzymatic degrading systems where the transport of enzyme into the gel is slower than the rate of enzymatic degradation (Lin and Metters, 2006). So, drug molecules remain entrapped in the matrix due to the low rate of diffusion, and therefore, the drug release is degradation-controlled (Peppas et al., 2000).

3 Drug release

Mathematical models Hopfenberg model (Hopfenberg, 1976) is the earliest one to propose a surface degrading system. His model was a semiempirical model describing drug release from degradable drug delivery systems having a release rate proportional to the time-dependent surface area of the device:  Mt ka t n ¼1 1 C0 a0 M∞

(43)

C0, k0, a0, and n are initial drug concentration, erosion kinetic rate constant, radius for spherical and cylindrical and half-thickness for slab geometry, and shape factor (1 for slab, 2 for cylindrical, and 3 for spherical geometry), respectively. ka can be calculated for different hydrogels through experimental data. In the case of slab, zero-order release rate can be achieved. Katzhendler et al. (1997) extended his approach for modeling eroding cylinder with different rate constant in vertical and radial axis (ka and kb for radial and vertical degradation constant, respectively) for a flat tablet with initial dimension of a0, b0:   Mt ka t 2 2kb t ¼ 1 1 1 C0 a0 C0 b0 M∞

(44)

Although the original models were developed for polymeric matrices, they were later applied to surface-eroding hydrogels with minor modifications and extensions for different geometries (Siepmann et al., 1999; Karasulu et al., 2000). In another work, Lee (1980) proposed an analytic solution to equations that describe diffusional release of drug in drug delivery systems with surface erosion with planar membrane geometry. This approach has not been considered in previous works dealing with surface erosion. He treated the erosion by applying an erosion front with constant velocity (S(t) ¼ a  Bt) in the model (S(t) in Fig. 5 that was initially in a position). Drug diffused from the undissolved portion (drug concentration (A) ≫ drug solubility (Cs)) to dissolved one through a diffusion front in the matrix that also moved in constant velocity. He assumed perfect sink condition in outside medium throughout the experiment. Fick’s law was solved with application of appropriate boundary condition as follows:  Mt B:a Cs 1 a3 ¼δ+ :τ  δ: : + M∞ A 2 6 D

(45)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   s   2   ffi A 1 Ba A A 1 Ba A 1 Ba A 1 1 1 a3 ¼ + δ  +δ  1  2δ Cs 2D Cs Cs 2D Cs 2D Cs (46) ðtÞ δ ¼ SðtÞR , a

τ ¼ Dt a2

where are relative separation between the diffusion and erosion fronts and dimensionless time, respectively. He obtained fractional drug release for different CAs and Ba D ratios.

255

256

CHAPTER 9 Mathematical models of drug release

FIG. 5 Scheme of the model developed by Lee: (A) Dispersed solute concentration profile for surface-eroding polymeric systems, (B) Fractional release from the system having different A/Cs ratio. From Lee, P.I., 1980. Diffusional release of a solute from a polymeric matrix—approximate analytical solutions. J. Membr. Sci. 7 (3), 255–275.

Aimetti et al. (2009) utilized the mass loss model to predict solute release from enzymatically degradable thin hydrogel disk. In this model, solute release is proportional to mass loss profile: Mt k0  %mass loss ¼ Rt ¼ t M∞ h0

(47)

0

h0 is the initial thickness of hydrogel disk. So, if k can be estimated from mass loss data, solute release profile also can be predicted.

Experimental system and validation Lee did not validate his approximate analytic solution with any experimental data. The analytic result predicted that as CAs increased (drug loading was much larger than drug solubility) while Ba D ¼ 1, release rate approached zero order. When erosion front velocity exceeded diffusion velocity in high extent, zero-order released rate can be achieved. The model developed by Aimetti et al. (2009) has been applied to enzymatically erodible hydrogel disks. Hydrogel network was formed by thiol-ene photopolymerization of four-armed PEG endcapped by norbornene with two-armed thiolated cross-linker. Each cross-linker molecule contained human neutrophil elastase (HNE)-sensitive peptides. Bovine serum albumin (BSA) (66 kD) and carbonic anhydrase (29 kD) were used as two different sized model drugs. Mass loss and also fractional release of two proteins from same hydrogel disks were measured. Two proteins were released at the same rate. It indicated that drug release rate was independent of

3 Drug release

solute size and was influenced solely by gel degradation kinetic. They also used developed model to accurately predict BSA release from three-layered disk of hydrogel, which consists of identical bottom and top layers and a different middle layer. The bimodal BSA release from three-layered gel predicted by the model matched with the experimental release profile.

Contributions and constraints Hopfenberg and its extended model are interesting due to the simplicity. Device geometry has a great effect on its erosion and subsequently drug release (Karasulu et al., 2000). The applicability of this model is limited because of its geometry constraint. The model was in fact written for one dimension as all except only one face in the device were ignored. Lee did not account for matrix swelling, the concentration dependence of the solute diffusion coefficient, and the external mass transfer resistance in his model. As mentioned, he validated his model by comparing the result for one case (dissolved matrix (CAs ¼ 1)) with Peterlin’s (Lee, 1980) in early release time before diffusion front met centerline. Peterlin provided exact solution for the sorption of solvent in a glassy polymer membrane undergoing case II swelling in which diffusion is much faster than polymer relaxation. Unknown parameters in his model are two erosion and diffusion front velocities, and it is not clear how they can be obtained. Although the model was interesting, it needs future experimental validation. The model developed by Aimetti group was able to predict the BSA and carbonic anhydrase in different gel formulation very well from the mass loss data. This model is based on the assumption of zero-order kinetic for mass loss in which surface area is kept constant. The result was in a good agreement with experimental data from very thin hydrogels.

3.3 Diffusion/degradation controlled release from bulk degrading networks As mentioned before, the relative rate of water or any substrate diffusion to the network and rate of bond cleavage determines the mode of degradation (Yao and Weiyuan, 2010). In bulk-degrading hydrogels, water can penetrate to the matrix at a much faster rate than bond cleavage and hydrates it homogenously. Then, bond cleavage occurs randomly throughout the matrix. This leads to hydrogel swelling, release of degrading fragments, and decrease in mechanical strength over time. Hydrogels containing hydrolyzable groups such as PLA and PGA undergo bulk degradation (Lin and Anseth, 2013). So, in these systems of reaction and diffusion, phenomena couple to govern drug release profile.

3.3.1 Statistical-kinetic model The basic idea in this model is to relate transport properties of solute inside hydrogel with altering network structural by using existing diffusion theories developed for nondegradable gels (mentioned in Section “Diffusion controlled”). Several reports

257

258

CHAPTER 9 Mathematical models of drug release

have shown the results for a series of hydrolytically degradable hydrogels (Anseth et al., 2002; Mason et al., 2001; Lu and Anseth, 2000).

Mathematical model To investigate solute release from bulk-degrading hydrogels, scaling laws (Metters et al., 2001b) and kinetic degradation model mentioned in Section “Mathematical models” are combined together. It is assumed that degradable units are hydrolyzed by pseudo first-order kinetics within a highly swollen hydrogel. For hydrogels made of multiarmed macromers in which degradable units are present in cross-linking macromers, the concentration of intact cross-linked chains reversely changes with chain molecular weight between cross-links as Mc ∝

1 1 ∝ ρc ð1  PÞð1  Pm1 Þ

(48)

where Mc , ρc, P, and m are molecular weight between cross-links, concentration of cross-linked chains, fraction of hydrolyzed bonds, and number of arms in the multiarmed macromer. If the macromer has two arms, then 0

Mc ∝ e2jk t

(49)

where j represents the number of degradable bonds per each arm of macromer. t is the degradation time. Using a simplified version of Flory-Rehner equation (Flory, 1953) in a highly swollen network (Qv > 10), the volumetric swelling ratio is proportional 3

to Mc 5 . To calculate mesh size of the hydrogel, Canal-Peppas equation (Canal and Peppas, 1989) is used; 1  1=2 1 1=2   7 ε ¼ v2, s  3 r02 ¼ Q3 M c ffi Mc 10

(50)

where ε, v2, s, r02 , and Qv are hydrogel mesh size, polymer volume fraction in equilibrium swollen state, the root-mean-squared end-to-end distance of the polymer chains in the unperturbed state, and volumetric swelling ratio, respectively. Among numerous theoretical models mentioned for nondegradable hydrogels aiming to relate network structure and solute diffusion coefficient, free volume theory presented by Lustig and Peppas (1988) was selected due to its widespread application for PEG-based hydrogels. With the application of Eqs. (49), (50), diffusion coefficient within the hydrogel as a function of degradation time can be scaled as follows: 1

7 0 Dg rs  7 ¼ ∝ Mc 10 ∝ e 5 jk t Do ε

(51)

where Dg the solute diffusivity is in a swollen gel, Do is the solute diffusivity in the swelling solvent, and rs is the radius of the solute. Therefore, as degradation proceeds, diffusivity of solute within the hydrogels increases in an exponential manner within swollen hydrogels. The typical strategy for drug release modeling consists of two steps: (i) calculating k0 by fitting the experimental data of the swelling ratio 3

6

0

through Q∝ Mc 5 ∝ e5jK t and (ii) using the time-dependent diffusion coefficient

3 Drug release

(Eq. 51) to predict drug release from a degrading hydrogel. For simplicity, a hydrogel disk is used in the experimental setup such that one-dimensional diffusional release equation can be applied. Many parameters such as degree of macromer functionality, macromer concentration, molecular weight of core molecule, and number of cleavable bonds in the macromer have been manipulated to elucidate their effects on the network degradation profile.

Experimental systems and validation Release of different molecular weight proteins (Anseth et al., 2002; Mason et al., 2001; Lu and Anseth, 2000) and dextran (77 kD) (Lu and Anseth, 2000) from PLA-b-PEG-b-PLA and PLA-g-PVA (Mason et al., 2001; Anseth et al., 2002) photopolymerized hydrogels were investigated. As Eq. (51) predicted, mesh size increased exponentially with degradation time, which leads to increase in diffusion coefficient of solute. Solution of Fick’s law for both nondegradable and degradable disk hydrogels, using constant and time-dependent diffusion coefficient, respectively, illustrated how degradation of cross-links could change the release profile and enhance the mobility of solute in degradable hydrogel. To validate the model, release of two solutes, BSA (rs ¼ 3.5 nm) and lysozyme (rs ¼ 1.6 nm) from same degrading hydrogels made of PLA-PEG-PLA was investigated. Effect of degrees of acrylation (cross-linkable end group) and initial macromer concentrations on the drug release profile was also investigated. For different hydrogel formulations, k0 was first estimated from the swelling ratio profile. Results showed that as the degree of acrylate functionalization increased, the initial concentration of cross-linked chains increased, which caused hydrogels to have lower initial swelling ratios. The apparent degradation rate of hydrogel decreased, which led to slower drug release from the hydrogel. For hydrogel with higher macromer concentration, the extent of cyclization decreased due to lower intramolecular reaction. It led to increase in kinetic chain length (Lu and Anseth, 2000; Mason et al., 2001), additional physical entanglements (Anseth et al., 2002), and lower initial swelling ratio. Changes in microstructure (local water concentration) as a result of macromer concentration altered the fitted k0 so that hydrogel at higher macromer concentration underwent lower swelling ratio at all the degradation time compared with the ones with lower concentration. Different BSA release profile from these hydrogels illustrated how the differences between their network structures could affect the shape of release curve. Although the observed shape in the experimental release curve has been followed by the model, there was a significant deviation between the predicted and experimental release profile.

Contributions and constraints There are some points that should be mentioned about the application of the above model for drug release. In the case of BSA release from hydrogels with two degrees of acrylation, the selected macromers differed from not only the degree of acrylation but also the macromer molecular weight and number of degradable bonds. Changes in these two parameters (molecular weight and number of degradable bonds) resulted

259

260

CHAPTER 9 Mathematical models of drug release

in different microstructure for hydrogels and therefore the changes in degradation rate. So, the statement that observed differences in the release profile was because of changes in only one parameter (acrylation degree) was not correct. Although the trend of predicted drug release for this effect was nearly correct, however, there was a statistically significant deviation between experimental and model results. In addition, the effect of these parameters was shown in one kinetic parameter, k0 , which was calculated by fitting to swelling ratio profile for each hydrogel, and they did not assign any actual mathematical representative parameter in the model for them. The model can predict the solute release from three hydrogels with different initial macromer concentration in a correct trend. They obtained a satisfactory fit if they would adjust the solute diffusivity in pure water for each gel. However, there is not any physical meaning in changing solute diffusivity in pure water. Failure of model for higher macromer concentration is much more pronounced. Therefore, the model was unable to predict accurately the experimental results. One reason can be that the assumed diffusion coefficient equation is not valid. To use of free volume theory, polymer volume fraction should be <0.1. This criterion could not be met by the experimental system for model validation (Mason et al., 2001). It means that initial structure has fundamental differences that the model cannot explain. Simply assigning different kinetic rate constants to different hydrogel formulation does not help to understand the physical picture. As Stokes-Einstein equation, applying the current model to solute with shape other than spherical is limited (Lu and Anseth, 2000). Slower release is predicted by the model for dextran polymer compared with experimental data, which can be attributed to the shape of dextran molecule being rodlike.

3.3.2 Monte-Carlo simulations Monte Carlo method is a computational algorithm with broad application in many fields. This method uses randomness concept to compute target functions by generating scenarios based on the probability functions (Hubbard, 2010). As degradation is a random process and it is difficult to predict the exact degradation time of a particular bond in a specific location, it is practical to use stochastic method to handle the random nature of this process. First-order Erlang probability function can be used to describe such events. Pioneers in this field are G€opferich and Langer (1995) and Siepmann et al. (2002). They used this approach to describe the stochastic nature of erosion in polymer-based drug delivery systems. The basic concept for all the applications is as follows. First, the system is discretized in location and time coordinate, and one randomly determined lifetime parameter is applied based on the kinetics of chemical or any other processes relevant to the system. General equation for lifetime (tlifetime) for each node based on the first-order Erlang probability function is given in Eq. (52): tlifetime ¼ taverage +

ð1Þε  ε  ln 1  100 k

(52)

3 Drug release

where taverage, k, and ε are average lifetime of nodes, degradation constant, and random number between 0 and 100, respectively. Drug release from hydrogels can be modeled through relating transport properties of solute to the resultant structural changes from Monte Carlo simulations.

Mathematical model The developed model by Vlugt-Wensink et al. (2006) based on the Monte Carlo concept was applied for hydrogel microspheres with micronsize diameter containing different sized proteins. According to Fig. 6, polymerization takes place, and proteins are trapped in microspheres when the protein size is larger than network mesh size. The model is built based on some assumptions. The microsphere is modeled as a cubic lattice in which each lattice site has a certain number of cross-links (NModel). There is no protein on the lattice surface. Just one protein occupies one lattice site, and there will be no cross-link in the lattice when it is occupied with protein. Protein release depends on its hydrodynamic radius (rh) and actual cross-linking density. Confinement of protein in lattice is modeled by defining NModel cross-link arranged on one line. When XModel out of NModel cross-links are hydrolyzed, lattice site will be opened, and protein can jump into neighboring lattice with a jump rate of DModel. Simulation was performed on the lattice with 50  50  50 dimension. So, each lattice site either contains a certain number of cross-links or a protein molecule. Authors chose NModel ¼ 2000 in order to have sufficient space for incorporating a large range of protein sizes. Simulation starts with selecting one event among all the possible events with different rate constants (Eq. 53): pl ¼

rl k X

(53) rj

j¼1

Possible events are a list of cross-links that can be hydrolyzed with rate constants of rj (events per time). k is the number of events. This particular event is executed, and simulation goes to the next time step (Eq. 54): τi + 1 ¼ τi +

j ln ðuÞj K X rj

(54)

j¼1

u is a uniformly distributed random number between 0 and 1. When all the XModel neighboring cross-links are broken in order to open the lattice site, time τ is recorded. To be more statistically accurate, the process should be repeated at least 106 times. Then, probability at which XModel out of NModel are hydrolyzed can be calculated through Eq. (55): M X

PðtÞ ¼

θ ðt  τ i Þ

i¼1

M

(55)

261

262

Radical Polymerization 1 2

se

d

6 7 8

3 4 5 6 7 8

C

lo

se

d

C

lo

1 2 3 4 5

n

M cycles

O

pe

Methacryloyl group

Oligomethacrylate (Closed) lattice site

Hydrolysis

(A)

Continue until open

Record time t

Hydrolysis Diffusion

(B)

(C)

Release (Open) lattice site

FIG. 6 Schematic 2-D cage of one protein in network (left) and (A) and (B) are 2-D representation of the procedure in which one lattice becomes open when NModel ¼ 8, XModel ¼ 4 , a cross shows intact cross-links and empty square shows hydrolyzed ones. (C) Possible choices of one protein for jumping to a open neighboring site (right) From Vlugt-Wensink, K.D.F., et al., 2006. Modeling the release of proteins from degrading crosslinked dextran microspheres using kinetic Monte Carlo simulations. J. Control. Release 111 (1–2), 117–127.

CHAPTER 9 Mathematical models of drug release

Dextran

3 Drug release

where M number at which the whole process is repeated (M ¼ 106 times), θ is usual Heaviside step function (θ(t) ¼ 1 for t > 0 and θ(t) ¼ 0 otherwise). At the beginning of simulation, τi should be calculated for each lattice site i, through setting Eq. (55) equal to u. After obtaining the distribution of τi for the entire grid, any solute can diffuse within lattice sites by Dmodel. Proteins are considered as released when they diffuse out of the lattice.

Experimental system and validation The model was examined for hydroxyethyl methacrylated dextran (dex-HEMA) microsphere hydrogels with different degree of substitution of HEMA on dextran chains (DS). Proteins of various sizes (BSA (monomer/dimer), immunoglobulin (IgG), human growth hormone (hGH), and liposomes) were dissolved in polymer solution prior to gelation. Loading amount was assigned based on the volume of protein to volume of microsphere. The work could be divided into three major parts. First, XModel and DModel were obtained for the releasing of BSA monomer and liposomes through comparing the simulated release curve and available experimental data in the literature. XModel and DModel were obtained through fitting for BSA monomer and liposomes. Simulation data revealed that the release curve was biphasic. In the first phase, no protein was released, and so, it was called time lag. Its duration was dependent on only XModel. Second phase strongly depended on DModel. The effect of higher loading on protein release was much more pronounced at higher DModel and caused slightly shorter time lag. The fitted parameters were used to establish a scaling relationship of XModel and DModel with the hydrodynamic radius (rh) of protein. The XModel scaled with rh α in which 0 < α < 1. The actual scaling relationship lies between two extreme conditions when (1) XModel ∝ rh in which all cross-links in the perimeter of the projected plane of protein needed to be hydrolyzed to open the lattice site or (2) XModel ∝ rh 2 in which all the cross-links at the surface of the projected plane of protein needed to be hydrolyzed to open the lattice site. These two cases are ideal (and not happening in real case) due to the random nature of hydrolysis. α was calculated to be 1.64 based on the release profile of liposomes. Last, scaling relationships XModel ∝ rh 1:64 and DModel ∝ rh 1 (Stokes-Einstein equation) were used to predict the release of IgG, hGH, and BSA dimer from microspheres.

Contributions and constraints An overview on the BSA monomer and dimer release curve reveals the fact that although second phase of release is a good fit but their simulation cannot follow burst release observed in the experimental data. On the other words, the model cannot follow the time-lag phase in the release profile. In addition, simulation results for IgG at later time for DS 3 and 6% are in relatively good agreement but are unable to predict exactly early release time. There are some deviations between simulation and experimental result for higher DS values for both phases of the release profile. In hGH case, the release profile in the presence of Tween 80 is accurately predicted by simulation, while in the absence of Tween 80, due to formation of some aggregation, release profile is deviated greatly from model prediction. Overall, first phase in nearly all the solute release could not be predicted by the model. This indicates existence of phenomenon

263

264

CHAPTER 9 Mathematical models of drug release

other than diffusion and degradation in the release that is not considered in the model. One can be swelling of microsphere that is neglected in the model. Swelling can have a substantial effect on the release especially early times. The model parameters NModel, XModel, and DModel—defined as the total and hydrolyzable number of cross-linked chains and jumping rate of protein—do not have real physical meaning and cannot be translated into measurable experimental parameters.

4 Disintegration time 4.1 Mathematical model This model (Xu et al., 2014) is developed for the networks formed through complementary reaction between two end groups of multiarmed macromer (Fig. 7). Different degradable linkers (labile or nearly stable) can be incorporated into the arms of macromers to render different degradation properties in the hydrogel networks. In a highly swollen state, cleavable bonds are hydrolyzed according to a pseudo first-order reaction (Eq. 56). Pintact is time-dependent fraction of remaining intact bonds as Pintact ¼

½linkaget ¼ ekd t ½linkage0

(56)

when Pintact reaches a previously mentioned critical conversion value (Pc), hydrogel network no longer exists as an infinite network resulting in gel dissolution. It is the

FIG. 7 Schematic representation of hydrogel network from its four-armed macromers with different degradable linkers within structure.

4 Disintegration time

same as critical gelation point during cross-linking process. So, disintegration time can be determined by following relation: tc ¼ ð ln Pc Þ=kd

(57)

Depending on the number of cleavable bonds with different hydrolysis rate, one can write the fraction of intact bonds over time. For example, for two different bonds within the network, we will have Pintact ¼

½linkaget ¼ rekd, f t + ð1  r Þekd, s t ½linkage0

(58)

Hydrogel contains r fraction of bonds with hydrolysis rate constant of kd, f and 1  r fraction of bonds with hydrolysis rate constant of kd, s. For four different types of bonds within the network: Pintact ¼

½linkaget ¼ ð1  rN3 Þð1  rDBCO Þ + ð1  rN3 ÞrDBCO ekd, DBCO t ½linkage0 + rN3 ð1  rDBCO Þekd, N3 t + rN3 rDBCO ekd, N3 + kd, DBCO Þt

(59)

Applying Flory theory for gelation point, the hydrogel made by step-growth polymerization of four-armed macromer was found to have Pc ¼ 1/3. Using Eq. (57) and experimentally measured disintegration time for hydrogels, authors were able to calculate the hydrolysis rate constant for two types of labile bonds (Xu et al., 2014). This allowed prediction of disintegration time for gels formed by different ratios of degradable and nondegradable macromers.

4.2 Experimental system and validations They (Xu et al., 2014) applied the model for hydrogels made of four-armed PEG formed by reacting azide (N3) and dibenzocyclooctyl (DBCO) end groups. As illustrated in Fig. 7, there were two kinds of macromers with N3 (Fig. 7A1 and A2) or DBCO (Fig. 7B1 and B2) end groups. Their difference is that one out of two formulations that ends to N3 or DBCO reactive group was degradable. So, four different types of hydrogels could be formed that contain zero, one, and two degradable bonds in the cross-linked chains within their structures. kd was fitted for hydrogel compositions that contained one labile linker through application of Eq. (57) with experimentally measured disintegration time. Then, fitted kd for each specific linker was used to predict disintegration times for hydrogels produced through combination of the macromers with different labile linkers at different ratios.

4.3 Contributions and constraints One necessary condition for their model to be valid is that the labile linkage should be positioned near cross-links such that the hydrolysis of labile bonds is a reverse process of gelation through cross-linking. The predicted disintegration time (tc) deviated from experimentally measured value when r ratio is lower. It may be because the

265

266

CHAPTER 9 Mathematical models of drug release

structure is different from the ideal fully reacted network. Flory theory for gelation point is developed for an ideal network formed through step-growth polymerization of multiarmed macromers. However, in real system, due to incomplete conversion or intramolecular reaction, there are some structural defects in the network.

5 Conclusion Proper design of network properties of biodegradable hydrogels is essential for their biomedical applications such as drug delivery and tissue engineering. To this aim, mathematical modeling has an important role. A successful model must be able to capture the physical details inside the system and account for the many interrelated complicated phenomena during degradation. Drug release profile from degrading hydrogels is affected by the initial properties of a hydrogel, including polymer concentration, cross-linking density, polymer molecular weight, chemical compositions, hydrogel dimensions (swelling ratios), the mesh size of the hydrogel network, and the hydrodynamic size of drug. Although many models were developed, the challenge remains to describe all the simultaneously changing parameters. Two different approaches have been explored to quantify the degradation profile and drug release: kinetic-statistical and Monte Carlo simulations. The first approach provides a semiquantitative picture of the hydrogel network and some useful insights, while the second one is more informative about the changing microstructure but is more complicated to be set up. In conclusion, this chapter provides a summary of the models reported in literature for different types of degradable hydrogels and discusses about their limitations and advantages in terms of ease of use (e.g., the availability of model input parameters) and the ease to apply the results (e.g., the physical interpretation of the model outputs).

References Aimetti, A.A., Machen, A.J., Anseth, K.S., 2009. Poly(ethylene glycol) hydrogels formed by thiol-ene photopolymerization for enzyme-responsive protein delivery. Biomaterials 30 (30), 6048–6054. Amsden, B., 1998. Solute diffusion within hydrogels. Mechanisms and models. Macromolecules 31, 8382–8395. Amsden, B., 1999. An obstruction-scaling model for diffusion in homogeneous hydrogels. Macromolecules 32 (3), 874–879. Anseth, K.S., Metters, A.T., Bryant, S.J., Martens, P.J., Elisseeff, J.H., Bowman, C.N., 2002. In situ forming degradable networks and their application in tissue engineering and drug delivery. J. Control. Release 78, 199–209. Buchholz, F., Peppas, N.A.E., 1994. Superabsorbent polymer. In: Science and Technology. 16, American Chemical Society, Washington, DC, pp. 193–202. Canal, T., Peppas, N.a., 1989. Correlation between mesh size and equilibrium degree of swelling of polymeric networks. J. Biomed. Mater. Res. 23 (10), 1183–1193.

References

Cheng, F., Choy, Y.B., Choi, H., Kim, K.K., 2011. Modeling of small-molecule release from crosslinked hydrogel microspheres: effect of crosslinking and enzymatic degradation of hydrogel matrix. Int. J. Pharm. 403 (1–2), 90–95. Chung, H.J., Park, T.G., 2009. Self-assembled and nanostructured hydrogels for drug delivery and tissue engineering. Nano Today 4 (5), 429–437. Cukier, R.I., 1984. Diffusion of Brownian spheres in semidilute polymer solutions. Macromolecules 17 (2), 252–255. DuBose, J.W., Cutshall, C., Metters, A.T., 2005. Controlled release of tethered molecules via engineered hydrogel degradation: model development and validation. J. Biomed. Mater. Res. A 74 (1), 104–116. Elbert, D.L., Pratt, A.B., Lutolf, M.P., Halstenberg, S., Hubbell, J.A., 2001. Protein delivery from materials formed by self-selective conjugate addition reactions. J. Control. Release 76 (1), 11–25. Fenoli, C.R., Bowman, C.N., 2014. The Thiol-Michael addition click reaction: a powerful and widely used tool in materials chemistry. Chem. Mater. 26 (1), 724–744. Flory, P.J., 1953. Principles of Polymer Chemistry First. Cornell University press, New York. Ganji, F., Vasheghani-Farahani, S., Vasheghani-Farahani, E., 2010. Theoretical description of hydrogel swelling: a review. Iran. Polym. J. 19 (5), 375–398. G€opferich, A., Langer, R., 1995. Modeling of polymer erosion in three dimensions: rotationally symmetric devices. AICHE J. 41 (10), 2292–2299. Hopfenberg, H.B., 1976. Paul, D.R., Haris, F.W. (Eds.), Controlled Release Polymeric Formulations. Vol. 33. American Chemical Society, Washington, DC, pp. 26–31. Hubbard, D.W., 2010. How to Measure Anything Finding the Value of “Intangibles” in Business, second ed. John Wiley & Sons, Inc., Hoboken. Huglin, M.R., 1989. Hydrogels in medicine and pharmacy Edited by N. A. Peppas, CRC Press Inc., Boca Raton, Florida, 1986 (Vol. l), 1987 (Vols 2 and 3). British Poly. J. 21 (2), 184. Kamaly, N., Yameen, B., Wu, J., Farokhzad, O.C., 2016. Degradable controlled-release polymers and polymeric nanoparticles: mechanisms of controlling drug release. Chem. Rev. 116 (4), 2602–2663. Karasulu, H.Y., Ertan, G., K€ose, T., 2000. Modeling of theophylline release from different geometrical erodible tablets. Eur. J. Pharm. Biopharm. 49 (2), 177–182. Katzhendler, I., Hoffman, A., Goldberger, A., Friedman, M., 1997. Modeling of drug release from erodible tablets. J. Pharm. Sci. 86 (1), 110–115. Khanlari, S., Dub
e, M.A., 2013. Bioadhesives: a review. Macromol. React. Eng. 7 (11), 573–587. Lao, L.L., Peppas, N.A., Boey, F.Y.C., Venkatraman, S.S., 2011. Modeling of drug release from bulk-degrading polymers. Int. J. Pharm. 418 (1), 28–41. Lee, P.I., 1980. Diffusional release of a solute from a polymeric matrix—approximate analytical solutions. J. Membr. Sci. 7 (3), 255–275. Li, X., Tsutsui, Y., Matsunaga, T., Shibayama, M., Chung, U.-i., Sakai, T., 2011. Precise control and prediction of hydrogel degradation behavior. Macromolecules 44 (9), 3567–3571. Lin, C.-C., Anseth, K.S., 2013. Chapter II.4.3. In: The Biodegradation of Biodegradable Polymeric Biomaterials. third ed. Elsevier. Lin, C.C., Metters, A.T., 2006. Hydrogels in controlled release formulations: network design and mathematical modeling. Adv. Drug Deliv. Rev. 58 (12 13), 1379–1408. Lu, S., Anseth, K.S., 2000. Release behavior of high molecular weight solutes from poly (ethylene glycol)-based degradable networks. Macromolecules 33 (7), 2509–2515.

267

268

CHAPTER 9 Mathematical models of drug release

Lustig, S.R., Peppas, N.a., 1988. Solute diffusion in swollen membranes. IX. Scaling laws for solute diffusion in gels. J. Appl. Polym. Sci. 36 (4), 735–747. Martens, P., Metters, A.T., Anseth, K.S., Bowman, C.N., 2001. A generalized bulkdegradation model for hydrogel networks formed from multivinyl cross-linking molecules. J. Phys. Chem. B 105 (22), 5131–5138. Martens, P.J., Bryant, S.J., Anseth, K.S., 2003. Tailoring the degradation of hydrogels formed from multivinyl poly(ethlene glycol) and poly(vinyl alcohol) macromers for cartilage tissue engineering. Biomacromolecules 4 (2), 283–292. Martens, P.J., Bowman, C.N., Anseth, K.S., 2004. Degradable networks formed from multifunctional poly(vinyl alcohol) macromers: comparison of results from a generalized bulkdegradation model for polymer networks and experimental data. Polymer 45 (10), 3377–3387. Masaro, L., Zhu, X.X., 1999. Physical models of diffusion for polymer solutions, gels and solids. Prog. Polym. Sci. 24 (5), 731–775. Mason, M.N., Metters, A.T., Bowman, C.N., Anseth, K.S., 2001. Predicting controlled-release behavior of degradable PLA- b -PEG- b -PLA hydrogels. Macromolecules 34 (13), 4630–4635. Metters, A., Hubbell, J., 2005. Network formation and degradation behavior of hydrogels formed by Michael-type addition reactions. Biomacromolecules 6 (1), 290–301. Metters, A.T., Bowman, C.N., Anseth, K.S., 2000a. A statistical kinetic model for the bulk degradation of PLA- b -PEG- b -PLA hydrogel networks. J. Phys. Chem. B 104 (30), 7043–7049. Metters, a.T., Anseth, K.S., Bowman, C.N., 2000b. Fundamental studies of a novel, biodegradable PEG-b-PLA hydrogel. Polymer 41 (11), 3993–4004. Metters, A.T., Anseth, K.S., Bowman, C.N., 2001a. A statistical kinetic model for the bulk degradation of PLA-b-PEG-b-PLA hydrogel networks: incorporating network non-idealities. J. Phys. Chem. B 105 (34), 8069–8076. Metters, A.T., Bowman, C.N., Anseth, K.S., 2001b. Verification of scaling Laws for degrading PLA- b -PEG- b -PLA hydrogels. AICHE J. 47 (6), 1432–1437. Park, H., Park, K., Shalaby, W.S.B., 2011. Biodegradable Hydrogels for Drug Delivery. CRC Press., Lancaster, Pennsylvania. Patel, R.H., Patel, P.K., Patel, D.A., Bharadia, P.D., Pandya, V.M., Modi, D.A., 2011. Hydrogel as a controlled drug delivery system. IJPI’s J. Pharmaceut. Cosmetol. 1 (5), 51–59. Peppas, N.A., Bures, P., Leobandung, W., Ichikawa, H., 2000. Hydrogels in pharmaceutical formulations. Eur. J. Pharm. Biopharm. 50 (1), 27–46. Reid, R., Sgobba, M., Raveh, B., Rastelli, G., Sali, A., Santi, D.V., 2015. Analytical and simulation-based models for drug release and gel-degradation in a tetra-PEG hydrogel drug-delivery system. Macromolecules 48 (19), 7359–7369. Shah, N.M., Pool, M.D., Metters, A.T., 2006. Influence of network structure on the degradation of photo-cross-linked PLA-b-PEG-b-PLA hydrogels. Biomacromolecules 7 (11), 3171–3177. Shih, H., Lin, C.C., 2012. Cross-linking and degradation of step-growth hydrogels formed by thiol-ene photoclick chemistry. Biomacromolecules 13 (7), 2003–2012. Siepmann, J., Siepmann, F., 2008. Mathematical modeling of drug delivery. Int. J. Pharm. 364 (2), 328–343. Siepmann, J., Kranz, R., Bodmeier, R.P.N.A., 1999. A new model combining diffusion, swelling, and dissolutions mechanisms and predicting the release kinetics. Pharm. Res. 16 (11), 1748–1756.

References

Siepmann, J., Faisant, N., Benoit, J., 2002. A new mathematical model quantifying drug release from bioerodible microparticles using Monte Carlo simulations. Pharm. Res. 19 (12). Tibbitt, M.W., Kloxin, A.M., Sawicki, L.A., Anseth, K.S., 2013. Mechanical properties and degradation of chain and step-polymerized photodegradable hydrogels. Macromolecules 46 (7), 2785–2792. Tzafriri, A.R., 2000. Mathematical modeling of diffusion-mediated release from bulk degrading matrices. J. Control. Release 63 (1–2), 69–79. Vashist, A., Vashist, A., Gupta, Y.K., Ahmad, S., 2014. Recent advances in hydrogel based drug delivery systems for the human body. J. Mater. Chem. B 2 (2), 147. Vlugt-Wensink, K.D.F., Vlugt, T.J.H., Jiskoot, W., Crommelin, D.J.A., Verrijk, R., Hennink, W.E., 2006. Modeling the release of proteins from degrading crosslinked dextran microspheres using kinetic Monte Carlo simulations. J. Control. Release 111 (1–2), 117–127. White, C.J., Tieppo, A., Byrne, M.E., 2011. Controlled drug release from contact lenses: a comprehensive review from 1965-present. J. Drug Deliv. Sci. Technol. 21 (5), 369–384. Xu, J., Feng, E., Song, J., 2014. Bioorthogonally cross-linked hydrogel network with precisely controlled disintegration time over a broad range. J. Am. Chem. Soc. 136 (11), 4105–4108. Yao, F., Weiyuan, J.K., 2010. Drug release kinetics and transport mechanisms of non- degradable and degradable polymeric delivery systems. Expert Opin. Drug Deliv. 7 (4), 429–444. Yu, Y., Chau, Y., 2012. One-step “click” method for generating vinyl sulfone groups on hydroxyl-containing water-soluble polymers. Biomacromolecules 13 (3), 937–942. Yu, Y., Lau, L.C.M., Lo, A.C.-Y., Chau, Y., 2015. Injectable chemically crosslinked hydrogel for the controlled release of bevacizumab in vitreous: A 6-month in vivo study. Translational vision science & technology 4 (2), 5.

269