Stochastic Modeling of Degradation Behavior of Hydrogels

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Stochastic Modeling of Degradation Behavior of Hydrogels Ghodsiehsadat Jahanmir,†,‡ Mohammad J. Abdekhodaie,‡ and Ying Chau*,† †

Department of Chemical and Biological Engineering, Academic building, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China ‡ Chemical and Petroleum Engineering Department, Sharif University of Technology, Azadi Ave., 11365-11155 Tehran, Iran S Supporting Information *

ABSTRACT: We describe here a theoretical framework to model the bulk degradation of hydrogels, which are prepared by chemical cross-linking of pendant functional groups on long polymer chains. The random order of the cleavage of degradable bonds was described by stochastic Monte Carlo simulation. The events of bond cleavage were related to the macroscopic changes of hydrogel by the Bray−Merrill equation. Next, the time for the gel to disintegrate was predicted by considering the relation between the recursive nature in breaking the cross-link nodes and gel-to-sol transition. To start the simulation, initial network properties including the number of active functional groups on the polymer chain and the concentration of polymer were employed as input, and the kinetic rate constant of bond cleavage was fitted for the swelling profile. No fitting parameter was required for disintegration time. A series of degradable hydrogels composed of dextran modified by methacrylate and thiol groups were synthesized and examined experimentally to verify the models. The measured mass swelling ratio and gel disintegration time matched with the model predictions. Correlation was found between the initial hydrogel network properties and the profiles of degradation. The results also revealed that degradable hydrogels with a wide range of disintegration times (from 3 days to 1 month) could be prepared by manipulating the hydrogel formulation (for example, polymer concentration and degree of modification) without altering the chemistry of the cleavable bond.

1. INTRODUCTION Hydrogels are cross-linked polymer networks that can be prepared by a large variety of hydrophilic monomers or polymers. They have the ability to retain a large fraction of water within their structure. This property makes them suitable materials for biomedical applications, such as tissue engineering and drug delivery.1,2 In many applications, degradable hydrogels are preferred as they negate the need of removal from the host. In order to achieve desired properties for certain application, which may include the mode of degradation, the time span of the material, and mechanical strength, the hydrogel should be designed by tuning important influencing parameters. These comprise backbone chemistry, molecular weight, polymer concentration, functionality, and degree of cross-linking. Ideally, hydrogel degradation behavior should be predictable through manipulating these parameters. In this research, the aim is to study and quantify the relationship between these controllable parameters and the degradation behavior of hydrogel, manifested in the dynamic change of © XXXX American Chemical Society

hydrogel swelling ratio and the gel disintegration time. We focus our effort in modeling the degradation of hydrogels made by cross-linking of long linear polymer chains containing pendant functional groups3−5 (Scheme 1), which we refer to as polymer−polymer hydrogels. Smaller mesh size can be attained, making this type of hydrogel more apt for long-term controlled release. Small molecular weight cross-linkers are not used, thus reducing the risk of unwanted cytotoxicity.6,7 This type of hydrogel may bear hydrolytically cleavable bonds to make them bulk degradable. To our knowledge, there is no proposed model for this type of hydrogel in the literature. The key in modeling degradable hydrogel behavior is to describe how the number of small chains between cross-link nodes changes over time, and it affects the hydrogel swelling ratio. An elegant mean-field statistical-co-kinetic model has Received: January 23, 2018 Revised: April 24, 2018

A

DOI: 10.1021/acs.macromol.8b00165 Macromolecules XXXX, XXX, XXX−XXX

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Scheme 1. Schematics of Degradable Polymer−Polymer Hydrogel, in Which Long Polymer Chains Are Cross-Linked by the Functional Pendent Groups on the Polymer Chaina

a

Formed cross-link nodes have one hydrolyzable ester bond. The example shown here and used in the current study is comprised of methacrylate and thiol groups as the cross-linking pairs.

forming the hydrogel of interest can have different concentration, degree of modifications, and molecular weight. A subtle and yet important feature of the polymer−polymer hydrogel is that the effect of cleavage of cross-link node on the change in number of chains between cross-link nodes is dependent on its position within the network structure (Figure S4). Thus, the mean value for one small portion of the network cannot represent the overall behavior of the hydrogel, and therefore it is not possible to directly apply the previous models. One viable strategy is to track the status of individual cross-linking nodes in the network. To account for the possible scenarios during degradation in polymer−polymer hydrogels, and the complexity of networks from heterogeneous polymer chains, we have implemented in this paper a stochastic microscopic method that is based on the consideration of the random nature of cross-linking bonds degradation. The Monte Carlo method is a computational algorithm with broad application in many fields. This method uses the randomness concept to compute target functions by generating scenarios based on the probability functions.19 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 the stochastic method to handle the random nature of this process. The first-order Erlang probability function can be used to describe such events. This approach has been taken to model the surface erosion of hydrophobic degradable polymers.20−22 The model accounted for changes in the porosity of the polymer matrix, which in turn influenced the kinetics of drug release. The Monte Carlo scheme was also used to model the protein release from degradable hydrogel microspheres, though without taking into account the swelling

been developed for predicting the bulk degradation behavior of hydrogels formed by photopolymerization of multiarmed macromers.8−14 In this model, one cross-linking macromer is selected at a time, and the probability of this macromer remaining intact is estimated. Scaling laws are used to relate the macroscopic properties (such as mass swelling ratio) with the molecular weight between cross-link nodes.8,10,15−17 This approach was later extended to describe another type of hydrogel comprising low molecular weight cross-linkers and multiarmed PEG macromers.18 The model was modified to accommodate the structural characteristics of this type of hydrogel. A binomial probability equation was proposed to account for the possible statuses of the multifunctional macromers. To estimate the change in the concentration of small chains between cross-link nodes, one can make a summation over the probabilities of cross-linking points remaining intact for each macromer and multiply the sum with the initial concentration of macromers. While this approach satisfactorily describes hydrogel networks formed from homogeneous macromers, it cannot be directly applicable to polymer−polymer hydrogels with pendant cross-linkable groups along the polymer chains.6,7 The microstructure of the network formed by polymer− polymer hydrogels is different from networks composed of cross-linking macromers. Namely, the number of cross-link nodes is different from the number of small chains between cross-link nodes in polymer−polymer hydrogels, whereas the two values are the same in the network formed by cross-linking macromers (Figure S3). The degree of heterogeneity and the number of possible scenarios happening in a polymer−polymer hydrogel network increase as the long constituent chains B

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Scheme 2. Simplified Two-Dimensional Initial Structure of Degradable Polymer−Polymer Hydrogel Network with Chemical Composition of Cross-Link Nodes

of hydrogel.23 Here, we used the Monte Carlo approach to capture the randomness of degradation events within a hydrogel network. Consider, for example, two structurally identical cross-link nodes adjacent to each other. Though having the same probability of cleavage, these two nodes may not be degraded simultaneously. With a random factor assigned to the lifetime of individual nodes, we were able to obtain a more realistic picture on how the cross-link nodes changed over time within the hydrogel network. In addition to the time course of cross-link nodes, many microstructural information as well as mass loss can be obtained for wide ranges of initial gel compositions. For example, hydrogels with heterogeneous distribution of polymeric length and number of active groups per chain can be modeled. Another phenomenon during hydrogel degradation that we seek to understand is the time it takes for the hydrogel to disintegrate. This knowledge is especially important when hydrogels are used as depots for the prolonged delivery of bioactive molecules. Gel disintegration can be considered as “reverse gelation”, and therefore insights can be obtained from methods for estimating the gelation point. Notably, a recursive approach together with conditional probability were used to predict the molecular weight of developing nonlinear molecules during polymerization.24,25 The onset of gelation was identified as the point when this model could not yield a finite solution for the molecular weight of the developing molecules or, in other words, when an “infinite” network was formed. Here, we employed this approach in a reverse manner to predict the disintegration time of polymer−polymer hydrogels. A similar strategy was used to predict the disintegration of hydrogel synthesized by mixed polymerization.14 The theoretical results were verified with experiments in this report. The experimental system consisted of dextran polymers modified by methacrylate and thiol groups along the linear chain. Polymers varying in concentration and degree of modifications were mixed to form hydrogels by chemical cross-linking via Michael addition. Different initial network structures were obtained. Degradation was made possible by the presence of ester linkage in each cross-link node. Model prediction on the hydrogel swelling ratio and disintegration time were compared to experimentally measured values for different hydrogel formulations.

2. MATERIALS AND METHODS 2.1. Experimental Setup. 2.1.1. Materials. Dextran 40 kDa was purchased from Sigma-Aldrich Co., Ltd. Weight-average molecular weights (Mw) were measured by the company using GPC coupled with MALLS and RI detectors. Dimethyl sulfoxide, divinyl sulfone (DVS, 97%), dithiothreitol (DTT), methacrylic anhydride (MA), triethylamine (TEA), and deuterium oxide (D2O) were purchased from Sigma Chemical Co., U.S.A. SH- and MA-dextran are dextran modified to contain thiol and methacrylate groups respectively, prepared using procedures described below. 2.1.2. Polymer Modification. SH-dextran were synthesized based on our previous reports.6,7,26 Briefly, dextran was first modified by adding DVS to dextran solution (5% w/v, pH = 12) at a molar ratio of 1.2× the hydroxyl groups of the polymer and reacting for 1−4 min depending on the degree of modification. Reaction was stopped by adjusting the pH to 4 with 6 M HCl. The product obtained was purified by dialysis using Spectra/Por dialysis membrane (MWCO: 3.5kD, Spectrum Laboratories, U.S.A.) against DI water (pH = ∼5.5) and then reacted with DTT in excess (at a molar ratio of 10) under N2 purging for 30 min in phosphate buffer to afford SH-dextran. SHdextran was purified by dialysis in acidic water and lyophilized subsequently and stored at −20 °C before use. MA-dextran was synthesized as follows.27 Dextran was dissolved in DMF/LiCl at ∼8% w/v at 90 °C in a round-bottom flask. After complete dissolution, methacrylate anhydride was added in different molar ratios of MA to hydroxyl groups on the polymer (1:0.5 to 1:1) for different degrees of modification, and the reaction was fixed at 60 °C. Then TEA was added in 0.001 molar ratio to hydroxyl groups, and the reaction continued overnight. The reaction mixture was then transferred to a dialysis tube and dialyzed for 2 days against DI water. MA-dextran was lyophilized and stored at −20 °C before use. The degree of modification (DM) of polymers is defined as the number of SH or MA units per 100 repeating units, expressed in percentage. It was determined according to 1H NMR spectra (acquired by a 300 MHz NMR spectrometer, Mercury VX 300, Varian, USA). 2.1.3. Hydrogel Formation. Hydrogels were formed by Michael addition reaction between the vinyl groups on MA-dextran and the thiol groups on SH-dextran in aqueous solution. MA- and SH-dextran were dissolved separately in 0.1 M phosphate buffer (PB) at pH 7.4 and vortexed until complete dissolution. Hydrogels were formed by mixing MA- and SH- polymer solution at different concentrations and DM values. They were kept in a humid chamber for overnight incubation (∼16 h) at room temperature to ensure complete gelation. 2.1.4. Hydrogel Degradation. Mass swelling measurements were performed according to reported methods.1 Hydrogels (each gel ∼ 50 μL) after overnight incubation were weighted to obtain the relaxed weight (Mr). They were put to swell in a large volume of 0.1 M PB (at least 10× the initial gel volume) at 37 °C. Wet equilibrium weight of C

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Macromolecules each gel sample after 24 h (M24 h,s) was measured. This value is marked as initial equilibrium swelling ratio for each gel and used to estimate cross-linking efficiency (eq 2). For each gel formulation, at least three samples immediately after gel formation and after equilibrium swelling were dried in oven and weighted. The dry weight of the gel at relaxed state (Md,r) and swollen state (Md,s) was obtained by subtracting the weight of the PB salt (∼17.3 mg/mL) from the weight measured after drying. After 24 h of swelling, gels continued to be submerged in the buffer and taken out at different time points to obtain the wet weight (Mt,s). Experimentally measured mass swelling ratio (Qm,exp) was calculated by dividing Mt,s by Md,s (eq 1).

Q m,exp(t ) =

Pintact =

(1)

2.2. Mathematical Model. 2.2.1. Initial Network Structure. The initial network structure of a degradable polymer−polymer hydrogel (Scheme 2) is defined at the starting point of the mathematical model. The degree of deviation from ideality of the experimental system is expressed by cross-linking efficiency (η) (eq 2).

η = ve,actual /ve,ideal

ε = 1 − e −k ′ t

ti , j =

(2)

v2̅ v1̅ (Q m − 1) + v2̅

ve,actual Vgel,r

=−

⎧1 if node is broken xi , j = ⎨ ⎩ 0 if node is intact

( )

( )

(4)

where ρxl, Vgel,r, v2,r, V1, and χ1 are the concentration of small chains between cross-link nodes in the network, the volume of the gel, the polymer volume fraction at the relaxed state, the molar volume of the swelling agent (which equals 18 mL/mol for water), and the Flory polymer−solvent interaction parameter (0.4630), respectively. 2.2.2. Hydrogel Degradation by the Monte Carlo Method. Hydrogels formed via the method described contain one hydrolytically labile ester bond per cross-link node. Degradation starts when the hydrogel is fully formed and is exposed to excess amount of aqueous solution. The macroscopic properties (e.g., hydrogel swelling) are related to the status of the many cross-link nodes with the hydrogel. In order to describe the lifetime of each particular ester bond within the macromolecular network using the kinetics parameter, we adopt a stochastic approach based on the Monte Carlo concept. Bond hydrolysis is assumed to follow pseudo-first-order kinetics (eqs 5 and 6). This assumption is valid when water concentration and medium pH are both constant.8−12,16,18,31−33 d[DB] = − k′[DB] dt

(8)

(9)

i and j are the number of active functional groups on the two types of precursor chains. They are obtained based on the degree of modification and concentration of polymer in the precursors’ solution and revised based on the procedure as described in the previous section to account for deviation from the ideal structure. Before starting the simulation, the lifetime of each node is calculated based on eq 8. Simulation starts at time zero. One relatively small time step (0.05 day for most cases is small enough) is considered for each iteration. The current time is compared with the lifetime (ti,j) of individual nodes. When the current time point is greater than a certain node’s lifetime ti,j, that node is expired, and the cross-link node is considered broken. Correspondingly, the value of xi,j is changed from 0 to 1. Summation over the status function (xi,j) for both polymers gives the number of broken bonds on each backbone chain. This information is stored in two separate variables and updated in each iteration. Once the status function of all nodes on one chain is changed to 1, that chain will be considered detached from the network. Based on the number of detached chains and the number of broken bonds one every polymer chain, the remaining mass bounded to the hydrogel and ve,actual can be calculated. The outputs include the number of moles and the mass of detached chains, the average number of intact nodes on both backbone chains, and consequently the number of moles of small chains between them. 2.2.3. Swelling Profile. The swelling of hydrogel is related to the ρxl.34 The result about the ve,actual after each time iteration is input to the Bray−Merrill equation (eq 4) to calculate the polymer volume fraction (v2,s). Mass swelling ratios can then be calculated using eq 3. To increase the accuracy and precision of the simulations, each run was repeated for at least 100 times, and the average was reported for each gel. The mathematical model was implemented on MATLAB (MATLAB R2014b). Codes can be found in the following link: https://hkustconnect-my.sharepoint.com/:b:/g/personal/gjahanmir_ connect_ust_hk/EYVmbI4Ufh9Cgs5X7s4yg9YBrPym20pI6Znzq_ 11TDtJvg?e=ZId8h. 2.2.4. Gel Disintegration Time. For polymer−polymer hydrogels, gelation occurs by cross-linking of long polymer chains through pendant group’s reaction (Scheme 2).34 Gel disintegration occurs when the cross-linked polymeric network is no longer a supermolecule with “infinite” molecular weight from the mathematical point of view. When this happens, the gel is turned into a polymer solution.9 This phenomenon can be considered the reverse of gelation point in the polymer cross-linking process, wherein soluble monomers are

(3)

2 1 ln(1 − v2,s) + v2,s + X1v2,s ⎞ V1 ⎛ v2,s 1/3 1 v ⎜ v − 2 v2,s ⎟ 2,r ⎠ ⎝ 2,r

−1 (ln(1 − ε) k′

We now define a new variable, xi,j, which describes the status of the cross-link node (i, j) in the following manner.

v1̅ and v2̅ are the specific volume of water (1 mL/g) and the specific volume of dextran (0.64 mL/g28), respectively. v2,s is input to Bray− Merrill equation29 (eq 4) to obtain ve,actual. Afterward, the actual number of either active functional groups per each chain is revised based on the cross-linking efficiency and used to obtain actual distribution of cross-link nodes within the network. This distribution is a starting point for modeling the degradation profile. ρxl =

(7)

After rearranging eq 7, one can calculate the lifetime of each arbitrary cross-link node (i, j) as a function of the random variable.

It is the ratio of actual (ve,actual) to ideal (ve,ideal) moles of small chains between cross-link nodes. The ideal network is the one in which all the active functional groups on the chains react at a 1:1 ratio and contribute to the initial concentration of small chains between crosslink nodes in the network (ρxl). ve,ideal is deduced from the mole number of SH and MA groups per polymer chain and the initial mole concentration of polymer in the gel. When DM and concentrations are different for polymers, ve,ideal is calculated based on the minimum value of available active groups between two chains. Experimentally measured initial equilibrium swelling after 24 h is used to calculate polymer volume fraction at equilibrium state (v2,s) (eq 3).

v2,s =

(6)

where t is time, k′ is the pseudo-first-order kinetics constant, [DB] is the concentration of degradable bonds in the network, [DB]0 is the initial concentration of degradable bonds, and Pintact is the probability that any degradable unit remaining intact. It is assumed that the environment for hydrolysis is homogeneous throughout the hydrogel network (that is, all the bonds are exposed to the aqueous medium simultaneously and crystalline region is absent). Because of the random nature of degradation, not all the bonds are cleaved at the same time, however. Therefore, a randomly distributed lifetime (ti,j) is assigned to each node. For the calculation of time at which each node is degraded, a sampling technique was used such that eq 6 is set to equal to ϵ, a random variable equally distributed in the interval (0, 1).

M t,s Md,s

[DB] = e −k ′ t [DB]0

(5) D

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Scheme 3. Schematic Representation of Recursive Process in Tracking the Molecular Weight of Developing Structure Resulting from Hydrogel Degradation for the Prediction of Its Disintegration Timea

a

1

2

3

4

Note: →, →, →, and → are directions dangling weights are tracked along to complete the loop. MA and SH are methacrylate and thiol groups grafted on the dextran backbone.

connected to form a network of “infinite” molecular weight. Macosko24,25 adopted a recursive approach to calculate the molecular weight of the developing structure in the cross-linking process. The law of conditional probability was used to capture the extent of reaction conversion. Mathematically, hydrogel can be described as an infinite network and is formed when the molecular weight diverges to an infinite value. Here, we use the same approach, but in a reverse manner, to track the molecular weight change of the hydrogel network. The time-dependent change is based on the probability of intactness of each degradable bond. Gel structure is turned into a soluble structure when the molecular weight converges to a finite value. Using this criterion, disintegration time of the hydrogel can be predicted. The hydrogel is composed of two polymer chains, with initial moles (nSH, nMA), molecular weight (Mw,SH and Mw,MA), and the number of active functional groups on each backbone (NSH, NMA) specified. Actual values for the mentioned parameters (NSH, NMA) (which reflect the actual distribution of cross-link nodes within network) are obtained after accounting for the cross-linking efficiency (eq 2). To calculate the molecular weight of the degrading hydrogel, one arbitrary set of arrows is defined to track the weights attached to cross-link nodes in a recursive manner. Imagine a cross-link node is picked up; the weight of the whole network (M̅ w) equals the weight dangling from the selected node in both directions (Scheme 3A). On one side is the weight attached via MA group (WMA), and on the other side is the weight attached via the SH group (WSH). To examine the weight attached to MA group, one starts by selecting one cross-link node ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ((MA)(SH) bond) randomly and examining the weight hanging to

E(W→1 ,MA ) = E(W→ 2, SH) × Pintact + 0 × (1 − Pintact)

Pintact is obtained from eq 6. E(W2⃗,SH) is the expected weight attached 2

to SH group in direction →. It equals the molecular weight of the SH chain, Mw,SH, plus the sum of the expected weight on each of the remaining (NSH − 1) node. The expected weight attached to SH group 3

on each node in the direction of → is E(W3⃗,SH) → E(W→ 2, SH) = M w,SH + (NSH − 1) × E(W 3, SH)

4

attached to MA group in direction → if the node is intact and zero otherwise. That is → E(W→ 3, SH) = E(W 4, MA ) × Pintact

(13)

Furthermore, the recursive equation can be derived, following similar logics as in eq 12: → E(W→ 4, MA ) = M w,MA + (NMA − 1) × E(W 1 ,MA )

(14)

For the network of interest, it takes four steps to complete the loop. Because the gel is combination of two types of polymers reacting to each other, the mass fraction (wi) of each backbone chain is included to calculate the molecular weight of the macromolecular network.

1

⎪

⎪

(12)

Similar to the calculation of E(W1⃗,MA), E(W3⃗,SH) equals to the weight

MA group in direction → (Scheme 3B).

⎧0 if node is broken ⎯⎯⎯⎯⎯⎯→ = ⎨ W1,MA ⎯⎯⎯⎯⎯→ if node is intact W ⎩ 2,SH

(11)

→ E(WMA ) = E(W→ 4, MA ) + E(W 1 ,MA )

(15)

→ E(WSH) = E(W→ 3, SH) + E(W 2, SH)

(16)

M̅ w = wMA × E(WMA ) + wSH × E(WSH)

(17)

(10) 1

W1⃗,MA is the weight hanging to MA group in direction →. It would be zero if that node is broken. If it is not broken, the attached weight in 1

direction → would be equal to the weight attached to the SH group in 2

direction →. So, expectation or the average value of molecular weight (E) is as follows:

wMA =

MWMA × nMA MWMA × nMA + MWSH × nSH

(18)

wSH =

MWSH × nSH MWMA × nMA + MWSH × nSH

(19)

Combining terms from eqs 11−19, one obtains

E

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M̅ w = wSH ×

(1 − Pintact 2((NMA − 1)(NSH − 1)))(M w,SH + NSHPintactM w,MA ) + (NMA − 1)(PintactM w,SH + Pintact 2(NSH − 1)M w,MA ) 1 − Pintact 2((NMA − 1)(NSH − 1))

+ wMA ×

PintactM w,SH + Pintact 2(NSH − 1)M w,MA + (PintactM w,SH + Pintact 2(NSH − 1)M w,MA )M w,MA (NMA − 1) 1 − Pintact 2((NMA − 1)(NSH − 1))

The developing molecular weights of MA and SH-dextran are influenced by the extent of individual bond cleavage (eq 20). As mentioned earlier, when gel is turned to polymer solution, it will have finite molecular weight. In mathematical language, eq 20 has a solution when the term 1 − Pintact2((NMA − 1)(NSH − 1)) in the denominator is a positive value. On the other hand, if the denominator is negative, molecular weight does not have a finite value (i.e., system is still a gel network). Therefore, when 1 − Pintact2((NMA − 1)(NSH − 1)) becomes zero, the gel network disintegrates to a polymer solution. Therefore, the probability of bonds remaining intact during onset of disintegration is reflected by

Pdis =

(20)

The effects of polymer concentration and degree of modification on the cross-linking efficiency were examined in the experimental system (Figure 2). The cross-linking efficiency

1 ((NMA − 1)(NSH − 1))

(21)

Eventually, the disintegration time, or onset time at which the network becomes soluble is

tdis

⎛ ln⎜ ⎝ ln(Pdis) =− =− k′

1 ((NMA − 1)(NSH − 1))

k′

⎞ ⎟ ⎠

Figure 1. Model prediction for the swelling ratio of hydrogel with [SH] = [MA] = 30% w/v and DM_SH = DM_MA = 10% with crosslinking efficiency of 1.0, 0.7, and 0.4 and tdis = 26.48, 23.13, and 18.81 days; k′ = 0.12 per day.

(22)

3. RESULTS AND DISCUSSION 3.1. Examination of Cross-Linking Efficiency in Initial Network. Dextran hydrogels with different initial polymer concentration and degree of modification are formed through chemical reaction between reactive pendant groups on the polymer chains. The initial value for (ve,ideal) was calculated based on the number of active functional groups on each polymer chain and the initial polymer concentration. Inputting ve,ideal into the Bray−Merrill relation (eq 4) allows one to obtain the polymer volume fraction and therefore the ideal theoretical mass swelling ratio (eq 3). When polymers are mixed at 1:1 ratio, they are likely to have unreacted functional groups, meaning that the actual ρxl is lower than an ideal, fully crosslinked network. This phenomenon was also observed previously in nondegradable hydrogels prepared with similar cross-linking reactions.6 Such deviation from ideal network can be more pronounced for “in situ” hydrogels, as they are formed by mixing the precursors in a target tissue, and the duration may not be sufficient to complete the reaction. As observed previously18,32 and as shown here in Figure 1, the initial ρxl has a significant effect on the degradation behavior of a hydrogel. To account for the effect of deviation from the ideal network, a term called cross-linking efficiency has been defined here (eq 2). The model predicts that for the same composition of hydrogel, lower cross-linking efficiency leads to higher initial swelling ratio, apparently faster degradation rate, and shorter time for the gel to disintegrate (Figure 1). This can be explained by the difference in the distribution of cross-link nodes in the hydrogel network. Thus, in order to make accurate prediction, it is essential to measure the cross-linking efficiency and use it as an input to the model.

was obtained via the measurement of initial equilibrium swelling ratio (Qm,exp(24 h)) and compared with the theoretical ideal value (eqs 3 and 4 where ve,ideal is input). As polymer concentration increases, cross-linking efficiency increases (Figure 2A). This trend was verified in two gel formulations with different ratio of functional groups. The measured initial equilibrium swelling ratio approached the ideal value with increasing polymer concentration (Supporting Information Figure S1). This is due to the higher probability for the complementary functional groups to meet each other at the entanglement points. As the degree of modification (DM) increases, crosslinking efficiency increases (Figure 2B). This trend was verified in gels at two polymer concentrations. The total number of certain functional groups equals the concentration of polymer carrying that type of functional groups multiplying by its DM. In these experiments, the DM of the polymer chain carrying the functional groups in excess is held constant, while the DM of polymer chain carrying the complementary functional groups varies. In Figure 2B, the DM of MA-dextran is kept constant. As DM of SH-dextran increases, the total number of cross-link nodes increases and so does the cross-linking efficiency. Gels with nonequimolar DM ratio and concentrations have higher cross-linking efficiency. Therefore, to achieve a network near ideal structure, higher concentration as well as higher DM ratio between precursors is necessary. This phenomenon also can be justified by decreasing number of unreacted active functional group. This number decreased due to increasing moles of one chain and increasing the probability of reaction between the unreacted functional groups which are located away from entanglement points.6 F

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Figure 2. Cross-linking efficiency of hydrogel varies with composition. (A) Effects of polymer concentration. (B) Effects of degree of modification.

Figure 3. Simulation results and experimental data showing the effect of polymer concentration on hydrogel degradation: (A) [SH] = [MA], DM_SH < DM_MA. (B) [SH] = [MA], DM_SH = DM_MA. (C) [SH] = 2[MA]. k′ unit is day−1; symbols are experimental data and lines are from simulation.

3.2. Modeling of Hydrogel Degradation. As the crosslink nodes are cleaved, the value of ve,actual decreases. This changes the equilibrium state between mechanical and thermodynamic forces, water is drawn into the hydrogel, and gel is increasingly swollen. Profiling swelling ratio is an indicator of hydrogel degradation. The dependence of hydrogel mass swelling ratio on the value of ve,actual is according to the Bray−Merrill equation (eq 4), and it is shown in Figure S2. The status of the cross-link nodes was predicted using the stochastic Monte Carlo approach. The resulting model prediction on hydrogel swelling was confirmed using experimental measurements, as detailed below. 3.2.1. Effect of Polymer Concentration on Hydrogel Degradation. Although hydrogel degradation starts from single node cleavage, but it is not practical to use only kinetic parameter to predict hydrogel overall behavior. As node degradation follows a stochastic pattern and these hydrogels contain numerous degradable bonds within their structure, it is not possible to predict lifetime of each particular ester bond located at a specific backbone chain within the macromolecular network. So, a stochastic approach based on Monte Carlo concept was utilized to relate hydrogel structural properties and

kinetic parameters to macroscopic properties, i.e., hydrogel swelling during bulk degradation. Although the comparison between previous models and the current one is out of the scope of this paper, by making some strong assumptions, we managed to apply the mean-field statistical-kinetic approach to the polymer−polymer hydrogels in our experimental system. The assumption was that long linear polymeric chains behave like small multiarmed macromers (data not shown). Although previous models work very well for hydrogels composed of cross-linked macromers, they do not generate a satisfactory fit to the experimental data for polymer−polymer hydrogels in our experiments. We reason previous models were not satisfactory able to capture some essential feature of polymer−polymer hydrogels. The number of chains between cross-link nodes (ve,actual) is different from the number of cleavable cross-link nodes in polymer−polymer hydrogels and is dependent on the distribution of cross-link nodes during gel formation and degradation (Figures S3 and S4). Also, the position of the cross-link node where cleavage occurs within the network is a random process; thus, the average value of ρxl could not be assumed to be similar for all locations within the networks. Error will be introduced if the status of one cross-linking G

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Figure 4. Simulation results showing the isolated effect of (A) polymer concentration, (B) degree of modification, and (C) kinetic rate constant of hydrolysis on hydrogel degradation. Hydrogel properties: (A) DM_MA = 17%, DM_SH = 9%, k′ = 0.1 day−1. (B) [SH] = [MA] = 15% w/v, k′ = 0.1 day−1. (C) [SH] = [MA] = 15% w/v, DM_MA = 17%, DM_SH = 9%. Cross-linking efficiency assumed to be 1 (η = 1) for all formulations.

Figure 5. Simulation results and experimental data showing the effect of polymer degree of modification on hydrogel degradation: (A) DM_SH = 9% and DM_SH = 5%, in both gels (B) DM_SH = 17%, DM_SH = 9%, DM_SH = 5%, k′ in 1/day; symbols are experimental data, and lines are from simulation.

hydrogels may start with similar values of swelling ratio, but as degradation proceeds, the profiles diverge more significantly. As mentioned in the previous section, polymer concentration affects the cross-linking efficiency in the initial network structure. This has been shown to influence the subsequent degradation profile (Figure 1). Higher polymer concentration leads to more ideal network. A more highly cross-linked network degrades at a slower rate. In addition, the fitted values of k′ obtained by the model are lower for hydrogels with higher polymer concentrations. Note that chemistry of the cleavable bonds at the cross-link nodes is identical for all these gels. Based on these results and others,9,12,13,15−17 k′ depends not only on the bond chemistry but also on the local microstructure (e.g., local water

macromer is orderly repeated to analyze the whole network of a polymer−polymer hydrogel. More discrepancy between the experiments and theoretical prediction occurs as the degree of heterogeneity (in terms of polymers with different DM and concentration) of the system increases. This is unlike in more simpler homogeneous network described by previous work (Figure S5).18 The model results accurately match the experimental measurements of hydrogels with different polymer concentration (Figure 3). Results show that hydrogels with lower polymer concentration experience faster degradation. A similar trend was observed by others in the hydrolytic degradation of PEG4NB-DTT hydrogels32 and the photopolymerized PLA− PEG−PLA hydrogels.9,12,13,15−17 At the beginning, these H

DOI: 10.1021/acs.macromol.8b00165 Macromolecules XXXX, XXX, XXX−XXX

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Figure 6. Disintegration time for degradable hydrogels (A) DM_MA = 45% and DM_SH = 17%, (B) DM_MA = 17% and DM_SH = 9%, (C) DM_MA = DM_SH = 9%, (D) DM_MA = 17%, [SH] = [MA] = 15% w/v. (E) DM_MA = 13%, [SH] = 10%, [MA] = 20% w/v.

to the pseudo-kinetic rate constant k′, polymer concentration, and its effect on the initial network structure. 3.2.2. Effect of Degree of Modifications Ratio on Hydrogel Degradation. The model results match the measured degradation profiles of hydrogels containing polymers with different degree of modification (Figure 5). In these experiments, DM_MA was kept constant, and the effect of changing DM_ SH dextran was investigated. Result show that both the initial swelling ratio and degradation rate are influenced by DM. As DM of one chain increases while the DM of the other one (carrying excess functional groups) is kept constant, the available number of active groups for participating in crosslinking reaction increases and thus enhances the chance at which active groups meet and react. The results in Figure 5 show that this leads to decreased degradation rate. As DM_SH increases, more cross-link nodes are formed, and hydrogel swells at a lower rate. Other than affecting the networking structure, the model revealed that DM influences the rate of cleavage of the ester bond. For gels in Figure 5A, in which the concentration of MA-dextran is different from that of SHdextran, the fitted value of k′ decreases from 0.27 to 0.23 day−1 when DM_SH dextran increases from 5% to 9%. For gels in Figure 5B, in which both MA-dextran and SH-dextran have the same concentration, k′ drops from 0.265 to 0.2 day−1 when DM values varies from 5% to 17% on the SH chain. So, these data confirm that changing the degree of modification of polymer not only influences the initial equilibrium swelling ratio but also has a considerable effect on the degradation rate of hydrogel. A similar result for changing the number of arms (which can be interpreted as the DM effect) has been also observed in the degradation of hydrogels formed from PEG4NB-DTT.32 In order to separate the effect of kinetic constant and degree of modification on swelling ratio profile, the model was used to predict the effect of degree of modification, keeping k′ constant at 0.1 day−1 (Figure 4B). The simulation results show that hydrogels of the same polymer concentration, but higher DM value, swell more slowly, meaning that they exhibit a lower apparent rate of degradation. Similar to the effect of polymer concentration, this result shows that even that if ester bonds in the cross-link nodes

concentration and presence of excess functional groups), which is in turn related to the polymer concentration. The gels in Figures 3A and 3B are different in the ratio of DM_SH versus DM_MA, and they have different cross-linking efficiency. Gels with DM_MA > DM_SH degrade at slightly higher rate than gels with DM_MA = DM_SH despite their higher cross-linking efficiencies for all three polymer concentrations. The fitted k′ is about 10% higher at all concentrations. This leads to slightly higher mass swelling ratios for same concentration. This phenomenon may be possibly explained by the presence of unreacted groups. Excess SH groups may form disulfide bonds at a very slow rate35 and lower the overall degradation rate. Therefore, the difference in the degradation profile expressed in terms of swelling ratio became more pronounced toward the later stage of degradation. Gels with 1:1 DM ratio (DM_MA = DM_SH) have lower cross-linking efficiency and have more unreacted groups compared to gels with different DM ratio (DM_MA > DM_SH). Excess methacrylate groups present on the polymer backbone may also affect the pseudo-first-order kinetic rate constant. Gels in Figure 3C can be treated as ideal network as calculated cross-linking efficiencies are near 1. The result for these two gels confirms the observations for the gels in Figure 3A,B. So, it can be concluded that polymer concentration has direct effect (discussed as change in k′) on the degradation rate other than its influence on the initial network structure. In fact, for gels in Figure 3A,B, concentration affects degradation rate through initial structure as well as direct effect on the kinetic rate constant, k′. In order to separate the effect of kinetic constant and polymer concentration on swelling ratio profile, the model was used to predict effect of polymer concentration, keeping k′ constant at 0.1 day−1. Figure 4A shows that with the same k′, the higher the polymer concentration, the slower the degradation rate. This result reveals what causes hydrogels to be degraded at different rates in the real experimental system is a combination of the effect of changing the kinetic constant and the polymer concentration (Figure 3). In other words, the apparent rate of degradation observed in Figure 3 is attributed I

DOI: 10.1021/acs.macromol.8b00165 Macromolecules XXXX, XXX, XXX−XXX

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hydrogels in reality deviate from the ideal fully cross-linked network. The initial equilibrium mass swelling ratio was measured experimentally to capture the cross-linking efficiency, allowing a correct input of the initial number of small chain between cross-link nodes into the stochastic model. The time course of mass swelling ratio predicted by the model matches with experimental measurements in hydrogels formed by MAdextran and SH-dextran of varying polymer concentrations and degree of modification. The fitted kinetics rate constants further revealed their dependence on hydrogel microenvironment. A theoretical framework was developed to predict the disintegration time of hydrogel as a function of the structural and kinetic parameters. The model was validated by experiments. These results show the significant role of structural parameters on the overall degradation behavior of hydrogels. The apparent rate of hydrogel degradation and the onset of disintegration can be controlled by varying the polymer concentration and degree of modification without changing the cleavable bond chemistry. The model will be useful for the design of degradable hydrogel for prolonged drug release and tissue regeneration.

are hydrolyzed at the same rate, the structural differences due to changes in DM have significant effects on the rate of the hydrogel degradation. In the actual experimental systems, the differences in the swelling ratio profile encompass both the effects of changing k′ and DM values. The implication is that very different hydrogel degradation rate can be achieved even with the same cleavable bond chemistry. 3.2.3. Gel Disintegration Time. The time at which a degradable hydrogel turns into polymer solution is called disintegration time (tdis). The ability to predict this time is desirable in biomedical applications because many processes such as drug release and tissue regeneration are affected by life span of the gel. To validate the model and to understand the effect of composition, a series of hydrogels with different concentration and degree of modification were prepared. There is a good match between experimental data and model prediction of disintegration time with discrepancy within the acceptable range (Figure 6). Disintegration time ranges from 4.5 days to more than 25 days in the formulations tested. This suggests that one can control another outcome of hydrogel degradation by manipulating the initial network (degree of modification and precursor concentration) without changing bond chemistry. This phenomenon is consistent with the meaning of eq 22. Figures 6A−C show the effect of polymer concentration while Figures 6D,E illustrate the effect of degree of modification on tdis. It is more sensitive to changes in the degree of modification. When the polymer concentration increases from 12 to 30% v/w, tdis increases by 17.8% according to model prediction (Figure 6A). On the other hand, tdis increases by 158% when DM_SH increases from 5% to 17% (Figure 6D). From eq 22, the disintegration time is not only determined by the hydrolysis rate constant but is also highly dependent on the number of active functional groups on each backbone chain. When DM is kept constant, the number of active functional groups on each backbone chain does not vary with polymer concentration in an ideal hydrogel network. In reality, one must consider how cross-linking efficiency is affected by polymer concentration (Figure 2A). Namely, higher polymer concentration leads to higher cross-linking efficiency and therefore higher number of active functional groups actually participating in the cross-link nodes. However, this effect is modest on disintegration time. For example, the hydrogel with the highest polymer concentration has 19% more active functional groups on each backbone chain relative to the one with the lowest concentration in Figure 6A. This difference does not cause any sharp increase in tdis. In comparison, the hydrogel with the highest DM has 300% more active functional groups on each backbone chain relative to the one with the lowest DM (Figure 6D). In summary, both the model and experiments indicate that a large range of disintegration time can be attained through initial network formulation without changing the cleavable bond chemistry. The effect of degree of modification is more prominent than polymer concentration.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00165. Relation between hydrogel mass swelling ratio and extent of degradation and effect of polymer concentration and degree of modification on the actual and ideal initial swelling ratio; comparison of different network structure during formation and degradation in polymer-polymer vs multi-armed crosslinked hydrogels (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(Y.C.) Phone (852) 2358-8935; Fax (852) 2358-0054; e-mail [email protected]. ORCID

Ghodsiehsadat Jahanmir: 0000-0003-3484-9697 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the Hong Kong General Research Fund (GRF 16322016). NOMENCLATURE t time, day DM degree of modification of polymer chain DM_SH degree of modification of polymer chain modified with thiol group DM_MA degree of modification of polymer chain modified with methacrylate group [SH] concentration of polymer modified by SH group, %w/v [MA] concentration of polymer modified by MA group, %w/v Mr wet polymer mass at relaxed state, g M24 h,s equilibrium hydrogel wet mass after 24 h, g Mt,s swollen hydrogel mass at different time points, g Md,r dried polymer mass at relaxed state, g

4. CONCLUSION The Monte Carlo based approach was applied to model the bulk degradation behavior of a class of hydrogels made by cross-linking reaction between pendant groups on long linear polymer chains. The model can predict the time-dependent quantity of small chains between cross-link nodes within the network and relate this to the macroscopic mass swelling ratio of hydrogel through the Bray−Merrill equation. Because of incomplete conversion and the presence of unreacted groups, J

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Md,s Qm ρxl

dried polymer mass at different time points, g mass swelling ratio concentration of small chains between cross-link nodes, mol/mL ve,ideal ideal mole number of small chains between cross-link nodes, mole ve,actual actual mole number of small chains between cross-link nodes, mole η cross-linking efficiency Mw or Mn dextran molecular weight, g/mol v1̅ specific volume of water, mL/g v2̅ specific volume of dextran, mL/g v2,s polymer volume fraction in swollen state v2,r polymer volume fraction in relaxed state k′ hydrolysis kinetic rate constant, day−1 [DB], [DB]0 current and initial degradable bond concentration ti,j lifetime of cross-link node(i, j) ϵ random number (0, 1) xi,j status function of cross-link node, %w/v (i, j) NSH number of SH groups on polymer chain NMA number of MA groups on polymer chain Pintact probability of one degradable bond to be intact tdis hydrogel disintegration time, day Pdis critical fraction of intact cross-link nodes E expectation value of a random variable MWi molecular weight of either MA or SH chains, g/ mol ni mole number of either SH or MA chains wi weight fraction of either SH or MA chains Wi ⃗ weight looking in the i ⃗ direction from the crosslinkage M̅ w average molecular weight of developing structure formed during hydrogel degradation, g/mol

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