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Tissue Engineering

Constitutive Framework for Biodegradable Polymers with Applications to Biodegradable Stents

Soares, João S.*; Moore, James E. Jr; Rajagopal, Kumbakonan R.*†

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doi: 10.1097/MAT.0b013e31816ba55a
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Abstract

The availability and utilization of synthetic biodegradable polymers has increased dramatically over the last 50 years, with applications ranging from the field of agriculture to biomedical devices. An early biomedical application of biodegradable polymers was absorbable sutures in the 1970s,1 and to date it remains as the most widespread use of this family of materials. Biodegradable polymers have also proven to be useful as a design material in orthopedics,2 effective carriers for local drug delivery,3 and are widely used in tissue engineering research.4

When used for prosthetic purposes in orthopedics, the contribution of the polymer is required for a limited period of time, the healing time, and the polymer can be engineered to degrade at a rate that will transfer the load to the healing bone.5 To accomplish this, one has to understand with clarity the load bearing capabilities of the device as well as the evolution of the properties of the material due to degradation over time. On the other hand, for drug delivery implants, attention is shifted to delivery kinetics and their changes during degradation.3

A highly desirable application for biodegradable polymers is endovascular biodegradable stents.6–8 A stent is a mechanical scaffold that is inserted inside an artery to counteract a disease-induced flow restriction. This application can potentially combine mechanical support with drug delivery. The stent must perform mechanically, maintaining the artery patent after deployment and during degradation, and at the same time be capable of effective drug delivery.

Besides leading to unpredictable complications, there are no demonstrable clinical concerns with a successful permanent stent.7 On the other hand, a temporary biodegradable stent has many advantages: 1) it will not be a potential nidus for infection or an obstacle for future treatments; 2) its controlled degradation allows prediction of failure, shifts the load from the stent to the healing wall, and enhances drug delivery; and 3) the characteristics of the polymeric material allows a nonchronic deployment and a response to pulsatile loading that progressively becomes closer to the physiological response of a healthy artery.

Polymer degradation is the irreversible chain scission process that breaks polymer chains down to oligomers and finally monomers. Extensive degradation leads to erosion, which is the process of material loss from the polymer bulk. Thus, degradation and erosion are distinct but related processes.2,9 The prevailing mechanism of biological degradation of synthetic aliphatic polyesters (the main class of biodegradable polymers used in biomedical applications) is scission of the hydrolytically unstable backbone chain by passive hydrolysis.10 The vast majority of the research effort on biodegradable polymers has been directed toward varying formulations and product development.11,12 There is a fair amount of experimental data concerning biodegradable polymers, ranging from molecular weight distribution (MWD) evolution, mass loss, and amount of drug eluted.13,14 Besides understanding the processes of degradation or erosion, the mechanical response is equally relevant, but the attempts to describe have been limited. Rajagopal and Wineman15 developed a model of a nonlinear elastic material model, which describes a network that is able to undergo scission and healing, forming new networks in new natural states. The framework was further developed by allowing for the effect of temperature, and the solutions of several boundary value problems16 have been carried out to account for different situations such as scission due to high temperatures.17

Our initial work in this area was directed at an axisymmetric annulus composed of a linear degradable material.18 In this study, we aim to address the natural extension of the previous model, i.e., nonlinear degradable materials that undergo finite deformations, and its application to polymeric materials. Uniaxial extension, while not directly relevant to the description of the response of a stent, assumes great relevance for the experimentalist interested in material characterization and parameter estimation. Degradation experiments with poly(L-lactic acid) fibers under tension are currently being performed by the authors, and hence, the choice for the particular boundary value problem addressed in this article.

The overall aim of this research is to provide material constitutive models that can be applied to stent design. These models should be capable of not only describing the response of the stent subjected to mechanical loads and degradation, but also of predicting the location and time of eventual structural failure.

Materials and Methods

Constitutive Model

We assume that the bodies under consideration, in the absence of degradation, are incompressible, isotropic hyperelastic solids. The most general representation for the Cauchy stress T for this class of elastic solids is given by19

where −pI is the indeterminate part of the stress due to the constraint of incompressibility, W is the stored energy function, I1 = tr C and I2 = 1/2[(tr C) − tr C2] are the first and second principal invariants of right Cauchy-Green stretch tensor C = FTF, and the left Cauchy-Green stretch tensor is B = FFT. The deformation gradient of the motion that maps the place X of a particle in the reference configuration to the corresponding place x in the current configuration at time t and is defined as F = ∂x/∂X.20

Measures of Degradation.

Central to the theory of deformation-induced degradation is the introduction of a scalar field that reflects the degradation. Microstructural changes take place as the body is deformed. The cause of these modifications may be the hydrolytic cleavage of the backbone chains that constitute the degradable polymer. These changes result in reduction in the molecular weight and consequent depreciation of elastic properties.

We assume properties of the body change with deformation and time, and we capture it by introducing a degradation parameter. We assume the existence of a positive finite measure of degradation, which is a measure of the number of broken bonds that may be assigned to a representative volume element at time t and location x. Moreover, there exists a scalar field d with values between zero and unity that expresses the fraction of broken bonds and quantifies the degree of local degradation at a given particle at place x at time t

The degradation parameter will always be nonnegative and with the upper bound being unity; 1 − d is a measure of the fraction of intact crosslinks in a representative volume element of the body. The value d = 0 will represent a virgin specimen and d → 1 corresponds to the state of maximum possible degradation, i.e., a network without any remaining crosslinked segments and unable to withstand any load.

Processes of Degradation.

Mechanical energy transferred to a polymeric system can be dissipated via two harmless (i.e., not inducing chemical changes) relaxation processes: the slippage of chains relative to surrounding molecules, and changes in chain conformation.9 In addition, scission of chemical bonds can occur. Bond rupture occurs when sufficient energy is concentrated in a certain segment of a macromolecule as a consequence of nonuniform distribution of internal stresses. Strain is a prerequisite and the probability of scission should increase as relaxation is impeded.21 Besides mechanical loads promoting bond rupture per se, microstructural changes can occur due to high temperature (thermal degradation), ultraviolet radiation (photo-degradation), or the presence of a small molecular weight reactant such as oxygen (oxidation) or water (hydrolysis).

The major mechanism for the degradation of aliphatic polyesters is by random attack by uptaken water. The rate of swelling of these polymers is usually higher than the rate of hydrolysis. Degradation is extensive through the polymer bulk and the common mode of erosion is bulk or homogeneous erosion.

Hydrolytic scission occurs spontaneously in the presence of the readily available water, and its rate may be influenced by the properties of the molecular network: 1) dramatic changes in network morphology occur as a consequence of chain alignment in response to uniaxial tensile strain22; 2) the crystalline phase is more resistant to degradation than the amorphous phase23; 3) Baek and Srinivasa developed a model that describes changes in the amount of swelling of a network due to deformation24; and finally 4) experimental evidence shows that deformation and degradation are a coupled process.25–27

We assume a rate equation, where the rate of change of degradation is

that is, the rate of change of degradation is assumed to be dependent on the state of deformation, the extent of degradation, and implicitly on both spatial location and time. We note that the current governing equation does not take into account thermal effects or other degradation-induced mechanisms.

Forms of Degradation.

We postulate that the degradation at a given particle and its consequent molecular weight reduction leads to depreciation in the mechanical properties of the corresponding particle as the number of effective crosslinks decrease due to degradation. The stored energy function will depend on the degradation parameter and assumes the general form

The form of the stored energy given above does not correspond to a hyperelastic body. One expects that as the material degrades, its ability to store energy will change, albeit at a fixed value of the degradation, the model does correspond to a hyperelastic body. We will restrict our attention to bodies that in their virgin state are hyperelastic, and at a fixed level of degradation, the body behaves as a hyperelastic material whose properties have depreciated due to the degradation.

Biodegradable Neo-Hookean Body

For the purposes of illustrating our ideas, we employed one of the simplest models for describing nonlinear elastic solids, the neo-Hookean model. It is widely used to describe the behavior of polymeric materials undergoing large deformations.28 This particular choice was made due to the simplicity it accords and the reasonable agreement with available experimental data. Any hyperelastic model with its corresponding stored energy function can be used.

The Stored Energy Function for the neo-Hookean Material is given by

where μ > 0 is the shear modulus.20 The Cauchy stress T in the material is given by

The shear modulus will be allowed to change due to degradation and is a material function instead of a material constant, μ = μ(d). Initially, we can assume a linear dependence of the shear modulus on degradation from the maximum value of the intact specimen, i.e.,

where μ0 is the shear modulus of the virgin specimen.

Material frame indifference and isotropy imply that

Dependence on other variables such as time derivatives of kinematical quantities is possible, but beyond the scope of this initial development. We expect that greater strains lead to greater degradation, and also, the same strains, when acting during longer periods of time, lead to greater degradation. This can be expressed by the relationship

where C is a constant. The rate of degradation varies linearly with the state of deformation, expressed in Equation 9 as the radius in I1I2 plane. Thus, when the material is in its undeformed state, I1 = I2 = 3 and no degradation occurs. Furthermore, the amount of degradation already present decreases the rate of increase of degradation, i.e. as the material approaches the state of maximum allowable degradation (d → 1), less bonds will be available for degradation and consequently its rate of degradation decreases. The choice Equation 9 is among the simplest ones possible, is adequate for the purpose of a preliminary study for assessing our approach and was obtained through phenomenological reasoning. Rajagopal et al.,29 after a constitutive framework that maximizes the rate of dissipation, obtained for deformation-induced degradation a governing equation of the form

where 𝔊 is a function of the “driving forces D” which in turn depend on the deformation gradient F through the invariants of B.

The balance of linear momentum (neglecting body forces) reduces to

where v is the velocity, defined as the material time derivative of the motion. Substituting Equations 7 and 9 yields

Equation 12 coupled with Equation 9 are the partial differential equations governing the response of a neo-Hookean-like degradable material. The indeterminate part of the stress is the reaction to the constraint of incompressibility. Initial conditions for the motion and the initial value for the degradation (the natural choice is a virgin specimen) must also be specified.

Uniaxial Extension.

We now illustrate the predictions of the model for homogeneous uniaxial extension. This choice of deformation is made in view of its simplicity and the fact that it has experimental relevance. Uniaxial extension is a common experimental mode to determine the mechanical properties of a material.

We shall seek a semi-inverse solution of the following motion:

where (X1, X2, X3) and (x1, x2, x3) represent the coordinates of a typical material particle in the undeformed and deformed configurations, respectively. In Equation 13, α(t) is a function of time representing the time dependent axial stretch. The motion above is isochoric. Also, degradation is assumed to be homogeneous and independent of position, i.e., d = d(t).

The Cauchy stress for a neo-Hookean body undergoing uniaxial extension is given by Equation 6 where the Lagrange multiplier is obtained from boundary conditions. If we restrict motions to be quasi-static, the inertial contribution on the stress field through the right hand side of Equation 11 becomes negligible. The lateral surfaces are traction free, and consequently, p will be independent of position and given by p = μ/α(t). The Cauchy stress tensor has only a single nonzero component, T33, that takes the form

This relationship is nonlinear, yielding generally three possible solutions for a given stress. α(t) must be real and positive, greater than unity for positive values of stress and less than one if the body is under a state of compression.

The degradation takes its toll on the shear modulus and in the view of Equation 7, Equation 14 becomes

where the degradation parameter is homogeneous due to the homogeneous deformation.

The first and second invariants of the left Cauchy-Green stretch tensor are given by

and Equation 9 yields

Equations 15 and 17 are the equations governing the uniaxial extension of an initially neo-Hookean material that degrades. They can be solved for any two of the following three quantities: α(t), d(t), or T(t). For the situation in which a controlled stretch is applied over time, the amount of degradation is obtained by integration of Equation 17, and the necessary stress to maintain the stretch given by Equation 15. On the other hand, for the case in which a stress T(t) is imposed, the stretch α(t) is obtained as one solution of Equation 15, and from it, the corresponding increase of degradation with Equation 17.

poly (L-lactic acid) (PLLA), widely used in biomedical applications and reported in the literature to possess a Young’s modulus of 3.5 GPa, was chosen to illustrate the predictions of the model. Although the model is based on nonlinear elasticity, the shear modulus of the virgin material μ0 was chosen to be 1/3 of the reported Young’s modulus. PLLA may be assumed to be incompressible.30,31

Results

With time, the stress necessary to maintain a given constant stretch decreases, i.e., stress relaxation is observed (Figure 1). Moreover, the decay is exponential, with its time scale dependent on the applied stretch. For larger stretches, the initial required stress is larger, but it relaxes faster. The degradation due to a constant stretch increases steeply during the initial stage of degradation. This rate is directly related to the amount of stretch, i.e., more stretch will lead to a faster degradation. The degradation tends asymptotically to its maximum value of unity as time tends to infinity (Figure 2).

Figure 1.
Figure 1.:
Nondimensional Cauchy stress T(t)/μ0 required to maintain a fixed constant stretch α (with C = 1). The material softens as it degrades and consequently less force is necessary to maintain a given deformation, i.e., stress relaxation occurs. When the material is fully degraded, the shear modulus and the required stress decay to zero. Higher initial stretching leads to a greater rate of degradation, and thus a shorter time to effective material breakdown.
Figure 2.
Figure 2.:
Degradation d(t) for several constant stretches α (with C = 1). The deformation on the right hand side of Equation 4.7 will be constant, and hence, due to the 1 − d(t) term, the degradation asymptotically tends to its maximum allowable value of unity. For greater stretches, the material degrades at a greater rate.

Under the influence of constant loads, the stretch consistently increases over time (Figure 3). The extent of this creep-like behavior is dependent on the amount of degradation and consequent softening of the material. The amount of degradation increases progressively (Figure 4), where the effects of greater softening lead to greater stretches and the final steep increase in degradation. Two of the cases considered reach the point of maximum allowable amount of degradation in the time interval considered. Close to this point, the modulus of the material decreases and the stretch increases dramatically.

Figure 3.
Figure 3.:
Stretch α(t) for several constant applied stresses T/μ0 (with C = 1). As T/μ0 increases, the impact of degradation on the mechanical properties is greater. Although the applied stress remains constant in each case, α(t) increases over time, i.e. the body exhibits creeps-like behavior.
Figure 4.
Figure 4.:
Degradation d(t) for several applied constant stresses T/μ0 (with C = 1). As the material degrades and softens, the applied stresses, although remaining constant, lead to greater deformations (cf. Figure 3) that consequently lead to greater increases in degradation, i.e. the degradation increases asymptotically and reaches its maximum allowable value. Greater stresses induce greater deformations that lead to different rates of increase of degradation.

Hysteresis loops are observed in the nondimensional Cauchy stress T(t)/μ0vs. stretch α(t) diagram when the stretch is cycled between 1.00 and 1.25 (Figure 5). Hysteresis is dependent on the stretching rate, with the area spanned by the hysteresis loop increasing as the rate of stretch decreases. The effects of degradation are indistinguishable during the initial stages of loading for the four stretch rates considered, but as degradation proceeds the curves deviate from each other. Lastly, no permanent set is induced due to the degradation: when the body is back to the original configuration, it is stress free (Figure 6).

Figure 5.
Figure 5.:
Nondimensional Cauchy stress T(t)/μ0 vs. stretch α(t) (C = 1). The stretch cycles between 1.00 and 1.25 with constant stretching rates. Hysteresis can be observed.
Figure 6.
Figure 6.:
Nondimensional Cauchy stress T(t)/μ0 (lines with markers) for a cycle in stretch α(t) (from 1.00 to 1.25, dashed lines) with constant stretching rate (with C = 1). As the rate of stretch decreases, the degradation mechanism is effective for a longer period of time and is responsible for greater softening.

Discussion

The ability to model the behavior of biodegradable polymers subjected to mechanical loads would enhance the predictive nature of biodegradable stent design. Several particularizations of the proposed model were considered in the previous sections and provide, qualitatively, phenomenological support. Degradation is quantified by a parameter that describes locally the amount of degradation and changes the mechanical properties through a constitutive equation that is dependent on the aforementioned degradation parameter. The body starts out in a virgin state and due to the imposed stretch or due to the imposed loads, it degrades. The results indicate an increase in degradation (Figures 2 and 4) and a consequent decrease in the shear modulus (not shown but following directly from Equation 7.

Another feature is “stress relaxation” in response to a constant stretch history (observed in Figure 1). Stress decreases as the material degrades – as the shear modulus decreases, less force is required to maintain a constant deformation. The exponential decay in stress results from the asymptotic increase in degradation (Figure 2).

On the other hand, when the force is maintained constant, an increase in stretch occurs due to degradation, i.e., the body creeps as it degrades under a constant load (Figure 3). Increases in stretch promote greater changes in degradation and this cycle is repeated until the material reaches a point where the maximum degradation and a zero modulus are achieved, allowing infinitely large deformations. Of course, structural failure probably occurs much sooner.

The material exhibits behavior similar to mechanical hysteresis when the stretch is cycled. The extent of hysteresis is dependent on degradation, as the four cases (shown in Figures 5 and 6) demonstrate. For a quick loading/unloading (on the left of Figure 6), degradation acts during a smaller period of time and the stress almost follows the corresponding loading and unloading. As the rate of loading/unloading decreases, the effects of degradation are more pronounced due to the fact that the deformation is occurring over a time scale in which significant degradation occurs.

All of these distinct features, while superficially similar to the response characteristics of a viscoelastic material, have origins that are quite different. Firstly, if there is no degradation the material is elastic; however, when degradation is taking place, the material exhibits stress relaxation, creep, and hysteresis. The response of this body is markedly different from similar phenomena exhibited by aging viscoelastic materials.32

Ogden and Roxburgh33,34 considered a body whose stored energy function depends on a damage parameter due to softening. Instead of the parameter being completely determined by the kinematics as in their study, here degradation (a parameter on which the stored energy function depends) is governed by a coupled partial differential equation describing the entropy producing mechanism responsible for softening of the material. Wineman and co-workers15 developed a model where scission and healing of crosslinks occurs continuously. The conversion is dependent on a state variable that quantifies the extent of deformation. The framework presented here is a related case, but where no new networks are formed and the degree of scission is given by a rate equation describing the kinetics of degradation, which in turn are dependent on deformation.

Furthermore, the integration of the governing equation for degradation (cf. Equation 3) yields

where the integrand can be seen as an intrinsic material clock function. This concept was originally proposed in Bernstein and Shokooh as a stress clock function for viscoelastic fluids35 and was further extended by Wineman and co-workers with a strain clock for viscoelastic solids.36,37 In this particular case, due to the sole dependence on kinematical quantities and the 1 − d(t) term (cf. Equation 9) yields a strain clock. The clock introduces a “degradation scale”, describing locally the evolution of degradation and providing a form to measure and relate different amounts of degradation.

Equation 4 is a reasonably general constitutive equation for a degradable material. It is worth observing: 1) uniaxial extension is a homogeneous deformation, and hence only homogeneous degradation will result from it; 2) the specific forms of Equations 7 and 9 describing this particular mechanism of deformation-induced degradation were chosen on the basis of the simplicity that they accord, therefore the results obtained from them should be regarded only as preliminary qualitative results; and 3) biodegradable polymers, like PLLA, are nonlinear elastic materials and in this context should perhaps be modeled as fully nonlinear viscoelastic materials.

Conclusions

A fully nonlinear model was developed to describe the mechanical response of degrading polymers. The results obtained for this model allows one to conclude that deformation-induced degradation can be modeled through a scalar field describing the local state of degradation and a partial differential equation governing the rate of increase of degradation coupled with the equations of motion. The constitutive equation of the material takes into account the effect of degradation on the mechanical properties.

The degradable material considered here shows stress relaxation, creep, and hysteresis, thus a dissipative process is associated with this particular degradation mechanism. The efficacy of this approach was substantiated by solving the governing equations for a simple but relevant problem, uniaxial extension. Furthermore, experiments towards the validation of the model presented here are currently being performed with poly(L-lactic acid) fibers being degraded in vitro subjected to various degrees of stress and hence the justification for the choice of this particular boundary problem.

The model will be extended to degradable materials that in the absence of degradation are viscoelastic, a better model to describe polymeric materials. In such a model, it is possible to distinguish and quantify the viscoelastic effects inherent to polymeric materials obtained with a proper viscoelastic model from parallel phenomena arising from degradation. The constitutive model will be integrated into a finite element software package in order to analyze more realistic geometries, such as an endovascular stent. A useful design tool for stents is the quantification of stress reduction achieved in a stented artery wall as the stiffness of the biodegradable stent decreases. It should also be possible to predict with some degree of accuracy the location and time of structural breakdown of the stent structure.

Acknowledgments

This work was partially funded by the Portuguese FCT – Fundação para a Ciência e Tecnologia (SFRH/BD/17060/2004) and NIH grant R01 EB000115.

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