Multiscale Models of Degradation and Healing of Bone Tissue Engineering Nanocomposite Scaffolds

Sodium-montmorillonite clay (SWy-2; Crook County, Wyoming) was purchased from the Clay Minerals Repository at the University of Missouri, Columbia, Missouri. Polycaprolactone (average molecular weight $M_n=\mathrm{80,000}$), 1,4 dioxane (anhydrous, 99.8%), and 5-aminovaleric acid were purchased from Sigma-Aldrich (St. Louis, Missouri). Sodium phosphate (${\mathrm{Na}}_2{\mathrm{HPO}}_4$) and calcium chloride (${\mathrm{CaCl}}_2$) were purchased from J. T. Baker (Phillipsburg, New Jersey) and EM Sciences (Hatfield, Pennsylvania), respectively. Human osteoblast cells (hFOB 1.19 cell line CRL 11372) were obtained from American Type Culture Collection (ATCC) (Manassas, Virginia). The cell culture medium used in the hOB cells culture was made of fetal bovine serum (FBS) from ATCC, a G418 antibiotic from JR Scientific (Woodland, California), and HyQ Dulbecco’s modified eagle’s medium (DMEM)-12 (1:1) from Hyclone (Logan, Utah).

PCL/in situ HAPclay scaffolds were prepared following the method described in the group’s previous study (Ambre et al. 2015). In summary, 3.6 g of PCL was added to 40 mL of 1,4 dioxane and stirred in a beaker until all the polymer was dissolved. A 0.4 g of in situ HAPclay (the preparation procedure is described in Ambre et al. 2011) was sonicated in 16 mL of 1,4 dioxane. The sonicated suspension was added to the PCL solution, and the resulting solution was further stirred for another two hours at room temperature. The final solution was transferred into the polypropylene (PP) centrifuge tubes, and these tubes were frozen overnight in an isopropyl alcohol bath at $-20°\mathrm{C}$. These cylindrical frozen PCL composite samples were removed from the tubes and transferred into absolute ethanol ($-20°\mathrm{C}$) for the solvent extraction. The absolute ethanol solution was replaced every 24 h. After four days, the frozen PCL composite samples were removed from the solution and dried at room temperature. Finally, these scaffolds were cut into the size 13 mm long and 13 mm in diameter, to ensure the uniform tissue growth in the cell culture well plate. Samples of the same size were used for consistency in all the studies presented in this paper.

A JEOL JSM-6490LV [Japan Electron Optics Laboratory (JOEL), Peabody, Massachusetts] scanning electron microscope was used to study the microstructure of the PCL/in situ HAPclay scaffolds. The SEM imaging of the adhesion of the hOB cells on the PCL/in situ HAPclay scaffolds was performed with the same SEM instrument. The seeded scaffolds were washed with phosphate buffer saline to remove the cell culture medium. For fixing the live hOB cells, the seeded scaffolds were first treated with glutaraldehyde (2.5%) and then dehydrated in an ethanol series (10% v/v, 30% v/v, 50% v/v, 70% v/v, and 100%). Then, hexamethyldisilazane was used to replace the dried 100% ethanol. Before their viewing in the SEM, all these samples were sputter coated with gold and mounted on an SEM sample stub.

Three samples of the PCL/in situ HAPclay scaffolds were scanned using a microcomputed tomography technique. The samples were attached to a glass rod using carbon tape and placed inside a GE Phoenix (General Electric, Boston, Massachusetts) v|tome|x s X-ray computed tomography system (μCT) equipped with an 180 kV high power nanofocus X-ray tube xs|180nf and a high contrast GE DXR250RT (General Electric, Boston, Massachusetts) flat panel detector. One thousand projections of the sample were acquired at 60 kV and 350 mA, using a molybdenum target. The detector timing was 1,000 ms and the total acquisition time was 1 h and 6 min. The sample magnification was 7.28× with the voxel size 24.38 μm. The acquired images were reconstructed into a volume data set using datos|x 3D computer tomography software version 2.2. A Diconde image series was then acquired from the reconstructed volume. Finally, the image analysis software Mimics was used for the three-dimensional reconstruction of the scaffold [Fig. 1(a)]. Cylinders of 1.5-mm height and 1.5-mm diameter from each sample were modeled.

To evaluate the mechanical properties of the PCL/in situ HAPclay scaffolds, the finite element (FE) models were created from the assembly of 3D STereoLithography (STL) geometric models of scaffolds. The small cylinders obtained after 3D reconstructions were then converted into a 3D tetrahedral mesh using 3-Matic software and used as FE models. The typical 3D FE model contains 134,929 nodes and 352,316 tetrahedral elements of approximately 10–25-μm length. The scaffold models were imported into Mentat software for the FE analysis simulations. An unconfined compressive loading condition was simulated to determine the mechanical properties of the scaffolds. Parallel plates were attached to the top and bottom surfaces of the scaffold. For the loading condition, the axial load was uniformly applied on the upper plate of the model, whereas the lower plate was constrained in all directions. The material parameters used in the FE analysis simulations were obtained from this group’s previous SMD study [Fig. 1(b)] (Sharma et al. 2015). The loading conditions for the scaffold are shown in Fig. 1(c). Finally, the stress–strain responses in the meshed scaffold were analyzed with Marc 13.0 FE analysis software. All FE analysis simulations were run on 256 processors (32 nodes; eight processors per node) at the Center for Computationally Assisted Science and Technology (CCAST) clusters at North Dakota State University (Fargo, North Dakota) using Marc 13.0 software. Figs. 2(a and b) show the SEM images of the scaffolds and the porosities and microporosities in the scaffold walls.

The mechanical degradation of the PCL/in situ HAPclay scaffolds was studied in alkaline conditions (0.1M NaOH). Scaffold samples were ultraviolet (UV) sterilized for 45 min and then immersed in 70% alcohol overnight. The samples were then washed with phosphate buffer saline (PBS) to prepare them for degradation studies. The PBS-washed scaffold samples were transferred to a 0.1M NaOH solution in separate glass vials. The scaffold samples in glass vials were placed at 37°C and 5% ${\mathrm{CO}}_2$ for degradation. Finally, the scaffolds were removed from the glass vials after each degradation period (1, 5, 7, 14, and 18 days) and washed with deionized water and dried at room temperature conditions.

The compressive mechanical tests were carried out on undegraded (0 days, control) and degraded (1, 5, 7, 14, and 18 days) PCL/in situ HAPclay composite scaffold samples using a material testing servo mechanical test frame (MTS 858, MTS Systems, Eden Prairie, Minnesota). Each sample was placed between the flat and smooth platens to perform the tests. A $1\mathrm{mm}/\min$ of constant deformation rate was applied for up to 10% strain to each test sample. The load and corresponding displacement data were recorded. The load-displacement data were used to construct the stress–strain response curves for each sample.

Cell culture experiments for cell viability and mechanical properties studies of the PCL/in situ HAPclay scaffolds were carried out using hOB cells. For this purpose, UV sterilized, 70% alcohol-immersed, and PBS-washed scaffolds samples were used. The PBS-washed scaffold samples were first immersed in a cell culture medium and kept at 37°C and 5% ${\mathrm{CO}}_2$ in the incubator. After 24 h, $1\times {10}^5$ hOB cells were seeded on each sample and 1 mL of cell culture medium was then added. This was followed by incubation of cell-seeded scaffolds at 37°C and 5% ${\mathrm{CO}}_2$ for 4, 7, 18, and 28 days. The cell culture medium was changed every 3 days.

The compressive mechanical tests of PCL/in situ HAPclay scaffolds seeded with hOB cells were carried out in wet conditions using a compression testing machine (Mechanical test system SATEC model 22 EMF, Instron, Norwood, Massachusetts). For wet testing, the scaffolds were directly removed from the cell culture medium (at 37°C) after 4, 7, 18, and 28 days and tested at room temperature. The samples were immediately placed between the flat and smooth platens of the mechanical testing machine and remained under wet condition during the entire experiments. Deformation control loading was applied at a constant rate of $1\mathrm{mm}/\min$ up to 10% strain. The load-displacement results were analyzed using Bluehill software v. 2.5 and used for stress–strain calculations.

An investigation of the viability of the hOB cells grown in the PCL/in situ HAPclay scaffolds was studied using WST-1 assay (Roche Applied Science, Mannheim, Germany). The cell viability analysis was performed on the hOB cells-seeded scaffolds for 4, 7, 18, and 28 days following the manufacturer protocol (Cell Proliferation Reagent WST-1 2011). The hOB cells-seeded scaffolds were removed from the culture medium and washed with PBS. Then, the washed scaffolds were placed in a 96-well plate filled with culture medium and WST-1 reagent and incubated at 37°C and 5% ${\mathrm{CO}}_2$. After 4 h of incubation, the scaffolds were removed from the solution and the absorbance of formazan-formed solution, which is directly related to the number of live cells, was read at 450 nm using a microplate spectrophotometer (Bio-Rad, Benchmark Plus, Bio-Rad Laboratories, Hercules, California).

An integrative FE modeling was performed to analyze the effect of the time-dependent process of accelerated degradation and the hOB cell culture on the mechanical behavior of the implant PCL/in situ HAPclay scaffold in critical size bone defects. The length and diameter of the FE model were 4 and 2.82 cm, respectively. The FE model was constructed using 1.5-mm tetrahedral elements. The element length was the same as the size of the FE model of the PCL/in situ HAPclay. The total numbers of nodes and elements of the FE model were 7,133 and 36,037, respectively. The bottom and top surfaces were attached to plates. The bottom plate was constrained in all three directions, and a uniform axial load in the $z$-direction was applied on the top plate. The time-dependent material parameters used in the FE simulations were obtained from the integration of the stress–strain response of the predicted FE model of the PCL/in situ HAPclay scaffolds and the proposed degradation function. Finally, the stress–strain responses were reported for the FE simulations of the implant scaffold in critical size bone defects. The FE model was carried out using the Marc/Mentat 13.0.

The accelerated degradation studies (0.1M NaOH maintained at 37°C) were carried out on the PCL/in situ HAPclay scaffolds. Fig. 3 shows the stress–strain plots of the undegraded and degraded samples. The compressive mechanical properties of the scaffolds decreased with time. The plot shows a gradual decrease in the first 5 days followed by a more rapid drop to day 18. The significant drop in the mechanical properties was observed between days 5 and 7.

In the present study, the authors proposed a new analytical mechanical degradation model of PCL/in situ HAPclay scaffolds. The model was developed to predict the mechanical properties of scaffolds with degradation. The proposed analytical modeling approach was inspired by the disturbed state concept (DSC) (Katti and Desai 1995). The DSC can be used to analyze and predict the mechanical degradation behavior of polymer scaffolds.

The DSC considers material to be completely intact at the relative intact (RI) state and completely deformed at the fully adjusted (FA) state, or vice versa. For degradation studies, the RI is the undegraded scaffold and the FA is the fully degraded scaffold. In the current work, the degradation was computedwhere $D$ = degradation; $S_t$ = strain energy density of the degraded sample at the time $t$; and $S_0$ = strain energy density of the undegraded sample. The strain energies are calculatedwhere $p$ = predetermined strain value used for calculating the strain energy density for mechanical experiments on intact samples and samples at various degradation times. Effectively, $S$ is the area under the stress–strain curves up to strain $p$. In the current study, $p$ was 10%. The value of $D$ ranged from $D=0$ for intact (the undegraded sample) to $D=1$ or a residual value for the fully degraded sample. As shown in Fig. 3, with increasing time, the mechanical response of the scaffold degraded so that the strain energy density $S_t$ decreased with time while the degradation $D$ increased with time.

A degradation time evolution function was proposed to capture the experimental behavior and was used as a predictive mathematical model. The authors proposed the use of a mathematical form used in damage mechanics similar to that was used in (Katti and Desai 1995)where $D$ = degradation; $D_u$ = residual degradation value if the degradation does not lead to a complete loss of mechanical properties; $t$ = time; and $A$ and $Z$ = degradation evolution parameters for the scaffolds obtained from the mechanical tests on scaffolds. The parameters $A$ and $Z$ can be affected by pressure and fluid flow rate. In the current work, all experiments were conducted at atmospheric pressure and at no flow conditions. $D_u=1$ in the current work. Consequently, Eq. (3) becomes

As subsequently described, that function was effective in modeling the degradable behavior of scaffolds used in the study.

The accelerated degradation, $D$, of the PCL/in situ HAPclay scaffolds at days 1, 5, 7, and 18 was obtained from the experimental compression tests results presented in Fig. 3. The values of $D$ were found by calculating the strain energy density of the scaffolds at 10% strain using Eq. (1). $S$ was found from the area under the stress–strain curve from the compression test results shown in Fig. 3. Fig. 4(a) shows the degradation values calculated using Eq. (1) versus time. As shown in Fig. 4(a), the mechanical degradation increases with time. Applying the natural log on the left-hand and right-hand sides of Eq. (4) leads to

Eq. (5) is a line in the $\ln [-\ln (1-D)]$ versus $\ln (t)$ space [Fig. 4(b)], where $\ln (A)$ = intercept and $z$ = slope of the line. The calculated degradation data at different days is found from Fig. 4(a). The authors did not use the data for day 14 for the calculation of parameters; the data was subsequently used for validation. By substituting the degradation values at different days, $A$ and $z$ were found by using the least-squares method. The values of $A$ and $z$ are 0.1205 and 0.8253, respectively [Fig. 4(b)]. Therefore, for the accelerated degradation of the PCL/in situ HAPclay scaffolds, the mechanical degradation function, $D$, in Eq. (4) can be written

The accelerated degradation analytical model proposed in Eq. (4) was validated by predicting the mechanical compression test results not used in finding the parameters. The model was validated with respect to the stress–strain response for the degraded PCL/in situ HAPclay scaffold at day 14. The degradation value for the degraded PCL/in situ HAPclay scaffold at day 14 was obtained from Eq. (6). The stress values were calculated by using following equation:where $\sigma$ = stress of the degraded sample; ${\sigma }_0$ = stress of the undegraded sample at the same strain; and $D$ = degradation function value at day $t$.

Fig. 4(c) shows the comparison of the stress–strain response, after 14 days’ degradation, of the predicted stress response obtained from the analytical model with the stress response from the compression tests. The comparison shows a good agreement between the test results and the predicted model for the stress–strain responses of the degraded PCL/in situ HAPclay scaffolds. The strain energy density of the 14 days degraded sample was back-predicted using the $D$ value calculated from the analytical model. Table 1 shows the comparison of the strain energy density, at 5 and 10% strains, of the predicted-from model with the compression test results. The difference of the strain energy density, at 5 and 10% strains, calculated from the model and compression tests, is 1.40 and 9.06%, respectively. The authors found a good agreement between the predicted model and the compression test results, indicating that the proposed model [Eq. (4)] provides a satisfactory prediction of the mechanical behavior of the degradation with time of the PCL/in situ HAPclay scaffolds.

The viability of hOB cells on the PCL/in situ HAPclay scaffolds at 4, 7, 18, and 28 days was quantitatively obtained using the WST-1 assay. In the WST-1 assay, the metabolic activity of living cells converts the WST-1 reagent into colored formazan. The number of cells can be estimated by measuring the intensity of the formazan, which is proportional to the number of living cells. The results showed a progressive increase in the hOB cell proliferation with the number of days [Fig. 5(a)].

The SEM micrographs in Figs. 5(b and c) show the behavior of hOB cells on the PCL/in situ HAPclay scaffolds for a cell culture period of 28 days. The hOB cells grown on these scaffolds showed attachment, spreading, and the formation of mineralized ECM. These images indicate that the PCL/in situ HAPclay scaffolds used in this study are biocompatible and have the potential for bone tissue regeneration.

Due to their complex structure, the biomaterials have shown different mechanical properties in dry and wet conditions (Ribeiro-Samy et al. 2008; Sobral et al. 2011). It has been reported that the mechanical properties of the biological materials decrease with an increase in temperature (Verma et al. 2015; Verma and Tomar 2014). Therefore, to understand the mechanical properties of the hOB cells–seeded PCL/in situ HAPclay scaffolds, the compressive tests were carried out in the wet condition. During the compression testing, scaffolds were soaked with cell culture medium at room temperature. Fig. 6(a) shows the stress–strain plot of the seeded scaffolds. Fig. 6(a) indicates that the compressive mechanical properties of seeded scaffolds increased with time. The increase in mechanical properties indicated the progressive (healing) effect of the hOB cells on the PCL/in situ HAPclay scaffolds.

Similar to the mechanical degradation model of the PCL/in situ HAPclay in 0.1M NaOH, the authors have proposed an analytical model of PCL/in situ HAPclay seeded with hOB cells. For hOB cells culture studies, the RI is unseeded scaffolds, and the FA is fully seeded scaffolds. The degradation, $D$, of the PCL/in situ HAPclay scaffolds seeded with hOB cells at days 4, 18, and 28 were obtained from compression test results [Fig. 6(a)]. The day 7 data as not used for parameter determination; it was used subsequently for validation. The values of $D$ were calculated using Eq. (1), as shown in Fig. 6(b). The $D$ values were found to decrease with time, indicating the improved mechanical properties that the authors refer to as healing. Parameters $A$ and $z$ were obtained from Fig. 6(c) using Eq. (5). The value of $A$ is 1.9644 and the value of $z$ is $-0.4786$ for the analytical model of scaffolds seeded with hOB cells. Therefore, for the hOB cells culture studies of the PCL/in situ HAPclay scaffolds, the mechanical degradation function, $D$, can be written

To validate the analytical model of the PCL/in situ HAPclay scaffolds seeded with hOB cells, the authors compared the stress–strain response and the strain energy density (at 5 and 10%) of the experimental compression tests obtained at day 7 with the predicted stress–strain response calculated using Eq. (8). The stress–strain response comparison is plotted in Fig. 6(d). Table 2 presents the comparison of the strain energy density, at 5 and 10% strains, predicted from the model and the compression test result. The difference of the strain energy density, at 5 and 10% strains, calculated from the model and the compression tests is 4.14 and 1.58%, respectively. It is evident from Fig. 6(d) and Table 2 that the proposed analytical model [Eq. (8)] exhibits a good prediction of the mechanical behavior of the PCL/in situ HAPclay scaffolds seeded with hOB cells.

The integrative FE analysis simulations were performed to predict the mechanical behavior of the PCL/in situ HAPclay scaffold in critical size bone defects with time in accelerated degradation (0.1 M NaOH) and hOB cells–seeded conditions. The authors constructed an FE model of a solid cylinder, 4 cm in diameter and 2.82 cm in height, to represent a critical bone defect. The details about the model and simulations are presented in the preceding sections on the materials and methods; the model is shown in Fig. 7(a). Each element of the multiscale integrative model [Fig. 7(a)] used in this study was a representative model of the PCL/in situ HAPclay scaffold [Fig. 1(c)] and thus represented the microstructure and material properties of the PCL/in situ HAPclay scaffold. The mechanical behavior of the scaffold evolved because of accelerated degradation and the interactions with the hOB cells. To obtain the time-dependent response of the critical bone defect implant model using the integrative FE model, the following approach was used: (1) the authors introduced the stress–strain response of the predicted FE model of the PCL/in situ HAPclay scaffolds as the materials response of the critical defect intact scaffold model material; and (2) to model the time dependent response, the degradation function $D$ from Eq. (6) is introduced in Eq. (7) for accelerated degradation simulations and degradation function $D$ from Eq. (8) is introduced in Eq. (7) for the hOB cell-seeded samples simulations. The results from the multiscale integrative simulations of the accelerated degradation and the hOB cells-seeded implant scaffold in critical size bone defects are presented in Figs. 7(b and c), respectively.

Fig. 7(b) shows the stress–strain response obtained from the FE simulations for the critical bone defect implant model at various days of soaking at accelerated degradation conditions. The figure shows that the stress–strain response degrades with time, as expected and as observed in this experimental work. Fig. 7(c) shows the stress–strain response obtained from the FE simulations on the hOB-seeded critical bone defect implant model with time. The mechanical properties of the cell-tissue scaffold implant improve with time. Figs. 7(b and c) show the time-dependent deformation profiles of the FE model in different degradation conditions. The same axial stress was applied to compare these deformation profiles.

Because of the highly complex and hierarchically organized nature of biological materials, the standard protocols for using the specific mechanical properties of materials such as Young’s modulus in mechanics models are not fully capable of capturing the mechanical behavior evolution of biological materials (Hellmich and Katti 2015). In the current work, a truly computational multiscale approach was demonstrated to predict the mechanical property evolution in biomaterials. The entire stress–strain material response was used rather than only the elastic modulus usually used in the literature. The authors constructed a representative FE model of the PCL/in situ HAPclay scaffold by accurately incorporating the real microstructure from a representative volume of the scaffold cylinder, using high-resolution μCT, and transforming the 3D image to FE models using the μCT-FE technique. The stress–strain response obtained from the steered molecular dynamics (SMD) simulations (Sharma et al. 2015) provided the material response in the FE analysis of the PCL/in situ HAPclay representative scaffold models. The 3D reconstructed FE model [Fig. 1(c)] contains the high porosity and pore interconnectivity in the PCL/in situ HAPclay observed in the μCT images. A significant portion of the pores is in the size range of 100–300 μm. The micropores in the scaffold walls (10–30 μm or smaller) could not be identified due to the limitation of the μCT resolution. Thus, the tetrahedral element mesh obtained from the μCT does not contain the micropores in the scaffold walls observed in the SEM images. The absence of the scaffold wall porosity in the FE analysis led to the overestimation of the stress–strain responses predicted from the FE simulations. The effect of wall porosity on the mechanical response of the scaffold was incorporated by introducing a reduction factor, $K$. The reduction factor, which considers the wall porosity, was calculated from the SEM imaging of the scaffolds. The FE model of the PCL/in situ HAPclay scaffold developed in this study was validated by comparing the results of the stress–strain responses of the scaffolds that were calculated with the compression test experiments on the scaffolds with the results of the FE simulation. The FE simulation results, which incorporate the nanocomposite mechanical properties obtained from the SMD simulations, compare well with an experimentally observed response [Fig. 2(d)].

The PCL polymer is known for its slow degradation rate (3–4 years) (Gunatillake and Adhikari 2003; Lam et al. 2008). Thus, alkaline conditions are often used as a catalytic substance to accelerate the hydrolytic degradation of PCL to allow study of the degradation mechanism in a short period of time. Fig. 3 shows the stress–strain curves obtained from compression tests of the PCL/in situ HAPclay scaffolds degraded in an accelerated condition (0.1M NaOH). The mechanical properties of the PCL/in situ HAPclay scaffolds decreased gradually for the first 5 days, and then a higher rate of change was observed to day 18. It has been reported that the inclusion of organomodified nanoclay particles disrupts the crystallinity of polymer nanocomposites (Sikdar et al. 2007, 2008a) and the presence of calcium phosphate–based nanoparticles affects the arrangement of polymer chains (Lam et al. 2008), and the presence of nanoclay and HAP could potentially impact degradation rates.

Figs. 5(a and b) show the SEM micrographs of the PCL/in situ HAPclay scaffolds containing proliferating hOB cells at 28 days. The hOB cells show good adhesion to the PCL/in situ HAPclay scaffolds and appear to have flat and rounded morphology. It was also observed that the attached hOB cells support the formation of mineralized ECM on the seeded scaffolds at 28 days. The proliferation of hOB cells on the PCL/in situ HAPclay scaffolds was evaluated at 4, 7, 18, and 28 days of in vitro cell culture. Fig. 5(a) shows that the number of live hOB cells increased over a period of 28 days. The results revealed that the proliferation of hOB cells significantly increased at 28 days compared with their proliferation at 4 days. The increased proliferation of hOB cells on the PCL/in situ HAPclay scaffolds with time indicated their ability to support the mineralization and formation of bone nodules. The SEM images and the proliferation study results revealed that the hOB cells could thrive and proliferate on the PCL/in situ HAPclay scaffolds. Similarly, this group’s previous studies of in situ HAPclay–containing polymer composite scaffolds showed enhanced cell adhesion and proliferation when seeded with hOB cells and hMSCs (Ambre et al. 2011; Katti et al. 2015a). The increase in mechanical properties of the seeded scaffolds corresponds to the increase in the cell proliferation on the scaffolds.

The polymer scaffolds can be designed to mimic the bone hierarchy with an appropriate porous microenvironment and adequate mechanical properties for cell adherence, proliferation, and subsequent tissue formation. However, the prediction of the time-dependent mechanical behavior of the polymer nanocomposite scaffolds during degradation and tissue regeneration is still a major challenge in bone tissue regeneration. Several material-degradation models are available for predicting the degradation of polymers. The authors developed damage mechanics–based degradation models of the PCL/in situ HAPclay scaffolds to predict the mechanical behavior during accelerated degradation and with hOB cells over time. The authors used the change in strain energy density to calculate the experiment degradation ($D$) values. The strain energy density calculations allowed capture of the complete mechanical response of the scaffolds to the predetermined strain. As shown by their respective stress–strain profiles [Figs. 3 and 6(a)], the $D$ values increased with time for the accelerated degradation and decreased with time when the hOB cells were introduced in the scaffolds. The parameters $A$ and $z$ were calculated from experiments and used in the analytical models. Both analytical models were validated using experimental data that were not used in the calculation of the degradation functions. Fig. 4(c) presents the validation of an analytical model for the degradation studies, and Fig. 6(d) presents the validation of an analytical model for the hOB cell culture studies. It appeared that in both cases, the stress–strain response from the model and from the experiments had a good correlation. Similarly, Tables 1 and 2 show that the strain energy density comparison between the model and the experiments showed a good agreement. The parameters of such analytical models are relatively easy to obtain.

No animals have been used in the study. Appropriate Institutional committee approvals (IBC-NDSU) have been obtained for the use of the human cell lines used in this study.