Modelling of silk-reinforced PDMS properties for soft tissue engineering applications

Abstract

BACKGROUND:

Polydimethylsiloxane (PDMS) is widely used in biomedical research and technology, but its mechanical properties should be tuned according to the desired product specifications. Mixing ratio of base polymer to curing agent or additives enables its mechanical properties to be manipulated and fit to mechanical properties of biological tissues.

OBJECTIVE:

In this paper, we analysed the effect of mechanical load on silk-reinforced PDMS depending on silk concentration.

METHODS:

We prepared cylinder-type PDMS samples with different silk concentrations and performed cyclic uniaxial compression tests with a fixed magnitude of applied strain. Next, we analysed the mechanical charascteristics of PDMS using computational modelling.

RESULTS:

The stress-strain data within the large-strain region of different PDMS cylinders without silk and with 1%, 5% and 10% silk concentrations was fitted to non-linear second order Mooney-Rivlin, and third-order Ogden models. The results show the equivalence of both models for investigated strain region of PDMS. On the other hand, PDMS cylinders with 10% silk concentration allowed the successful fitting of experimental data just for the second-order Mooney-Rivlin model, while all numerical probes to find an appropriate fitting parameters for third-order Ogden models were unsuccessful.

CONCLUSIONS:

The second-order Mooney-Rivlin model is preferable for analysing the properties of silk-reinforced PDMS over the entire measurement range.

Keywords

Biomaterials tissue engineering PDMS silk nonlinear materials

1. Introduction

Polydimethylsiloxane (PDMS) is a silicone elastomer that is highly biocompatible, biostable and supports extracellular matrix deposition. Its suitability for microporous scaffold fabrication has been demonstrated [1, 2]. These characteristics together with its gas permeability, good optical transparency and low autofluorescence make PDMS an attractive candidate for tissue engineering applications as a scaffold material. However, tissues in the organism depending on their location are exposed to various mechanical forces such as compression, fluid shear stress, hydrostatic pressure, and stretching [3]. Numerous studies have demonstrated how scaffold mechanical properties affect cell adhesion, cytoskeletal arrangement, motility [4], proliferation [5], and differentiation [6].

PDMS resistance to mechanical loading such as pressure or external force is limited. On the other hand, mechanical properties of the elastomer can be optimised by varying curing temperature and mixing ratio or adding other materials. Young’s modulus, ultimate tensile strength, compressive modulus, ultimate compressive strength and hardness dependency on curing temperature were characterized by Johnston et al. [7]. To modulate PDMS properties, Quake and his colleagues proposed the structural bonding of relatively flexible vinyl PDMS and rigid Si-H PDMS [8]. Kim et al. used this method to demonstrate the possibility of fabricating many devices [9]. As a composite component to stabilize PDMS, silk fibroin, a natural biomaterial produced by silkworms and spiders, can be used. Because of its unique mechanical properties, biocompatibility and the ability to support cell differentiation, silk is favourable for a numerous tissue engineering applications, especially in bone tissue engineering [10, 11, 12]. Moreover, silk may be combined synergistically with other biomaterials to form composites of desired physical and biological features.

It is widely agreed that it is very difficult to predict and evaluate the nearly incompressible mechanical behaviour of PDMS structures [9]. Therefore, computational modelling is needed in order to understand and choose optimal mechanical properties for bioapplications [13, 14, 15, 16, 17, 18, 19, 20]. In this paper, we analyse the effect of mechanical load on silk-reinforced PDMS depending on silk concentration. The nonlinear material properties of PDMS were modelled using both the second-order Mooney-Rivlin and the third-order Ogden models.

2. Experimental methods

2.1 Silk-reinforced PDMS sample fabrication

The silk preparation procedure was adopted from the protocol developed by Rockwood et al. [21]. Silkworm (Bombyx mori) cocoons were fragmented using scissors and subsequently boiled for 1 h in a 0.02 M solution of $\mathrm{Na}_{2}\mathrm{CO}_{3}$ that was gently stirred. Cocoon leftovers were removed from the solution, and the remaining silk was washed three times with deionised water for 1 h each. The silk fibers were then pressed to remove excess water and left to dry overnight.

PDMS (Sylgard 184, Dow Corning) was thoroughly mixed with the proprietary thermoinitiator at a mass-to-mass ratio of 10:1. Silk was subsequently added to the mixture at 1%, 5%, or 10% mass ratio, and a pure PDMS-thermoinitiator mixture was used as the reference. The mixtures were poured into wells of 24-well polystyrene tissue culture plates (cylinders with diameters of 15.6 mm, heights of 17.8 mm, and volumes of approximately 3.4 mL) and left in a refrigerator (4 ${}^{\circ}$ C) for 1 hour for the air bubbles to evaporate. The plate with PDMS-silk mixtures was then heated for 1 h at 100 ${}^{\circ}$ C. Finally, the polystyrene plate was mechanically crushed to remove the samples.

2.2 Experimental setup

The compression tests were performed using a Mecmesin MultiTest 2,5-i micro-compression machine (Mecmesin Limited., Slinfold, UK) in a Mecmesin AFG25 load cell with controlled load of measured accuracy $\pm$ 0.01 mm, compression force in range 2–2500 N and accuracy $\pm$ 0.1% where croohead speed accuracy of indicated speed was $\pm$ 0.1%. After installing a sample, cyclic loading was applied (Fig. 1 shows photographs of the system and PDMS sample during cyclic compression). The displacement and applied force during the test were recorded automatically. The measurements were performed at room temperature.

Figure 1.

PDMS sample during the cyclic compression test.

The loading was controlled by measured the displacement until the maximum displacement value of $\Delta L=$ 4 mm for all specimens (Fig. 2) within a compression speed value of 0.167 mm/s.

3. Modelling approach of material properties

3.1 Different models of hyperelasticity

Each hyperelastic model defines its own energy deformation function with material constants that should be obtained by fitting experimental data with model curves. In essence, these are phenomenological models that treat the problem from the perspective of mechanical continuity and stress-deformation behaviour, ignoring the microscopic properties of the structure. However, if appropriate, the revised physics models should consider the characteristics of the microscopic properties. The following section presents a brief overview of hyperelastic models used in this study and an explanation of the algorithm that performs a nonlinear fit of the constants in the models.

3.2 Full polynomial model

The polynomial model for isotropic and compressible rubber is described as follows [22]:

$\displaystyle U=\sum\limits_{i,j=0}^{N}C_{ij}(\bar{I}_{1}-3)^{i}(\bar{I}_{2}-3% )^{j}+\sum_{i=1}^{N}D_{i}(J_{el}-1)^{2i}$ (1)

Here, $N$ represents the number of terms in strain energy function, $C_{ij}$ represent the material constants that control the shear behaviour and can be determined from uniaxial, biaxial and planar tests and $\bar{I}_{i}$ for $i=1,2$ represent the first and second strain invariants divided by $J_{el}^{-2/3}$ and $J_{el}^{-4/3}$ , respectively. $D_{i}$ represents a material constant that controls bulk compressibility and is equal to zero for fully incompressible rubber. It is non-zero in volumetric tests. Finally, $J_{el}$ represents the elastic volume ratio.

3.3 Ogden model

This model was proposed in 1972 by Ogden [23] as a phenomenological model that was based on the use of principal stretches instead of invariants. The model can describe upturn (stiffening) of stress-strain curve and is accurate in models of rubber for large ranges of deformation. However, the use of this model is limited. (e.g. only when the tension is uniaxial).

$\displaystyle U=\sum_{i=1}^{N}{\displaystyle\frac{2\mu_{i}}{\alpha_{i}^{2}}}(% \lambda_{1}^{\alpha_{i}}+\lambda_{2}^{\alpha_{i}}+\lambda_{3}^{\alpha_{i}}-3)+% \sum_{i=1}^{N}D_{i}(J_{el}-1)^{2i}$ (2)

where $\lambda_{i}$ represents the deviatoric principal stretch and $\mu_{i}$ and $\alpha_{i}$ represent temperature-dependent material properties, as presented in the full polynomial model.

3.4 Nonlinear curve fitting

The nonlinear curve fitting is based on finding one or more parameters of a model equation $\sigma(\lambda,C)$ so that the residual of experimental curve $\sigma_{M}(\lambda)$ and model equation is as small as possible. In this study, the stress-strain curves for compression tests must be approximated. The stress-strain curve in compression is used to explain the regression analysis process. The final goal is creating a fitting model equation resulting from regression analysis so that the equation yields the same stress values as the curve obtained using measured data:

$\displaystyle|\sigma(\lambda,C)-\sigma_{M}(\lambda)|<\varepsilon$ (3)

where $\sigma(\lambda,C)$ represents the model function, which depends on stretch parameters, and $\sigma_{M}(\lambda)$ denotes the measured stress-stretch curve and $\varepsilon$ represents the maximum fitting residuals, which should be as much as possible. The model equation for compression can be written in the following form:

$\displaystyle\sigma=f(\lambda,C_{10},C_{01},C_{20},C_{02},C_{11})$ (4)

The material constants $C_{i}j$ must be determined through regression analysis by solving equations of equality to zero of partial derivatives of each constant in the regression equation, which linearly appears in the compression model equation. Finally, an overdetermined system of linear equations should be solved [24] in such a way that would yield a satisfactory level of fit.

4. Results

4.1 Preparation of silk-reinforced PDMS samples

Three samples were fabricated in each of the following categories: control PDMS samples without silk, samples with 1% silk, samples with 5% silk and samples with 10% silk (Fig. 2). The samples with silk tended to have air bubbles formed within the materials due to insufficient degassing as compared to pure PDMS, and this might have influenced the mechanical properties of the samples. In addition, the silk dispersed throughout the samples randomly, possibly yielding varying degrees of stiffness within and between the samples.

Figure 2.

Pictures of PDMS samples with varying concentrations of silk.

4.2 Experimental results of compression tests

Compression tests were performed on all samples. The stress-strain curves averaged from measured displacement-load curves of different PDMS samples are shown in Fig. 3A–D. Hysteresis was observed during the cyclic loading and relaxing paths of the stress-strain curves. The results retain initial series measured within the limits of accuracy; therefore, fully nonlinear elasticity behaviour was observed during the experiment.

Figure 3.

Experimental stress-strain relations. A – PDMS, B – PDMS $+$ 1% silk, C – PDMS $+$ 5% silk, D – PDMS $+$ 10% silk.

The results presented in Fig. 3 show that for PDMS with 10% silk, the maximum stress was 2144 MPa; for PDMS with 5% silk, the maximum stress was 1324 MPa; for PDMS with 1% silk, the maximum stress was 832 MPa; and in pure PDMS, the maximum stress was the smallest and equal to 752 MPa. The changes in $\Delta L/L$ in the hysteresis loops to determine the widest areas for each material were $-$ 0.009 (PDMS with 1% silk), 0.0201 (pure PDMS), 0.0299 (PDMS with 5% silk), and 0.0637 (PDMS with 10% silk). The hysteresis loops show that the $\Delta L/L$ (when PDMS samples were loaded and unloaded) was the smallest in PDMS samples with 1% silk and the largest in PDMS samples with 10% silk.

These results show that the mechanical properties of silk-reinforced PDMS can be controlled. The hysteresis can be reduced by changing the proportions of the composite components. These results provide a qualitative understanding of silk-reinforced PDMS.

4.3 Uncertainties measurement of material properties

The experimental data for the cyclic compression tests of PDMS with different silk concentrations were modelled (Fig. 3). Two different material models, Ogden and Polynomial, were examined. The quality of nonlinear fitting was measured by calculating the mean absolute percentage error (MAPE) as follows:

$\displaystyle\text{MAPE}=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{A_{i}-F_{i}}{A% _{i}}\right|$ (5)

where $A_{i}$ represent the experimental values and $F_{i}$ represent the model values. Obtained MAPE values are summarized in Fig. 4. The y-axis of the chart presents MAPE values in percentages, and $x$ axis denotes the percentage concentrations of silk in the PDMS test specimens. The material constants of each model for silk-reinforced PDMS samples are summarized in Table 1.

As shown in Fig. 4, MAPE ranges from 1.36% to 1.58% for the polynomial $N=$ 2 coefficient fitting and ranges from 5.36% to 6.35% for PDMS with 0%, 1%, and 5% silk. In contrast, MAPE increases until 8.35% for Polynomial material, and the fitting with Ogden $N=$ 3 model results in failure. Similar results were obtained for root mean squired error (RMSE). RMSE values are in range from 0.92% to 1.12% for the polynomial $N=$ 2 coefficient fitting and ranges from 3.95% to 4.87% for PDMS with 0%, 1%, and 5% silk for Ogden $N=$ 3 model.

Combined uncertainties of measured material properties in compression tests is expressed as follow

$\displaystyle u_{c}=\sqrt{u_{E}^{2}+u_{\textit{CoMa}}^{2}+u_{\sigma}^{2}+u_{% \textit{Califer}}^{2}+V_{i}}$ (6)

where $u_{E}$ is uncertainty of tangent curve value of nonlinear stress-strain relations, $u_{\textit{CoMa}}$ is uncertainty measurement of compression machine, $u_{\sigma}$ uncertainty of of tangent curve value of non linear stress-strain relations due to stress variation, $u_{\textit{Califer}}$ is uncertainty of micrometer and $V_{i}$ is mean squared error of stress-strain(stretch) curve fitting. Analysis shows that $V_{i}$ term is much more bigger than other terms, so expanded uncertainty $U_{E}$ on measured parameters of PDMS with and without silk inclusions can be expressed as follow

$\displaystyle U_{E}=k\cdot\sqrt{V_{i}}$ (7)

where $k$ is depending on the desired level of confidence ( $k=2$ for 95% confidence).

Table 1

Material models of silk-reinforced PDMS samples and their corresponding material constants obtained from experiments

Material	Material	PDMS $+$ 0% silk		PDMS $+$ 1% silk		PDMS $+$ 5% silk		PDMS $+$ 10% silk
model	constants	[MPa]		[MPa]		[MPa]		[MPa]
Poly,	$C_{10}$	2	.239	$-$ 1	.554	$-$ 3	.202	$-$ 11	.96
$N=$ 2	$C_{01}$	$-$ 1	.864	1	.868	3	.844	14	.39
	$C_{20}$	0	.06726	0	.05517	0	.1167	0	.4030
	$C_{11}$	$-$ 0	.3478	$-$ 0	.2859	$-$ 0	.6030	$-$ 2	.113
	$C_{02}$	1	.006	0	.8344	1	.736	6	.326
Ogden,	$\mu_{1}$	$-$ 0	.01457	$-$ 0	.01486	$-$ 0	.02185	–
$N=$ 3	$\mu_{2}$	0	.00277	0	.00280	0	.00413	–
	$\mu_{3}$	0	.01761	0	.01769	0	.02569	–
	$\alpha_{1}$	2	.078e-9	2	.045e-6	2	.028e-6	$-$
	$\alpha_{2}$	3	.967e-9	3	.974e-6	3	.978e-6	–
	$\alpha_{3}$	$-$ 2	.356e-9	$-$ 2	.229e-6	$-$ 2	.166e-6	–

Figure 4.

MAPE and RMSE for different materials.

Figure 5 represents the nonlinear fitting results for silk-reinforced PDMS samples. Two material models, second-order Mooney and second-order Ogden models, were evaluated. Considering the number of coefficients (presented in Table 1) and calculation time required in finite element method simulations, both models are suitable for describing the small strain region of silk-reinforced PDMS. In the nonlinear and large strain region, the second-order Mooney-Rivlin model defined the mechanical behaviour of the silk-reinforced PDMS elastomer better than the second-order Ogden model. The silk-reinforced PDMS samples were too stiff or flexible in the measurement range for the second-order Ogden model. In particular, the nonlinear model of the second-order Mooney-Rivlin model matched the measured strain-stress curves for load regime in Fig. 3 well. Consequently, the second-order Mooney-Rivlin model is preferable to analyse silk-reinforced PDMS structures over the entire measurement range.

Figure 5.

Comparison of Ogden and polynomial curves with experimental data. A – PDMS, B – PDMS $+$ 1% silk, C – PDMS $+$ 5% silk, D – PDMS $+$ 10% silk.

5. Discussion

The nonlinear mechanical properties of silk-reinforced PDMS were obtained from a nonlinear fit of the experimental stress-strain curves of the silk-reinforced PDMS samples. The modelling results show that both the second-order Mooney-Rivlin model and the third-order Ogden model were suitable for the small strain region of silk-reinforced PDMS. However, in the nonlinear and large strain regions, only the second-order Mooney-Rivlin model characterized the mechanical behaviour of silk-reinforced PDMS. Because of stiffness and flexibility in the measurement range, the third-order Odgen model was not functional enough. Hence we may conclude that the second-order Mooney Rivlin model is preferable for modelling the nonlinear behaviour of silk-reinforced PDMS over the entire measurement range.

The results obtained in this work can serve as a basis in constructing silk-reinforced PDMS scaffolds for soft tissue engineering applications, where it is essential to finely tune the mechanical properties of the artificial cell niches to both integrate with the surrounding tissues and to guide cellular differentiation towards the desired lineages. Future work should be focussed on the investigation of cellular responses to such silk-reinforced materials.

6. Conclusions

PDMS-silk composites were prepared, and cyclic compression tests were performed. The materials showed varying degrees of hysteresis, with $\Delta L/L$ ranging from 0.009 (PDMS with 1% silk), 0.0201 (PDMS), 0.0299 (PDMS with 5% silk) to 0.0637 (PDMS with 10% silk). The maximum obtained values of stress in silk-reinforced PDMS increased with increasing silk reinforcement (752 MPa in the case of pure PDMS and 2144 MPa in the case of PDMS with 10% silk).

Footnotes

Acknowledgments

This research was funded by a grant (No. SEN-13/2015) from the Research Council of Lithuania. E. B. would also like to acknowledge Dr. John G. Hardy from Lancaster University (Lancaster, United Kingdom) for generously providing the silk used in these experiments.

Conflict of interest

None to report.

References

Ozbolat

Dey

Ayan

Povilianskas

Demirel

Ozbolat

. 3D printing of PDMS improves its mechanical and cell adhesion properties. ACS Biomaterials Science & Engineering2018; 4: 682-693.

Chuah

Koh

Lim

Menon

Kang

. Simple surface engineering of polydimethylsiloxane with polydopamine for stabilized mesenchymal stem cell adhesion and multipotency. Scientific Reports2015; 5: 18162.

Stoltz

Dumas

Wang

Payan

Mainard

Paulus

Maurice

Netter

Muller

. Influence of mechanical forces on cells and tissues. Biorheology.2000; 37(1-2): 3-14.

Yeung

Georges

Flanagan

Marg

Ortiz

Funaki

Zahir

Ming

Weaver

Janmey

. Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion. Cell Motility and the Cytoskeleton2005; 60: 24-34.

Hadjipanayi

Mudera

Brown

. Close dependence of fibroblast proliferation on collagen scaffold matrix stiffness. Journal of Tissue Engineering and Regenerative Medicine2009; 3: 77-84.

Engler

Sen

Sweeney

Discher

. Matrix elasticity directs stem cell lineage specification. Cell2006; 126: 677-689.

Johnston

McCluskey

Tan

CKL

Tracey

. Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering. Journal of Micromechanics and Microengineering2014; 24: 035017.

Unger

Chou

Thorsen

Scherer

Quake

. Monolithic microfabricated valves and pumps by multilayer soft lithography. Science2000; 288: 113-116.

Kim

Jeong

. Measurement of nonlinear mechanical properties of PDMS elastomer. Microelectronic Engineering2011; 88(8): 1982-1985.

10.

Yang

Yin

Shi

Han

Zhang

Jin

Nie

Wang

Hao

Fan

Zhou

. In vitro and in vivo characterization of silk fibroin/gelatin composite scaffolds for liver tissue engineering. Journal of Digestive Diseases2012; 13: 168-178.

11.

Bhattacharjeea

Kundu

Naskar

Kim

H-W

Maiti

Bhattacharya

Kunduc

. Silk scaffolds in bone tissue engineering: An overview. Acta Biomaterialia2017; 63: 1-17.

12.

Melke

Midhac

Ghosh

Ito

Hofmanna

. Silk fibroin as biomaterial for bone tissue engineering. Acta Biomaterialia2016; 31: 1-16.

13.

Belanger

Marois

. Hemocompatibility, biocompatibility, inflammatory and in vivo studies of primary reference materials low-density polyethylene and polydimethylsiloxane: a review. Journal of Biomedical Materials Research2001; 58: 467-77.

14.

Piruska

Nikcevic

Lee

Ahn

Heineman

Limbach

Seliskar

. The autofluorescence of plastic materials and chips measured under laser irradiation. Lab on a Chip2005; 5: 1348-54.

15.

Hua

Sun

Gaur

Meitl

Bilhaut

Rotkina

Wang

Geil

Shim

Rogers

. Polymer Imprint Lithography with Molecular-Scale Resolution. Nano Letters2004; 4: 2467-2471.

16.

du Roure

Saez

Buguin

Austin

Chavrier

Silberzan

Ladoux

. Force mapping in epithelial cell migration. PNAS2005; 102: 2390-2395.

17.

Charati

Stern

. Diffusion of Gases in Silicone Polymers: Molecular Dynamics Simulations. Macromolecules1998; 31: 5529-5535.

18.

Leclerc

Sakai

Fujii

. Cell Culture in 3-Dimensional Microfluidic Structure of PDMS (polydimethylsiloxane). Biomedical Microdevices2003; 5: 109-114.

19.

Paguirigan

Beebe

. From the cellular perspective: exploring differences in the cellular baseline in macroscale and microfluidic cultures. Integrative Biology2009; 1: 182-195.

20.

Mukhopadhyay

. When PDMS isn’t the best. Analytical Chemistry2007; 79: 3248-3253.

21.

Rockwood

Preda

Yucel

Wang

Lovett

Kaplan

. Materials fabrication from Bombyx mori silk fibroin. Nature Protocols2011; 6: 1612-1631.

22.

Rivlin

Saunders

. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London A1951; 243(865): 251-288.

23.

Ogden

. Large Deformation Isotropic Elasticity – On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences1972; 326: 565-584.

24.

Hartmann

. Numerical studies on the identification of the material parameters of Rivlin’s hyperelasticity using tension-torsion tests. Acta Mechanica2001; 148: 129-155.