Tunable elastomer materials with vascular tissue-like rupture mechanics behavior

Purpose. Laboratory models of human arterial tissues are advantageous to examine the mechanical response of blood vessels in a simplified and controllable manner. In the present study, we investigated three silicone-based materials for replicating the mechanical properties of human arteries documented in the literature. Methods. We performed uniaxial tensile tests up to rupture on Sylgard184, Sylgard170 and DowsilEE-3200 under different curing conditions and obtained their True (Cauchy) stress-strain behavior and Poisson’s ratios by means of digital image correlation (DIC). For each formulation, we derived the constitutive parameters of the 3-term Ogden model and designed numerical simulations of tubular models under a radial pressure of 250 mmHg. Results. Each material exhibits evident non-linear hyperelasticity and dependence on the curing condition. Sylgard184 is the stiffest formulation, with the highest shear moduli and ultimate stresses at relative low strains (μ 184 = 0.52–0.88 MPa, σ 184 = 15.90–16.54 MPa, ε 184 = 0.72–0.96). Conversely, Sylgard170 and DowsilEE-3200 present significantly lower shear moduli and ultimate stresses that are closer to data reported for arterial tissues (μ 170 = 0.33–0.7 MPa σ 170 = 2.61–3.67 MPa, ε 170 = 0.69–0.81; μ dow = 0.02–0.09 MPa σ dow = 0.83–2.05 MPa, ε dow = 0.91–1.05). Under radial pressure, all formulations except DowsilEE-3200 at 1:1 curing ratio undergo circumferential stresses that remain in the elastic region with values ranging from 0.1 to 0.18 MPa. Conclusion. Sylgard170 and DowsilEE-3200 appear to better reproduce the rupture behavior of vascular tissues within their typical ultimate stress and strain range. Numerical models demonstrate that all three materials achieve circumferential stresses similar to human common carotid arteries (Sommer et al ), making these formulations suited for cylindrical laboratory models under physiological and supraphysiological loading.


Introduction
Cardiovascular diseases (CVD) have represented the leading cause of death in the last three decades and the rate of mortality has steadily increased since 1990 (Roth et al 2020, World Health Organization 2020. Unfortunately, this trend is not expected to change any time soon. The underlying pathology of CVD is atherosclerosis, which is initiated by cholesterol buildup beneath the endothelium of artery blood vessels, which ultimately evolves into an atherosclerotic plaque (Burke et al 2014). The rupture of this plaque can result in major clinical outcomes, including myocardial features that may influence the biomechanics of vascular tissue rupture and the stability of atherosclerotic plaques. However, performing these experiments on blood vessels presents several challenges. The specimens require delicate handling, and their manipulation can potentially alter the outcome of the experiments. In addition, it can be difficult to isolate the effect of individual aspects of vessel's morphology and composition on plaques stability given the heterogeneous and multifactorial nature of arterial tissues. Therefore, the development of tissue-mimicking materials with similar mechanical properties as human blood vessels to study the behavior of atheroma-like phantoms with reproducible and tunable mechanical properties comprise an alternative to the challenges posed by experimental testing of blood vessels. The use of a laboratory model reduces the complexity of the problem at hand and allows to better examine the significance of specific factors that may otherwise be very difficult to study.
One of the most adopted elastomers in the field is polydimethylsiloxane (PDMS), in particular the two types Sylgard184 and Sylgard170 (Dow Corning Corporation). These two silicones have a simple manufacturing process, present a nonlinear behavior and Sylgard184 material properties are known to vary with both curing and operational temperatures (Mata et al 2005, Kim et al 2011, Johnston et al 2014, Choi, Dae-Woong Park 2014. Sylgard184 is also biocompatible, which makes this silicone a good substrate for cell culture and mechanotransduction studies (Fischer et al 2017, Hecker et al 2005, Colombo et al 2010. The hyperelastic properties of these materials within the low stretch range (30%-50% stretch) resemble the typical stress-hardening behavior of soft tissues, which results from stretching the constitutive elastic fibers of the extracellular tissue, in particular collagen (Payan andOhayon 2017, Fratzl 2008). For this reason, Sylgard184 and Sylgard170 have been previously used to prepare laboratory models of healthy arteries and aortic aneurysm (Colombo et al 2010, Doyle et al 2010. While the mechanical tunability of Sylgard 184 has been extensively investigated (Mata et al 2005, Liu 2007, Kim et al 2011, Johnston et al 2014, the studies on mechanical properties of Sylgard170 are scarce. Studies on the behavior of Sylgard170 have only been reported for the mix curing ratio recommended by the provider (Doyle et al 2009(Doyle et al , 2010; however, there exist the potential for tuning its mechanical properties when modifying the base (pre-polymer) to curing agent (cross-linker) ratio. Importantly, the ultimate strength (i.e. maximal stress at rupture) of Syl-gard184 seems to be too high compared to that reported for any type of artery, while the ultimate strain seems comparable to that of arterial tissues (Mata et al 2005, Doyle et al 2009, Walsh et al 2014. This makes Syl-gard184 less suited for replicating the rupture mechanism of vascular tissues within their typical stress and strain range to rupture. On the other hand, the encapsulant DowsilEE-3200 (Dow Corning Corporation) appears to show a significantly lower ultimate strength (0.55 MPa, based on the provider's documentation) while maintaining a hyperelastic response under high deformations. Hence, DowsilEE-3200 could be a potential candidate for arterial tissue laboratory models. To the best of our knowledge, there is no literature of a mechanical analysis of DowsilEE-3200 or Sylgard 170 under different curing conditions to develop atheroma-like phantoms that can replicate the biomechanics of rupture of atheromatous plaque tissues.
Therefore, the purpose of this study is to provide an accurate characterization of the material properties of Sylgard184, Sylgard170 and DowsilEE-3200 under different curing formulations by combining uniaxial tensile tests, digital image correlation (DIC) and numerical simulations. This analysis is intended to be used as a reference for future application of these materials as vascular laboratory models to investigate the biomechanics of plaque rupture.

Samples preparation
Samples were manufactured based on 80%-scaled down ASTM D412-Type C geometry to replicate typical dimensions of arterial tissues. Each specimen had a gauge region with a length, width, and thickness of 6.6, 1.2 and 0.6 mm respectively (figure 1). To prepare our samples, we injected the material into a custom-designed molding system consisting of two 0.5 in (1.27 cm)-thick plates. The bottom plate was carved with a total of 12 sample shapes using a milling machine. The top plate was then used to cover the bottom part and the system was tightly closed by a set of screws and nuts (figure 1). The two plates were made of Polyetherimide, which is an amber-transparent thermoplastic with a glass transition temperature of 217°C (Kyriacos 2017).
The three elastomer materials are supplied as a two-part kit, consisting of a pre-polymer and a crosslinker (Sylgard184) or PartA and PartB (Sylgard170 and Dowsil EE-3200). Sylgard184: Samples were prepared using 15:1, 10:1 (recommended by the manufacturer) and 5:1 base-to-curing agent ratios. For each group, the PDMS was thoroughly mixed, degassed for about 30 min, poured into the molds, cured at 30°C for 16 h and then cured again at 100°C for 4 h. Syl-gard170: We tested the material at 1:1 (recommended by the manufacturer), 5:1 and 1:5 Part A-Part B ratios. Samples were prepared following the same protocol as for Sylgard184. Dowsil EE-3200: For this material, the mixtures considered were 1:1 (recommended by the manufacturer), 1:1.5 and 1:2.5 ratios. The manufacturing process was the same as for the other materials except the final curing temperature was set to 50°C, as the provider recommends not to heat the material above 60°C. For every formulation, each ratio between the two mixing parts was considered by weight.
After the curing process, samples were carefully removed from their cast and visually inspected using a stereomicroscope (SteREO Discovery.V12, Zeiss). Specimens that presented air bubbles or imperfections at the boundaries were discarded. Finally, the laboratory models were sprayed with water-based ink to perform digital image correlation (DIC). The use of an air brush enabled us to control with accuracy the speckles sizes (3-5 pixels) and the speckles density (20%-40% of the sample surface) (Jones et al 2018).

Testing protocol
We tested the samples using a custom-made micro material testing system equipped with real-time control and acquisition software (LabVIEW, v. 2018, National Instruments). The machine was equipped with a load cell with a capacity of 10N and sensitivity of 10 mN. An extensive description of our testing system can be found in (Corti et al 2022). The experiments consisted of applying 10-cylces preconditioning stretch at 10% strain, followed by one-single pull to rupture. The loading cycle was governed by a ramp waveform under displacement control at a constant strain rate of 1.5 mm/s. Throughout the test, the reaction force and displacement were measured by the system and images of the sample were recorded by a FLIR Blackfly S high-resolution camera (Teledyne FLIR) equipped with a Tamron M111FM50 lens (IDS Imaging).

Data processing
To perform DIC analyses we used the GOM Correlate software (ZEISS Group, v.2019) which allowed us to obtain the True (Cauchy) stress and strain for each sample as well as the material Poisson's ratio, ν. The true stress values were calculated by measuring the reduction in cross-sectional area from the change in width in the gauge region throughout the test, assuming the material as isotropic. The average Poisson's ratio for each formulation was obtained considering the material response up to 10% strain and it was computed as (1): where e trans is the transverse strain (perpendicular to the tensile direction) and e axial is the axial strain (parallel to the tensile direction). The individual and average true stress and strain curves and the Poisson's ratio were obtained combining the data from the tensile system and the DIC analysis in a custom made MATLAB script. By knowing the nominal (engineering) stress and strain, and the Poisson's ratio, we were also able to determine the constitutive coefficients of the 3-term Ogden model for each tested elastomer formulation. The inverse analysis was carried out using the material evaluation capability in Abaqus (Dassault Systemes, v.2019). The 3-term Ogden constitutive model has been widely used to replicate the hyperelastic behavior of arterial tissues ( respectively. The fitting accuracy of the Ogden description was tested by replicating our experiments in finite element method (FEM) simulations in Abaqus. Each material was considered, and the sample was stretched up to the average ultimate displacement of the corresponding formulation. Finally, the Ogden description was used to simulate the expansion of 30 mm-long cylindrical tubes by a uniform and linearly increasing pressure up to 250 mmHg. Here, the two axial extremes of the vessels were free to move in the radial direction only. These geometries were designed to replicate the average radius (R=4.15 mm) and thickness (H=1.17 mm) of common carotid arteries walls reported by (Sommer et al 2010).

Statistical analysis
Data are reported as mean ± SD. Analysis of variance was performed by one-way ANOVA test followed by post-hoc pairwise comparison of the mean strength between the groups. The null hypothesis was rejected if p < 0.05.

Tensile response and rupture threshold
The average true stress and strain curves for each group (figure 2) depict the evident hyperelastic nonlinear behavior of the elastomers up to rupture, and the influence that different base to curing agent ratios have on this behavior. Our results show that Syl-gard184 is the stiffest of the materials considered as it experiences the highest shear modulus and ultimate stress at relative low strain levels. On the other hand, Sylgard170 and DowsilEE-3200 present significantly lower shear moduli and ultimate stress levels that are closer to data reported for arterial tissues (Holzapfel et al 2005, Teng et al 2009. Importantly, Dowsil is the most compliant material, with the lowest shear modulus among these elastomers, and with ultimate stress and strain values very close to arterial tissue. The right panels in figure 2 offer a closer view of the behavior of each formulation up to a strain of 0.5, which is a common range of stretch in arteries under physiological and supraphysiological loadings. The average ultimate stress and stretch ratio values as well as the Poisson's ratio of each formulation are reported in table 1. For comparison purposes, we have Overall, sylgard 184 is the stiffest elastomer, followed by Sylgard 170 and the most complaint was found to be Dowsil EE-3200. The stress-strain behavior was found tunable (dependent on the polymer to curing agent ratio) in all formulations. also included the longitudinal and circumferential Cauchy stresses and stretch ratios at rupture that have been reported in the literature for carotid and coronary arteries (Holzapfel et al 2005, Teng et al 2009. In the first study, Teng et al analyzed the mechanical behavior under tensile tension of human carotid arteries presenting sections with fatty streak or pre-atheroma with extracellular lipid pools (lesion II-III). The authors were able to test both intact samples as well as dissected media and adventitia layers. In the second study, Holzapfel et al considered dissected layers intima, media and adventitia of non-stenotic human coronary arteries presenting intimal thickening. Our data show that Sylgard184 is the material with the highest values of ultimate stress, which are far from those showed by human arterial tissues. Differently, Sylgard170 and DowsilEE-3200 experience significantly lower stresses at rupture, with Dowsil having the highest extensibility. The statistical analysis on the effect of mix ratios for each formulation is illustrated in figure 3. For Sylgard184, our results indicate similar stresses at rupture among the three formulations, while there is a significant difference between R15:1 ultimate strain and the other two ratios. Reducing the relative amount of curing agent from the recommended mix ratio increases the extensibility of Syl-gard184. In the case of Sylgard170, each ratio presents statistically different ultimate stresses and R5:1 is the only formulation that shows a meaningful shift in strain levels. A higher concentration of PartA makes Syl-gard170 stiffer. The influence of the mix ratio is also evident for DowsilEE-3200. Each formulation manifests  The strain deviates only when considering a higher concentration of PartB than the amount recommended by the provider.

Stress to stretch ratio response of laboratory models and atherosclerotic carotid tissues
To validate the mechanical behavior of our laboratory models, we compared their stress versus stretch ratio (where the stretch ratio λ=strain + 1) response to published data of carotid plaques after endarterectomy ). In addition, we included for comparison the tensile test data of one sample of intact atherosclerotic carotid artery from Teng et al (Teng et al 2009). In this analysis, we considered the formulation of each material that could better fit the arterial data ( and Sylgard 170 could be used for modeling the adventitia and/or the media layer(s) of the laboratory model, but they will not be suited for replicating the rupture of the intima layer due to the significant different ultimate stress and stress levels shown by these materials when compared to blood vessels. However, the only material that is able to reproduce similar ultimate stress and stretch values to rupture of the intima and media layers is DowsilEE-3200, as Sylgard184 and Sylgard170 reaches stress levels that are far beyond the rupture threshold of carotid samples.

Material fitting and numerical simulations
The nominal stress and strain data as well as the Poisson's ratio of each material were imported into Abaqus and the Ogden coefficients were evaluated. Every formulation respected Ducker's stability criteria (Drucker 1959) for all ranges of tension and compression strains and the coefficients values are reported in table 2. After obtaining the constitutive description, we performed numerical simulations on samples with the same geometry as our laboratory models. In these finite element analyses, the samples were stretched until reaching similar levels of ultimate strain as the test data. The axial stress and strain in the gauge region were then exported and graphed against the average stress and strain curves of the corresponding formulation. The fitted Ogden model accurately replicates the mechanical response of all material, as can be seen in figure 5(A). The strain distribution computed through DIC is also similar when compared with that of FEM models, as shown in figure 5(B). The mechanical behavior of each material under radial pressure is illustrated in figure 6, which represents the average values of 30 mesh elements per data set that were selected along the thickness of the vessel. The only formulation that reaches the hyperelastic regime is the Dowsil R1:1, which is the most compliant material considered in this study and undergoes significant degrees of expansion ( figure 6). It approaches a circumferential stress of 0.5 MPa, which corresponds to high levels of strains (0.5-0.9 strain according to figure 5(A)). The other materials exhibit similar stress profiles that remain in the elastic region with values that range from 0.1 to 0.18 MPa.

Discussion
Given Sylgard184 wide range of applications, the tunability of its material properties has been previously studied in terms of morphological features, experimental procedure, curing blends and temperatures. Liu et al  documented the effect of the sample thickness on Sylgard184 mechanical properties. The authors demonstrated that a thickness of 200 μm represents the transition point from bulk to thickness-dependent behavior, below which Syl-gard184 membranes exhibit 2 to 5-fold increase in stress levels. In this study, we considered 600 μm-thick samples and the stress levels that we obtained are close to those reported by Liu et al for membranes governed by a bulk behavior. Other researchers have focused on the effect of varying curing temperatures on the  Table 2. List of the constitutive Ogden coefficients obtained from experimental stress-strain data of each formulation of Sylgard184, Sylgard170 and DowsilEE-3200.
Formulation  (Johnston et al 2014) demonstrated that that the material Young's modulus (linear stress-strain behavior at low strains) is directly proportional to the curing temperature, while the maximum strain seems to be inversely proportional to such curing temperature. In the present study, we only focused on one curing temperature for every formulation, but the influence of this factor on Sylgard170 and Dowils EE-3200 might be relevant and worth investigating in the future. In another study, Kim et al (Kim et al 2011) performed uniaxial loadings up to rupture on Syl-gard184 samples prepared with different mix ratios. The authors considered blends of 5:1, 10:1 and 15:1 and observed higher stresses and lower strains when increasing the amount of curing agent. The change in strain agrees with what we obtained for ratio 15:1. However, our data show no significant change in ultimate stress regardless of the formulation. We believe this difference originates from the type of stress that is calculated from the raw data. Kim et al reported the nominal (engineering) stress, which doesn't take into consideration the change in cross-sectional area under stretch, while we were able to derive the True (Cauchy) stress in the sample making use of the DIC technique. Different stress measurements and distinct experimental protocols can potentially be the reason why published data for Sylgard184 fluctuate widely among various studies (table 3). Indeed, the method used to estimate the strain of the sample gauge region plays a relevant role on the accuracy of the results. In the case of dumbbell-shaped geometries, Schneider et al (Schneider et al 2008) showed that the deformation of the shouldered sides of the sample has a significant influence on the total strain progression. The authors calculated that the total strain measured automatically by the tensile machine was more than a factor of 2 larger than the strain in the test section. This difference appears to remain constant up to 40% strain and so it can be addressed by applying a correction factor. However, for larger deformation regimes this assumption is not valid, and the strain of the sample gauge region should be measured directly, either manually or through video elastography. Besides Sylgard184, we extended the analysis of varying curing conditions to Sylgard170 and Dowsi-lEE-3200 which show stress and strain levels much similar to those of arterial tissues. We demonstrated that the mechanical behavior of both materials can be controlled by tuning the amount of the base polymer component and cross-linker. In the case of Sylgar170, the relative amount of PartA appears to govern the stiffness and ultimate strength of the material. For DowsilEE-3200, increasing the PartB concentration leads to higher stresses and lower strains. These results reveal the possibility to adjust the curing ratios of these materials to replicate the mechanical properties of specific vascular tissues or arterial layers (i.e. intima, media and adventitia). Multiple research groups have studied the anisotropic response of arterial tissues   (Holzapfel et al 2004) for carotid and coronary arteries, given their relevance in cardiovascular diseases. These reported values from actual anisotropic vascular tissues in the axial and circumferential directions serve as lower and upper reference limits for comparison against our equivalent isotropic laboratory materials. While the isotropic character of elastomers is a limitation when compared to actual anisotropic tissues, isotropic laboratory models are valuable for a broad range of studies in vascular mechanics research. Future studies may extend the use of the Sylgar170 or DowsilEE-3200 with fibers to transform these materials into anisotropic ones. In addition, we focused our attention on the circumferential stresses and strains, as they are often considered as determinants of plaque vulnerability. Sylgard184 exhibits ultimate stretch ratios that are similar to those of carotid and coronary tissues. However, its ultimate strength is too high when compared with the published human tissue data. Every formulation of this material reaches levels of stresses that offset the ultimate strength of arterial layers by 5-15 times. This difference makes Sylgard184 an inappropriate laboratory model when trying to reproduce the mechanical failure of arterial tissues under similar ranges of stresses and strains. Sylgard170 also manifests strain levels similar to the data of Teng and Holzapfel. Specifically, ratios 5:1 is the material that most closely approaches the circumferential stretch values of coronary layers. The ultimate stresses are much lower compared to Syl-gard184, but they moderately exceed the average values reported for human arteries. However, these levels of stresses appear to lie on the set of values shown by porcine coronary arteries (Lally et al 2004, Noble et al 2016. DowsilEE-3200 is the most compliant elastomer in this study. It experiences the lowest values of stresses while reaching higher deformations. For this material, Dowsil 1:1.5 is the formulation that best approaches the stress and strain values of the arteries considered. It shows ultimate stresses that are very close to those of carotid intact wall and layers as well as coronary adventitia. However, the average stretch ratio at rupture lies outside the range of arterial tissues, with values that are up to 40% higher than carotid intact wall samples and 10%-30% higher than coronary layers. From the stress-strain data, we also derived the material constants of the Ogden strain energy function for each formulation. These values can be used to perform numerical studies of different nature, as they guarantee computational stability for all tensile and compressive strains. We used the constitutive description of the materials to investigate the mechanical response of each formulation under radial pressure by means of FEM simulations. We compared the numerical results of our models to those of common carotid arteries studied by Sommer et al (Sommer et al 2010). We found that all material but Dowsil 1:1 exhibit a linear increase in stresses up to a pressure of 250 mmHg, with maximum values in the range of 100-180 kPa. These stresses closely replicate those in (Sommer et al 2010), which span between 100 and 175 kPa depending on the axial pre-stretch. These results demonstrate that cylindrical tubes made of Sylgard184, Sylgard170 and DowsilEE-3200 (except R1:1) undergo similar intramural stress distributions as common carotid arteries under physiological and supraphysiological hydraulic pressures.

Conclusion
In the present study, we performed a mechanical characterization and comparative analysis of three potential vascular tissue mimicking-materials, Syl-gard184, Sylgard170 and Dowsil EE-3200 under different composition ratio and curing conditions. The average mechanical behavior for each material was then curve-fitted against a constitutive material description based on the 3-term Ogden hyperelastic model, to obtain the parameters characterizing such constitutive model. Numerical simulations were performed to analyze the behavior of the material in a simplified blood vessel under the action of blood pressure. The tensile behavior to rupture was compared against published data of carotid and coronary arteries. On one hand, we demonstrated that Syl-gard184 is significantly stiffer than blood vessels, and highly exceed the ultimate tensile strength shown by these blood vessels. Even though Sylgard184 is among the elastomer materials most commonly used for vascular laboratory models, it does not well replicate the tensile properties to rupture of these tissues. On the other hand, Sylgard170 and DowsilEE-3200 are able to better replicate the typical range of stresses and strains of the arteries being considered. We conclude that Sylgard170 and DowsilEE-3200 are easy to use, mechanically tunable, isotropic elastomer materials that can replicate the hyperelastic behavior to rupture of carotid and coronary arteries, to be used as vascular tissue mimicking laboratory models. and in the decision to submit the article for publication.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Declaration of interest
none.