Mechanical Assessment and Hyperelastic Modeling of Polyurethanes for the Early Stages of Vascular Graft Design.

The material design of vascular grafts is required for their application in the health sector. The use of polyurethanes (PUs) in vascular grafts intended for application in the body appears to be adequate due to the fact that native tissues have similar properties as PUs. However, the influence of chemical structure on the biomechanics of PUs remains poorly described. The use of constitutive models, together with numerical studies, is a powerful tool for evaluating the mechanical behavior of materials under specific physiological conditions. Therefore, the aim of this study was to assess the mechanical properties of different PU mixtures formed by polycaprolactone diol, polyethylene glycol, and pentaerythritol using uniaxial tensile, strain sweep, and multistep creep-recovery tests. Evaluations of the properties were also recorded after samples had been soaked in phosphate-buffer saline (PBS) to simulate physiological conditions. A hyperelastic model based on the Mooney-Rivlin strain density function was employed to model the performance of PUs under physiological pressure and geometry conditions. The results show that the inclusion of polyethylene glycol enhanced viscous flow, while polycaprolactone diol increased the elastic behavior. Furthermore, tensile tests revealed that hydration had an important effect on the softening phenomenon. Additionally, after the hydration of PUs, the ultimate strength was similar to those reported for other vascular conduits. Lastly, hyperelastic models revealed that the compliance of the PUs showed a cyclic behavior within the tested time and pressure conditions and is affected by the material composition. However, the compliance was not affected by the geometry of the materials. These tests demonstrate that the materials whose compositions are 5-90-5 and 46.3-46.3-7.5 could be employed in the designs of vascular grafts for medical applications since they present the largest value of compliance, ultimate strength, and elongation at break in the range of reported blood vessels, thus indicating their suitability. Moreover, the polyurethanes were revealed to undergo softening after hydration, which could reduce the risk of vascular trauma.


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model. This constitutive model formulation was used to simulate the behavior of the PUs over time as a vascular conduit under physiological conditions. As such, the material was analyzed under sinusoidal pressure ranges at different graft radii and thicknesses. The effect of the PU mixture composition was evaluated to broaden the comprehension of the material's mechanics and its performance for application in the early stages of vascular graft design, allowing for the selection of materials with the greatest potential to be used in further studies and designs.
Before continuing, the reader should know that the results presented here are limited by the conditions of the test performed. Only uniaxial load results are presented and the modeling supposes an incompressible material. As mentioned, the ultimate aim was to identify the advantages of each material and select the best ones for further design and numerical analyses.
Pentaerythritol (PE) was obtained from Alfa Aesar (Heysham, UK), and phosphate-buffered saline (PBS) was obtained from VWR (Randor, PA). In this study, PEG and PCL played the role of polyols, while PE and IPDI comprised the hard segment. DMF was used as the reaction solvent.

Synthesis of PUs
PUs were synthesized as previously reported [21]. Briefly, PCL and PEG were dissolved in DMF at 70 °C. IPDI was then added to the polyol blend and allowed to react for 15 min at 70 °C. Next, the second solution of PE in DMF was added, and the solvent was evaporated for at least 5 h. Finally, the solution was poured onto a glass surface, and thin films were made with the help of an Elcometer 3580 casting knife film applicator (Elcometer Ltd., UK) with a gap of 150 µm. The PU was cured for 12 h at 110 °C. Four blends were synthesized with the compositions listed in Table 1. These blends were selected based on results from our previous work showing they had better phase mixing and damping behavior of those tested [21]. For the tensile tests, a rate control program was set up with a strain ramp of 10 mm/min ( Figure   1A). Engineering stress-strain curves were obtained, and the ultimate strength and elongation at break were calculated. Strain sweep tests were performed using an oscillation program at a frequency of 1 Hz and a strain range of 1.0%-50.0%, with 1.0% increments ( Figure 1B) Figure 1C describes the loads applied to each PU.

Hyperelastic modeling
For the hyperelastic model, a strain energy function W(C) of the right Cauchy-Green tensor, defined as C = F T F, was chosen so that a constitutive equation could be found, where F is the deformation gradient tensor. For the present study, the Mooney-Rivlin model was chosen [23][24][25] such that W= f (I1, I2,I3). I1, I2, and I3 are the three principal strain invariants of C and are defined in terms of the stretch ratios in the principal direction as = + + (1) According to Rivlin [26] and considering the conventional assumption of incompressibility for .
Based on the suggestion by Kumar and Rao [28], a three-parameter Mooney-Rivlin function was used. Additionally, tests were performed under uniaxial conditions. Therefore, the following is true: As previously reported, mechanical models can be used to determine the correlation of compliance with uniaxial test results [29]. Therefore, parameters were estimated from the uniaxial tensile test results. An average stress-strain curve of the three tested samples for each PU was calculated, and nonlinear regression was used to calculate the model parameters. The engineering stress (σ Eng ) and strain (ε Eng ) were transformed into the true strain and stress, as follows: = + . (8)

Modeling vascular grafts under physiological conditions
The behaviors of the PU-based vascular grafts were studied under simulated physiological conditions (hydration, temperature, and simulated pressure). A sinusoidal arterial blood pressure model was used [30]: where is the frequency in hertz; t is the time in seconds; ϵ = P s P m ; P s is the amplitude (10 mmHg) of the sinusoidal pressure; and P m is the mean pressure, which was calculated by (10) as shown in previous reports [12]: The maximum (P max ) and minimum (P min ) pressure was 180 and 40 mmHg, respectively, so P m was 86.67 mmHg.
The circumferential stresses (σ θθ ) were estimated based on the thick wall theory for vessels reported elsewhere [12,31].
Here, r 0 is the graft inner radius under undeformed conditions, h is the graft thickness, and P(t) is the pressure as a function of time, as previously described.
To determine the change in radius, σ θθ was used to calculate the stretch from the hyperelastic model, and the radius was calculated as follows: ( ) = ( − ) + . (12) Compliance was calculated as follows:

Statistics and modeling
Mechanical properties from stress-strain curves were analyzed using ANOVA and groups were compared using two different post hoc tests. Sidak's test was used to compare hydration states while Tukey's test was employed to compare compositions. Before ANOVA, a Shapiro-Wilks test was performed to evaluate the normality (alpha = 0.05). The root mean square error (RMSE) and Lin's concordance correlation coefficient (CCC) were used to evaluate the goodness of fit and reliability of the hyperelastic models.
The parameter estimation and mathematic modeling were performed in MATLAB 2019b (MathWorks, MA). The algorithms used in this work are presented in the supplementary material.

Results and Discussion
3.1. Mechanical assessment Figure  2 shows the performances of the PUs in tensile experiments. Figure 2A-D displays the engineering stress and strain curves for each PU composition with their characteristic elastomeric shape. Initially, a small region of elasticity at low deformation was observed, which resulted from hydrogen bonding. Then, the polymer chains uncoiled, producing a moderate stress increase with deformation. Finally, after the strain increased to more than approximately 200%, a "stress-induced crystallization" phenomenon was observed. In that case, the strengthening produced by the chain orientation could ease the formation of new hydrogen bonds [32,33]. This behavior has previously been described for elastomer materials [34][35][36]. In particular,   [43,44]. Such softening, mediated by water absorption, could provide advantages for medical applications [14]. Particularly in the case of cardiovascular devices, it could reduce patient discomfort and the risk of vascular trauma [18]. Strain sweep testing is a tool used to study the viscoelastic behaviors of materials under dynamic conditions [45].  (15) where σ, ε, and δ are the stress, strain, and phase angle, respectively.
In general, the PUs did not reveal linear viscoelastic regions. Only 5-90-5 (see Figure 3A) exhibited a short linear region at strains lower than 3% and 2% for the non-hydrated and hydrated PUs, respectively. Thus, the PUs had a strong stress and strain dependency, revealing a nonlinear behavior.      The information presented at this point describes the mechanical behavior of the polyurethanes and the influence of the composition. Moreover, the effect of water uptake was observed for the same mechanical properties. The water swelling reduced the ultimate strength and the elongation at break, and the viscous flow and permanent deformation were reached at lower forces. This points to the relevance of the study of water in the polymeric matrix, which has been poorly described in the literature for vascular graft applications. Materials employed for vascular grafts are placed in contact with physiological fluid and, therefore, swelling could take place, jeopardizing graft performance.

Hyperelastic modeling
An elastomeric behavior and large extensions were observed in the previously mentioned mechanical assessments of the PUs. The PUs demonstrated a significant strain dependency even at low strain values, and the elastic response and viscous flow were regulated by the PU composition.
In this way, the large extension of rubber-like materials and nonlinear elastic behaviors have been expressed using hyperelastic models [47]. Additionally, the hydration of the PUs produced a softening phenomenon. Hence, hyperelastic modeling and further simulations were performed with the hydrated materials, which had behaviors closer to those of an in vivo application.
In the past, PUs have been considered for vascular graft design [9,48] because of their hyperelastic behavior, which can support repeated stress, similarly as a native blood vessel [49]. In this way, a hyperelastic model based on the Mooney-Rivlin strain energy density function was used to address the stress and strain behaviors.
The coefficients of the average curves were calculated using three samples of each PU. According to Cook et al., biomechanical models constructed with average values do not produce average results [50]. Hyperelastic models based on averages tend to fail when any coefficient underwent nonlinear change [51], which was the case with the Mooney-Rivlin function used in this work.
A physical interpretation of Mooney-Rivlin parameters was provided by Kumar and Rao [28].
Parameter C10, from Eq. (5), can be used to calculate the crosslinked density of the polymers following the next equation: where R is the universal gas constant and T is the temperature.
The crosslinked density (ѵ) is reported in Table 2  values. However, the RMSE is scale-dependent, which made it impossible to compare the PU compositions. Likewise, the CCC was used as a reliability index. The CCC is a modification of the Pearson correlation coefficient and is used to assess either how close the data are to the line for the best fit or how far that line is from the 45° line through the origin. This makes it possible to evaluate the correlation and agreement between methods [52]. Additionally, the CCC is scale-independent.
The models obtained in this work show great goodness of fit, with value of CCC close to one. This indicated that the models are reliable and represent the behaviors of the Pus well. Vascular grafts based on the PUs were studied under simulated physiological mechanical conditions. The frequency, vascular graft inner radius, and thickness were studied in terms of the compliance, and the results are presented in Figure 7. Figure 7 shows that compliance has a cyclic behavior with pressure frequency variation. The PUs show cyclic variations, but with the same minimum and maximum values for each frequency.
These results illustrate the capacity of the PUs to withstand physiological conditions, which is one of the requirements for vascular grafts and is in agreement with the cyclic behavior shown in the work of Valdez-Jasso et al. [53]. Additionally, it shows that for each composition, the radius and thickness of the vascular graft do not affect the compliance behavior. However, there was variation in the compliance values of the graft compositions. These results suggest that for each PU composition, the stiffness of the material does not change under different graft geometries, but can be controlled and modified by adjusting the composition. Regarding this, Stewart and Lyman [6] reported that mismatches in both graft compliance and dimensions with the native vessel must be avoided to prevent any possible adverse effect on the performance.. Therefore, the obtained results indicate that these PUs, which retained their stiffness when their geometry was changed, are potential materials that could be used to address the compliance mismatch problem for vascular vessels with different diameters and thicknesses.  A limitation of the present study was that it did not consider the anisotropy of the vascular vessel, where the heterogeneity of strain affects the stress distribution [54]. This anisotropy is associated with the vessel wall composition, which includes various layers. Recent studies have reported computational models for bilayered grafts with different geometrical constraints and material properties [55]. Characterization of the anisotropic properties of the material requires in situ observations that were beyond the scope of this work, but that could be achieved using the experimental methods mentioned here combined with multiphoton microscopy techniques [56].
Additionally, the second limitation of this paper is the coefficient source. The coefficients presented in this work were only derived from a uniaxial test. Therefore, further studies on different configurations of the mechanical test (e.g., planar and biaxial) should be added to validate the parameters. However, this study aimed to evaluate the hyperelasticity of a group of materials to select the best for vascular graft design. The methods presented in this work can be used to select material for further use in the early stage of vascular graft design.

Conclusions
Mechanical assessments of the synthesized PUs showed a loss of mechanical properties after hydration, as measured in terms of the ultimate strength and elongation at break. However, softening could reduce the risk of vascular trauma and patient discomfort. The values for the ultimate strength and elongation at break were similar to those reported for coronary arteries and saphenous veins.
The non-hydrated and hydrated PUs both showed significant strain dependencies, revealing nonlinear viscoelasticity. Additionally, creep-recovery tests revealed significant elastic behaviors for PUs with large PCL concentrations, while PEG addition enhanced the viscous flow and reduced the elastic performance. Moreover, hydration reduced the elastic region.
The material was modeled using a nonlinear hyperelastic model based on a Mooney-Rivlin strain density function. The RMSE and CCC revealed the good fit and reliability of the experimental data and model. Under simulated physiological conditions, the PUs showed cyclic behaviors, indicating their capacity to sustain physiological conditions. Moreover, the compliance did not change with the radius and thickness, suggesting enhancement of the compliance and vascular geometry mismatch.