Beyond Linear Elastic Modulus: Viscoelastic Models for Brain and

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Beyond Linear Elastic Modulus: Viscoelastic Models for Brain and Brain Mimetic Hydrogels Mark Calhoun, Sarah A. Bentil, Eileen Elliott, Jose J. Otero, Jessica O. Winter, and Rebecca B. Dupaix ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.8b01390 • Publication Date (Web): 23 Apr 2019 Downloaded from http://pubs.acs.org on April 29, 2019

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ACS Biomaterials Science & Engineering

Beyond Linear Elastic Modulus: Viscoelastic Models for Brain and Brain Mimetic Hydrogels Mark A. Calhoun1, Sarah A. Bentil2, Eileen Elliott3, Jose J. Otero4, Jessica O. Winter1,3, Rebecca B. Dupaix5,* 1Department

of Biomedical Engineering, The Ohio State University, Columbus, OH

2Department

3William

of Mechanical Engineering, Iowa State University, Ames, IA

G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, OH

4Department

5Department

of Pathology, College of Medicine, The Ohio State University, Columbus, OH

of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH Correspondence: Prof. Rebecca Dupaix Department of Mechanical and Aerospace Engineering, The Ohio State University, 1 ACS Paragon Plus Environment

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E310 Scott Lab 201 W 19th Ave., Columbus, OH, 43210, USA [email protected]

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Abstract

With their high degree of specificity and investigator control, in vitro disease models provide a natural complement to in vivo models. Especially in organs such as brain, where anatomical limitations make in vivo experiments challenging, in vitro models have been increasingly used to mimic disease pathology. However, brain mimetic models may not fully replicate the mechanical environment in vivo, which has been shown to influence a variety of cell behaviors. Specifically, many disease models consider only the linear elastic modulus of brain, which describes the stiffness of a material with the assumption that mechanical behavior is independent of loading rate. Here, we characterized porcine brain tissue using a modified stress relaxation test, and across a panel of viscoelastic models, showed that stiffness depends on loading rate. As such, the linear elastic modulus does not accurately reflect the viscoelastic properties of native brain. Among viscoelastic models, the Maxwell model was selected for further analysis because of its simplicity and excellent curve fit (R2 = 0.99 ± 0.0006). Thus, mechanical response of native brain and hydrogel mimetic models was analyzed using the Maxwell model and the linear elastic model to evaluate the effects of strain rate, time post mortem, region, tissue type (i.e. bulk brain vs white matter), and, in brain mimetic models, hydrogel composition, on observed mechanical properties. In comparing the Maxwell and linear elastic models, linear elastic modulus is consistently lower than the Maxwell elastic modulus across all brain regions. Additionally, the Maxwell model is sensitive to changes in viscosity and small changes in elasticity, demonstrating improved fidelity. These findings demonstrate the insufficiency of linear elastic modulus as a primary mechanical characterization for brain mimetic materials and provide quantitative



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information toward the future design of materials that more closely mimic mechanical features of brain.

Keywords:

Viscoelasticity,

Stress

Relaxation,

Maxwell

Model,

Brain,

Hydrogels,

Mechanobiology


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1. Introduction The effects of the mechanical microenvironment on cell behavior have been widely studied in a variety of disease models. Cells are thought to probe their mechanical environment by generating traction forces, which in turn generate a conformational change in focal adhesion proteins, initiating signal transduction1-2. Seminal mechanobiology experiments utilizing different stiffness polyacrylamide hydrogels have shown that basic cell functions, such as proliferation3, survival4, differentiation5, and chemokine expression6 are all modulated by mechanics in diverse cell types, including tumor cells, stem cells, and macrophages. These studies, along with many follow-up studies, primarily rely on substrate stiffness, or the linear elastic (LE) modulus, to characterize the mechanical environment. For viscoelastic materials, such as biological tissues and hydrogels mimicking those tissues, mechanical characterization would likely benefit from the inclusion of a viscosity parameter, most often described by a time constant (i.e. tau). The need for a time constant is dictated by the way in which cells interact with their substrate. Cells generate and maintain traction forces with their substrate, with tumor cells generating more traction7-8. This contractility, by which the cell loads, then holds, its substrate, engages the mechanical environment in a manner similar to stress relaxation experiments. Additionally, the rate of loading at these focal adhesions is assumed to be different for faster and slower migrating cells. For example, faster migrating cells would be expected to impart a higher displacement rate on the substrate than slower migrating cells. This displacement rate plays an important role in the mechanical response of viscoelastic tissues. The addition of a time constant would thus permit deeper probing into mechanobiology, with the expectation that elasticity and viscosity likely affect cell phenotype in fundamentally different ways. 5 ACS Paragon Plus Environment


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Further, whereas a myriad of biomolecular assays are available to probe biological response, the mechanical environment is comparably under-characterized. The mechanical environment is often evaluated using models that are either motivated by a mechanical analog, utilizing Hookean springs and Newtonian dashpots, or purely phenomenological. The most common models yield a single characterizing parameter, such as the LE modulus (i.e., Young’s modulus). Young’s modulus is determined from data collected only during the loading phase of a uniaxial test and does not account for changes in mechanical response with time, which may be significant for viscoelastic materials. In practice, this leads to a philosophical difference in analytical approach. Researchers are tempted to employ Young’s modulus, which is simple and easy to use, but which fails to capture a large portion of mechanical data. Mechanical response could likely be better described using multiple parameters, such as stiffnesses and time constants, available in viscoelastic models. Conversely, non-linear viscoelastic models (e.g., quasi-linear viscoelastic model9) have been developed to capture mechanical responses as a function of time. However, these models can be difficult to implement, requiring advanced mathematical understanding. Such models are typically employed by mathematicians and computational modelers; however, they are less commonly employed by experimental tissue engineers. The gap between these approaches necessitates development of mechanical models that can capture time dependent mechanical behavior, thereby increasing fidelity of the mechanical characterization, but which are easily implemented and analytically approachable. Here, we use a porcine brain model to evaluate the fitness of various mechanical models at different strain rates. We employed a modified stress relaxation test with a distinct, noninstantaneous load phase and evaluated the mechanical response with Maxwell (MW), power 6 ACS Paragon Plus Environment

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law (PL), standard linear solid (SLS), and generalized Maxwell (GMW) models and compared these viscoelastic models to the responses given by the linear elastic model. Brain tissue is one of the most mechanically complex tissues in the body, with relatively high deformability and a high ratio of cells to the extracellular matrix10. It is therefore an ideal model against which to test the fidelity of our model. Further, we explore the applicability of these models to brain mimetic hydrogels. Because of the complexity of brain tissue and mechanical similarity, hydrogels are often used to mimic the brain in vitro. These hydrogels are often composed of collagen I/hyaluronic acid (col I/HA) composites11-16 and are typically evaluated using the LE modulus. The inclusion of viscosity effects in mechanical analysis (i.e. time constant), of both biological tissue and hydrogel, provides a foundation for improved and approachable mechanical characterization methods that can be used to enhance understanding of cell response to their substrates.

2. Methods 2.1 Brain Tissue Isolation Brains were removed from 25 freshly slaughtered swine obtained from the Ohio State University Meat Laboratory and a local abattoir. Swine were between the ages of 6-10 months, with an average weight of 250 lb, and fed similar diets. After removal of the entire brain, tissue was harvested by region or by tissue type (i.e., bulk brain or white matter). The latter was supervised by a board-certified neuropathologist. For tissue classified by region, a cylindrical coring tool was used to generate 15 mm diameter specimens of the cerebrum from the approximate locations of the frontal, temporal, and occipital lobes (Figure 1). Tissue samples harvested by region were tested at 6 hr, 24 hr, 3 days, or 1 week following dissection and were 7 ACS Paragon Plus Environment


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stored in 1% physiological saline solution at 4°C until testing. For tissue classified by type, white matter was dissected from bulk brain tissue using a smaller, 6 mm diameter coring tool. Grey matter was not isolated because sample size would be too small for accurate analysis on the testing apparatus employed.

Figure 1. Regions of brain tissue isolation. Porcine brain tissue samples were harvested from the occipital (A), temporal (B), and frontal (C) lobes. Reproduced with permission from ref 17. Copyright 2013. Sarah A Bentil. 2.2 Hydrogel Synthesis Hydrogels were synthesized in cell culture media using varying concentrations of thiolmodified hyaluronic acid (Glycosil, ESI Bio) (3.7-4.3 mg/ml), thiol-reactive polyethylene diacrylate (PEGDA) crosslinker (Extralink, ESI Bio) (0.9-3.7 mg/ml), and either thiol-modified gelatin (Gelin-S, ESI Bio) (3.7 mg/ml) or unmodified collagen I (PureCol, Sigma) (4.3-5.3


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mg/ml). Hydrogels were allowed to gel overnight; then, were punched with a 10 mm arch punch in preparation for mechanical testing at a final height of 3.01 ± 0.55 mm.

2.3 Tissue Mechanical Characterization After isolation, samples were subjected to parallel plate unconfined compression testing (RSA III, New Castle, DE). The load cell capacity for the RSA III is 350 gf. Samples were compressed at a displacement rate of 1.0 mm/min and 5.0 mm/min at room temperature. The stress-free contact point was determined visually by contact with the top surface and numerically by a nonzero force readout on the RSA III. Samples were subjected to a two phase deformation, in which each was compressed to ~10% strain, then held at that strain for 15 s, or 10 s for the hydrogel samples.

2.4 Curve Fitting Force, displacement, and time data were obtained for each two-phase deformation. Crosssectional area and gauge length were used with these data to generate stress-strain and stresstime curves. The slope of the regression line over the compression portion of each stress-strain curve was used to determine the LE modulus. Parameters for the Maxwell model were generated using the ‘fminsearch’ function in the commercially available software Matlab, which is based on the simplex search method of Lagarias et al.18. Parameters were optimized using the inverse of R2 with four different initial guesses to reduce the possibility of finding local minima. Three of the 234 samples were removed from analysis because the parameters did not converge to a minimum.



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2.5 Statistical Analysis For each set of experiments, a hierarchical statistical model was used to test factors and corresponding responses in JMP statistical software package. A multivariate ANOVA (MANOVA) was initially used to test for significant effects at α = 0.05 for all parameters simultaneously. Then, post-hoc ANOVA tests were run on each response demonstrating significant effects in the MANOVA. Non-significant effects were removed from the model for each ANOVA. A Bonferroni correction was applied to the alpha level of the individual AVOVA tests according to Equation 1 to keep the total alpha level of the hierarchical model at α = 0.05. Comparisons between the LE and MW models had three responses (i.e., α = 0.017 for each of the three ANOVA tests). Comparisons between all models employed had thirty responses ((i.e., α = 0.0017 for each of the thirty ANOVA tests). Finally, Tukey’s HSD T-tests or Student’s T-tests, where appropriate, were run to test significance at individual levels for each factor. To compare differences in responses for each sample, a paired t-test was used at α = 0.05. All data is displayed as mean ± standard error of the mean.

𝛼𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 =

𝛼𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙

(1)

𝑚

3. Linear Viscoelasticity Theory Several experimental techniques are commonly used to quantify viscoelasticity, including stress relaxation tests, creep tests, and dynamic/cyclic loading tests. Cyclic loading tests are fundamentally different and have their own set of caveats. Here, we employ a modified stress relaxation experiment with a distinct, non-instantaneous loading phase followed by a hold phase. Theoretically, this testing protocol is reminiscent of how a cell generates traction forces and 10 ACS Paragon Plus Environment

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maintains contractility on its substrate19. In a true stress relaxation experiment, the loading is applied instantaneously, and the relaxation modulus of a given mechanical model is fit to the stress response. The quality of this curve fit ideally identifies which viscoelastic model is most appropriate to capture the behavior of the material. Examples of the relaxation modulus, E(t), for selected models are given in Table 1.

Table 1. Selected Mechanical Models of Viscoelasticity. Model

Schematic

𝐸(𝑡)

Simple Power Law

Empirical

𝐸(𝑡) = 𝐸𝑡𝑛

Maxwell 𝐸(𝑡) = 𝐸𝑚𝑒

Standard Linear Solid (Maxwell Form)

Material Properties E, n

𝑡 𝜏𝑚

𝐸(𝑡) = 𝐸1 + 𝐸2𝑒

𝐸𝑚, 𝜏𝑚 (or 𝜂𝑚 = 𝐸𝑚 ∗ 𝜏 𝑚) 𝑡 𝜏

𝐸1, 𝐸2, 𝜏 (𝜂 = 𝐸2 ∗ 𝜏)

𝑡 𝜏1

𝑡 𝐸𝑒, 𝐸𝑡1,𝜏1, 𝜏2 𝜏𝑛 𝐸(𝑡) = 𝐸𝑒 + 𝐸1𝑒 + 𝐸2𝑒 + … + 𝐸𝑛𝑒 𝐸2,𝜏2,…,

Generalized Maxwell

𝐸𝑛, 𝜏𝑛



The relatively slow loading in our testing protocol obliges a nontrivial amount of relaxation during the loading phase. To account for this relaxation, we fit the mechanical model to the entire load-hold stress response for a corresponding ramp-hold strain input (Supplementary Figure 1). The mathematical representation of this strain input is given by Equation 2.

{

𝜀𝑡 , 0 < 𝑡 < 𝑡𝑜 𝑡 ≥ 𝑡𝑜

𝜀(𝑡) = 𝜀 , 𝑜

(2)

where ε is strain, t is time, εo is the strain at the beginning of the hold phase, to is the time at εo, 𝜀 is the strain rate (i.e. εo/to) and T is the integration variable. The constitutive relationship for a linear viscoelastic material can be rewritten to include the relaxation modulus, E(t), as in Equation 3. 𝑑𝜀

𝑡

𝜎 = ∫0𝐸(𝑡 ― 𝑇)𝑑𝑇𝑑𝑇

(3)

By inserting the derivative of Equation 2 into Equation 3, the general piecewise stress response for a given relaxation modulus can be obtained, as in Equation 4. The final equations for each model used for curve fitting are provided in Supplementary Table 1. 𝜎(𝑡) =

{

𝑡

𝜀∫0𝐸(𝑡 ― 𝑇)𝑑𝑇, 𝑡 < 𝑡𝑜 𝑡

𝜀∫0𝑜𝐸(𝑡 ― 𝑇)𝑑𝑇, 𝑡 ≥ 𝑡𝑜

(4)

4. Results 4.1 Mechanical Models of Viscoelasticity Evaluated Using a Representative Bulk Brain Sample The mechanical response of biological tissues can be modeled in a variety of ways. Most often, an LE or linear viscoelastic model is employed, though more complicated models exist that 12 ACS Paragon Plus Environment

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represent stress relaxation using a spectrum of relaxation times or that invoke nonlinear elasticity. As an example of standard testing methodologies, we applied the LE model to bulk brain samples. The LE model is based on Hooke’s law, utilizes the load phase of a mechanical test, and applies a linear fit to the stress-strain curve (Figure 2A). In biomaterials literature, this slope is most often referred to as Young’s modulus, elastic modulus, or substrate stiffness, and is referred to herein as the LE modulus. Alternatively, linear viscoelastic models can be fit to the entire load-hold mechanical response, compensating for the effects of stress relaxation that occurs during the loading phase (Figure 2B). This is done using Equation 4 such that the constants for each model are based on both the loading and stress relaxation phases. As a comparison to the LE model, we employed Maxwell (MW), Power Law (PL), Standard Linear Solid (SLS), and a two-element Generalized Maxwell (GMW) model of linear viscoelasticity. These common models (Table 1) are based on series and parallel arrangements of Hookean springs and Newtonian dashpots (e.g., Maxwell, SLS, GMW, etc.) or empirical fits of the data (i.e., Power law). Each of these viscoelastic models has an excellent curve fit (R2 > 0.99) for the representative bulk brain sample shown, with the number of parameters required for the fit ranging from two to five. The difference between observed and theoretical values are shown in residual plots for each curve fit (Figure 2C). These show that each fit consistently initially over-predicts and then underpredicts the load phase, represented by negative and positive residuals, respectively. Also, the MW, SLS, and GMW consistently over-predict and then under-predict the hold phase. In contrast, the PL model generally under-predicts and then over-predicts the hold phase. However, when each residual is expressed as a percent of its predicted value (Figure 2D), it is clear that the deviations during the hold phase are relatively small (< ~7%) for each model. 13 ACS Paragon Plus Environment


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Thus, each of the linear viscoelastic models was able to fit a representative bulk brain sample comparably. Interestingly, even those with only two parameters, i.e., MW and PL models, were as effective as the GMW model that has 5 elements. In comparing the former two models, the MW model has the benefits of being simple and of having parameters that relate to material properties (i.e. elasticity, viscosity). For these reasons, the MW model was selected for continued study and subjected to further curve fit analysis.

Figure 2. Linear elastic and linear viscoelastic models applied to a representative bulk brain specimen tested at a strain rate of 40.2%/min. The linear elastic model (A) and Maxwell, Power Law, Standard Linear Solid, and two-element Generalized Maxwell models (B) are fit to the stress response. Residuals (observed value minus theoretical value) in the form of Pascals (C) and percent of theoretical value (D) are plotted against time. 14 ACS Paragon Plus Environment

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4.2 Comparison of Elasticity in the Linear Elastic and Maxwell Models Next, we compared the mechanical elasticity values derived from the MW model to those obtained from the more commonly used LE model. The LE modulus (slope of the loading portion of the stress-strain curve) is influenced by the combined viscous and elastic material response during the load phase. In contrast, the MW modulus is a mathematical representation of the elastic response during loading and holding that has been decoupled from the viscosity. Predicted stress-time curves are presented for three loading scenarios (instantaneous loading, 1 mm/min, and 5 mm/min) using the average MW parameters from data at 6 hr (Figure 3A,B). The peak stress values show a large drop in magnitude with loading rate because of the influence of viscosity. Only the peak stress at instantaneous loading is a true representation of the elasticity and is directly related to the MW modulus. We obtain this MW modulus by fitting the MW model to the entire load-hold curve from test data at 1 mm/min or 5 mm/min. As can be seen, an LE modulus taken from either of the non-instantaneous loading rates will underestimate the elasticity because the peak stress values are lower than would be seen in an instantaneous loading scenario. Indeed on average, these data show that the MW modulus is greater than the LE modulus by 392.38 ± 17.03 Pa, which is a ~20% difference, across a range of samples (p

Beyond Linear Elastic Modulus: Viscoelastic Models for Brain and

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