Energy dissipation in quasi-linear viscoelastic tissues, cells, and extracellular matrix

Abstract

Characterizing how a tissue’s constituents give rise to its viscoelasticity is important for uncovering how hidden timescales underlie multiscale biomechanics. These constituents are viscoelastic in nature, and their mechanics must typically be assessed from the uniaxial behavior of a tissue. Confounding the challenge is that tissue viscoelasticity is typically associated with nonlinear elastic responses. Here, we experimentally assessed how fibroblasts and extracellular matrix (ECM) within engineered tissue constructs give rise to the nonlinear viscoelastic responses of a tissue. We applied a constant strain rate, “triangular-wave, ” loading, and interpreted responses using the Fung quasi-linear viscoelastic (QLV) material model. Although the Fung QLV model has several well-known weaknesses, it was well suited to the behaviors of the tissue constructs, cells, and ECM tested. Cells showed relatively high damping over certain loading frequency ranges. Analysis revealed that, even in cases where the Fung QLV model provided an excellent fit to data, the the time constant derived from the model was not in general a material parameter. Results have implications for design of protocols for the mechanical characterization of biological materials, and for the mechanobiology of cells within viscoelastic tissues.

1. Introduction

Connective tissue structures throughout the body are constantly under dynamic loading (Butler et al., 2000; Ditsios et al., 2002; Kim et al., 2008). The responses of these tissues to dynamic loads are therefore central to their structural function, and data are needed to fit and validate the many multiscale models of how these responses arise from those of a tissue’s constituents (Genin et al., 2017). The broad literature of techniques for characterizing viscoelastic behavior of tissues focuses on uniaxial tests such as creep or relaxation testing, due in part to the aligned, fibrous structure common amongst tissues (Purslow et al., 1998; Jamison et al., 1968; Hudnut et al., 2017; Castile et al., 2016). Because nonlinearity can arise from both time-independent and time-dependent sources, isolating strain-rate dependence of viscoelasticity is challenging (Sakamoto et al., 2017; Lokshin & Lanir, 2009; Li et al., 2017).

Here, we developed and applied the quasi-linear viscoelastic (QLV) framework for a material subjected to a triangular-wave strain loading pulse (Figure 1) to study the damping of a collagenous tissue construct. The Fung QLV material model is known to be highly limited. Although available data indicate that the Fung QLV model’s separation of nonlinear elastic effects from linear, strain independent relaxation effects is often acceptable over a certain range of conditions (Babaei et al., 2015a), no known material follows Fung QLV behavior faithfully across all loadings (Lakes & Vanderby, 1999; Provenzano et al., 2001, 2002; Ciarletta et al., 2006; Duenwald et al., 2010). Many fixes and extensions of the Fung QLV model exist (Provenzano et al., 2001; Pryse et al., 2003; Nekouzadeh et al., 2007). However, the Fung QLV model is nearly a standard starting point for the analysis of nonlinear biological materials (Zou & Zhang, 2011; Thomopoulos & Genin, 2012; Giles et al., 2007; Bischoff, 2006; Doehring et al., 2004; Kwan et al., 1993; Carew et al., 1999; Sverdlik & Lanir, 2002). We therefore focused the treatment of the subject on the Fung QLV model itself.

Figure 1.

Annular engineered tissue constructs (ETCs) were stretched in a custom stretching apparatus using a triangular-wave (sawtooth) strain profile. Portions of this were figure reproduced with permission from Elson & Genin (2016).

Although a very general framework exists for modeling nonlinear viscoelastic behavior (Coleman & Noll, 1961), a general framework for specific biological tissues remains elusive. Fung’s QLV model is attractive as a starting point because it identifies a class of quasi-linearity that is appropriate for many biological tissues (Nekouzadeh et al., 2007). When applying it, one must bear in mind the restrictions it imposes due to the “box-shaped” temporal relaxation function that Fung proposed in his text (Fung, 2013). The width of this spectrum typically cannot be fit with confidence, which complicates the comparison of materials (Babaei et al., 2015a).

To determine whether the Fung QLV is a reasonable model, a spectral approach can be used such as the discrete Fung QLV (DQLV) approach of Babaei et al. (2015a). This identifies ranges of discrete time constants over which the Fung QLV model is a reasonable approximation to the material response. Identifying when the box-shaped relaxation function is inadequate poses difficulty (Thomopoulos et al., 2003; Sauren & Rousseau, 1983) because, with this box spectrum, the Fung QLV model can fit relaxation data for materials whose responses to dynamic loading it would fail to predict (Iatridis et al., 1997; Anderson et al., 1991). However, in the current study, we observed that for each triangle loading cycle, only one time constant and one corresponding damping coefficient was expressed strongly in the specimens, reducing the modeling complexity required for interpreting the data. The current testing protocol enabled samples to relax adequately between loading cycles. Both the Fung QLV and DQLV models were able to fit the data properly, and we therefore focussed on the former, and on a range of loading rates that are relevant to slower physiological processes.

Theories describing instantaneous stress response and energy dissipation of a standard linear viscoelastic solid subject to such loadings are well established, although the definitive works of Tschoegl (1981) and Yang & Chen (1982) are surprisingly recent. The application of this and other methods of estimating hysteresis in biological structures is widespread (Nyman et al., 2007; Koolstra et al., 2007; LaCroix et al., 2013). However, even in the linear sense, the sensitivity of hysteretic energy dissipation for biological structures across the spectrum of loading periods has not been explored in detail. Given the time constants of connective tissues, knowledge of dissipated energy as a function of loading time can provide a better understanding of, for example, the loading conditions at which injury resistance might be optimal, or those at which cells might be more sensitive to mechanical stimuli. We and others have observed that hysteresis observed in loading/unloading curves measured on engineered tissue constructs is not well represented in terms of linear viscoelasticity (Nekouzadeh et al., 2007).

From a clinical perspective, models of connective tissue-based support structures are needed to understand physiological degeneration, especially at the slow loading rates associated with viscoelastic relaxation responses of tissues (Karamanidis & Arampatzis, 2005; Hajdu et al., 1990; Rotsch & Radmacher, 2000). In tissue engineering, such models are needed to engineer tissues that mimic structures within the body (Vunjak-Novakovic et al., 1999). Mechanical standards to evaluate the performance of artificially grown structures prior to and after implantation represent a longstanding need (Butler et al., 2000), especially given the range of work focused on altering the viscoelastic properties of engineered tissue structures by manipulating scaffolding material, varying matrix protein compositions, and external static and dynamic stresses on tissues during formation (Feng et al., 2006; Courtney et al., 2006; von der Mark et al., 2010; Thomopoulos et al., 2011; Huang et al., 2017; Yeh et al., 2017; Li & Zhang, 2014). However, assessing and predicting how these structures will perform requires further study of the dynamic responses to different loading patterns and how this behavior changes as a function of the loading period.

From the standpoint of cellular mechanobiology, key questions are the degree to which cells and extracellular matrix (ECM) contribute to the overall mechanical function of a tissue, including its hysteretic behavior, and how mechanical signals travel through ECM to reach cells (Shakiba et al., 2017). Estimates of the viscoelastic responses of cells have been made in using artificial tissue constructs, but these have been limited to linear viscoelasticity (Babaei et al., 2016). In this work, we extended these results to distinguish the Fung QLV behaviors of cells and ECMs in such constructs.

The work focused on triangular-wave stretching of tissue constructs, rather than frequently-used sinusoidal stretching, for two reasons. First, whereas for a sinusoidal excitation, the nominal strain rate alters sinusoidally over each loading cycle, the magnitude of the nominal strain rate is constant for a triangular-wave excitation. Thus, to study the effect of the strain rate on the dynamic responses of a biological material, a triangular-wave excitation is a better choice. Second, because biological materials typically behave non-linearly (Storm et al., 2005; Dhume & Barocas, 2017), these varying strain rates must be convolved into model fitting for something other than infinitesimal straining. Although these nonlinear responses can be estimated by a test protocol with a series of infinitesimal relaxation increments (Pryse et al., 2003), we show that the triangular waveform has certain advantages. Therefore, we offered a mathematical approach for modeling the viscoelastic response to triangular-wave excitation, which allows practical experimentation under a constant strain rate. We then applied this model to simulate and interpret the behavior of engineered tissue constructs (ETCs), including the dependence of energy dissipation on loading rate in ETCs, remodeled collagen ECMs, and fibroblasts.

2. Theory

We studied the response of a Fung QLV material to a single triangular-wave stretching (Fig. 1). An assumption was that the material constants do not change substantially from one loading cycle to the next. Although this is never fully the case for a biological tissue, it can be achieved to a reasonable approximation for the tissues of interest through application of the appropriate preconditioning protocols (Wagenseil et al., 2003; Cohen et al., 2008; Marquez et al., 2006b). In this section, we begin by reviewing linear elastic, linear viscoelastic, and quasi-linear material responses, then apply these to develop expressions for the damping of these materials when subjected to a triangular-wave loading pulse.

2.1. Background: hereditary viscoelastic materials

The response of a hereditary or non-aging viscoelastic material to a one-dimensional straining can be represented by the Stieltjes convolution integral (Stieltjes, 1894; Lockett, 1972):

$$\sigma (t)={\int }_0^t\varphi (t-u)\frac{\partial \epsilon }{\partial u}\mathit{\text{du}}$$ (2.1)

where σ(t) is engineering stress, t is time, ε is linearized strain, u is a dummy parameter, and ϕ(t) is a material modulus function. ϕ(t) can represent a linear elastic solid with the choice ϕ = E₀, in which E₀ is the solid’s elastic modulus; it can represent a Newtonian fluid with the choice ϕ(t) = ηδ(t), where δ(t) is Dirac’s delta function and η is the fluid’s viscosity; and it can represent a viscoelastic material possessing both elastic and viscous properties with ϕ(t) chosen as a generalized function that captures the whole spectrum of material behaviors. A common form for representing this is the generalized Maxwell (Maxwell-Weichert) linear viscoelastic material, comprised of a linear spring in parallel with n Maxwell elements. Each Maxwell element consists of a linear spring of modulus E_i in series with a linear dashpot of viscosity η_i, so that (Wang et al., 2018; Wineman, 2009):

$$\varphi (t)=E_0+\sum _{i=1}^n{E}_i{\mathrm{e}}^{(-t/{\tau }_i)},$$ (2.2)

where τ_i = E_i/η_i, the relaxation time constant of the i^th Maxwell element, governs the rate of viscoelastic relaxation in that element.

Applying linear elastic and linear viscoelastic (Maxwell) models to our experimental data shows their inadequacy (supplemental material) and motivates the application of the Fung QLV model. Whereas the stress-strain behavior of the tissues we studied showed a transition from a concave down to a concave up shape, linear viscoelasticity could predict only a concave-down response.

In the Fung QLV theory of Fung et al. (1972), a nonlinear elastic term is combined with a linear, viscoelastic relaxation function. Fung treated the case of a material that is nonlinear due only to its elasticity, with this nonlinearity completely decoupled from the (normalized) temporal relaxation of the material. He additionally specialized the relaxation to be represented by a specific spectral function, so that:

$$\varphi (t;\epsilon (t))=\mathit{\text{AB }}{\mathrm{e}}^{B\epsilon (t)}\frac{1+H{\mathrm{e}}^{(-t/\tau )}}{1+H},$$ (2.3)

where A and B represent the magnitude and non-linearity of the nonlinear elastic stress response, respectively; H is a dimensionless damping coefficient; and τ is the time constant. Although more advanced spectral representations of ϕ(t; ε(t)) show that equation (2.3) is a poor approximation for the entire spectrum of a tissue’s responses, it is a reasonable approximation for many special cases of loading (Babaei et al., 2015a).

Although assigning physical, microstructural meaning to the parameters of the phenomenological Fung QLV model is notoriously difficult (Sopakayang & De Vita, 2011), the current analysis in terms of energy dissipation provides insight, despite the fact that the Fung QLV model has limitations including poor ability to model the experimentally observed dependence of damping on strain level. The coefficient A, which scales the tangent elastic modulus at a given strain level, also scales the energy dissipation linearly. A general trend, observed commonly both in the current results and in viscoelastic materials in general (Stewart et al., 2016; Perepelyuk et al., 2016; Hudnut et al., 2017), is that the elastic modulus decreases with decreasing strain rate (or, here, as the loading period increased). The coefficient B represents non-linearity within the system, which did not vary with loading rate. In the context of energy dissipation, B affects material behavior predominantly by being the factor that scales the effect of strain rate on material response. Additionally, these effects are tempered by an inverse scaling: B attenuates the effect of A on the rate-independent portion of the energy dissipation. The dimensionless parameter H has the physical meaning of an amplitude of relaxation. As is evident from equation (2.3), H = 0 corresponds to an elastic material, and H = ∞ corresponds to a Fung QLV version of a viscoelastic fluid. From the perspective of energy dissipation, H serves many roles, including to act with B to attenuate the effect of elastic modulus on dissipation. The time constant, τ, determines the time scales of relaxation. As will be shown, the loading rate dependence of a Fung QLV material’s energy dissipation nearly collapses to a single curve when T is normalized by τ.

The fitting algorithm was implemented in uncompiled Matlab (The Mathworks, Natick, MA). The Fminsearchbnd function was used to find the minimum of the constrained multivariable function using derivative-free approach. Bounds were applied internally, and initial guesses were needed. In this study, following boundaries were set for fitting: 0 ≤ A ≤ 1000 Pa, 0 ≤ B ≤ 100, 0 ≤ H ≤ 100, and 0 ≤ τ ≤ 100 s. The uniqueness of the four estimated material constants was validated across different iterations and initial guesses.

2.2. Response to a triangular-wave (sawtooth) strain history

A complete mathematical linear viscoelastic theory for stress responses and energy dissipation during a triangular-wave strain loading (Fig. 1) was not established until the 1980’s in the work of Yang & Chen (1982). A great many extensions and applications of that work exist, but the original paper by Yang & Chen (1982) lays the foundation for our extension to the analysis of a Fung QLV solid. The loading studied is a linear ramp stretching over the time interval 0 ≤ t ≤ T/2, followed immediately by an unloading ramp to the specimen’s reference configuration over the time interval T/2 ≤ t ≤ T, so that:

$$\epsilon (t)=\{\begin{array}{cc}{\dot{\epsilon }}_{0}t=2{\epsilon }_0\frac{t}{T},\hfill & 0\le t\le \frac{T}{2}\hfill \\ {\dot{\epsilon }}_0(T-t)=2{\epsilon }_0(1-\frac{t}{T}),\hfill & \frac{T}{2}\le t\le T\hfill \end{array}$$ (2.4)

The stress response to this loading can be found from equation (2.1). For a generalized Maxwell material, in the situation in which the viscoelastic relaxation is sufficiently fast that the specimen does not become slack, this response is:

$$\sigma (t)=\{\begin{array}{cc}{E}_0{\dot{\epsilon }}_{0}t+\sum _{i=1}^n{E}_i{\tau }_i{\dot{\epsilon }}_0(1-\text{exp}(-t/{\tau }_i)),\hfill & 0\le t\le \frac{T}{2}\hfill \\ E_0{\dot{\epsilon }}_0(T-t)+\sum _{i=1}^n{E}_i{\tau }_i{\dot{\epsilon }}_0\left(2\text{ exp }\right(\frac{T-2t}{2{\tau }_i})-1-\text{exp }(-\frac{t}{{\tau }_i}\left)\right),\hfill & \frac{T}{2}\le t\le T\hfill \end{array}$$ (2.5)

where the linear elastic response is recovered for cases in which E_i = 0 for i > 0, and for cases where the τ_i are all so large that over the measured time range the terms in the summation vanish. For the quasi-linear viscoelastic loading cycle, the stress response in the loading phase (0 ≤ t ≤ T/2) is:

$$\sigma (t)=\frac{A(e^{\mathit{\text{Bt}}{\dot{\epsilon }}_0}(1+B(1+H){\dot{\epsilon }}_0\tau )-1-B{\dot{\epsilon }}_0\tau (1+H{\mathrm{e}}^{-\frac{t}{\tau }}\left)\right)}{(1+H)(1+B{\dot{\epsilon }}_0\tau )}$$ (2.6)

and that for the unloading phase (T/2 ≤ t ≤ T) is:

$$\sigma (t)=\frac{A{\mathrm{e}}^{-t(B{\dot{\epsilon }}_0+\frac{1}{\tau })}}{(1+H)(-1+B{\dot{\epsilon }}_0\tau )(1+B{\dot{\epsilon }}_0\tau )}(-2B{\mathrm{e}}^{\frac{1}{2}(B(2t+T){\dot{\epsilon }}_0+\frac{T}{\tau })}H{\dot{\epsilon }}_0\tau -B{\mathrm{e}}^{\mathit{\text{Bt}}{\dot{\epsilon }}_0}H{\dot{\epsilon }}_0\tau (-1+B{\dot{\epsilon }}_0\tau )+{\mathrm{e}}^{\mathit{\text{BT}}{\dot{\epsilon }}_0+\frac{t}{\tau }}(1+B{\dot{\epsilon }}_0\tau )(-1+B(1+H){\dot{\epsilon }}_0\tau )+{\mathrm{e}}^{t(B{\dot{\epsilon }}_0+\frac{1}{\tau })}(1-B^2{\dot{\epsilon }}_0^2{\tau }^2))$$ (2.7)

2.3. Energy dissipation

The energy dissipation per unit volume associated with a loading cycle is:

$$W_{\mathit{\text{diss}}}=\oint \sigma (\epsilon ;t)d\epsilon =\oint \sigma (t)\dot{\epsilon }(t)\mathit{\text{dt}}$$ (2.8)

For the loading phase of the cycle:

$$W_{\mathit{\text{load}}}={\int }_0^{T/2}\sigma (t)(+{\dot{\epsilon }}_0)\mathit{\text{ dt}},$$ (2.9)

and that associated with the unloading phase is:

$$W_{\mathit{\text{unload}}}={\int }_{T/2}^T\sigma (t)(-{\dot{\epsilon }}_0)\mathit{\text{ dt}}.$$ (2.10)

The energy dissipated in a full loading cycle is therefore:

$$W_{\mathit{\text{diss}}}=|W_{\mathit{\text{load}}}|-|W_{\mathit{\text{unload}}}|.$$ (2.11)

For a generalized Maxwell linear viscoelastic material, these expressions can be written in a simple closed form:

$$W_{\mathit{\text{diss}}}=|{\int }_0^{\frac{T}{2}}{\sigma }_{\mathit{\text{load}}}(t){\epsilon }_{\mathit{\text{load}}}(t)\mathit{\text{ dt}}|-|{\int }_{\frac{T}{2}}^T{\sigma }_{\mathit{\text{unload}}}(t){\epsilon }_{\mathit{\text{unload}}}(t)\mathit{\text{ dt}}|$$ (2.12)

$$=\sum _{i=1}^N{E}_i{\dot{\epsilon }}_0^2{\tau }_i^2(-3-{\mathrm{e}}^{-\frac{T}{{\tau }_i}}+4{\mathrm{e}}^{-\frac{T}{2{\tau }_i}}+\frac{T}{{\tau }_i})$$ (2.13)

For the Fung QLV model, the expression is too lengthy to transcribe, but scales as:

$$W_{\mathit{\text{diss}}}=\frac{A}{2B(1+H)}f(B{\dot{\epsilon }}_0,\frac{T}{\tau },H,{\epsilon }_0)$$ (2.14)

Two features of the equations offer insight into the responses of viscoelastic materials to triangular-wave displacement pulses. The first is that both Generalized Maxwell and Fung QLV materials dissipate energy according to a Deborah number that compares the loading period T to the time constant(s) τ. Note that in equation (2.14), the quantity ε̇₀T is the peak strain ε₀, and the quantity ε̇₀τ = 2ε₀τ/T is proportional to the Deborah number τ/T. The second is that, in both equations (2.14) and (2.13), energy dissipation scales linearly with the elastic coefficients E_i or A. Expressions can be written for multiple, consecutive loading cycles, but these offer little additional insight into these responses.

3. Experimental methods

3.1. Engineered tissue construct (ETC) preparation

ETCs were synthesized using procedures previously described in detail Babaei et al. (2016). Briefly, human dermal fibroblasts (Lonza, Allendale, NJ, USA) were cultured in Dulbecco’s modified Eagle’s medium (DMEM, Gibco) at 37°C and 5% CO₂. Media were changed every 3–4 days and the cells were split when the cell confluence reached > 80% of the dish surface. The cells were used for culturing ETCs at the 7th–10th passage. 0.5 million cells were mixed with DMEM and type I rat tail collagen (Cultrex, USA) at a concentration of 0.5 mg/ml. The starting volume of each specimen was 1 ml. The pH of this mixture was brought to neutrality using 0.1 N NaOH. 0.5 ml of this mixture was poured into hollow, cylindrical Teon molds; the molds contained a central rod to create an annular well with outer and inner diameters of 14.9 mm and 9.5 mm, respectively (Figure 1). The final mixture was incubated at 37°C with 5% CO₂ for 30 minutes to allow the collagen to polymerize. Then, the molds were filled with DMEM supplemented with 5% fetal bovine serum (FBS) and were kept in an incubator for 72 hours to allow the cells to remodel the collagen. Three specimens were prepared for the experiments.

3.2. Saw-tooth testing apparatus and protocol

Saw-tooth tests were performed on three ring-shaped ETCs. The ETCs were mounted within glass organ baths filled with HEPES-buffered DMEM (pH 7.4) and 5% FBS, both kept at 37°C. One end of the ETC was attached to a force transducer, and the other was attached to an actuator connected to a stepper motor, as described elsewhere (Wakatsuki et al., 2000a) (Figure 1). Tissues were allowed to accommodate to the new media for 30 minutes before the test, during which time the fibroblasts contracted, causing pre-stress in the ETCs. The protocol started with 15 minutes of force monitoring to establish a baseline. This monitoring was followed by tissue preconditioning, including 5 cycles of ε = 4% strain and a 1 hour recovery interval; this protocol was chosen because it enables similar ETCs to reach a state at which subsequent stretching has little effect on their mechanical responses. Then, the rings were stretched to ε = 4% in 3 constant-rate, triangular-wave loading cycles over periods of T=2 s, 20 s or 200 s, with each of the stretches followed by a one hour recovery interval (Figure 2). Specimens were returned to their approximately 15 mm initial reference lengths at the same strain rate according to the triangular-wave excitation profile. Force data were recorded at 50 Hz. Inertial effects can skew results testing of viscoelastic and nonlinear media, introducing both wave motion and strain localization (Nekouzadeh et al., 2005; Massouros et al., 2014). For this reason, all loading rates were kept well below the wavespeed in collagen.

Figure 2.

Schematic of the triangular-wave displacement profile applied to engineered tissue constructs (ETCs). The protocol began with five triangular-wave preconditioning pulses with a period of 5s, followed by a one hour recovery interval. Then, single triangular-wave pulses were applied over periods of 2 s, 20s, and 200 s, each separated by one hour rest intervals.

3.3. Deoxycholate treatment

Following the above testing protocol, the specimens were returned to their baseline configurations and allowed to recover for 30 minutes to prepare for testing the contribution of the remodeled ECM to viscoelastic behavior of the ring constructs. For this purpose, DMEM+HEPES was replaced with 0.05% w/v deoxycholate in PBS (pH 7.4), and the ETCs were allowed to incubate in the new medium for 1 hour. The triangular-wave protocol was then repeated. Deoxycholate was chosen over inhibitors such as cytochalasin D and latrunculin because it both lyses cells and leaves the mechanics of the remaining porous ECM unaltered (Marquez et al., 2006b).

3.4. Measurement of ETC dimensions

After the end of each experiment, specimens were mounted on spacers and stretched to their reference length, then fixed in 4% formaldehyde for 20 minutes at room temperature. Afterwards, specimens were cut into two equal pieces and placed within four-well plates filled with PBS. The width and thickness of each tissue were measured using Confocal microscopy (LSM 510, Zeiss). The thickness was measured near the upper and lower borders as well as in the middle of the tissue. The cross sectional area of the tissue was calculated as the mean of the three measurements. All measurements were conducted by the same person. Stress data were inferred from the recorded force data divided by the cross-sectional areas measured for specimens.

3.5. Application of the model to ETCs

The linear elastic, linear viscoelastic and quasi-linear viscoelastic models were fitted to the experimentally collected stress response data of identically made engineered tissue constructs (ETCs) under triangular-wave strain loading. Material constants were estimated by minimizing the error between the data and the predicted model fit. Fung QLV fitting was performed numerically using standard procedures in Matlab (The Mathworks, Natick, MA). By simulating sample loadings at different loading frequencies, we estimated the material constants (A, B, H, and τ).

3.6. Statistics

Comparisons amongst parameters for cells, ECM, and ETCs and amongst responses at different loading periods T were made using paired Student’s T-tests, with Bonferroni correction. The level of significance for elimination of false detection was set at p < 0.05.

4. Results

4.1. Force-displacement curves can be partitioned into contributions from cells and ECM through biochemical inhibition

When ETCs were treated with deoxycholate to lyse cells and then subjected to a triangular-wave stretching pulse, the behavior of the remaining ECM differed substantially from the behavior of an intact ETC (Figure 3). This was the case for all three loading rates considered. Because ε̇₀ = 2ε₀/T for the loading and ε̇₀ = −2ε₀/T for the unloading phase, the corresponding strain rates for T = 2 s, 20 s, and 200 s were |ε̇₀|=1 s⁻¹, 0.1 s⁻¹, and 0.01 s⁻¹, respectively. The loading response of the ECM was less stiff than that of the ETC, and was concave-up, indicating nonlinear behavior. The peak stress was lower for ETCs treated with deoxycholate than for untreated ETCs. The baseline stress was eliminated by deoxycholate treatment, ndicating that its presence requires intact cells.

Figure 3.

The stress-strain responses for ETCs could all be fit by the Fung QLV model, as could the responses of the deoxycholate-treated ETCs (ECM), and the difference between the ETC and ECM responses (cells), for all three loading periods studied: (a) T = 2s, (b) T = 20s and T=200s. The data were averaged for all three samples. Corresponding material parameters are respresented in Table 1.

Estimates of the cell contribution to ETC mechanics are commonly made by comparing responses of the intact tissue construct to those of the ECM that remains after treatment with deoxycholate (Wakatsuki et al., 2000b; Marquez et al., 2005a,b). Because deoxycholate at the treatment levels used lyses and removes the cell entirely without affecting ECM mechanics (Marquez et al., 2006b), the difference between the ETC and ECM responses yields an estimate of the cell response. Results showed that this deoxycholate-sensitive component of the ETC contribution dominated over the ECM response at lower strains.

4.2. Cells and ECM mechanical responses varied with loading rate

The Fung QLV properties of the cells, ECM, and ETCs varied about 30% over the very large range of loading rates, as could be seen by fitting the material constants A, B, H, and τ, to experimental data using equations. 2.1, 2.3, and 2.4. (Figure 3). The difference in the elastic coefficient A between T = 2 s and T = 200 s was significant statistically, as was the difference in damping coefficients for cells between T = 2 s and T = 200 s. For all specimens A_tissue > A_cells > A_ECM. On the other hand, nonlinearity was clearly dominated by the ECM, with B_ECM > B_tissue > B_cells. In contrast, cells had the highest effect on damping, with H_cells > H_tissue > H_ECM. The Fung QLV time constant τ increased linearly with the loading period, T, in a statistically significant way, and was not significantly different for ETCs, ECM, and cells (τ_tissue ≈ τ_cells ≈ τ_ECM). The linear increase of τ with T followed approximately τ(T)/τ(T = 2s) = 0.5(1 + T/(2s)) (R² > 0.99)

4.3. Energy dissipation

To explore the reasons why Fung QLV fitting predicted time constants that were identical for cells, ECM, and ETCs, we ran sets of simulated experiments while varying the Fung QLV material parameters A, B, H and τ (Figure 5). For each set of simulated experiments, all material coefficients were kept constant except for one, thereby enabling independent depiction of the relationship between each Fung QLV material parameter and dissipated energy. We used material constants of the ETCs, represented in Table 1 and Fig.4. As will be discussed in the discussion section, these simulated experiments provided insight into bias associated with estimating τ.

Figure 5.

The effects on energy dissipation of Fung QLV material parameters. (a) Elasticity coefficient, A. (b) Nonlinearity coefficient, B. (c) Damping coefficient, H. (d) Time constant, τ. (e) Energy dissipation was a strong function of the Deborah number, De = τ/T. Energy dissipation is strongest for a Deborah number just less than one.

Table 1.

Average ETC material constants for ETCs for the loadings with T = 2 s, 20 s and 200 s (refer to Fig. 4)

T(s)	A(Pa)	B	H	τ (s)
2s	267	40.3	1.69	0.166
20s	271	36.7	2.10	1.07
200s	211	37.0	2.22	8.47

Figure 4.

Estimated Fung QLV parameters for three samples. Although parameter A, B, and H showed no statistically significant trends, the energy dissipated by ETCs, ECM, and cells varied substantially with loading period T. (a) Elasticity coefficient, A. (b) Nonlinearity coefficient, B. (c) Damping coefficient, H. (d) Time constant, τ, which varied linearly with the loading period. (e) Energy dissipation.

The dissipated energy, W, scaled linearly with the elasticity coefficient, A, for all loading periods T. Energy dissipation diverged for large values of the nonlinearity coefficient,B (Figure 5b), and reached an asymptote for sufficiently large damping coefficient, H (Figure 5c).

The ratio of the Fung QLV time constant to the loading time presented a dimensionless parameter analogous to a Deborah number, De = τ/T. The magnitude of damping was a function of τ, but the time constant at which the peak damping occurred was a function of the dimensionless Deborah number (Figure 5d).

5. Discussion

Results revealed several interesting aspects of the Fung QLV behavior of cells, ETCs, and ECM, and also several interesting features of Fung QLV fitting. From the perspective of Fung QLV fitting of ETC mechanical responses, inspection of stress-strain curves (Figure 1 in Supplemental material) revealed that ETCs were neither linear elastic nor linear viscoelastic: the shift from concave-down to concave-up in the stress-strain curves could be explained only by nonlinearity. Although a range of other nonlinear viscoelastic models is capable of doing the same, this shows that the Fung QLV model is capable of replicating the responses of ETCs to this loading regime in a way that no elastic or linear viscoelastic model can.

As shown by others (Wakatsuki et al., 2000b; Thomopoulos et al., 2006; Babaei et al., 2016, 2015b), the cells and ECM both contribute to the shape of the stress-strain curves. The initially concave down portion of the stress-strain curves disappeared when the cells were lysed by deoxycholate (Figure 3). The cells imparted a pre-stress upon the ETCs, and then exhibited concave-down behavior over the first 2% strain before switching to concave-up behavior. Note that this concave-up response of the cells has been shown to reverse to a concave-down response at around 20% strain (Wakatsuki et al., 2000b). The ECM response remaining after the cells were lysed became increasingly dominant after 2–3% strain. Cells damped energy disproportionately, suggesting perhaps that cellular remodeling might dominate over ECM responses in determining the degree to which an ETC absorbs energy (Figure 4). Another possibility is that some of this dissipation might be linked to ATP-dependent actin-myosin interactions that are activated by stress. Note that quantifying these responses in terms of the behaviors of ensembles of cells and individual cells requires an additional level of integrated modeling (Zahalak et al., 2000; Chandran & Barocas, 2004; Marquez et al., 2010, 2006a, 2005a,b; Sakamoto et al., 2017), as reviewed elsewhere (Elson & Genin, 2016; Genin & Elson, 2014a,b; Elson & Genin, 2013; Genin et al., 2011). However, these results serve to show the degree to which the Fung QLV model can capture the qualitative mechanical responses of cells, ECM, and ETCs. Note that deoxycholate at the concentrations used completely removes cells from the ECM (Marquez et al., 2006b; Babaei et al., 2016). Collapse of the resulting cavities would be expected to appear as a new time constant or set of time constants in the relaxation spectrum. However, time constants associated with cells or cell/ECM interactions disappeared, and no new time constants appeared.

Although a perfect material model would predict loading-rate independent parameters, some variations of the Fung QLV parameters for cells, ECM, and ETCs as a function of loading period T were evident (Figure 4). Notable amongst the variations with respect to loading period T was the Fung QLV time constant τ, the only variable that varied significantly with T. As discussed below, this might be due to the fast relaxing components dying out during the course of stretching for long T, which would then leave the relaxation behavior dominated by the relatively longer τs.

In subsequent analysis of these responses, fitting of the data to the Fung QLV model was performed and the constants, A, B, H, and τ were studied. To assess the roles of the Fung QLV parameters on energy dissipation as a function of loading period, T, a series of parametric numerical analyses was performed. Energy dissipation for a 4% triangular-wave stretching scaled linearly with the Fung QLV elastic coefficient, A, as expected from equation 2.14, with a slope that increased with decreasing T. Energy dissipation also scaled with the Fung QLV nonlinearity coefficient, B, in a way that was nearly exponential, as expected from the exponential form of the quasi-static stress-strain relation in the Fung QLV model; in all cases dissipation decreased with increasing T. Energy dissipation reached an asymptote with increasing damping coefficient, H, with the asymptote being lower and achieved more rapidly with increasing T. The Fung QLV time constant, τ, affected energy dissipation according to the Deborah number.

These Deborah number dependent peaks in energy absorption showed that triangular-wave excitations with lower strain rates (or longer period T), dissipated lower energy. This finding has important implications for Fung QLV model fitting: ETCs exhibit a spectrum of time constants at different loading frequencies, and the linear relationship between the observed Fung QLV time constant τ and loading frequency T arises from this phenomenon. At a specific strain rate, the time constants that are activated do not represent the full spectrum of time constants for the ETC. Using the analogy of nonlinear springs and dashpots for a Fung QLV material, this indicates that only viscoelastic elements that are well-tuned to a specific loading rate participate in dissipation process. This is consistent with earlier observations that the Fung QLV model can fit viscoelastic relaxation data even with poor approximations to the viscoelastic relaxation spectrum of a material (Babaei et al., 2015a). Note that errors associated with this become apparent in Fung QLV predictions of responses to repeated cyclical loadings (Babaei et al., 2017).

This is a limitation of Fung QLV fitting of materials using the protocol studied, but is also a feature of Fung QLV materials that is interesting in the context of mechanobiology. Cells within a tissue are well-known to respond differently to different loading frequencies and amplitudes, with higher strain rates capable of leading to depolymerization of the actin cytoskeleton(Nekouzadeh et al., 2008; Krishnan et al., 2009; Hsu et al., 2010; Lee et al., 2012; Ronan et al., 2014). In the contexts of a Fung QLV ECM and a Fung QLV cell, the results show that frequency bands of mechanical communication are possible, a result that is interesting from the perspective of rate-dependent energy absorption observed in connective tissues (Babaei et al., 2017).

The Fung QLV model has well-known limitations in modeling of biological tissues, and the characterization of biological tissues at higher strain rates using the Fung QLV model presents a well-known challenge. Gimbel et al. (Gimbel et al., 2004) established that much of the literature must be called into question because of the sensitivity of Fung QLV parameter estimation to mild “overshooting” of a desired strain level in relaxation testing. Given these limitations of the Fung QLV model, how confident can one be in the material property estimates, especially when extrapolating to strain rates that are not accessible by the methods used?

Results such as our observation that the damping coefficient, H, is independent of loading rate at the strain level and strain rates studied would suggest at first glance that the damping coefficient should not change much for a decade or so outside of the testing range. However, extrapolation is limited in accuracy not only because of the nature of fitting to the Fung QLV model but also because of the nature of the Fung model itself. The fittings of all Fung QLV parameters show relatively broad confidence intervals that might mask trends. Within this limitation, the key to assessing the accuracy of a fit lies in the energy dissipation frequency sensitivity plot of Figure 5d. This plot shows that the peak dissipation for a Fung QLV material occurs for loading at a material time constant τ that is on the order of the loading time constant T. The sensitivity dies off over about two decades of time constant, and nearly collapses onto a single master curve when the material time constant is normalized by the loading rate. This indicates that the damping coefficient measured might indeed be trusted for an extrapolation of a decade or so outside of the loading frequency, but that contributions to the measured energy dissipation from material time constants outside of this loading range might be too far attenuated to measure using the approach we applied.

Cells and ECM each made distinct contributions to the overall mechanical responses of the ETCs. The greatest difference between the contributions of cells and ECM was the disproportionate damping coefficient of the cells. This high damping has not been characterized before, but is not unexpected from work on the role of mechanics in stress fiber kinetics (Elson & Genin, 2013). The dynamic responses of actin stress fibers within a cell’s cytoskeleton show depolymerization and repolymerization responses over timescales associated with the loading rates tested here (Nekouzadeh et al., 2008; Nekouzadeh & Genin, 2011; Krishnan et al., 2009; Kaunas & Hsu, 2009; Kaunas et al., 2006; Lee et al., 2012). These time constants arise from the intrinsic lifetime of stress fiber phosphorylation (Kaunas et al., 2006; Kaunas & Hsu, 2009), and from the dynamics of cell contractility interacting with upper and lower stress limits for stress fiber stability (Deshpande et al., 2006; Nekouzadeh et al., 2008; Krishnan et al., 2009; Chen et al., 2010; Ronan et al., 2012). The strong damping of cells over the range of loading rates studied is consistent with the observation that cytoskeletal repolymerization events occur on the scale of 1–100 seconds (Nekouzadeh et al., 2008).

This result is also consistent with the observation that cells and ECM possess different intrinsic mechanisms of viscoelastic dissipation. Within the ECM, dissipation arises from a broad range of timescales and from interactions between cells and ECM (Gautieri et al., 2012; Liu et al., 2016). These include microstructural rearrangements (Ghavanloo, 2017; Silver et al., 2001) and intra- and inter-fibril relaxation (Gupta et al., 2010; Shen et al., 2011). An additional factor at the molecular level is poroelasticity, with the hydration layer surrounding the covalently bonded tropocollagen molecules possibly playing a mechanical role through modulation of intra-molecular hydrogen bonds, and also affecting fibril packing (Chapman et al., 1971; Gautieri et al., 2011; Mogilner et al., 2002; Wachtel & Maroudas, 1998). Crosslinking of collagen should also affect stress relaxation, although the mechanisms and extent of this effect are unclear (Xu et al., 2013; Feng et al., 2003; Babaei et al., 2016).

We note that the testing set-up used in these experiments has a range of limitations. Foremost amongst these is a limitation in the loading rate that can be applied without the specimen losing contact with the loading bars during the unloading phase of a force-displacement cycle. In such cases, the stress-strain curve should dip beneath the value of σ = 0, but, because compression is not possible, the stress-strain curve returns to the origin along the axis σ = 0. Compressive stress will offer further energy dissipation, but this cannot be captured in this experimental set-up.

6. Conclusion

A triangular-wave strain loading pulse is a useful protocol for the study of strain-dependent dynamic viscoelastic responses owing to the simplicity of the protocol. Analysis of material responses using this protocol is simplified because the magnitude of the strain rate, |ε̇|, can be kept constant. This is in contrast to sinusoidal excitation tests, where the strain rate varies over loading cycles, complicating analysis of rate-dependent viscoelasticity. The Fung QLV model is effective for modeling the concave-up stress-strain curves of ETCs and their constituents.

Applying this protocol and analysis framework to engineered tissue constructs showed that the fibroblasts studied have higher contributions to the damping coefficient of the ETCs studied than did the ECM. However, analysis of this loading protocol revealed a fundamental challenge with Fung QLV modeling, namely that during a triangular-wave straining cycle, only a fraction of a material’s time constants are activated, within an order of magnitude of the loading period. The fitting of Fung QLV parameters therefore yielded estimates of the Fung QLV time constant τ that were proportional to the loading period T. This feature of energy dissipation by Fung QLV materials is of interest from the perspective of mechanobiology as a potential mechanism for differential, frequency-banded mechanical communication to cells through the ECM.