An experimental and computational investigation of the effects of volumetric 1 boundary conditions on the compressive mechanics of passive skeletal muscle 2

19 Computational modeling, such as finite element analysis, is employed in a range of biomechanics 20 specialties, including impact biomechanics and surgical planning. These models rely on accurate material 21 properties for skeletal muscle, which comprises roughly 40% of the human body. Due to surrounding 22 tissues, compressed skeletal muscle in vivo likely experiences a semi-confined state. Nearly all previous 23 studies investigating passively compressed muscle at the tissue level have focused on muscle in 24 unconfined compression. The goals of this study were to (1) examine the stiffness and time-dependent 25 material properties of skeletal muscle subjected to both confined and unconfined compression (2) develop 26 a model that captures passive muscle mechanics under both conditions and (3) determine the extent to 27 which different assumptions of volumetric behavior affect model results. Muscle in confined compression 28 exhibited stiffer behavior, agreeing with previous assumptions of near-incompressibility. Stress relaxation 29 was found to be faster under unconfined compression, suggesting there may be different mechanisms that 30 support load these two conditions. Finite element calibration was achieved through nonlinear optimization 31 (normalized root mean square error <6%) and model validation was strong (normalized root mean square 32 error <17%). Comparisons to commonly employed assumptions of bulk behavior showed that a simple 33 one parameter approach does not accurately simulate confined compression. We thus recommend the use 34 of a properly calibrated, nonlinear bulk constitutive model for modeling of skeletal muscle in vivo . Future 35 work to determine mechanisms of passive muscle stiffness would enhance the efforts presented here.

It is likely that in vivo muscle experiences a variation between confined and unconfined volumetric boundary conditions [3]- [5], [21], where semi-confinement is created by tissues surrounding the muscle.
Nearly all previous studies investigating passively compressed muscle at the tissue level have focused on muscle in unconfined compression (UC) [3]- [5], [22]- [24], where the sample is free to expand laterally when loaded.One group has investigated muscle under anisotropic semi-confined compression, and the specific confinement was found to affect both muscle structural deformation and mechanical response [14], [15].Traditionally, muscle has been modelled as a nearly incompressible hyperelastic material [14], [24]- [26].However, to the best of the authors' knowledge, there have been no investigations of skeletal muscle in fully confined compression (CC), where volumetric strain is applied and the assumption of near incompressibility can be directly tested.This gap in understanding the effects of volumetric boundary conditions (UC and CC) on the compressive properties of skeletal muscle affect the subsequent models derived to predict skeletal muscle and whole body behavior.
The time dependent nature of muscle can be observed in significant stress relaxation following compressive deformation [3]- [5], [16], [17].Stress relaxation tests have been thus used extensively to characterize the stress-strain and stress-time behavior of the tissue, and are typically accompanied by various viscoelastic modelling approaches [3], [4], [17], [27]- [29].Inverse finite element methods are effective in determining material properties through parameter optimization to experimental data [27], [30]- [32].In previous continuum mechanics based modeling of skeletal muscle, the assumption of near incompressibility leads to a decoupling of the volumetric (volume changing) and isochoric (shape changing) responses of the hyperelastic model [23], [33].We developed a non-linear hyper-viscoelastic finite element model to simulate both UC and CC testing conditions concurrently that was calibrated using inverse finite element analysis through a nonlinear optimization protocol.This model was then used to investigate various assumptions about muscle compressibility, and what the most appropriate modelling approaches may be for passively compressed skeletal muscle.
The goals of this study were to (1) examine the stiffness and time-dependent material properties of skeletal muscle subjected to two boundary conditions (UC and CC) (2) develop a computational model that captures the behavior of muscle subject to these different volumetric boundary conditions and (3) determine the extent to which different assumptions of volumetric behavior affect model results.We hypothesize that the material properties of skeletal muscle differs in confined versus unconfined compression.Since in vivo muscle behavior is likely to be semi-confined, we also believe that by considering both conditions we improve the accuracy of finite element models of skeletal muscle.

Sample Preparation
Whole tibialis anterior (TA) muscles were isolated from seven female porcine hind limbs following sacrifice [14], [15], [23], [24], [26], [34].Participants did not handle live animals as all tissue was acquired from a local abattoir.Connective tissues and fat were removed from TA using standard dissection instruments.Two sample geometries were identified for experimentation in this study: transverse oriented cuboids for UC and transverse cylindrical plugs for CC.The cuboids (height = 7.2 ± 0.9 mm, cross-sectional area=149.6±25.7 mm 2 , n = 15 for fast compression and n = 14 for slow compression) [14] were acquired using scalpels and a custom tissue slicer.Sample height was measured via micrometer and area was calculated through image analysis prior to testing.The cylindrical plugs (height = 7.1 ± 0.6 mm, n = 16 for fast compression and n = 15 for slow compression) were obtained using metal hole punch (ø = 10mm) and scalpels.Since passive skeletal muscle is an incredibly soft tissue, using a larger metal punch ensured that the sample tightly fit into the compression well.Four to five cuboids and cylindrical plugs were excised from each muscle and the samples were not paired between unconfined and confined compression.All samples were taken from muscle midbelly and were kept hydrated by phosphate buffered saline throughout testing [24], [26], [34].To limit effects of rigor mortis, all testing was completed within eight hours of sacrifice [3]- [5], [17], [29], [35].Tissue damage was controlled at two points in experimental protocol.
Firstly, during dissection, a custom slicer and high-profile histology blades or surgical scalpels were used to cut the sample as few times as possible.Secondly, after each UC and CC tests, the sample was visually checked for damage, and any damaged sample was discarded.

Data Analysis and Viscoelastic Modelling
First Piola-Kirchhoff (PK) stress , nominal strain , and peak stress were determined through original specimen dimensions (Equation 1, where  is the measured load and   is the original specimen crosssectional area) [17], [28].Peak modulus (    ⁄ ) was also calculated.Three relaxation ratios (1-5s, 6-105s, 106-400s) were determined to evaluate the amount of relaxation associated with various relaxation times (Equation 2, where  is the relaxation ratio and   and   are the first PK stress at time points  and ) [29].These three time periods were chosen as they generally characterized "short", "medium", and "long" term relaxation for the samples tested in this study, and are similar to time periods previously used for skeletal muscle in tension [29].
Here stress  is calculated from the convolution integral (Equation 3), which includes the Prony series reduced relaxation function (), nominal strain , and an integration parameter .This includes three relaxation coefficients   , three time constants   , and the instantaneous modulus  0 .As the purpose of this analysis was to compare relaxation behavior only,  0 = 1 was fixed and all data were normalized.The model also accounted for the experimental overshoot in strain applied by the Instron during fast compression.Parameter determination was performed in two steps: a Monte Carlo simulation followed by a nonlinear least-squares deterministic optimization (lsqnonlin in MATLAB) [27], [28], [30], [42]- [45].In the Monte Carlo simulation, the six parameters (  1−3 and  1−3 ) were randomly varied for 100,000 simulations, ensuring 0 <  1 +  2 +  3 < 1 [30].The set of parameters minimizing percent error between normalized model and experimental hold stress was used as initial guesses for the deterministic optimization, which optimized percent difference between normalized model and experimental hold stresses.This approach used the global stochastic Monte Carlo method in conjunction with the precision of a local deterministic approach.All modelling was performed in MATLAB (The Mathworks, Inc.).

Finite Element Modelling
Two finite element models of UC and CC geometries were developed and simulated with an implicit finite element approach in Abaqus/Standard (Dassault Systèmes).(Figure 1C-D).The UC geometry was reduced to a quarter of the sample by symmetry, with 175 first-order 8-node hexahedral elements with a cubic volume of 1 mm 3 (type C3D8RH).This model was compressed by coupling the top surface to a reference node and displacing the node to follow experimental displacement.The reaction force on the reference point was divided by initial area to acquire first Piola-Kirchhoff stress.The CC geometry was reduced to a two-dimensional axisymmetric model of cylinder with 96 first-order 4-node quadrilateral elements with a rectangular area of 3.75 mm 2 (type CAX4RH).A convergence study was performed by doubling and halving the mesh densities, and the model outputs were virtually identical to the outputs of the original models.Displacement was prescribed for top surface and as no lateral expansion occurred, first Piola-Kirchhoff stress was determined directly from model axial stress.Displacements in both fast compression models simulated the slight experimental overshoot applied by the Instron.Four boundary conditions were applied to the UC model to ensure quarter symmetry while leaving exposed faces to expand due to the Poisson effect (Figure 1C).Since very little sample sliding was noticed during pilot testing or experimentation, sliding was not controlled and was not included in any of the models.An axisymmetric rectangular model was developed to simulate the cylindrical CC testing geometry with restricted exterior faces to simulate the impermeable steel well (Figure 1D).Any initial lateral pressure exerted by the walls of the CC well during insertion and prior to loading were small compared to the loading experienced by the sample during tests.Thus, it was deemed appropriate to model the CC well as a fixed boundary with no pre-stress.
Here  0 and   are material parameters that characterize the isochoric and volumetric responses, respectively. ̅ 1 is defined as  ̅ 1 =  ̅ 1 +  ̅ 2 +  ̅ 3 where  ̅  =  − 1 3   (  are the principle stretches), and  is the volume ratio.Due to the nonlinearity of the stress-strain curves for UC and CC data, three  0 terms and two   terms were used.A Prony series viscoelastic model (Equation 7) was applied to the decoupled responses in Equation 4.Here () is the time dependent bulk modulus and () is the time dependent shear modulus. ∞ and  ∞ model long-term bulk and shear moduli, respectively.   and    are time The finite element model consisted of thirteen parameters (five hyperelastic, eight viscoelastic).Model calibration (determination of parameters) was again achieved in two steps: a Monte Carlo simulation followed by a nonlinear least-squares deterministic optimization (lsqnonlin in MATLAB) [27], [28], [30], [42]- [45].For computational efficiency, the Monte Carlo simulation made 5000 random guesses for the eight viscoelastic parameters (all hyperelastic parameters set to 1) and the error function (Equation 8) was used to calculate the weighted difference between normalized model and experimental fast-compression stress relaxation data for both UC and CC models simultaneously.This function assigns greater weight around the peak region where there are fewer time points, thus improving the fit throughout the optimization procedure.
A second set of UC and CC finite element models with the same constitutive formulation were calibrated using only the UC fast compression data.This is to reflect the approach of assuming near-incompressibility with a single-parameter bulk hyperelastic term, as is most common in finite element models of skeletal muscle [23], [25], [33], [49], [50].The volumetric parameters (  ) were assumed to be three, four, and five orders of magnitude larger than the isochoric parameters ( 0 ) to reflect a range of assumptions.The time dependent bulk (  ) and shear (  ) moduli were assumed to the same.This model represents the typical approach for finite element modeling of skeletal muscle and was later used to predict the CC fast compression data.
Finally, a semi-confined compression model (SC) was developed by surrounding the quarter-brick UC model with a generic linear elastic material to create a quarter disk (Figure 2A).The outer boundary of this disk was restricted laterally, thus while the whole structure was subject to confined compression, modulating the Young's modulus of this outer material enabled a semi-confined state for the muscle geometry.This modulation then enabled the simulation of a transition from unconfined (Figure to 2B) to confined (Figure 2C) compression during a 40% compressive strain ramp to mimic the UC model strain.
As the Young's Modulus of the disk was varied, the SC model stress and volume ratio from the peak were recorded.This parametric analysis was performed for the optimized parameters and parameters acquired from the three models calibrated using UC fast compression data.the observed incompressibility of passive skeletal muscle.Specifically, the initial Poisson's ratio  was determined for each case based on the initial shear and bulk moduli (Equation 9) [51].

Statistical Analysis
All statistical comparisons between groups were performed using two-sample two-tailed t-tests, with significance set to p<0.05.The goodness of fit (GoF) for all fits were evaluated with the goodnessOfFit function in MATLAB (Equation 10) [30].Here    and    are the model and experimental stress values, respectively, at the i th data point and N is the total number of data points.Fits range from -∞ (worst) to 1 (perfect).The overall percent error, peak stress percent error, and normalized root mean square error (NRMSE) were also determined (Equation 11) [30].

Results
Despite lower strain levels, muscle in CC exhibited stiffer behavior than muscle in UC in both fast and slow compression.Muscle in CC showed ~1200% higher peak modulus (p-value<0.001)(Figures 3-4) (mean UC fast peak modulus = 0.0524 ± 0.0340 MPa, mean CC fast peak modulus = 1.856 ± 0.908 MPa).Muscle in CC slow compression exhibited a ~860% higher peak stress than muscle in UC slow compression (p-value<0.001)(Figures 3-4) (mean UC slow peak modulus = 0.041 ± 0.020 MPa, mean CC slow peak modulus = 1.058 ± 0.623 MPa).Muscle in CC and UC fast relaxation showed different relaxation behavior (Figure 5A).The three relaxation ratios (RR1 for 1-5s, RR2 for 6-105s, and RR3 for 106-400s) showed that differences in CC and UC fast compression time dependence was more apparent at the early stage of relaxation.RR1 for UC (0.648±0.043) was approximately twice RR1 for CC (0.344±0.191) (p<0.001).RR2 for UC (0.519±0.076) was larger than RR2 for CC (0.403±0.180) (p=0.029), while RR3 was not different (p=0.521)(Figure 5B).The global stochastic Monte Carlo simulation in conjunction with the deterministic optimization yielded excellent fits between the three-term linear Prony series viscoelastic model and normalized experimental stress data (average percent error = 1.06 ± 0.13%, average NRMSE = 0.016 ± 0.002, average GoF = 0.998 ± 0.0002).Comparisons of fast compression viscoelastic parameters showed  1 for UC (0.902±0.036) was larger than that for CC (0.604±0.192) (p<0.001), 2 for UC (0.043±0.015) was smaller than that for CC (0.103±0.038) (p<0.001), and  3 for UC (0.033±0.019) was smaller than that for CC (0.141±0.078) (p<0.001)(Figure 6A).Additionally,  1 for UC (0.132±0.029 s) was smaller than that for CC (0.331±0.190 s) (p<0.001), 2 for UC (7.612 ±1.667 s) was smaller than that for CC (12.751±0.427s) (p<0.001), and  3 for UC (96.824±20.134s) was smaller than that for CC (183.419±90.805s) (p<0.001)(Figure 6B).The global stochastic Monte Carlo in conjunction with the deterministic optimization again yielded strong concurrent fitting between the finite element model and experimental UC and CC fast data (Table 1) (Figure 7A-B).The initial shear and bulk modulus were calculated using the optimized hyperelastic parameters in Table 2 as 0.0445 kPa and 18.89 kPa, respectively [52].The model also exhibited very strong predictions for non-linear ramp and relaxation for the slow compression data for both UC and CC (Table 1) (Figure 7C-D).The finite element model with the   parameters derived from the  0 and the bulk and shear coefficients equal gave strong fits to the UC data (Table 3).However, these models provided poor predictions for the CC fast data, specifically missing the non-linear ramp, stiffness, and the relaxation behavior (Figure 8A-B).The peak stress was seen to increase rapidly with decreasing volume ratio for the semiconfined (SC) model derived from the full optimized parameters (Figure 8C).However, the peak stresses for the three, four, and five orders of magnitude parameters increased more linearly and were thus comparatively too stiff at low strains or too soft at high strains (Figure 8C-D).The four sets of bulk and shear moduli calculated from the four new fits (Figure 7 A and B) show the spread in the experimentally found moduli values have only a small effect on initial Poisson's ratio (Table 4).Even with the variability observed and investigated in this study, muscle exhibited nearly-incompressible behavior.

Discussion
This study aimed to characterize and compare the stress relaxation behavior of skeletal muscle subject to two volumetric boundary conditions (unconfined compression or UC and confined compression or CC) by comparing stress relaxation data and computational models.We chose to employ both UC and CC conditions as in vivo muscle is likely to experience semi-confined compression (SC) where the muscle is partially restricted by surrounding hard and soft tissues.While previous viscoelastic approaches successfully modelled stress relaxation of muscle in UC alone [3], [4], this study simultaneously fit stress relaxation data of muscle in CC and UC, thus providing greater accuracy for the volumetric behavior of skeletal muscle.

Experimental Findings
Van Loocke et al. found that muscle exhibited a Cauchy stress of ~4 kPa in UC in cross-fiber direction at strain of 30% applied at quasi-static rate of 0.05% s -1 [3].This stress value is comparable to the First Piola-Kirchhoff stress values from UC following relaxation (<5 kPa).Palevski et al. measured the short term and long-term shear modulus of porcine gluteus muscle in vitro by rapid indentation tests [53].They found the shear modulus to be ~700±300 Pa.The hyperelastic finite element model developed in this study yielded a shear modulus of 445 Pa, which agrees well with these previously published data.A later Van Loocke et al. study used a non-linear viscoelastic model and found the shear modulus to be 523 Pa, which is also in close agreement with the shear modulus found in this study [4].
The linear viscoelastic model shows that there is approximately 97% relaxation of muscle instantaneous modulus associated relaxation (Figure 6) for UC.This finding agrees with the findings of Van Loocke et al., whose viscoelastic model showed that muscle experiences about 80% relaxation in first 100 seconds of the hold phase of stress relaxation tests.Wheatley et al. also found that muscle exhibits up to 99% relaxation in UC at long relaxation times, and also found muscle exhibits initial relaxation of ~75%, which is consistent with what our viscoelastic model shows (Figure 6) [28].On the other hand, muscle in CC shows nearly half of initial relaxation that muscle in UC exhibited (Figure 5).This is supported by statistical comparisons of viscoelastic parameters (Figure 6).In all, the time dependent behavior of skeletal muscle has been shown to depend on the loading condition.
Muscle has been known to be ~70-80% incompressible fluid [54] and some studies suggest this fluid plays a significant role in the mechanical properties of the tissue [55]- [58].In UC, when muscle is compressed, fluid is free to redistribute within the tissue while maintaining a nearly constant volume and exude from the sample from the pores on the lateral sides.Thus, it is unlikely to directly bear a significant portion of the load, leaving solid muscle constituents such as the extracellular matrix and myofibrils to perhaps resist compression directly.When load is applied to muscle in CC, the fluid cannot escape laterally, and must flow through the porous indenter.Fluid could then be retained in interstitial and intracellular space, thus pressurizing and supporting a greater load than in UC.Since fluid is incompressible, this effect can lead to drastic differences in observed stiffness between UC and CC such as those observed here.This fluid pressurization may also contribute the differences in relaxation behavior between UC and CC, as muscle has been shown to have a non-negligible permeability [57].However, this hypothesis remains untested and future work should be completed to directly investigate fluid pressurization's effect on viscoelastic behavior of muscle in different loading conditions.
Two sample geometries were identified for experimentation in this study: transverse oriented cuboids for UC and transverse cylindrical plugs for CC.To minimize tissue damage, high profile histology blades and surgical scalpels were used for dissection.These geometries, in accordance with previous literature [59], [60] were used due to the constraints from the experimental apparatus used (Figure 1 A and B).Specifically, the differences in tissue stiffness and required boundary conditions based on testing condition made using a single sample geometry between tests unreasonable.While previous work has shown that sample size can affect the observed compressive modulus of passive skeletal muscle [61] sample size and dimension are not likely to explain the major differences in compressive stiffness observed in this study.Additionally, the size of muscle fibers (~50-100 µm) relative to the sample size used here (multiple mm) suggest that the specimens used in this study are representative of bulk muscle tissue.
While the stress-time data and standard deviation presented here shows that not all samples exhibited identical passive material properties, this is not uncommon for biological soft tissues and in particular skeletal muscle [3], [4] sources of variability may be tissue hydration, although all samples were stored soaked in phosphate buffered saline prior to testing to limit this effect.
One limitation of this work is that muscle samples in UC and CC were compressed to different strain levels.
The strain levels of 40% in UC and 15% in CC ensured that enough load was applied to each sample without damaging the samples.These strain levels were determined through extensive pilot stress-relaxation testing to investigate strain level and tissue damage.Due to the highly soft nature of passive skeletal muscle under unconfined compression and the relatively stiff response in confined compression, these two strain levels gave more comparable data than similar strain levels would.As muscle has been shown be nonlinearly viscoelastic in unconfined compression, the differences in relaxation behavior may vary somewhat with strain.However, the major differences in tissue stiffness between testing conditions suggest different mechanisms that support load under UC and CC.Additionally, the effectiveness of bulk and shear Prony series viscoelastic terms employed in this study further support the notion of different mechanisms driving the stiffness and time dependent responses in UC and CC.Future work to test muscle in UC and CC at different strain levels would further clarify how relaxation behavior depends on strain level under these conditions.

Model Findings
Two modeling approaches were used in this study: an analytical linear Prony series viscoelastic model and an uncoupled Yeoh/Prony hyper-viscoelastic model.Both of these approaches had a similar two-step optimization method, but each model served a different purpose.The relatively simple viscoelastic analytical model generated sets of parameters for each individual sample that could be used to compare relaxation behavior of muscle between unconfined and confined compression.In comparison, the hyperviscoelastic finite element model was developed to concurrently characterize the behavior of both testing conditions.Together, these two modeling approaches enabled statistical comparison of testing conditions as well as comprehensive characterization of tissue stress relaxation behavior.The finite element model was calibrated using the fast compression data and used to predict the slow compression data as the fast data encompasses a more comprehensive time dependent data set; this is also a common practice in viscoelastic modelling [4], [28].Moreover, the current study aimed to characterize the larger differences between the testing conditions and not the effect of more specific factors like strain rate, which could be investigated in future studies.
Blemker et al. used a decoupled strain energy formulation to model the biceps branchii in which the volumetric or bulk parameter is assumed to be five orders of magnitude larger than the isochoric or shear parameters [33].Similarly, Calvo et al. and Grasa et al. take the only volumetric parameter to be between two and three orders of magnitude larger than the isochoric parameters [25], [66].In this study we collect volumetric compression data (CC) and span the assumptions made by Blemker et al. and Calvo et al to predict the CC data.The predictions are very poor (Figure 8), showing that this simple assumption is not appropriate for representing the nonlinear stress-strain behavior under highly confined compression.We present a finite element model in which the volumetric and isochoric responses are concurrently optimized.
Our models still agree that muscle is what would generally be considered to be nearly-incompressible (initial shear modulus of muscle is ~3 orders of magnitude smaller than initial bulk modulus), but that the volumetric response is nonlinear as shown by the strong predictions of a two-term volumetric function (Figure 7).We thus recommend using a higher-order volumetric term to better characterize compressed muscle.
We believe that in vivo muscle experiences loading that is most similar to semi-confined compression.This loading could vary between conditions that approach unconfined or confined compression among different muscles in the body, or even in different regions on the same muscle.This is supported by previous magnetic resonance imaging of passively stretched human tibialis anterior that observed volumetric strains as high as 20% in one region and nearly 0% in another [67].The three models representing literature methods and the model that we concurrently optimized using UC and CC data provide very different results for semi-confinement, particularly regarding nonlinearity (Figure 8B).This may be a concern as passive stiffness nonlinearity could act as a mechanism to prevent damage of bone and other tissues during high impact loads in vivo.Additionally, the use of a single-term volumetric formulation is likely to be either too stiff at low volumetric strains or too soft at high volumetric strains.
The study presents a finite element model that can concurrently characterize the unconfined and confined compression conditions.The modeling and optimization approach employed here fit thirteen hyperviscoelastic parameters.This number of parameters were used because (1) the ramp phase for both unconfined and confined compression are highly non-linear (2) muscle specimens relaxed for a four hundred seconds following compresison at a relatively fast rate (up to 40%/sec), thus enacting a wide range of time dependence.Future work could reduce the number of optimized or varied parameters by locking Prony terms or coupling terms together similar to previous work [30].Alternatively, the model provides strong predictions and future efforts could combine direction-dependent, contractile, and tensile mechanics as well to create a more comprehensive model.It has previously been shown that fitting models to average experimental data yields different parameters than fitting model individually to tests and then averaging the parameters [68].Since goal of the study was to characterize the broader differences between the testing conditions, the authors found it appropriate to only fit the models to averaged data and acquire one set of parameters.
One limitation of the finite element model provided in this study is that it cannot capture the comprehensive properties exhibited by passive skeletal muscle such as tension-compression asymmetry, anisotropy, and contractile properties.It is generally assumed, however, that skeletal muscle is primarily compressed in the transverse direction in vivo, thus this is the direction of importance when considering in vivo muscle deformation.A comprehensive model is likely to have quite a large number of parameters, thus increasing the model complexity.However, previous studies [30] have explored how parameter coupling can reduce the number of parameters in a model while still utilizing nonlinear optimization and statistical interpretation of model results.In the future, this approach could be employed to implement a model that not only characterizes differences between muscle in unconfined and confined compression but also has other established properties of skeletal muscle.We also chose not to use a biphasic or poroelastic constitutive approach in this work because viscoelasticity what is commonly used to characterize skeletal muscle stress relaxation and is a computationally efficient and stable approach to modeling time dependence.Fluid pressurization, however, requires the solution of an additional condition (either pressure equilibrium or conservation of mass of the fluid) and is more unstable at high strain rates.Future studies exploring the mechanisms involved in tension and compression and could employ mechanistic anisotropic models that include components such as tension only fibers and saturating fluid.This type of a model would be greatly beneficial to the field of passive muscle mechanics.
In all, the study found that muscle in CC exhibits stiffer compressive behavior despite lower strains and muscle in UC exhibits greater and faster relaxation.This study also showed that concurrently fitting isochoric and volumetric hyper-viscoelastic parameters with these data improves model predictions, and is recommended for cases where semi-confinement is likely.Future work to better understand the mechanisms of force transmission in compressed skeletal muscle would greatly benefit the field.

Figure 1 :
Figure 1: Schematics showing experimental set up for (A) unconfined compression and (B) confined

Figure 2 :
Figure 2: Finite element geometry and boundary conditions of semiconfined compression.The muscle

Figure 3 :
Figure 3: Average experimental stress relaxation curves with standard deviation for (A) unconfined fast

Figure 4 :
Figure 4: Peak moduli of muscle samples for the four testing conditions and t-test p-values.Unconfined

Figure 5 :
Figure 5: (A) Average normalized experimental stress relaxation curves for the hold phase plotted on

Figure 6 :
Figure 6: (A) Relaxation parameters  1−3 for unconfined and confined fast compression.(B) Time constants  1−3 for unconfined and confined fast compression.Unconfined compression is represented by circles and confined compression by triangles.

Figure 7 :
Figure 7: Average fast experimental data with standard deviation, finite element model calibrations, and

Figure 8 :
Figure 8: Average confined compression fast experimental data with standard deviation (dashed red) and , [62].These differences are often explained by natural variability of structure and content of constituents such as collagen in the extracellular matrix, fiber/fascicle size and organization, and fluid content from animal to animal and muscle to muscle.Despite the fact that clear and consistent structure-function mechanisms in passive skeletal muscle are not fully understood, both the extracellular matrix and muscle fibers are involved in passive load transmission in skeletal muscle [63]-[65].Other The set of bulk-viscoelastic parameters minimizing the CC error and set of shear-viscoelastic parameters minimizing UC error were used as initial guesses for the deterministic optimization.Initial guesses for all hyperelastic parameters were set to a value of one.Following calibration, UC and CC slowcompression data were predicted by this optimized model as a means of validation.Due to the simple and symmetric model geometries in this work, mesh convergence analysis showed virtually no difference in model behavior as a function of element size.

Table 1 :
Overall percent error, peak error, normalized root mean square error (NRMSE), and goodness of fit (GoF) values for finite element models calibrated to fast unconfined and confined compression data concurrently and validated against slow unconfined and confined compression data concurrently.

Table 2 :
Hyperelastic and viscoelastic parameters of the finite element model calibrated using unconfined and confined fast compression data concurrently.

Table 3 :
Overall percent error, peak error, normalized root mean square error (NRMSE), and goodness of fit (GoF) calibration values for finite element models fit with fast unconfined compression data only.

Table 4 .
Normalized root mean square error (NRMSE), initial bulk and shear moduli, bulk-to-shear modulus ratio, and initial Poisson's Ratio for the additional four fits to the mean experimental data plus or minus standard deviation.