Development of fibroblast-seeded collagen gels under planar biaxial mechanical constraints: a biomechanical study

DOI 10.1007/s10237-012-0448-x

O R I G I NA L PA P E R

Development of fibroblast-seeded collagen gels under planar

biaxial mechanical constraints: a biomechanical study

Jin-Jia Hu· Yen-Ching Liu · Guan-Wen Chen · Mei-Xuan Wang· Pei-Yuan Lee

Received: 20 July 2012 / Accepted: 9 October 2012 / Published online: 25 October 2012 © Springer-Verlag Berlin Heidelberg 2012

Abstract Prior studies indicated that mechanical loading influences cell turnover and matrix remodeling in tissues, suggesting that mechanical stimuli can play an active role in engineering artificial tissues. While most tissue culture studies focus on influence of uniaxial loading or constraints, effects of multi-axial loading or constraints on tissue devel-opment are far from clear. In this study, we examined the biaxial mechanical properties of fibroblast-seeded collagen gels cultured under four different mechanical constraints for 6 days: free-floating, equibiaxial stretching (with three dif-ferent stretch ratios), strip-biaxial stretching, and uniaxial stretching. Passive mechanical behavior of the cell-seeded gels was also examined after decellularization. A contin-uum-based two-dimensional Fung model was used to quan-tify the mechanical behavior of the gel. Based on the model, the value of stored strain energy and the ratio of stiffness in the stretching directions were calculated at prescribed strains for each gel, and statistical comparisons were made Electronic supplementary material The online version of this article (doi:10.1007/s10237-012-0448-x) contains supplementary material, which is available to authorized users.

J.-J. Hu (

B

Department of Biomedical Engineering, National Cheng Kung University, #1 University Rd., Tainan 701, Taiwan

e-mail: [email protected]

J.-J. Hu

Medical Device Innovation Center, National Cheng Kung University, Tainan, Taiwan

Y.-C. Liu

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan

P.-Y. Lee

Department of Orthopedics Surgery, Show Chwan Memorial Hospital, Changhua, Taiwan

among the gels cultured under the various mechanical con-straints. Results showed that gels cultured under the free-floating and equibiaxial stretching conditions exhibited a nearly isotropic mechanical behavior, while gels cultured under the strip-biaxial and uniaxial stretching conditions developed a significant degree of mechanical anisotropy. In particular, gels cultured under the equibiaxial stretching con-dition with a greater stretch ratio appeared to be stiffer than those with a smaller stretch ratio. Also, a decellularized gel was stiffer than its non-decellularized counterpart. Finally, the retained mechanical anisotropy in gels cultured under the strip-biaxial stretching and uniaxial stretching conditions after cell removal reflected an irreversible matrix remodeling. Keywords Fibroblast-seeded collagen gels·

Tissue development· Mechanical constraints · Mechanical anisotropy· Biaxial mechanical testing · Constitutive modeling

1 Introduction

The extracellular matrix (ECM) provides vital structural integrity for various kinds of tissues. Particularly for load-bearing tissues such as blood vessels, heart valve leaf-lets, tendons, and ligaments, their load-bearing capacity depends heavily on the organization of the ECM (Kjaer

2004; Wagenseil and Mecham 2009; Lincoln et al. 2006).

It is intriguing that most load-bearing tissues develop struc-tural and mechanical anisotropy that reflects their functions. Mechanical conditioning has been used to engineer the load-bearing tissues aiming at better functionality (Seliktar et al.

2000;Mol et al. 2005;Grodzinsky et al. 2000;Mauck et al.

2000;Shearn et al. 2007). In most of these cases, however, the

engineered tissue is subjected to a mechanical environment that replicates some aspects of the physiological conditions;

it is generally not feasible to correlate the findings (e.g., struc-tural or mechanical anisotropy) in these studies to precise, quantitative mechanical loading or constraints.

Fibroblasts, among the many types of cells within the ECM, play a particularly important role by producing, maintaining, reorganizing, and degrading components of the ECM, in particular collagen. Fibroblast-seeded colla-gen gels have thus been used as a model system to study cell–matrix interactions in tissue morphogenesis and wound healing (Ehrlich 1988;Bell et al. 1979;Harris et al. 1981). Specifically in tissue engineering, the property of directed collagen contraction in uniaxially constrained, rectangular, fibroblast-seeded collagen gels has been used to engineer highly anisotropic load-bearing tissues (Feng et al. 2006;

Grenier et al. 2005;Shi and Vesely 2003).Knezevic et al.

(2002) were the first to study biaxially loaded cell-seeded collagen gels. They developed a novel culture system in which square cell-seeded collagen gels can be subjected to isotonic biaxial loading.Thomopoulos et al.(2005) subse-quently used the system to study the development of struc-tural and mechanical anisotropy in the gels and found that gels cultured under a constant uniaxial force developed struc-tural and mechanical anisotropy, whereas gels cultured under constant equibiaxial forces remained mechanically isotropic. Quite a few tissues in the body (e.g., blood vessels, heart valve leaflets, skin, etc.), however, are subjected to neither uniax-ial nor equibiaxuniax-ial loading. Also, to the best of our knowl-edge, effects of the extent of equibiaxial stretching on tissue development have not been studied. The information is fun-damental for increasing our understanding of native tissue mechanics as well as our ability to engineer functional tissue equivalents.

Our novel culture system allowed the application of sophisticated biaxial mechanical constraints (e.g., strip-biaxial stretching) to cruciform-shaped fibroblast-seeded collagen gels (Hu et al. 2009). The previous study, which focused on microstructural changes in fibroblast-seeded col-lagen gels cultured under defined planar biaxial mechan-ical constraints, demonstrated that the microstructure of the fibroblast-seeded collagen gels can be manipulated by the biaxial mechanical constraints (Hu et al. 2009). It is unclear, however, how well the microstructural findings can be correlated to mechanical properties. As the functionality of load-bearing tissues is more related to their mechanical properties, in this study we examined the biaxial mechanical properties of cruciform-shaped fibroblast-seeded collagen gels cultured under various mechanical constraints. We used tension-controlled protocols to examine the biaxial mechan-ical properties of the gels. Passive mechanmechan-ical properties of the gels, which were determined after decellularization, were also examined. Finally, a continuum-based two-dimensional (2-D) Fung model was used to quantify the mechanical behavior of the decellularized gels, allowing calculation of

the value of stored strain energy as well as the ratio of stiffness in the stretching directions of the gels. Specifically, for eval-uation of the predictive capability of the model, the model with material parameters determined by fitting proportional tension data of a decellularized gel was used to simulate the mechanical behavior of the gel under equibiaxial tension.

2 Methods

2.1 Preparation of cruciform-shaped fibroblast-seeded collagen gels

3T3 fibroblasts were cultured in DMEM containing 10 % calf serum and antibiotics and passaged for every 6–7 days. Con-fluent cells were detached by 0.05 % trypsin-EDTA, counted, and resuspended in culture medium at 2.3 × 106cells/mL to be incorporated into collagen gels. Cell-seeded collagen gels were prepared on ice to avoid premature gelation. Specifi-cally, 0.45 mL of reconstituted buffer (10×) was mixed with 0.9 mL of DMEM solution (5×) and 1.2 mL of concentrated collagen type I (9.37 mg/mL; BD Biosciences, San Jose, CA), mixed thoroughly, and then neutralized by∼110 mL of 0.1 N sodium hydroxide before adding 1.95 mL of the cell suspension (or DMEM solution (1×) for acellular gels). The final densities of cell and collagen were 1× 106 cells/mL and 2.68 mg/mL, respectively. This mixture was poured into a cruciform mold (Fig. 1a) consisting of a silicone gasket

Cell-suspended collagen soln.

Remove the mold and culture for 1 day Mount the gel and culture for 6 days

Silicone mold 1 2 λ2=1.2 unconstrained λ2=1.2 λ1=1.01 Allow to gel in the mold at 37°C

Porous bar Glass dish λ=1.1 λ=1.1 A B C D

Fig. 1 Schematic diagram showing how a cruciform-shaped fibro-blast-seeded collagen gel is made and clamped to the biaxial tissue culture chamber (a). The cell-seeded gels are constrained equibiaxially (b), strip-biaxially (c), or uniaxially (d) in the chamber

Fig. 2 (a) Top view and side

view of the custom-made planar

biaxial mechanical tester with detailed illustration of the custom-made force transducer. (b) A photograph showing a decellularized gel clamped to the force transducers and holders with tracking markers on its surface. (c) Schematic drawing showing the axial forces and the undeformed transverse lengths that are used to determine the first

Piola–Kirchhoff tensions Holder Collagen gel Lens Step motor Force transducer Displacement sensor Be-Cu cantilever Collagen gel Ring illumination Porous bar A B f₂ f₂ f₁ f₁ L₁ L₂ C 1 2

placed within a 150× 20 (diameter×height, mm) glass Petri dish. Porous polyethylene bars (Small Parts, Miramar, FL) were positioned at each end of the four arms of the mold before adding the collagen–cell mixture. Cell-seeded colla-gen gels were formed in the mold at 37◦C in a humidified CO2incubator. The silicone gasket was removed after 30 min

of gelation, and the collagen gel was cultured in 40 mL of cul-ture media within an incubator for an additional 24 h before subjecting to a specific mechanical constraint.

2.2 Biaxial tissue culture chamber and culturing conditions Cell-seeded collagen gels were cultured under four differ-ent static mechanical constraints: free-floating, equibiaxial stretching (λ1:λ2= 1.01:1.01, 1.1:1.1, and 1.2:1.2, where λ1

andλ2are the global stretch ratios in the 1 and 2 directions,

respectively), strip-biaxial stretching (λ1:λ2 = 1.01:1.2),

and uniaxial stretching (λ2 = 1.2) (Fig.1b). For the

free-floating condition, cruciform-shaped cell-seeded collagen gels were simply cultured in a Petri dish at 37 ◦C in a humidified CO2incubator for 6 days. For all the other

con-ditions, a custom biaxial tissue culture chamber was used to impart the defined planar biaxial mechanical constraints to the cell-seeded collagen gel (Humphrey et al. 2008). Briefly,

the chamber was sterilized with autoclave before use. Under aseptic conditions, the cell-seeded collagen gel was clamped within the chamber via the porous polyethylene bars embed-ded at the end of its arms and stretched uniaxially or biaxially. The chamber was then covered and placed in a 37◦C humid-ified CO2incubator for 6 days.

2.3 Planar biaxial mechanical testing

A modified version of the biaxial test system described by

Wells et al.(2006) was employed. Briefly, the system

con-sists of four loading assemblies, a CCD camera with a zoom lens, two custom-made, highly sensitive force transducers, a specimen chamber, and a PC (Fig.2). The four loading assem-blies are driven by PC-controlled step motors to cyclically stretch the cruciform-shaped gel along its arms. The deforma-tion within the central region of the gel is monitored by the CCD camera, and the tensile force in each axis is measured by the force transducer. Specifically, the force transducer is composed of a cantilever beam (Be–Cu alloy) to which a grip-per is attached, and a displacement sensor (EX-110V, Key-ence, Japan) for measuring the deflection of the cantilever. The stiffness of the Be–Cu alloy as well as the dimensions of the cantilever was selected to minimize the deflection of

the cantilever at the target sensing range (0–5 mN). The force transducer was calibrated with known weights within the sens-ing range; the relationship between the weight and the deflec-tion appeared to be linear within the range. The modeflec-tion control, vision, and data acquisition of the system are integrated by LabVIEW (National Instruments, Austin, TX). In this study tension-controlled protocols were used, in which the ratio of the axial first Piola–Kirchhoff tensions (i.e., T11:T22) was kept

constant by two independent virtual PID controllers (one for each axis) during the loading and unloading cycle. The first Piola–Kirchhoff tensions are defined as T₁₁Exp = f1/L2and T₂₂Exp = f2/L1where f1and f2 are the measured tensile

forces in the 1 and 2 directions, respectively, and L1and L2

are the undeformed transverse length in the 1 and 2 directions, respectively (Fig.2c). Our preliminary experiments showed that the mechanical response of the gel was dependent on the strain rate (see Supplemental Figure 1 for an example). Thus, quasi-static loading was used in the test. Note that tension (also called stress resultant or membrane stress), instead of stress, was used to characterize the mechanical behavior of the gel for at least two reasons. First, the gel is thin with negligible bend-ing stiffness; it can thus be approximated as a membrane. Sec-ond, as the gel is highly hydrated, it is difficult to measure its thickness reproducibly; the accuracy of the calculated stress is compromised. Note, furthermore, that most water in the gel is not confined, and the total volume of the gel may change dur-ing the test. The incompressibility may not be used to deter-mine the out-of-plane strain; it is thus not possible to derive the three-dimensional constitutive model of the gel from our planar biaxial data.

After 6 days of culture, the chamber with the gel was removed from the incubator and mounted onto the biaxial test system. One holder in each axis was replaced by the force transducer. A number of 50-micron markers were delivered onto the surface of a central region of the gel and allowed to affix to the gel; at least nine of these markers were video-tracked to determine the 2-D deformation gradient tensor F2Dof the region throughout the test (seeHu et al.(2007)

for details). The gel was preconditioned equibiaxially for 3–5 times, and then the unloaded reference configuration was recorded. Subsequently, the gel was tested using an equibi-axial tension protocol (T11:T22= 1:1). After the mechanical

contribution of cell traction was removed via osmotic shock, the decellularized gel was subjected to the same testing pro-cedures for measuring its passive mechanical properties. In addition, proportional tension protocols (T11:T22 = 0.5:1,

0.75:1, 1:0.75, 1:0.5) were performed on the decellularized gel. Note that the equibiaxial tension protocol was performed after the equibiaxial preconditioning and repeated following the proportional tension protocols. The results of the two equibiaxial tension tests were compared to ensure that no mechanical damage occurred. For all of the gels tested in this

study, the results of the two tests were similar, indicating the loadings were within the elastic range of the gels. Note also that the kinematic analysis for a non-decellularized gel and its decellularized counterpart was based on distinct, unloaded reference configurations.

In this study the first Piola–Kirchhoff tension and the stretch ratio were used to characterize the mechanical behav-ior of the gel. Note that the first Piola–Kirchhoff tension is conjugate to the deformation gradient and hence to the stretch ratio in this case. The stretch ratios in the 1 and 2 direc-tions are extracted from the deformation gradient assum-ing that in-plane shears are negligible. That is, λ1 = F11

andλ2 = F22. As an important strain measure for

model-ing mechanical behavior, the 2-D Green strain tensor was also calculated using E2D = 1₂



FT_2D· F2D− I



. In addi-tion, the rigid body rotation angleφ was determined using φ = tan−1F12−F21

F11+F22 

(Mcculloch et al. 1987). Based on the tension-stretch data, mechanical anisotropy of the gel was quantified by an anisotropy index, defined as the ratio of the difference in stretch |λ2− λ1| to the mean stretch (λ1+ λ2) /2 under 0.1 N/m equibiaxial tension (Langdon

et al. 1999). Note that the anisotropy index is not valid as a measure of anisotropy for proportional tension protocols. A perfectly isotropic gel has an anisotropy index of zero. The greater the anisotropy index, the stronger the mechani-cal anisotropy of the gel.

2.4 Constitutive modeling

The nonlinear, anisotropic pseudo-elastic mechanical behavior of cell-seeded collagen gels was quantified by a hy-perelastic model. Consider a general nonlinear, anisotropic membrane subjected to an in-plane biaxial stretching wherein the deformation gradient is, in matrix form, as follows: [F] = ⎡ ⎣F11F21 F12F22 00 0 0 λ3(= h/H) ⎤ ⎦

where h and H are the thickness of the membrane in its deformed and undeformed configurations, respectively. A general 2-D constitutive relation between the 2-D Cauchy stressσ2Dand strain energyw, defined per unit undeformed

area, can be derived as follows (Humphrey et al. 1992): σ2D= 1 h J2DF2D· ∂w ∂E2D · F T 2D

where J2D= F11F22−F12F21is the determinant of F2D. The

2-D first Piola–Kirchhoff stress tensor is determined using P2D= _H1 _∂E2D∂w · FT2D. The in-plane first Piola–Kirchhoff

ten-sion tensor is then as follows:

T= HP2D= ∂w

∂E2D

· FT 2D.

Table 1 The maximum shear strain and maximum rigid body rotation angle in the central region of acellular gels and cell-seeded gels cultured under the various mechanical constraints throughout the test

Acellular Free floating 1.01 × 1.01 1.1 × 1.1 1.2 × 1.2 1.01 × 1.2 Uniaxial_1.2

(N= 6) (N= 5) (N= 6) (N= 5) (N= 5) (N= 5) (N= 5) Non-decellularized Max|E12| – 0.02 ± 0.00 0.02 ± 0.01 0.02 ± 0.02 0.00 ± 0.00 0.01 ± 0.01 0.01 ± 0.01 Max|φ| – 0.57 ± 0.24 0.48 ± 0.27 0.49 ± 0.41 0.23 ± 0.08 0.27 ± 0.24 0.33 ± 0.31 Decellularized Max|E12| 0.01 ± 0.01 0.00 ± 0.00 0.00 ± 0.01 0.00 ± 0.00 0.01 ± 0.01 0.00 ± 0.00 0.00 ± 0.00 Max|φ| 0.33 ± 0.23 0.23 ± 0.10 0.27 ± 0.20 0.25 ± 0.19 0.24 ± 0.22 0.15 ± 0.09 0.12 ± 0.06 The angles are displayed in degrees. Results are presented as mean± standard deviation

To fit the biaxial tension-strain data of the gel before or after decellularization, we used a 2-D Fung-type exponential strain energy functionw of the form:

w = c 2



expQ−1 

where Q= c1E₁₁2 +c2E₂₂2 +2c3E11E22and c, c1, c2and c3

are material parameters, and E11and E22are components of

the 2-D Green strain tensor in the 1 and 2 directions, respec-tively. Note, to satisfy physical and mathematical (convexity) constraints, c> 0, c1 > 0, c2 > 0, c3 > 0 and c1c2 > c2₃

(Humphrey 1999;Holzapfel et al. 2000). The nonzero com-ponents of T are as follows:

T₁₁Theory= cF11expQ(c1E11+ c3E22) , and T₂₂Theory= cF22expQ(c2E22+ c3E11) .

Best-fit material parameters for each gel were determined using a nonlinear regression. This was accomplished using a modified fminsearch function (Matlab) to minimize the objective function (fitting error):

e= N i=1  T₁₁Theory− T₁₁Exp 2 i +  T₂₂Theory− T₂₂Exp 2 i  . The goodness of fit for the model was assessed by the coef-ficient of determination (r2).

Given the material parameters, the value of stored strain energy of the gel, which was used as a stiffness index, was calculated at E11 = E22 = 0.05. The higher the stored strain

energy, the stiffer the gel. The material stiffness tensor of the gel, defined as K = _∂E2D∂S2D where S2D = _∂E2D∂w is the 2-D

second Piola–Kirchhoff tension tensor, was also determined. Particularly, the ratio of stiffness in the 1 and 2 directions (i.e.,(K )1111/K2222) was calculated at E11 = E22 = 0.05

and used as another anisotropy index. Note that the second Piola–Kirchhoff tension is conjugate to the Green strain.

Finally, for evaluation of the predictive capability of the model, results of proportional tension protocols (excluding the equibiaxial tension protocol) from a decellularized gel were fit for the material parameters. The material parameters

were then used to simulate the mechanical behavior of the gel under equibiaxial tension.

2.5 Statistical analysis

Differences before and after decellularization were assessed via paired t test. For comparisons among the various mechan-ical constraints, one-way ANOVA with Holm–Sidak post hoc testing (significance level p= 0.05) was used. Holm–Sidak testing is useful to correct for false positives enumerated by the Bonferroni inequality and adequate for the relatively few number of comparisons in this study. Results are reported as mean± standard deviation.

3 Results

3.1 Validation of the biaxial mechanical testing

For all of the acellular gels and cell-seeded gels examined in this study, the shear strain (E12) and the rigid body rotation

angle (φ) in the central region were negligible throughout the test (Table1and Supplemental Figure 2).

3.2 Biaxial mechanical properties of the cell-seeded collagen gels

The anisotropy indices of the gels, determined based on the tension-stretch data, are shown in Fig.3. Although gels cul-tured under the free-floating condition had a greater variation in anisotropy index than those subjected to the equibiaxial stretching conditions, the anisotropy indices of all of these gels as well as acellular gels were close to zero. On the other hand, gels cultured under the strip-biaxial stretching condi-tion had a moderate anisotropy index, while gels cultured under the uniaxial stretching condition had a higher anisot-ropy index. Moreover, the anisotanisot-ropy indices of the gels did not change significantly after decellularization for all of the

Acellular Free floating1.01x1.01 1.1x1.1 1.2x1.21.01x1.2 Uniaxial_1.2 Anisotropy index 0.0 0.1 0.2 0.3 * * * * * * * * * * * * * *

Fig. 3 Anisotropy indices of acellular gels and cell-seeded gels cul-tured under the various mechanical constraints. Solid bars represent non-decellularized gels, while open bars represent acellular and decell-ularized gels. Asterisk means p< 0.05

mechanical constraints except the 1.01 × 1.2 strip-biaxial stretching condition.

As acellular gels and cell-seeded gels cultured under the free-floating and equibiaxial stretching conditions appeared to be isotropic, it seems reasonable to combine tension-stretch data of the two orthogonal axes for analysis; it is in fact not practical to define the 1 and 2 directions in these cases as the mechanical constraints in both directions are no different. Figure4 shows that for cell-seeded gels cultured under the equibiaxial stretching conditions, the gels became stiffer as the stretch ratio increased. Moreover, compared with the acellular gels, all of the cell-seeded gels appeared to be less stiff. Also, a decellularized gel was stiffer than its non-decellularized counterpart. Note that the stretches were calculated based on different unloaded reference configura-tions for a non-decellularized gel and its decellularized coun-terpart. Again, gels cultured under the free-floating condi-tion exhibited a great variacondi-tion in the tension-stretch curves. After decellularization, the tension-stretch curves of the gels appeared to be more linear. Although the decellularized gels appeared stiffer in the case of the 1.2×1.2 equibiaxial stretch-ing condition, the difference in tension-stretch curves among the five groups was not significant.

Figure5 shows that gels cultured under the strip-biax-ial stretching condition developed a moderate degree of mechanical anisotropy while gels cultured under the uniaxial stretching condition developed a strong degree of mechani-cal anisotropy. Both groups of the gels became stiffer after decellularization. The mechanical anisotropy after decellu-larization appeared to be preserved well in the case of the uniaxial stretching condition and less well in the case of the strip-biaxial stretching condition.

1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Stretch ratio 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Acellular A Free floating B λ=1.01 λ=1.01 C λ=1.1 λ=1.1 D λ=1.2 λ=1.2 E

Fig. 4 Equibiaxial tension data for acellular gels (a) and cell-seeded gels cultured under free-floating (b) and equibiaxial stretching (c 1.01× 1.01, d 1.1 × 1.1, e 1.2 × 1.2) conditions. Solid symbols represent non-decellularized gels, while open symbols represent non-decellularized gels

1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Stretch ratio 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Stretch ratio 1.0 1.1 1.2 1.3 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 λ2=1.2 λ1=1.01 λ2=1.2 unconstrained A B C D

Fig. 5 Equibiaxial tension data for cell-seeded gels cultured under strip-biaxial stretching (a and c 1.01 × 1.2) and uniaxial stretching (b and d unconstrained× 1.2) conditions. Solid symbols represent

non-decellularized gels (a and b), while open symbols represent decellu-larized gels (c and d). Filled circle and open circle T11− E11, filled

inverted triangle and open inverted triangle T22− E22

3.3 Constitutive modeling

The Fung model was used to quantify the mechanical charac-teristics of the gels. Best-fit values of the material parameters obtained by fitting the Fung model to equibiaxial tension data of each cell-seeded gel and its decellularized counterpart are listed in Tables2and3, respectively. The Fung model pro-vided great fitting to the equibiaxial tension data, and the coefficient of determination (r2) for each fitting was greater than 0.98. Nevertheless, if the data of each gel in the same group were combined and fitted for the Fung model, the r2 for each fitting was significantly worse.

The value of stored strain energy for each gel was cal-culated based on the material parameters in Tables2and3. Figure6shows the statistical comparisons of the values of stored strain energy for acellular gels and cell-seeded gels cultured under the free-floating and equibiaxial stretching conditions. There was significant difference between the 1.2 × 1.2 equibiaxial stretching and other groups before decellularization. After decellularization, gels cultured under the 1.2 × 1.2 equibiaxial stretching and 1.01 × 1.01 equibi-axial stretching conditions remained significantly different. Furthermore, gels cultured under the 1.2 × 1.2 equibiaxial stretching condition stored significantly greater strain energy than acellular gels at the prescribed strains. Note that all of the groups showed significant difference in the value of stored

strain energy before and after decellularization except the 1.2 × 1.2 equibiaxial stretching condition.

Also calculated based on the modeling results, the ratio of stiffness in the stretching directions was served as another index for mechanical anisotropy. The comparisons of the ratios of stiffness among the seven groups are shown in

Fig.7. Consistent with the findings in Fig.3, the ratios of

stiffness were close to one for gels cultured under free-float-ing and equibiaxial stretchfree-float-ing conditions, while those for gels cultured under the strip-biaxial stretching and uniaxial stretching conditions were less than one. There was signif-icant difference between the uniaxial stretching and other groups before decellularization except the pair of the uniax-ial and strip-biaxuniax-ial stretching conditions. After decellular-ization, significant differences were observed between the uniaxial stretching and all of the other groups. No significant difference in the ratio of stiffness was found before and after decellularization for each group, however.

Best-fit values of the material parameters obtained by fit-ting the Fung model to proportional tension data of each acellular gel and decellularized gel are listed in Table 4. Fair fittings of the proportional tension data were achieved for the Fung model, and the mean r2 _{for each group was}

ranged from 0.88 to 0.99. Again, if the data of each gel in the same group were combined and fitted for the Fung model, the r2 _{for each fitting was significantly worse.}

Table 2 Best-fit values of the material parameters and the goodness of fit for the Fung model for each cell-seeded gel cultured under the various mechanical constraints (non-decellularized gels)

Specimen Material parameters Goodness

of fit c (mN/m) c1 c2 c3 r2 Free floating 1 152.94 0.81 0.75 0.78 0.999 2 7.00 12.97 12.49 11.71 0.996 3 45.19 2.59 3.62 3.06 0.995 4 7.24 4.15 4.10 4.12 0.998 5 14.18 7.58 9.22 8.36 0.996 Group 170,161.61 0.00 0.00 0.00 0.753 1.01 × 1.01 1 3.70 14.36 12.98 12.95 0.999 2 3.85 21.02 21.07 21.04 0.997 3 30.60 5.57 5.01 0.81 0.998 4 4.41 18.63 16.88 17.73 0.999 5 16.00 5.05 5.36 5.20 0.997 6 5.63 9.45 9.88 8.43 0.998 Group 32.87 0.52 0.52 0.52 0.829 1.1 × 1.1 1 10.77 11.39 12.26 11.06 0.998 2 28.59 5.72 5.51 5.43 0.999 3 13.84 5.72 5.66 5.69 0.995 4 17.35 15.04 14.39 10.40 0.997 5 7.89 21.72 20.36 16.70 0.999 Group 99,680.24 0.00 0.00 0.00 0.784 1.2 × 1.2 1 4.65 75.23 69.30 16.92 0.996 2 11.85 59.89 61.72 60.80 0.993 3 13.11 74.48 73.69 74.09 0.988 4 4.99 73.23 72.50 72.86 0.999 5 8.33 33.61 35.03 34.31 0.999 Group 83.13 9.77 0.68 0.72 0.790 1.01 × 1.2 1 2.01 13.86 13.21 13.53 0.994 2 15.13 19.94 22.51 21.18 0.993 3 10.20 13.99 13.89 13.94 0.990 4 5.97 20.08 24.71 22.28 0.999 5 5.02 38.61 48.42 43.24 0.999 Group 59,011.98 0.00 0.00 0.00 0.779 Uniaxial_1.2 1 5.56 10.37 43.42 10.93 0.999 2 4.70 24.22 48.87 17.89 0.999 3 2.36 30.78 73.18 23.16 0.999 4 10.04 28.84 38.48 33.31 0.991 5 11.59 29.08 36.08 32.39 0.986 Group 160,841.13 0.00 0.00 0.00 0.673

Table 3 Best-fit values of the material parameters and the goodness of fit for the Fung model for each acellular gel and cell-seeded gel cultured under the various mechanical constraints (decellularized gels)

Specimen Material parameters Goodness

of fit c (mN/m) c1 c2 c3 r2 Acellular 1 2.22 121.81 122.90 121.57 0.999 2 3.27 84.71 81.10 65.71 0.999 3 1.34 77.92 75.03 71.22 0.999 4 3.70 48.81 49.78 49.29 0.994 5 0.87 122.09 123.57 122.83 0.993 6 11.13 23.43 24.01 23.72 0.998 Group 63.38 0.79 0.78 0.79 0.843 Free floating 1 44.16 11.20 10.82 11.01 0.996 2 5.48 148.48 159.84 113.30 0.999 3 7.45 72.86 68.59 67.36 0.999 4 2.17 72.82 72.42 72.62 0.998 5 14.28 69.86 69.28 69.57 0.997 Group 88.17 0.80 0.71 0.75 0.728 1.01 × 1.01 1 5.66 37.71 36.97 37.34 0.999 2 3.39 113.49 109.49 107.21 0.999 3 578.12 0.91 0.93 0.92 0.994 4 6.88 48.07 43.59 36.13 0.999 5 10.21 24.38 25.96 25.16 0.999 6 5.96 37.47 37.20 37.33 0.999 Group 64.57 0.76 0.80 0.78 0.885 1.1 × 1.1 1 2,342.75 0.81 0.80 0.80 0.982 2 377,132.85 0.00 0.00 0.00 0.997 3 30.90 43.03 43.99 43.29 0.999 4 16.63 45.12 45.49 45.30 0.988 5 9.54 61.31 57.28 46.63 0.998 Group 140.34 0.80 0.73 0.76 0.818 1.2 × 1.2 1 13.30 155.31 145.13 121.02 0.997 2 4.70 113.66 114.05 113.85 0.983 3 14.57 242.55 259.84 1.03 0.996 4 142.15 9.62 9.41 9.52 0.993 5 5.70 79.61 81.67 80.63 0.999 Group 132.72 0.85 0.70 0.77 0.70 1.01 × 1.2 1 5.66 59.23 80.76 50.66 0.999 2 5.64 128.50 136.47 132.43 0.999 3 4.20 86.07 87.69 86.87 0.999 4 4.24 97.43 110.42 97.71 0.999 5 7.79 105.94 112.02 108.94 0.990 Group 104.92 0.76 0.80 0.78 0.825

Table 3 Continued

Specimen Material parameters Goodness

of fit c (mN/m) c1 c2 c3 r2 Uniaxial_1.2 1 8.51 80.73 116.53 73.74 0.999 2 4.53 117.73 127.35 122.44 0.998 3 2.70 119.09 199.01 110.52 0.999 4 3.87 94.89 109.44 101.91 0.993 5 2.61 191.05 313.60 222.10 0.999 Group 100.67 0.84 1.00 0.91 0.897 Acellular Free floating 1.01x1.01 1.1x1.1 1.2x1.2

Stored strain energy (J/m

2 ) 0.00 0.01 0.02 * * * * * * * *

Fig. 6 Values of stored strain energy of acellular gels and cell-seeded gels cultured under free-floating and equibiaxial stretching conditions.

Solid bars represent non-decellularized gels, while open bars represent

acellular and decellularized gels. Asterisk means p< 0.05

Acellular Free floating1.01x1.01 1.1x1.1 1.2x1.21.01x1.2 Uniaxial_1.2 K1111 /K 2222 0.0 0.5 1.0 1.5 2.0 * * _* * * _* * * * *

Fig. 7 Ratios of stiffness in the stretching directions of acellular gels and cell-seeded gels cultured under the various mechanical constraints.

Solid bars represent non-decellularized gels, while open bars represent

acellular and decellularized gels. Asterisk means p< 0.05

Figures8,9,10,11,12,13,14show the proportional ten-sion data of acellular gels and decellularized gels that were cultured under the various mechanical constraints and their fits by the Fung model. The Fung model appeared to fit

Table 4 Best-fit values of the material parameters and the goodness of fit for the Fung model for each acellular gel or cell-seeded gel cultured under the various mechanical constraints (Decellularized gels)

Specimen Material parameters Goodness

of fit c (mN/m) c1 c2 c3 r2 Acellular 1 2.10 189.13 161.99 85.02 0.982 2 3.02 86.81 111.34 38.32 0.909 3 2.17 80.04 84.50 31.94 0.929 4 375.77 2.47 1.95 0.99 0.853 5 339,872.30 0.00 0.00 0.00 0.798 6 29.71 33.58 27.38 11.51 0.874 Group 17,459.75 0.00 0.00 0.00 0.828 Free floating 1 21.68 32.06 28.02 10.88 0.997 2 3.32 278.78 371.18 92.29 0.986 3 513,065.07 0.00 0.00 0.00 0.759 4 2.83 88.96 236.45 57.70 0.948 5 14.01 78.03 133.65 68.22 0.984 Group 67,837.15 0.00 0.00 0.00 0.701 1.01 × 1.01 1 8.75 58.53 49.41 23.20 0.984 2 3.66 218.06 156.85 92.19 0.993 3 452,563.69 0.00 0.00 0.00 0.995 4 3.71 104.00 83.83 43.53 0.985 5 6.91 43.16 61.14 16.46 0.994 6 5.78 71.18 76.10 19.76 0.989 Group 6,472.02 0.01 0.01 0.01 0.864 1.1 × 1.1 1 50.90 85.03 119.82 69.21 0.970 2 280.43 5.80 6.21 2.77 0.988 3 23.64 59.24 86.13 41.71 0.957 4 304.56 8.72 8.98 4.47 0.969 5 61.37 20.73 21.27 7.86 0.921 Group 30,449.79 0.01 0.01 0.00 0.758 1.2 × 1.2 1 349,332.41 0.01 0.01 0.00 0.887 2 720,146.66 0.00 0.00 0.00 0.684 3 30,907.94 0.37 0.37 0.30 0.956 4 23.17 79.35 90.22 47.08 0.941 5 110,067.69 0.02 0.02 0.01 0.939 Group 32,673.88 0.01 0.00 0.00 0.682 1.01 × 1.02 1 3.68 85.91 170.93 50.99 0.991 2 3.14 210.19 307.08 95.22 0.974 3 3.91 117.90 153.26 43.31 0.984 4 4.03 138.10 190.75 86.42 0.967 5 2.18 241.48 508.55 182.95 0.992 Group 8,718.09 0.01 0.01 0.01 0.828

Table 4 Continued

Specimen Material parameters Goodness

of fit c (mN/m) c1 c2 c3 r2 Unload_1.02 1 3.94 131.15 353.94 78.64 0.989 2 4.15 126.46 327.47 78.47 0.983 3 2.55 118.55 303.23 77.41 0.990 4 3.00 124.48 205.27 121.84 0.903 5 4.73 140.84 295.85 157.10 0.931 Group 0.62 23.54 39.19 21.61 0.836

the nonlinear mechanical behavior of the gels well, partic-ularly the gels cultured under strip-biaxial stretching and uniaxial stretching conditions. Inconsistent fittings of propor-tional tension data were observed, however, if the mechanical behavior of the gels appeared to be linear and isotropic. Note that if an exceptional large c was obtained, the fitting came along with negligible c1, c2, and c3.

3.4 Predictive capability of the constitutive model

The material parameters for each gel in Table4were used to simulate the mechanical behavior of the gel under equibiaxial tension. Figure15illustrates the experimental and simulated equibiaxial tension data of a representative acellular gel and representative decellularized gels that were cultured under the various mechanical constraints. It appeared that the pre-dictive capability of the Fung model for the gels was fair.

4 Discussion

The microstructure of the cell-seeded collagen gels devel-oped under the various mechanical constraints was investi-gated previously (Hu et al. 2009). Changes in mechanical properties due to the mechanical constraints were identified in this study. Similar to the microstructural findings, the biax-ial mechanical properties of the gels appeared to be influ-enced by the mechanical constraints. Notably, the mechanical isotropy or anisotropy observed in this study appeared to be correlated well with the irreversible collagen fiber alignment shown in the previous study (Hu et al. 2009).

Specifically in this study, we quantified and compared, for the first time, the biaxial mechanical properties of the cell-seeded collagen gels cultured under different extents of equibiaxial stretching. Contrary to previous findings that cell-seeded gels become stiffer when they are allowed to contract (Chen et al. 2008;Berry et al. 2009), cell-seeded gels cul-tured under the 1.2 × 1.2 equibiaxial stretching condition displaced the stiffest mechanical properties among the other

equibiaxial stretching conditions as well as the free-floating condition, particularly before the gels were decellularized. Uniaxially constrained, rectangular cell-seeded collagen gels tend to contract laterally over time in culture, resulting in an axially stiffer gel (Feng et al. 2006;Grenier et al. 2005;Shi and Vesely 2003). We further showed that applied uniaxial stretch expedited and enhanced the structural remodeling if the gel was laterally unconstrained (Hu et al. 2009). The stiff-ness increase in the uniaxially constrained gels is mainly due to the enhanced fiber alignment, which, however, may not be responsible for the stiffness increase in the equibiaxially constrained gels as their microstructure remained isotropic. The stiffness increase in gels cultured under the 1.2 × 1.2 equibiaxial stretching condition may be in part attributed to a mechanism underlying the so-called plastic compression. Plastic compression is essentially an irreversible process of water expulsion accomplished by unconfined compression of the collagen gel (Brown et al. 2005). The collagen gel is a random mesh of collagen fibers embedded in a signifi-cant amount of fluid (99 %). The excess fluid in the gel is a result of the casting, rather than any inherent swelling prop-erty of the collagen (Brown et al. 2005). This unique propprop-erty may allow the applied equibiaxial stretching to enhance the mechanical properties of the collagen gel without the par-ticipation of cells. Note, however, that in preliminary experi-ments biaxial mechanical properties of the gel were examined before and after enforcing the various mechanical constraints to ensure that the extent of each mechanical constraint did not cause irreversible mechanical changes. Therefore, all of the mechanical changes in the gels after 6 days of culture might be well attributed to a cell-mediated matrix remodeling pro-cess. Although the 1.2 × 1.2 equibiaxial stretching is still within the elastic range of the gels, the applied equibiaxial stretching might enhance the cell-mediated gel contraction, facilitating irreversible water expulsion and resulting in a stiffer gel. It is interesting to note thatThomopoulos et al. (2005) reported that acellular gels also yielded irreversible mechanical anisotropy when subjected to a constant uniaxial force (200 mg); that is, the presence of cells is not responsible for the change. In that case the applied uniaxial force may exceed the elastic limit of the gel and cause plastic deforma-tion. Nevertheless, our preliminary experiments showed that acellular gels cultured under the uniaxial stretching condi-tion did not develop mechanical anisotropy over time, impli-cating the importance of cell-mediated matrix remodeling. Note also that gels were cultured under isotonic mechanical loadings in their studies, whereas gels were cultured under isometric mechanical constraints in our studies.

In the biaxial mechanical testing of the gels, the resistance due to interstitial fluid flow is negligible compared to that for stretching the solid network phase (Chandran and Barocas 2004). Also, quasi-static loading that was used during the test minimized the viscous effect of the gel on its mechanical

Acellular Tii:Tjj=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Tii:Tjj=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 F C B E D A

Fig. 8 Proportional tension data for acellular gels (a Tii:Tj j = 0.5:1,

d Tii:Tj j = 0.75:1; i, j are the orthogonal directions).

Representa-tive proportional tension data (solid symbols) from an acellular gel and the fits (open symbols) by the Fung model (b T11:T22 = 0.5:1, c T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Filled

cir-cle and open circir-cle T11− E11, filled inverted triangle and open inverted

triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

behavior. Indeed, in order to fully characterize the mechani-cal behavior of cell-seeded collagen gels, the biphasic charac-teristics of the gel and the viscoelastic properties of collagen fibers must be taken into account. Nevertheless, the purpose of mechanical testing in this study was to probe the tensile behavior of the solid network phase of the gel as our focus is on the remodeling of the solid network phase due to the various mechanical constraints.

In this study, sequential measurements of mechanical properties were performed on each gel before and after decellularization. Cell traction is believed to participate in reversible collagen fiber alignment. Removal of cells or deac-tivation of cell traction thus reveals the passive mechanical properties that may have been irreversibly changed.

Gen-erally, irreversible collagen fiber alignment involves mech-anisms such as formation of chemical bonds that entrench alignment or matrix deposition at preferred orientations.

Marenzana et al.(2006) used a transducer to monitor directly

the forces generated by fibroblast-populated collagen gels over 60 h of culture and showed that tension results from a combination of cell-mediated traction and a gradually increasing “residual matrix tension.” They suggested that this residual matrix tension, which accounted for∼50 % of mea-sured force after 60 h of culture, indicated “a time-dependent shortening of the collagen network, progressively stabilized into a built-in tension within the matrix.” The irreversible mechanical anisotropy found in this study is consistent with the existence of a residual matrix tension that is

presum-Free floating Tii:Tjj=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Tii:Tjj=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 A D B E F Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 C

Fig. 9 Proportional tension data for decellularized gels that were cul-tured under the free-floating condition (a Tii:Tj j = 0.5:1, d Tii:Tj j =

0.75:1; i, j are the orthogonal directions). Representative proportional tension data (solid symbols) from a gel of the group and the fits (open

symbols) by the Fung model (b T11:T22= 0.5:1, c T11:T22= 1:0.5,

e T11:T22= 0.75:1, f T11:T22= 1:0.75). Filled circle and open circle

T11− E11, filled inverted triangle and open inverted triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

ably related to a tendency toward tensional homeostasis. A similar “stable remodeling” of collagen gels by fibroblasts was reported bySawhney and Howard(2002), who empha-sized the importance of fibroblast-generated traction fields in remodeling the matrix as reemphasized byDahlmann-Noor

et al.(2007).

The mechanical properties of gels cultured under the free-floating condition had a greater variability than the other gels that were cultured under the mechanical constraints. In the free-floating condition, the porous bars at the end of the arms of the gel were actually floated over the medium while the gel was suspended in the medium. Although efforts were made to make the gel and porous bars at the same level, the shape of the gel after 6 days of culture generally deviated from

the original cruciform shape. This may lead to a regionally heterogeneous microstructure and result in the great variation in the mechanical properties.

Two methods of attaching a specimen to a loading system have proven particularly useful in biaxial mechanical testing of planar soft tissues: the use of arrays of sutures to couple square specimens to the system and the use of end-clamps to couple cruciform specimens to the system (Humphrey et al. 2008). Although neither experimental configuration is per-fect, the limited suture retention strength of collagen gels suggests the use of cruciform samples wherein ends can be secured well for loading.Waldman and Lee(2005) showed in native pericardium, for example, that mechanical prop-erties can be estimated consistently using cruciform-shaped

1.01x1.01 Tii:Tjj=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Tii:Tjj=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 F C B E A D

Fig. 10 Proportional tension data for decellularized gels that were cul-tured under the 1.01×1.01 equibiaxial stretching condition (a Tii:Tj j=

0.5:1, d Tii:Tj j= 0.75:1; i, j are the orthogonal directions).

Represen-tative proportional tension data (solid symbols) from a gel of the group and the fits (open symbols) by the Fung model (b T11:T22= 0.5:1, c

T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Filled circle and open circle T11− E11, filled inverted triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

samples wherein the arms are 50–100 % of the length of the central region, but samples with arms only 5 % of the length of the central region result in an overly constrained mate-rial with an artificially high stiffness. Whereas

Thomopou-los et al.(2005) used biaxial specimens with very short arms,

our cruciform specimens had arms 200 % of the length of the central region. Also note thatWaldman and Lee(2005) cau-tioned that “any non-symmetrical fiber arrangement about the material test axes should produce a centralized sample rotation toward the direction of stretch under equibiaxial loading.” For all of the gels examined in this study, both the shear strain and the rigid body rotation angle in the central region were negligible throughout the test, suggesting that the

collagen fibers in the central region were either randomly oriented or symmetrically aligned to one of the material test axes. This, in fact, is consistent with microstructural findings reported in the previous study (Hu et al. 2009). Note that it is important not to infer shear information from planar biaxial tests because of the inability to impose defined shear stresses (Holzapfel and Ogden 2009).

In this study, the cell seeding density was twice as much as that used in the previous study. Perhaps limited by the sensitivity of the force transducer, our preliminary experi-ments showed that mechanical properties of the gel were not significantly changed at day 3 or if the original (i.e., half of the current study) cell seeding density was used. These

1.1x1.1 Tii:Tjj=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Tii:Tjj=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 F C B E A D

Fig. 11 Proportional tension data for decellularized gels that were cul-tured under the 1.1 × 1.1 equibiaxial stretching condition (a Tii:Tj j=

0.5:1, d Tii:Tj j= 0.75:1; i, j are the orthogonal directions).

Represen-tative proportional tension data (solid symbols) from a gel of the group and the fits (open symbols) by the Fung model (b T11:T22 = 0.5:1,

c T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Filled

cir-cle and open circir-cle T11− E11, filled inverted triangle and open inverted

triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

indicated the important role of cells and culturing time in the remodeling of the gels. Note that although the sensitiv-ity of the force transducer can be further increased, it may complicate the mechanical testing as the determination of the unloaded, reference state of the gel, which is the most technically difficult part in this study, could become more challenging.

It was found that planar biaxial constraints, in con-trast to the uniaxial constraint, prohibit development of an extreme structural anisotropy due to gel compaction and that structural anisotropy imposed initially in a strip-biax-ially stretched region of a gel became irreversible after 6 days of culture (Hu et al. 2009). It is interesting to note

that the structural anisotropy was not obvious until embed-ded cells were removed; that is, cell traction appeared to interfere in the observation of irreversible fiber re-orienta-tion. The mechanical anisotropy developed by the strip-biax-ial stretching became less obvious after decellularization, however, suggesting that the embedded cells by themselves may be positively involved in the mechanical anisotropy and cell traction may not influence the mechanical anisot-ropy of the gel. Also note that 6 days may not be suf-ficient to obtain consistent, fully irreversible mechanical anisotropy for the gels. As the biaxial mechanical proper-ties of the gels were examined at day 6 only, it remains to be clarified whether the mechanical anisotropy induced by

1.2x1.2 Tii:Tjj=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Tii:Tjj=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 F C B E A D

Fig. 12 Proportional tension data for decellularized gels that were cul-tured under the 1.2 × 1.2 equibiaxial stretching condition (a Tii:Tj j=

0.5:1, d Tii:Tj j= 0.75:1; i, j are the orthogonal directions).

Represen-tative proportional tension data (solid symbols) from a gel of the group and the fits (open symbols) by the Fung model (b T11:T22 = 0.5:1,

c T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Filled

cir-cle and open circir-cle T11− E11, filled inverted triangle and open inverted

triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

culturing under the strip-biaxial stretching condition will be enhanced or remain unchanged beyond 6 days of cul-ture. Note also that cell-seeded collagen gels, like other native or artificial tissues that contain cells, are capable of remodeling themselves to adapt environments. Lee et al. (2008) demonstrated that anisotropic cell-seeded collagen gels that are initially developed under mechanical constraints undergo structural and mechanical remodeling in response to altered loading conditions. As pointed out by Lee et al., it needs to be emphasized that cell-seeded tissue substitutes can undergo growth and remodeling and their structural and mechanical properties may change over time after implanta-tion.

Although the Fung model appeared to fit the nonlinear ten-sion-strain curves well, it did not perform comparably well on the linear tension-strain curves probably due to the nature of the exponential function in the model. Nevertheless, quantifi-cation of the biaxial mechanical properties by the model pro-vided valuable indices for statistical comparisons. Note also that if the data of each gel in the same group were combined for fitting, the r2of all of the groups were poor. Although it may in part be explained by the variations in the mechan-ical behavior of the gels, it could be attributed to the use of an inappropriate model. Finally, the Fung model is purely phenomenological, involves no structural information, and has limited physical meaning. Structure-motivated

1.01x1.2 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 H D G C B F A E

Fig. 13 Proportional tension data for decellularized gels that were cul-tured under 1.01 × 1.2 strip-biaxial stretching condition (a T11:T22= 0.5:1, b T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Rep-resentative proportional tension data (solid symbols) from a gel of the group and the fits (open symbols) by the Fung model (c T11:T22= 0.5:1,

d T11:T22= 1:0.5, g T11:T22= 0.75:1, h T11:T22= 1:0.75). Filled

cir-cle and open circir-cle T11− E11, filled inverted triangle and open inverted

triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

phenomenological constitutive models (e.g., Lanir 1983; Humphrey and Yin 1987) that may help elucidate the under-lying cause of mechanical behavior will be evaluated in fitting the experimental data in a future study.

Considering extraction of information from the data, it would be ideal to have a single growth and remodeling model that accounts for the outcomes of the various mechanical con-straints. Development of such a model could provide more

Uniaxial_1.2 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 A E T11:T22=1:0.75 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 B F -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.75:1 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=0.5:1 G C Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.75 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 T11:T22=1:0.5 H D

Fig. 14 Proportional tension data for decellularized gels that were cul-tured under 1.2 uniaxial stretching condition (a T11:T22 = 0.5:1, b

T11:T22= 1:0.5, e T11:T22= 0.75:1, f T11:T22= 1:0.75). Representa-tive proportional tension data (solid symbols) from a gel of the group and the fits (open symbols) by the Fung model (c T11:T22 = 0.5:1,

d T11:T22= 1:0.5, g T11:T22= 0.75:1, h T11:T22= 1:0.75). Filled

cir-cle and open circir-cle T11− E11, filled inverted triangle and open inverted

triangle T22− E22. Note that the best-fit material parameters were used to predict the mechanical behavior of the gel under equibiaxial tension in Fig.15

insight into the underlying mechanobiology. Nevertheless, additional experiments are definitely required to achieve the goal. For example, time courses of the structural and

mechan-ical changes of gels need to be examined. Also, an afore-mentioned structure-based model would be more appropriate serving as a base for such a model.

-0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 A F B G C -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 D Green strain -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 Tension (N/m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 E

Fig. 15 Prediction of equibiaxial tension data for a representative acel-lular gel a and representative decelacel-lularized gels that were cultured under the various mechanical constraints (b free-floating, c 1.01×1.01, d 1.1 × 1.1, e 1.2 × 1.2, f 1.01 × 1.2, g Uniaxial_1.2). Filled circle

(experimental data) and solid line (predicted values) T11− E11, filled

inverted triangle (experimental data) and dash line (predicted values) T22− E22

The application of the static mechanical constraints in this study represents a single mechanical stimulus; its effects may decay over time. Cyclic stretching, on the other hand, represents a continuous mechanical stimu-lus. In fact, most cells in the body are under dynamic mechanical stimuli in physiological conditions. We have

thus designed and built a planar biaxial dynamic stretcher, onto which our culturing chamber can be mounted. The influence of cyclic biaxial stretching on these gels will be examined, which likely will lead to better understanding of the role of mechanical conditioning on tissue develop-ment.

5 Conclusions

This study demonstrated that the various mechanical constraints have different effects on the biaxial mechanical properties of the cell-seeded collagen gels. The nonlinear pseudo-elastic properties of the gels were quantified by the Fung model. Statistical comparisons of stiffness and mechan-ical anisotropy for the various mechanmechan-ical constraints were made. Culturing under the strip-biaxial stretching condition resulted in the gels with a modest degree of mechanical anisotropy while the uniaxial stretching induced a strong degree of mechanical anisotropy in the gels. Although the gels cultured under the equibiaxial stretching condition remained mechanically isotropic, increased extent of equi-biaxial stretching increased the stiffness of the gel signifi-cantly.

Acknowledgments This research was supported by a grant from the National Science Council in Taiwan (NSC-97-2218-E-006-289-MY2).

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