Biomech Model Mechanobiol DOI 10.1007/s10237-014-0644-y
ORIGINAL PAPER
Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering Jin-Jia Hu
Received: 6 October 2014 / Accepted: 16 December 2014 © Springer-Verlag Berlin Heidelberg 2015
Abstract In this study, we sought to model the mechanical behavior of an electrospun tubular scaffold previously reported for vascular tissue engineering with hyperelastic constitutive equations. Specifically, the scaffolds were made by wrapping electrospun polycaprolactone membranes that contain aligned fibers around a mandrel in such a way that they have microstructure similar to the native arterial media. The biaxial stress-stretch data of the scaffolds made of moderately or highly aligned fibers with three different off-axis fiber angles α (30◦ , 45◦ , and 60◦ ) were fit by a phenomenological Fung model and a series of structurally motivated models considering fiber directions and fiber angle distributions. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was also added to the structurally motivated models to examine its contribution. The two forms of fiber strain energy did not result in significantly different goodness of fit for most groups of the scaffolds. The absence of the neo-Hookean term in the structurally motivated model led to obvious nonlinear stressstretch fits at a greater axial stretch, especially when fitting data from the scaffolds with a small α. Of the models considered, the Fung model had the overall best fitting results; Electronic supplementary material The online version of this article (doi:10.1007/s10237-014-0644-y) 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
its applications are limited because of its phenomenological nature. Although a structurally motivated model using the nonlinear fiber stress–strain relation with the neo-Hookean term provided fits comparably as good as the Fung model, the values of its model parameters exhibited large within-group variations. Prescribing the dispersion of fiber orientation in the structurally motivated model, however, reduced the variations without compromising the fits and was thus considered to be the best structurally motivated model for the scaffolds. It appeared that the structurally motivated models could be further improved for fitting the mechanical behavior of the electrospun scaffold; fiber interactions are suggested to be considered in future models. Keywords Constitutive modeling · Hyperelasticity · Electrospun scaffolds · Fung model · Structurally motivated models · Mechanical properties
1 Introduction Scaffolds, cells and stimulating signals, generally referred to as the tissue engineering triad, are the key components for making a functional tissue-engineered construct. Specifically, the three-dimensional, porous scaffolds not only provide a surface for cells to adhere and grow, but also serve as a template to guide matrix formation. Manipulation of the microstructure of the scaffolds thus has potential to control the microstructure of resulting tissues and hence their mechanical properties. The latter are particularly important for the functionality of load-bearing tissues. Among the numerous polymer processing techniques, electrospinning has been widely used to fabricate scaffolds. Electrospun scaffolds are attractive because of their high porosity, large surface area-to-volume ratio, and nano-scale
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fibrous structure, similar to the features of native extracellular matrix. In general, only a simple, inexpensive setup is required for electrospinning (Ramakrishna 2005). Although the orientations of electrospun fibers are typically random, aligned electrospun fibers can be achieved with special collecting mechanisms (Teo and Ramakrishna 2006). The aligned electrospun fibers are capable of providing contact guidance to cultured cells (Bashur et al. 2006; Wang et al. 2011; Yang et al. 2005) and can be used to control the microstructure of engineered tissues. Very few tubular scaffolds that have helically oriented fibers were found in the literature although the helical organization is often observed in native arteries (Rhodin 1977) and appears to reflect on their complex mechanical behaviors. We recently made such a tubular scaffold by wrapping electrospun membranes that contain aligned fibers around a mandrel (Hu et al. 2012). In this study, we sought to model the biaxial stress-stretch data of the tubular scaffold made of moderately or highly aligned fibers with different off-axis fiber angles using hyperelastic constitutive equations. There have been only a few attempts to establish constitutive relations of electrospun scaffolds (Courtney et al. 2006; De Vita et al. 2006; Nerurkar et al. 2007, 2008). For the interest of better scaffold design, there is a need to model their mechanical behavior with constitutive equations. A proper constitutive equation with predictive capability can serve as a guideline for making scaffolds that have desired mechanical properties. We first evaluated a two-dimensional (2-D) phenomenological Fung model. As the fibrous structure of the tubular scaffold suggests that their mechanical behavior can be quantified by structural approaches previously applied for soft tissues (Billiar and Sacks 2000; Dahl et al. 2008; Lanir 1979, 1983), a series of structurally motivated models considering fiber directions and fiber angle distributions were used to fit the data. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was added to the structurally motivated models to examine its contribution. Finally, the influence of prescribing the dispersion of fiber orientation on the modeling was examined.
2 Methods 2.1 Electrospun tubular scaffolds The details of construction and characterization of the tubular scaffold are described in Hu et al. (2012). Briefly, scaffold membranes consisting of moderately aligned fibers and highly aligned fibers, denoted as membrane MA and membrane HA, respectively, were prepared by collecting electrospun fibers on a grounded rotating drum at linear tangential
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A α
α
B
α
Fig. 1 Schematic diagram showing how a tubular scaffold with an offaxis fiber angle α was constructed
velocity of 5.3 and 8 m/s, respectively. Rectangular pieces of membrane MA (or HA) were cut in such a way that the angle between the predominant fiber direction and the long axis (i.e., α) is 30◦ , 45◦ , or 60◦ . Two pieces with the same fiber angle were stacked so that the fiber array of each piece aligned symmetrically along the long axis of the rectangle, forming an axisymmetric membrane. The axisymmetric membrane was then wrapped around a mandrel, resulting in a tubular scaffold with an off-axis fiber angle α (either 30◦ , 45◦ or 60◦ ; see Fig. 1). In total, six groups of five tubular scaffolds each, denoted as Tube MA-30 (membrane-α), Tube MA-45, Tube MA-60, Tube HA-30, Tube HA-45, and Tube HA-60, were constructed and mechanically characterized (Hu et al. 2012). The pressure-diameter and the pressure-axial load data previously reported were used along with the unloaded dimensions of the scaffold to calculate the mean circumferential and axial Cauchy stresses σθθ and σzz , respectively, as, σθθ =
Pa F + πa 2 P , σzz = h π h(2a + h)
(1)
where P is the applied transmural pressure, F is the axial load imposed by extending the scaffold (Hu et al. 2012), a and h are the deformed inner radius and wall thickness, respectively. Note that the scaffolds were treated as a thinwalled tube and thus σθθ σrr and σzz σrr . 2.2 Hyperelastic constitutive modeling The biaxial mechanical properties of the tubular scaffolds were quantified by hyperelastic constitutive models. Consis-
Constitutive modeling of an electrospun tubular scaffold
tent with a 2-D analysis, the mean Cauchy stresses can also be determined constitutively using a strain energy function W with an assumption that shearing is negligible during inflation of the scaffold (Hu et al. 2007; Humphrey 2002), namely σθθ = λ2θ
∂W ∂W , σzz = λ2z ∂ E θθ ∂ E zz
(2)
or alternatively, σθθ = λθ
∂W ∂W , σzz = λz . ∂λθ ∂λz
(3)
In this study, two specific forms of W commonly applied for soft tissues were considered. First, a 2-D phenomenological Fung model (Fung 1990) (4)
2 + c E 2 + 2c E E where Q = c1 E zz 2 θθ 3 zz θθ and c0 , c1 , c2 and c3 are model parameters, and E zz and E θθ are components of the Green strain tensor in directions z and θ , respectively. Note, to satisfy physical and mathematical (convexity) constraints, c0 > 0, c1 > 0, c2 > 0, c3 > 0 and c1 c2 > c32 (Holzapfel et al. 2000; Humphrey 1999a). Second, a structurally motivated model in which W is assumed to be the sum of the strain energies of the individual fibers (Lanir 1983)
W =
π 2
− π2
R (θ ) w f
k2 > 0 are material parameters related to the fiber modulus and the degree of stress-stretch nonlinearity of the fiber, respectively (Nerurkar et al. 2008). Herein, the structurally motivated model with linear fibers and that with nonlinear fibers are denoted as LFM and NLFM, respectively. Derived from Eq. (2), the mean Cauchy stresses for the Fung model are: 2 2 + c2 E θθ σθθ = c0 λ2θ (c2 E θθ + c3 E zz ) exp c1 E zz + 2c3 E zz E θθ )
1 W = c0 e Q − 1 2
the value of k with that of k1 introduced later. On the other hand, we used the fiber strain energy proposed by Holzapfel et al. (2000) fiber stress–strain relation,
for a nonlinear 2 k1 − 1 , where k1 > 0 and w f = 4k2 exp k2 λ2f − 1
λ f (θ ) dθ
σzz =
c0 λ2z (c1 E zz
+ 2c3 E zz E θθ ) .
(6)
(7)
Derived from Eq. (3), the mean Cauchy stresses for the LFM are: σθθ =
λ2θ
σzz =
λ2z
(5)
π 2
− π2 π 2
− π2
R (θ ) k λ2f − 1 sin2 θ dθ
(8)
R (θ ) k λ2f − 1 cos2 θ dθ ,
(9)
Similarly, for the NLFM,
where w f is the fiber strain energy function that is dependent
on the fiber stretch ratio λ f = λ2z cos2 θ + λ2θ sin2 θ and R (θ ) represents the fiber angle distribution function, subπ jected to a normalization constraints −2 π R (θ )dθ = 1. We 2 used Cauchy distribution (Courtney et al. 2006) in this study as it fit experimentally measured fiber distributions well (see Supplemental Figure 1) and is efficient in computing. That is, 2 −1
where α is the predominant R (θ ) = π d 1 + θ−α d angle of the fibers and d represents the dispersion of fiber orientation. Two forms of w f for a linear and a nonlinear fiber stress– strain relation, respectively, were examined. The fiber strain energy used herein for a linear fiber stress–strain relation 2 is w f = k E 2f = k4 λ2f − 1 , where E f is the fiber Green strain, and k > 0 the material parameter related to fiber modulus; it was derived from a linear second Piola–Kirchhoff stress–Green strain relation (i.e., s f (= dw f /dE f ) = 2k E f , where s f is the second Piola-Kirchhoff stress in the fiber) (Humphrey and Yin 1987). It will become clear upon comparing Eqs. (8–9) with Eqs. (10–11) that the fiber strain energy defined in this way enables us to directly compare
2 2 + c3 E θθ ) exp c1 E zz + c2 E θθ
σθθ = λ2θ
π 2
− π2
R (θ ) k1 λ2f − 1
2 2 2 × exp k2 λ f − 1 sin θ dθ σzz = λ2z
π 2
− π2
(10)
R (θ ) k1 λ2f − 1
2 2 2 cos θ dθ. × exp k2 λ f − 1
(11)
Note that for structurally motivated models, w f λ f (θ ) was set to zero when λ f = λ2z cos2 θ + λ2θ sin2 θ ≤ 1 as fibers cannot support compressive stress. We also added an isotropic neo-Hookean strain energy function to the LFM and the NLFM π to examine its contri bution. That is, W = 2c (I1 − 3) + −2 π R (θ ) w f λ f (θ ) dθ 2
where c > 0 is a material parameter and I1 = λ2θ +λ2z +λr2 = −2 λ2θ + λ2z + λ−2 θ λz is the first principal invariant of the right Cauchy–Green tensor. The two resulting models are denoted as c_LFM and c_NLFM, respectively.
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Correspondingly, for the c_LFM, 1 2 σθθ = cλθ 1 − 4 2 λθ λ z π 2 R (θ ) k λ2f − 1 sin2 θ dθ + λ2θ
σzz
(12)
− π2
1 = 1− 4 2 λ z λθ π 2 R (θ ) k λ2f − 1 cos2 θ dθ , + λ2z cλ2z
(13)
− π2
and for the c_NLFM, π 2 1 2 2 σθθ = cλθ 1 − 4 2 + λθ R (θ ) k1 λ2f − 1 λθ λ z − π2 2 2 2 sin θ dθ (14) exp k2 λ f − 1 π 2 1 R (θ ) k1 λ2f − 1 σzz = cλ2z 1 − 4 2 + λ2z λ z λθ − π2 2 2 2 cos θ dθ. (15) exp k2 λ f − 1 Finally, the influence of prescribing the value of d in the c_NLFM on the goodness of fit was examined. For the Tube MA and the Tube HA, d = 0.3 and d = 0.1 were prescribed, respectively. The best-fit values of the parameters of each model for each scaffold were determined using nonlinear regression of its biaxial data. This was accomplished by using a modified fminsearch function (Matlab) to minimize the objective function:
e=
⎡⎛
⎞2 Theory Exp − σ σ ⎢⎝ θ θ ⎠ ⎣ Exp mean σθ i=1 i
N
⎛ +
⎤
⎞2 Exp − σz ⎠ ⎥ ⎦ Exp mean σz i
Theory ⎝ σz
(16) where N is the number of data points, and superscripts Theory and Exp denote theoretically calculated and experimentally determined values, respectively. The goodness of fit is presented as the coefficient of determination, r 2 . Given the deformation and the values of the model parameters of each model for each scaffold, the stored strain energy in the scaffold can be determined. The value of stored strain energy at λθ = 1.04 & λz = 1.00, which can be used as a circumferential stiffness index, was calculated. The higher the stored strain energy, the stiffer the scaffolds in the circumferential direction. Also, the stiffness in the 2 ) = ∂∂ EW2 and stretching directions, defined as K θθθθ (= ∂∂ ESθθ θθ K zzzz (= ∂∂ ESzzzz ) =
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∂2W 2 , where Sθθ ∂ E zz
θθ
and Szz are components of
the second Piola–Kirchhoff stress tensor in directions θand z, respectively, was determined and the ratio of the two i.e., zzzz was calculated at λθ = λz = 1.02 as , KKθθθθ max KKθθθθ zzzz an anisotropy index. The definition of stiffness was selected for the simplicity of derivation; note that the second Piola– Kirchhoff stress and the Green strain are work conjugate. The biaxial stretches, at which the stored strain energy and the stiffness ratio were calculated, were chosen simply to be within the range of stretching in the mechanical testing as a constitutive model is considered valid only for the conditions under which it is derived. For the Fung model, in particu3 c2 +c3 , lar, another anisotropy index defined as min cc21 +c +c3 c1 +c3 (Bellini et al. 2011) was also calculated. 2.3 Statistical analysis Because all of the constitutive models were used to fit the data of each scaffold, one-way repeated-measures ANOVA with Holm–Sidak post hoc testing was used for comparing the goodness of fit among models. Results are reported as mean ± standard deviation.
3 Results Figure 2 illustrates representative biaxial stress-stretch data from each group of the scaffolds and the corresponding fits of the Fung model. The Fung model fits the biaxial stress-stretch data well for all groups of the scaffolds. The best-fit values of the parameters of the Fung model and the associated goodness of fit for each group of the scaffolds are listed in Table 1; details for each scaffold can be found in Supplemental Table 1. Figure 3 shows the same representative biaxial stressstretch data from each group and the corresponding fits of the LFM, the NLFM, and the c_LFM. Note that the scales in the plots were changed to better illustrate the differences between models. The fitting curves for the 3-parameter LFM and the 4-parameter NLFM were close to each other. Of particular interest, the LFM and the NLFM led to obvious nonlinear stress-stretch fits, especially for the Tube MA-30 and the Tube HA-30 that were subjected to axial stretching. The nonlinear fits were not observed for the c_LFM, however. The best-fit values of the parameters of the LFM, the NLFM, and the c_LFM and the associated goodness of fit for each group of the scaffolds are listed in Table 2; details for each scaffold can be found in Supplemental Tables 2–4. The estimated value of k in the LFM increased as α increased for both the Tube MA and the Tube HA. Similarly, the estimated value of k1 in the NLFM increased with increasing α. In particular, the value of k in the LFM or the value of k1 in the NLFM estimated by fitting data from the Tube MA was greater than
Constitutive modeling of an electrospun tubular scaffold 1400 800
Fung model Exp
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
600
400
200
0 0.96
1.00
1.02
1.04
Fung model Exp
600
400
200
0.98
1.00
1.02
1.04
200
1200
0.98
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1.02
1.04
1.06
Fung model Exp
1000 800 600 400 200 0 0.96
1.06
0.98
1400 800
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
0.96
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
800
Fung model Exp
600
400
200
0 0.96
0.98
1.00
1.02
1.04
1200
Fung model Exp
1000 800 600 400 200 0 0.96
1.06
D
0.98
1400 800
Fung model Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
1.06
C
Tube HA-30
800
1400
0 0.96
Tube MA-60
1000
0 0.98
B
Tube MA-45
Fung model Exp
1200
600
400
200
0 0.96
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
1200
Fung model Exp
1000 800 600 400 200 0 0.96
0.98
1.00
Circ. Stretch
Fig. 2 Fits of the Fung model (circle) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
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J.-J. Hu 1400
600
400
200
0 0.96
0.98
Circ. Cauchy Stress (kPa)
1.00
1.02
1.04
1200 1000 800 600 400 200 0 0.96
1.06
Fung model Exp
0.98
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1400
F
Tube HA-60
Axial Cauchy Stress (kPa)
Fung model Exp
800
Fung model Exp
800
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
600
400
200
0 0.96
0.98
1.00
1.02
1.04
1200 1000 800 600 400 200 0 0.96
1.06
Fung model Exp
0.98
Circ. Stretch
Circ. Stretch
Fig. 2 continued Table 1 Best-fit values of the parameters of the Fung model and the goodness of fit for each group of the scaffolds Specimen type
Model parameters c0 (Pa)
Goodness of fit c1
c2
c3
r2
362,700 ± 196,314
81.06 ± 56.48
64.20 ± 39.67
27.39 ± 15.43
Tube MA-45
778,616 ± 904,116
78.74 ± 79.89
144.64 ± 121.98
42.31 ± 41.34
0.975 ± 0.009
Tube MA-60
1,493,720 ± 1,581,935
17.65 ± 15.52
81.20 ± 78.01
12.51 ± 12.64
0.942 ± 0.026
Tube MA-30
0.992 ± 0.006
Tube HA-30
167,551 ± 65,250
105.03 ± 44.22
58.96 ± 23.00
44.97 ± 14.80
0.987 ± 0.007
Tube HA-45
57,250 ± 54,848
281.34 ± 140.24
500.34 ± 256.10
243.29 ± 114.50
0.977 ± 0.011
Tube HA-60
264,352 ± 480,152
143.93 ± 116.44
883.30 ± 671.74
177.31 ± 143.94
0.948 ± 0.030
Data are presented as mean ± standard deviation
the value of the corresponding k or k1 estimated by fitting data from the Tube HA given the same α. The estimated values of k2 (i.e., nonlinearity) in the NLFM was found to be in the order: Tube MA-30 ∼ = Tube MA-45 > Tube MA-60 and Tube HA-45 > Tube HA-30 > Tube HA-60. For both the LFM and the NLFM, the estimated values of α were consistently greater than the experimentally prescribed angles. The inclusion of the neo-Hookean term in the LFM altered the estimated values of d among groups dramatically and affected the estimated values of k as well. Also, the neoHookean term improved the fits of c_LFM for the scaf-
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folds with α = 45◦ and 60◦ ; no significant difference was found, though. In general, the within-group variations in the model parameters of the three constitutive models for the Tube HA were less than those for the Tube MA. Figure 4 shows the same representative biaxial stressstretch data from each group and the corresponding fits of the c_NLFM and the c_NLFM with d prescribed. The fits of the c_NLFM and the c_NLFM with d prescribed were better than those of the previous three structurally motivated models. The best-fit values of the parameters of the c_NLFM and
Constitutive modeling of an electrospun tubular scaffold 1400 LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
400
200
0 1.00
1.02
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
400
200
200 1.00
1.02
1.04
LFM NLFM c_LFM Exp
1200 1000 800 600 400 200 0
1.00
1.01
1.02
0.99
C
1.00
1.01
1.02
1000 LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
1400 LFM NLFM c_LFM Exp
800
0.99
400
200
0
LFM NLFM c_LFM Exp
800 600 400 200 0
0.990
0.995
1.000
1.005
1.010
0.990
D
0.995
1.000
1.005
1.010
1400 800
600
LFM NLFM c_LFM Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
0.98
0
Tube HA-30
800
1.04
B
Tube MA-60
1000
0 0.98
Tube MA-45
LFM NLFM c_LFM Exp
1200
400
200
0
1200 1000
LFM NLFM c_LFM Exp
800 600 400 200 0
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
0.98
1.00
1.02
1.04
1.06
Circ. Stretch
Fig. 3 Fits of the LFM (inverted triangle), the NLFM (triangle), and the c_LFM (diamond) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
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J.-J. Hu
LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
400
200
0 1.00
1.01
1.02
400
200
1.03
LFM NLFM c_LFM Exp
800
600
0.99
Axial Cauchy Stress (kPa)
F Circ. Cauchy Stress (kPa)
600
0 0.99
Tube HA-60
LFM NLFM c_LFM Exp
800
400
200
0
500
1.00
1.01
1.02
1.03
LFM NLFM c_LFM Exp
400 300 200 100 0
0.990
0.995
1.000
1.005
1.010
Circ. Stretch
0.990
0.995
1.000
1.005
1.010
Circ. Stretch
Fig. 3 continued
the c_NLFM with d prescribed and the associated goodness of fit for each group of the scaffolds are also listed in Table 2; details for each scaffold can be found in Supplemental Table 5–6. Similar to the finding of the c_LFM, the inclusion of the neo-Hookean term in the NLFM altered the estimated values of d among groups and affected the estimated values of k1 and k2 as well. In particular, there were large within-group variations in the model parameters of the c_NLFM. Note that the within-group variations in the model parameters of the c_LFM were relatively small. Interestingly, prescribing the value of d reduced the within-group variations aforementioned for c_NLFM and still resulted in fits comparable to those of the c_NLFM (i.e., no significant difference in fits between the c_NLFM and the c_NLFM with d prescribed) for all groups of the scaffolds. Furthermore, the c_NLFM with d prescribed involves only four parameters. Prescribing the value of d slightly reduced the goodness of fit for scaffolds with α = 45◦ and 60◦ ; the difference was not significant, though. Figure 5 compares the fits of the six constitutive models for each group of the scaffolds via one-way repeatedmeasures ANOVA. For fitting the data of the Tube MA, there was no significant difference among the structurally motivated models. Nevertheless, the Fung model was signif-
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icantly different from the LFM, the NLFM, and the c_LFM. On the other hand, for fitting the data of the Tube HA, the 5-parameter c_NLFM had the best fits among the structurally motivated models, particularly for the Tube HA30; it was the only structurally motivated model that had fits as good as the Fung model for all of the scaffolds tested. Note, particularly, that the Tube HA-30 was the only group for which the NLFM provided better fits than the LFM. For both the Tube MA and the Tube HA, the smaller the α, the better the fitting for all the constitutive models considered. Based on the constitutive models, the stored strain energy for each group of the scaffolds was calculated at λθ = 1.04 & λz = 1.00 and the results are listed in Table 3; details for each scaffold can be found in Supplemental Table 7. For all the constitutive models, the stored strain energy of the Tube MA-30 was significantly greater than that of the Tube HA-30. That is, the Tube MA-30 was circumferentially stiffer than the Tube HA-30. There was no significant difference in the value of stored strain energy between the Tube MA and the Tube HA with α = 45◦ and 60◦ , however. Note that the c_NLFM generated greater within-group variations for the scaffolds with α = 45◦ and 60◦ probably due to the extreme estimated values of k2 in these groups of the scaffolds.
– – – – – – 13,797,600 ± 1,998,995 15,881,400 ± 1,137,240 21,438,600 ± 2,208,973 9,597,160 ± 995,080 11,935,740 ± 2,133,627 19,753,400 ± 3,999,904 23 ± 18 22 ± 31 9 ± 19 47 ± 17 262 ± 154 39 ± 62 41.9 ± 4.9 68.8 ± 13.7 81.0 ± 5.6 38.3 ± 1.5 52.6 ± 2.1 65.8 ± 4.6 0.629 ± 0.288 0.700 ± 0.305 0.414 ± 0.093 0.160 ± 0.017 0.080 ± 0.042 0.135 ± 0.046 0.984 ± 0.007 0.957 ± 0.015 0.916 ± 0.039 0.983 ± 0.003 0.950 ± 0.016 0.907 ± 0.035
0.982 ± 0.007 0.957 ± 0.015 0.913 ± 0.043 0.974 ± 0.006 0.936 ± 0.021 0.904 ± 0.034
NLFM
– – – – – – 14,796,200 ± 2,454,972 16,517,000 ± 1,211,643 21,528,200 ± 2,137,937 11,007,200 ± 600,977 15,245,000 ± 1,501,535 20,142,000 ± 3,645,557 – – – – – – 36.4 ± 11.9 73.6 ± 16.7 81.0 ± 5.6 38.1 ± 1.8 51.5 ± 2.0 65.9 ± 4.6 0.893 ± 0.608 0.822 ± 0.348 0.424 ± 0.083 0.214 ± 0.038 0.160 ± 0.045 0.155 ± 0.048
LFM
Constitutive models
Data are presented as mean ± standard deviation
r2
d
α (◦ )
k2
Tube MA-30 -45 -60 Tube HA-30 -45 -60
Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60
c (Pa)
k (Pa) or k1 (Pa)
Specimen type
Parameters
0.987 ± 0.010 0.966 ± 0.017 0.929 ± 0.035 0.991 ± 0.004 0.977 ± 0.009 0.938 ± 0.037
2,277,680 ± 1,730,840 1,828,660 ± 2,236,482 2,556,880 ± 1,514,470 1,965,000 ± 299,213 1,638,011 ± 1,028,857 1,778,820 ± 250,972 7,469,820 ± 2,968,626 10,378,080 ± 6,350,751 13,373,200 ± 4,000,616 4,018,840 ± 1,345,613 7,117,080 ± 3,494,922 13,214,420 ± 5,204,275 59 ± 45 353 ± 588 324 ± 403 131 ± 39 591 ± 328 508 ± 677 43.7 ± 26.6 76.0 ± 14.0 77.0 ± 7.0 34.5 ± 1.9 52.8 ± 3.0 65.9 ± 3.8 1.012 ± 0.893 0.604 ± 0.543 0.129 ± 0.248 0.063 ± 0.015 0.022 ± 0.024 0.001 ± 0.001
2,154,146 ± 1,963,160 1,612,840 ± 2,068,162 3,533,520 ± 490,392 1,395,688 ± 407,129 2,475,888 ± 850,148 1,943,200 ± 245,219 9,316,660 ± 3,323,500 12,097,020 ± 4,481,981 11,786,100 ± 1,840,743 7,370,180 ± 1, 272, 902 8,600,340 ± 3,578,895 14,689,000 ± 3,707,885 – – – – – – 28.2 ± 14.4 82.7 ± 9.8 76.8 ± 5.9 35.6 ± 2.3 54.2 ± 3.6 67.1 ± 3.8 1.020 ± 0.772 0.893 ± 0.527 7.92e − 09 ± 8.53e − 09 0.137 ± 0.052 0.024 ± 0.033 1.15e − 08 ± 1.00e − 08 0.985 ± 0.009 0.963 ± 0.013 0.928 ± 0.034 0.974 ± 0.007 0.948 ± 0.027 0.927 ± 0.038
c_NLFM
c_LFM
Table 2 Best-fit values of the parameters of each structurally motivated model and the goodness of fit for each group of the scaffolds
0.984 ± 0.013 0.956 ± 0.016 0.916 ± 0.040 0.991 ± 0.004 0.961 ± 0.008 0.919 ± 0.033
3,344,940 ± 1,001,133 3,199,400 ± 881,668 1,205,358 ± 708,982 1,748,220 ± 252,281 883,584 ± 652,322 899,114 ± 342,603 4,154,026 ± 3,564,142 6,241,500 ± 3,622,385 17,936,200 ± 2,932,934 4,583,400 ± 1,397,210 10,534,400 ± 2,419,855 17,178,800 ± 4,451,203 196 ± 213 179 ± 94 17 ± 17 116 ± 38 197 ± 92 59 ± 78 35.2 ± 24.5 77.5 ± 11.8 81.3 ± 5.8 35.0 ± 2.0 53.5 ± 2.8 66.6 ± 4.1 0.3 0.3 0.3 0.1 0.1 0.1
c_NLFM; given d
Constitutive modeling of an electrospun tubular scaffold
123
J.-J. Hu 1400 c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
600
400
200
0 1.02
c_NLFM c_NLFM; given d Exp
800
600
400
200
200
0.98
1.00
1.01
1.00
1.02
1.04
1200 c_NLFM c_NLFM; given d Exp
1000 800 600 400 200
1.02
0.99
C
1.00
1.01
1.02
1000 c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
0 0.99
600
400
200
0
c_NLFM c_NLFM; given d Exp
800 600 400 200 0
0.990
0.995
1.000
1.005
1.010
0.990
D
0.995
1.000
1.005
1.010
1400 800
c_NLFM c_NLFM; given d Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
1.04
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
1.00
0
Tube HA-30
800
1400
B
Tube MA-60
c_NLFM c_NLFM; given d Exp
1000
0
0.98
Tube MA-45
1200
600
400
200
0
1200
c_NLFM c_NLFM; given d Exp
1000 800 600 400 200 0
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
0.98
1.00
1.02
1.04
1.06
Circ. Stretch
Fig. 4 Fits of the c_NLFM (inverted triangle) and the c_NLFM with d prescribed (diamond) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
123
Constitutive modeling of an electrospun tubular scaffold
c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
600
400
200
0
F
1.00
1.01
1.02
400
200
1.03
c_NLFM c_NLFM; given d Exp
800
0.99
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
0 0.99
Tube HA-60
c_NLFM c_NLFM; given d Exp
800
600
400
200
0
1.00
1.01
500
1.02
1.03
c_NLFM c_NLFM; given d Exp
400 300 200 100 0
0.990
0.995
1.000
1.005
1.010
0.990
Circ. Stretch
0.995
1.000
1.005
1.010
Circ. Stretch
Fig. 4 continued
The results of anisotropy analysis for each group of the scaffolds are summarized in Table 4; details for each scaffold can be found in Supplemental Table 8. Interestingly, the stiffness ratios for all groups of the scaffolds calculated at λθ = λz = 1.02 were greater than one and their Fung model anisotropy index deviated from one; the scaffolds with α = 30◦ were stiffer in the axial direction whereas the scaffolds with α = 45◦ and 60◦ were stiffer in the circumferential direction regardless of the fiber alignment in the scaffolds. In particular, although both the Tube MA-45 and the Tube HA-45 are structurally isotropic along the two stretching directions when unloaded, their stiffness ratios at λθ = λz = 1.02 were also greater than one (1.4–2.3) and their Fung model anisotropy index consistently deviated from one (∼0.7). Note, also, that the extent of anisotropy for the scaffolds with α = 60◦ was significantly greater than that for the scaffolds with α = 30◦ despite their similar structural anisotropy. The mechanical anisotropy of each group of the scaffolds appeared not correlated well with their structural anisotropy. On the other hand, if comparisons were made between the Tube MA and the Tube HA that have the same α, the extent of anisotropy of the Tube HA was greater than that of the Tube MA for α = 30◦ and 60◦ but the differ-
ence between the Tube MA-45 and the Tube HA-45 was not significant.
4 Discussion Although the Fung model provided the overall best fit to the experimental data, it is phenomenological in nature; its parameters have little physical meaning and its applications are limited. Structurally motivated models, on the other hand, have physically significant parameters and can potentially offer better predictive capability. That is, given experimentally prescribed structural parameters, a successful structurally motivated model can accurately predict the mechanical response of a scaffold with previously determined material parameters. Structurally motivated models can thus be used as a guideline to design a scaffold that has mechanical properties closer to the tissue to be replaced. Moreover, a structurally motivated model is preferred than a phenomenological model in the development of a growth and remodeling model that can describe the maturation process of a tissue-engineered construct (Humphrey and Rajagopal 2002); the growth and remodeling model can potentially
123
J.-J. Hu
Tube MA
Tube HA
1.10
1.10
1.05
*
1.05
2
1.00
r
2
** r
30
*
*
**
1.00
0.95
0.95
0.90
0.90
1.10
1.10
* **
**
* **
*
2
1.00
0.95
0.90
0.90
1.10
1.10
r2
r2
**
1.00
0.95
0.95
0.90
Fu
*
*
*
1.00
*
*
1.05
*
*
*
*
1.00
0.95
1.05
60
1.05
r
2
r
45
*
*
**
*
* 1.05
** *
0.90
n
gm
e od
l
M LF
N
M LF
d M M en LF LF c_ c_N ; giv FM NL c_
n Fu
o gm
de
l
M
LF
FM
NL
c_
M nd FM NL give _ ; c FM NL _ c
LF
Fig. 5 Comparisons of the goodness of fit, r 2 , among the six constitutive models for each group of the scaffolds using one-way repeated-measures ANOVA. Note that the Tube HA-60 was analyzed by repeated-measures ANOVA on ranks due to failure of the equal variance test. * p < 0.05 and ** p < 0.001
provide guidelines for culturing application-specific tissueengineered constructs (Guilak et al. 2014; Niklason et al. 2010). There exists a widely used structurally motivated constitutive model considering the helical organization of arteries (Holzapfel et al. 2000); very few tubular scaffolds with such microstructure were found in the literature, though. In this study, the electrospun tubular scaffold to be constitutively modeled not only has the helical organization but its microstructure including fiber directions and fiber angle distributions is controllable. The biaxial mechanical properties
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of the scaffolds may thus be used to validate structurally motivated models and serve as a tool for the development of such models for arteries or tissue-engineered vascular grafts. Often one problem associated with a comparative study is that each different model is based on data from different specimens. In this study, the Fung model and the structurally motivated models were compared for each scaffold based on the same set of data, and one-way repeated-measures ANOVA was used to examine the potential differences among the models. Therefore, the role of each individual model parameter was able to be studied.
Constitutive modeling of an electrospun tubular scaffold Table 3 Stored strain energy calculated based on the best-fit values of the parameters of each model for each group of scaffolds Specimen type
Tube MA-30
Stored strain energy @ λθ = 1.04 & λz = 1.00 (J/m3 ) Fung
LFM
NLFM
c_LFM
c_NLFM
15,108 ± 742
15,723 ± 680
15,577 ± 890
15,876 ± 592
c_NLFM; given d
15,732 ± 809
15,372 ± 1,396
Tube MA-45
28,146 ± 4,656
26,111 ± 3,337
26,884 ± 3,133
25,622 ± 2,976
37,602 ± 16,484
31,566 ± 3,949
Tube MA-60
47,972 ± 6,793
45,998 ± 4,916
47,430 ± 6,002
45,521 ± 4,798
272,489 ± 333,617
47,695 ± 4,942
Tube HA-30
7,602 ± 411
8,703 ± 208
7,562 ± 472
8,793 ± 256
8,069 ± 371
7,973 ± 317
Tube HA-45
24,308 ± 3,190
20,051 ± 1,498
25,347 ± 3,121
19,635 ± 1, 659
28,376 ± 5,000
23,908 ± 1,202
Tube HA-60
96,459 ± 53,879
41,724 ± 7,905
44,933 ± 6,551
40,634 ± 6,805
464,929 ± 788,243
46,661 ± 5,672
Data are presented as mean ± standard deviation Table 4 Ratio of stiffness in the stretching directions and the Fung model anisotropy indexa calculated based on the best-fit values of the parameters of each model for each group of scaffolds Specimen type
Stiffness ratio @ λθ = 1.02 & λz = 1.02
Anisotropy Fung
Fung
LFM
NLFM
c_LFM
c_NLFM
c_NLFM; given d
Tube MA-30
1.26 ± 0.27
1.35 ± 0.31
1.25 ± 0.26
1.21 ± 0.16
1.20 ± 0.16
1.20 ± 0.16
Tube MA-45
1.99 ± 0.54
1.83 ± 0.48
1.92 ± 0.41
1.63 ± 0.32
2.34 ± 0.78
1.93 ± 0.36
0.64 ± 0.11
Tube MA-60
4.50 ± 1.21
4.34 ± 0.73
4.54 ± 1.08
2.75 ± 0.47
9.92 ± 9.21
3.92 ± 1.19
0.35 ± 0.07
Tube HA-30
1.82 ± 0.22
1.73 ± 0.20
1.82 ± 0.22
1.53 ± 0.16
1.55 ± 0.16
1.56 ± 0.16
0.70 ± 0.06
0.87 ± 0.11
Tube HA-45
1.93 ± 0.35
1.79 ± 0.29
2.34 ± 0.40
1.44 ± 0.15
2.25 ± 0.35
2.07 ± 0.29
0.70 ± 0.08
Tube HA-60
8.19 ± 2.67
6.05 ± 1.35
6.59 ± 1.10
3.74 ± 0.75
9.51 ± 5.67
5.50 ± 1.18
0.31 ± 0.06
the Fung model only. Data are presented as mean ± standard deviation c2 +c3 zzzz 3 > KKθθθθ and cc21 +c For Tube MA-30 and Tube HA-30, KKθθθθ +c3 > c1 +c3 whereas for Tube MA-45, 60 and Tube HA-45, 60, zzzz
a For
c1 +c3 c2 +c3
ORIGINAL PAPER
Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering Jin-Jia Hu
Received: 6 October 2014 / Accepted: 16 December 2014 © Springer-Verlag Berlin Heidelberg 2015
Abstract In this study, we sought to model the mechanical behavior of an electrospun tubular scaffold previously reported for vascular tissue engineering with hyperelastic constitutive equations. Specifically, the scaffolds were made by wrapping electrospun polycaprolactone membranes that contain aligned fibers around a mandrel in such a way that they have microstructure similar to the native arterial media. The biaxial stress-stretch data of the scaffolds made of moderately or highly aligned fibers with three different off-axis fiber angles α (30◦ , 45◦ , and 60◦ ) were fit by a phenomenological Fung model and a series of structurally motivated models considering fiber directions and fiber angle distributions. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was also added to the structurally motivated models to examine its contribution. The two forms of fiber strain energy did not result in significantly different goodness of fit for most groups of the scaffolds. The absence of the neo-Hookean term in the structurally motivated model led to obvious nonlinear stressstretch fits at a greater axial stretch, especially when fitting data from the scaffolds with a small α. Of the models considered, the Fung model had the overall best fitting results; Electronic supplementary material The online version of this article (doi:10.1007/s10237-014-0644-y) 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
its applications are limited because of its phenomenological nature. Although a structurally motivated model using the nonlinear fiber stress–strain relation with the neo-Hookean term provided fits comparably as good as the Fung model, the values of its model parameters exhibited large within-group variations. Prescribing the dispersion of fiber orientation in the structurally motivated model, however, reduced the variations without compromising the fits and was thus considered to be the best structurally motivated model for the scaffolds. It appeared that the structurally motivated models could be further improved for fitting the mechanical behavior of the electrospun scaffold; fiber interactions are suggested to be considered in future models. Keywords Constitutive modeling · Hyperelasticity · Electrospun scaffolds · Fung model · Structurally motivated models · Mechanical properties
1 Introduction Scaffolds, cells and stimulating signals, generally referred to as the tissue engineering triad, are the key components for making a functional tissue-engineered construct. Specifically, the three-dimensional, porous scaffolds not only provide a surface for cells to adhere and grow, but also serve as a template to guide matrix formation. Manipulation of the microstructure of the scaffolds thus has potential to control the microstructure of resulting tissues and hence their mechanical properties. The latter are particularly important for the functionality of load-bearing tissues. Among the numerous polymer processing techniques, electrospinning has been widely used to fabricate scaffolds. Electrospun scaffolds are attractive because of their high porosity, large surface area-to-volume ratio, and nano-scale
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J.-J. Hu
fibrous structure, similar to the features of native extracellular matrix. In general, only a simple, inexpensive setup is required for electrospinning (Ramakrishna 2005). Although the orientations of electrospun fibers are typically random, aligned electrospun fibers can be achieved with special collecting mechanisms (Teo and Ramakrishna 2006). The aligned electrospun fibers are capable of providing contact guidance to cultured cells (Bashur et al. 2006; Wang et al. 2011; Yang et al. 2005) and can be used to control the microstructure of engineered tissues. Very few tubular scaffolds that have helically oriented fibers were found in the literature although the helical organization is often observed in native arteries (Rhodin 1977) and appears to reflect on their complex mechanical behaviors. We recently made such a tubular scaffold by wrapping electrospun membranes that contain aligned fibers around a mandrel (Hu et al. 2012). In this study, we sought to model the biaxial stress-stretch data of the tubular scaffold made of moderately or highly aligned fibers with different off-axis fiber angles using hyperelastic constitutive equations. There have been only a few attempts to establish constitutive relations of electrospun scaffolds (Courtney et al. 2006; De Vita et al. 2006; Nerurkar et al. 2007, 2008). For the interest of better scaffold design, there is a need to model their mechanical behavior with constitutive equations. A proper constitutive equation with predictive capability can serve as a guideline for making scaffolds that have desired mechanical properties. We first evaluated a two-dimensional (2-D) phenomenological Fung model. As the fibrous structure of the tubular scaffold suggests that their mechanical behavior can be quantified by structural approaches previously applied for soft tissues (Billiar and Sacks 2000; Dahl et al. 2008; Lanir 1979, 1983), a series of structurally motivated models considering fiber directions and fiber angle distributions were used to fit the data. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was added to the structurally motivated models to examine its contribution. Finally, the influence of prescribing the dispersion of fiber orientation on the modeling was examined.
2 Methods 2.1 Electrospun tubular scaffolds The details of construction and characterization of the tubular scaffold are described in Hu et al. (2012). Briefly, scaffold membranes consisting of moderately aligned fibers and highly aligned fibers, denoted as membrane MA and membrane HA, respectively, were prepared by collecting electrospun fibers on a grounded rotating drum at linear tangential
123
A α
α
B
α
Fig. 1 Schematic diagram showing how a tubular scaffold with an offaxis fiber angle α was constructed
velocity of 5.3 and 8 m/s, respectively. Rectangular pieces of membrane MA (or HA) were cut in such a way that the angle between the predominant fiber direction and the long axis (i.e., α) is 30◦ , 45◦ , or 60◦ . Two pieces with the same fiber angle were stacked so that the fiber array of each piece aligned symmetrically along the long axis of the rectangle, forming an axisymmetric membrane. The axisymmetric membrane was then wrapped around a mandrel, resulting in a tubular scaffold with an off-axis fiber angle α (either 30◦ , 45◦ or 60◦ ; see Fig. 1). In total, six groups of five tubular scaffolds each, denoted as Tube MA-30 (membrane-α), Tube MA-45, Tube MA-60, Tube HA-30, Tube HA-45, and Tube HA-60, were constructed and mechanically characterized (Hu et al. 2012). The pressure-diameter and the pressure-axial load data previously reported were used along with the unloaded dimensions of the scaffold to calculate the mean circumferential and axial Cauchy stresses σθθ and σzz , respectively, as, σθθ =
Pa F + πa 2 P , σzz = h π h(2a + h)
(1)
where P is the applied transmural pressure, F is the axial load imposed by extending the scaffold (Hu et al. 2012), a and h are the deformed inner radius and wall thickness, respectively. Note that the scaffolds were treated as a thinwalled tube and thus σθθ σrr and σzz σrr . 2.2 Hyperelastic constitutive modeling The biaxial mechanical properties of the tubular scaffolds were quantified by hyperelastic constitutive models. Consis-
Constitutive modeling of an electrospun tubular scaffold
tent with a 2-D analysis, the mean Cauchy stresses can also be determined constitutively using a strain energy function W with an assumption that shearing is negligible during inflation of the scaffold (Hu et al. 2007; Humphrey 2002), namely σθθ = λ2θ
∂W ∂W , σzz = λ2z ∂ E θθ ∂ E zz
(2)
or alternatively, σθθ = λθ
∂W ∂W , σzz = λz . ∂λθ ∂λz
(3)
In this study, two specific forms of W commonly applied for soft tissues were considered. First, a 2-D phenomenological Fung model (Fung 1990) (4)
2 + c E 2 + 2c E E where Q = c1 E zz 2 θθ 3 zz θθ and c0 , c1 , c2 and c3 are model parameters, and E zz and E θθ are components of the Green strain tensor in directions z and θ , respectively. Note, to satisfy physical and mathematical (convexity) constraints, c0 > 0, c1 > 0, c2 > 0, c3 > 0 and c1 c2 > c32 (Holzapfel et al. 2000; Humphrey 1999a). Second, a structurally motivated model in which W is assumed to be the sum of the strain energies of the individual fibers (Lanir 1983)
W =
π 2
− π2
R (θ ) w f
k2 > 0 are material parameters related to the fiber modulus and the degree of stress-stretch nonlinearity of the fiber, respectively (Nerurkar et al. 2008). Herein, the structurally motivated model with linear fibers and that with nonlinear fibers are denoted as LFM and NLFM, respectively. Derived from Eq. (2), the mean Cauchy stresses for the Fung model are: 2 2 + c2 E θθ σθθ = c0 λ2θ (c2 E θθ + c3 E zz ) exp c1 E zz + 2c3 E zz E θθ )
1 W = c0 e Q − 1 2
the value of k with that of k1 introduced later. On the other hand, we used the fiber strain energy proposed by Holzapfel et al. (2000) fiber stress–strain relation,
for a nonlinear 2 k1 − 1 , where k1 > 0 and w f = 4k2 exp k2 λ2f − 1
λ f (θ ) dθ
σzz =
c0 λ2z (c1 E zz
+ 2c3 E zz E θθ ) .
(6)
(7)
Derived from Eq. (3), the mean Cauchy stresses for the LFM are: σθθ =
λ2θ
σzz =
λ2z
(5)
π 2
− π2 π 2
− π2
R (θ ) k λ2f − 1 sin2 θ dθ
(8)
R (θ ) k λ2f − 1 cos2 θ dθ ,
(9)
Similarly, for the NLFM,
where w f is the fiber strain energy function that is dependent
on the fiber stretch ratio λ f = λ2z cos2 θ + λ2θ sin2 θ and R (θ ) represents the fiber angle distribution function, subπ jected to a normalization constraints −2 π R (θ )dθ = 1. We 2 used Cauchy distribution (Courtney et al. 2006) in this study as it fit experimentally measured fiber distributions well (see Supplemental Figure 1) and is efficient in computing. That is, 2 −1
where α is the predominant R (θ ) = π d 1 + θ−α d angle of the fibers and d represents the dispersion of fiber orientation. Two forms of w f for a linear and a nonlinear fiber stress– strain relation, respectively, were examined. The fiber strain energy used herein for a linear fiber stress–strain relation 2 is w f = k E 2f = k4 λ2f − 1 , where E f is the fiber Green strain, and k > 0 the material parameter related to fiber modulus; it was derived from a linear second Piola–Kirchhoff stress–Green strain relation (i.e., s f (= dw f /dE f ) = 2k E f , where s f is the second Piola-Kirchhoff stress in the fiber) (Humphrey and Yin 1987). It will become clear upon comparing Eqs. (8–9) with Eqs. (10–11) that the fiber strain energy defined in this way enables us to directly compare
2 2 + c3 E θθ ) exp c1 E zz + c2 E θθ
σθθ = λ2θ
π 2
− π2
R (θ ) k1 λ2f − 1
2 2 2 × exp k2 λ f − 1 sin θ dθ σzz = λ2z
π 2
− π2
(10)
R (θ ) k1 λ2f − 1
2 2 2 cos θ dθ. × exp k2 λ f − 1
(11)
Note that for structurally motivated models, w f λ f (θ ) was set to zero when λ f = λ2z cos2 θ + λ2θ sin2 θ ≤ 1 as fibers cannot support compressive stress. We also added an isotropic neo-Hookean strain energy function to the LFM and the NLFM π to examine its contri bution. That is, W = 2c (I1 − 3) + −2 π R (θ ) w f λ f (θ ) dθ 2
where c > 0 is a material parameter and I1 = λ2θ +λ2z +λr2 = −2 λ2θ + λ2z + λ−2 θ λz is the first principal invariant of the right Cauchy–Green tensor. The two resulting models are denoted as c_LFM and c_NLFM, respectively.
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J.-J. Hu
Correspondingly, for the c_LFM, 1 2 σθθ = cλθ 1 − 4 2 λθ λ z π 2 R (θ ) k λ2f − 1 sin2 θ dθ + λ2θ
σzz
(12)
− π2
1 = 1− 4 2 λ z λθ π 2 R (θ ) k λ2f − 1 cos2 θ dθ , + λ2z cλ2z
(13)
− π2
and for the c_NLFM, π 2 1 2 2 σθθ = cλθ 1 − 4 2 + λθ R (θ ) k1 λ2f − 1 λθ λ z − π2 2 2 2 sin θ dθ (14) exp k2 λ f − 1 π 2 1 R (θ ) k1 λ2f − 1 σzz = cλ2z 1 − 4 2 + λ2z λ z λθ − π2 2 2 2 cos θ dθ. (15) exp k2 λ f − 1 Finally, the influence of prescribing the value of d in the c_NLFM on the goodness of fit was examined. For the Tube MA and the Tube HA, d = 0.3 and d = 0.1 were prescribed, respectively. The best-fit values of the parameters of each model for each scaffold were determined using nonlinear regression of its biaxial data. This was accomplished by using a modified fminsearch function (Matlab) to minimize the objective function:
e=
⎡⎛
⎞2 Theory Exp − σ σ ⎢⎝ θ θ ⎠ ⎣ Exp mean σθ i=1 i
N
⎛ +
⎤
⎞2 Exp − σz ⎠ ⎥ ⎦ Exp mean σz i
Theory ⎝ σz
(16) where N is the number of data points, and superscripts Theory and Exp denote theoretically calculated and experimentally determined values, respectively. The goodness of fit is presented as the coefficient of determination, r 2 . Given the deformation and the values of the model parameters of each model for each scaffold, the stored strain energy in the scaffold can be determined. The value of stored strain energy at λθ = 1.04 & λz = 1.00, which can be used as a circumferential stiffness index, was calculated. The higher the stored strain energy, the stiffer the scaffolds in the circumferential direction. Also, the stiffness in the 2 ) = ∂∂ EW2 and stretching directions, defined as K θθθθ (= ∂∂ ESθθ θθ K zzzz (= ∂∂ ESzzzz ) =
123
∂2W 2 , where Sθθ ∂ E zz
θθ
and Szz are components of
the second Piola–Kirchhoff stress tensor in directions θand z, respectively, was determined and the ratio of the two i.e., zzzz was calculated at λθ = λz = 1.02 as , KKθθθθ max KKθθθθ zzzz an anisotropy index. The definition of stiffness was selected for the simplicity of derivation; note that the second Piola– Kirchhoff stress and the Green strain are work conjugate. The biaxial stretches, at which the stored strain energy and the stiffness ratio were calculated, were chosen simply to be within the range of stretching in the mechanical testing as a constitutive model is considered valid only for the conditions under which it is derived. For the Fung model, in particu3 c2 +c3 , lar, another anisotropy index defined as min cc21 +c +c3 c1 +c3 (Bellini et al. 2011) was also calculated. 2.3 Statistical analysis Because all of the constitutive models were used to fit the data of each scaffold, one-way repeated-measures ANOVA with Holm–Sidak post hoc testing was used for comparing the goodness of fit among models. Results are reported as mean ± standard deviation.
3 Results Figure 2 illustrates representative biaxial stress-stretch data from each group of the scaffolds and the corresponding fits of the Fung model. The Fung model fits the biaxial stress-stretch data well for all groups of the scaffolds. The best-fit values of the parameters of the Fung model and the associated goodness of fit for each group of the scaffolds are listed in Table 1; details for each scaffold can be found in Supplemental Table 1. Figure 3 shows the same representative biaxial stressstretch data from each group and the corresponding fits of the LFM, the NLFM, and the c_LFM. Note that the scales in the plots were changed to better illustrate the differences between models. The fitting curves for the 3-parameter LFM and the 4-parameter NLFM were close to each other. Of particular interest, the LFM and the NLFM led to obvious nonlinear stress-stretch fits, especially for the Tube MA-30 and the Tube HA-30 that were subjected to axial stretching. The nonlinear fits were not observed for the c_LFM, however. The best-fit values of the parameters of the LFM, the NLFM, and the c_LFM and the associated goodness of fit for each group of the scaffolds are listed in Table 2; details for each scaffold can be found in Supplemental Tables 2–4. The estimated value of k in the LFM increased as α increased for both the Tube MA and the Tube HA. Similarly, the estimated value of k1 in the NLFM increased with increasing α. In particular, the value of k in the LFM or the value of k1 in the NLFM estimated by fitting data from the Tube MA was greater than
Constitutive modeling of an electrospun tubular scaffold 1400 800
Fung model Exp
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
600
400
200
0 0.96
1.00
1.02
1.04
Fung model Exp
600
400
200
0.98
1.00
1.02
1.04
200
1200
0.98
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1.02
1.04
1.06
Fung model Exp
1000 800 600 400 200 0 0.96
1.06
0.98
1400 800
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
0.96
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
800
Fung model Exp
600
400
200
0 0.96
0.98
1.00
1.02
1.04
1200
Fung model Exp
1000 800 600 400 200 0 0.96
1.06
D
0.98
1400 800
Fung model Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
1.06
C
Tube HA-30
800
1400
0 0.96
Tube MA-60
1000
0 0.98
B
Tube MA-45
Fung model Exp
1200
600
400
200
0 0.96
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
1200
Fung model Exp
1000 800 600 400 200 0 0.96
0.98
1.00
Circ. Stretch
Fig. 2 Fits of the Fung model (circle) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
123
J.-J. Hu 1400
600
400
200
0 0.96
0.98
Circ. Cauchy Stress (kPa)
1.00
1.02
1.04
1200 1000 800 600 400 200 0 0.96
1.06
Fung model Exp
0.98
1.00
1.02
1.04
1.06
1.00
1.02
1.04
1.06
1400
F
Tube HA-60
Axial Cauchy Stress (kPa)
Fung model Exp
800
Fung model Exp
800
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
600
400
200
0 0.96
0.98
1.00
1.02
1.04
1200 1000 800 600 400 200 0 0.96
1.06
Fung model Exp
0.98
Circ. Stretch
Circ. Stretch
Fig. 2 continued Table 1 Best-fit values of the parameters of the Fung model and the goodness of fit for each group of the scaffolds Specimen type
Model parameters c0 (Pa)
Goodness of fit c1
c2
c3
r2
362,700 ± 196,314
81.06 ± 56.48
64.20 ± 39.67
27.39 ± 15.43
Tube MA-45
778,616 ± 904,116
78.74 ± 79.89
144.64 ± 121.98
42.31 ± 41.34
0.975 ± 0.009
Tube MA-60
1,493,720 ± 1,581,935
17.65 ± 15.52
81.20 ± 78.01
12.51 ± 12.64
0.942 ± 0.026
Tube MA-30
0.992 ± 0.006
Tube HA-30
167,551 ± 65,250
105.03 ± 44.22
58.96 ± 23.00
44.97 ± 14.80
0.987 ± 0.007
Tube HA-45
57,250 ± 54,848
281.34 ± 140.24
500.34 ± 256.10
243.29 ± 114.50
0.977 ± 0.011
Tube HA-60
264,352 ± 480,152
143.93 ± 116.44
883.30 ± 671.74
177.31 ± 143.94
0.948 ± 0.030
Data are presented as mean ± standard deviation
the value of the corresponding k or k1 estimated by fitting data from the Tube HA given the same α. The estimated values of k2 (i.e., nonlinearity) in the NLFM was found to be in the order: Tube MA-30 ∼ = Tube MA-45 > Tube MA-60 and Tube HA-45 > Tube HA-30 > Tube HA-60. For both the LFM and the NLFM, the estimated values of α were consistently greater than the experimentally prescribed angles. The inclusion of the neo-Hookean term in the LFM altered the estimated values of d among groups dramatically and affected the estimated values of k as well. Also, the neoHookean term improved the fits of c_LFM for the scaf-
123
folds with α = 45◦ and 60◦ ; no significant difference was found, though. In general, the within-group variations in the model parameters of the three constitutive models for the Tube HA were less than those for the Tube MA. Figure 4 shows the same representative biaxial stressstretch data from each group and the corresponding fits of the c_NLFM and the c_NLFM with d prescribed. The fits of the c_NLFM and the c_NLFM with d prescribed were better than those of the previous three structurally motivated models. The best-fit values of the parameters of the c_NLFM and
Constitutive modeling of an electrospun tubular scaffold 1400 LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
400
200
0 1.00
1.02
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
400
200
200 1.00
1.02
1.04
LFM NLFM c_LFM Exp
1200 1000 800 600 400 200 0
1.00
1.01
1.02
0.99
C
1.00
1.01
1.02
1000 LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
1400 LFM NLFM c_LFM Exp
800
0.99
400
200
0
LFM NLFM c_LFM Exp
800 600 400 200 0
0.990
0.995
1.000
1.005
1.010
0.990
D
0.995
1.000
1.005
1.010
1400 800
600
LFM NLFM c_LFM Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
0.98
0
Tube HA-30
800
1.04
B
Tube MA-60
1000
0 0.98
Tube MA-45
LFM NLFM c_LFM Exp
1200
400
200
0
1200 1000
LFM NLFM c_LFM Exp
800 600 400 200 0
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
0.98
1.00
1.02
1.04
1.06
Circ. Stretch
Fig. 3 Fits of the LFM (inverted triangle), the NLFM (triangle), and the c_LFM (diamond) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
123
J.-J. Hu
LFM NLFM c_LFM Exp
800
600
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
400
200
0 1.00
1.01
1.02
400
200
1.03
LFM NLFM c_LFM Exp
800
600
0.99
Axial Cauchy Stress (kPa)
F Circ. Cauchy Stress (kPa)
600
0 0.99
Tube HA-60
LFM NLFM c_LFM Exp
800
400
200
0
500
1.00
1.01
1.02
1.03
LFM NLFM c_LFM Exp
400 300 200 100 0
0.990
0.995
1.000
1.005
1.010
Circ. Stretch
0.990
0.995
1.000
1.005
1.010
Circ. Stretch
Fig. 3 continued
the c_NLFM with d prescribed and the associated goodness of fit for each group of the scaffolds are also listed in Table 2; details for each scaffold can be found in Supplemental Table 5–6. Similar to the finding of the c_LFM, the inclusion of the neo-Hookean term in the NLFM altered the estimated values of d among groups and affected the estimated values of k1 and k2 as well. In particular, there were large within-group variations in the model parameters of the c_NLFM. Note that the within-group variations in the model parameters of the c_LFM were relatively small. Interestingly, prescribing the value of d reduced the within-group variations aforementioned for c_NLFM and still resulted in fits comparable to those of the c_NLFM (i.e., no significant difference in fits between the c_NLFM and the c_NLFM with d prescribed) for all groups of the scaffolds. Furthermore, the c_NLFM with d prescribed involves only four parameters. Prescribing the value of d slightly reduced the goodness of fit for scaffolds with α = 45◦ and 60◦ ; the difference was not significant, though. Figure 5 compares the fits of the six constitutive models for each group of the scaffolds via one-way repeatedmeasures ANOVA. For fitting the data of the Tube MA, there was no significant difference among the structurally motivated models. Nevertheless, the Fung model was signif-
123
icantly different from the LFM, the NLFM, and the c_LFM. On the other hand, for fitting the data of the Tube HA, the 5-parameter c_NLFM had the best fits among the structurally motivated models, particularly for the Tube HA30; it was the only structurally motivated model that had fits as good as the Fung model for all of the scaffolds tested. Note, particularly, that the Tube HA-30 was the only group for which the NLFM provided better fits than the LFM. For both the Tube MA and the Tube HA, the smaller the α, the better the fitting for all the constitutive models considered. Based on the constitutive models, the stored strain energy for each group of the scaffolds was calculated at λθ = 1.04 & λz = 1.00 and the results are listed in Table 3; details for each scaffold can be found in Supplemental Table 7. For all the constitutive models, the stored strain energy of the Tube MA-30 was significantly greater than that of the Tube HA-30. That is, the Tube MA-30 was circumferentially stiffer than the Tube HA-30. There was no significant difference in the value of stored strain energy between the Tube MA and the Tube HA with α = 45◦ and 60◦ , however. Note that the c_NLFM generated greater within-group variations for the scaffolds with α = 45◦ and 60◦ probably due to the extreme estimated values of k2 in these groups of the scaffolds.
– – – – – – 13,797,600 ± 1,998,995 15,881,400 ± 1,137,240 21,438,600 ± 2,208,973 9,597,160 ± 995,080 11,935,740 ± 2,133,627 19,753,400 ± 3,999,904 23 ± 18 22 ± 31 9 ± 19 47 ± 17 262 ± 154 39 ± 62 41.9 ± 4.9 68.8 ± 13.7 81.0 ± 5.6 38.3 ± 1.5 52.6 ± 2.1 65.8 ± 4.6 0.629 ± 0.288 0.700 ± 0.305 0.414 ± 0.093 0.160 ± 0.017 0.080 ± 0.042 0.135 ± 0.046 0.984 ± 0.007 0.957 ± 0.015 0.916 ± 0.039 0.983 ± 0.003 0.950 ± 0.016 0.907 ± 0.035
0.982 ± 0.007 0.957 ± 0.015 0.913 ± 0.043 0.974 ± 0.006 0.936 ± 0.021 0.904 ± 0.034
NLFM
– – – – – – 14,796,200 ± 2,454,972 16,517,000 ± 1,211,643 21,528,200 ± 2,137,937 11,007,200 ± 600,977 15,245,000 ± 1,501,535 20,142,000 ± 3,645,557 – – – – – – 36.4 ± 11.9 73.6 ± 16.7 81.0 ± 5.6 38.1 ± 1.8 51.5 ± 2.0 65.9 ± 4.6 0.893 ± 0.608 0.822 ± 0.348 0.424 ± 0.083 0.214 ± 0.038 0.160 ± 0.045 0.155 ± 0.048
LFM
Constitutive models
Data are presented as mean ± standard deviation
r2
d
α (◦ )
k2
Tube MA-30 -45 -60 Tube HA-30 -45 -60
Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60 Tube MA-30 -45 -60 Tube HA-30 -45 -60
c (Pa)
k (Pa) or k1 (Pa)
Specimen type
Parameters
0.987 ± 0.010 0.966 ± 0.017 0.929 ± 0.035 0.991 ± 0.004 0.977 ± 0.009 0.938 ± 0.037
2,277,680 ± 1,730,840 1,828,660 ± 2,236,482 2,556,880 ± 1,514,470 1,965,000 ± 299,213 1,638,011 ± 1,028,857 1,778,820 ± 250,972 7,469,820 ± 2,968,626 10,378,080 ± 6,350,751 13,373,200 ± 4,000,616 4,018,840 ± 1,345,613 7,117,080 ± 3,494,922 13,214,420 ± 5,204,275 59 ± 45 353 ± 588 324 ± 403 131 ± 39 591 ± 328 508 ± 677 43.7 ± 26.6 76.0 ± 14.0 77.0 ± 7.0 34.5 ± 1.9 52.8 ± 3.0 65.9 ± 3.8 1.012 ± 0.893 0.604 ± 0.543 0.129 ± 0.248 0.063 ± 0.015 0.022 ± 0.024 0.001 ± 0.001
2,154,146 ± 1,963,160 1,612,840 ± 2,068,162 3,533,520 ± 490,392 1,395,688 ± 407,129 2,475,888 ± 850,148 1,943,200 ± 245,219 9,316,660 ± 3,323,500 12,097,020 ± 4,481,981 11,786,100 ± 1,840,743 7,370,180 ± 1, 272, 902 8,600,340 ± 3,578,895 14,689,000 ± 3,707,885 – – – – – – 28.2 ± 14.4 82.7 ± 9.8 76.8 ± 5.9 35.6 ± 2.3 54.2 ± 3.6 67.1 ± 3.8 1.020 ± 0.772 0.893 ± 0.527 7.92e − 09 ± 8.53e − 09 0.137 ± 0.052 0.024 ± 0.033 1.15e − 08 ± 1.00e − 08 0.985 ± 0.009 0.963 ± 0.013 0.928 ± 0.034 0.974 ± 0.007 0.948 ± 0.027 0.927 ± 0.038
c_NLFM
c_LFM
Table 2 Best-fit values of the parameters of each structurally motivated model and the goodness of fit for each group of the scaffolds
0.984 ± 0.013 0.956 ± 0.016 0.916 ± 0.040 0.991 ± 0.004 0.961 ± 0.008 0.919 ± 0.033
3,344,940 ± 1,001,133 3,199,400 ± 881,668 1,205,358 ± 708,982 1,748,220 ± 252,281 883,584 ± 652,322 899,114 ± 342,603 4,154,026 ± 3,564,142 6,241,500 ± 3,622,385 17,936,200 ± 2,932,934 4,583,400 ± 1,397,210 10,534,400 ± 2,419,855 17,178,800 ± 4,451,203 196 ± 213 179 ± 94 17 ± 17 116 ± 38 197 ± 92 59 ± 78 35.2 ± 24.5 77.5 ± 11.8 81.3 ± 5.8 35.0 ± 2.0 53.5 ± 2.8 66.6 ± 4.1 0.3 0.3 0.3 0.1 0.1 0.1
c_NLFM; given d
Constitutive modeling of an electrospun tubular scaffold
123
J.-J. Hu 1400 c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Tube MA-30
Circ. Cauchy Stress (kPa)
A
600
400
200
0 1.02
c_NLFM c_NLFM; given d Exp
800
600
400
200
200
0.98
1.00
1.01
1.00
1.02
1.04
1200 c_NLFM c_NLFM; given d Exp
1000 800 600 400 200
1.02
0.99
C
1.00
1.01
1.02
1000 c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
400
0 0.99
600
400
200
0
c_NLFM c_NLFM; given d Exp
800 600 400 200 0
0.990
0.995
1.000
1.005
1.010
0.990
D
0.995
1.000
1.005
1.010
1400 800
c_NLFM c_NLFM; given d Exp
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
1.04
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
1.00
0
Tube HA-30
800
1400
B
Tube MA-60
c_NLFM c_NLFM; given d Exp
1000
0
0.98
Tube MA-45
1200
600
400
200
0
1200
c_NLFM c_NLFM; given d Exp
1000 800 600 400 200 0
0.98
1.00
1.02
Circ. Stretch
1.04
1.06
0.98
1.00
1.02
1.04
1.06
Circ. Stretch
Fig. 4 Fits of the c_NLFM (inverted triangle) and the c_NLFM with d prescribed (diamond) in the circumferential (left panels) and the axial (right panels) directions to representative experimental stress-stretch data (plus symbol) from each group of the scaffolds
123
Constitutive modeling of an electrospun tubular scaffold
c_NLFM c_NLFM; given d Exp
800
Axial Cauchy Stress (kPa)
Tube HA-45
Circ. Cauchy Stress (kPa)
E
600
400
200
0
F
1.00
1.01
1.02
400
200
1.03
c_NLFM c_NLFM; given d Exp
800
0.99
Axial Cauchy Stress (kPa)
Circ. Cauchy Stress (kPa)
600
0 0.99
Tube HA-60
c_NLFM c_NLFM; given d Exp
800
600
400
200
0
1.00
1.01
500
1.02
1.03
c_NLFM c_NLFM; given d Exp
400 300 200 100 0
0.990
0.995
1.000
1.005
1.010
0.990
Circ. Stretch
0.995
1.000
1.005
1.010
Circ. Stretch
Fig. 4 continued
The results of anisotropy analysis for each group of the scaffolds are summarized in Table 4; details for each scaffold can be found in Supplemental Table 8. Interestingly, the stiffness ratios for all groups of the scaffolds calculated at λθ = λz = 1.02 were greater than one and their Fung model anisotropy index deviated from one; the scaffolds with α = 30◦ were stiffer in the axial direction whereas the scaffolds with α = 45◦ and 60◦ were stiffer in the circumferential direction regardless of the fiber alignment in the scaffolds. In particular, although both the Tube MA-45 and the Tube HA-45 are structurally isotropic along the two stretching directions when unloaded, their stiffness ratios at λθ = λz = 1.02 were also greater than one (1.4–2.3) and their Fung model anisotropy index consistently deviated from one (∼0.7). Note, also, that the extent of anisotropy for the scaffolds with α = 60◦ was significantly greater than that for the scaffolds with α = 30◦ despite their similar structural anisotropy. The mechanical anisotropy of each group of the scaffolds appeared not correlated well with their structural anisotropy. On the other hand, if comparisons were made between the Tube MA and the Tube HA that have the same α, the extent of anisotropy of the Tube HA was greater than that of the Tube MA for α = 30◦ and 60◦ but the differ-
ence between the Tube MA-45 and the Tube HA-45 was not significant.
4 Discussion Although the Fung model provided the overall best fit to the experimental data, it is phenomenological in nature; its parameters have little physical meaning and its applications are limited. Structurally motivated models, on the other hand, have physically significant parameters and can potentially offer better predictive capability. That is, given experimentally prescribed structural parameters, a successful structurally motivated model can accurately predict the mechanical response of a scaffold with previously determined material parameters. Structurally motivated models can thus be used as a guideline to design a scaffold that has mechanical properties closer to the tissue to be replaced. Moreover, a structurally motivated model is preferred than a phenomenological model in the development of a growth and remodeling model that can describe the maturation process of a tissue-engineered construct (Humphrey and Rajagopal 2002); the growth and remodeling model can potentially
123
J.-J. Hu
Tube MA
Tube HA
1.10
1.10
1.05
*
1.05
2
1.00
r
2
** r
30
*
*
**
1.00
0.95
0.95
0.90
0.90
1.10
1.10
* **
**
* **
*
2
1.00
0.95
0.90
0.90
1.10
1.10
r2
r2
**
1.00
0.95
0.95
0.90
Fu
*
*
*
1.00
*
*
1.05
*
*
*
*
1.00
0.95
1.05
60
1.05
r
2
r
45
*
*
**
*
* 1.05
** *
0.90
n
gm
e od
l
M LF
N
M LF
d M M en LF LF c_ c_N ; giv FM NL c_
n Fu
o gm
de
l
M
LF
FM
NL
c_
M nd FM NL give _ ; c FM NL _ c
LF
Fig. 5 Comparisons of the goodness of fit, r 2 , among the six constitutive models for each group of the scaffolds using one-way repeated-measures ANOVA. Note that the Tube HA-60 was analyzed by repeated-measures ANOVA on ranks due to failure of the equal variance test. * p < 0.05 and ** p < 0.001
provide guidelines for culturing application-specific tissueengineered constructs (Guilak et al. 2014; Niklason et al. 2010). There exists a widely used structurally motivated constitutive model considering the helical organization of arteries (Holzapfel et al. 2000); very few tubular scaffolds with such microstructure were found in the literature, though. In this study, the electrospun tubular scaffold to be constitutively modeled not only has the helical organization but its microstructure including fiber directions and fiber angle distributions is controllable. The biaxial mechanical properties
123
of the scaffolds may thus be used to validate structurally motivated models and serve as a tool for the development of such models for arteries or tissue-engineered vascular grafts. Often one problem associated with a comparative study is that each different model is based on data from different specimens. In this study, the Fung model and the structurally motivated models were compared for each scaffold based on the same set of data, and one-way repeated-measures ANOVA was used to examine the potential differences among the models. Therefore, the role of each individual model parameter was able to be studied.
Constitutive modeling of an electrospun tubular scaffold Table 3 Stored strain energy calculated based on the best-fit values of the parameters of each model for each group of scaffolds Specimen type
Tube MA-30
Stored strain energy @ λθ = 1.04 & λz = 1.00 (J/m3 ) Fung
LFM
NLFM
c_LFM
c_NLFM
15,108 ± 742
15,723 ± 680
15,577 ± 890
15,876 ± 592
c_NLFM; given d
15,732 ± 809
15,372 ± 1,396
Tube MA-45
28,146 ± 4,656
26,111 ± 3,337
26,884 ± 3,133
25,622 ± 2,976
37,602 ± 16,484
31,566 ± 3,949
Tube MA-60
47,972 ± 6,793
45,998 ± 4,916
47,430 ± 6,002
45,521 ± 4,798
272,489 ± 333,617
47,695 ± 4,942
Tube HA-30
7,602 ± 411
8,703 ± 208
7,562 ± 472
8,793 ± 256
8,069 ± 371
7,973 ± 317
Tube HA-45
24,308 ± 3,190
20,051 ± 1,498
25,347 ± 3,121
19,635 ± 1, 659
28,376 ± 5,000
23,908 ± 1,202
Tube HA-60
96,459 ± 53,879
41,724 ± 7,905
44,933 ± 6,551
40,634 ± 6,805
464,929 ± 788,243
46,661 ± 5,672
Data are presented as mean ± standard deviation Table 4 Ratio of stiffness in the stretching directions and the Fung model anisotropy indexa calculated based on the best-fit values of the parameters of each model for each group of scaffolds Specimen type
Stiffness ratio @ λθ = 1.02 & λz = 1.02
Anisotropy Fung
Fung
LFM
NLFM
c_LFM
c_NLFM
c_NLFM; given d
Tube MA-30
1.26 ± 0.27
1.35 ± 0.31
1.25 ± 0.26
1.21 ± 0.16
1.20 ± 0.16
1.20 ± 0.16
Tube MA-45
1.99 ± 0.54
1.83 ± 0.48
1.92 ± 0.41
1.63 ± 0.32
2.34 ± 0.78
1.93 ± 0.36
0.64 ± 0.11
Tube MA-60
4.50 ± 1.21
4.34 ± 0.73
4.54 ± 1.08
2.75 ± 0.47
9.92 ± 9.21
3.92 ± 1.19
0.35 ± 0.07
Tube HA-30
1.82 ± 0.22
1.73 ± 0.20
1.82 ± 0.22
1.53 ± 0.16
1.55 ± 0.16
1.56 ± 0.16
0.70 ± 0.06
0.87 ± 0.11
Tube HA-45
1.93 ± 0.35
1.79 ± 0.29
2.34 ± 0.40
1.44 ± 0.15
2.25 ± 0.35
2.07 ± 0.29
0.70 ± 0.08
Tube HA-60
8.19 ± 2.67
6.05 ± 1.35
6.59 ± 1.10
3.74 ± 0.75
9.51 ± 5.67
5.50 ± 1.18
0.31 ± 0.06
the Fung model only. Data are presented as mean ± standard deviation c2 +c3 zzzz 3 > KKθθθθ and cc21 +c For Tube MA-30 and Tube HA-30, KKθθθθ +c3 > c1 +c3 whereas for Tube MA-45, 60 and Tube HA-45, 60, zzzz
a For
c1 +c3 c2 +c3
