constitutive modeling of an electrospun tubular scaffold

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ArticleTitle Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering

Article Sub-Title

Article CopyRight Springer-Verlag Berlin Heidelberg

(This will be the copyright line in the final PDF)

Journal Name Biomechanics and Modeling in Mechanobiology

Corresponding Author Family Name Hu

Particle

Given Name Jin-Jia

Suffix

Division Department of Biomedical Engineering

Organization National Cheng Kung University

Address #1 University Rd., Tainan, 701, Taiwan

Division Medical Device Innovation Center

Organization National Cheng Kung University

Address Tainan, Taiwan

Email [email protected]

Schedule

Received 6 October 2014

Revised

Accepted 16 December 2014

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 (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 stressstrain

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 stress-stretch fits at a greater

axial stretch, especially when fitting data from the scaffolds with a small . Of the models considered, the

Fung model had the overall best fitting results; its applications are limited because of its phenomenological

nature. Although a structurally motivated model using the nonlinear fiber stressstrain relation with theneo-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 (separated by '-') Constitutive modeling - Hyperelasticity - Electrospun scaffolds - Fung model - Structurally motivated

models - Mechanical properties

Footnote Information 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.


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Biomech Model Mechanobiol

DOI 10.1007/s10237-014-0644-y

O R I G I N A L PA P E R

Constitutive modeling of an electrospun tubular scaffoldused for vascular tissue engineering

Jin-Jia Hu

Received: 6 October 2014 / Accepted: 16 December 2014

Springer-Verlag Berlin Heidelberg 2014

Abstract In this study, we sought to model the mechan-1

ical behavior of an electrospun tubular scaffold previously2

reported for vascular tissue engineering with hyperelastic3

constitutive equations. Specifically, the scaffolds were made4

by wrapping electrospun polycaprolactone membranes that5

contain aligned fibers around a mandrel in such a way that6

they have microstructure similar to the native arterial media.7

The biaxial stress-stretch data of the scaffolds made of mod-8

erately or highly aligned fibers with three different off-axis9

fiber angles (30, 45, and 60) were fit by a phenom-10

enological Fung model and a series of structurally motivated11

models considering fiber directions and fiber angle distri-12

butions. In particular, two forms of fiber strain energy in13

the structurally motivated model for a linear and a nonlin-14

ear fiber stressstrain relation, respectively, were tested. An15

isotropic neo-Hookean strain energy function was also added16

to the structurally motivated models to examine its contri-17

bution. The two forms of fiber strain energy did not result18

in significantly different goodness of fit for most groups of19

the scaffolds. The absence of the neo-Hookean term in the20

structurally motivated model led to obvious nonlinear stress-21

stretch fits at a greater axial stretch, especially when fitting22

data from the scaffolds with a small . Of the models con-23

sidered, the Fung model had the overall best fitting results;24

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 25

nature. Although a structurally motivated model using the 26

nonlinear fiber stressstrain relation with the neo-Hookean 27

term provided fits comparably as good as the Fung model, the 28

values of its model parameters exhibited large within-group 29

variations. Prescribing the dispersion of fiber orientation in 30

the structurally motivated model, however, reduced the varia- 31

tions without compromising the fits and was thus considered 32

to be the best structurally motivated model for the scaffolds. 33

It appeared that the structurally motivated models could be 34

further improved for fitting the mechanical behavior of the 35

electrospun scaffold; fiber interactions are suggested to be 36

considered in future models. 37

Keywords Constitutive modeling Hyperelasticity 38

Electrospun scaffolds Fung model Structurally motivated 39

models Mechanical properties 40

1 Introduction 41

Scaffolds, cells and stimulating signals, generally referred 42

to as the tissue engineering triad, are the key components 43

for making a functional tissue-engineered construct. Specif- 44

ically, the three-dimensional, porous scaffolds not only pro- 45

vide a surface for cells to adhere and grow, but also serve 46

as a template to guide matrix formation. Manipulation of 47

the microstructure of the scaffolds thus has potential to con- 48

trol the microstructure of resulting tissues and hence their 49

mechanical properties. The latter are particularly important 50

for the functionality of load-bearing tissues. 51

Among the numerous polymer processing techniques, 52

electrospinning has been widely used to fabricate scaffolds. 53

Electrospun scaffolds are attractive because of their high 54

porosity, large surface area-to-volume ratio, and nano-scale 55

1 3Journal: 10237 MS:0644 TYPESET DISK LE CP Disp.:2014/12/22 Pages:17 Layout:Large
http://dx.doi.org/10.1007/s10237-014-0644-yhttp://dx.doi.org/10.1007/s10237-014-0644-yhttp://dx.doi.org/10.1007/s10237-014-0644-y


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J.-J. Hu

fibrous structure, similar to the features of native extracel-56

lular matrix. In general, only a simple, inexpensive setup is57

required for electrospinning (Ramakrishna 2005). Although58

the orientations of electrospun fibers are typically random,59

aligned electrospun fibers can be achieved with special60

collecting mechanisms (Teo and Ramakrishna 2006). The61

aligned electrospun fibers are capable of providing contact62

guidance to cultured cells(Bashur et al. 2006;Wang et al.63

2011; Yang et al. 2005) and can be used to control the64

microstructure of engineered tissues.65

Very few tubular scaffolds that have helically oriented66

fibers were found in the literature although the helical orga-67

nization is often observed in native arteries (Rhodin 1977)68

and appears to reflect on their complex mechanical behav-69

iors. We recently made such a tubular scaffold by wrapping70

electrospun membranes that contain aligned fibers around a71

mandrel(Hu et al. 2012). In this study, we sought to model72

the biaxial stress-stretch data of the tubular scaffold made73

of moderately or highly aligned fibers with different off-axis74

fiber angles using hyperelastic constitutive equations. There75

have been only a few attempts to establish constitutive rela-76

tions of electrospun scaffolds(Courtney et al. 2006;De Vita77

etal. 2006; Nerurkar et al.2007,2008). Forthe interest of bet-78

ter scaffold design, there is a need to model their mechanical79

behavior with constitutive equations. A proper constitutive80

equation with predictive capability can serve as a guideline81

for making scaffolds that have desired mechanicalproperties.82

We first evaluated a two-dimensional (2-D) phenomeno-83

logical Fung model. As the fibrous structure of the tubular84

scaffold suggests that their mechanical behavior canbe quan-85

tified by structural approaches previously applied for soft tis-86

sues (Billiar and Sacks 2000;Dahl et al. 2008;Lanir 1979,87

1983), a series of structurally motivated models considering88

fiber directions and fiber angle distributions were used to fit89

the data. In particular, two forms of fiber strain energy in the90

structurally motivated model for a linear and a nonlinear fiber91

stressstrain relation, respectively, were tested. An isotropic92

neo-Hookean strain energy function was added to the struc-93

turally motivated models to examine its contribution. Finally,94

the influence of prescribing the dispersion of fiber orientation95

on the modeling was examined.96

2 Methods97

2.1 Electrospun tubular scaffolds98

The details of construction and characterization of the tubu-99

lar scaffold are described inHu et al.(2012). Briefly, scaf-100

fold membranes consisting of moderately aligned fibers and101

highly aligned fibers, denoted as membrane MA and mem-102

brane HA, respectively, were prepared by collecting electro-103

spun fibers on a grounded rotating drum at linear tangential104

A

B

Fig. 1 Schematic diagram showing how a tubular scaffold with an off-axis fiber angle was constructed

velocity of 5.3 and 8 m/s, respectively. Rectangular pieces of 105

membrane MA(or HA) were cut in such a way that the angle 106

between the predominant fiber direction and the long axis 107

(i.e.,) is 30, 45, or 60. Two pieces with the same fiber 108

angle were stacked so that thefiber array of each piecealigned 109

symmetrically along the long axis of the rectangle, forming 110

an axisymmetric membrane. The axisymmetric membrane 111

was then wrapped around a mandrel, resulting in a tubular 112

scaffold with an off-axis fiber angle(either 30, 45 or60; 113

see Fig.1). In total, six groups of five tubular scaffolds each, 114

denoted as Tube MA-30 (membrane-), Tube MA-45, Tube 115

MA-60, Tube HA-30, Tube HA-45, and Tube HA-60, were 116

constructed and mechanically characterized (Hu et al. 2012). 117

The pressure-diameter and the pressure-axial load data pre- 118

viously reported were used along with the unloaded dimen- 119

sions of the scaffold to calculate the mean circumferential 120

and axial Cauchy stresses andzz , respectively, as, 121

=Pa

h, zz =

F+ a2 P

h(2a + h)(1) 122

where P is the applied transmural pressure, F is the axial 123

load imposed by extending the scaffold (Hu et al. 2012), 124

a and h are the deformed inner radius and wall thickness, 125

respectively. Note that the scaffolds were treated as a thin- 126

walled tube and thus rr andzz rr. 127

2.2 Hyperelastic constitutive modeling 128

The biaxial mechanical properties of the tubular scaffolds 129

were quantified by hyperelastic constitutive models. Consis- 130



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Correspondingly, for the c_LFM,207

= c2

1

1

42z

208

+ 2

2

2

R( )

k2f 1

sin2

d (12)209

zz = c2z

1 1

4z2

210

+ 2z

2

2

R( )

k2f 1

cos2

d, (13)211

and for the c_NLFM,212

= c2

1

1

42z

+ 2

2

2

R( )

k1

2f 1

213

exp

k2

2f 1

2sin2

d (14)214

zz = c2z

1 1

4z2

+ 2z

2

2

R( )

k12f 1

215

exp

k2

2f 1

2cos2

d. (15)216

Finally, the influence of prescribing the value ofdin the217

c_NLFM on the goodness of fit was examined. For the Tube218

MAand the Tube HA, d= 0.3 and d= 0.1 were prescribed,219

respectively.220

The best-fit values of the parameters of each model for221

each scaffold were determined using nonlinear regression of222

its biaxial data. This was accomplished by using a modi-223

fied fminsearch function (Matlab) to minimize the objective224

function:225

e =

Ni=1

Theory Exp

mean

Exp

2

i

+

Theoryz Expz

mean

Expz

2

i

(16)226

whereNis the number of data points, and superscripts The-227

ory and Exp denote theoretically calculated and experimen-228

tally determined values, respectively. The goodness of fit is229

presented as the coefficient of determination, r

2

.230

Given the deformation and the values of the model para-231

meters of each model for each scaffold, the stored strain232

energy in the scaffold can be determined. The value of stored233

strain energy at = 1.04 & z = 1.00, which can be234

used as a circumferential stiffness index, was calculated.235

The higher the stored strain energy, the stiffer the scaffolds236

in the circumferential direction. Also, the stiffness in the237

stretching directions, defined as K (=SE

) = 2 W

E2and238

Kzz zz (=SzzEzz

) = 2 W

E2zz, whereSandSzz are components of239

the second PiolaKirchhoff stress tensor in directionsand 240

z, respectively, was determined and the ratio of the two

i.e., 241

max

Kzzz zK

, KKzzz z

was calculated at = z = 1.02 as 242

an anisotropy index. The definition of stiffness was selected 243

for the simplicity of derivation; note that the second Piola 244

Kirchhoff stress and the Green strain are work conjugate. 245

The biaxial stretches, at which the stored strain energy and 246the stiffness ratio were calculated, were chosen simply to be 247

within the range of stretching in the mechanical testing as a 248

constitutive model is considered valid only for the conditions 249

under which it is derived. For the Fung model, in particu- 250

lar, another anisotropy index defined as min

c1+c3c2+c3

, c2+c3

c1+c3

251

(Bellini et al. 2011) was also calculated. 252

2.3 Statistical analysis 253

Because all of the constitutive models were used to fit the 254

data of each scaffold, one-way repeated-measures ANOVA 255

with HolmSidak post hoc testing was used for comparing 256

the goodness of fit among models. Results are reported as 257

mean standard deviation. 258

3 Results 259

Figure2 illustrates representative biaxial stress-stretch data 260

from each group of the scaffolds and the corresponding fits of 261

theFung model.The Fung model fitsthe biaxial stress-stretch 262

data well for all groups of the scaffolds. The best-fit values of 263

the parameters of the Fung model and the associated good- 264

ness of fit for each group of the scaffolds are listed in Table 1; 265

details for each scaffold can be found in Supplemental 266

Table 1. 267

Figure 3 shows the same representative biaxial stress- 268

stretch data from each group and the corresponding fits of 269

the LFM, the NLFM, and the c_LFM. Note that the scales 270

in the plots were changed to better illustrate the differences 271

between models. The fitting curves for the 3-parameter LFM 272

and the 4-parameter NLFM were close to each other. Of par- 273

ticular interest, the LFM and the NLFM led to obvious non- 274

linear stress-stretch fits, especially for the Tube MA-30 and 275

the Tube HA-30 that were subjected to axial stretching. The 276

nonlinear fits were not observed for the c_LFM, however. 277

The best-fit values of the parameters of the LFM, the NLFM, 278

and the c_LFM and the associated goodness of fit for each 279

group of the scaffolds are listed in Table 2; details for each 280

scaffold can be found in Supplemental Tables 24. The esti- 281

mated value ofkin the LFMincreased asincreased for both 282

theTubeMA andthe Tube HA.Similarly, the estimated value 283

ofk1in the NLFM increased with increasing . In particular, 284

the value ofk in the LFM or the value ofk1 in the NLFM 285

estimated by fitting data from the Tube MA was greater than 286



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Constitutive modeling of an electrospun tubular scaffold

0.96 0.98 1.00 1.02 1.04 1.06

Circ.

Ca

uchyStress(kPa)

0

200

400

600

800 Fung modelExp

0.96 0.98 1.00 1.02 1.04 1.06

AxialC

auchyStress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

0.96 0.98 1.00 1.02 1.04 1.06

Circ.

Cau

chyStress(kPa)

0

200

400

600

800 Fung modelExp

0.96 0.98 1.00 1.02 1.04 1.06

AxialCa

uchyStress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

0.96 0.98 1.00 1.02 1.04 1.06

Circ.

CauchyStress(kPa)

0

200

400

600

800 Fung modelExp

0.96 0.98 1.00 1.02 1.04 1.06

AxialCauchyStress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

A

Tube

MA-30

Tube

MA-45

Tube

MA-60

B

C

Circ. Stretch

AxialCauchy

Stress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

Circ. Stretch

Circ.

Cauchy

Stress(kPa)

0

200

400

600

800 Fung modelExp

0.96 0.98 1.00 1.02 1.04 1.060.96 0.98 1.00 1.02 1.04 1.06

D

Tube

HA-30

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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Circ.

Ca

uchyStress(kPa)

0

200

400

600

800 Fung modelExp

AxialC

auchyStress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

Circ. Stretch

Circ.

Cauc

hyStress(kPa)

0

200

400

600

800 Fung modelExp

Circ. Stretch

0.96 0.98 1.00 1.02 1.04 1.06 0.96 0.98 1.00 1.02 1.04 1.06

0.96 0.98 1.00 1.02 1.04 1.06 0.96 0.98 1.00 1.02 1.04 1.06

AxialCauc

hyStress(kPa)

0

200

400

600

800

1000

1200

1400

Fung model

Exp

E

F

Tube

HA-45

Tube

HA-60

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 Goodness of fit

c0 (Pa) c1 c2 c3 r2

Tube MA-30 362,700 196,314 81.06 56.48 64.20 39.67 27.39 15.43 0.992 0.006

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 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 fitting287

data from the Tube HA given the same. The estimated val-288ues ofk2 (i.e., nonlinearity) in the NLFM was found to be289

in the order: Tube MA-30 = Tube MA-45> Tube MA-60290

and Tube HA-45> Tube HA-30> Tube HA-60. For both291

the LFM and the NLFM, the estimated values ofwere con-292

sistently greater than the experimentally prescribed angles.293

The inclusion of the neo-Hookean term in the LFM altered294

the estimated values of damong groups dramatically and295

affected the estimated values of k as well. Also, the neo-296

Hookean term improved the fits of c_LFM for the scaf-297

folds with = 45 and 60, and no significant differ- 298

ence was found though. In general, the within-group vari- 299ations in the model parameters of the three constitutive 300

models for the Tube HA were less than those for the 301

Tube MA. 302

Figure 4 shows the same representative biaxial stress- 303

stretch data from each group and the corresponding fits of 304

the c_NLFM and the c_NLFM withdprescribed. The fits of 305

the c_NLFM and the c_NLFM withdprescribed were better 306

than those of the previous three structurally motivated mod- 307

els. The best-fit values of the parameters of the c_NLFM and 308



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A

Tube

MA-30

Tube

MA-45

Tube

MA-60

B

C

0.98 1.00 1.02 1.04

Circ.C

auchyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

0.98 1.00 1.02 1.04

AxialC

auchyStress(kPa)

0

200

400

600

800

1000

1200

1400

LFM

NLFMc_LFMExp

0.99 1.00 1.01 1.02

Circ.

Ca

uchyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

0.99 1.00 1.01 1.02

AxialCau

chyStress(kPa)

0

200

400

600

800

1000

1200

1400

LFM

NLFMc_LFMExp

0.990 0.995 1.000 1.005 1.010

Circ.

CauchyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

0.990 0.995 1.000 1.005 1.010


0

200

400

600

800

1000

LFM

NLFMc_LFMExp

D

Tube

HA-30

Circ. Stretch

0.98 1.00 1.02 1.04 1.06

Circ.

Cauch

yStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

Circ. Stretch

0.98 1.00 1.02 1.04 1.06


0

200

400

600

800

1000

1200

1400

LFM

NLFMc_LFMExp

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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E

F

Tube

HA-45

Tube

HA-60

0.99 1.00 1.01 1.02 1.03

Circ.C

auchyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

0.99 1.00 1.01 1.02 1.03

AxialC

auchyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

Circ. Stretch

0.990 0.995 1.000 1.005 1.010

AxialCau

chyStress(kPa)

0

100

200

300

400

500 LFMNLFMc_LFMExp

Circ. Stretch

0.990 0.995 1.000 1.005 1.010

Circ.

Cau

chyStress(kPa)

0

200

400

600

800 LFMNLFMc_LFMExp

Fig. 3 continued

the c_NLFM withdprescribed and the associated goodness309

of fit for each group of the scaffolds are also listed in Table 2;310

details for each scaffold can be found in Supplemental Table311

56. Similar to the finding of the c_LFM, the inclusion of the312

neo-Hookean term in the NLFM altered the estimated values313

ofd among groups and affected the estimated values ofk1314

andk2 as well. In particular, there were large within-group315

variations in the model parameters of the c_NLFM. Note316

that the within-group variations in the model parameters of317

the c_LFM were relatively small. Interestingly, prescribing318

the value ofd reduced the within-group variations aforemen-319

tioned for c_NLFM and still resulted in fits comparable to320

those of the c_NLFM (i.e., no significant difference in fits321

between the c_NLFM and the c_NLFM with dprescribed)322

for all groups of the scaffolds. Furthermore, the c_NLFM323

withdprescribed involves only four parameters. Prescribing324

the value ofdslightly reduced the goodness of fit for scaf-325

folds with = 45 and60, thedifference was notsignificant326

though.327

Figure 5 compares the fits of the six constitutive mod-328

els for each group of the scaffolds via one-way repeated-329

measures ANOVA. For fitting the data of the Tube MA,330

there was no significant difference among the structurally331

motivated models. Nevertheless, the Fung model was signif-332

icantly different from the LFM, the NLFM, and the c_LFM. 333

On the other hand, for fitting the data of the Tube HA, the 334

5-parameter c_NLFM had the best fits among the struc- 335

turally motivated models, particularly for the Tube HA- 336

30; it was the only structurally motivated model that had 337

fits as good as the Fung model for all of the scaffolds 338

tested. Note, particularly, that the Tube HA-30 was the only 339

group for which the NLFM provided better fits than the 340

LFM. For both the Tube MA and the Tube HA, the smaller 341

the , the better the fitting for all the constitutive models 342

considered. 343

Based on the constitutive models, the stored strain energy 344

for each group of the scaffolds was calculated at = 345

1.04 &z = 1.00 and the results are listed in Table3;details 346

for each scaffold can be found in Supplemental Table 7. For 347

all the constitutive models, the stored strain energy of the 348

Tube MA-30 was significantly greater than that of the Tube 349

HA-30. That is, the Tube MA-30was circumferentially stiffer 350

than the Tube HA-30. There was no significant difference in 351

the value of stored strain energy between the Tube MA and 352

the Tube HA with = 45 and 60, however. Note that the 353

c_NLFM generated greater within-group variations for the 354

scaffolds with = 45 and 60 probably due to the extreme 355

estimated values ofk2 in these groups of the scaffolds. 356



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J.-J. Hu

A

Tube

MA-30

Tube

MA-45

Tube

MA-60

B

C

Circ.CauchyStress(kPa)

0

200

400

600

800 c_NLFM

c_NLFM; given dExp


0

200

400

600

800

1000

1200

1400

c_NLFM

c_NLFM; given dExp


0

200

400

600

800

1000

1200

1400

c_NLFM

c_NLFM; given dExp

Circ.

CauchyStress(kPa)

0

200

400

600

800 c_NLFM

c_NLFM; given dExp

Circ.

Cau

chyStress(kPa)

0

200

400

600

800 c_NLFMc_NLFM; given dExp

AxialCauc

hyStress(kPa)

0

200

400

600

800

1000

c_NLFM

c_NLFM; given dExp

D

Tube

HA-30

Circ. Stretch

Circ.

Cauch

yStress(kPa)

0

200

400

600


Circ. Stretch

AxialCauchy

Stress(kPa)

0

200

400

600

800

1000

1200

1400

c_NLFM

c_NLFM; given dExp

0.98 1.00 1.02 1.04 0.98 1.00 1.02 1.04

0.99 1.00 1.01 1.020.99 1.00 1.01 1.02

0.990 0.995 1.000 1.005 1.010 0.990 0.995 1.000 1.005 1.010

0.98 1.00 1.02 1.04 1.06 0.98 1.00 1.02 1.04 1.06

Fig. 4 Fits of the c_NLFM (inverted triangle) and the c_NLFM with dprescribed (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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E

F

Tube

HA-45

Tube

HA-60

Circ.C

auchyStress(kPa)

0

200

400

600

800 c_NLFM

c_NLFM; given dExp

AxialC

auchyStress(kPa)

0

200

400

600


Circ. Stretch

Circ.

Cauc

hyStress(kPa)

0

200

400

600

800 c_NLFM

c_NLFM; given dExp

Circ. Stretch

0.99 1.00 1.01 1.02 1.03 0.99 1.00 1.01 1.02 1.03

0.990 0.995 1.000 1.005 1.010 0.990 0.995 1.000 1.005 1.010

AxialCauch

yStress(kPa)

0

100

200

300

400


Fig. 4 continued

The results of anisotropy analysis for each group of the357

scaffolds are summarized in Table4;details for each scaf-358

fold can be found in Supplemental Table 8. Interestingly,359

the stiffness ratios for all groups of the scaffolds calculated360

at = z = 1.02 were greater than one and their Fung361

model anisotropy index deviated from one; the scaffolds362

with = 30 were stiffer in the axial direction whereas363

the scaffolds with = 45 and 60 were stiffer in the cir-364

cumferential direction regardless of the fiber alignment in365

the scaffolds. In particular, although both the Tube MA-45366

and the Tube HA-45 are structurally isotropic along the two367

stretching directions when unloaded, their stiffness ratios at368

= z = 1.02 were also greater than one (1.42.3) and369

theirFung modelanisotropy index consistently deviatedfrom370

one (0.7). Note, also, that the extent of anisotropy for the371

scaffolds with = 60 was significantly greater than that372

for the scaffolds with = 30 despite their similar structural373

anisotropy. The mechanical anisotropy of each group of the374

scaffolds appeared not correlated well with their structural375

anisotropy. On the other hand, if comparisons were made376

between the Tube MA and the Tube HA that have the same377

, the extent of anisotropy of the Tube HA was greater than378

that of the Tube MA for = 30 and 60 but the differ-379

ence between the Tube MA-45 and the Tube HA-45 was not 380

significant. 381

4 Discussion 382

Although the Fung model provided the overall best fit to the 383

experimental data, it is phenomenological in nature; its para- 384

meters have little physical meaning and its applications are 385

limited. Structurally motivated models, on the other hand, 386

have physically significant parameters and can potentially 387

offer better predictive capability. That is, given experimen- 388

tally prescribedstructural parameters, a successful struc- 389

turally motivated model can accurately predict the mechani- 390

cal response of a scaffold with previously determined mate- 391

rial parameters. Structurally motivated models can thus be 392

used as a guideline to design a scaffold that has mechan- 393

ical properties closer to the tissue to be replaced. More- 394

over, a structurally motivated model is preferred than a phe- 395

nomenological model in the development of a growth and 396

remodeling model that can describe the maturation process 397

of a tissue-engineered construct(Humphrey and Rajagopal 398

2002); the growth and remodeling model can potentially 399



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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 Stored strain energy @= 1.04 &z = 1.00 (J/m3)

Fung LFM NLFM c_LFM c_NLFM c_NLFM; givend

Tube MA-30 15,108 742 15,723 680 15,577 890 15,876 592 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 1659 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 thestretchingdirectionsand theFung model anisotropy indexa calculated based on thebest-fit values of theparameters

of each model for each group of scaffolds

Specimen type Stiffness ratio @= 1.02 &z = 1.02 Anisotropy

Fung

Fung LFM NLFM c_LFM c_NLFM c_NLFM; givend

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 0.87 0.11Tube 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

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

a For the Fung model only. Data are presented as mean standard deviation

For Tube MA-30 and Tube HA-30, Kzzz z

K > K

Kzzz zand c1+c3

c2+c3> c2

+c3c1+c3

whereas for Tube MA-45, 60 and Tube HA-45, 60, Kzzz zK

< K Kzzz z

andc1+c3c2+c3

< c2+c3

c1+c3

The estimated values of the fiber modulus-related para-422

meter,kin the LFM ork1in the NLFM, were expected to be423

the same for either the Tube MA or the Tube HA; the Tube424

MA and the Tube HA were made, respectively, of scaffold425

membranes with distinct fiber angle distributions. Both kand426

k1 were found to increase with increasing of the scaffolds,427

however. The dependence of estimated fiber modulus on the428

in both the LFM and the NLFM indicated that there exists429

some factor whose influence is dependent on the and was430

not accounted for in the two models. One reasonable factor is431

the resistance due to fiber interactions.Croisier et al.(2012)432

observed three orders of magnitude difference in the moduli433

of fiber scaffolds versus single fibers and attributed the dif-434

ference to the lacunar and random structure of the scaffolds.435

The fiber modulus estimated from the fit for the Tube MA436

was also found to be greater than that for the Tube HA for437

the same . This may be well attributed to the higher den-438

sity of fiber interconnections in the Tube MA; the Tube MA,439

compared to the Tube HA, might be subjected to a greater440

resistance due to fiber interactions when inflated.Courtney441

et al.(2006) related fiber alignment in an electrospun mem-442

brane, which was manipulated by controlling the rotation443

speed of a grounded mandrel, to the mechanical behavior444

of the membrane based on a similar structurally motivated445

model. In contrast to our finding, they found that the esti- 446

mated fiber stiffness is greater if the fiber alignment is better. 447

The contradictory findings may be due to different geometry 448

of specimens, different protocols of mechanical testing, and 449

different scaffolding materials. Note that a positive correla- 450

tion between the crystallinity of the polymers in the fibers, 451

which was shown to affect their tensile properties (Lim et al. 452

2008), and the mandrel velocity was also reported in their 453

study. In addition tokor k1, that the value of estimated by 454

both the LFM and the NLFM was in general greater than the 455

experimentally prescribed also suggested that some fac- 456

tor other than those considered in the two models was active 457

in increasing circumferential stiffness during inflation of the 458

scaffolds. 459

In this study, electrospun fibers were collected on a 460

rapidly rotating, grounded drum. As crimped fibers were not 461

observed in the electrospun membranes that were used to 462

fabricate the scaffolds, we did not consider those forms of 463

fiber strain energy function that account for fiber undula- 464

tion (Hill et al. 2012) or fiber curvature (Pai et al. 2011). 465

Nevertheless, we tested a strain energy function that consid- 466

ers nonlinear fiber stressstrain relation. Although the fits 467

of the 3-parameter LFM were comparable to those of the 468

4-parameter NLFM for most groups of the scaffolds, the 469



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J.-J. Hu

performance of the NLFM in fitting data from the Tube470

HA-30 was significantly better than the LFM. The improve-471

ment of fitting due to the introduction of fiber nonlinearity472

may be attributed to gradually increasing fiber interactions473

upon inflation since these aligned polycaprolactone fibers474

likely exhibit linear stress-stretch behavior. Note that the475

fiber nonlinearity-related parameter, k2, in the NLFM was476

also found to be dependent on the of the scaffold; the esti-477

mated nonlinearity increased from = 30 to 45 and then478

decreased from = 45 to 60. The contribution of fiber479

interactions might be increasing upon inflation for the scaf-480

folds with a small but remain steady during inflation for481

the scaffolds with a large .482

Although all of the scaffolds exhibited fair linear stress-483

stretch behavior (only representative data are shown), non-484

linear stress-stretch fits by the LFM and the NLFM were485

observed, which were particularly obvious when the scaffold486

is subjected to axial stretching (e.g.,z = 1.04). This is inter-487

esting but not surprising as in both the LFM and the NLFM,488

fiber interactions were not explicitly considered. In the two489

models, circumferentially oriented fibers in the scaffold can490

freely become crimped when the scaffold is solely axially491

stretched (without inflation) assuming the incompressibility492

of the scaffold. Note that the crimped fibers( f < 1)have493

no mechanical contribution in the two models. The gradually494

straightening of the crimped fibers upon inflation can thus495

lead to the nonlinear fits of the two models. The nonlinear496

fits could also be attributed to the affine motion of fibers or497

fiber re-orientation upon inflation in structurally motivated498

models. The contribution may not be as significant due to499

the small deformation and the axial constraint; nonlinear fits500

were not observed when the inflated scaffold was not axi-501

ally stretched (i.e., z = 1.00). The inclusion of the neo-502

Hookean term in the LFM and the NLFM, however, allevi-503

ated the nonlinearity. The neo-Hookean term was introduced504

in structurally motivated models as a strain energy function505

associated with isotropic deformation of amorphous elastic506

constituents(Holzapfel et al. 2000;Humphrey 1999b). The507

neo-Hookean term is arguably phenomenological (Hollan-508

der et al.2011), and specific quantitative information about509

the term is hardly obtained (from histological and structural510

analyses). Nevertheless, the inclusion of the neo-Hookean511

term did improve the fits of the c_LFM and the c_NLFM512

when compared to the LFM and the NLFM, respectively. We513

suggested that the neo-Hookean term may, in part, account514

for the resistance due to fiber interactions. Note that the tubu-515

larscaffolds that were constitutivelymodeled in this study are516

made of electrospun fibers and contain no isotropic matrix.517

In our preliminary tests, a LFM with a fiber angle dis-518

tribution function that includes explicitly the portion of519

isotropically oriented fibers (i.e., R( ) = x + (1 x)520

d

1 +

d

21, where x is the mass fraction of521

isotropically oriented fibers) wastested. The estimatedvalues 522

ofxfor both the Tube MA and the Tube HA, in most cases, 523

were found to be negligible, which was consistent with the 524

finding ofCourtney et al. (2006). For most of the scaffolds, 525

the fits of this 4-parameter LFM were very similar to those of 526

the 3-parameter LFM; the nonlinear stress-stretch fits were 527

also observed. That is, the additionalparameter,x, inthefiber 528

angle distribution function wasnot effective in improving the 529

fitting. Note that all fibers including those isotropically ori- 530

ented fibers were assumed to share the same fiber modulus in 531

the 4-parameter LFM, and therefore, the neo-Hookean term, 532

which is independent on the fiber modulus, may not be fully 533

compensated by the mass fraction of isotropically oriented 534

fibers or by a different fiber angle distribution function. 535

The potential role of fiber interactions in the mechani- 536

cal behavior of electrospun scaffolds was identified in this 537

study. Although some model parameters, e.g., c, k (or k1), 538

and k2, in the structurally motivated models could some- 539

how account for the resistance of fiber interactions, an 540

isolated strain energy function specific for fiber interac- 541

tions is certainly more desirable. Palmer (2008) proposed 542

a cross-link torsional strain energy to account for the stiff- 543

ness of fiber junction, which was then used to model 544

the mechanical properties of electrospun nonwoven fiber 545

meshes (Pai et al. 2011). The inclusion of torsional strain 546

energy function, however, did not perform as good as the 547

inclusion of the neo-Hookean term in our preliminary tests 548

(data not shown). On the other hand, multi-scale network 549

model that incorporates all the structural information of the 550

specimen was developed to describe the mechanical behavior 551

of electrospun scaffolds (Argento et al. 2012). The network 552

model may be more helpful in accounting for the fiber inter- 553

actions than structural continuum models. Their accuracy, 554

however, depends on how detailed and accurate the micro- 555

mechanics of the network was captured. 556

Both the Fung model and the structurally motivated mod- 557

els fit the mechanical behavior of the scaffolds with a small 558

betterthan thescaffolds with a large.Thismaybeduetoless 559

deformation of the latter during inflation, and the influence of 560

inherent experimental errors became more dominated. Fur- 561

thermore, upon inflation, the two symmetric arrays of fibers 562

in the latter might become closer to each other (i.e., merging 563

into one group) and the structurally motivated models could 564

fail; indeed, the estimated model parameters of the struc- 565

turally motivated models for the Tube MA-60 and the Tube 566

HA-60 became not as unique as those for the Tube MA-30 567

and the Tube HA-30. This certainly constitutes a limitation 568

of the structurally motivated models. 569

The model parameters of an ideal structure-based con- 570

stitutive relation must be independent; that is, the mechani- 571

cal contribution of one model parameter cannot be compen- 572

sated by the other. The estimated value of a model parame- 573

ter can thus represent the physical significance of the para- 574



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