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On the deformation behavior of human amnion
Journal of Biomechanics 46 (2013) 1777–1783
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On the deformation behavior of human amnion Wilfried Buerzle a,n, Edoardo Mazza a,b a b
Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland Swiss Federal Laboratories for Materials Science and Technology, EMPA Dubendorf, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Accepted 20 May 2013
Recently renewed interest for the mechanical behavior of fetal membranes is related to the problem of iatrogenic preterm rupture, limiting the effectiveness and applicability of minimally invasive fetal surgery. This study aimed at characterizing and modeling the deformation behavior of the amnion layer, the highly deformable and tough membrane that surrounds the amniotic fluid and the growing fetus in the uterine cavity. Uniaxial tension tests have been performed on samples obtained immediately after cesarean section, and the deformation field has been analyzed by digital image correlation. The results show that the kinematic response of human amnion is highly reproducible and that the incremental Poisson's ratio is, with a value of up to 8, higher than any previously reported value for biological or synthetic materials. This unique behavior is related to the characteristic architecture of amnion's microstructure and can be rationalized by mechanisms of rotation, stretching and buckling of collagen fibers. Simple constitutive equations have been selected based on this interpretation, which lead to a model with excellent predictive capabilities for the uniaxial and equibiaxial mechanical response of human amnion. Relevant insights were gained on the role of collagen fibers in determining the deformability and toughness of soft biological tissue. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Fetal membrane Poisson's ratio Kinematics Network Constitutive model
1. Introduction The fetal membrane (FM) surrounds and protects the fetus during pregnancy and is adjacent to the uterine wall. The FM is composed of two main layers, called amnion and chorion. During pregnancy, it has to resist deformations due to internal pressure and fetal movements. Rupture of FM is a natural event for term delivery but has serious complications when it happens prior to term. The so called spontaneous preterm premature rupture of FM (PPROM) affects 3% of all pregnancies worldwide (Calvin and Oyen, 2007; Mercer, 2003) and is associated with a higher risk of perinatal morbidity and mortality. Moreover, preterm rupture after fetal surgery, called iatrogenic PPROM (iPPROM), is the Achilles heel for developments in this field and occurs in about 30% of all treatments (Beck et al., 2011). The development of methods to avoid FM preterm rupture requires a better understanding of their mechanical behavior. Only few attempts were reported for the determination of a constitutive model of FM (Miller et al., 1979; Prevost, 2004; Jabareen et al., 2009; Joyce, 2009). Most previous studies focused on the extraction of parameters to quantitatively characterize FM rupture and on the effects leading to PPROM (Artal et al., 1976; Lavery and Miller, 1979; Al-Zaid et al., 1980). Other investigations were aimed at determining differences in mechanical properties of amnion and n Corresponding author at: Swiss Federal Institute of Technology, Center of Mechanics, Tannenstrasse 3, 8092 Zurich, Switzerland. Tel.: +41 44 632 77 91; fax: +41 44 632 11 45. E-mail address: [email protected] (W. Buerzle).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.05.018
chorion (Polishuk et al., 1962; Oxlund et al., 1990; Helmig et al., 1993). It is known from these foregoing studies that amnion is stiffer and stronger than chorion and is therefore considered the mechanically dominant layer of FM. Oyen has investigated the time and history dependence of amnion's deformation behavior (Oyen et al., 2004, 2005) and has shown that amnion is more sensitive to chemical and mechanical changes that occur during gestation (Oyen et al., 2006). Amnion is the inner layer of the FM, facing the amniotic fluid. Membranes at term have an amnion layer with a thickness in the range of 100 μm (Ilancheran et al., 2009). Next to the amniotic epithelium and the basement membrane, the so called “compact layer” is made predominately of collagens type I and III that form a dense network (Malak et al., 1993; Parry and Strauss, 1998). The adjacent fibroblast layer is the thickest layer of amnion and is characterized by collagens forming a looser network. The outermost layer of amnion is the “intermediate layer”, which contains a nonfibrillar network of mostly type III collagen (Parry and Strauss, 1998), it lies between amnion and chorion and accommodates relative movement between the layers. Fig. 1 shows scanning electron micrograph (SEM) images of a FM sample. It illustrates the crimped collagen fibres, forming a random network in the membrane plane, with no distinct directionality. Chorioamniotic membrane separation leads to a premature rupture of the membranes in 63% of the cases after surgery (Sydorak et al., 2002). The separation of the FM layers has been observed in puncture experiments (Arikat et al., 2006). We also consistently observed a separation of amnion and chorion in uniaxial tension tests on intact FM, see Fig. 2. In this case
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Chorion
Amnion
Fig. 2. Image of an amnion–chorion FM sample during tension test. The constitutive layers of the FM start to separate at a certain level of deformation. The dotted line indicates the border of the amnion layer after separation. protocol approved by the ethical committee of the district of Zurich (study Stv22/ 2006). The patients were randomly selected after thorough serologic testing to exclude HIV, hepatitis B, and streptococcus B. The selected patients had no history of diabetes, connective tissue disorders, or chromosomal abnormalities. After cesarean section the FM was cut approximately 2 cm away from the placental disk and stored in saline solution until mechanical testing, within a few hours after delivery. Amnion and chorion were separated manually by gently pulling the layers apart. Amnion was spread on a lubricated plastic mat and a die cutter (see Hollenstein et al., 2010) was used to create rectangular samples with the dimension of 80 15 mm. Small markers of India Ink were applied on the surface of the sample with the aid of a brush, which allows better optical tracking of the deformation field.
2.2. Mechanical experiments
Fig. 1. Scanning electron micrograph of a human fetal membrane sample. Crosssectional view (a) and top view (b) show an irregular network of singe collagen fibers with no distinct directionality. With permission from Hollenstein et al. (2010).
In order to test the FM samples in a uniaxial stress configuration, a custom built experimental setup (Hollenstein et al., 2010) was used. This consists mainly of computer controlled hydraulic actuators equipped with force sensors (20 N capacity) and a CCD camera system which allows to record image series during the experiment. The tensile tests were performed as monotonic tension to failure with an elongation rate of 0.5% nominal strain per second, starting from an initial free length of 60 mm. No preconditioning was applied. All tests were performed at room temperature in a bath filled with saline solution to avoid drying or swelling of the sample. Data on the response of amnion in a biaxial state of stress were obtained using a custom built inflation device (Haller, 2012). The inflation setup, the experimental protocol, and the procedures for extraction of biaxial tension–stretch curves are described in Buerzle et al. (2012).
2.3. Analysis of the deformation field in uniaxial experiments
separation is caused by different contraction behavior (Poisson's effect) of the two layers, with amnion displaying a much greater lateral contraction. Unusually large contraction of amnion in tensile tests had never been reported. This motivated a series of tensile tests on human amnion with quantitative determination of lateral contraction. Experimental observations along with considerations on mechanisms of rotation, stretching and buckling of fibers in a random network (Stylianopoulos and Barocas, 2007; Kabla and Mahadevan, 2007; Picu, 2011) led to the selection of a constitutive model formulation. Model parameters were determined to simultaneously match tension–stretch and contraction–elongation curves of tensile tests. Model response to equibiaxial tension was finally compared with corresponding data from previous measurements (Haller, 2012). 2. Experimental methods 2.1. Sample harvesting and preparation Fetal membranes were collected from patients with single child pregnancies who underwent elective cesarean section between 37 and 39 weeks of gestation. Patients were recruited for this study with informed written consent according to a
The recorded image series serves as basis for the determination of the kinematic response of the amnion layer. The digital image correlation (DIC) software VEDDAC 4.0 (Chemnitzer Werkstoffmechanik GmbH, Chemnitz, Germany) is used to trace a set of user defined measurement points corresponding to the markers in the central region of the sample, see Fig. 3. The resulting discrete displacement filed is used for the approximation of the in-plane stretches by the use of element based interpolation functions, as applied in the finite element (FE) method. The displacements of the measurement points are considered as nodal displacements of a linear four node element. This approach allows calculating the in-plane components of the element specific deformation gradient Fel as a function of the associated nodal displacements u^ and the linear shape functions N 4
ðaÞ Fel ¼ I þ ∑ u^ ⊗ a¼1
∂N ðaÞ : ∂X
ð1Þ
All common measures of deformation can be calculated based on the deformation gradient. For the load cases considered in this work, only the two in-plane principal stretches λ1 and λ2 are determined. Averaging over the entire region, in this case a 2 3 element scheme, provides the kinematic response of the amnion layer. The typical variability of elongation values in different elements was in the range of 5%. The Poisson's ratio is defined as the ratio of lateral strain (contraction, in direction 2) and longitudinal strain (elongation, in direction 1). It is strictly defined only for infinitesimal deformations. For large strain behavior, an incremental Poisson's ratio or Poisson's function, according to the definitions of Alderson et al. (1997) and Smith et al. (1999) can be used, which is based on the ratio of
W. Buerzle, E. Mazza / Journal of Biomechanics 46 (2013) 1777–1783
μ 0 ¼ μ0 h0
ð4Þ
1.4
g ¼ g 2 ðβ1 Þ þ g 3 ðλI Þ
ð5Þ
1.35
g 2 ðβ1 Þ ¼ m2 ðβ1 −3Þ
ð6Þ
1.3
g 3 ðλI Þ ¼ ∑
X Position [pixel] 0
0
100
200
300
400
500
600
700
800
900
1000
100
300
500 600
I
1.25 1.2
Stretch λ1
Y Position [pixel]
200
400
1.15
700 1.1 800 900 1000
1.05 1
Fig. 3. Illustration of the image analysis procedure. Image shows the transparent amnion layer with markers of India Ink which are used to trace a set of user defined measurement points (red squares) by DIC method. A finite element scheme is used for the determination of the principal stretches. The camera field of view is 30 30 mm and the typical element size is 5 mm (x-direction) and 2 mm (y-direction). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) derivatives of logarithmic strains in the lateral and axial directions. νincr 12 ¼ −
λ1 ∂λ2 λ2 ∂λ1
ð2Þ
This incremental Poisson's ratio might be regarded as akin to the definition of a tangent modulus for stiffness.
ð7Þ
β1 ¼ trðBÞ is the first invariant of the right Cauchy–Green deformation tensor B ¼ FFT , and λI ¼ jmI j is the stretch of the I-th family of fibers, with mI ¼ FMI . The directions of the fibers in the reference configuration are defined by the unit vector MI . Very important for the present application, the fibers are considered not to bear compression loads, as indicated by the use of the Macaulay brackets 〈x〉 ¼ 1=2ðx þ jxjÞ. m2, m3, m4, μ 0 , and q are model parameters which have to be determined from an inverse analysis of the experimental data. Due to the high water content (up to 90% Halaburt et al., 1989), volume preservation (J ¼1) was assumed in this case. Note that at present this assumption cannot be verified experimentally, since the thickness or the water content of the test pieces cannot be monitored during the tests. 3.2. Model parameters determination The Cauchy stress tensor r is obtained from derivatives of the strain energy function with respect to the corresponding deformation tensor. The membrane is in the 1–2 plane and the deformation gradient is (for all cases considered here) of the form 2 3 λ1 0 0 6 7 F ¼ 4 0 λ2 0 5 ð8Þ 0 0 λ3
λ3 ¼
3.1. Model formulation Fig. 1 shows the collagen microstructure of amnion. Thin, randomly oriented collagen fibers characterize this tissue and are expected to determine the mechanical response of amnion. A corresponding phenomenological model has to include terms representative of the mechanisms of stretching, rotation and buckling of the fibers. In this work the constitutive model formulation developed by Rubin and Bodner (2002) was chosen to model the response of the amnion layer. The model allows considering (families of) fibers embedded in an isotropic matrix. It was shown in previous investigations to provide useful representation of the uniaxial (Jabareen et al., 2009) stress response of intact FM but underestimated the biaxial stress response (Buerzle et al., 2012), if formulated without fibers. The Rubin–Bodner model includes terms to also describe dissipative deformation behavior. Our experimental data do not provide information on time and history dependence of amnion's response. Thus, only the elastic part of the equations was considered in this work. The corresponding strain energy function, ψ M , is a composition of different contributions, describing the pure distortional and dilatational parts and the stretch of the single families of fibers. The original equations have been adapted for a membrane model formulation, i.e. ψ M ¼ ψh0 is the strain energy per unit reference area of the membrane midplane surface with h0 being the membrane thickness in the reference configuration and ψ the strain energy per unit reference volume (note that the original Rubin–Bodner model is formulated in terms of energy per unit mass). μ 0 qg ½e −1 2q
m3 〈λI −1〉2m4 m4
with
3. Constitutive model
ψM ¼
1779
ð3Þ
1 : λ1 λ2
ð9Þ
Following the definition of a membrane formulation according to Section 3.1, the membrane tension tensor t (in the current configuration) reads h0 m2 μ 1 t¼r ¼ B− ðB IÞI λ1 λ2 3 λ1 λ2 m3 μ 1 ph 〈λI −1〉2m4 −1 ðmI ⊗mI Þ− 0 I λ λ λ λ I 1 2 1 λ2 I¼1 8
þ ∑
ð10Þ
μ ¼ μ 0 eqg
ð11Þ
The membrane tension tensor t must fulfill two boundary conditions in a state of uniaxial tension. First, the stress vector in thickness direction must vanish, which can be achieved by the corresponding determination of the hydrostatic pressure p, which is a general function of position and time due to the incompressibility constraint. Second, also the stress vector in lateral (inplane) direction must vanish. This condition for the 22 component of t t 22 ¼ 0 ¼
m3 8 1 1 ∑ 〈λI −1〉2m4 −1 ðmI ⊗mI Þ22 þ λ22 − 2 2 λI m2 I ¼ 1 λ1 λ2
ð12Þ
determines simultaneously the kinematic response of the constitutive model, since it provides the stretch λ2 , as a function of λ1 and λI . The kinematic response in the uniaxial stress state depends only on two model parameters, i.e. the ratio m3/m2 and the parameter m4. The other model parameters do not affect the kinematic response. This circumstance can be used to split up the inverse analysis in two steps. First, only the parameter-ratio m3/m2 and the parameter m4 is determined in order to match the experimental kinematic response (contraction–elongation curve).
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Next, the remaining parameters q, μ 0 , and m2 (or m3) are determined to match the tension–stretch response, while keeping m3/m2 and m4 unchanged. The best fit model parameters are determined by the use of the Nelder–Mead simplex method, fminsearch, in MATLAB 2010a (The MathWorks, Inc., Natick, MA, USA).
Cauchy Tension [N/mm]
1
4. Results 4.1. Uniaxial tests
4.2. Constitutive model parameters The model response should be (i) nearly isotropic in the membrane plane. In fact, no preferential directions have been reported in any previous study. It should display the typical (ii) j-shaped tension–stretch curve, and, in particular, reproduce the observed (iii) very large lateral contraction. Simultaneous fulfillment of all these conditions is a challenging task. The Rubin–Bodner model (Rubin and Bodner, 2002) has been implemented considering eight identical families of fibers in the initial directions of f0; 722:5; 7 45; 767:5; 90g1, thus providing a quasi-isotropic behavior in the membrane plane. Recent work on a discrete fiber model based on fibers oriented in the directions of opposing vertices of a regular icosahedron shows that this model is not exactly isotropic (Flynn et al., 2010; Flynn and Rubin, 2012). Fig. 5 shows the results of the two step optimization procedure (explained in Section 3.2) applied to the averaged uniaxial measurements. The model very well matches both, the tension–stretch response as well as the contraction–elongation curves. Since comparison between model and measurements is performed for the membrane tension (and not for stress), there is no need to determine a representative initial thickness of the “average” amnion sample. The “best fit” model parameters for the averaged amnion response are reported in Table 2. The key for representing the observed kinematic response is the fact that the fibers do not resist compressions (no stress for λI o 1). Fig. 6 illustrates the expected rotation of two fibers for a prescribed elongation in direction e1 . Fiber A has an initial orientation (dashed line) with a small angle with respect to e1 , whereas Fiber B has an initial large angle deviation from e1 . Deformation leads to fibers stretching and rotation. Fiber A
0.6 0.4 0.2 0 1
1.1
1.2
1.3
1.4
Stretch λ1 [−] 1
0.8
Stretch λ2 [−]
Uniaxial tension tests have been performed on 12 samples from six different membranes. The clamped amnion sample is initially slack. A low force threshold (5 mN in this case) was applied to define the initial configuration. Fig. 4(a) shows the tension–stretch curves for each single amnion sample and Fig. 4(b) illustrates the corresponding in-plane kinematic responses, i.e. λ2 versus λ1 . It can be seen in Fig. 4(a) that the tension–stretch curves demonstrate a typical nonlinear behavior with low initial stiffness followed by a transition into a region of higher stiffness. Moreover, the tension–stretch data show a pronounced (but common) scatter. On the other hand, the kinematic response in Fig. 4(b) is highly reproducible and shows a very large lateral contraction. Table 1 reports values of membrane tension and lateral contractions at λ1 ¼ 1:1, and maximum values of incremental Poisson's ratio. The corresponding standard deviations reported in the table highlights the reproducibility of contraction data, as opposed to the large variability of tension values. The maximum values of the Poisson's ratio show values up to 8.2. For completeness, values of membrane tension and longitudinal engineering strain at rupture are reported in Table 1. However, these values cannot be considered as material properties, since rupture often occurs at the clamping site.
0.8
0.6
0.4
0.2 1
1.1
1.2
1.3
1.4
Stretch λ1 [−] Fig. 4. Tension–stretch (a) and corresponding contraction–elongation curves (b) for all tested amnion samples.
Table 1 Mechanical parameters of maximum incremental Poisson's ratio νmax incr , maximum membrane tension tmax, and maximum longitudinal engineering strain ϵmax . The values membrane tension and lateral contraction denoted byn have been evaluated at λ1 ¼ 1:1. Sample
t n ðN=mmÞ
λn2 (–)
νmax incr (–)
t max ðN=mmÞ
ϵmax ð%Þ
1 2 3 4 5 6 7 8 9 10 11 12
6.3E−2 6.1E−3 6.7E−2 1.6E−2 3.2E−2 6.9E−3 6.3E−3 6.5E−3 1.4E−2 2.7E−2 1.1E−1 5.0E−2
0.66 0.68 0.60 0.63 0.68 0.66 0.67 0.63 0.69 0.65 0.57 0.55
6.1 5.4 7.9 6.4 6.4 5.7 5.6 6.8 6.4 8.2 7.3 7.7
0.60 0.17 0.48 0.84 0.51 0.34 0.32 0.15 0.31 0.46 0.37 0.47
23 28 19 28 20 28 27 22 23 20 15 22
Avg7 Std
(3.4 7 3.3)E−2
0.64 70.04
6.7 7 0.9
0.42 7 0.19
23 74
requires large contraction in order to minimize its stretch, whereas Fiber B minimizes its deformation (shortening) in case of low contraction. This opposite requirement would lead to a “balanced” contraction behavior when the resistance to lengthening (tension) and shortening (compression) would be comparable. In case fibers are unable to resist contraction, Fiber A dominates and induces a large contraction, leading to minimized lengthening of Fiber A but significant shortening of Fiber B. Fig. 7 illustrates the comparison of the Rubin–Bodner model response for two cases: fibers with and without compressive stiffness. There is only a minor difference in the tension–stretch response between the two cases but the kinematic response
W. Buerzle, E. Mazza / Journal of Biomechanics 46 (2013) 1777–1783
e2
Cauchy Tension [N/mm]
0.5 Experiment Model
0.4
B
0.3
A
0.2
e1 0.1 0
1
1.1
1.2
1.3
1.4
Stretch λ1 [−]
Fig. 6. Schematic representation of fiber's kinematics. Fiber A, on the right, is representative for fibers initially oriented in a direction close to e1 . Fiber B, on the left, represents fibers closer to the perpendicular orientation. Initial configurations are shown with dashed lines, the deformed (prescribed elongation in direction e1 ) is indicated with solid lines.
1
0.4
Experiment
zero compression stiffness
Model
finite compression stiffness
0.8
Tension [N/mm]
Stretch λ2 [−]
1781
0.6
0.4
0.2
1
1.1
1.2
1.3
1.4
0.3
0.2
0.1
0 1
Stretchλ1 [−]
1.05
1.1
1.15
1.2
Stretch λ1 [−]
Fig. 5. Optimization results for averaged amnion response. (a) Tension–stretch and (b) kinematic response.
1
Model parameter
m2 (–)
m3 (–)
m4 (–)
μ 0 ðN=mmÞ
q (–)
“Best fit” value
6.55E−4
3.15
1.45
0.447
1.36
0.9 zero compression stiffness
Stretch λ2 [−]
Table 2 Best fit model parameters of the Rubin–Bodner model, optimized for the averaged uniaxial amnion response. The model includes eight identical families of fibers in the directions of f0; 7 22:5; 7 45; 7 67:5; 90g1 in the reference configuration.
0.8
finite compression stiffness isotropic incompressible
0.7 0.6 0.5
changes completely, leading to a behavior close to the one expected for a isotropic incompressible material ðν ¼ 0:5Þ for the case with compressive stiffness. Note that the original form of the Rubin–Bodner model already includes the assumption of zero fiber stiffness under compression. The obtained descriptive capabilities are thus related to the explicit representation in this model of the mechanisms of stretch, rotation and buckling of a random fiber network.
0.4 1
1.05
1.1
1.15
1.2
Stretch λ1 [−] Fig. 7. Comparison of model predictions for (a) tension–stretch and (b) kinematic response depending on the assumption of the fiber behavior under compression. Note that this comparison is only possible if the parameter m4 is a positive integer. Corresponding model parameters are: m2 ¼ 6E−3, m3 ¼ 3.0, m4 ¼1.0, μ 0 ¼ 0:1 N=mm, and q ¼1.5.
5. Discussion 4.3. Constitutive model verification 5.1. Experiments Averaged biaxial tension–stretch data obtained from inflation experiments on amnion samples (Haller, 2012; Buerzle et al., 2012) are used to evaluate the predictive capabilities of the proposed model equations, i.e. their performance is evaluated without changing any of the five model parameters reported in Section 4.2. The biaxial tension–stretch data are normalized with respect to the same level of initial membrane tension as the uniaxial data in order to obtain a comparable reference configuration. Fig. 8 compares the model with experimental results and illustrates the remarkable quality of the prediction.
For the first time, image analysis based determination of longitudinal and lateral strains allowed reliable determination of the state of deformation of human amnion in uniaxial stress experiments. Local deformation analysis is very important in these experiments. Due to the fact that the sample is clamped at the two extremities, reliable analysis of the lateral contraction requires a large aspect ratio, in order to obtain a uniaxial state of stress in the central region of the sample. In fact, differences of up to 25% were observed between nominal and actual elongations. The major
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0.4 uniaxial experiment
Tension [N/mm]
uniaxial fit biaxial experiment
0.3
biaxial prediction
0.2
0.1
0
1
1.1
1.2
1.3
1.4
Stretch [−] Fig. 8. Comparison of uniaxial and biaxial experimental data with the corresponding model response. Biaxial model prediction has been calculated without changing any of the model parameters determined based on the averaged uniaxial data.
advantage of the deformation analysis is the availability of quantitative data on lateral contraction, which was essential in this case for constitutive model development. Reliable identification of the “zero stress state” is required for the mechanical characterization of materials. This state is difficult to attain for soft biological tissues because of their low initial stiffness, which results in an associated uncertainty in the horizontal position of the tension–stretch curves (Miller, 2001; Gao and Desai, 2010). A low force threshold (less than two orders lower than the maximum force) was introduced in the current study in order to normalize the curves. This approach allowed the comparison of uniaxial and equibiaxial tension data with respect to the same level of initial membrane tension. The observed values and the scatter of membrane tension are in line with previously reported experimental results (Oxlund et al., 1990; Helmig et al., 1993; Prevost, 2004). The observed variability in tension data is common for biological tissues, and contrasts with the high reproducibility of the kinematic response of the same samples, see Fig. 4(b) and values in Table 1. This indicates that the amount and distribution of collagens type I, III, IV, V and VI in each sample varies to a significant extent (Malak et al., 1993), whereas the mechanism of deformation of the fiber network is a characteristic of this type of tissue. The resistance of fibers to elongation leads to rotation and to buckling of the fibers initially oriented at a larger angle with respect to the direction of tension. This causes remarkable lateral contraction. This interpretation is in line with the findings of Kabla and Mahadevan (2007) for soft fibrous networks and with results obtained using random fibers network models (Picu, 2011). The maximum observed incremental Poisson's ratio of amnion is 8.2, which is, to our knowledge, larger than any previously reported values for biological or synthetic materials. There are two recent studies (Vader et al., 2009; Lake and Barocas, 2011) in which the Poisson's ratio has been determined for collagen gels, reporting values of up to 5. Other studies measuring the Poisson's ratio of anisotropic biological tissues such as human annulus fibrosus, human hip joint capsule ligaments, and sheep flexural tendon (Elliott and Setton, 2001; Hewitt et al., 2001; Lynch et al., 2003) report values in the range of 0.6–1.8, 1.4, and 3, respectively. 5.2. Constitutive modeling Key for the rationalization of amnion's behavior is the formulation of a (i) transversally isotropic model (quasi-isotropic in-plane fiber distribution) with (ii) fibers active in tension only (no stiffness for shortening, finite stiffness for lengthening). Consideration of distributed families of fibers in a homogeneous matrix is
common to several proposed models for soft biological tissue (Holzapfel et al., 2000; Sacks, 2003; Chen et al., 2011). The particular aspect highlighted in this work was not investigated in previous publications, mainly for the two following reasons: (1) little attention has been paid so far to the kinematic response of uniaxial tension tests, focusing on the stress-strain behavior; (2) when fibers are introduced often pronounced anisotropy is described, so that the contraction perpendicular to the preferred fiber direction is dominated by the matrix (typically ν≈0:5). Physically motivated selection of the model formulation made it possible to achieve excellent predictive capabilities with only five model parameters. Note that this is the same number of material constants as required to describe the behavior of a transversally isotropic material in linear elasticity. Furthermore, determination of model parameters was particularly efficient, using the split between parameters related to kinematic response and those determining the tension–stretch behavior. The determination of the parameters related to the kinematic response (m3/m2 and m4) was shown to lead to the same results for a broad range of initial values. Note, on the contrary, that the solution for the parameters related to the tension–stretch response is not unique. The current model predicts an increase of the thickness for the uniaxial experiments. This behavior cannot be verified in the experimental setup utilized for the present work. Current efforts in our laboratory are toward in-situ measurements in a multiphoton microscope in order to monitor thickness changes for different states of in-plane deformation. 5.3. Implications of large lateral contractions Our findings show that human amnion possesses an exceptionally high Poisson's ratio. This might be a key property for fulfillment of its function. In fact, large lateral contraction in a state of uniaxial stress (such as at hole- or crack-like defects in a membrane) causes stress redistribution and local stiffening, thus providing higher toughness and making amnion a defect-tolerant membrane. This interpretation is in line with the findings of Koh and Oyen (2012) who have observed in numerical simulations of fibrous networks corresponding fiber reorientations at the crack tip and crack blunting, this representing an important toughening mechanism. In addition, differences in contraction behavior of amnion with respect to chorion and maternal deciduas (the layer interfacing to the uterine wall) facilitate mechanisms of local relative sliding of the layers. This relative movement is considered a prominent mechanism to avoid amniotic fluid leakage in case of localized FM defects (Gratacós et al., 2006).
Conflict of interest statement None of the authors has any conflict of interest.
Acknowledgments The authors gratefully acknowledge financial support by the Swiss National Science Foundation project no. 3200B-124925/1. We thank the team of Prof. Zimmermann at the University Hospital Zurich for their invaluable support and A.G. Bittermann at the Center for Microscopy and Image Analysis, University of Zurich. References Al-Zaid, N.S., Bou-Resli, M.N., Ibrahim, M.E., 1980. Study of the connective tissue of human fetal membranes. Journal of Reproduction & Fertility 59, 383–386.
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Alderson, K.L., Alderson, A., Evans, K.E., 1997. The interpretation of the straindependent Poisson's ratio in auxetic polyethylene. Journal of Strain Analysis for Engineering 32, 201–212. Arikat, S., Novince, R.W., Mercer, B.M., Kumar, D., Fox, J.M., Mansour, J.M., Moore, J. J., 2006. Separation of amnion from choriodecidua is an integral event to the rupture of normal term fetal membranes and constitutes a significant component of the work required. American Journal of Obstetrics and Gynecology 194, 211–217. Artal, R., Sokol, R.J., Neuman, M., Burstein, A.H., Stojkov, J., 1976. The mechanical properties of prematurely and non-prematurely ruptured membranes. American Journal of Obstetrics and Gynecology 125, 655–659. Beck, V., Lewi, P., Gucciardo, L., Devlieger, R., 2011. Preterm prelabor rupture of membranes and fetal survival after minimally invasive fetal surgery: a systematic review of the literature. Fetal Diagnosis and Therapy 31, 1–9. Buerzle, W., Haller, C., Jabareen, M., Egger, J., Mallik, A., Ochsenbein-Koelble, N., Ehrbar, M., Mazza, E., 2012. Multiaxial mechanical behavior of human fetal membranes and its relationship to microstructure. Biomechanics and Modeling in Mechanobiology, Epub ahead of print. Calvin, S., Oyen, M., 2007. Microstructure and mechanics of the chorioamnion membrane with an emphasis on fracture properties. Annals of the New York Academy of Sciences, Blackwell Publishing, England. pp. 166–185. Chen, H., Liu, Y., Zhao, X., Lanir, Y., Kassab, G.S., 2011. A micromechanics finite-strain constitutive model of fibrous tissue. Journal of the Mechanics and Physics of Solids 59, 1823–1837. Elliott, D.M., Setton, L.A., 2001. Anisotropic and inhomogeneous tensile behavior of the human anulus fibrosus: experimental measurement and material model predictions. Journal of Biomechanical Engineering-T ASME 123, 256. Flynn, C., Rubin, M.B., 2012. An anisotropic discrete fibre model based on a generalised strain invariant with application to soft biological tissues. International Journal of Engineering Science 60, 66–76. Flynn, C., Rubin, M.B., Nielsen, P., 2010. A model for the anisotropic response of fibrous soft tissues using six discrete fibre bundles. International Journal for Numerical Methods in Biomedical Engineering 27, 1793–1811. Gao, Z., Desai, J.P., 2010. Estimating zero-strain states of very soft tissue under gravity loading using digital image correlation. Medical Image Analysis 14, 126–137. Gratacós, E., Sanin-Blair, J., Lewi, L., Toran, N., Verbist, G., Cabero, L., Deprest, J., 2006. A histological study of fetoscopic membrane defects to document membrane healing. Placenta 27, 452–456. Halaburt, J.T., Uldbjerg, N., Helmig, R., Ohlsson, K., 1989. The concentration of collagen and the collagenolytic activity in the amnion and the chorion. European Journal of Obstetrics & Gynecology and Reproductive Biology 31, 75–82. Haller, C., 2012. Evaluation of Human Fetal Membrane Sealing. Ph.D. Thesis. University Hospital Zurich. Helmig, R., Oxlund, H., Petersen, U., 1993. Different biomechanical properties of human fetal membranes obtained before and after delivery. European Journal of Obstetrics & Gynecology and Reproductive Biology 48, 183–189. Hewitt, J., Guilak, F., Glisson, R., Vail, T.P., 2001. Regional material properties of the human hip joint capsule ligaments. Journal of Orthopaedic Research 19, 359–364. Hollenstein, M., Ehret, A.E., Itskov, M., Mazza, E., 2010. A novel experimental procedure based on pure shear testing of dermatome-cut samples applied to porcine skin. Biomechanics and Modeling in Mechanobiology 10, 651–661. Holzapfel, G.A., Gasser, T.C., Ogden, R.A.Y.W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity 61, 1–48. Ilancheran, S., Moodley, Y., Manuelpillai, U., 2009. Human fetal membranes: a source of stem cells for tissue regeneration and repair?. Placenta 30, 2–10. Jabareen, M., Mallik, A.S., Bilic, G., Zisch, A.H., Mazza, E., 2009. Relation between mechanical properties and microstructure of human fetal membranes: an attempt towards a quantitative analysis. European Journal of Obstetrics & Gynecology and Reproductive Biology 144 (Suppl), S134–S141.
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Joyce, E., 2009. Micromechanical Mechanisms of Fetal Membrane Failure. Ph.D. Thesis. University of Pittsburgh. Kabla, A., Mahadevan, L., 2007. Nonlinear mechanics of soft fibrous networks. Journal of the Royal Society Interface 4, 99–106. Koh, C.T., Oyen, M.L., 2012. Branching toughens fibrous networks. Journal of the Mechanical Behavior of Biomedical 12, 74–82. Lake, S.P., Barocas, V.H., 2011. Mechanical and structural contribution of nonfibrillar matrix in uniaxial tension: a collagen–agarose co-gel model. Annals of Biomedical Engineering 39, 1891–1903. Lavery, J., Miller, C., 1979. Deformation and creep in the human chorioamniotic sac. American Journal of Obstetrics and Gynecology 134, 366–375. Lynch, H.A., Johannessen, W., Wu, J.P., Jawa, A., Elliott, D.M., 2003. Effect of fiber orientation and strain rate on the nonlinear uniaxial tensile material properties of tendon. Journal of Biomechanical Engineering-T ASME 125, 726. Malak, T.M., Ockleford, C.D., Bell, S.C., Dalgleish, R., Bright, N., Macvicar, J., 1993. Confocal immunofluorescence localization of collagen types i, iii, iv, v and vi and their ultrastructural organization in term human fetal membranes. Placenta 14, 385–406. Mercer, B., 2003. Preterm premature rupture of the membranes. Obstetrics & Gynecology 101, 178–193. Miller, C., Lavery, J., Donnelly, T., 1979. Determination of elastic parameters for human fetal membranes. Journal of Rheology 23, 57–78. Miller, K., 2001. How to test very soft biological tissues in extension? Journal of Biomechanics 34, 651–657. Oxlund, H., Helmig, R., Halaburt, J.T., Uldbjerg, N., 1990. Biomechanical analysis of human chorioamniotic membranes. European Journal of Obstetrics & Gynecology and Reproductive Biology 34, 247–255. Oyen, M., Calvin, S., Landers, D., 2006. Premature rupture of the fetal membranes: is the amnion the major determinant?. American Journal of Obstetrics and Gynecology 195, 510–515. Oyen, M., Cook, R., Stylianopoulos, T., Barocas, V., Calvin, S., Landers, D., 2005. Uniaxial and biaxial mechanical behavior of human amnion. Journal of Materials Research 20, 2902–2909. Oyen, M.L., Calvin, S.E., Cook, R.F., 2004. Uniaxial stress–relaxation and stress–strain responses of human amnion. Journal of Materials Science Materials in Medicine 15, 619–624. Parry, S., Strauss, J., 1998. Premature rupture of the fetal membranes. New England Journal of Medicine 338 (10), 663–670. Picu, R.C., 2011. Mechanics of random fiber networks—a review. Soft Matter 7, 6768. Polishuk, W., Peranio, A., Kohane, S., 1962. The physical properties of fetal membranes. Obstetrics & Gynecology 20, 204–210. Prevost, T.P., 2004. Biomechanics of the Human Chorioamnion. Master Thesis. Department of Materials Science and Engineering, Massachusetts Institute of Technology. Rubin, M.M., Bodner, S.R., 2002. A three-dimensional nonlinear model for dissipative response of soft tissue. International Journal of Solids and Structures 39, 5081–5099. Sacks, M.S., 2003. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. Journal of Biomechanical Engineering-T ASME 125, 280. Smith, C.W., Wootton, R.J., Evans, K.E., 1999. Interpretation of experimental data for Poisson's ratio of highly nonlinear materials. Experimental Mechanics 39, 356–362. Stylianopoulos, T., Barocas, V., 2007. Volume-averaging theory for the study of the mechanics of collagen networks. Computer Methods in Applied Mechanics 196, 2981–2990. Sydorak, R.M., Hirose, S., Sandberg, P.L., Filly, R.A., Harrison, M.R., Farmer, D.L., Albanese, C.T., 2002. Chorioamniotic membrane separation following fetal surgery. Journal of Perinatology 22, 407–410. Vader, D., Kabla, A., Weitz, D., Mahadevan, L., 2009. Strain-induced alignment in collagen gels. PLoS ONE 4, 12.
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
On the deformation behavior of human amnion Wilfried Buerzle a,n, Edoardo Mazza a,b a b
Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland Swiss Federal Laboratories for Materials Science and Technology, EMPA Dubendorf, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Accepted 20 May 2013
Recently renewed interest for the mechanical behavior of fetal membranes is related to the problem of iatrogenic preterm rupture, limiting the effectiveness and applicability of minimally invasive fetal surgery. This study aimed at characterizing and modeling the deformation behavior of the amnion layer, the highly deformable and tough membrane that surrounds the amniotic fluid and the growing fetus in the uterine cavity. Uniaxial tension tests have been performed on samples obtained immediately after cesarean section, and the deformation field has been analyzed by digital image correlation. The results show that the kinematic response of human amnion is highly reproducible and that the incremental Poisson's ratio is, with a value of up to 8, higher than any previously reported value for biological or synthetic materials. This unique behavior is related to the characteristic architecture of amnion's microstructure and can be rationalized by mechanisms of rotation, stretching and buckling of collagen fibers. Simple constitutive equations have been selected based on this interpretation, which lead to a model with excellent predictive capabilities for the uniaxial and equibiaxial mechanical response of human amnion. Relevant insights were gained on the role of collagen fibers in determining the deformability and toughness of soft biological tissue. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Fetal membrane Poisson's ratio Kinematics Network Constitutive model
1. Introduction The fetal membrane (FM) surrounds and protects the fetus during pregnancy and is adjacent to the uterine wall. The FM is composed of two main layers, called amnion and chorion. During pregnancy, it has to resist deformations due to internal pressure and fetal movements. Rupture of FM is a natural event for term delivery but has serious complications when it happens prior to term. The so called spontaneous preterm premature rupture of FM (PPROM) affects 3% of all pregnancies worldwide (Calvin and Oyen, 2007; Mercer, 2003) and is associated with a higher risk of perinatal morbidity and mortality. Moreover, preterm rupture after fetal surgery, called iatrogenic PPROM (iPPROM), is the Achilles heel for developments in this field and occurs in about 30% of all treatments (Beck et al., 2011). The development of methods to avoid FM preterm rupture requires a better understanding of their mechanical behavior. Only few attempts were reported for the determination of a constitutive model of FM (Miller et al., 1979; Prevost, 2004; Jabareen et al., 2009; Joyce, 2009). Most previous studies focused on the extraction of parameters to quantitatively characterize FM rupture and on the effects leading to PPROM (Artal et al., 1976; Lavery and Miller, 1979; Al-Zaid et al., 1980). Other investigations were aimed at determining differences in mechanical properties of amnion and n Corresponding author at: Swiss Federal Institute of Technology, Center of Mechanics, Tannenstrasse 3, 8092 Zurich, Switzerland. Tel.: +41 44 632 77 91; fax: +41 44 632 11 45. E-mail address: [email protected] (W. Buerzle).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.05.018
chorion (Polishuk et al., 1962; Oxlund et al., 1990; Helmig et al., 1993). It is known from these foregoing studies that amnion is stiffer and stronger than chorion and is therefore considered the mechanically dominant layer of FM. Oyen has investigated the time and history dependence of amnion's deformation behavior (Oyen et al., 2004, 2005) and has shown that amnion is more sensitive to chemical and mechanical changes that occur during gestation (Oyen et al., 2006). Amnion is the inner layer of the FM, facing the amniotic fluid. Membranes at term have an amnion layer with a thickness in the range of 100 μm (Ilancheran et al., 2009). Next to the amniotic epithelium and the basement membrane, the so called “compact layer” is made predominately of collagens type I and III that form a dense network (Malak et al., 1993; Parry and Strauss, 1998). The adjacent fibroblast layer is the thickest layer of amnion and is characterized by collagens forming a looser network. The outermost layer of amnion is the “intermediate layer”, which contains a nonfibrillar network of mostly type III collagen (Parry and Strauss, 1998), it lies between amnion and chorion and accommodates relative movement between the layers. Fig. 1 shows scanning electron micrograph (SEM) images of a FM sample. It illustrates the crimped collagen fibres, forming a random network in the membrane plane, with no distinct directionality. Chorioamniotic membrane separation leads to a premature rupture of the membranes in 63% of the cases after surgery (Sydorak et al., 2002). The separation of the FM layers has been observed in puncture experiments (Arikat et al., 2006). We also consistently observed a separation of amnion and chorion in uniaxial tension tests on intact FM, see Fig. 2. In this case
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Chorion
Amnion
Fig. 2. Image of an amnion–chorion FM sample during tension test. The constitutive layers of the FM start to separate at a certain level of deformation. The dotted line indicates the border of the amnion layer after separation. protocol approved by the ethical committee of the district of Zurich (study Stv22/ 2006). The patients were randomly selected after thorough serologic testing to exclude HIV, hepatitis B, and streptococcus B. The selected patients had no history of diabetes, connective tissue disorders, or chromosomal abnormalities. After cesarean section the FM was cut approximately 2 cm away from the placental disk and stored in saline solution until mechanical testing, within a few hours after delivery. Amnion and chorion were separated manually by gently pulling the layers apart. Amnion was spread on a lubricated plastic mat and a die cutter (see Hollenstein et al., 2010) was used to create rectangular samples with the dimension of 80 15 mm. Small markers of India Ink were applied on the surface of the sample with the aid of a brush, which allows better optical tracking of the deformation field.
2.2. Mechanical experiments
Fig. 1. Scanning electron micrograph of a human fetal membrane sample. Crosssectional view (a) and top view (b) show an irregular network of singe collagen fibers with no distinct directionality. With permission from Hollenstein et al. (2010).
In order to test the FM samples in a uniaxial stress configuration, a custom built experimental setup (Hollenstein et al., 2010) was used. This consists mainly of computer controlled hydraulic actuators equipped with force sensors (20 N capacity) and a CCD camera system which allows to record image series during the experiment. The tensile tests were performed as monotonic tension to failure with an elongation rate of 0.5% nominal strain per second, starting from an initial free length of 60 mm. No preconditioning was applied. All tests were performed at room temperature in a bath filled with saline solution to avoid drying or swelling of the sample. Data on the response of amnion in a biaxial state of stress were obtained using a custom built inflation device (Haller, 2012). The inflation setup, the experimental protocol, and the procedures for extraction of biaxial tension–stretch curves are described in Buerzle et al. (2012).
2.3. Analysis of the deformation field in uniaxial experiments
separation is caused by different contraction behavior (Poisson's effect) of the two layers, with amnion displaying a much greater lateral contraction. Unusually large contraction of amnion in tensile tests had never been reported. This motivated a series of tensile tests on human amnion with quantitative determination of lateral contraction. Experimental observations along with considerations on mechanisms of rotation, stretching and buckling of fibers in a random network (Stylianopoulos and Barocas, 2007; Kabla and Mahadevan, 2007; Picu, 2011) led to the selection of a constitutive model formulation. Model parameters were determined to simultaneously match tension–stretch and contraction–elongation curves of tensile tests. Model response to equibiaxial tension was finally compared with corresponding data from previous measurements (Haller, 2012). 2. Experimental methods 2.1. Sample harvesting and preparation Fetal membranes were collected from patients with single child pregnancies who underwent elective cesarean section between 37 and 39 weeks of gestation. Patients were recruited for this study with informed written consent according to a
The recorded image series serves as basis for the determination of the kinematic response of the amnion layer. The digital image correlation (DIC) software VEDDAC 4.0 (Chemnitzer Werkstoffmechanik GmbH, Chemnitz, Germany) is used to trace a set of user defined measurement points corresponding to the markers in the central region of the sample, see Fig. 3. The resulting discrete displacement filed is used for the approximation of the in-plane stretches by the use of element based interpolation functions, as applied in the finite element (FE) method. The displacements of the measurement points are considered as nodal displacements of a linear four node element. This approach allows calculating the in-plane components of the element specific deformation gradient Fel as a function of the associated nodal displacements u^ and the linear shape functions N 4
ðaÞ Fel ¼ I þ ∑ u^ ⊗ a¼1
∂N ðaÞ : ∂X
ð1Þ
All common measures of deformation can be calculated based on the deformation gradient. For the load cases considered in this work, only the two in-plane principal stretches λ1 and λ2 are determined. Averaging over the entire region, in this case a 2 3 element scheme, provides the kinematic response of the amnion layer. The typical variability of elongation values in different elements was in the range of 5%. The Poisson's ratio is defined as the ratio of lateral strain (contraction, in direction 2) and longitudinal strain (elongation, in direction 1). It is strictly defined only for infinitesimal deformations. For large strain behavior, an incremental Poisson's ratio or Poisson's function, according to the definitions of Alderson et al. (1997) and Smith et al. (1999) can be used, which is based on the ratio of
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μ 0 ¼ μ0 h0
ð4Þ
1.4
g ¼ g 2 ðβ1 Þ þ g 3 ðλI Þ
ð5Þ
1.35
g 2 ðβ1 Þ ¼ m2 ðβ1 −3Þ
ð6Þ
1.3
g 3 ðλI Þ ¼ ∑
X Position [pixel] 0
0
100
200
300
400
500
600
700
800
900
1000
100
300
500 600
I
1.25 1.2
Stretch λ1
Y Position [pixel]
200
400
1.15
700 1.1 800 900 1000
1.05 1
Fig. 3. Illustration of the image analysis procedure. Image shows the transparent amnion layer with markers of India Ink which are used to trace a set of user defined measurement points (red squares) by DIC method. A finite element scheme is used for the determination of the principal stretches. The camera field of view is 30 30 mm and the typical element size is 5 mm (x-direction) and 2 mm (y-direction). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) derivatives of logarithmic strains in the lateral and axial directions. νincr 12 ¼ −
λ1 ∂λ2 λ2 ∂λ1
ð2Þ
This incremental Poisson's ratio might be regarded as akin to the definition of a tangent modulus for stiffness.
ð7Þ
β1 ¼ trðBÞ is the first invariant of the right Cauchy–Green deformation tensor B ¼ FFT , and λI ¼ jmI j is the stretch of the I-th family of fibers, with mI ¼ FMI . The directions of the fibers in the reference configuration are defined by the unit vector MI . Very important for the present application, the fibers are considered not to bear compression loads, as indicated by the use of the Macaulay brackets 〈x〉 ¼ 1=2ðx þ jxjÞ. m2, m3, m4, μ 0 , and q are model parameters which have to be determined from an inverse analysis of the experimental data. Due to the high water content (up to 90% Halaburt et al., 1989), volume preservation (J ¼1) was assumed in this case. Note that at present this assumption cannot be verified experimentally, since the thickness or the water content of the test pieces cannot be monitored during the tests. 3.2. Model parameters determination The Cauchy stress tensor r is obtained from derivatives of the strain energy function with respect to the corresponding deformation tensor. The membrane is in the 1–2 plane and the deformation gradient is (for all cases considered here) of the form 2 3 λ1 0 0 6 7 F ¼ 4 0 λ2 0 5 ð8Þ 0 0 λ3
λ3 ¼
3.1. Model formulation Fig. 1 shows the collagen microstructure of amnion. Thin, randomly oriented collagen fibers characterize this tissue and are expected to determine the mechanical response of amnion. A corresponding phenomenological model has to include terms representative of the mechanisms of stretching, rotation and buckling of the fibers. In this work the constitutive model formulation developed by Rubin and Bodner (2002) was chosen to model the response of the amnion layer. The model allows considering (families of) fibers embedded in an isotropic matrix. It was shown in previous investigations to provide useful representation of the uniaxial (Jabareen et al., 2009) stress response of intact FM but underestimated the biaxial stress response (Buerzle et al., 2012), if formulated without fibers. The Rubin–Bodner model includes terms to also describe dissipative deformation behavior. Our experimental data do not provide information on time and history dependence of amnion's response. Thus, only the elastic part of the equations was considered in this work. The corresponding strain energy function, ψ M , is a composition of different contributions, describing the pure distortional and dilatational parts and the stretch of the single families of fibers. The original equations have been adapted for a membrane model formulation, i.e. ψ M ¼ ψh0 is the strain energy per unit reference area of the membrane midplane surface with h0 being the membrane thickness in the reference configuration and ψ the strain energy per unit reference volume (note that the original Rubin–Bodner model is formulated in terms of energy per unit mass). μ 0 qg ½e −1 2q
m3 〈λI −1〉2m4 m4
with
3. Constitutive model
ψM ¼
1779
ð3Þ
1 : λ1 λ2
ð9Þ
Following the definition of a membrane formulation according to Section 3.1, the membrane tension tensor t (in the current configuration) reads h0 m2 μ 1 t¼r ¼ B− ðB IÞI λ1 λ2 3 λ1 λ2 m3 μ 1 ph 〈λI −1〉2m4 −1 ðmI ⊗mI Þ− 0 I λ λ λ λ I 1 2 1 λ2 I¼1 8
þ ∑
ð10Þ
μ ¼ μ 0 eqg
ð11Þ
The membrane tension tensor t must fulfill two boundary conditions in a state of uniaxial tension. First, the stress vector in thickness direction must vanish, which can be achieved by the corresponding determination of the hydrostatic pressure p, which is a general function of position and time due to the incompressibility constraint. Second, also the stress vector in lateral (inplane) direction must vanish. This condition for the 22 component of t t 22 ¼ 0 ¼
m3 8 1 1 ∑ 〈λI −1〉2m4 −1 ðmI ⊗mI Þ22 þ λ22 − 2 2 λI m2 I ¼ 1 λ1 λ2
ð12Þ
determines simultaneously the kinematic response of the constitutive model, since it provides the stretch λ2 , as a function of λ1 and λI . The kinematic response in the uniaxial stress state depends only on two model parameters, i.e. the ratio m3/m2 and the parameter m4. The other model parameters do not affect the kinematic response. This circumstance can be used to split up the inverse analysis in two steps. First, only the parameter-ratio m3/m2 and the parameter m4 is determined in order to match the experimental kinematic response (contraction–elongation curve).
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Next, the remaining parameters q, μ 0 , and m2 (or m3) are determined to match the tension–stretch response, while keeping m3/m2 and m4 unchanged. The best fit model parameters are determined by the use of the Nelder–Mead simplex method, fminsearch, in MATLAB 2010a (The MathWorks, Inc., Natick, MA, USA).
Cauchy Tension [N/mm]
1
4. Results 4.1. Uniaxial tests
4.2. Constitutive model parameters The model response should be (i) nearly isotropic in the membrane plane. In fact, no preferential directions have been reported in any previous study. It should display the typical (ii) j-shaped tension–stretch curve, and, in particular, reproduce the observed (iii) very large lateral contraction. Simultaneous fulfillment of all these conditions is a challenging task. The Rubin–Bodner model (Rubin and Bodner, 2002) has been implemented considering eight identical families of fibers in the initial directions of f0; 722:5; 7 45; 767:5; 90g1, thus providing a quasi-isotropic behavior in the membrane plane. Recent work on a discrete fiber model based on fibers oriented in the directions of opposing vertices of a regular icosahedron shows that this model is not exactly isotropic (Flynn et al., 2010; Flynn and Rubin, 2012). Fig. 5 shows the results of the two step optimization procedure (explained in Section 3.2) applied to the averaged uniaxial measurements. The model very well matches both, the tension–stretch response as well as the contraction–elongation curves. Since comparison between model and measurements is performed for the membrane tension (and not for stress), there is no need to determine a representative initial thickness of the “average” amnion sample. The “best fit” model parameters for the averaged amnion response are reported in Table 2. The key for representing the observed kinematic response is the fact that the fibers do not resist compressions (no stress for λI o 1). Fig. 6 illustrates the expected rotation of two fibers for a prescribed elongation in direction e1 . Fiber A has an initial orientation (dashed line) with a small angle with respect to e1 , whereas Fiber B has an initial large angle deviation from e1 . Deformation leads to fibers stretching and rotation. Fiber A
0.6 0.4 0.2 0 1
1.1
1.2
1.3
1.4
Stretch λ1 [−] 1
0.8
Stretch λ2 [−]
Uniaxial tension tests have been performed on 12 samples from six different membranes. The clamped amnion sample is initially slack. A low force threshold (5 mN in this case) was applied to define the initial configuration. Fig. 4(a) shows the tension–stretch curves for each single amnion sample and Fig. 4(b) illustrates the corresponding in-plane kinematic responses, i.e. λ2 versus λ1 . It can be seen in Fig. 4(a) that the tension–stretch curves demonstrate a typical nonlinear behavior with low initial stiffness followed by a transition into a region of higher stiffness. Moreover, the tension–stretch data show a pronounced (but common) scatter. On the other hand, the kinematic response in Fig. 4(b) is highly reproducible and shows a very large lateral contraction. Table 1 reports values of membrane tension and lateral contractions at λ1 ¼ 1:1, and maximum values of incremental Poisson's ratio. The corresponding standard deviations reported in the table highlights the reproducibility of contraction data, as opposed to the large variability of tension values. The maximum values of the Poisson's ratio show values up to 8.2. For completeness, values of membrane tension and longitudinal engineering strain at rupture are reported in Table 1. However, these values cannot be considered as material properties, since rupture often occurs at the clamping site.
0.8
0.6
0.4
0.2 1
1.1
1.2
1.3
1.4
Stretch λ1 [−] Fig. 4. Tension–stretch (a) and corresponding contraction–elongation curves (b) for all tested amnion samples.
Table 1 Mechanical parameters of maximum incremental Poisson's ratio νmax incr , maximum membrane tension tmax, and maximum longitudinal engineering strain ϵmax . The values membrane tension and lateral contraction denoted byn have been evaluated at λ1 ¼ 1:1. Sample
t n ðN=mmÞ
λn2 (–)
νmax incr (–)
t max ðN=mmÞ
ϵmax ð%Þ
1 2 3 4 5 6 7 8 9 10 11 12
6.3E−2 6.1E−3 6.7E−2 1.6E−2 3.2E−2 6.9E−3 6.3E−3 6.5E−3 1.4E−2 2.7E−2 1.1E−1 5.0E−2
0.66 0.68 0.60 0.63 0.68 0.66 0.67 0.63 0.69 0.65 0.57 0.55
6.1 5.4 7.9 6.4 6.4 5.7 5.6 6.8 6.4 8.2 7.3 7.7
0.60 0.17 0.48 0.84 0.51 0.34 0.32 0.15 0.31 0.46 0.37 0.47
23 28 19 28 20 28 27 22 23 20 15 22
Avg7 Std
(3.4 7 3.3)E−2
0.64 70.04
6.7 7 0.9
0.42 7 0.19
23 74
requires large contraction in order to minimize its stretch, whereas Fiber B minimizes its deformation (shortening) in case of low contraction. This opposite requirement would lead to a “balanced” contraction behavior when the resistance to lengthening (tension) and shortening (compression) would be comparable. In case fibers are unable to resist contraction, Fiber A dominates and induces a large contraction, leading to minimized lengthening of Fiber A but significant shortening of Fiber B. Fig. 7 illustrates the comparison of the Rubin–Bodner model response for two cases: fibers with and without compressive stiffness. There is only a minor difference in the tension–stretch response between the two cases but the kinematic response
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e2
Cauchy Tension [N/mm]
0.5 Experiment Model
0.4
B
0.3
A
0.2
e1 0.1 0
1
1.1
1.2
1.3
1.4
Stretch λ1 [−]
Fig. 6. Schematic representation of fiber's kinematics. Fiber A, on the right, is representative for fibers initially oriented in a direction close to e1 . Fiber B, on the left, represents fibers closer to the perpendicular orientation. Initial configurations are shown with dashed lines, the deformed (prescribed elongation in direction e1 ) is indicated with solid lines.
1
0.4
Experiment
zero compression stiffness
Model
finite compression stiffness
0.8
Tension [N/mm]
Stretch λ2 [−]
1781
0.6
0.4
0.2
1
1.1
1.2
1.3
1.4
0.3
0.2
0.1
0 1
Stretchλ1 [−]
1.05
1.1
1.15
1.2
Stretch λ1 [−]
Fig. 5. Optimization results for averaged amnion response. (a) Tension–stretch and (b) kinematic response.
1
Model parameter
m2 (–)
m3 (–)
m4 (–)
μ 0 ðN=mmÞ
q (–)
“Best fit” value
6.55E−4
3.15
1.45
0.447
1.36
0.9 zero compression stiffness
Stretch λ2 [−]
Table 2 Best fit model parameters of the Rubin–Bodner model, optimized for the averaged uniaxial amnion response. The model includes eight identical families of fibers in the directions of f0; 7 22:5; 7 45; 7 67:5; 90g1 in the reference configuration.
0.8
finite compression stiffness isotropic incompressible
0.7 0.6 0.5
changes completely, leading to a behavior close to the one expected for a isotropic incompressible material ðν ¼ 0:5Þ for the case with compressive stiffness. Note that the original form of the Rubin–Bodner model already includes the assumption of zero fiber stiffness under compression. The obtained descriptive capabilities are thus related to the explicit representation in this model of the mechanisms of stretch, rotation and buckling of a random fiber network.
0.4 1
1.05
1.1
1.15
1.2
Stretch λ1 [−] Fig. 7. Comparison of model predictions for (a) tension–stretch and (b) kinematic response depending on the assumption of the fiber behavior under compression. Note that this comparison is only possible if the parameter m4 is a positive integer. Corresponding model parameters are: m2 ¼ 6E−3, m3 ¼ 3.0, m4 ¼1.0, μ 0 ¼ 0:1 N=mm, and q ¼1.5.
5. Discussion 4.3. Constitutive model verification 5.1. Experiments Averaged biaxial tension–stretch data obtained from inflation experiments on amnion samples (Haller, 2012; Buerzle et al., 2012) are used to evaluate the predictive capabilities of the proposed model equations, i.e. their performance is evaluated without changing any of the five model parameters reported in Section 4.2. The biaxial tension–stretch data are normalized with respect to the same level of initial membrane tension as the uniaxial data in order to obtain a comparable reference configuration. Fig. 8 compares the model with experimental results and illustrates the remarkable quality of the prediction.
For the first time, image analysis based determination of longitudinal and lateral strains allowed reliable determination of the state of deformation of human amnion in uniaxial stress experiments. Local deformation analysis is very important in these experiments. Due to the fact that the sample is clamped at the two extremities, reliable analysis of the lateral contraction requires a large aspect ratio, in order to obtain a uniaxial state of stress in the central region of the sample. In fact, differences of up to 25% were observed between nominal and actual elongations. The major
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0.4 uniaxial experiment
Tension [N/mm]
uniaxial fit biaxial experiment
0.3
biaxial prediction
0.2
0.1
0
1
1.1
1.2
1.3
1.4
Stretch [−] Fig. 8. Comparison of uniaxial and biaxial experimental data with the corresponding model response. Biaxial model prediction has been calculated without changing any of the model parameters determined based on the averaged uniaxial data.
advantage of the deformation analysis is the availability of quantitative data on lateral contraction, which was essential in this case for constitutive model development. Reliable identification of the “zero stress state” is required for the mechanical characterization of materials. This state is difficult to attain for soft biological tissues because of their low initial stiffness, which results in an associated uncertainty in the horizontal position of the tension–stretch curves (Miller, 2001; Gao and Desai, 2010). A low force threshold (less than two orders lower than the maximum force) was introduced in the current study in order to normalize the curves. This approach allowed the comparison of uniaxial and equibiaxial tension data with respect to the same level of initial membrane tension. The observed values and the scatter of membrane tension are in line with previously reported experimental results (Oxlund et al., 1990; Helmig et al., 1993; Prevost, 2004). The observed variability in tension data is common for biological tissues, and contrasts with the high reproducibility of the kinematic response of the same samples, see Fig. 4(b) and values in Table 1. This indicates that the amount and distribution of collagens type I, III, IV, V and VI in each sample varies to a significant extent (Malak et al., 1993), whereas the mechanism of deformation of the fiber network is a characteristic of this type of tissue. The resistance of fibers to elongation leads to rotation and to buckling of the fibers initially oriented at a larger angle with respect to the direction of tension. This causes remarkable lateral contraction. This interpretation is in line with the findings of Kabla and Mahadevan (2007) for soft fibrous networks and with results obtained using random fibers network models (Picu, 2011). The maximum observed incremental Poisson's ratio of amnion is 8.2, which is, to our knowledge, larger than any previously reported values for biological or synthetic materials. There are two recent studies (Vader et al., 2009; Lake and Barocas, 2011) in which the Poisson's ratio has been determined for collagen gels, reporting values of up to 5. Other studies measuring the Poisson's ratio of anisotropic biological tissues such as human annulus fibrosus, human hip joint capsule ligaments, and sheep flexural tendon (Elliott and Setton, 2001; Hewitt et al., 2001; Lynch et al., 2003) report values in the range of 0.6–1.8, 1.4, and 3, respectively. 5.2. Constitutive modeling Key for the rationalization of amnion's behavior is the formulation of a (i) transversally isotropic model (quasi-isotropic in-plane fiber distribution) with (ii) fibers active in tension only (no stiffness for shortening, finite stiffness for lengthening). Consideration of distributed families of fibers in a homogeneous matrix is
common to several proposed models for soft biological tissue (Holzapfel et al., 2000; Sacks, 2003; Chen et al., 2011). The particular aspect highlighted in this work was not investigated in previous publications, mainly for the two following reasons: (1) little attention has been paid so far to the kinematic response of uniaxial tension tests, focusing on the stress-strain behavior; (2) when fibers are introduced often pronounced anisotropy is described, so that the contraction perpendicular to the preferred fiber direction is dominated by the matrix (typically ν≈0:5). Physically motivated selection of the model formulation made it possible to achieve excellent predictive capabilities with only five model parameters. Note that this is the same number of material constants as required to describe the behavior of a transversally isotropic material in linear elasticity. Furthermore, determination of model parameters was particularly efficient, using the split between parameters related to kinematic response and those determining the tension–stretch behavior. The determination of the parameters related to the kinematic response (m3/m2 and m4) was shown to lead to the same results for a broad range of initial values. Note, on the contrary, that the solution for the parameters related to the tension–stretch response is not unique. The current model predicts an increase of the thickness for the uniaxial experiments. This behavior cannot be verified in the experimental setup utilized for the present work. Current efforts in our laboratory are toward in-situ measurements in a multiphoton microscope in order to monitor thickness changes for different states of in-plane deformation. 5.3. Implications of large lateral contractions Our findings show that human amnion possesses an exceptionally high Poisson's ratio. This might be a key property for fulfillment of its function. In fact, large lateral contraction in a state of uniaxial stress (such as at hole- or crack-like defects in a membrane) causes stress redistribution and local stiffening, thus providing higher toughness and making amnion a defect-tolerant membrane. This interpretation is in line with the findings of Koh and Oyen (2012) who have observed in numerical simulations of fibrous networks corresponding fiber reorientations at the crack tip and crack blunting, this representing an important toughening mechanism. In addition, differences in contraction behavior of amnion with respect to chorion and maternal deciduas (the layer interfacing to the uterine wall) facilitate mechanisms of local relative sliding of the layers. This relative movement is considered a prominent mechanism to avoid amniotic fluid leakage in case of localized FM defects (Gratacós et al., 2006).
Conflict of interest statement None of the authors has any conflict of interest.
Acknowledgments The authors gratefully acknowledge financial support by the Swiss National Science Foundation project no. 3200B-124925/1. We thank the team of Prof. Zimmermann at the University Hospital Zurich for their invaluable support and A.G. Bittermann at the Center for Microscopy and Image Analysis, University of Zurich. References Al-Zaid, N.S., Bou-Resli, M.N., Ibrahim, M.E., 1980. Study of the connective tissue of human fetal membranes. Journal of Reproduction & Fertility 59, 383–386.
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