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The biomechanics of arterial elastin
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Research paper
The biomechanics of arterial elastin Namrata Gundiah a , Mark B. Ratcliffe c,d , Lisa A. Pruitt a,b,∗ a Department of Mechanical Engineering, University of California, Berkeley, CA, USA b Department of Bioengineering, University of California, Berkeley, CA, USA c Department of Surgery, University of California, San Francisco, USA d San Francisco Veterans Affairs Medical Center, San Francisco, CA, USA
A R T I C L E
I N F O
A B S T R A C T
Article history:
Uniaxial mechanical experiments have shown that a neo-Hookean/Gaussian model is
Received 27 June 2008
suitable to describe the mechanics of arterial elastin networks [Gundiah, N., Ratcliffe, M.B.,
Received in revised form
Pruitt, L.A., 2007. Determination of strain energy function for arterial elastin: Experiments
14 October 2008
using histology and mechanical tests. J. Biomech. 40, 586–594]. Based on the three-
Accepted 24 October 2008
dimensional elastin architecture in arteries, we have proposed an orthotropic material
Published online 6 November 2008
symmetry for arterial elastin consisting of two orthogonally oriented and symmetrically placed families of mechanically equivalent fibers. In this study, we use these results to describe the strain energy function for arterial elastin, with dependence on a reduced subclass of invariants, as W = W(I1 , I4 ). We use previously published equations for this dependence [Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990a. Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Eng. 112, 333–339], in combination with a theoretical guided Rivlin–Saunders framework [Rivlin, R.S., Saunders, D.W., 1951. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phil. Trans. R. Soc. A 243, 251–288] and biaxial mechanical experiments, to obtain the form of this dependence. Using mechanical equivalence of elastin in the circumferential and longitudinal directions, we add a term in I6 to W that is similar to the form in I4 . We propose a semi-empirical model for arterial elastin 2 2 given by W = c0 I1 − 3 + c1 I4 − 1 + c2 I6 − 1 , where c0 , c1 and c3 are unknown coefficients. We used the Levenberg–Marquardt algorithm to fit theoretically calculated and experimentally determined stresses from equibiaxial experiments on autoclaved elastin tissues and obtain c0 = 73.96 ± 22.51 kPa, c1 = 1.18 ± 1.79 kPa and c2 = 0.8 ± 1.26 kPa. Thus, the entropic contribution to the strain energy function, represented by c0 , is a dominant feature of elastin mechanics. Because there are no significant differences in the coefficients corresponding to invariants I4 and I6 , we surmise that there is an equal distribution of fibers in the circumferential and axial directions. c 2008 Elsevier Ltd. All rights reserved.
∗ Corresponding address: Department of Mechanical Engineering, 5134 Etcheverry Hall, Mailstop 1740, University of California Berkeley, Berkeley, CA 94720-1740, USA. Tel.: +1 510 642 2595; fax: +1 510 642 6163. E-mail address: [email protected] (L.A. Pruitt). c 2008 Elsevier Ltd. All rights reserved. 1751-6161/$ - see front matter doi:10.1016/j.jmbbm.2008.10.007
J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M AT E R I A L S
1.
Introduction
Mammalian arteries undergo several billion cycles of pulsatile motion, without appreciable fatigue, over the typical lifetime of an organism. The structural integrity of blood vessels depends critically on passive structural proteins, embedded in a ground matrix, along with active smooth muscle cells. Elastin and collagen comprise the main passive structural proteins in arterial walls, and are arranged in a definite sequence within vertebrate arteries (Wolinsky and Glagov, 1967). Their relative composition within arterial walls directly influence the non-linear, stress–strain behavior of composite arteries (Sage and Gray, 1977). Elastin networks dictate the stereotypical lower arterial modulus at small stretches, whereas collagen networks drive the steeper response at larger strains (Gosline et al., 2002). The characteristic nonlinear arterial response proves crucial for blood circulation because it enables the pulsatile walls to reduce mechanical load on the heart, while preventing the vessels from ballooning under high pressure conditions (Roach and Burton, 1957). Because the mechanics of arterial tissue depends crucially on the micro-structure of underlying fibrillar networks, one approach to study the composite and non-linear arterial mechanics is to develop constitutive models based on material properties of individual networks and consider their mutual interactions. This work focuses on the biomechanics of elastin networks isolated from porcine arteries. Most existing studies on the mechanical properties of elastin are based on uniaxial experiments with little emphasis on multiaxial data (Gundiah et al., 2007; Lillie et al., 1996; Lillie and Gosline, 1990). Although uniaxial tests are important to delineate tissue characteristics, they are however not sufficient to model the in vivo multiaxial conditions. In a recent study, we conducted uniaxial mechanical tests on isolated elastin networks and showed that a generalized Mooney–Rivlin form of strain energy function is inappropriate to describe arterial elastin because the coefficients do not satisfy the Baker–Ericksen inequalities (Gundiah et al., 2007). Physically, the Baker–Ericksen criteria ensure that the greater principal stress always occurs in the direction of greater principal stretch. However, a neo-Hookean form, equivalent to a Gaussian model for randomly oriented long-chain molecules, was suitable to model uniaxial data on elastin. Equibiaxial mechanical experiments on autoclaved elastin demonstrated the same stiffness in the circumferential and axial directions, suggesting equal number of layers in these directions. We also demonstrated that the arterial microstructure consists of axially oriented elastin in the intima and the adventitial layers whereas circumferentially oriented elastin is present in the medial layers. Based on this elastin architecture, we proposed an orthotropic material symmetry consisting of orthogonally oriented, mechanically equivalent fiber layout to describe the properties of the elastin networks in arteries. This study investigates the biaxial mechanics of elastin networks, isolated from porcine arteries, using a continuum mechanical framework. The experimental protocol used here is inspired by the theoretical principles developed by Rivlin and Saunders in their study of the constitutive properties of isotropic rubber (Rivlin and Saunders, 1951), and previous
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studies of the constitutive properties of the myocardium quantified using transversely isotropic symmetry (Humphrey et al., 1990a,b). The goal of our study was to use orthotropic material symmetry in arterial elastin to obtain a form of the strain energy function.
2.
Methods
2.1. Strain energy function for porcine elastin networks based on material symmetry We briefly summarize the basic kinematics and constitutive equations for the description of an orthotropic, finite ∂x is the gradient associated with elastic material. F = ∂X the deformation of material points X in the reference configuration into x in the current configuration. We assume that elastin networks are incompressible, and constrain all deformations to be isochoric. Thus, det(F) = 1.
(1)
We use the notation W to designate the strain energy per unit volume for arterial elastin. For an orthotropic material, W is a function of seven invariants that are described using the right Cauchy–Green tensor C =FT F , and unit vectors M and M0 along the two fiber directions (Holzapfel, 2000). Because it is experimentally difficult to determine all seven unknowns, for simplicity, we express W in terms of a restricted set of invariants. Assuming a linear additive decomposition of the invariants (Qiu and Pence, 1997), W is given by ˆ I1 , I4 , I6 W=W I1 = tr (C) ,
I4 = C · (M ⊗ M) ,
and I6 = C · M0 ⊗ M0
(2)
Ii , i = 1, 4, 6 are the invariants of C. The first invariant I 1 represents the neo-Hookean contribution whereas I4 =λ22 and I6 =λ21 characterize the square of stretches in each of the two fiber directions. ⊗ is the notation used to denote tensorial product. For a material described by Eq. (2) and subjected to biaxial deformations, the components of Cauchy stress are given by T11 = −p + 2W1 λ21 + 2W4 λ21 T22 = −p + 2W1 λ22 + 2W6 λ22 (3) T33 = −p + 2W1 λ23 = 0 T12 = T23 = T13 = 0. Here, p is a scalar pressure-like Lagrange multiplier, present due to the assumption of material incompressibility, and λi ; i = 1 : 3 denote stretches in the principal material directions. We have used the notation Wi = ∂W ∂I ; i = 1, 4, 6 i
for convenience. Assuming mechanical symmetry in the two principal fiber directions, we have equivalence in I4 and I6 . Thus, λ22 − 21 2 T11 − λ21 − 21 2 T22 λ1 λ2 λ λ 1 2 W4 = (4a) 1 2 2 2λ1 λ2 − 2 2 λ1 λ2
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and W1 =
T22
2 λ22 − 21 2 λ1 λ2
.
(4b)
The equations are now similar to those derived by Humphrey et al. for a transversely isotropic material (Humphrey et al., 1990a,b).
2.2. The Rivlin–Saunders protocol applied to elastin networks A constitutive approach based on invariants was first pioneered by Rivlin and Saunders in their influential paper on the properties of rubber (Rivlin and Saunders, 1951). For an incompressible and isotropic material, the strain ˜ is described using the Mooney–Rivlin energy function (W) function (Mooney, 1940; Treloar, 1943), given in terms of strain invariants I1 = (λ21 + λ22 + λ−2 λ−2 ) and I2 = (λ−2 + λ−2 + λ21 λ22 ) 1 2 1 2 as ˜ = c01 I1 − 3 + c10 I2 − 3 W (5) c01 and c10 are constants. The first term in Eq. (5) is called the neo-Hookean term and has dependence on I1 alone whereas the second term is purely phenomenological. The exact form ˜ is next determined by varying the stretches in the two of W directions such that either I1 or I2 is maintained constant at each time. In a biaxial set-up to study elastin, described using Eq. (2), constant I1 experiments require that the tissue be simultaneously subjected to loading in one direction and unloading in the orthogonal direction. This criterion violates the assumption of pseudoelasticity for biological tissues like arteries (Fung, 1993; Sacks and Sun, 2003). However, it is meaningful to conduct constant I4 =λ22 experiments to evaluate the dependence of the partial derivatives of W on I1 and I4 , respectively. In this study, we have used constant I4 experiments to obtain the form of strain energy function, as was also done by Humphrey et al. to model the mechanics of passive myocardial tissues (Humphrey et al., 1990a). Constant I4 condition involves keeping one set of arms at a constant stretch (‘constant axis’) while varying the other over a range of deformation (‘dynamic axis’). Next, we used the entire data set from the constant I4 experiments for each sample to reconstruct the results for constant I1 condition. For a given value of each stretch corresponding to the constant axis, we selected stresses in the dynamic axis for the stretch values that satisfied the constant I1 criterion. These data were used to calculate W1 and W4 in Eq. (4) for the constant I1 condition. Using this method, we generate a parameter map in λ1 –λ2 space that satisfied the constant I1 condition. Although fewer data points could be generated using this method, we were able to address the unloading problems associated with the constant I1 condition.
2.3.
Experimental protocol
Porcine thoracic aortas, obtained from the local abattoir, were cleaned and stored in saline at 4 ◦ C for the mechanical experiments (n = 6). To isolate elastin, one can use either
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the autoclaving or hot alkali treatment. Hot alkali treatment is known to cause fragmentation of peptide chains (Lansing et al., 1952). In a previous study, we showed using histology and uniaxial tests that these changes are accompanied with a loss in the structural integrity of purified elastin networks (Gundiah et al., 2007). In contrast, the autoclaving treatment (Partridge et al., 1955) leaves behind higher collagen and proteoglycan impurities but is a relatively conservative alternative for elastin isolation. In these experiments, we used the autoclaving method to isolate the elastin from porcine arteries. The protocol includes repeated autoclaving cycles of tissue samples for six periods of 1 h each in distilled water, a 24 h treatment in 6M guanidine hydrochloride followed by two additional cycles of autoclaving. The treatment will guanidine hydrochloride solubilizes the denatured collagen and proteoglycans present in the purified elastin segments. Isolated elastin samples were stored in 80% ethanol and all mechanical testes were completed within 3 days after acquiring the aorta samples. Thoracic aorta, stored in 80% ethanol, were thoroughly cleaned and rehydrated in distilled water for at least two hours prior to mechanical testing (Lillie et al., 1996). We measured the tissue thickness, using calipers, by gently sandwiching the specimen between two glass slides (h0 ). The thickness of glass slides alone was subtracted from h0 to obtain sample thickness (h). Each tested elastin specimen was 25.4 mm × 25.4 mm in dimension and oriented in the principal circumferential and axial tissue directions. A custom-built stretcher (Fig. 1(A)) was used for all biaxial experiments and is described in detail elsewhere (Gundiah et al., 2007). Briefly, four independently actuated arms comprised the main units of the stretcher and were attached to a linear positioning table (Model 402004LN, Parker Hannifin Corporation Daedal Division, Harrison City, PA). The arms were driven by four microstepper motors, with attached shaft encoders that were used to assess the linear displacement of each arm (Model OEM750, Parker Hannifin Corporation Compumotor Division, Rohnert Park, CA). The distal end of each arm was clamped to a tissue edge. Miniature load cells (Model 31/3672-02, Honeywell Sensotec Inc., Columbus, OH, 1000 g) were attached to the proximal end of two orthogonal stretcher arms, and these measured forces on the tissue during stretching. To maintain tissue hydration we placed the specimen in an acrylic bath tray containing distilled water at 37 ◦ C (Fig. 1(A)). The stepper motors moved the linear arms in a synchronized manner to deform the specimen. A video camera, placed orthogonally over the specimen (Fig. 1(A)), tracked the motion of ceramic marker beads glued to the tissue surface. These marker displacements were converted to stretches in the circumferential and axial directions (λ1 and λ2 ) using undeformed tissue lengths (L10 and L20 ) and the Green strains (E11 , E22 ) in the circumferential and axial directions were calculated as 1 2 E11 = λ1 − 1 (6a) 2 and E22 =
1 2 λ2 − 1 . 2
(6b)
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Fig. 1 – (A) Custom built biaxial stretcher is used to measure the stress–strain properties of biological tissues. The tissue sample is placed between the orthogonal arms equipped with load cells to measure stress. A camera placed over the sample records the deformation by tracking the movement of marker beads placed on the tissue surface. The deformation is converted to Green strains (B) using minimum encoder position corresponding to the resting tissue length and forces from load cells were converted to Cauchy stresses (C).
Forces from the load cells in the two orthogonal directions (F1 and F2 ) were amplified using an analog amplifier (Model 6600, Gould Amplifier, Valley View, OH). Cauchy stresses (T11 , T22 ) in the circumferential and axial directions were calculated as T11 = λ1
F1 hL20
(7a)
F2 . hL10
(7b)
and T22 = λ2
In the experiments reported here, samples were preconditioned at 10% stretch for 10 cycles in a triangular waveform at 0.05 Hz (Zhou and Fung, 1997) in both circumferential and axial directions. Constant I4 experiments were performed by stretching the tissue along one direction for different amounts (Table 1). In this method, the tissue was held stretched at a constant value in one direction and simultaneously the orthogonal set of arms were stretched out. This procedure was carried out for various incremental constant stretches for each sample to get a series of stress–strain data. All the data reported in this study correspond to the first loading cycle following preconditioning.
2.4.
Statistical analysis
Students’ paired t tests were performed to test for differences in the coefficients to the newly proposed strain energy function. A statistical level of p < 0.05 was considered significant. Square of the Pearson product moment correlation coefficient was also used in regression analysis for the different elastin samples.
3.
Results
3.1.
Constant I4 experiments
Fig. 1(B) shows a sample data set for an experiment in which one set of arms were maintained at a constant imposed stretch and the orthogonal stretcher arms moved out to a maximum displacement of 20% of the tissue length in the dynamic axis. Cauchy stresses recorded along the constant axis have non-zero values while those in the dynamic axis change from zero to a fixed maximum. Fig. 2 shows the
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Table 1 – Slopes of the stress–strain curves corresponding to the moving axis from all stretches on the sample. Specimen 4b Constant/ Slope (kPa) Dynamic (%)
Specimen 6a Constant/ Slope (kPa) Dynamic (%)
Specimen 7a Constant/ Slope (kPa) Dynamic (%)
Specimen 9b Constant/ Slope (kPa) Dynamic (%)
2/ 20 5/ 10 10/ 10 15/ 15
870.07 896.51 947.78 865.38
2/ 10 5/ 10 10/ 15 15/ 20 20/ 25
692.22 751.02 818.29 888.13 913.22
2/ 15 5/ 15 10/ 20 10/ 25
581.95 588.18 676.17 624.07
Equi-biax:5 Avg ± Stdev
897.95 895.54 ± 32.76 or 3.6%
Equi-biax:10
1106.8 861.61 ± 145.80 or 16.9%
Equi-biax:10
633.93 620.86 ± 38.15 or 6.1%
2/ 15 5/ 15 10/ 20 15/ 20 20/ 25 25/ 30 Equi-biax:10
576.73 605.87 630.72 686.03 678.3 681.3 774.28 661.89 ± 64.84 or 9.8%
Table 2 – Coefficients c0 , c1 and c2 in the new strain energy function (Eq. (8)) obtained using the Levenberg–Marquardt algorithm for all purified elastin samples in the study.
Fig. 2 – Data from experiments performed at various constant values of I4 shown for a representative arterial elastin sample (Specimen 9b). The different symbols in the figure are shown for various constant I4 experiments given in Table 2. Also shown in the table are the slopes of the curves.
Cauchy Stress vs. Green strain plots from various constant I4 experiments, each shown using a different symbol, for one representative elastin sample. The tissue underwent slight deformations (<5% of tissue length for a maximum stretch) along the axis of constant imposed stretch. All tissues showed a robust and repeatable response at these high values of strain for several different deformations. Additionally, the slopes of the Cauchy stress–Green strain curves corresponding to the dynamic axis for each sample were similar for various combinations of constant and varying imposed stretches (Table 1) indicating that there was minimal damage accumulation in the elastin tissue during different stretching protocols. As expected due to inherent biological variation, the slopes showed some variance between different samples. To estimate the partial derivatives of the strain energy function Wi , i = 1, 4, we used stress–strain data from various constant I4 stretching protocols. Fig. 3(A) represents a parameter map of constant I4 values for a representative sample. This plot also illustrates the dependence of Wi , i = 1, 4 on invariant I1 (n = 4). Different symbols are used
Sample
c0 (kPa)
c1 (kPa)
c2 (kPa)
4b 6a 7a 9b 11 11a Average Stdev
84.22 90.29 40.69 65.50 102.36 60.69 73.96 22.51
0.37 4.27 0 2.45 0.31 0.55 1.18 1.79
0 2.75 0 2.08 0 0.03 0.80 1.26
for experiments corresponding to a constant I4 stretching protocol for the same sample. From Fig. 3(A), we see that λ1 , λ2 → 1, W1 → ∞ (also see Eq. (4b)). For ease of plotting, some of these values are not shown in the graph. Also, independent of the actual value at which I4 was maintained, W1 approached a constant value with increasing I1 . As I4 increased, values of W4 became increasingly negative (Fig. 3(B)). Further, W4 values increased to a steady value, relatively independent of I1 , for a given I4 . Thus, as a first approximation we may assume that W has a linear dependence on invariant I1 ; that is W contains a neo-Hookean form with a linear dependence on the invariant I1 . The experimental data from the dynamic stretches corresponding to each constant λ2 experiment were used to estimate constant I1 values. Stresses and stretches for these values were used to calculate the partial derivatives of W with respect to invariants I1 and I4 . Fewer I1 values could however be obtained using this method as compared to the constant I4 experiments. W1 and W4 are shown plotted for constant I1 values in Fig. 4 for a representative sample. As I4 approaches the value of 1, I1 → 3.0 and the W1 plots approach very large numbers. There is a trend for linear relationship between W4 and I4 at constant I4 . However, the sparse data points obtained by this method could only be used to infer trends in the data. We will hence use data from constant I4 experiments to obtain the nature of this dependence. From the W4 plots in Fig. 3, we assume that the last point in each experiment used for the calculation of W4 represented the asymptotic value. This value is plotted as a function of I4 in Fig. 5. W4 shows a linear dependence on I4 for each elastin sample tested in the study, with values
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Fig. 3 – The dependence of W1 and W4 , calculated using Eqs. (4a) and (4b), shown as a function of I1 from various constant I4 experiments for specimen 9b. Each of the constant: moving stretch values are shown, using different symbols, for the various experiments performed on the same elastin sample.
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293
Fig. 5 – W4 as a function of I4 for all samples in the study. The final value of W4 was used in obtaining the plot. r2 values are shown to reflect the dependence of W4 on I4 (p < 0.005). elastin in arterial walls: 2 2 W = c0 I1 − 3 + c1 I4 − 1 + c2 I6 − 1 .
(8)
This equation presents a semi-empirical form of the strain energy function modified to include the biaxial properties of elastin networks.
3.2.
Estimation of constitutive coefficients
For planar materials described by Eq. (8), the Cauchy stresses in principal directions are given by T11 = 2c0 λ21 − λ23 + 4c1 λ41 (9a) T22 = 2c0 λ22 − λ23 + 4c2 λ42 . In equibiaxial tests, λ1 = λ2 = λ.
(9b)
For an incompressible material, we have from Eq. (1) 1 . (9c) λ2 We used data from equibiaxial studies on autoclaved elastin, presented in an earlier study (Gundiah et al., 2007), to determine the unknown coefficients to the new model for arterial elastin. To arrive at an optimal set of coefficients for the above non-linear multivariate equations, we used the Levenberg–Marquardt algorithm in MATLAB (The Mathworks, v7.0.1, Natick, MA). The algorithm allowed us to identify bestfit solutions to functions, obtained as a difference between experimental data and theoretically calculated stress in Eq. (9a). Fig. 6 shows a converged solution for a representative sample in the study and Table 2 shows a full set of the coefficients for all samples. From these studies, the coefficients are given by (average ± stdev) λ3 =
Fig. 4 – Results for the variation of W1 and W4 from the constant I1 condition for the same specimen 9b.
of r2 ranging from 0.909 to 0.999. Thus, the strain energy function has a second order dependence on I4 . Furthermore, because the orthogonal fibers of elastin have essentially the same mechanical behavior, we may extend the results from constant I4 to describe I6 ; thus incorporating the assumption of symmetry and mechanical equivalence of these two terms of the strain energy function. Finally, we must also impose the constraint that W = 0 for the unstretched case i.e. when I1 = 3, I4 = 1 and I6 = 1. From these results and constraints, we propose the following equation to represent the strain energy function of
c0 = 73.96c ± 22.51 kPa, c1 = 1.18 ± 1.79 kPa and c2 = 0.8 ± 1.26 kPa
(10)
c0 is an order of magnitude larger than coefficients c1 and c2 corresponding to invariants I4 and I6 , respectively. The difference in Student’s paired t-test between coefficients c1 and c2 (p = 0.68) is statistically insignificant.
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Fig. 6 – Fit of equibiaxial tension data (shown by symbols) for a representative elastin tissue using the new constitutive equation (8). The data were fit using a Levenberg–Marquardt algorithm for non-linear multivariate functions.
4.
Discussion
The data presented in this study investigates the material properties of elastin in the arterial networks for porcine arteries. To the best of our knowledge, these are the first comprehensive biaxial data on purified elastin networks and represent a step towards generating a constitutive model for arteries incorporating tissue microstructure. We have performed experiments within a theoretical framework (Rivlin and Saunders, 1951) to obtain an empirical form of the strain energy function for arterial elastin based on the orthotropic material symmetry of elastin fibers within arterial walls (Gundiah et al., 2007).
4.1.
The biomechanics of aortic elastin
Elastic fibers are present in static and dynamic tissues including skin, arteries and lung. These tissues, owing mainly to the presence of elastin networks, are able to endure several million cycles of cyclic deformation without undergoing appreciable fatigue (Gosline et al., 2002). During development, elastin fibers are deposited as tropoelastin monomers on microfibrillar scaffolds where they assemble and undergo cross-linking into amorphous, insoluble and hydrophobic elastin networks (Banga, 1966). The mechanics of elastin networks have been described using Gaussian rubber models (Hoeve and Flory, 1974). Local structurefunctional relationships within elastin have been explained using molecular equilibria phenomena (Debelle and Alix, 1999). Elastin networks have been modeled as random chain networks with a Gaussian distribution of end-to-end chain lengths between the cross-links (Hoeve and Flory, 1958). More recently, Urry and coworkers (Urry et al., 2002) used atomic force microscopy and acoustic absorption on single chain elastin molecules to show that elastin consists of regularly repeating ordered blocks of amino acids, with lysine derived cross-links, that deform on extension using entropic
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elasticity mechanisms. No study has to date investigated the biomechanics of elastin networks incorporating their underlying material symmetry. Uniaxial mechanical tests show that elastin is a protein rubber (Lillie and Gosline, 1990) with the modulus of single elastin fibers approaching 1.2 MPa (Aaron and Gosline, 1980). Using uniaxial mechanical data on porcine elastin, we showed in an earlier study that a neo- Hookean strain energy function is suitable to describe the mechanics of elastin networks whereas a more general Mooney–Rivlin form is not applicable. In this study, our biaxial experiments on elastin networks isolated from porcine arteries also show that a neo-Hookean term dominates the strain energy function for arterial elastin. Because the neo-Hookean model is the continuum mechanics equivalent of a material described by a Gaussian distribution, the constitutive model proposed here essentially agrees with the data from previous studies on elastin. In addition to the neo-Hookean term to the strain energy function, our experiments indicate the presence of two additional terms that arise from the distribution of elastic fibers in the artery. However, the contributions of the two terms resulting from material symmetry are not as substantial as the neo-Hookean term; demonstrating that the entropic contribution to the stored energy forms a dominant mechanism of elasticity in elastin networks.
4.2.
An isotropic neo-Hookean model for arterial elastin
Because purified elastin is very fragile, repetitive biaxial experiments were often difficult to perform. Nevertheless, repeated experimental protocols on the same specimen leave the slope of the stress–strain curves essentially unchanged (Fig. 4) thus confirming the robustness of elastin networks and elastin fibers. These experiments provide a strong empirical basis to models of elastin networks that use Gaussian statistics to describe the elasticity of the networks. A quadratic form of the strain energy function, referred as a standard reinforcing model, has frequently been employed to describe the reinforcing contribution of fiber reinforced materials (Qiu and Pence, 1997; Triantafyllidis and Abeyaratne, 1983). The mechanical results from our study also indicate that such a term would be appropriate to describe the mostly linear stress–strain curves exhibited by arterial elastin. However, results from our study show that the contributions to the strain energy function of terms in I4 and I6 , describing the actions of orthogonal fibers, are much smaller than the neo-Hookean term. Thus, the entropic contribution to the strain energy function is a dominant feature of elastin mechanics. Furthermore, because the I4 and I6 terms are not statistically different from one another, these suggest that the distributions of elastic fibers, and perhaps their cross-linking density, are the same in the circumferential and axial directions. Thus, the arterial microstructure aids in the equal distribution of stresses in the circumferential and axial directions during vessel loading. Based on the new strain energy function proposed here, we measured a modulus of 523 kPa for arterial elastin. This value is similar to the previously reported values of 900.8 ± 687 kPa for autoclaved arterial elastin (Gundiah et al., 2007). Using the coefficients in the Holzapfel model (Holzapfel et al., 1996)
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we get an elastin modulus of 308 kPa, and a value of 337 kPa using coefficients in the Zulliger et al. model (Zulliger et al., 2004). These latter results were however obtained by fitting equibiaxial data from arteries to a specific form of the strain energy function. Both these values are significantly smaller than other uniaxial results published for arterial elastin (Aaron and Gosline, 1980; Gosline, 1980; Gosline et al., 2002; Gundiah et al., 2007). In a recent study, Lillie and Gosline compared the moduli of autoclaved porcine elastin networks purified from the proximal and distal regions of the aorta (Lillie and Gosline, 2007). They suggest, based on higher moduli from the distal segments, that elastin networks are progressively more anisotropic away from the heart. Because collagen contents increase in distally located segments, a higher proportion of collagen remaining in the tested segments may be an unaccounted source of error in that study. In this study, we have not explored the regional differences in the biaxial mechanics of isolated elastin. Such experiments, performed over the entire range of tissue deformation, may however be used to quantify any such regional differences in the layout of elastin in arterial segments. However, results from this study indicate that any regional microstructural differences in the circumferential and axial layout may however not be as significant as the dominant neo-Hookean term.
4.3.
Limitations
Kinematical constraints like incompressibility used in this study help to reduce the total number of invariants by one. Earlier studies have shown that elastin has a hydrophobic core from which aqueous solvent is excluded (Robert et al., 1970). Dry elastin is brittle and plasticizing effects due to water are necessary to the network elasticity. Hydrophobic domains present between cross-linking domains in elastin exhibit high mobility, contributing greatly to the network entropy. Extensions of the elastin network hence cause the hydrophobic regions to come into contact with water. Consequently, the network absorbs water and disrupts the hydrophobic interactions (Gosline, 1980). It may hence be surmised that elastin does not deform at constant volume, an important premise in tissue biomechanics for samples in the physiological loading range. In this study, the phenomenon of water absorption of elastin fibers during stretching is considered a local microscopic event, which does not affect the continuum, and the incompressibility criterion is hence a first approximation. From a materials characterization perspective, Merodio and Ogden have shown that it is advantageous, from the point of ellipticity of the strain energy function, to neglect the contribution of invariant I8 that incorporates the contribution between the two fiber families from the constitutive equation (Merodio and Ogden, 2006). Although the microstructure may undergo slight deviations from orthotropy in the arteries (Clark and Glagov, 1985), it may however be advantageous to neglect the terms corresponding to I8 . We have used the results from Humphrey et al. (1990a,b) to reduce the dependence of the strain energy function from six to three invariants (I1 , I4 and I6 ). Nevertheless, the explicit form of W, dependent on three invariants, cannot be
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determined from biaxial experiments alone and additional experiments are required to assess the full functional relationship. In the study outlined here, we assume the mechanical equivalence of the two orthogonal fiber families based on earlier studies on porcine arterial elastin (Gundiah et al., 2007). We have hence assumed equivalence in terms I4 and I6 in this study. Any cross interaction terms in I4 and I6 or the combination with other invariants in the strain energy function cannot be assessed using this method and additional mechanical experiments will be necessary to study the applicability of the proposed function. The standard deviations obtained in the values of the coefficients c1 and c2 in the new constitutive model (Eqs. (8) and (10)) are very large (Table 2). These may be a result of any possible regional differences in the elastin samples and also inter sample variability. All samples were excised staring from 1 to 5 cm away from the heart. Future studies would benefit in accounting for any such variations due to the position of elastin along the arterial tree. Nevertheless, such terms are much smaller than the dominant neo-Hookean term in the model and any additional terms representing the regional differences may not be as important. We have assumed a stress-free ‘unloaded’ reference state for the tissues in this study. Residual strains, measured using a measure of the opening angle, are however present in arteries and would result as an addition to the strains in the elastin segments. Results from our lab on the measurement of residual stresses in purified elastin show that residual strains in purified elastin is negative causing the elastin ring to close in, rather than move away from the radial cut as is normal for arterial rings (Wang, 2004).
5.
Conclusions
We present a revised constitutive model for arterial elastin based on material symmetry and theoretically guided experiments and show that arterial elastin behaves essentially as a neo-Hookean material with added I4 and I6 terms in the constitutive equation due elastin layout in arteries. Our experiments suggest that the macroscopic elastic networks behave essentially as a neo-Hookean material, as has also been suggested by earlier models, albeit with minor modifications. The neo-Hookean term measured here is significantly higher than previously reported values obtained by the fitting of mechanical data to an assumed form of the strain energy function (Holzapfel et al., 1996; Zulliger et al., 2004).
Acknowledgements We would like to thank Dr. Sanjay Sane for many useful discussions and to Dr. Kay Sun for critically reading the manuscript. This study was funded by the NSF grant CMS0106010 to UC Berkeley and by NIH 2RO1 HL063348-05 to UC San Francisco.
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REFERENCES
Aaron, B.B., Gosline, J.M., 1980. Optical properties of single elastic fibers indicate random protein conformation. Nature 287, 865–867. Banga, I., 1966. Structure and Function of Elastin and Collagen. Akademiai Kiado, Budapest. Clark, J.M., Glagov, S., 1985. Transmural organization of the arterial media: The lamellar unit revisited. Arteriosclerosis 5, 19–34. Debelle, L., Alix, A.J.P., 1999. The structures of elastin and their function. Biochemie 81, 981–994. Fung, Y.C., 1993. Biomechanics: Mechanical Properties of Living Tissues. Springer Verlag, New York. Gosline, J.M., 1980. The Elastic Properties of Rubber-like Proteins and Highly Extensible Tissues. Cambridge University Press. Gosline, J.M., Lillie, M., Carrington, E., Guerette, P., Ortlepp, C., Savage, K., 2002. Elastic proteins: Biological roles and mechanical properties. Phil. Trans. R. Soc. B 357, 121–132. Gundiah, N., Ratcliffe, M.B., Pruitt, L.A., 2007. Determination of strain energy function for arterial elastin: Experiments using histology and mechanical tests. J. Biomech. 40, 586–594. Hoeve, C.A.J., Flory, P.J., 1958. The elastic properties of elastin. J. Am. Chem. Soc. 80, 6523–6526. Hoeve, C.A.J., Flory, P.J., 1974. Elastic properties of elastin. Biopolymers 13, 677–686. Holzapfel, G.A., 2000. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, West Sussex, England. Holzapfel, G.A., Eberlein, R., Wriggers, P., Weizsacker, H.W., 1996. A new axisymmetrical membrane element for anisotropic, finite strain analysis of arteries. Commun. Numer. Methods Eng. 12, 507–517. Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990a. Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Eng. 112, 333–339. Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990b. Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J. Biomech. Eng. 112, 340–346. Lansing, A.I., Rosenthal, T.B., Alex, M., Dempsey, W., 1952. The structure and chemical characterization of elastin fibers as revealed by elastase and by electron microscopy. Anat. Rec. 114, 555–575. Lillie, M.A., Chalmers, G.W.G., Gosline, J.M., 1996. Elastin dehydration through the liquid and the vapor phase — A comparison of osmotic stress models. Biopolymers 39 (5), 627–639. Lillie, M.A., Gosline, J.M., 1990. The effects of hydration on the dynamic mechanical properties of elastin. Biopolymers 29 (8–9), 1147–1160.
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Lillie, M.A., Gosline, J.M., 2007. Mechanical properties of elastin along the thoracic aorta in the pig. J. Biomech. 40 (10), 2214–2221. Merodio, J., Ogden, R.W., 2006. The influence of the invariant I8 on the stress-deformation and ellipticity characteristics of doubly-fiber reinforced non-linearly elastic solids. Internat. J. Non-linear Mech. 41, 556–563. Mooney, M., 1940. A theory of large elastic deformation. J. Appl. Phys. 11, 582–592. Partridge, S.M., Davis, H.F., Adair, G.S., 1955. The chemistry of connective tissues. 2. Soluble proteins derived from partial hydrolysis of elastin. Biochem. J. 61, 11–21. Qiu, G.Y., Pence, T.J., 1997. Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids. J. Elast. 49, 1–30. Rivlin, R.S., Saunders, D.W., 1951. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phil. Trans. R. Soc. A 243, 251–288. Roach, M.R., Burton, A.C., 1957. The reason for the shape of the distensibility curves of arteries. Can. J. Biochem. Physiol. 35, 181–190. Robert, L., Robert, B., Robert, A.M., 1970. Molecular biology of elastin as related to aging and atherosclerosis. Exp. Gerontol. 5, 339–356. Sacks, M.S., Sun, W., 2003. Multiaxial mechanical behavior of biological materials. Annu. Rev. Biomed. Eng. 5, 251–284. Sage, E., Gray, W., 1977. Evolution of elastin structure. Adv. Exp. Med. Biol. 79, 291–312. Treloar, L., 1943. The elasticity of a network of long-chain molecules- II. Trans. Faraday Soc. 39, 241–246. Triantafyllidis, N., Abeyaratne, R., 1983. Instabilities of a finitely deformed fiber-reinforced elastic material. J. Appl. Mech. 50, 149–156. Urry, D.W., Hugel, T., Seitz, M., Gaub, H.E., Sheiba, L., Dea, J., Xu, J., Parker, T., 2002. Elastin: A representative ideal protein elastomer. Philos. Trans. R. Soc. London B 357, 169–184. Wang, J., 2004. The contribution of elastin, collagen and smooth muscle cells to the biomechanics of large elastic arteries. In: Bioengineering, Berkeley, Ph.D. University of California Berkeley. Wolinsky, H., Glagov, S., 1967. A lamellar unit of aortic medial structure and function in mammals. Circ. Res. 20, 99–111. Zhou, J., Fung, Y.C., 1997. The degree of nonlinearity and anisotropy of blood vessel elasticity. Proc. Natl. Acad. Sci. USA 94 (26), 14255–14260. Zulliger, M.A., Fridez, P., Hayashi, K., Stergiopulos, N., 2004. A strain energy function for arteries accounting for wall composition and structure. J. Biomech. 37, 989–1000.
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available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jmbbm
Research paper
The biomechanics of arterial elastin Namrata Gundiah a , Mark B. Ratcliffe c,d , Lisa A. Pruitt a,b,∗ a Department of Mechanical Engineering, University of California, Berkeley, CA, USA b Department of Bioengineering, University of California, Berkeley, CA, USA c Department of Surgery, University of California, San Francisco, USA d San Francisco Veterans Affairs Medical Center, San Francisco, CA, USA
A R T I C L E
I N F O
A B S T R A C T
Article history:
Uniaxial mechanical experiments have shown that a neo-Hookean/Gaussian model is
Received 27 June 2008
suitable to describe the mechanics of arterial elastin networks [Gundiah, N., Ratcliffe, M.B.,
Received in revised form
Pruitt, L.A., 2007. Determination of strain energy function for arterial elastin: Experiments
14 October 2008
using histology and mechanical tests. J. Biomech. 40, 586–594]. Based on the three-
Accepted 24 October 2008
dimensional elastin architecture in arteries, we have proposed an orthotropic material
Published online 6 November 2008
symmetry for arterial elastin consisting of two orthogonally oriented and symmetrically placed families of mechanically equivalent fibers. In this study, we use these results to describe the strain energy function for arterial elastin, with dependence on a reduced subclass of invariants, as W = W(I1 , I4 ). We use previously published equations for this dependence [Humphrey, J.D., Strumpf, R.K., Yin, F.C.P., 1990a. Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Eng. 112, 333–339], in combination with a theoretical guided Rivlin–Saunders framework [Rivlin, R.S., Saunders, D.W., 1951. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phil. Trans. R. Soc. A 243, 251–288] and biaxial mechanical experiments, to obtain the form of this dependence. Using mechanical equivalence of elastin in the circumferential and longitudinal directions, we add a term in I6 to W that is similar to the form in I4 . We propose a semi-empirical model for arterial elastin 2 2 given by W = c0 I1 − 3 + c1 I4 − 1 + c2 I6 − 1 , where c0 , c1 and c3 are unknown coefficients. We used the Levenberg–Marquardt algorithm to fit theoretically calculated and experimentally determined stresses from equibiaxial experiments on autoclaved elastin tissues and obtain c0 = 73.96 ± 22.51 kPa, c1 = 1.18 ± 1.79 kPa and c2 = 0.8 ± 1.26 kPa. Thus, the entropic contribution to the strain energy function, represented by c0 , is a dominant feature of elastin mechanics. Because there are no significant differences in the coefficients corresponding to invariants I4 and I6 , we surmise that there is an equal distribution of fibers in the circumferential and axial directions. c 2008 Elsevier Ltd. All rights reserved.
∗ Corresponding address: Department of Mechanical Engineering, 5134 Etcheverry Hall, Mailstop 1740, University of California Berkeley, Berkeley, CA 94720-1740, USA. Tel.: +1 510 642 2595; fax: +1 510 642 6163. E-mail address: [email protected] (L.A. Pruitt). c 2008 Elsevier Ltd. All rights reserved. 1751-6161/$ - see front matter doi:10.1016/j.jmbbm.2008.10.007
J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M AT E R I A L S
1.
Introduction
Mammalian arteries undergo several billion cycles of pulsatile motion, without appreciable fatigue, over the typical lifetime of an organism. The structural integrity of blood vessels depends critically on passive structural proteins, embedded in a ground matrix, along with active smooth muscle cells. Elastin and collagen comprise the main passive structural proteins in arterial walls, and are arranged in a definite sequence within vertebrate arteries (Wolinsky and Glagov, 1967). Their relative composition within arterial walls directly influence the non-linear, stress–strain behavior of composite arteries (Sage and Gray, 1977). Elastin networks dictate the stereotypical lower arterial modulus at small stretches, whereas collagen networks drive the steeper response at larger strains (Gosline et al., 2002). The characteristic nonlinear arterial response proves crucial for blood circulation because it enables the pulsatile walls to reduce mechanical load on the heart, while preventing the vessels from ballooning under high pressure conditions (Roach and Burton, 1957). Because the mechanics of arterial tissue depends crucially on the micro-structure of underlying fibrillar networks, one approach to study the composite and non-linear arterial mechanics is to develop constitutive models based on material properties of individual networks and consider their mutual interactions. This work focuses on the biomechanics of elastin networks isolated from porcine arteries. Most existing studies on the mechanical properties of elastin are based on uniaxial experiments with little emphasis on multiaxial data (Gundiah et al., 2007; Lillie et al., 1996; Lillie and Gosline, 1990). Although uniaxial tests are important to delineate tissue characteristics, they are however not sufficient to model the in vivo multiaxial conditions. In a recent study, we conducted uniaxial mechanical tests on isolated elastin networks and showed that a generalized Mooney–Rivlin form of strain energy function is inappropriate to describe arterial elastin because the coefficients do not satisfy the Baker–Ericksen inequalities (Gundiah et al., 2007). Physically, the Baker–Ericksen criteria ensure that the greater principal stress always occurs in the direction of greater principal stretch. However, a neo-Hookean form, equivalent to a Gaussian model for randomly oriented long-chain molecules, was suitable to model uniaxial data on elastin. Equibiaxial mechanical experiments on autoclaved elastin demonstrated the same stiffness in the circumferential and axial directions, suggesting equal number of layers in these directions. We also demonstrated that the arterial microstructure consists of axially oriented elastin in the intima and the adventitial layers whereas circumferentially oriented elastin is present in the medial layers. Based on this elastin architecture, we proposed an orthotropic material symmetry consisting of orthogonally oriented, mechanically equivalent fiber layout to describe the properties of the elastin networks in arteries. This study investigates the biaxial mechanics of elastin networks, isolated from porcine arteries, using a continuum mechanical framework. The experimental protocol used here is inspired by the theoretical principles developed by Rivlin and Saunders in their study of the constitutive properties of isotropic rubber (Rivlin and Saunders, 1951), and previous
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studies of the constitutive properties of the myocardium quantified using transversely isotropic symmetry (Humphrey et al., 1990a,b). The goal of our study was to use orthotropic material symmetry in arterial elastin to obtain a form of the strain energy function.
2.
Methods
2.1. Strain energy function for porcine elastin networks based on material symmetry We briefly summarize the basic kinematics and constitutive equations for the description of an orthotropic, finite ∂x is the gradient associated with elastic material. F = ∂X the deformation of material points X in the reference configuration into x in the current configuration. We assume that elastin networks are incompressible, and constrain all deformations to be isochoric. Thus, det(F) = 1.
(1)
We use the notation W to designate the strain energy per unit volume for arterial elastin. For an orthotropic material, W is a function of seven invariants that are described using the right Cauchy–Green tensor C =FT F , and unit vectors M and M0 along the two fiber directions (Holzapfel, 2000). Because it is experimentally difficult to determine all seven unknowns, for simplicity, we express W in terms of a restricted set of invariants. Assuming a linear additive decomposition of the invariants (Qiu and Pence, 1997), W is given by ˆ I1 , I4 , I6 W=W I1 = tr (C) ,
I4 = C · (M ⊗ M) ,
and I6 = C · M0 ⊗ M0
(2)
Ii , i = 1, 4, 6 are the invariants of C. The first invariant I 1 represents the neo-Hookean contribution whereas I4 =λ22 and I6 =λ21 characterize the square of stretches in each of the two fiber directions. ⊗ is the notation used to denote tensorial product. For a material described by Eq. (2) and subjected to biaxial deformations, the components of Cauchy stress are given by T11 = −p + 2W1 λ21 + 2W4 λ21 T22 = −p + 2W1 λ22 + 2W6 λ22 (3) T33 = −p + 2W1 λ23 = 0 T12 = T23 = T13 = 0. Here, p is a scalar pressure-like Lagrange multiplier, present due to the assumption of material incompressibility, and λi ; i = 1 : 3 denote stretches in the principal material directions. We have used the notation Wi = ∂W ∂I ; i = 1, 4, 6 i
for convenience. Assuming mechanical symmetry in the two principal fiber directions, we have equivalence in I4 and I6 . Thus, λ22 − 21 2 T11 − λ21 − 21 2 T22 λ1 λ2 λ λ 1 2 W4 = (4a) 1 2 2 2λ1 λ2 − 2 2 λ1 λ2
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and W1 =
T22
2 λ22 − 21 2 λ1 λ2
.
(4b)
The equations are now similar to those derived by Humphrey et al. for a transversely isotropic material (Humphrey et al., 1990a,b).
2.2. The Rivlin–Saunders protocol applied to elastin networks A constitutive approach based on invariants was first pioneered by Rivlin and Saunders in their influential paper on the properties of rubber (Rivlin and Saunders, 1951). For an incompressible and isotropic material, the strain ˜ is described using the Mooney–Rivlin energy function (W) function (Mooney, 1940; Treloar, 1943), given in terms of strain invariants I1 = (λ21 + λ22 + λ−2 λ−2 ) and I2 = (λ−2 + λ−2 + λ21 λ22 ) 1 2 1 2 as ˜ = c01 I1 − 3 + c10 I2 − 3 W (5) c01 and c10 are constants. The first term in Eq. (5) is called the neo-Hookean term and has dependence on I1 alone whereas the second term is purely phenomenological. The exact form ˜ is next determined by varying the stretches in the two of W directions such that either I1 or I2 is maintained constant at each time. In a biaxial set-up to study elastin, described using Eq. (2), constant I1 experiments require that the tissue be simultaneously subjected to loading in one direction and unloading in the orthogonal direction. This criterion violates the assumption of pseudoelasticity for biological tissues like arteries (Fung, 1993; Sacks and Sun, 2003). However, it is meaningful to conduct constant I4 =λ22 experiments to evaluate the dependence of the partial derivatives of W on I1 and I4 , respectively. In this study, we have used constant I4 experiments to obtain the form of strain energy function, as was also done by Humphrey et al. to model the mechanics of passive myocardial tissues (Humphrey et al., 1990a). Constant I4 condition involves keeping one set of arms at a constant stretch (‘constant axis’) while varying the other over a range of deformation (‘dynamic axis’). Next, we used the entire data set from the constant I4 experiments for each sample to reconstruct the results for constant I1 condition. For a given value of each stretch corresponding to the constant axis, we selected stresses in the dynamic axis for the stretch values that satisfied the constant I1 criterion. These data were used to calculate W1 and W4 in Eq. (4) for the constant I1 condition. Using this method, we generate a parameter map in λ1 –λ2 space that satisfied the constant I1 condition. Although fewer data points could be generated using this method, we were able to address the unloading problems associated with the constant I1 condition.
2.3.
Experimental protocol
Porcine thoracic aortas, obtained from the local abattoir, were cleaned and stored in saline at 4 ◦ C for the mechanical experiments (n = 6). To isolate elastin, one can use either
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the autoclaving or hot alkali treatment. Hot alkali treatment is known to cause fragmentation of peptide chains (Lansing et al., 1952). In a previous study, we showed using histology and uniaxial tests that these changes are accompanied with a loss in the structural integrity of purified elastin networks (Gundiah et al., 2007). In contrast, the autoclaving treatment (Partridge et al., 1955) leaves behind higher collagen and proteoglycan impurities but is a relatively conservative alternative for elastin isolation. In these experiments, we used the autoclaving method to isolate the elastin from porcine arteries. The protocol includes repeated autoclaving cycles of tissue samples for six periods of 1 h each in distilled water, a 24 h treatment in 6M guanidine hydrochloride followed by two additional cycles of autoclaving. The treatment will guanidine hydrochloride solubilizes the denatured collagen and proteoglycans present in the purified elastin segments. Isolated elastin samples were stored in 80% ethanol and all mechanical testes were completed within 3 days after acquiring the aorta samples. Thoracic aorta, stored in 80% ethanol, were thoroughly cleaned and rehydrated in distilled water for at least two hours prior to mechanical testing (Lillie et al., 1996). We measured the tissue thickness, using calipers, by gently sandwiching the specimen between two glass slides (h0 ). The thickness of glass slides alone was subtracted from h0 to obtain sample thickness (h). Each tested elastin specimen was 25.4 mm × 25.4 mm in dimension and oriented in the principal circumferential and axial tissue directions. A custom-built stretcher (Fig. 1(A)) was used for all biaxial experiments and is described in detail elsewhere (Gundiah et al., 2007). Briefly, four independently actuated arms comprised the main units of the stretcher and were attached to a linear positioning table (Model 402004LN, Parker Hannifin Corporation Daedal Division, Harrison City, PA). The arms were driven by four microstepper motors, with attached shaft encoders that were used to assess the linear displacement of each arm (Model OEM750, Parker Hannifin Corporation Compumotor Division, Rohnert Park, CA). The distal end of each arm was clamped to a tissue edge. Miniature load cells (Model 31/3672-02, Honeywell Sensotec Inc., Columbus, OH, 1000 g) were attached to the proximal end of two orthogonal stretcher arms, and these measured forces on the tissue during stretching. To maintain tissue hydration we placed the specimen in an acrylic bath tray containing distilled water at 37 ◦ C (Fig. 1(A)). The stepper motors moved the linear arms in a synchronized manner to deform the specimen. A video camera, placed orthogonally over the specimen (Fig. 1(A)), tracked the motion of ceramic marker beads glued to the tissue surface. These marker displacements were converted to stretches in the circumferential and axial directions (λ1 and λ2 ) using undeformed tissue lengths (L10 and L20 ) and the Green strains (E11 , E22 ) in the circumferential and axial directions were calculated as 1 2 E11 = λ1 − 1 (6a) 2 and E22 =
1 2 λ2 − 1 . 2
(6b)
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Fig. 1 – (A) Custom built biaxial stretcher is used to measure the stress–strain properties of biological tissues. The tissue sample is placed between the orthogonal arms equipped with load cells to measure stress. A camera placed over the sample records the deformation by tracking the movement of marker beads placed on the tissue surface. The deformation is converted to Green strains (B) using minimum encoder position corresponding to the resting tissue length and forces from load cells were converted to Cauchy stresses (C).
Forces from the load cells in the two orthogonal directions (F1 and F2 ) were amplified using an analog amplifier (Model 6600, Gould Amplifier, Valley View, OH). Cauchy stresses (T11 , T22 ) in the circumferential and axial directions were calculated as T11 = λ1
F1 hL20
(7a)
F2 . hL10
(7b)
and T22 = λ2
In the experiments reported here, samples were preconditioned at 10% stretch for 10 cycles in a triangular waveform at 0.05 Hz (Zhou and Fung, 1997) in both circumferential and axial directions. Constant I4 experiments were performed by stretching the tissue along one direction for different amounts (Table 1). In this method, the tissue was held stretched at a constant value in one direction and simultaneously the orthogonal set of arms were stretched out. This procedure was carried out for various incremental constant stretches for each sample to get a series of stress–strain data. All the data reported in this study correspond to the first loading cycle following preconditioning.
2.4.
Statistical analysis
Students’ paired t tests were performed to test for differences in the coefficients to the newly proposed strain energy function. A statistical level of p < 0.05 was considered significant. Square of the Pearson product moment correlation coefficient was also used in regression analysis for the different elastin samples.
3.
Results
3.1.
Constant I4 experiments
Fig. 1(B) shows a sample data set for an experiment in which one set of arms were maintained at a constant imposed stretch and the orthogonal stretcher arms moved out to a maximum displacement of 20% of the tissue length in the dynamic axis. Cauchy stresses recorded along the constant axis have non-zero values while those in the dynamic axis change from zero to a fixed maximum. Fig. 2 shows the
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Table 1 – Slopes of the stress–strain curves corresponding to the moving axis from all stretches on the sample. Specimen 4b Constant/ Slope (kPa) Dynamic (%)
Specimen 6a Constant/ Slope (kPa) Dynamic (%)
Specimen 7a Constant/ Slope (kPa) Dynamic (%)
Specimen 9b Constant/ Slope (kPa) Dynamic (%)
2/ 20 5/ 10 10/ 10 15/ 15
870.07 896.51 947.78 865.38
2/ 10 5/ 10 10/ 15 15/ 20 20/ 25
692.22 751.02 818.29 888.13 913.22
2/ 15 5/ 15 10/ 20 10/ 25
581.95 588.18 676.17 624.07
Equi-biax:5 Avg ± Stdev
897.95 895.54 ± 32.76 or 3.6%
Equi-biax:10
1106.8 861.61 ± 145.80 or 16.9%
Equi-biax:10
633.93 620.86 ± 38.15 or 6.1%
2/ 15 5/ 15 10/ 20 15/ 20 20/ 25 25/ 30 Equi-biax:10
576.73 605.87 630.72 686.03 678.3 681.3 774.28 661.89 ± 64.84 or 9.8%
Table 2 – Coefficients c0 , c1 and c2 in the new strain energy function (Eq. (8)) obtained using the Levenberg–Marquardt algorithm for all purified elastin samples in the study.
Fig. 2 – Data from experiments performed at various constant values of I4 shown for a representative arterial elastin sample (Specimen 9b). The different symbols in the figure are shown for various constant I4 experiments given in Table 2. Also shown in the table are the slopes of the curves.
Cauchy Stress vs. Green strain plots from various constant I4 experiments, each shown using a different symbol, for one representative elastin sample. The tissue underwent slight deformations (<5% of tissue length for a maximum stretch) along the axis of constant imposed stretch. All tissues showed a robust and repeatable response at these high values of strain for several different deformations. Additionally, the slopes of the Cauchy stress–Green strain curves corresponding to the dynamic axis for each sample were similar for various combinations of constant and varying imposed stretches (Table 1) indicating that there was minimal damage accumulation in the elastin tissue during different stretching protocols. As expected due to inherent biological variation, the slopes showed some variance between different samples. To estimate the partial derivatives of the strain energy function Wi , i = 1, 4, we used stress–strain data from various constant I4 stretching protocols. Fig. 3(A) represents a parameter map of constant I4 values for a representative sample. This plot also illustrates the dependence of Wi , i = 1, 4 on invariant I1 (n = 4). Different symbols are used
Sample
c0 (kPa)
c1 (kPa)
c2 (kPa)
4b 6a 7a 9b 11 11a Average Stdev
84.22 90.29 40.69 65.50 102.36 60.69 73.96 22.51
0.37 4.27 0 2.45 0.31 0.55 1.18 1.79
0 2.75 0 2.08 0 0.03 0.80 1.26
for experiments corresponding to a constant I4 stretching protocol for the same sample. From Fig. 3(A), we see that λ1 , λ2 → 1, W1 → ∞ (also see Eq. (4b)). For ease of plotting, some of these values are not shown in the graph. Also, independent of the actual value at which I4 was maintained, W1 approached a constant value with increasing I1 . As I4 increased, values of W4 became increasingly negative (Fig. 3(B)). Further, W4 values increased to a steady value, relatively independent of I1 , for a given I4 . Thus, as a first approximation we may assume that W has a linear dependence on invariant I1 ; that is W contains a neo-Hookean form with a linear dependence on the invariant I1 . The experimental data from the dynamic stretches corresponding to each constant λ2 experiment were used to estimate constant I1 values. Stresses and stretches for these values were used to calculate the partial derivatives of W with respect to invariants I1 and I4 . Fewer I1 values could however be obtained using this method as compared to the constant I4 experiments. W1 and W4 are shown plotted for constant I1 values in Fig. 4 for a representative sample. As I4 approaches the value of 1, I1 → 3.0 and the W1 plots approach very large numbers. There is a trend for linear relationship between W4 and I4 at constant I4 . However, the sparse data points obtained by this method could only be used to infer trends in the data. We will hence use data from constant I4 experiments to obtain the nature of this dependence. From the W4 plots in Fig. 3, we assume that the last point in each experiment used for the calculation of W4 represented the asymptotic value. This value is plotted as a function of I4 in Fig. 5. W4 shows a linear dependence on I4 for each elastin sample tested in the study, with values
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Fig. 3 – The dependence of W1 and W4 , calculated using Eqs. (4a) and (4b), shown as a function of I1 from various constant I4 experiments for specimen 9b. Each of the constant: moving stretch values are shown, using different symbols, for the various experiments performed on the same elastin sample.
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Fig. 5 – W4 as a function of I4 for all samples in the study. The final value of W4 was used in obtaining the plot. r2 values are shown to reflect the dependence of W4 on I4 (p < 0.005). elastin in arterial walls: 2 2 W = c0 I1 − 3 + c1 I4 − 1 + c2 I6 − 1 .
(8)
This equation presents a semi-empirical form of the strain energy function modified to include the biaxial properties of elastin networks.
3.2.
Estimation of constitutive coefficients
For planar materials described by Eq. (8), the Cauchy stresses in principal directions are given by T11 = 2c0 λ21 − λ23 + 4c1 λ41 (9a) T22 = 2c0 λ22 − λ23 + 4c2 λ42 . In equibiaxial tests, λ1 = λ2 = λ.
(9b)
For an incompressible material, we have from Eq. (1) 1 . (9c) λ2 We used data from equibiaxial studies on autoclaved elastin, presented in an earlier study (Gundiah et al., 2007), to determine the unknown coefficients to the new model for arterial elastin. To arrive at an optimal set of coefficients for the above non-linear multivariate equations, we used the Levenberg–Marquardt algorithm in MATLAB (The Mathworks, v7.0.1, Natick, MA). The algorithm allowed us to identify bestfit solutions to functions, obtained as a difference between experimental data and theoretically calculated stress in Eq. (9a). Fig. 6 shows a converged solution for a representative sample in the study and Table 2 shows a full set of the coefficients for all samples. From these studies, the coefficients are given by (average ± stdev) λ3 =
Fig. 4 – Results for the variation of W1 and W4 from the constant I1 condition for the same specimen 9b.
of r2 ranging from 0.909 to 0.999. Thus, the strain energy function has a second order dependence on I4 . Furthermore, because the orthogonal fibers of elastin have essentially the same mechanical behavior, we may extend the results from constant I4 to describe I6 ; thus incorporating the assumption of symmetry and mechanical equivalence of these two terms of the strain energy function. Finally, we must also impose the constraint that W = 0 for the unstretched case i.e. when I1 = 3, I4 = 1 and I6 = 1. From these results and constraints, we propose the following equation to represent the strain energy function of
c0 = 73.96c ± 22.51 kPa, c1 = 1.18 ± 1.79 kPa and c2 = 0.8 ± 1.26 kPa
(10)
c0 is an order of magnitude larger than coefficients c1 and c2 corresponding to invariants I4 and I6 , respectively. The difference in Student’s paired t-test between coefficients c1 and c2 (p = 0.68) is statistically insignificant.
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Fig. 6 – Fit of equibiaxial tension data (shown by symbols) for a representative elastin tissue using the new constitutive equation (8). The data were fit using a Levenberg–Marquardt algorithm for non-linear multivariate functions.
4.
Discussion
The data presented in this study investigates the material properties of elastin in the arterial networks for porcine arteries. To the best of our knowledge, these are the first comprehensive biaxial data on purified elastin networks and represent a step towards generating a constitutive model for arteries incorporating tissue microstructure. We have performed experiments within a theoretical framework (Rivlin and Saunders, 1951) to obtain an empirical form of the strain energy function for arterial elastin based on the orthotropic material symmetry of elastin fibers within arterial walls (Gundiah et al., 2007).
4.1.
The biomechanics of aortic elastin
Elastic fibers are present in static and dynamic tissues including skin, arteries and lung. These tissues, owing mainly to the presence of elastin networks, are able to endure several million cycles of cyclic deformation without undergoing appreciable fatigue (Gosline et al., 2002). During development, elastin fibers are deposited as tropoelastin monomers on microfibrillar scaffolds where they assemble and undergo cross-linking into amorphous, insoluble and hydrophobic elastin networks (Banga, 1966). The mechanics of elastin networks have been described using Gaussian rubber models (Hoeve and Flory, 1974). Local structurefunctional relationships within elastin have been explained using molecular equilibria phenomena (Debelle and Alix, 1999). Elastin networks have been modeled as random chain networks with a Gaussian distribution of end-to-end chain lengths between the cross-links (Hoeve and Flory, 1958). More recently, Urry and coworkers (Urry et al., 2002) used atomic force microscopy and acoustic absorption on single chain elastin molecules to show that elastin consists of regularly repeating ordered blocks of amino acids, with lysine derived cross-links, that deform on extension using entropic
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elasticity mechanisms. No study has to date investigated the biomechanics of elastin networks incorporating their underlying material symmetry. Uniaxial mechanical tests show that elastin is a protein rubber (Lillie and Gosline, 1990) with the modulus of single elastin fibers approaching 1.2 MPa (Aaron and Gosline, 1980). Using uniaxial mechanical data on porcine elastin, we showed in an earlier study that a neo- Hookean strain energy function is suitable to describe the mechanics of elastin networks whereas a more general Mooney–Rivlin form is not applicable. In this study, our biaxial experiments on elastin networks isolated from porcine arteries also show that a neo-Hookean term dominates the strain energy function for arterial elastin. Because the neo-Hookean model is the continuum mechanics equivalent of a material described by a Gaussian distribution, the constitutive model proposed here essentially agrees with the data from previous studies on elastin. In addition to the neo-Hookean term to the strain energy function, our experiments indicate the presence of two additional terms that arise from the distribution of elastic fibers in the artery. However, the contributions of the two terms resulting from material symmetry are not as substantial as the neo-Hookean term; demonstrating that the entropic contribution to the stored energy forms a dominant mechanism of elasticity in elastin networks.
4.2.
An isotropic neo-Hookean model for arterial elastin
Because purified elastin is very fragile, repetitive biaxial experiments were often difficult to perform. Nevertheless, repeated experimental protocols on the same specimen leave the slope of the stress–strain curves essentially unchanged (Fig. 4) thus confirming the robustness of elastin networks and elastin fibers. These experiments provide a strong empirical basis to models of elastin networks that use Gaussian statistics to describe the elasticity of the networks. A quadratic form of the strain energy function, referred as a standard reinforcing model, has frequently been employed to describe the reinforcing contribution of fiber reinforced materials (Qiu and Pence, 1997; Triantafyllidis and Abeyaratne, 1983). The mechanical results from our study also indicate that such a term would be appropriate to describe the mostly linear stress–strain curves exhibited by arterial elastin. However, results from our study show that the contributions to the strain energy function of terms in I4 and I6 , describing the actions of orthogonal fibers, are much smaller than the neo-Hookean term. Thus, the entropic contribution to the strain energy function is a dominant feature of elastin mechanics. Furthermore, because the I4 and I6 terms are not statistically different from one another, these suggest that the distributions of elastic fibers, and perhaps their cross-linking density, are the same in the circumferential and axial directions. Thus, the arterial microstructure aids in the equal distribution of stresses in the circumferential and axial directions during vessel loading. Based on the new strain energy function proposed here, we measured a modulus of 523 kPa for arterial elastin. This value is similar to the previously reported values of 900.8 ± 687 kPa for autoclaved arterial elastin (Gundiah et al., 2007). Using the coefficients in the Holzapfel model (Holzapfel et al., 1996)
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we get an elastin modulus of 308 kPa, and a value of 337 kPa using coefficients in the Zulliger et al. model (Zulliger et al., 2004). These latter results were however obtained by fitting equibiaxial data from arteries to a specific form of the strain energy function. Both these values are significantly smaller than other uniaxial results published for arterial elastin (Aaron and Gosline, 1980; Gosline, 1980; Gosline et al., 2002; Gundiah et al., 2007). In a recent study, Lillie and Gosline compared the moduli of autoclaved porcine elastin networks purified from the proximal and distal regions of the aorta (Lillie and Gosline, 2007). They suggest, based on higher moduli from the distal segments, that elastin networks are progressively more anisotropic away from the heart. Because collagen contents increase in distally located segments, a higher proportion of collagen remaining in the tested segments may be an unaccounted source of error in that study. In this study, we have not explored the regional differences in the biaxial mechanics of isolated elastin. Such experiments, performed over the entire range of tissue deformation, may however be used to quantify any such regional differences in the layout of elastin in arterial segments. However, results from this study indicate that any regional microstructural differences in the circumferential and axial layout may however not be as significant as the dominant neo-Hookean term.
4.3.
Limitations
Kinematical constraints like incompressibility used in this study help to reduce the total number of invariants by one. Earlier studies have shown that elastin has a hydrophobic core from which aqueous solvent is excluded (Robert et al., 1970). Dry elastin is brittle and plasticizing effects due to water are necessary to the network elasticity. Hydrophobic domains present between cross-linking domains in elastin exhibit high mobility, contributing greatly to the network entropy. Extensions of the elastin network hence cause the hydrophobic regions to come into contact with water. Consequently, the network absorbs water and disrupts the hydrophobic interactions (Gosline, 1980). It may hence be surmised that elastin does not deform at constant volume, an important premise in tissue biomechanics for samples in the physiological loading range. In this study, the phenomenon of water absorption of elastin fibers during stretching is considered a local microscopic event, which does not affect the continuum, and the incompressibility criterion is hence a first approximation. From a materials characterization perspective, Merodio and Ogden have shown that it is advantageous, from the point of ellipticity of the strain energy function, to neglect the contribution of invariant I8 that incorporates the contribution between the two fiber families from the constitutive equation (Merodio and Ogden, 2006). Although the microstructure may undergo slight deviations from orthotropy in the arteries (Clark and Glagov, 1985), it may however be advantageous to neglect the terms corresponding to I8 . We have used the results from Humphrey et al. (1990a,b) to reduce the dependence of the strain energy function from six to three invariants (I1 , I4 and I6 ). Nevertheless, the explicit form of W, dependent on three invariants, cannot be
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determined from biaxial experiments alone and additional experiments are required to assess the full functional relationship. In the study outlined here, we assume the mechanical equivalence of the two orthogonal fiber families based on earlier studies on porcine arterial elastin (Gundiah et al., 2007). We have hence assumed equivalence in terms I4 and I6 in this study. Any cross interaction terms in I4 and I6 or the combination with other invariants in the strain energy function cannot be assessed using this method and additional mechanical experiments will be necessary to study the applicability of the proposed function. The standard deviations obtained in the values of the coefficients c1 and c2 in the new constitutive model (Eqs. (8) and (10)) are very large (Table 2). These may be a result of any possible regional differences in the elastin samples and also inter sample variability. All samples were excised staring from 1 to 5 cm away from the heart. Future studies would benefit in accounting for any such variations due to the position of elastin along the arterial tree. Nevertheless, such terms are much smaller than the dominant neo-Hookean term in the model and any additional terms representing the regional differences may not be as important. We have assumed a stress-free ‘unloaded’ reference state for the tissues in this study. Residual strains, measured using a measure of the opening angle, are however present in arteries and would result as an addition to the strains in the elastin segments. Results from our lab on the measurement of residual stresses in purified elastin show that residual strains in purified elastin is negative causing the elastin ring to close in, rather than move away from the radial cut as is normal for arterial rings (Wang, 2004).
5.
Conclusions
We present a revised constitutive model for arterial elastin based on material symmetry and theoretically guided experiments and show that arterial elastin behaves essentially as a neo-Hookean material with added I4 and I6 terms in the constitutive equation due elastin layout in arteries. Our experiments suggest that the macroscopic elastic networks behave essentially as a neo-Hookean material, as has also been suggested by earlier models, albeit with minor modifications. The neo-Hookean term measured here is significantly higher than previously reported values obtained by the fitting of mechanical data to an assumed form of the strain energy function (Holzapfel et al., 1996; Zulliger et al., 2004).
Acknowledgements We would like to thank Dr. Sanjay Sane for many useful discussions and to Dr. Kay Sun for critically reading the manuscript. This study was funded by the NSF grant CMS0106010 to UC Berkeley and by NIH 2RO1 HL063348-05 to UC San Francisco.
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