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Mechanical anisotropy of inflated elastic tissue from the pig aorta
Journal of Biomechanics 43 (2010) 2070–2078
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Mechanical anisotropy of inflated elastic tissue from the pig aorta M.A. Lillie n, R.E. Shadwick, J.M. Gosline Department of Zoology, University of British Columbia, Vancouver, B.C., Canada V6T 1Z4
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 9 April 2010
Uniaxial and biaxial mechanical properties of purified elastic tissue from the proximal thoracic aorta were studied to understand physiological load distributions within the arterial wall. Stress–strain behaviour was non-linear in uniaxial and inflation tests. Elastic tissue was 40% stiffer in the circumferential direction compared to axial in uniaxial tests and 100% stiffer in vessels at an axial stretch ratio of 1.2 or 1.3 and inflated to physiological pressure. Poisson’s ratio vyz averaged 0.2 and vzy increased with circumferential stretch from 0.2 to 0.4. Axial stretch had little impact on circumferential behaviour. In intact (unpurified) vessels at constant length, axial forces decreased with pressure at low axial stretches but remained constant at higher stretches. Such a constant axial force is characteristic of incrementally isotropic arteries at their in vivo dimensions. In purified elastic tissue, force decreased with pressure at all axial strains, showing no trend towards isotropy. Analysis of the force–length–pressure data indicated a vessel with vyz E 0.2 would stretch axially 2–4% with the cardiac pulse yet maintain constant axial force. We compared the ability of 4 mathematical models to predict the pressure-circumferential stretch behaviour of tethered, purified elastic tissue. Models that assumed isotropy could not predict the stretch at zero pressure. The neo-Hookean model overestimated the non-linearity of the response and two non-linear models underestimated it. A model incorporating contributions from orthogonal fibres captured the non-linearity but not the zero-pressure response. Models incorporating anisotropy and non-linearity should better predict the mechanical behaviour of elastic tissue of the proximal thoracic aorta. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Elastin Mechanical properties Anisotropy Poisson’s ratio Constitutive model
1. Introduction Forces on an artery wall are borne primarily by collagen and elastic fibres, the latter comprised of elastin and microfibrils. The three-dimensional load partitioning amongst these elements is closely modulated through arterial growth and remodeling; so accurately representing the contributions of each element improves the predictive ability of computer models and hence our understanding of mechanobiology. Arteries are generally considered anisotropic. This property is usually attributed to collagen (Holzapfel et al., 2000; Zulliger et al., 2004; Gasser et al., 2006), though both isotropic (Gundiah et al., 2009) and anisotropic (Sherebrin, 1983; Zou and Zhang, 2009) behaviours have been reported for elastic tissue. However, it is important to differentiate between inherent anisotropy due to structural features of an unstressed material, such as an unequal partitioning of orthogonal fibres, and induced anisotropy due to superimposed stresses on a non-linear material in which moduli depend on loading conditions (Dobrin, 1978). Arteries approach incremental isotropy under physiological loads (Dobrin, 1986; Weizsacker and Pinto, 1988), and therefore it is important to n
Corresponding author. Tel.: + 1 604 822 2373; fax: +1 604 822 2416. E-mail address: [email protected] (M.A. Lillie).
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.04.014
assess the elastic tissue under physiological conditions. In this paper we assessed the mechanical behaviour of purified elastic tissue over a range of strains, including physiological. For the proximal pig thoracic aorta, physiological circumferential stretch is 1.2–1.26 (Stergiopulos et al., 2001; Guo and Kassab, 2004), and axial stretch is around 1.2 (Han and Fung, 1995). We used uniaxial tests to identify inherent properties to establish the importance of tissue structure, and inflation tests to study the sensitivity to orthogonal loading conditions. We found elastic tissue from the proximal pig thoracic aorta was anisotropic under physiological strains and attribute this in part to the inherent structure of elastic tissue.
2. Methods Descending thoracic aortas from 100 kg pigs (5–6 month old) were collected from an abattoir, transported in iced phosphate buffered saline, PBS, and frozen in PBS at 20 1C. Thawed aortas were cleaned of loosely adhering tissue. Tissue more than 40 mm distal to the first intercostal artery was excluded. Elastic tissue was purified by 8 h autoclaving (Lillie et al., 1994). 2.1. Uniaxial tests 2.1.1. Compressibility Although elastic fibres are incompressible (Gosline, 1978), purified elastic tissue is porous and may collapse radially on deformation. We therefore examined
M.A. Lillie et al. / Journal of Biomechanics 43 (2010) 2070–2078 volume changes in circumferential strips of elastic tissue under tensile loads using an Instron 5500 apparatus. Reference marks were placed midway on the strip for determining circumferential (ly), axial (lz) and radial stretch (lr). Strips were mounted in air beside a mirror positioned at 451 for simultaneous viewing of the y–z and y r surfaces. Vessels were photographed with a 10 Mpix camera at 5% extension increments. Relative volume was V ¼ lylzlr. Radial deformation under compression was compared in rings of intact (not purified) and elastic tissue mounted around two bars and stretched circumferentially to compress the tissue at the bars (inset Fig. 1B). Radial deformation was measured at one bar using a custom-built displacement transducer. 2.1.2. Poisson’s ratio Strips of elastic tissue were stretched circumferentially. Incremental Poisson’s ratio was calculated as nzy ¼ Dez =Dey , where De is the incremental strain. For a generic length, ‘, De ¼ 2ð‘2 ‘1 Þ=ð‘2 þ ‘1 Þ, where the subscripts represent different load levels.
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Photos were digitized using ImageJ (Abramoff et al., 2004) to obtain tissue dimensions. Wall thickness, H, was calculated from rings, assuming circular crosssection. The mid-wall radius in the inflated vessel, r, was calculated assuming incompressibility qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ r ¼ ro2 ðR2o R2 Þ=lz where ro and Ro are the outer radii, the upper case referring to the unloaded dimension and the lower case to the deformed. lz ¼ l/L is the axial stretch at the midpoint of each segment, where L and l are the local segment lengths. We assumed circumferential strain was uniform around the segment and varied linearly across the wall. Mid-wall stretch ratio was ly ¼r/R. For the analysis we assumed the aortas were thin-walled, orthotropic, pseudoelastic and homogeneous, and that the axial stress was uniform. Circumferential and axial residual strains for purified elastic tissue from the proximal aorta are under 1% (Lillie and Gosline, 2006) and have been neglected. Circumferential and axial Cauchy stresses were calculated as Pri h P pri2 þ F sz ¼ pðro2 ri 2 Þ
sy ¼
2.1.3. Anisotropy Elastic tissue was cut into a 25 mm square, or a cross with 8 mm wide arms aligned with the orthogonal axes and held in 8 mm wide grips. Samples were preconditioned and then tested four times in alternating directions. Neither the sample shape nor the testing order had an impact on anisotropy. Stress, s, was force divided by the deformed cross-sectional area, A. Incremental modulus was calculated as E ¼ Ds/De at 2% stretch increments.
where ri is the inner radius. We assumed the tissue behaved linearly over small strains, and calculated incremental circumferential and axial strains following Dobrin and Doyle (1970):
2.2. Inflation tests
Dey ¼
Straight segments from 14 autoclaved aortas were tested. Of these, 4 were also tested intact. A grid was painted along each aorta for measuring local strain. The ends of the aorta were cannulated, and the aorta was mounted horizontally in PBS at 37 1C. The proximal end was fixed in position and connected to a pressure reservoir. The distal end was either free floating or fixed via a low-torsion coupling to a force transducer to measure axial load, F. The position of the distal mount could be varied to set the axial strain. The segment was preconditioned and then inflated to 14 kPa in increments of 1.3 kPa with a total loading time of about 2 min. The aortas were photographed at each pressure, P. Tests were run with the aorta untethered and tethered at 4–17 axial stretches between 1.02 and 1.46. After the inflation tests, a ring was cut from the middle of each vessel and photographed in PBS at 37 1C to measure circumference and wall thickness.
Dez ¼
Dsy Ey
Dsz Ez
nyz nzy
ð2a; bÞ
Dsz Ez
Dsy
ð3a; bÞ
Ey
Radial stresses averaged 5% of the circumferential stresses and were omitted from Eq. (3). Paired orthogonal moduli, Ey and Ez, were determined under conditions of same mean stress. The process is illustrated in Fig. 2. Curves were taken from 2 inflation cycles, in this example one at lz ¼ 1.19 and one at lz ¼ 1.2. We selected two points that fell at the same sz (P1 and P2). Since each point represents a specific sy and associated ey, P1 and P2 are separated by Dsy and Dey. High order polynomials were fit to the stress–strain data to obtain each ey. When the data are selected such that Dsz ¼ 0, Eq (3a) yields Ey ¼
Dsy Dey
ð4aÞ
Fig. 1. (A) Compressibility of strips of elastic tissue under circumferential tension. Volume decreased by 2.7% at the initial extension (P 50.001), but remained constant at higher extensions. (B) Relative compressibility of intact and elastic tissue under radial compression. Inset shows tissue ring loaded by movement of lower mounting bar with thickness measured at upper bar. Both data sets start at a radial stretch of 1.00 at a compressive load of zero. Between 1 and 40 kPa, the compression of the elastic tissue paralleled that of the intact. This is shown by the solid line, which gives the elastic tissue data shifted down by 3% for clarity. Broken lines show the response of rabbit thoracic aortas under uniaxial compression from Chuong and Fung (1984).
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Radial deformation was directly measured in intact (n ¼10) and elastic tissue (n ¼12; Fig. 1B). Our loading protocol had two limitations: stress distribution in the tissue was complex (not uniaxial compression), and there was likely friction between the tissue and supporting bar. However, the response of our intact tissue was comparable to that of intact rabbit aorta under uniaxial compression (Chuong and Fung, 1984) (Fig. 1B), indicating the loading at the point of measurement was essentially compressive. Further, the friction coefficients for intact and elastic tissue are likely similar; so although the absolute radial deformations may have been constrained, the relative response in these tissues should be largely unaffected. The elastic tissue thickened under minimal radial loads ( 1 kPa; Fig. 1B), and we cannot explain this response. However its deformation between 1 and 40 kPa paralleled that of the intact tissue, indicating equal incremental radial stiffness in intact and elastic tissue. The data in Fig. 1A and 1B show that elastic tissue can be treated as incompressible. 3.1.2. Poisson’s ratio The mean incremental Poisson’s ratio, nzy, measured in circumferential strips stretched to ly ¼1.2 was 0.337 0.02 (n ¼23).
Fig. 2. Calculation of moduli using Eq (4). Axial vs. circumferential stress data are shown for inflation at two axial stretches, lz ¼ 1.19 (closed circles) and 1.20 (open). Intermediate values were obtained by interpolation with a fitted polynomial (lines). Ey was evaluated at points P1 and P2 (closed squares), which are at the same axial stress but at different circumferential stresses and associated with different circumferential strains ey1 and ey2 . The strains ey1 and ey2 were obtained from stress–strain plots. The corresponding Ez was evaluated at points P3 and P4 (open squares), which are at the same circumferential stress but at different axial stresses and associated with different axial strains, ez3 and ez4. Mean stresses for P1 and P2 equal those for P3 and P4, shown by ‘‘x’’.
Similarly for points P3 and P4 in Fig. 2, for which Dsy ¼0, Eq (3b) yields Ez ¼
Dsz Dez
3.2. Inflation tests ð4bÞ
Poisson’s ratios in axially tethered vessels were calculated following Dobrin and Doyle (1970):
nyz ¼
Dsz Dsy
nzy ¼ nyz
Ey Ez
3.1.3. Anisotropy Behaviour of elastic tissue in uniaxial tests is shown in Fig. 3. Each sample was tested circumferentially and axially. Fig. 3A shows the force-stretch response for one sample undergoing a load/unload cycle. Resistance to deformation was greater along the circumferential axis compared to axial. Both circumferential and axial moduli increased with strain (Fig. 3B). The rapid increase in moduli between stretches of 1.0 and 1.1 corresponds to the initial toe region visible in Panel A. Beyond the toe region, the mean anisotropy, defined as Ey/Ez at equal stretches, was 1.4170.04 (n ¼15).
ð5Þ
ð6Þ
2.3. Statistical analysis Statistics were done with SigmaStat 2.03. A value for P of 0.05 was considered significant. Results are given as mean 7 SE.
3. Results Results are for purified elastic tissue unless specified for intact. 3.1. Uniaxial tests 3.1.1. Compressibility Relative volume of strips of elastic tissue under circumferential tension was calculated from the three orthogonal stretch ratios. There was a significant 2.770.3% loss of volume at the initial extension (Fig 1A), but the flat regression line indicates there was no further volume change at higher extensions (P50.001 linear regression constant, n ¼206, 22 samples).
Dimensions of the unloaded vessel were measured at the start and end of the tests to check for damage. Post-test length was 0.99570.007 of pretest, and post-test diameter was 1.00570.009. Neither changed significantly (paired t-test). 3.2.1. Untethered vessels Untethered vessels increased in both length and circumference when inflated, stretching about 60% more in the circumferential direction compared with the axial (Fig. 4). At a pressure of 13.3 kPa, ly averaged 1.2870.01 and lz 1.16 70.01 (n¼14). For intact vessels, ly averaged 1.23 70.02 and lz 1.18 70.01 (n ¼4). 3.2.2. Tethered vessels The mechanical properties of vessels tethered at lz ¼1.2 (n ¼14) and 1.3 (n ¼4) are shown in Figs. 5–8. Circumferential stretch increased non-linearly with pressure (Fig. 5). At zero pressure the circumferential stretch for lz ¼1.2 averaged 0.97, larger than the 0.91 predicted for isotropy (i.e., ly ¼1.2 0.5). In vessels tethered at lZ ¼1.2 and pressurized to 13.3 kPa, circumferential stretch averaged 1.26 70.01, which is near the physiological value, and at lZ ¼1.3 circumferential stretch averaged 1.3270.02. The circumferential modulus increased with inflation, but the axial modulus was essentially constant (Fig. 6). Increasing the axial stretch from 1.2 to 1.3 increased both moduli; around physiological circumferential stretches, the moduli increased 9%, which is accounted for by an 8% drop in wall
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Fig. 3. Behaviour of elastic tissue in uniaxial tests. (A) Force-stretch response for a single sample undergoing one load/unload cycle showing a greater resistance to deformation along the circumferential axis compared to axial. (B) Circumferential and axial moduli (mean 7SE) demonstrating non-linear behaviour. The tissue averaged 41% greater stiffness in the circumferential direction compared to axial.
Fig. 4. Inflation of untethered elastic tissue segments. Vessels increased in both diameter and length. Values are mean 7SE. Axial stretch is about 60% of circumferential.
Fig. 5. Inflation of elastic tissue segments tethered at an axial stretch of 1.2 (closed symbols) or 1.3 (open). The circumferential stretch at zero pressure is 0.97 for an axial stretch of 1.2, which is larger than the 0.91 predicted for an isotropic material. Values are mean 7 SE.
thickness. At very low pressure, the tissue was close to transverse isotropy, but it became anisotropic with further inflation (Figs. 6 and 7). At lZ ¼ ly ¼1.20, anisotropy averaged 1.870.1, higher than
the uniaxial value of 1.41 (P¼ 0.003). Both Poisson’s ratios increased with circumferential stretch: vyz remained near 0.2, but vzy increased from below 0.2 to over 0.4 (Fig. 8).
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Fig. 6. Moduli calculated from Eq (4) for inflated elastic tissue segments tethered at an axial stretch of 1.2 (circles) or 1.3 (squares). Axial modulus, Ez, was relatively constant while circumferential modulus, Ey, increased with inflation. Increasing axial stretch increased both moduli. Values are mean 7 SE.
Fig. 8. Poisson’s ratios for inflated elastic tissue segments. Solid data points with solid regression line show nyz for vessels tethered at lz ¼ 1.2 calculated from Eq (5). Open data points with broken regression line show nyz for vessels tethered at lz ¼ 1.3. Line with capped error bars shows nzy for lz ¼ 1.2 calculated with Eq (6), and line with non-capped error bars shows nzy for lz ¼1.3. The square at ly ¼ 1.2 shows the mean Poisson’s ratio, nzy, measured in the uniaxial tests.
Increasing the axial stretch increased the starting axial force but had little effect on curve shape: the curves were largely congruent when offset vertically to set the force at zero pressure to the arbitrary value of 2 N (Fig. 9B), and they displayed no trend towards the horizontal with increasing lz. This response was observed in all elastic tissue vessels tested, up to a maximum stretch of 1.46, and contrasts with that of intact aortas, illustrated in Fig. 9D by 19 curves from a single vessel. Nine curves at the bottom at axial stretches from 1.05–1.19 are indistinguishable, but from lZ ¼1.20–1.39 the slope progressively diminished. In the 4 intact aortas, force became independent of pressure around lZ ¼1.35. This is the ‘‘crossover’’ point in a force–length plot and generally corresponds to the axial stretch in vivo (see Discussion).
4. Discussion 4.1. Anisotropy of aortic elastic tissue
Fig. 7. Anisotropy of elastic tissue segments tethered at an axial stretch of 1.2 (closed circles) and 1.3 (open circles). The vessels were transversely isotropic at low circumferential stretches but became progressively more anisotropic with further inflation. Increasing axial stretch had no impact on anisotropy. At a circumferential stretch of 1.2, the mean anisotropy was 1.8 in the inflation tests (triangle), larger than the 1.4 obtained in the uniaxial tests (square).
3.2.3. Applied axial force The applied axial force on the elastic tissue vessels decreased with pressure. Fig. 9A shows the response in one vessel tested from lZ ¼1.03 (bottom) to 1.36 (top) in 2–3% increments.
Intact arteries are anisotropic, but because their behaviour is non-linear they can attain transverse isotropy (Ey ¼Ez aEr) at certain pairings of axial and circumferential stretch and can approach full isotropy (Ey ¼Ez ¼Er) near physiological stretches (Cox, 1975; Cox, 1978; Dobrin, 1986; Weizsacker and Pinto, 1988; Weizsacker and Kampp, 1990). By contrast, we found elastic tissue was anisotropic under most loading conditions. Our uniaxial tests showed elastic tissue possessed an inherent anisotropy, with a circumferential stiffness 1.4 times axial (Fig. 3). Similar values were obtained in uniaxial tests of alkalipurified elastic tissue from dog and sheep aortas (Sherebrin, 1983). Elastic tissue behaviour was non-linear in uniaxial tests (Fig. 3B), and therefore the inherent anisotropy exhibited under uniaxial loading can change under biaxial loading. Zou and Zhang (2009) found anisotropy persisted under biaxial loading, but Gundiah et al. (2009) obtained transverse isotropy. We also obtained transverse isotropy in our inflation tests of vessels
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Fig. 9. Dependence of applied axial force on pressure and axial strain in an elastic tissue segment (A–C) and in an intact (unpurified) segment (D). (A) The data show behaviour from lz ¼ 1.03 (bottom) to 1.36 (top) in 2–3% increments. Increasing axial stretch shifted the curves up with little change in shape. (B) Each curve in A has been shifted vertically to set the force at zero pressure to the arbitrary value of 2 N. The curves are nearly congruent and show no trend towards the horizontal with increasing lz. (C) The data from A have been replotted as pressure–axial stretch curves. Each line gives behaviour at a constant axial force, shown in 0.5 N increments. For example, the solid squares in panels A and C show combinations of axial stretch and pressure that will produce an axial force of 4 N. Lines are largely linear with steeper slopes at lower axial stretches. For a physiological pressure pulse of 5.3 kPa, the axial force would remain constant if the vessel stretched 2% (lower lz) to 4% (higher lz). (D) Data from an intact aorta from lz ¼ 1.05 (bottom) to 1.39 (top). The curves have been shifted vertically to set the force at zero pressure to 2 N. Slope is constant for 9 curves from lz ¼1.05–1.19. At higher axial strains, slope becomes less negative and is horizontal by lz ¼ 1.35.
tethered at axial stretches of 1.2–1.3, but only at the lowest circumferential stretches (Figs. 6 and 7). Around physiological stretch (ly 1.25), total anisotropy (inherent and induced) averaged 2 (Fig. 7). Increasing the axial stretch from 1.2 to 1.3 had no impact on the anisotropy. We have not measured axial stretches in vivo, but Han and Fung (1995) measured stretches around 1.2 in situ in unpressurized proximal thoracic aortas, and
our force–pressure data from intact vessels indicate a value around 1.35 (Fig. 9D). Thus, the stretches examined in our inflation tests are germane to physiological behaviour, and our results indicate elastic tissue is significantly anisotropic near physiological stretches. The striking difference between intact and elastic tissue is that at higher axial stretches intact tissue progresses towards full
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isotropy, but elastic tissue does not. We did not measure anisotropy in our intact tissues, but a shift to isotropy produces a distinctive pattern in the axial force–pressure curves: at low axial stretches the curves slope downwards, but with increased axial stretch the slopes tend to level off, and they become horizontal near the in vivo axial stretch (Van Loon et al., 1977; Weizsacker et al., 1983). This pattern was exhibited by intact aortas in this study (Fig. 9D), but not in elastic tissue (Fig. 9B), even at stretches up to 1.46. Constant axial force at physiological stretch occurs where nyz ¼ 0.5, which is its value at isotropy (Weizsacker and Pinto, 1988; Brossollet and Vito, 1995). The dependence of axial force on pressure can be calculated from Eqs. (2), (3b) and (6): F ¼ Ez pðro2 ri2 Þez þ Ppri2 ð2nyz 1Þ
ð7Þ
A similar equation is given by Brossollet and Vito (1995) for an isotropic vessel. For a fixed-length vessel, the first term on the right is constant, and it can be readily verified that force will remain constant with pressure if nyz ¼0.5 and fall if nyz o0.5. For elastic tissue, nyz remained around 0.2 (Fig. 8) and forces fell with pressure (Fig. 9B). The similarity of the force–pressure curves in Fig. 9B (elastic tissue) and 9D (intact tissue) indicates that nyz in intact aortic tissue must also be around 0.2 at axial stretches from lz ¼1.05–1.19. Axial forces are more complex in a variable-length vessel. At least in the dog, the thoracic aorta is not fixed in length but exhibits pressure-driven axial strains of 1–5% (Patel et al., 1961; Patel and Fry, 1964). Such strains would raise the axial force in an isotropic vessel, and we can use the data in Fig. 9A to estimate their impact on the axial force in an anisotropic, elastic tissue vessel. We assume the behaviour of purified elastic tissue represents its behaviour in vivo. Axial force depends positively on lz but negatively on pressure. We recast the force–pressure data in Fig. 9A as pressurestretch data in 9C, where each curve represents a constant axial force. The curves appear linear, and their slopes show that an aorta that stretched 2–4% under a pressure pulse of 5.3 kPa would experience a constant axial force. Thus with nyz E0.2, an anisotropic vessel could stretch axially yet maintain a constant axial force. Interestingly, Patel et al. (1969) found the dog thoracic aorta behaved anisotropically in vivo, with nyz E0.27. The in vivo response of the pig aorta depends on how the collagen and perivascular tissue modify the elastic tissue behaviour, and this must be determined. A constant axial force at in vivo length appears to be a fundamental property of arteries, basic to the premise that arteries are isotropic at physiological loads (Brossollet and Vito, 1995). Patel’s work on the thoracic aorta and our analysis of aortic elastic tissue indicate that axial forces may also remain constant in anisotropic arteries of varying length if nyz is low. 4.2. Modeling the inflation response of arterial elastic tissue Both neo-Hookean and non-linear strain energy functions, C, have been used to model arterial elastic tissue behaviour. Here we assess the ability of four models to predict the pressure-stretch response in Fig. 5: Cn-H, the neo-Hookean model; CZ, a widely used non-linear model proposed by Zulliger et al., (2004); CD, an exponential model introduced by Demiray (1972) and suggested by Ogden and Saccomandi (2007) to model elastic tissue with limiting chain extensibility, and CGRP, a semi-empirical model that incorporates specific contributions from orthogonal fibres (Gundiah et al., 2009).
CnH ¼ cnH ðI1 3Þ CZ ¼ cZ ðI1 3Þ1:5
cD exp bðI1 3Þ 1 b CGRP ¼ c0 ðI1 3Þ þ c1 ðI4 1Þ2 þc2 ðI6 1Þ2
CD ¼
ð8a2dÞ
where cn–H, cZ, cD, c0, c1, and c2 are material constants, b is a nondimensional parameter, and I1, I4, and I6 are the strain invariants of the right Cauchy-Green tensor. Assuming incompressibility, 2
2
2 2
I1 ¼ ly þ lz þ ly lz 2
I4 ¼ lz
ð9a2cÞ
2
I6 ¼ ly
The pressure-stretch relationships predicted by each model have been derived using Eqs. (8) and (9) and P¼
1 H @C ly lz R @ly
ð10Þ
given by Ogden and Saccomandi (2007). ! 2cnH H 1 1 4 2 PnH ¼ Rlz ly lz ! 3cZ H 1 0:5 ðI1 3Þ 1 4 2 PZ ¼ Rlz ly lz ! 2cD H 1 exp bðI1 3Þ 1 4 2 PD ¼ Rlz ly lz ! " # 2H 1 c0 1 4 2 þ 2c2 ðI6 1Þ PGRP ¼ Rlz l l
ð11a2dÞ
y z
The lines in Fig. 10 show the pressures predicted by Eq. 11(a–d). Parameter values were selected using least squares to fit the curvature of the data (symbols) between 0 and 5 kPa. The neoHookean, Zulliger, and Demiray models assume isotropy and therefore predict a Poisson’s ratio vyz near 0.5 and hence a zero pressure at ly ¼0.91. Our data start at ly ¼0.97, in accord with the Poisson’s ratio nyz around 0.15 shown in Fig. 8, and so these models fail to capture the response of the tissue. The neo-Hookean model overestimated the curvature of the data, while the Zulliger model underestimated it. The response predicted by these two models for elastic tissue under uniaxial loading showed similar deviations from experimental data (Watton et al., 2009). The Demiray model predicts neo-Hookean behaviour for small values of parameter b (not shown), which overestimates curvature, and higher b values (Fig. 10) introduced a convexity at higher strains that was absent in the inflation data. Convexity is present in single elastic fibres at high stretches (Aaron and Gosline, 1981). The Gundiah model does not assume isotropy, and the predicted stretch at zero pressure depends on parameter values. Gundiah et al. (2009) obtained a low empirical value for c2 (c0 ¼74, c2 ¼0.8), and since the model reverts to the neo-Hookean for c2 ¼0, they concluded neo-Hookean behaviour dominated the elastic tissue response. In Fig. 10 we allowed a much greater contribution from the I6 term by using c0 ¼33 and c2 ¼15. This allowed the Gundiah model to capture the curvature of our data, although the anisotropy remained insufficient to predict the zero-pressure response. None of these models adequately predicts the response of the elastic tissue from the proximal thoracic aorta. The mismatch between the observed and modeled responses visible in the zero pressure radii is by definition a mismatch between the observed and predicted Poisson’s ratios, nyz —the decrease in radius obtained on increase in length under a uniaxial load. Residual stresses and strains remain in an uncut artery under no load, but it is unlikely that they could account for the observed low Poisson’s ratio. First, circumferential, axial and radial residual strains in purified elastic tissue from the proximal aorta are small: 0.006, 0, and 0.008, respectively (Lillie and Gosline, 2006), and referring the measured stretches to the zero stress rather than the no load state marginally reduces the Poisson’s ratio further. Second, the value of nyz is largely insensitive to applied strain, whether circumferential (ly ¼0.95–1.35, Fig. 8) or axial
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indicate mesoscopic tissue organization is important. Third, elastic tissue behaviour differs substantially with position, even within the thoracic aorta (Lillie and Gosline, 2007; Zou and Zhang, 2009). Elucidating the physical and physiological underpinnings of these positional differences will further our understanding of arterial mechanobiology and our ability to model the behaviour of arteries from all sources.
Conflict of interest statement There is no conflict of interest regarding the publication of the manuscript ‘‘Mechanical Anisotropy of Inflated Elastic Tissue from the Pig Aorta’’.
Acknowledgements This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council to J.M.G. and by a Discovery Accelerator Grant from the Natural Sciences and Engineering Research Council to R.E.S. References Fig. 10. Modeling the circumferential response to inflation of elastin segments tethered at lz ¼1.2. The symbols show the data from Fig. 5. Lines show the pressures predicted by 4 models given in Eq (11a–d). Parameter values were selected to fit the curvature of the data between 0 and 5 kPa. At zero pressure the mean experimental value for ly was 0.97. The neo-Hookean, Zulliger, and Demiray models assume isotropy, which predicts zero pressure at ly ¼ 0.91, and so these models fail to capture that behaviour of the tissue. The Gundiah model does not assume isotropy and predicts a slightly better fit at zero pressure. The Gundiah model captured the curvature of the response shown by the data. The neoHookean model overestimates the curvature while the Zulliger and Demiray models underestimate it. The mean value for H/R for the tissues in our study was 0.21. Model parameter values were cn–H ¼42 kPa, cZ ¼77 kPa, cD ¼ 33 kPa, b ¼1.6, c0 ¼ 33 kPa and c2 ¼ 15 kPa.
(lz ¼1.2–1.3, Fig. 8 and lz ¼1.03–1.46, inferred from Fig. 9B in Discussion), and it should therefore remain unaffected by small residual strains. Third, the value of nyz at low axial stretches in the intact aorta (Fig. 9D) and in other arteries (Patel et al., 1969; Weizsacker et al., 1983; Dobrin, 1986) also falls around 0.2, even though residual strains are substantially greater in intact arteries (Greenwald et al., 1997; Zeller and Skalak, 1998; Lillie and Gosline, 2006). Thus the low values for nyz observed in both intact arteries and purified elastic tissue over a range of loading conditions arise from the structure of the elastic tissue. In an incompressible material nyz depends on the orthogonal moduli and its value will be low when Ey 4Er 1 1 þ0:5: ð12Þ nyz ¼ 0:5Ez Ey Er (Weizsacker and Pinto, 1988). Thus in addition to the anisotropy in the y z plane demonstrated in Fig. 7, there is also elastic tissue anisotropy in the y r plane. Future research must identify what physical parameters need be experimentally measured and incorporated into the models to improve their predictive ability. First, they require a more general treatment to accommodate the three-dimensional anisotropy, perhaps by incorporating elastic fibre orientation, as is done for veins (Rezakhaniha and Stergiopulos, 2008), or elastin chain orientation (Zou and Zhang, 2009). Second, the basis of the non-linearity must be identified. The Demiray and Zou models focus on the molecular level, but differences between single fibre (Aaron and Gosline, 1981) and elastic tissue behaviour
Aaron, B.B., Gosline, J.M., 1981. Elastin as a random-network elastomer—a mechanical and optical analysis of single elastin fibers. Biopolymers 20 (6), 1247–1260. Abramoff, M., Magelhaes, P.J., Ram, S.J., 2004. Image processing with image. Journal of Biophotonics International 11 (7), 36–42. Brossollet, L.J., Vito, R.P., 1995. An alternate formulation of blood-vessel mechanics and the meaning of the in-vivo property. Journal of Biomechanics 28 (6), 679–687. Chuong, C.J., Fung, Y.C., 1984. Compressibility and constitutive equation of arterialwall in radial compression experiments. Journal of Biomechanics 17 (1), 35–40. Cox, R.H., 1975. Anisotropic properties of canine carotid-artery invitro. Journal of Biomechanics 8 (5), 293–300. Cox, R.H., 1978. Passive mechanics and connective-tissue composition of canine arteries. American Journal of Physiology —Heart & Circulatory Physiology 234 (5), H533–H541. Demiray, H., 1972. Elasticity of soft biological tissues. Journal of Biomechanics 5 (3), 309–311. Dobrin, P.B., 1978. Mechanical-properties of arteries. Physiological Reviews 58 (2), 397–460. Dobrin, P.B., 1986. Biaxial anisotropy of dog carotid artery: estimation of circumferential elastic modulus. Journal of Biomechanics 19 (5), 351–358. Dobrin, P.B., Doyle, J.M., 1970. Vascular smooth muscle and anisotropy of dog carotid artery. Circulation Research 27 (1), 105–119. Gasser, T.C., Ogden, R.W., Holzapfel, G.A., 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the Royal Society Interface 3 (6), 15–35. Gosline, J., 1978. Hydrophobic interaction and a model for the elasticity of elastin. Biopolymers 17 (3), 677–695. Greenwald, S.E., Moore Jr., J.E., Rachev, A., Kane, T.P., Meister, J.J., 1997. Experimental investigation of the distribution of residual strains in the artery wall. Journal of Biomechanical Engineering 119 (4), 438–444. Gundiah, N., Ratcliffe, M.B., Pruitt, L.A., 2009. The biomechanics of arterial elastin. Journal of the Mechanical Behavior of Biomedical Materials 2 (3), 288–296. Guo, X., Kassab, G.S., 2004. Distribution of stress and strain along the porcine aorta and coronary arterial tree. American Journal of Physiology—Heart & Circulatory Physiology 286 (6), H2361–H2368. Han, H.C., Fung, Y.C., 1995. Longitudinal strain of canine and porcine aortas. Journal of Biomechanics 28 (5), 637–641. Holzapfel, G.A., Gasser, T.C., Ogden, R.W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity 61 (1–3), 1–48. Lillie, M.A., Chalmers, G.W.G., Gosline, J.M., 1994. The effects of heating on the mechanical properties of arterial elastin. Connective Tissue Research 31, 23–35. Lillie, M.A., Gosline, J.M., 2006. Tensile residual strains on the elastic Lamellae along the porcine thoracic aorta. Journal of Vascular Research 43, 587–601. Lillie, M.A., Gosline, J.M., 2007. Mechanical properties of elastin along the thoracic aorta in the pig. Journal of Biomechanics 40 (10), 2214–2221. Ogden, R.W., Saccomandi, G., 2007. Introducing mesoscopic information into constitutive equations for arterial walls. Biomechanics and Modeling in Mechanobiology 6 (5), 333–344.
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Patel, D.J., Fry, D.L., 1964. In situ pressure-radius-length measurements in ascending aorta of anesthetized dogs. Journal of Applied Physiology 19 (3), 413–416. Patel, D.J., Janicki, J.S., Carew, T.E., 1969. Static anisotropic elastic properties of aorta in living dogs. Circulation Research 25 (6), 765–779. Patel, D.J., Mallos, A.J., Fry, D.L., 1961. Aortic mechanics in the living dog. Journal of Applied Physiology 16, 293–299. Rezakhaniha, R., Stergiopulos, N., 2008. A structural model of the venous wall considering elastin anisotropy. Journal of Biomechanical Engineering—Transactions of the ASME 130 (3), 031017. Sherebrin, M.H., 1983. Mechanical anisotropy of purified elastin from the thoracic aorta of dog and sheep. Canadian Journal of Physiology and Pharmacology 61 (6), 539–545. Stergiopulos, N., Vulliemoz, S., Rachev, A., Meister, J.J., Greenwald, S.E., 2001. Assessing the homogeneity of the elastic properties and composition of the pig aortic media. Journal of Vascular Research 38 (3), 237–246. Van Loon, P., Klip, W., Bradley, E.L., 1977. Length-force and volume-pressure relationships of arteries. Biorheology 14 (4), 181–201.
Watton, P.N., Ventikos, Y.A., Holzapfel, G.A., 2009. Modelling the mechanical response of elastin for arterial tissue. Journal of Biomechanics 42, 1320–1325. Weizsacker, H.W., Kampp, T.D., 1990. Passive elastic properties of the rat aorta. Biomedizinische Technik 35 (10), 224–234. Weizsacker, H.W., Lambert, H., Pascale, K., 1983. Analysis of the passive mechanical-properties of rat carotid arteries. Journal of Biomechanics 16 (9), 703–715. Weizsacker, H.W., Pinto, J.G., 1988. Isotropy and anisotropy of the arterial-wall. Journal of Biomechanics 21 (6), 477–487. Zeller, P.J., Skalak, T.C., 1998. Contribution of individual structural components in determining the zero-stress state in small arteries. Journal of Vascular Research 35 (1), 8–17. Zou, Y., Zhang, Y.H., 2009. An experimental and theoretical study on the anisotropy of elastin network. Annals of Biomedical Engineering 37 (8), 1572–1583. Zulliger, M.A., Fridez, P., Hayashi, K., Stergiopulos, N., 2004. A strain energy function for arteries accounting for wall composition and structure. Journal of Biomechanics 37 (7), 989–1000.
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Mechanical anisotropy of inflated elastic tissue from the pig aorta M.A. Lillie n, R.E. Shadwick, J.M. Gosline Department of Zoology, University of British Columbia, Vancouver, B.C., Canada V6T 1Z4
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 9 April 2010
Uniaxial and biaxial mechanical properties of purified elastic tissue from the proximal thoracic aorta were studied to understand physiological load distributions within the arterial wall. Stress–strain behaviour was non-linear in uniaxial and inflation tests. Elastic tissue was 40% stiffer in the circumferential direction compared to axial in uniaxial tests and 100% stiffer in vessels at an axial stretch ratio of 1.2 or 1.3 and inflated to physiological pressure. Poisson’s ratio vyz averaged 0.2 and vzy increased with circumferential stretch from 0.2 to 0.4. Axial stretch had little impact on circumferential behaviour. In intact (unpurified) vessels at constant length, axial forces decreased with pressure at low axial stretches but remained constant at higher stretches. Such a constant axial force is characteristic of incrementally isotropic arteries at their in vivo dimensions. In purified elastic tissue, force decreased with pressure at all axial strains, showing no trend towards isotropy. Analysis of the force–length–pressure data indicated a vessel with vyz E 0.2 would stretch axially 2–4% with the cardiac pulse yet maintain constant axial force. We compared the ability of 4 mathematical models to predict the pressure-circumferential stretch behaviour of tethered, purified elastic tissue. Models that assumed isotropy could not predict the stretch at zero pressure. The neo-Hookean model overestimated the non-linearity of the response and two non-linear models underestimated it. A model incorporating contributions from orthogonal fibres captured the non-linearity but not the zero-pressure response. Models incorporating anisotropy and non-linearity should better predict the mechanical behaviour of elastic tissue of the proximal thoracic aorta. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Elastin Mechanical properties Anisotropy Poisson’s ratio Constitutive model
1. Introduction Forces on an artery wall are borne primarily by collagen and elastic fibres, the latter comprised of elastin and microfibrils. The three-dimensional load partitioning amongst these elements is closely modulated through arterial growth and remodeling; so accurately representing the contributions of each element improves the predictive ability of computer models and hence our understanding of mechanobiology. Arteries are generally considered anisotropic. This property is usually attributed to collagen (Holzapfel et al., 2000; Zulliger et al., 2004; Gasser et al., 2006), though both isotropic (Gundiah et al., 2009) and anisotropic (Sherebrin, 1983; Zou and Zhang, 2009) behaviours have been reported for elastic tissue. However, it is important to differentiate between inherent anisotropy due to structural features of an unstressed material, such as an unequal partitioning of orthogonal fibres, and induced anisotropy due to superimposed stresses on a non-linear material in which moduli depend on loading conditions (Dobrin, 1978). Arteries approach incremental isotropy under physiological loads (Dobrin, 1986; Weizsacker and Pinto, 1988), and therefore it is important to n
Corresponding author. Tel.: + 1 604 822 2373; fax: +1 604 822 2416. E-mail address: [email protected] (M.A. Lillie).
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.04.014
assess the elastic tissue under physiological conditions. In this paper we assessed the mechanical behaviour of purified elastic tissue over a range of strains, including physiological. For the proximal pig thoracic aorta, physiological circumferential stretch is 1.2–1.26 (Stergiopulos et al., 2001; Guo and Kassab, 2004), and axial stretch is around 1.2 (Han and Fung, 1995). We used uniaxial tests to identify inherent properties to establish the importance of tissue structure, and inflation tests to study the sensitivity to orthogonal loading conditions. We found elastic tissue from the proximal pig thoracic aorta was anisotropic under physiological strains and attribute this in part to the inherent structure of elastic tissue.
2. Methods Descending thoracic aortas from 100 kg pigs (5–6 month old) were collected from an abattoir, transported in iced phosphate buffered saline, PBS, and frozen in PBS at 20 1C. Thawed aortas were cleaned of loosely adhering tissue. Tissue more than 40 mm distal to the first intercostal artery was excluded. Elastic tissue was purified by 8 h autoclaving (Lillie et al., 1994). 2.1. Uniaxial tests 2.1.1. Compressibility Although elastic fibres are incompressible (Gosline, 1978), purified elastic tissue is porous and may collapse radially on deformation. We therefore examined
M.A. Lillie et al. / Journal of Biomechanics 43 (2010) 2070–2078 volume changes in circumferential strips of elastic tissue under tensile loads using an Instron 5500 apparatus. Reference marks were placed midway on the strip for determining circumferential (ly), axial (lz) and radial stretch (lr). Strips were mounted in air beside a mirror positioned at 451 for simultaneous viewing of the y–z and y r surfaces. Vessels were photographed with a 10 Mpix camera at 5% extension increments. Relative volume was V ¼ lylzlr. Radial deformation under compression was compared in rings of intact (not purified) and elastic tissue mounted around two bars and stretched circumferentially to compress the tissue at the bars (inset Fig. 1B). Radial deformation was measured at one bar using a custom-built displacement transducer. 2.1.2. Poisson’s ratio Strips of elastic tissue were stretched circumferentially. Incremental Poisson’s ratio was calculated as nzy ¼ Dez =Dey , where De is the incremental strain. For a generic length, ‘, De ¼ 2ð‘2 ‘1 Þ=ð‘2 þ ‘1 Þ, where the subscripts represent different load levels.
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Photos were digitized using ImageJ (Abramoff et al., 2004) to obtain tissue dimensions. Wall thickness, H, was calculated from rings, assuming circular crosssection. The mid-wall radius in the inflated vessel, r, was calculated assuming incompressibility qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ r ¼ ro2 ðR2o R2 Þ=lz where ro and Ro are the outer radii, the upper case referring to the unloaded dimension and the lower case to the deformed. lz ¼ l/L is the axial stretch at the midpoint of each segment, where L and l are the local segment lengths. We assumed circumferential strain was uniform around the segment and varied linearly across the wall. Mid-wall stretch ratio was ly ¼r/R. For the analysis we assumed the aortas were thin-walled, orthotropic, pseudoelastic and homogeneous, and that the axial stress was uniform. Circumferential and axial residual strains for purified elastic tissue from the proximal aorta are under 1% (Lillie and Gosline, 2006) and have been neglected. Circumferential and axial Cauchy stresses were calculated as Pri h P pri2 þ F sz ¼ pðro2 ri 2 Þ
sy ¼
2.1.3. Anisotropy Elastic tissue was cut into a 25 mm square, or a cross with 8 mm wide arms aligned with the orthogonal axes and held in 8 mm wide grips. Samples were preconditioned and then tested four times in alternating directions. Neither the sample shape nor the testing order had an impact on anisotropy. Stress, s, was force divided by the deformed cross-sectional area, A. Incremental modulus was calculated as E ¼ Ds/De at 2% stretch increments.
where ri is the inner radius. We assumed the tissue behaved linearly over small strains, and calculated incremental circumferential and axial strains following Dobrin and Doyle (1970):
2.2. Inflation tests
Dey ¼
Straight segments from 14 autoclaved aortas were tested. Of these, 4 were also tested intact. A grid was painted along each aorta for measuring local strain. The ends of the aorta were cannulated, and the aorta was mounted horizontally in PBS at 37 1C. The proximal end was fixed in position and connected to a pressure reservoir. The distal end was either free floating or fixed via a low-torsion coupling to a force transducer to measure axial load, F. The position of the distal mount could be varied to set the axial strain. The segment was preconditioned and then inflated to 14 kPa in increments of 1.3 kPa with a total loading time of about 2 min. The aortas were photographed at each pressure, P. Tests were run with the aorta untethered and tethered at 4–17 axial stretches between 1.02 and 1.46. After the inflation tests, a ring was cut from the middle of each vessel and photographed in PBS at 37 1C to measure circumference and wall thickness.
Dez ¼
Dsy Ey
Dsz Ez
nyz nzy
ð2a; bÞ
Dsz Ez
Dsy
ð3a; bÞ
Ey
Radial stresses averaged 5% of the circumferential stresses and were omitted from Eq. (3). Paired orthogonal moduli, Ey and Ez, were determined under conditions of same mean stress. The process is illustrated in Fig. 2. Curves were taken from 2 inflation cycles, in this example one at lz ¼ 1.19 and one at lz ¼ 1.2. We selected two points that fell at the same sz (P1 and P2). Since each point represents a specific sy and associated ey, P1 and P2 are separated by Dsy and Dey. High order polynomials were fit to the stress–strain data to obtain each ey. When the data are selected such that Dsz ¼ 0, Eq (3a) yields Ey ¼
Dsy Dey
ð4aÞ
Fig. 1. (A) Compressibility of strips of elastic tissue under circumferential tension. Volume decreased by 2.7% at the initial extension (P 50.001), but remained constant at higher extensions. (B) Relative compressibility of intact and elastic tissue under radial compression. Inset shows tissue ring loaded by movement of lower mounting bar with thickness measured at upper bar. Both data sets start at a radial stretch of 1.00 at a compressive load of zero. Between 1 and 40 kPa, the compression of the elastic tissue paralleled that of the intact. This is shown by the solid line, which gives the elastic tissue data shifted down by 3% for clarity. Broken lines show the response of rabbit thoracic aortas under uniaxial compression from Chuong and Fung (1984).
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Radial deformation was directly measured in intact (n ¼10) and elastic tissue (n ¼12; Fig. 1B). Our loading protocol had two limitations: stress distribution in the tissue was complex (not uniaxial compression), and there was likely friction between the tissue and supporting bar. However, the response of our intact tissue was comparable to that of intact rabbit aorta under uniaxial compression (Chuong and Fung, 1984) (Fig. 1B), indicating the loading at the point of measurement was essentially compressive. Further, the friction coefficients for intact and elastic tissue are likely similar; so although the absolute radial deformations may have been constrained, the relative response in these tissues should be largely unaffected. The elastic tissue thickened under minimal radial loads ( 1 kPa; Fig. 1B), and we cannot explain this response. However its deformation between 1 and 40 kPa paralleled that of the intact tissue, indicating equal incremental radial stiffness in intact and elastic tissue. The data in Fig. 1A and 1B show that elastic tissue can be treated as incompressible. 3.1.2. Poisson’s ratio The mean incremental Poisson’s ratio, nzy, measured in circumferential strips stretched to ly ¼1.2 was 0.337 0.02 (n ¼23).
Fig. 2. Calculation of moduli using Eq (4). Axial vs. circumferential stress data are shown for inflation at two axial stretches, lz ¼ 1.19 (closed circles) and 1.20 (open). Intermediate values were obtained by interpolation with a fitted polynomial (lines). Ey was evaluated at points P1 and P2 (closed squares), which are at the same axial stress but at different circumferential stresses and associated with different circumferential strains ey1 and ey2 . The strains ey1 and ey2 were obtained from stress–strain plots. The corresponding Ez was evaluated at points P3 and P4 (open squares), which are at the same circumferential stress but at different axial stresses and associated with different axial strains, ez3 and ez4. Mean stresses for P1 and P2 equal those for P3 and P4, shown by ‘‘x’’.
Similarly for points P3 and P4 in Fig. 2, for which Dsy ¼0, Eq (3b) yields Ez ¼
Dsz Dez
3.2. Inflation tests ð4bÞ
Poisson’s ratios in axially tethered vessels were calculated following Dobrin and Doyle (1970):
nyz ¼
Dsz Dsy
nzy ¼ nyz
Ey Ez
3.1.3. Anisotropy Behaviour of elastic tissue in uniaxial tests is shown in Fig. 3. Each sample was tested circumferentially and axially. Fig. 3A shows the force-stretch response for one sample undergoing a load/unload cycle. Resistance to deformation was greater along the circumferential axis compared to axial. Both circumferential and axial moduli increased with strain (Fig. 3B). The rapid increase in moduli between stretches of 1.0 and 1.1 corresponds to the initial toe region visible in Panel A. Beyond the toe region, the mean anisotropy, defined as Ey/Ez at equal stretches, was 1.4170.04 (n ¼15).
ð5Þ
ð6Þ
2.3. Statistical analysis Statistics were done with SigmaStat 2.03. A value for P of 0.05 was considered significant. Results are given as mean 7 SE.
3. Results Results are for purified elastic tissue unless specified for intact. 3.1. Uniaxial tests 3.1.1. Compressibility Relative volume of strips of elastic tissue under circumferential tension was calculated from the three orthogonal stretch ratios. There was a significant 2.770.3% loss of volume at the initial extension (Fig 1A), but the flat regression line indicates there was no further volume change at higher extensions (P50.001 linear regression constant, n ¼206, 22 samples).
Dimensions of the unloaded vessel were measured at the start and end of the tests to check for damage. Post-test length was 0.99570.007 of pretest, and post-test diameter was 1.00570.009. Neither changed significantly (paired t-test). 3.2.1. Untethered vessels Untethered vessels increased in both length and circumference when inflated, stretching about 60% more in the circumferential direction compared with the axial (Fig. 4). At a pressure of 13.3 kPa, ly averaged 1.2870.01 and lz 1.16 70.01 (n¼14). For intact vessels, ly averaged 1.23 70.02 and lz 1.18 70.01 (n ¼4). 3.2.2. Tethered vessels The mechanical properties of vessels tethered at lz ¼1.2 (n ¼14) and 1.3 (n ¼4) are shown in Figs. 5–8. Circumferential stretch increased non-linearly with pressure (Fig. 5). At zero pressure the circumferential stretch for lz ¼1.2 averaged 0.97, larger than the 0.91 predicted for isotropy (i.e., ly ¼1.2 0.5). In vessels tethered at lZ ¼1.2 and pressurized to 13.3 kPa, circumferential stretch averaged 1.26 70.01, which is near the physiological value, and at lZ ¼1.3 circumferential stretch averaged 1.3270.02. The circumferential modulus increased with inflation, but the axial modulus was essentially constant (Fig. 6). Increasing the axial stretch from 1.2 to 1.3 increased both moduli; around physiological circumferential stretches, the moduli increased 9%, which is accounted for by an 8% drop in wall
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Fig. 3. Behaviour of elastic tissue in uniaxial tests. (A) Force-stretch response for a single sample undergoing one load/unload cycle showing a greater resistance to deformation along the circumferential axis compared to axial. (B) Circumferential and axial moduli (mean 7SE) demonstrating non-linear behaviour. The tissue averaged 41% greater stiffness in the circumferential direction compared to axial.
Fig. 4. Inflation of untethered elastic tissue segments. Vessels increased in both diameter and length. Values are mean 7SE. Axial stretch is about 60% of circumferential.
Fig. 5. Inflation of elastic tissue segments tethered at an axial stretch of 1.2 (closed symbols) or 1.3 (open). The circumferential stretch at zero pressure is 0.97 for an axial stretch of 1.2, which is larger than the 0.91 predicted for an isotropic material. Values are mean 7 SE.
thickness. At very low pressure, the tissue was close to transverse isotropy, but it became anisotropic with further inflation (Figs. 6 and 7). At lZ ¼ ly ¼1.20, anisotropy averaged 1.870.1, higher than
the uniaxial value of 1.41 (P¼ 0.003). Both Poisson’s ratios increased with circumferential stretch: vyz remained near 0.2, but vzy increased from below 0.2 to over 0.4 (Fig. 8).
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Fig. 6. Moduli calculated from Eq (4) for inflated elastic tissue segments tethered at an axial stretch of 1.2 (circles) or 1.3 (squares). Axial modulus, Ez, was relatively constant while circumferential modulus, Ey, increased with inflation. Increasing axial stretch increased both moduli. Values are mean 7 SE.
Fig. 8. Poisson’s ratios for inflated elastic tissue segments. Solid data points with solid regression line show nyz for vessels tethered at lz ¼ 1.2 calculated from Eq (5). Open data points with broken regression line show nyz for vessels tethered at lz ¼ 1.3. Line with capped error bars shows nzy for lz ¼ 1.2 calculated with Eq (6), and line with non-capped error bars shows nzy for lz ¼1.3. The square at ly ¼ 1.2 shows the mean Poisson’s ratio, nzy, measured in the uniaxial tests.
Increasing the axial stretch increased the starting axial force but had little effect on curve shape: the curves were largely congruent when offset vertically to set the force at zero pressure to the arbitrary value of 2 N (Fig. 9B), and they displayed no trend towards the horizontal with increasing lz. This response was observed in all elastic tissue vessels tested, up to a maximum stretch of 1.46, and contrasts with that of intact aortas, illustrated in Fig. 9D by 19 curves from a single vessel. Nine curves at the bottom at axial stretches from 1.05–1.19 are indistinguishable, but from lZ ¼1.20–1.39 the slope progressively diminished. In the 4 intact aortas, force became independent of pressure around lZ ¼1.35. This is the ‘‘crossover’’ point in a force–length plot and generally corresponds to the axial stretch in vivo (see Discussion).
4. Discussion 4.1. Anisotropy of aortic elastic tissue
Fig. 7. Anisotropy of elastic tissue segments tethered at an axial stretch of 1.2 (closed circles) and 1.3 (open circles). The vessels were transversely isotropic at low circumferential stretches but became progressively more anisotropic with further inflation. Increasing axial stretch had no impact on anisotropy. At a circumferential stretch of 1.2, the mean anisotropy was 1.8 in the inflation tests (triangle), larger than the 1.4 obtained in the uniaxial tests (square).
3.2.3. Applied axial force The applied axial force on the elastic tissue vessels decreased with pressure. Fig. 9A shows the response in one vessel tested from lZ ¼1.03 (bottom) to 1.36 (top) in 2–3% increments.
Intact arteries are anisotropic, but because their behaviour is non-linear they can attain transverse isotropy (Ey ¼Ez aEr) at certain pairings of axial and circumferential stretch and can approach full isotropy (Ey ¼Ez ¼Er) near physiological stretches (Cox, 1975; Cox, 1978; Dobrin, 1986; Weizsacker and Pinto, 1988; Weizsacker and Kampp, 1990). By contrast, we found elastic tissue was anisotropic under most loading conditions. Our uniaxial tests showed elastic tissue possessed an inherent anisotropy, with a circumferential stiffness 1.4 times axial (Fig. 3). Similar values were obtained in uniaxial tests of alkalipurified elastic tissue from dog and sheep aortas (Sherebrin, 1983). Elastic tissue behaviour was non-linear in uniaxial tests (Fig. 3B), and therefore the inherent anisotropy exhibited under uniaxial loading can change under biaxial loading. Zou and Zhang (2009) found anisotropy persisted under biaxial loading, but Gundiah et al. (2009) obtained transverse isotropy. We also obtained transverse isotropy in our inflation tests of vessels
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Fig. 9. Dependence of applied axial force on pressure and axial strain in an elastic tissue segment (A–C) and in an intact (unpurified) segment (D). (A) The data show behaviour from lz ¼ 1.03 (bottom) to 1.36 (top) in 2–3% increments. Increasing axial stretch shifted the curves up with little change in shape. (B) Each curve in A has been shifted vertically to set the force at zero pressure to the arbitrary value of 2 N. The curves are nearly congruent and show no trend towards the horizontal with increasing lz. (C) The data from A have been replotted as pressure–axial stretch curves. Each line gives behaviour at a constant axial force, shown in 0.5 N increments. For example, the solid squares in panels A and C show combinations of axial stretch and pressure that will produce an axial force of 4 N. Lines are largely linear with steeper slopes at lower axial stretches. For a physiological pressure pulse of 5.3 kPa, the axial force would remain constant if the vessel stretched 2% (lower lz) to 4% (higher lz). (D) Data from an intact aorta from lz ¼ 1.05 (bottom) to 1.39 (top). The curves have been shifted vertically to set the force at zero pressure to 2 N. Slope is constant for 9 curves from lz ¼1.05–1.19. At higher axial strains, slope becomes less negative and is horizontal by lz ¼ 1.35.
tethered at axial stretches of 1.2–1.3, but only at the lowest circumferential stretches (Figs. 6 and 7). Around physiological stretch (ly 1.25), total anisotropy (inherent and induced) averaged 2 (Fig. 7). Increasing the axial stretch from 1.2 to 1.3 had no impact on the anisotropy. We have not measured axial stretches in vivo, but Han and Fung (1995) measured stretches around 1.2 in situ in unpressurized proximal thoracic aortas, and
our force–pressure data from intact vessels indicate a value around 1.35 (Fig. 9D). Thus, the stretches examined in our inflation tests are germane to physiological behaviour, and our results indicate elastic tissue is significantly anisotropic near physiological stretches. The striking difference between intact and elastic tissue is that at higher axial stretches intact tissue progresses towards full
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isotropy, but elastic tissue does not. We did not measure anisotropy in our intact tissues, but a shift to isotropy produces a distinctive pattern in the axial force–pressure curves: at low axial stretches the curves slope downwards, but with increased axial stretch the slopes tend to level off, and they become horizontal near the in vivo axial stretch (Van Loon et al., 1977; Weizsacker et al., 1983). This pattern was exhibited by intact aortas in this study (Fig. 9D), but not in elastic tissue (Fig. 9B), even at stretches up to 1.46. Constant axial force at physiological stretch occurs where nyz ¼ 0.5, which is its value at isotropy (Weizsacker and Pinto, 1988; Brossollet and Vito, 1995). The dependence of axial force on pressure can be calculated from Eqs. (2), (3b) and (6): F ¼ Ez pðro2 ri2 Þez þ Ppri2 ð2nyz 1Þ
ð7Þ
A similar equation is given by Brossollet and Vito (1995) for an isotropic vessel. For a fixed-length vessel, the first term on the right is constant, and it can be readily verified that force will remain constant with pressure if nyz ¼0.5 and fall if nyz o0.5. For elastic tissue, nyz remained around 0.2 (Fig. 8) and forces fell with pressure (Fig. 9B). The similarity of the force–pressure curves in Fig. 9B (elastic tissue) and 9D (intact tissue) indicates that nyz in intact aortic tissue must also be around 0.2 at axial stretches from lz ¼1.05–1.19. Axial forces are more complex in a variable-length vessel. At least in the dog, the thoracic aorta is not fixed in length but exhibits pressure-driven axial strains of 1–5% (Patel et al., 1961; Patel and Fry, 1964). Such strains would raise the axial force in an isotropic vessel, and we can use the data in Fig. 9A to estimate their impact on the axial force in an anisotropic, elastic tissue vessel. We assume the behaviour of purified elastic tissue represents its behaviour in vivo. Axial force depends positively on lz but negatively on pressure. We recast the force–pressure data in Fig. 9A as pressurestretch data in 9C, where each curve represents a constant axial force. The curves appear linear, and their slopes show that an aorta that stretched 2–4% under a pressure pulse of 5.3 kPa would experience a constant axial force. Thus with nyz E0.2, an anisotropic vessel could stretch axially yet maintain a constant axial force. Interestingly, Patel et al. (1969) found the dog thoracic aorta behaved anisotropically in vivo, with nyz E0.27. The in vivo response of the pig aorta depends on how the collagen and perivascular tissue modify the elastic tissue behaviour, and this must be determined. A constant axial force at in vivo length appears to be a fundamental property of arteries, basic to the premise that arteries are isotropic at physiological loads (Brossollet and Vito, 1995). Patel’s work on the thoracic aorta and our analysis of aortic elastic tissue indicate that axial forces may also remain constant in anisotropic arteries of varying length if nyz is low. 4.2. Modeling the inflation response of arterial elastic tissue Both neo-Hookean and non-linear strain energy functions, C, have been used to model arterial elastic tissue behaviour. Here we assess the ability of four models to predict the pressure-stretch response in Fig. 5: Cn-H, the neo-Hookean model; CZ, a widely used non-linear model proposed by Zulliger et al., (2004); CD, an exponential model introduced by Demiray (1972) and suggested by Ogden and Saccomandi (2007) to model elastic tissue with limiting chain extensibility, and CGRP, a semi-empirical model that incorporates specific contributions from orthogonal fibres (Gundiah et al., 2009).
CnH ¼ cnH ðI1 3Þ CZ ¼ cZ ðI1 3Þ1:5
cD exp bðI1 3Þ 1 b CGRP ¼ c0 ðI1 3Þ þ c1 ðI4 1Þ2 þc2 ðI6 1Þ2
CD ¼
ð8a2dÞ
where cn–H, cZ, cD, c0, c1, and c2 are material constants, b is a nondimensional parameter, and I1, I4, and I6 are the strain invariants of the right Cauchy-Green tensor. Assuming incompressibility, 2
2
2 2
I1 ¼ ly þ lz þ ly lz 2
I4 ¼ lz
ð9a2cÞ
2
I6 ¼ ly
The pressure-stretch relationships predicted by each model have been derived using Eqs. (8) and (9) and P¼
1 H @C ly lz R @ly
ð10Þ
given by Ogden and Saccomandi (2007). ! 2cnH H 1 1 4 2 PnH ¼ Rlz ly lz ! 3cZ H 1 0:5 ðI1 3Þ 1 4 2 PZ ¼ Rlz ly lz ! 2cD H 1 exp bðI1 3Þ 1 4 2 PD ¼ Rlz ly lz ! " # 2H 1 c0 1 4 2 þ 2c2 ðI6 1Þ PGRP ¼ Rlz l l
ð11a2dÞ
y z
The lines in Fig. 10 show the pressures predicted by Eq. 11(a–d). Parameter values were selected using least squares to fit the curvature of the data (symbols) between 0 and 5 kPa. The neoHookean, Zulliger, and Demiray models assume isotropy and therefore predict a Poisson’s ratio vyz near 0.5 and hence a zero pressure at ly ¼0.91. Our data start at ly ¼0.97, in accord with the Poisson’s ratio nyz around 0.15 shown in Fig. 8, and so these models fail to capture the response of the tissue. The neo-Hookean model overestimated the curvature of the data, while the Zulliger model underestimated it. The response predicted by these two models for elastic tissue under uniaxial loading showed similar deviations from experimental data (Watton et al., 2009). The Demiray model predicts neo-Hookean behaviour for small values of parameter b (not shown), which overestimates curvature, and higher b values (Fig. 10) introduced a convexity at higher strains that was absent in the inflation data. Convexity is present in single elastic fibres at high stretches (Aaron and Gosline, 1981). The Gundiah model does not assume isotropy, and the predicted stretch at zero pressure depends on parameter values. Gundiah et al. (2009) obtained a low empirical value for c2 (c0 ¼74, c2 ¼0.8), and since the model reverts to the neo-Hookean for c2 ¼0, they concluded neo-Hookean behaviour dominated the elastic tissue response. In Fig. 10 we allowed a much greater contribution from the I6 term by using c0 ¼33 and c2 ¼15. This allowed the Gundiah model to capture the curvature of our data, although the anisotropy remained insufficient to predict the zero-pressure response. None of these models adequately predicts the response of the elastic tissue from the proximal thoracic aorta. The mismatch between the observed and modeled responses visible in the zero pressure radii is by definition a mismatch between the observed and predicted Poisson’s ratios, nyz —the decrease in radius obtained on increase in length under a uniaxial load. Residual stresses and strains remain in an uncut artery under no load, but it is unlikely that they could account for the observed low Poisson’s ratio. First, circumferential, axial and radial residual strains in purified elastic tissue from the proximal aorta are small: 0.006, 0, and 0.008, respectively (Lillie and Gosline, 2006), and referring the measured stretches to the zero stress rather than the no load state marginally reduces the Poisson’s ratio further. Second, the value of nyz is largely insensitive to applied strain, whether circumferential (ly ¼0.95–1.35, Fig. 8) or axial
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indicate mesoscopic tissue organization is important. Third, elastic tissue behaviour differs substantially with position, even within the thoracic aorta (Lillie and Gosline, 2007; Zou and Zhang, 2009). Elucidating the physical and physiological underpinnings of these positional differences will further our understanding of arterial mechanobiology and our ability to model the behaviour of arteries from all sources.
Conflict of interest statement There is no conflict of interest regarding the publication of the manuscript ‘‘Mechanical Anisotropy of Inflated Elastic Tissue from the Pig Aorta’’.
Acknowledgements This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council to J.M.G. and by a Discovery Accelerator Grant from the Natural Sciences and Engineering Research Council to R.E.S. References Fig. 10. Modeling the circumferential response to inflation of elastin segments tethered at lz ¼1.2. The symbols show the data from Fig. 5. Lines show the pressures predicted by 4 models given in Eq (11a–d). Parameter values were selected to fit the curvature of the data between 0 and 5 kPa. At zero pressure the mean experimental value for ly was 0.97. The neo-Hookean, Zulliger, and Demiray models assume isotropy, which predicts zero pressure at ly ¼ 0.91, and so these models fail to capture that behaviour of the tissue. The Gundiah model does not assume isotropy and predicts a slightly better fit at zero pressure. The Gundiah model captured the curvature of the response shown by the data. The neoHookean model overestimates the curvature while the Zulliger and Demiray models underestimate it. The mean value for H/R for the tissues in our study was 0.21. Model parameter values were cn–H ¼42 kPa, cZ ¼77 kPa, cD ¼ 33 kPa, b ¼1.6, c0 ¼ 33 kPa and c2 ¼ 15 kPa.
(lz ¼1.2–1.3, Fig. 8 and lz ¼1.03–1.46, inferred from Fig. 9B in Discussion), and it should therefore remain unaffected by small residual strains. Third, the value of nyz at low axial stretches in the intact aorta (Fig. 9D) and in other arteries (Patel et al., 1969; Weizsacker et al., 1983; Dobrin, 1986) also falls around 0.2, even though residual strains are substantially greater in intact arteries (Greenwald et al., 1997; Zeller and Skalak, 1998; Lillie and Gosline, 2006). Thus the low values for nyz observed in both intact arteries and purified elastic tissue over a range of loading conditions arise from the structure of the elastic tissue. In an incompressible material nyz depends on the orthogonal moduli and its value will be low when Ey 4Er 1 1 þ0:5: ð12Þ nyz ¼ 0:5Ez Ey Er (Weizsacker and Pinto, 1988). Thus in addition to the anisotropy in the y z plane demonstrated in Fig. 7, there is also elastic tissue anisotropy in the y r plane. Future research must identify what physical parameters need be experimentally measured and incorporated into the models to improve their predictive ability. First, they require a more general treatment to accommodate the three-dimensional anisotropy, perhaps by incorporating elastic fibre orientation, as is done for veins (Rezakhaniha and Stergiopulos, 2008), or elastin chain orientation (Zou and Zhang, 2009). Second, the basis of the non-linearity must be identified. The Demiray and Zou models focus on the molecular level, but differences between single fibre (Aaron and Gosline, 1981) and elastic tissue behaviour
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