Biomech Model Mechanobiol (2009) 8:487–498 DOI 10.1007/s10237-009-0152-7
ORIGINAL PAPER
Viscoelastic and failure properties of spine ligament collagen fascicles Scott R. Lucas · Cameron R. Bass · Jeff R. Crandall · Richard W. Kent · Francis H. Shen · Robert S. Salzar
Received: 25 August 2008 / Accepted: 26 February 2009 / Published online: 24 March 2009 © Springer-Verlag 2009
Abstract The microstructural volume fractions, orientations, and interactions among components vary widely for different ligament types. If these variations are understood, however, it is conceivable to develop a general ligament model that is based on microstructural properties. This paper presents a part of a much larger effort needed to develop such a model. Viscoelastic and failure properties of porcine posterior longitudinal ligament (PLL) collagen fascicles were determined. A series of subfailure and failure tests were performed at fast and slow strain rates on isolated collagen fascicles from porcine lumbar spine PLLs. A finite strain quasi-linear viscoelastic model was used to fit the fascicle experimental data. There was a significant strain rate effect S. R. Lucas (B) Exponent Failure Analysis Associates, Biomechanics Practice, 3401 Market Street, Suite 300, Philadelphia, PA 19104, USA e-mail:
[email protected] C. R. Bass Duke University, Durham, USA J. R. Crandall · R. W. Kent · R. S. Salzar Department of Mechanical and Aerospace Engineering, Center for Applied Biomechanics, University of Virginia, Charlottesville, USA J. R. Crandall Department of Biomedical Engineering, Center for Applied Biomechanics, University of Virginia, Charlottesville, USA R. W. Kent Department of Emergency Medicine, Center for Applied Biomechanics, University of Virginia, Charlottesville, USA F. H. Shen Department of Orthopedic Surgery, University of Virginia, Charlottesville, USA
in fascicle failure strain (P < 0.05), but not in failure force or failure stress. The corresponding average fast-rate and slowrate failure strains were 0.098 ± 0.062 and 0.209 ± 0.081. The average failure force for combined fast and slow rates was 2.25 ± 1.17 N. The viscoelastic and failure properties in this paper were used to develop a microstructural ligament failure model that will be published in a subsequent paper. Keywords Spine · Ligament · Fascicle · Viscoelasticity · Failure · Biomechanics
1 Introduction Computational models can be used to assess injury scenarios, to understand injury mechanisms, and to help develop ways to mitigate injuries. For accurate reconstructions of a human response to dynamic events using computational models, it is crucial to include accurate material properties of the internal structures, such as connective tissues, of the body region that is modeled. Further, as connective tissues are viscoelastic, their properties must be valid for the range of simulated loading conditions, including strain magnitude and strain rate. Ligaments exhibit rate-sensitive failure properties (Yoganandan et al. 1989; Pioletti et al. 1998; Bass et al. 2007) and some ligaments exhibit rate-sensitive failure modes (Van Dommelen et al. 2005a). The explanation of these ratesensitive properties is currently speculative. To better understand the response of a ligament or tendon, it is helpful to examine the ligament or tendon microstructure. Ligaments are composed of varying amounts of the same microstructural components. Thus, it is conceivable to develop a material model based on the microstructure that can be used to predict the force response of an arbitrary ligament-type based on its microstructural composition and
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the microstructural material behavior. In as early as the nineteenth century, anatomical investigations (Carpenter 1876) identified human connective tissue to be organized in an hierarchical manner from macroscopic organized structures to microscopic molecules. More recently, there is a considerable amount of research on the hierarchical collagen structure in tendons (Bear 1952; Viidik 1972; Lees and Davidson 1977; Kastelic 1978; Gathercole and Keller 1991). Kastelic (1978) recognized the collagen fascicle and deemed the fascicle as the fundamental structural unit of tendons. Within ligaments and tendons, collagen fascicles have a wavy appearance (Hansen et al. 2002) and are comprised of collagen fibrils, elastin, and a hydrated matrix of proteoglycans, known as the ground substance matrix. In histological images (Niven et al. 1982), polarized transmission microscopy (Niven et al. 1982), optical microscopy (Rigby et al. 1959; Viidik and Ekholm 1968; Wilmink 1992), and scanning electron microscopy (Zervakis et al. 2005; Hurschler et al. 2003), the fibrils also have a wavy appearance within the fascicles. The wavy configuration of the fascicles and fibrils contributes to the mechanical response of ligaments and tendons. In developing a microstructural model, the progression of recruited fascicles under tensile loading must be accounted for in addition to the associated fascicle material response once recruited. The material and failure properties of collagen fascicles are not understood at fast loading rates, which prompted the research that is presented. This paper determines the material and failure properties of the spine ligament collagen fascicle at slow and fast loading rates. This research is a necessary part of a ligament microstructural failure model (Lucas 2008a) that will be presented in a subsequent paper.
2 Methodology 2.1 Fascicle isolation and test preparation Isolated collagen fascicle segments were removed from seven porcine lumbar posterior longitudinal ligament (PLL) bone– ligament–bone complexes and prepared for mechanical testing (Lucas 2008b; see Fig. 1). Under a 10–60× magnification, microsurgery tools were used to isolate individual fascicles and to cut segments from the isolated fascicles. Each fascicle segment was mounted on a 10 mm × 20 mm plastic coupon (Mylar, 76.2 µm thick) with a diamond cut out in the center that varied depending on the desired fascicle initial length. There were 3 fascicles tested with a 2 mm× 2 mm diamond coupon, 8 fascicles tested with a 4 mm × 2 mm diamond, and 30 fascicles with an 8 mm × 2 mm diamond. A small hole was punched above and below the diamond. Under a dissection microscope (National DC5-420TH, Carlsbad, CA, 10, 60× magnification), the collagen fascicle was aligned in a scored groove on the centerline of the coupon
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Fig. 1 Isolated PLL fascicle. Two other fascicles were already removed from this bundle. This picture was taken under the microscope at 45× magnification. The hook used in the dissection procedure was used only to isolate fascicles and was nondestructive to individual fascicles. Each tick on the ruler is 1 mm
and the ends of the fascicle were pushed through the holes and looped through the coupon. Small drops of cyanoacrylic adhesive were placed on the top of the coupon. One additional coupon was then pressed on the coupon that contained the fascicle, so that the diamonds in the center of the coupons were aligned. Under the microscope at 45× magnification equipped with a linear-type reticle, it was observed that the fascicles were approximately circular in cross-section and each fascicle diameter was measured in five locations along the length of the fascicle. The average diameter of the three middle locations was deemed the initial fascicle diameter, φι . The initial cross-sectional area, Ai , was calculated by assuming that the fascicle has a circular cross-section, which was based on observation and supporting literature (c.f. Robinson et al. 2004): 2 φi . (1) Ai = π ∗ 2 2.2 Fascicle test protocol The plastic coupons with the secured fascicle were held by two aluminum grips and mounted on a universal force test machine (Instron, model #8874, Norwood, MA, USA). An environmental chamber was used for temperature and humidity control (c.f. Lucas et al. 2008; Bass et al. 2007; see Fig. 2). To avoid stress concentrations at the grips, the grips held the plastic and not the fascicle itself. Once the coupon was mounted in the grips, the sides of the coupon were cut, as indicated by the dotted lines in Fig. 2, so that the fascicle was completely isolated for testing. Force was recorded by a 10N capacity load cell (Sensotec, Columbus, OH, USA)
Viscoelastic and failure properties of spine ligament collagen fascicles
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Fig. 2 Schematic of test fixture enclosed in an environmental chamber. The positioning table is fixed and the input displacement is applied at the top. The grips, plastic coupon, and fascicle are enlarged on the right side of the picture
that was mounted beneath the bottom grip, and displacement was recorded at the Instron actuator. All data was recorded at 10,000 Hz. The load cell was mounted on an X–Y positioning table, which was used for alignment. The tests were recorded via high-speed video (Phantom, Vision Research, Wayne, NJ, USA) at 1,000 frames per second. For magnification, three +5 diopter adapter lenses were mounted in series on the 50 mm high-speed camera lens. In a preliminary test series, there was no preconditioning before each test. Each fascicle was subjected to a ramp displacement until failure while measuring the corresponding force, displacement, and acceleration. The preliminary dataset includes six “fast” tests at strain rates equal to or greater than 25 s−1 and five “slow” tests at strain rates between 0.25 and 0.50 s−1 . The input ramp displacement rates were adjusted based on the assigned strain rate. For the 0.25 and 25 s−1 tests, the corresponding input displacement rates were 1 and 100 mm/s. To achieve the maximum possible strain rates, the preliminary data set includes tests faster than 100 mm/s. To maintain consistency in the fast failure tests after the preliminary test series, 25 s−1 was the fastest desired strain rate. In the subsequent test series, mechanical tests were conducted within a framework of applied elongations, where fascicle elongation, λ A, f , is defined as the ratio of input displacement, l, to initial (gage) fascicle length, l0, f . There was no preload applied to the fascicle. The zero strain state was defined as the strain after securing the fascicle to the plastic coupon. The fascicle initial length, l0, f , was defined as the length of the diamond on the plastic coupon. For each test, the input displacement was determined by the necessary input engineering strain. Each ligament was preconditioned with a 2% λ A, f sinusoidal input at 2 Hz for 20 cycles (test P2 ). Then, each ligament was subjected to tensile ramp-hold inputs to elongations of 2% (test R2 ) and 4% (test R4 ), and an oscillatory sinusoid input to 4% elongation (test V4 ), as shown in Fig. 3. Each ramp was held for 10 s with 100 s of recovery time between each test. The corresponding rise time for the R2 and R4 tests ranged from approximately 10–60 ms.
Fig. 3 Input elongation for each fascicle. The preconditioning and validation tests are included in detail within the figure set to a smaller time scale. Following the four tests in this figure was a tensile ramp test until failure
The V4 test was conducted at 0.5 Hz for 3 cycles, immediately followed by a sinusoid at 3 Hz for 12 cycles. The strain levels in tests P2 , R2 , R4 , and V4 , as determined from preliminary testing, were less than failure strain levels. In the V4 test, two frequencies were chosen to test the validity of the material model at different frequencies of oscillation. After the V4 test, each fascicle was subjected to a tensile ramp test until failure. To test the effect of strain rate on failure properties, the input displacement rate of the failure test varied for each fascicle. The fascicle test matrix, including the preliminary and subsequent test series, is shown in Table 1. Test numbers 1–11 included the preliminary failure tests. Test numbers 12–41 included the test battery shown in Fig. 3. Within tests 12–41, the subfailure test protocol (tests P2 , R2 , R4 , and V4 ) was identical; however, the strain rate of the subsequent failure test varied and not all subfailure tests included a subsequent failure test.
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490 Table 1 Fascicle test matrix
The gage length variation reflects the variation in the cut-out diamond upon which the fascicle was mounted. If a failure test was performed, the failure strain rate is included
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S. R. Lucas et al. Test number
Gage length (mm)
Subfailure test protocol?
Failure test?
Failure test strain rate (s−1 )
1
2
N
Y
84.6
2
4
N
Y
19.0
3
2
N
Y
0.50
4
2
N
Y
73.8
5
4
N
Y
26.6
6
4
N
Y
57.9
7
4
N
Y
22.3
8
4
N
Y
0.26
9
4
N
Y
0.25
10
4
N
Y
0.25
11
4
N
Y
0.25
12
8
Y
N
–
13
8
Y
N
–
14
8
Y
N
–
15
8
Y
Y
16.8
16
8
Y
Y
16.0
17
8
Y
N
–
18
8
Y
N
–
19
8
Y
N
–
20
8
Y
N
–
21
8
Y
N
–
22
8
Y
N
–
23
8
Y
N
–
24
8
Y
Y
12.0
25
8
Y
N
–
26
8
Y
N
–
27
8
Y
N
–
28
8
Y
N
–
29
8
Y
N
–
30
8
Y
N
–
31
8
Y
Y
1.54
32
8
Y
N
–
33
8
Y
N
–
34
8
Y
Y
11.8
35
8
Y
Y
18.9
36
8
Y
N
–
37
8
Y
Y
10.9
38
8
Y
N
–
39
8
Y
Y
17.2
40
8
Y
N
–
41
8
Y
Y
16.4
Viscoelastic and failure properties of spine ligament collagen fascicles
2.3 Fascicle data analysis 2.3.1 Fascicle material model Including the preliminary test series, the entire fascicle test series included preconditioning, subfailure ramp-holds, subfailure oscillations, and failure tests. There were 30 subfailure tests and 20 failure tests. Among the 20 failure tests, 14 were considered fast rate, five were slow rate, and one was medium rate. Among the total subfailure and failure tests, there were nine tests that were from the same fascicle. The fascicle subfailure tests were slower than the fascicle fast failure tests. To obtain fast rate material properties, the fast failure tests were included in the material property analysis. The nine matched subfailure tests were grouped with the six additional fast failure and six additional subfailure tests. For this grouping, the force response, F(λ, t), was analyzed in terms of the fascicle elongation using a hereditary integral for nonlinear displacement-time separable viscoelastic responses [quasi-linear viscoelasticity (QLV) theory (Fung 1981)]: t F(λ, t) =
G red (t − t )
dF e dλ dt . dλ dt
(2)
0
where G red is the reduced relaxation function, F e is the instantaneous elastic force, λ is the elongation, t is the time, and t is a dummy variable for integration. The force response during the strain ramp onset was included in this analysis. An exponential expression was used for the instantaneous elastic force (Fung 1981): F e (λ) = A[e Bλ − 1]
(3)
where A and B are the instantaneous elastic parameters. The reduced relaxation function was expressed as a summed exponential function as follows: G red (t) = G ∞ +
4
G n e−t/τn
n=1
G1 + G2 + G3 + G4 + G∞ = 1
(4)
where G ∞ is the steady-state relaxation coefficient, and τn are the time constants corresponding to each of the relaxation coefficients, G n . In this study, five relaxation coefficients were used. The time constants, τn , were constrained to decade values (τ1 = 1 s, τ2 = 100 ms, τ3 = 10 ms, τ4 = 1 ms) (c.f. Goh et al. 2003; Kauer et al. 2002; Sklavos et al. 2006; Van Dommelen et al. 2005b). The range of time constants (1 ms to 1 s) was chosen to encompass all rates exhibited in the test methodology, including an order of magnitude faster than the strain ramp onset to the end of the force relaxation data that were included in the model. A single hereditary integral (Eq. 1) was numerically integrated (Darvish et al.
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1999) for all of the grouped tests simultaneously in terms of instantaneous elastic parameters, A and B, and relaxation coefficients, G n and G ∞ . A generalized reduced gradient technique was used to find an optimal solution for A, B, G n , and G ∞ (Solver from Microsoft Excel 2003). The objective function of the optimization was the cumulative summation of the squared error between the numerically integrated model force, FM , and the experimentally measured force, FE . For the subfailure tests, the squared error was exponentially weighted as a function of the time constants in the relaxation function to place less emphasis on the slow rate behavior. No weighting was used on the failure tests, which placed more emphasis on the fast rate behavior. In the subfailure tests, the squared error was calculated as: E 2 = ( · (FM − FE ))2
(5)
where is the weighting function: =
(e−t/τ 1 + e−t/τ 2 + e−t/τ 3 + e−t/τ 4 ) . 4
(6)
In the failure tests, the squared error was simply: E 2 = (FM − FE )2 .
(7)
Following the analysis of the subfailure and failure grouped tests, the QLV numerical analysis was repeated for the 30 subfailure tests. The fast rate coefficient, G 4 , which was found by fitting the grouped tests, was used and held constant in the analysis of the 30 subfailure tests. The remaining relaxation coefficients, G 1 , G 2 , G 3 , and G ∞ , and instantaneous elastic parameters, A and B, were then optimized as outlined in the previous paragraph. For each of the 30 subfailure fascicle tests, all viscoelastic parameters were determined from the R4 experimental data and were validated by predicting a response for the R2 test and the V4 test. 2.3.2 Fascicle failure properties The failure properties calculated from each failure test include force, Lagrangian stress, and elongation. Failure force was defined as the peak force during the failure test. Lagrangian stress and elongation were calculated at the time of the peak force. Lagrangian stress (σF ) was defined as the ratio of fascicle failure force (FF ) and fascicle cross-sectional area (CSAF ): σF =
FF . CSAF
(8)
Fascicle failure elongation (λF ) was defined as the ratio of the displacement at the time of peak force (d F ) and the initial length (l0, f ): λF =
dF . l0, f
(9)
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To provide insight into the ligament failure mechanisms, the fascicle failure force and stress were compared with ligament failure force and stress. The fascicle force, stress, and elongation were used to investigate rate effects and gage length effects on fascicle properties.
3 Results 3.1 Fascicle material model The grouped 15 subfailure and 15 failure tests with the single QLV model fit are shown in Fig. 4. The QLV model fit represents a single QLV model (Eqs. 22–24) simultaneously, which was fit to the grouped subfailure and failure tests. The corresponding optimal G 1 , G 2 , G 3 , G 4 , G ∞ , A, and B were 0.03, 0.02, 0.05, 0.59, 0.31, 6242 N, and 0.0047. There is a relatively large variation in the force response, as expected in part by the variation in fascicle cross-sectional area, but mostly by the variation in strain rate. In some of the fascicle tests, there is evidence of strain softening before failure. The single QLV model predicts the average force response among the 15 grouped tests. The G 4 = 0.59 from the model shown in Fig. 4 was used and held constant in the subsequent model fit for the 30 subfailure tests. A QLV model was fit for each of the 30 subfailure R4 tests and the average (±1 SD) G 1 , G 2 , G 3 , Fig. 4 Subfailure and failure tests with the model fit. The model fit was determined by simultaneously fitting all of the subfailure and failure tests. The thickest solid line is the model fit. a Subfailure tests. b Failure tests. Note that the small oscillation in the model force is a result of noise in the strain data
Fig. 5 Representative QLV fit to a fascicle subfailure force response while using G 4 = 0.59 and optimizing for G 1 , G 2 , G 3 , A, and B
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and G ∞ were 0.05 ± 0.05, 0.04 ± 0.04, 0.05 ± 0.05, and 0.27 ± 0.05. These coefficients, along with G 4 = 0.59, depict the fascicle relaxation behavior. The average A and B, as found by minimizing the error between the model average instantaneous elastic force and the average instantaneous elastic force, were 6.89 N and 8.15. A representative subfailure model fit is shown in Fig. 5. Based on observation and goodness of fit (R 2 value), the model fits the experimental data very well. As a goodness of fit measure for the 30 subfailure tests, the average (±1 SD) R 2 value for the R4 tests was 0.98 ± 0.02. To assess the applicability of the material model over different strain levels, the model parameters were determined for the 30 subfailure R2 and R4 tests. A comparison of the R2 and R4 average (±1 SD) relaxation function is shown in Fig. 6. The average relaxation functions are similar among strain levels and the upper standard deviation is nearly indistinguishable. This indicates that there is very little strain dependence on the fascicle relaxation behavior, which supports the QLV fascicle material model assumption. A comparison of the R2 and R4 average (±1 SD) instantaneous elastic function is shown in Fig. 7. The average and standard deviations of the instantaneous elastic functions are nearly identical among strain levels. A representative force response prediction of V4 test is included in Fig. 8. It should be noted that the viscoelastic model developed in this paper is a tension model and does
Viscoelastic and failure properties of spine ligament collagen fascicles
493
Reduced Relaxation Function, G(t)F
red
1.0
R2 Avg
0.9
R2 Stdev
0.8
R4 Avg
0.7
R4 Stdev
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00 01
0.001
0.01
0.1
1
Time, t (s)
Fig. 6 Comparison of the fascicle reduced relaxation function between the 30 subfailure R2 and R4 tests. The high rate coefficient, G 4 = 0.59, remained constant for all tests
Fig. 8 Representative prediction of the V4 test using the R4 model. This figure corresponds to test number 40
3.2 Fascicle failure properties The fascicle failure results include the 20 failure tests, 14 of which are “fast” tests, five of which are “slow” tests, and one of which is a “medium (med)” test. For all of the fascicle data reported in this paper, the fascicles failed within the diamond cut-out in the specimen mounting plastic. The force–time and force–strain responses for all 20 failure tests are shown in Figs. 9 and 10. Figure 9 includes the force–time data on two different time scales to view the fast and slow tests separately. There is a clear distinction between the fast tests and the slow tests. The fast tests have a higher failure force and stress and a lower failure strain, where the peak force and stress are deemed the failure force and stress, and the failure strain is the strain at the failure stress (Table 2).
Fig. 7 Comparison of the fascicle instantaneous elastic function between the 30 subfailure R2 and R4 tests. The R2 instantaneous elastic function was extended to 0.04 Lagrangian strain for comparison with the R4 instantaneous elastic function
not predict force in compression; therefore, any compressive force imparted on the load cell during the experimental oscillatory displacement test was not predicted. For this reason, there is a discrepancy in Fig. 8 between the model force and the experimental compression force. In preliminary experiments, the behavior exhibited in Fig. 8 was repeatable, indicating that the force relaxation is an inherent viscoelastic property and not the result of fibril damage.
4 Discussion To the authors’ knowledge, the only other study that developed a fascicle viscoelastic model was that by Elliot et al. (2003). In their study, mouse tail tendon fascicles were displaced in tension at a strain rate of 0.005 s−1 in 0.05 mm increments until failure with 10 min of relaxation between each increment. A QLV model was fit to the stressrelaxation data while optimizing for the material parameters and time constants. They found the optimal time constants to be 0.082 ± 0.064 s and 323 ± 71 s. Since the applied strain rate was slow, it is difficult to interpret the relaxation response with a ∼100 ms time constant. While the discrepancy in strain rate and optimal time constant exists, the fast relaxation in the mouse tail tendon fascicle is consistent with the findings in this paper. The Elliot et al. (2003) exponential instantaneous
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Fig. 9 Force–time results from fascicle failure tests. There are 14 “fast” tests, 5 “slow” tests, and 1 “medium” test in this figure. a Time scale chosen to emphasize the “slow” tests. b Time scale chosen to emphasize the “fast” tests
a
Fast (14 tests) Med (1 test) Slow (5 tests)
5
b
4
Force, FF (N)
4
Force, FF (N)
Fast (14 tests) Med (1 test) Slow (5 tests)
5
3 2
3 2 1
1
0
0 0.0
.1
0.2
0.000
5 Fast (14 tests) Med (1 test) Slow (5 tests)
Force, FF (N)
4
3
2
1
0 0.0
0.1
0.2
Engineering Strain,
0.3
F
Fig. 10 Force–strain results from fascicle failure tests. There are 14 “fast” tests, 5 “slow” tests, and 1 “medium” test in this figure
elastic function appeared linear, which was also apparent in this study (Fig. 7). They attributed this behavior to the preload in their experimentation and suggested that the force response was outside the toe region. There was no preload in the fascicle methodology in this paper, however, suggesting that the nearly linear viscoelastic behavior may be exhibited from the slack condition through deformation. Similar to Elliot et al., a QLV model was chosen in this paper to approximate the fascicle material behavior. A QLV model assumes that the shape of the reduced relaxation function is independent of strain level. If the relaxation were dependent on time
0.005
0.010
Time, t (s)
Time, t (s)
and strain, a fully nonlinear model would be necessary to describe the fascicle behavior. The QLV assumption in this paper was justified by the similarity in reduced relaxation behavior (Fig. 6) and by the good model fit to experimental data (Figs. 5, 8). Currently there are no linear or nonlinear viscoelastic fascicle models in literature; however, there exist ligament viscoelastic models, which are useful in comparison. The current paper and the Lucas et al. (2008) results indicate that QLV is appropriate for both spinal ligaments and spinal ligament fascicles under fast rate deformations. Some researchers have reported that QLV is not appropriate to model ligaments (c.f. Provenzano et al. 2002), some have reported that QLV is only appropriate to certain strain levels (c.f. Funk et al. 2000), and some have reported that QLV is appropriate at fast strain rates (Lucas et al. 2008; Van Dommelen et al. 2005a,b). All three arguments are justified under the loading conditions described in the respective papers. In addition to the response to disparate loading conditions in literature, the microstructural material behavior presented in this paper provides insight into the nature of ligament viscoelasticity. Among the fascicle QLV coefficients, G 4 was the largest in magnitude, followed by the steady-state coefficient, G ∞ . The sum of the magnitudes of G 4 and G ∞ is approximately 0.85. The summation of the relaxation coefficients is one and the remaining 0.15 is approximately divided equally among G 1 , G 2 , and G 3 . This behavior is similar to spinal ligament relaxation at fast strain rate deformations. In a previous study on subfailure cervical spine viscoelastic properties (Lucas et al. 2008), a viscoelastic analysis was performed using a similar test battery and the same number of
Table 2 Summarized failure properties (average ± SD) from fascicle failure test series Force (N)
Lag stress (MPa)
Elongation
Stiffness (N/mm)
Fast
2.43 ± 1.34
65.9 ± 36.8
0.10 ± 0.06
10.4 ± 13.4
Slow
1.98 ± 0.37
33.0 ± 11.8
0.21 ± 0.08
3.57 ± 1.74
Ratio of fast/slow
1.23
2.00
0.47
2.90
Only the fast and slow tests are included in this comparison. The medium test is excluded
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Reduced Relaxation Function, G(t)F
red
1.0
ALL PLL LF Fascicle
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0001
0.001
0.01
0.1
1
Time, t (s)
Fig. 11 Reduced relaxation function comparison between the fascicle model developed in this study and the spinal ligament model developed in a previous experiment. The ALL, PLL, and LF data are from Lucas et al. (2008)
instantaneous elastic and relaxation parameters with the same time constants. For the ALL, PLL, and LF, the average G 4 and G ∞ were 0.65 and 0.25, 0.69 and 0.21, and 0.61 and 0.19. For each ligament, G 1 , G 2 , and G 3 comprised the remaining portion of the reduced relaxation function and were similar in magnitude. For comparison, the reduced relaxation functions are shown in Fig. 11 for the collagen fascicle, ALL, PLL, and LF. It should be noted that the viscoelastic model developed for the spinal ligaments did not use failure tests, in contrast to the fascicle viscoelastic model in this paper. The strain rates in the spinal ligament tests, however, were of the same order of magnitude as the strain rates used in the fascicle failure tests. As shown in Fig. 11, the fascicle and ligament relaxation functions are similar in shape and magnitude, although there is a slight difference in early relaxation behavior. The fascicle fast rate coefficient in this paper (G 4 = 0.59) is smaller in magnitude than the average ALL, PLL, and LF fast rate relaxation coefficients. Thus, the ligament relaxation functions exhibit a larger decrease in magnitude early in the relaxation time history. It has been suggested that ligament viscoelasticity is due partly to fluid flow and interactions among collagen fascicles and ground substance (Atkinson et al. 1997; Butler et al. 1997; Chimich et al. 1992; Weiss et al. 2002). While this assertion may have merit for slow rate deformations, the comparable collagen fascicle and spinal ligament relaxation functions suggest that, at fast rate deformations, ligament viscoelasticity may be attributed more to the intrinsic viscoelastic properties of the collagen fascicle. Further research is necessary to investigate fascicle and ligament relaxation behavior at varying deformation rates and loading conditions.
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The viscoelastic analysis in this paper was unconventional because the failure tests were included in the development of the material model. Including the failure tests was advantageous in providing fast-rate relaxation behavior that was missed by analyzing the slower subfailure tests. In addition, including the failure tests strengthens the assumption that the fascicle QLV material model is valid until fascicle failure. It was shown that the fascicle QLV model accurately predicts the fascicle behavior at 0.02 and 0.04 strain, and at varying oscillatory frequencies. Other than that the fascicle model fits to failure data from which the model was developed, there is no analytical evidence showing that the fascicle QLV model is valid until fascicle failure; however, there is precedent from the ligament and tendon literature to support this notion. Researchers have reported QLV to be valid up to finite strains of approximately 0.15 (Fung 1981; Funk et al. 2000), which is beyond the fascicle failure strain observed in the fast fascicle tests in this study. This supports the assertion that QLV is valid for a ligament until the initial recruited fascicles fail. The literature of existing studies on collagen fascicle failure properties (Butler et al. 1986; Derwin and Soslowsky 1999; Yamamoto et al. 1999, 2000; Robinson et al. 2004; Heraldsson et al. 2005) includes tests performed at strain rates ranging between 0.0001 (Yamamoto et al. 1999) and 1 s−1 (Butler et al. 1986). A summary of the failure properties found in these tests is included in Table 3. The ultimate tensile strength ranged between 17.2 ± 4.1 (Yamamoto et al. 1999) and 76.0±9.5 MPa (Heraldsson et al. 2005). The strain at failure ranged between 0.109 ± 0.016 (Yamamoto et al. 1999) and 0.150 ± 0.008 (Butler et al. 1986). Yamamoto et al. (1999) investigated strain rate effects on failure properties of collagen fascicles from rabbit patellar tendons and found a significant difference in failure stress, but not in failure strain. Robinson et al. (2004) reported strain rate effects on mouse tail tendon fascicle failure stress, elastic modulus, and failure strain. In this paper, the porcine PLL fascicle failure Lagrangian stress and strain was 33.0 ± 11.8 and 0.21 ± 0.08 MPa for the slow strain rate tests (0.25–0.50 s−1 ) and 65.9 ± 36.8 and 0.10 ± 0.06 MPa for the fast strain rate tests (10.9–84.6 s−1 ). The failure data was recorded at the peak force. In most of the fascicle failure tests, the force decreased abruptly just beyond the peak force. In some tests, it was evident that the fascicle force decreased slightly just after the peak force until ultimate failure. As in ligaments, progressive fascicle damage likely occurs before ultimate failure. To improve the fascicle model accuracy up to failure and to improve the fast rate data used in the material response, fascicle subfailure damage should be investigated. To this end, fibril experimentation is necessary at the rates used in this paper. Robinson et al. (2004) acknowledged that the “fast strain rate of 0.50 s−1 used in their study may not be as fast as
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Table 3 Summary of existing fascicle failure data from tensile failure experiments Primary author, year
Specimen
Strain rate, (s−1 ) Failure lagrangian stress, (MPa) Failure strain
Butler et al. (1986)
Human ACL, PCL, LCL, PT
1.0
Ligs: 36.4 ± 2.5
Ligs: 0.15 ± 0.01
PT: 68.5 ± 6.0
PT: 0.14 ± 0.01
Derwin and Soslowsky (1999) Mouse tail tendon
0.005
26.4 ± 1.9
0.11 ± 0.02
Yamamoto et al. (1999)
0.0001
14.0 ± 1.3
0.10 ± 0.03a
0.01
19.1 ± 2.8
0.11 ± 0.02a
Rabbit PT
± 3.5a
Yamamoto et al. (2000)
Rabbit PT
0.017
20.0
Robinson et al. (2004)
Mouse tail tendon
0.005
28.0 ± 4.0a
0.50
58.0
± 3.0a
0.13 ± 0.02a NR NR
Heraldsson et al. (2005)
Human PT: anterior and posterior bundle 0.0037
Ant: 76.0 ± 9.5
Ant:0.12 ± 0.01
Post: 38.5 ± 3.9
Post: 0.13 ± 0.01
Current study
Porcine PLL
0.25–0.50
33.0 ± 11.8
0.21 ± 0.08
10.9–84.6
65.9 ± 36.8
0.10 ± 0.06
NR not reported; ACL anterior cruciate ligament; PCL posterior cruciate ligament; LCL lateral collateral ligament; PT patellar tendon; PLL posterior longitudinal ligament a Estimated from bar charts in respective papers
the loading rates experienced during high impact activities (e.g. running, jumping) and that results from higher rates may differ from those reported here”. Compared to automobile crashes and military impacts, even running and jumping are considered slow strain rate events. For example, spinal ligaments could exhibit a strain rate of 5 s−1 or more in a frontal car crash at 48 km/h (calculated from data in Forman et al. 2006) and 25 s−1 or more in military aircraft ejections (calculated from data in Bolton 2002). The fast strain rates of fascicles tested in this paper were 10.9–84.6 s−1 and the strain rates in the previous cervical spine ligament failure tests were 70–150 s−1 . Summarizing, the spinal ligament strain rates in the present paper are faster than the rates exhibited in a frontal car crash at 48 km/h and comparable to the rates in military aircraft ejections. Of the two existing studies on fascicle strain rate sensitivity, Yamamoto et al. (1999) reported strain rate effects on failure stress, but not on failure strain, and Robinson et al. (2004) reported strain rate effects on both stress and strain. These two studies encompassed strain rates between 0.0001 and 0.50 s−1 . This paper tested fascicles until failure between 0.25 and 85 s−1 . The results indicated that collagen fascicle failure Lagrangian stress, force, and elongation may be strain rate sensitive. The strain rate effects on strain are more apparent than on stress because of the large standard deviation in force and stress. A general linear ANOVA (significance set at 0.05) on the failure results in this paper revealed that the strain rate effects are significant (P < 0.05) for elongation, but are not significant for force (P = 0.48) or stress (P = 0.07). Including these results with the Robinson et al. study, there is now evidence to suggest that fascicle failure elongation is strain rate sensitive among strain rates ranging from 0.5 to 85 s−1 . The lack of significance for failure force and stress
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does not coincide with the Yamamoto et al. and Robinson et al. studies; however, the results in this paper are ambiguous with a large standard deviation. As a comparison, the spinal ligaments and knee ligaments are strain rate sensitive for failure stress (spine: Bass et al. 2007; Yoganandan et al. 1989, knee: Yamamoto et al. 2003; Kerrigan et al. 2003; Weiss et al. 2002). A possible reason for the large standard deviation in failure data could be the range in fascicle gage lengths tested in this study. Haut (1986) reported that the rat tail tendon fascicle failure strain decreases as gage length increases; however, there was no supporting statistical relevance. The trend reported by Haut (1986) was also observed in this paper; however, a general linear ANOVA (significance set at 0.05) revealed that there is no significant difference due to gage length in failure elongation (P = 0.051), failure force (P = 0.25), or failure stress (P = 0.76). Without statistical significance, there was no reason to isolate the gage length groups in the fascicle failure elongation analysis. The results from this paper were used in the development of a microstructural model that will be published in a subsequent paper. This is part of a greater effort needed to develop a general ligament model that can be used to predict the response of a uniaxial arbitrary ligament-type based solely on its composition. Further research is needed pertaining to ligament response due to fluid interactions, friction between microstructural components, and orientation. Acknowledgements This study was supported by the US Office of Naval Research, the Naval Air Systems Command and the University of Virginia School of Engineering and Applied Science. This material was presented in part at the 2008 American Society of Mechanical Engineering Summer Bioengineering Conference in Marco Island, Florida.
Viscoelastic and failure properties of spine ligament collagen fascicles
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